Range-Based Component Models for

Conditional Volatility and Dynamic Correlations

A Thesis Submitted to the Committee on Graduate Studies in Partial Fulfillment of the Requirements for the Degree of

Master of Science in the Faculty of Arts and Science

TRENT UNIVERSITY

Peterborough, Ontario, Canada

© Copyright by Stephen Swanson 2017

Applied Modelling & Quantitative Methods M.Sc. Graduate Program

May 2017

ii

ABSTRACT

Range-Based Component Models for Conditional Volatility

and Dynamic Correlations

Stephen Swanson

Volatility modelling is an important task in the financial markets. This paper first

evaluates the range-based DCC-CARR model of Chou et al. (2009) in modelling

larger systems of assets, vis-à-vis the traditional return-based DCC-GARCH.

Extending Colacito, Engle and Ghysels (2011), range-based volatility specifications

are then employed in the first-stage of DCC-MIDAS conditional covariance

estimation, including the CARR model of Chou et al. (2005). A range-based

analog to the GARCH-MIDAS model of Engle, Ghysels and Sohn (2013) is also

proposed and tested - which decomposes volatility into short- and long-run

components and corrects for microstructure biases inherent to high-frequency price-

range data. Estimator forecasts are evaluated and compared in a minimum-

variance portfolio allocation experiment following the methodology of Engle and

Colacito (2006). Some consistent inferences are drawn from the results, supporting

the models proposed here as empirically relevant alternatives. Range-based DCC-

MIDAS estimates produce efficiency gains over DCC-CARR which increase with

portfolio size.

Keywords: Dynamic correlations, Component models, Mixed Frequency, High-

low range, Forecasting, Covariance, Volatility, MIDAS, DCC, CARR, GARCH,

Asset allocation, Portfolio risk management

Acronyms: Dynamic conditional correlations (DCC), Conditional Autoregressive

Range (CARR), Generalized-ARCH (GARCH), Mixed Data Sampling (MIDAS)

iii

Acknowledgements

I would first like to express sincere gratitude towards my supervisors, Dr. Bruce Cater and Dr. Marco Pollanen, for their invaluable guidance and support throughout the research process. Any errors contained in this paper are entirely my own. Besides my supervisors, I would like to also thank my external examiner, Dr. Marcos Escobar-Anel, and the rest of my examining committee, for the insightful and helpful feedback they provided. I am also grateful to Dr. Hang Qian for his continued support of the MIDAS MATLAB Toolbox, and Dr. Ric Colacito, whose willingness to respond to inquiries and to provide clarifications to a student like myself was humbling. This research was also made possible, in part, thanks to financial support provided by the Trent University School of Graduate Studies, Patricia Southern, Wally Macht, Scotiabank and the CFA Institute. Last but not least, I thank my family; my brother and my mother, most of all. If it were not for the continuous love, support and encouragement they provide, this journey would not have been possible. I dedicate this thesis to them.

iv

Contents

Abstract .................................................................................................................. ii

Acknowledgements .............................................................................................. iii

List of Figures ....................................................................................................... vi

List of Tables ....................................................................................................... vii

List of Abbreviations ......................................................................................... viii

1 Introduction ..................................................................................................... 1

2 Models and Background

2.1 Historical Volatility Measures ................................................................................ 8

Realized Volatility ................................................................................................. 9

Realized Range ................................................................................................... 10

2.2 Return- and Range-Based Models of Conditional Volatility and Correlation

2.2.1 Univariate Conditional Variance Models ................................................. 12

Autoregressive Conditional Heteroscedasticity (ARCH) ........................ 13

Generalized ARCH (GARCH) ............................................................... 14

Spline-GARCH ........................................................................................ 15

2.2.2 Range-Based Conditional Volatility Models ............................................ 17

Conditional Autoregressive Range (CARR) ........................................... 18

2.2.3 Multivariate Conditional Volatility Models ............................................. 21

Conditional Correlation (CC) models .................................................... 22

Dynamic Conditional Correlation (DCC) .............................................. 23

2.3 Mixed Data Sampling (MIDAS) Methods

2.3.1 GARCH-MIDAS ..................................................................................... 25

2.3.2 DCC-MIDAS ........................................................................................... 27

v

3 Range-Based Component Models for Conditional Volatility

and Dynamic Correlations ................................................................................ 29

3.1 Range-based DCC-MIDAS ................................................................................. 30

3.2 CARR-MIDAS .................................................................................................... 34

4 Empirical Application .................................................................................. 37

4.1 Methodology and Data ....................................................................................... 39

4.2 Results and Discussion ....................................................................................... 48

5 Conclusion ...................................................................................................... 69

References ............................................................................................................. 72

Appendix A: Resulting Simulated Portfolio Variances

(raw-output), by Portfolio ........................................................................... 76

Appendix B: p-Values Pertaining to Tests of Predictive Accuracy

(raw-output), by Portfolio ........................................................................... 97

Appendix C: Efficiency Gains and Diebold Mariano Test Results

(vs. DCC-CARR benchmark), by Portfolio ............................................... 118

Appendix D: MATLAB Code ............................................................................ 155

vi

List of Figures

Figures 1 - 8 Log-Return and Log-Range Time-Series of Major International Stock Indices (In-Sample):

(1) S&P500, (2) S&P/TSX, (3) DAX30 ............................ 45

(4) FTSE100, (5) CAC40, (6) FTSEMIB .......................... 46

(7) NIKKEI225, (8) RTSI .................................................. 47

Figure 9 Average Efficiency Gains across Simulated Portfolios (vs. DCC-

CARR benchmark), by Portfolio Size ......................................... 65

vii

List of Tables

Table 1 Summary Statistics for Log-Return Time-Series of Major International Stock Indices (In-Sample) .............................................. 49

Table 2 Summary Statistics for Absolute Log-Return Time-Series of Major International Stock Indices (In-Sample) .................................... 50

Table 3 Summary Statistics for Log-Range Time-Series of Major International Stock Indices (In-Sample) .............................................. 51

Table 4 Summary of Accuracy Test Results from Appendix B, by Portfolio Size ........................................................................................ 54

Table 5 Average and Maximum Efficiency Gains across Simulated Portfolios (vs. DCC-CARR Benchmark), by Portfolio Size ................................. 60

viii

List of Abbreviations

AIC - Akaike Information Criterion ACF - Autocorrelation Function ARCH - Autoregressive Conditional Heteroscedasticity ARMA - Autoregressive Moving Average BIC - Bayesian Information Criterion BLUE - Best Linear Unbiased Estimator CAC 40 - Cotation Assistée en Continu 40 Index (“Continuous Assisted

Quotation”) (French Stock Index) CARR - Conditional Autoregressive Range CC - Conditional Correlation CCC - Constant Conditional Correlation DAX 30 - Deutscher Aktienindex 30 (German Stock Index) DCC - Dynamic Conditional Correlation EGARCH - Exponential GARCH FTSE 100 - Financial Times Stock Exchange 100 Index (UK Stock Index) FTSE MIB - Financial Times Stock Exchange Milano Italia Borsa

(Italian Stock Index) FX - Foreign Exchange GARCH - Generalized Autoregressive Conditional Heteroscedasticity GARCH-PARK-R - Parkinson Range GARCH GJR-GARCH - Glosten-Jagannathan-Runkle GARCH IGARCH - Integrated GARCH IS - abbrev. “In-Sample” IID - Independent and Identically Distributed LBQ - Ljung-Box Q-Test MGARCH - Multivariate GARCH MIDAS - Mixed-Data Sampling NASDAQ - NASDAQ (“National Association of Securities Dealers Automated

Quotations”) Composite Index (US Stock Index) NIKKEI 225 - Nikkei 225 Index (Japanese Stock Index)

ix

OLS - Ordinary Least Squares OOS - abbrev. “Out-of-Sample” REGARCH - Range-based Exponential GARCH RGARCH - Range-based GARCH RR - Realized Range RTSI - Russia Trading System Index (Russian Stock Index) RV - Realized Variance (or ‘Realized Volatility’) S&P500 - Standard & Poors 500 Index (US Stock Index) S&P/TSX - Standard & Poors / Toronto Stock Exchange Composite Index

(Canadian Stock Index) Std.Dev. - abbrev. “Standard Deviation” SV - Stochastic Volatility T-bond - Treasury Bond VCC - Time-Varying Conditional Correlation

1

1 Introduction

Financial volatility characterizes dispersion in the second-moments of asset

returns. This represents a primary consideration for risk- or loss-averse investors,

since the probability of recognizing price erosion or portfolio loss is increased for

assets or portfolios that exhibit greater dispersion. Measures of volatility and

comovement are key inputs for numerous decision tools that are regularly applied

by investors, and so volatility modelling represents an important consideration for

many, and with far-reaching implications across the financial markets. Better

estimates of these inputs allow for better decisions to be made across many areas of

finance including, but not limited to, portfolio selection, hedging and risk

management.

In practice, investors have the option to invest in individual securities or some

combination of assets to form a portfolio. There are a number of approaches to

‘optimally’ select securities for inclusion in a portfolio and how to allocate funds

between them, depending upon an investor’s unique investment objectives - such as

maximizing return at a tolerable level of risk, minimizing risk subject to a required

return constraint, maximizing the Sharpe ratio, etc. In any case, the ability to

accurately identify the complete set of investment opportunities available, such

that the best course of action can be identified, depends upon an accurate

understanding of expected returns, volatility and correlations. These variables

change through time, and so the efficient frontier is dynamic. Volatility and

correlations are latent variables governed by stochastic processes, and so the

question of how to best estimate these measures, so as to allow for better

investment decisions to be made, remains an area of ongoing research.

2

Simple historical approaches include the estimation of sample standard

deviation and correlation, the realized volatility model, etc. These represent some

of the most basic, but also some of most widely understood, estimators of

dispersion or comovement in asset returns. Some more advanced volatility

estimation methods accommodate for various stylized empirical facts that emerge

from the statistical analysis of price variations in various types of financial markets,

and they can be used to provide better estimates of these latent volatility processes.

Stochastic Volatility (SV) models, for instance, represent a family of models that

are capable of capturing time-varying variance - a widely accepted, stylized fact of

financial time series. Conditional Volatility models can also capture this

heteroskedastic behavior, but they are specified to further accommodate for

temporal dependence in these second-order moments of asset returns, by

deterministically modelling their changes through time. Estimators belonging to

this class of models build upon the seminal ARCH and GARCH conditional

variance specifications, as proposed by Engle (1982) and Bollerslev (1986), to

accommodate for variety of other features inherent to financial time series data.

One of the most valuable extensions of the conditional volatility framework was

inspired by the observed tendency for volatilities to move together over time,

across assets and markets. The Multivariate GARCH (MGARCH) framework has

spurred a number of conditional correlation estimation techniques, with perhaps

the most ubiquitous of which being the Dynamic Conditional Correlation (DCC)

model of Engle (2002). The DCC-GARCH is a conditional covariance matrix

estimator that employs a two-stage specification and estimation strategy; cross-

products of the standardized residuals resulting from first-stage conditional

variance estimation are subsequently utilized in the second-stage estimation of

conditional correlations, which are specified to follow GARCH-like dynamics.

3

In the effort to produce improved estimates, further extension and

adaptation of existing conditional volatility and conditional correlation estimation

methods represent promising areas of ongoing research and development. Recent

examples of such extensions integrate historical measures volatility and correlation

into the conditional volatility and correlation specifications of the GARCH and

DCC-GARCH, using Mixed-Data Sampling (MIDAS) and estimation methods.

The GARCH-MIDAS model of Engle, Ghysels, and Sohn (2013) employs a

specification that decomposes conditional variance into two, mixed-frequency

components: a short-run ‘transient’ component, and a long-run ‘secular’ component

of volatility. The high-frequency component exhibits mean-reverting GARCH-like

dynamics, while the low-frequency component is determined upon a lagged history

of MIDAS-weighted realized volatility or macroeconomic measures. The DCC-

MIDAS model of Colacito, Engle and Ghysels (2011) represents a multivariate

extension of the GARCH-MIDAS model for dynamic correlations. Like the DCC,

this model employs a two-stage conditional covariance specification and estimation

strategy, but with second-stage conditional correlations being decomposed into two-

parts: a short-run component of conditional correlations that exhibit GARCH- or

GARCH-MIDAS-like dynamics, and a long-run component of correlation

determined upon a lagged history of MIDAS-weighted sample autocorrelation

measures. Empirical applications from the literature suggest that covariance

matrix forecasts produced by this latter model are comparatively more accurate

than estimates produced by the conventional DCC. Within a portfolio risk

management framework - Colacito, Engle and Ghysels (2011) demonstrate that use

of this mixed-frequency correlation specification in place of the DCC yields gains to

efficiency that increase with system size. And so, the DCC-MIDAS could

4 potentially represent a preferable alternative to the conventional DCC in modelling

larger asset systems; in particular, for applications of mid- to high-dimensional

portfolio allocation.

Another promising area of ongoing research is in the development of

advanced volatility estimators, which integrate various combinations of daily price

information into estimation. Opening, closing, daily-high, and daily-low prices are

widely available for most publicly traded securities, with the information contained

in these prices having been utilized in Japanese candlestick charting techniques,

and other technical analysis indicators, for several decades. A particular subset of

advanced volatility models are estimated on the basis of an asset’s daily high-low

price range. Range-based volatility models are estimated upon two distinct price

points representing the maximum and minimum prices observed across discrete

measurements of an asset’s intraday price path, taken at high-frequency intervals.

By contrast, conventional return-based volatility models are estimated solely on the

basis of an asset’s closing prices, which are comparatively arbitrary in the sense

that intraday price path is effectively ignored. Especially during turbulent days,

with drops and recoveries in the markets, return-based volatility models tend to

produce estimates which underestimate daily volatility. Some range-based historical

volatility models, which have been found to be more efficient and better able to

characterize daily volatilities versus realized variance measures, include the

Realized Range estimators of Parkinson (1980) and Martens and Van Dijk (2006).

Attempts have also been made to integrate range-based estimation

approaches into the conditional volatility framework. For instance, the Conditional

Autoregressive Range (CARR) model of Chou (2005) specifies the conditional range

of an asset’s price, within a fixed time interval, to evolve according to GARCH-like

dynamics. With their conditional specifications being nearly identical in form and

5 function, the CARR model could be thought of as a range-based analog of the

GARCH; any statistical software capable of estimating the latter can be used to

estimate the former. However, perhaps an obvious, but important, distinction that

should be addressed is as follows. While the GARCH model produces estimates of

the conditional variance in an asset’s daily returns, the CARR model produces

estimates of the conditional mean of an asset’s daily price range.

Estimates of ‘variance’ or ‘standard deviation’ are certainly the most

familiar and well understood representations of ‘volatility’, so much so that these

terms are often, but naïvely, used interchangeably. Failing to understand volatility

as a fundamental concept - that is, as a latent representation of dispersion in the

second-moments of asset returns - could lead one to be immediately skeptical of the

utility of a price range estimator in characterizing financial volatilities. But in fact,

it is a well-established statistical property that range is an estimator of the

standard deviation for a random variable. This concept is supported by both

Parkinson (1980) and Lo (1991), whose results demonstrate the range of any

distribution to be proportional to its standard deviation. It is based on these

principles that estimates produced by a conditional range model, such as the

CARR, could also be regarded to estimate conditional volatility.

A multivariate extension of the CARR model has also been proposed, to

estimate dynamic correlations. Exploiting the two-stage specification and

estimation strategy of the DCC, as well as the statistical and asymptotic

similarities between the GARCH and CARR models, Chou et al. (2009) propose

the DCC-CARR. As the name implies, the DCC-CARR represents a range-based

analog of the DCC-GARCH model, wherein CARR replaces GARCH in first-stage

conditional volatility estimation. In theory, use of a more efficient, range-based

estimation method in this first-stage could result in less-noisy error processes, and

6 translate to improved second-stage conditional correlation estimation. While Chou

et al. (2009) postulate that their DCC-CARR could be as readily applied to the

modelling of large asset systems as the DCC-GARCH has been - the empirical

application presented in that seminal paper, and all applications which have

followed, have been limited in scale. The results of Chou and Liu (2011) and

Lahiani and Guesmi (2014) empirically demonstrate the relative benefits of the

DCC-CARR for minimum-variance hedging applications. The original study, as

well as those exercises presented in Chou, Wu, and Liu (2009) and Chou and Liu

(2010), represent attempts to establish the economic value of the DCC-CARR for

volatility timing strategies - but with these applications being limited to three-asset

portfolios. The superiority of the DCC-CARR in modelling bivariate and trivariate

systems of assets vis-à-vis its return-based counterpart has been well-established.

But since investors typically form portfolios to gain exposure to multiple assets,

and to reap the rewards of diversification - a pertinent issue that is investigated in

this study, which the existing literature fails to address, is how the DCC-CARR

performs in modelling larger asset systems, and in more practical applications of

higher-dimension portfolio allocation.

In contribution to the ongoing development of advanced estimation methods,

this paper also introduces a pair of novel, range-based component models for the

estimation of conditional volatility and dynamic correlations. The inspiration

behind each model is straightforward, and follows naturally from the progression of

advanced return- and range-based estimation methods developed in the literature.

The multivariate, range-based component model presented in this paper is

distinguished by the fact that it employs range-based conditional volatility models

in the first-stage of estimation and decomposes conditional correlations into short-

and long-run components. This model could be perceived simply as a range-based

7 analog of the DCC-MIDAS or, alternatively, as a mixed-frequency component

model extension of the DCC-CARR. In theory, the conditional correlation

specification of this model has been formulated to combine the gains to efficiency

that can be recognized through estimation on range-based error processes, with

robustness to the estimation of large systems that seems to be obtained through

use of a mixed-frequency component specification. This dynamic conditional

correlation specification is developed, at least in part, to address anticipated

shortcomings of the DCC-CARR, which are formally established in this paper, in

applications to larger-systems and high-dimensional portfolio allocation.

A univariate, range-based component model is also proposed in this paper;

one which can be thought of as a range-based analog of the GARCH-MIDAS. In

the conditional volatility specification of this model we seek to simply replace each

of the mechanisms through which the individual components of conditional

variance are estimated, with theoretically more efficient range-based alternatives.

Essentially, the CARR model is adapted to replace the GARCH-like specification

that governs the short-run component of volatility, and the long-run component of

volatility is estimated upon temporally aggregated measures of the scaled Realized

Range, as proposed by Martens and Van Dijk (2006), rather than the realized

variance. This model is proposed as an alternative to the CARR; one which could

also be employed in the first-stage conditional volatility estimation of our

multivariate range-based component model.

The large-scale, comparative empirical study presented in this paper

evaluates the forecasting accuracy of various return- and range-based conditional

correlation estimation techniques in modelling low- and higher-dimension systems

of risky assets. Each estimator is tasked with providing one-day-ahead forecasts

across 247 simulated portfolios, which consist of various combinations of the major

8 international equity indices of G8 constituent countries, and range in size from 2 to

8 assets. The economic value of each estimator is evaluated within a minimum-

variance portfolio risk management framework, following the methodology of Engle

and Colacito (2006). The objective of this study is twofold. First, the suitability

of the DCC-CARR in modelling larger systems of assets, and for dictating higher-

dimensional portfolio allocations vis-à-vis the DCC-GARCH, is explored. Secondly,

economic value of the novel estimation methods proposed in this paper are

evaluated, across low- and higher-dimension systems, relative to the existing return,

and range-based DCC models.

The remainder of this paper is structured as follows. Section 2 presents a

literature review of existing volatility and correlation estimation methods which are

either directly employed in this study, or which provide inspiration for the novel

methods proposed in this paper. Section 3 presents a pair of range-based

component models for estimating conditional volatility or dynamic correlations.

The motivations for each model are described, specifications are provided, and

practical estimation issues are discussed. A comparative empirical study between

the estimation methods proposed in this paper, the DCC-GARCH, and the DCC-

CARR is presented in Section 4. Concluding remarks are presented in Section 5.

2 Models and Background

2.1 Historical Measures of Volatility

Historical models of volatility are capable of producing simple estimates of

an asset’s dispersion over a given reference period. Historical estimates of monthly

or yearly volatility, for instance, can be calculated using series consisting of 22 or

9 252 consecutive daily closing prices, respectively. Daily measures of realized or

historical volatility require intraday data to be calculated - data which is not freely

available.

The most widely understood and applied statistic in measuring or

representing dispersion in financial time series is the standard deviation (or

variance), which calculates the average (squared) deviation in an asset’s price from

its average price throughout a reference period. Another common historical

measure of volatility is the ‘realized variance’ or Realized Volatility (RV) model,

defined simply as the sum of squared returns observed over a given reference period.

The RV of a given asset on day t, for instance, can be calculated using a series of

squared intraday returns measured in ‘I’ equally-spaced intervals:

𝑅𝑉𝑡 = 𝑟𝑡−1+𝑖2𝐼

𝑖=1

(1)

Note that variance or standard deviation and realized volatility are each estimated

using the closing prices (or close-to-close returns). Since a large number of samples

are often required to obtain good estimates of historical volatility - the use of a

large number of closing values can obscure short-term changes in volatility. Some

other advanced volatility models have been developed to more fully utilize daily

price quote data which is typically made freely available alongside a stock’s closing

prices and often overlooked in conventional modelling. These models make use of

various combinations of a stock’s opening and closing, daily high- and low-prices,

and are designed to theoretically capture volatility which close-to-close models

10 could ignore, such as intraday or after-hours price movements, or both.1 Parkinson

(1980) first proposed a scaled high-low range estimator for the daily variance:

(log𝐻𝑡 − log 𝐿𝑡)4 log 2

2

(2)

where 𝐻𝑡 and 𝐿𝑡 represent the high- and low-prices of an asset throughout trading

day t - prices which are typically published for every stock. Parkinson

demonstrates the efficiency of his estimator versus traditional estimates of daily

volatility, and shows that his model could achieve a given level of accuracy in

predicting daily variance using about 80% less data (and thus, a smaller time

interval). More recently, Martens and Van Dijk (2006) introduced the Realized

Range (RR) model for estimating daily volatility. They show that just as the scaled

high-low range estimator improves over the daily squared return, so too does the

RR over the RV obtained by summing squared returns for intraday intervals (as

demonstrated through simulation and empirical experiments). The Realized Range

(RR) is defined as:

𝑅𝑅𝑡 = 1

4 log 2log𝐻𝑡,𝑖 − log 𝐿𝑡,𝑖

2𝐼

𝑖=1

(3)

Like measures of realized variance estimated on high-frequency data, the Realized

Range estimate is expected to suffer from market microstructure effects, such as

Bid-Ask bounce.2 For instance, assuming prices were continuously observed the

high-price during any given interval is likely to be an ‘ask’ while the low-price is 1 This is an extensive and ongoing area of research. While the focus of this paper is on (H-L) range-based volatility estimators, for some intriguing models which utilize all (OHLC) prices to accommodate drift, and in some cases overnight jumps, see Garman and Klass (1980), Rogers and Satchell (1991), Yang and Zhang (2000). 2 See Hansen and Lunde (2006) and Bandi and Russell (2006) for a thorough introduction and discussion of the effects of microstructure noise on RV estimates.

11 likely to be a ‘bid’ such that the observed range overestimates the true range by an

amount equal to the Bid-Ask Spread. As such, Realized Range estimates could be

subject to substantial upward bias for higher sampling frequencies (i.e. intraday).

To address this upward bias, Brandt and Diebold (2006) propose a simple

correction procedure wherein the Bid-Ask Spread is subtracted from intraday

measures of price range. However, this may not always be appropriate considering

prices are not observed continuously in practice and so there is a certain

probability, which depends upon trading intensity, that the ‘high’ price observed

over a given reference period could actually be a bid. Furthermore, and as

addressed by scaling-procedures proposed by Rogers and Satchell (1991) and

Christensen and Podolskij (2005), downward bias in the range can be induced by

infrequent trading - wherein the ticker quoted low- and high-prices are likely to

over- and underestimate their true values, respectively. To adjust for both up- and

downward biases in Realized Range estimates, Martens and Van Dijk (2006)

propose the following bias-correction procedure:

𝑅𝑅𝑆,𝑡 = ∑ 𝑅𝑅𝑅𝑅𝑅𝑡−1𝑄∗𝑙=1

∑ 𝑅𝑅𝑡−1𝑄∗𝑙=1

𝑅𝑅𝑡 (4)

where 𝑅𝑅𝑆,𝑡 represents the scaled measure of Realized Range in period t. The

Realized Range 𝑅𝑅𝑡, as calculated in (3), is scaled by the ratio of the average level

of the daily range and the average level of the Realized Range over the previous 𝑄∗

trading days. If trading intensity and bid-ask spread were to remain constant,

accuracy could be gained by setting 𝑄∗ as high as possible. However, since the

magnitude of these microstructure frictions tend to vary over time, only a recent

price history should be used and so 𝑄∗ should not be set too large. Selecting an

appropriate number of trading days 𝑄∗ to use in computing the scaling factor is a

12 decision the authors leave to the practitioner. The ‘scaled’ Realized Range can be

thought of as a more efficient, range-based analog of the realized volatility model,

and will be factored into the extreme-value component model of conditional

volatility proposed in Section 3.2 of this paper.

2.2 Return- and Range-Based Models of Conditional

Volatility and Correlation

2.2.1 Univariate Conditional Variance Models

Although time series of asset returns widely appear to follow martingale

processes, there is clear evidence to suggest temporal dependence in the second-

order moments of these observations across most major asset classes. Mandelbrot

(1963) was the first to report on a frequently observed phenomenon in finance, now

referred to as ‘volatility clustering’, where:

“large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes”.

Along with this heteroscedastic behavior, daily asset returns typically exhibit

heavy-tailed distributions which are far from being IID; the likelihood of observing

significant deviations from the mean are greater than in the case of the normal

distribution. These stylized facts represent violations to the Gauss-Markov

assumptions and so OLS linear regression cannot provide the BLUE. In contrast to

models of historical or realized volatility, conditional volatility models consider

issues of temporal dependence and are used to specify or filter a latent data

generation process (DGP) in volatility estimates.

13

The Autoregressive Conditional Heteroscedasticity (ARCH) model of Engle

(1982), who first used it to model inflation rates, accommodates for these time-

varying behaviors and allows for valid coefficients to be obtained by including a

parameterization for the variance of error terms. For demonstrative purposes,

suppose the return to asset i observed at time period t can be denoted as 𝑟𝑖,𝑡. An

ARCH(p) model can be specified as:

𝑟𝑖,𝑡 = µ + 𝜀𝑖,𝑡 (5)

where 𝜀𝑖,𝑡|𝐼𝑡−1 ~ 𝑁0,ℎ𝑖,𝑡 and 𝑖 = 1

and ℎ𝑖,𝑡 = 𝜔𝑖 + ∑ 𝛼𝑖,𝑛𝜀𝑖,𝑡−𝑛2𝑝𝑛=1 (6)

where 𝜔𝑖 > 0 and 𝛼𝑖,𝑛 ≥ 0 for n = 1,2,…,p where 𝜀𝑖,𝑡 denotes the return residuals, or the error terms of the ARCH model with

respect to some mean process µ. The residuals are conditionally normally

distributed, with a mean of zero and a time-dependent variance ℎ𝑖,𝑡, given the

information set represented by time process I up to time t-1. An ARCH(p)

conditional variance specification includes a mean variance parameter 𝜔𝑖 which

describes the long-term persistence of variance around the mean, and evolves

according to the series and sequence of squared return residuals observed over the

previous ‘p’ periods, ∑ 𝜀𝑖,𝑡−𝑛2𝑝𝑛=1 . Corresponding alpha coefficients 𝛼𝑖,𝑛 characterize

the ARCH-effects, or the degree to which volatility shocks observed n-periods ago

are transmitted through into the volatility of asset i today. Parameters in the

conditional variance specification are constrained as shown in (6) to avoid negative

ℎ𝑖,𝑡 values.

14

Bollerslev (1986) built upon the seminal work of Engle by introducing the

Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. The

distinguishing characteristic of the GARCH vis-à-vis the ARCH model is that the

former allows for conditional variance to evolve according to past values of itself, in

addition to volatility shocks observed over previous time-periods. Whereas its

predecessor is deterministically specified, the GARCH model is capable of capturing

volatility clustering and spillovers which are stochastic in nature and timing.

Following the framework outlined above by (5), the conditional variance

specification of a GARCH(p,q) model is:

𝜀𝑖,𝑡|𝐼𝑡−1 ~ 𝑁0,ℎ𝑖,𝑡 where 𝑖 = 1

and ℎ𝑖,𝑡 = 𝜔𝑖 + ∑ 𝛼𝑖,𝑛𝜀𝑖,𝑡−𝑛2𝑞𝑛=1 + ∑ 𝛽𝑖,𝑚ℎ𝑖,𝑡−𝑚

𝑝𝑚=1

(7)

where 𝜔𝑖 > 0

𝛼𝑖,𝑛 ≥ 0 for n = 1,2,…,q

𝛽𝑖,𝑚 ≥ 0 for m = 1,2,…,p. where return residuals 𝜀𝑖,𝑡 are conditionally drawn from a Gaussian distribution

with a mean of zero and a time-varying variance ℎ𝑖,𝑡, given the information set

represented by time process I up to time t-1. The GARCH(p,q) conditional

variance specification includes mean variance parameter 𝜔𝑖 and ‘q’ ARCH-terms

∑ 𝛼𝑖,𝑛𝜀𝑖,𝑡−𝑛2𝑞𝑛=1 as in (6), but further incorporates lagged estimates of its own value

over the past ‘p’ periods, ∑ ℎ𝑖,𝑡−𝑚𝑝𝑚=1 , into its evolutionary process. Corresponding

beta coefficients 𝛽𝑖,𝑚 characterize GARCH-effects, or the degree to which the

influence of volatility shocks observed in past periods decay over time. The

GARCH model is similar to EWMA in the sense that both models employ

exponential smoothing and weight recent information more heavily in their

15 determination. To ensure the conditional variance of 𝜀𝑖,𝑡 is stationary, parameters

𝜔𝑖,𝛼𝑖,1, … ,𝛼𝑖,𝑞 ,𝛽𝑖,1, … ,𝛽𝑖,𝑝 must be restricted such that ∑ 𝛼𝑖,𝑛𝑞𝑛=1 + ∑ 𝛽𝑖,𝑚 < 1𝑝

𝑚=1 ,

and ℎ𝑖,𝑡 > 0 for all t.

The sum 𝛼𝑖,𝑛 + 𝛽𝑖,𝑚 measures the persistence of the conditional variance to

volatility shocks observed over previous periods; shocks become increasingly

persistent as this measure approaches ‘1’ from below. In fact, Bollerslev and Engle

(1986) describe a restricted form of the GARCH model wherein ∑ 𝛼𝑖,𝑛𝑞𝑛=1 +

∑ 𝛽𝑖,𝑚 = 1𝑝𝑚=1 . This Integrated GARCH (IGARCH) model introduces a unit-root

into the GARCH specification and can be used to model highly persistent, non-

stationary time series. Other parametric specifications of GARCH-type models

exist to accommodate for other common features of financial time series - such as

leptokurtosis (through conditional non-normality of error process 𝜀𝑖,𝑡), asymmetric

dynamics in ARCH/GARCH processes (captured by GJR-GARCH and EGARCH

models, respectively), and more - but these are not pertinent to the analysis that

follows and so will not be covered here.

Engle and Rangel (2008) modify the GARCH to accommodate for a long-

term trend in the volatility process of returns. Their Spline-GARCH model

decomposes volatility into high- and low-frequency components - represented by a

mean-reverting unit GARCH and a slowly varying deterministic component,

respectively. Supposing again that the return to asset i observed at time period t

can be denoted as 𝑟𝑖,𝑡, the Spline-GARCH model can be specified as:

𝑟𝑖,𝑡 = µ + 𝜀𝑖,𝑡 (8)

where 𝜀𝑖,𝑡 = ℎ𝑖,𝑡𝜏𝑡𝑧𝑡 and 𝑖 = 1

16

and ℎ𝑖,𝑡 = 𝜔𝑖 + 𝛼𝑖𝜀𝑖,𝑡−12 + 𝛽𝑖ℎ𝑖,𝑡−1 where 𝜔𝑖,𝛼𝑖 ,𝛽𝑖 > 0 (9)

and 𝜏𝑖,𝑡 = 𝑅𝑒𝑒 ∑ 𝜙𝑖,𝑘𝑡𝑖 − 𝑡𝑖,𝑘2𝐾

𝑘=1 (10)

where the idiosyncratic part of returns, 𝜀𝑖,𝑡, are multiplicatively decomposed into

two parts. The high-frequency component of volatility, ℎ𝑖,𝑡, is specified as a

GARCH(1,1) process and describes transitory volatility behavior that, while

perhaps persistent, has no impact on long-term levels of market volatility.3 The

low-frequency component of volatility, 𝜏𝑡, is intended to capture lower frequency

variations in volatility, such as trends and seasonalities. This measure is

approximated non-parametrically using a quadratic spline with k knots, or equally

spaced intervals 𝑡1, 𝑡2, … , 𝑡𝐾, over time horizon T. This number of knots, k,

governs the cyclical pattern in the low-frequency trend of volatility - which can be

appropriately chosen according to information criterion. Low values of k imply less

frequent cycles and result in smoother low-frequency volatility processes. The

‘sharpness’ of each cycle is governed by its respective spline coefficient 𝜙. In fact,

the GARCH(1,1) could be expressed as a restricted version of the Spline-GARCH

in which 𝜙𝑖,1 = 𝜙𝑖,2 = ⋯ = 𝜙𝑖,𝑘 = 0. Parameter 𝑧𝑡 represents an independent error

term.

Engle and Rangel drew upon the cross-sectional behavior of the spline

component 𝜏𝑖,𝑡 across fifty countries (and up to fifty years of data) to show that

volatility in macroeconomic factors such as GDP, inflation, growth and interest

rates are important determinants of equity market volatility. The Spline-GARCH

3 Though less common, the specification in (9) can be generalized to account for more lags in the conditional

variance. The Spline-GARCH(p,q) specifies ℎ𝑖,𝑡 as a GARCH(p,q) [ see: equation (7) ]

17 component specification serves as inspiration, and provides a framework, for the

GARCH-MIDAS model introduced in Subsection 2.3.1.

2.2.2 Range-based Conditional Volatility Models

An area of ongoing research is in the use of range-based volatility proxies in

various traditionally return-based conditional volatility models. The intuition

behind these model adaptions is that the range data represents a less-noisy proxy of

volatility, in theory, compared to measures of variance computed on close-to-close

returns. Alizadeh, Brandt, and Diebold (2002) first estimated stochastic volatility

models using daily FX range-data with favorable results, and also showed that the

log-range is approximately Gaussian and robust to microstructure noise. Other

researchers have since adapted the conditional variance specifications of various

GARCH-type models to utilize price range data and estimate measures of

conditional volatility.

Brandt and Jones (2006) propose the Range-based EGARCH (REGARCH)

model which, as the name implies, is a modification of the conventional EGARCH

model. The REGARCH employs the square root of the intraday price range in

place of absolute returns and has been demonstrated to provide greater predictive

accuracy, compared to its return-based counterpart, in out-of-sample forecasts of

volatility in the S&P500 index.

Both Mapa (2003) and Molnar (2012) individually modified the GARCH(p,q)

model to utilize the scaled high-low range estimator of Parkinson (1980), in place of

asset return-based variance, to generate efficient measures of daily volatility. Mapa

(2003) proposes the GARCH Parkinson Range (GARCH-PARK-R) as a less-costly

alternative to the realized volatility model in generating estimates and forecasts of

daily volatility since, with respect to data requirements, use of the latter model

18 necessitates access to intraday price data for input. Mapa shows that the GARCH-

PARK-R outperforms realized volatility in forecasting the daily Philippine Peso-

U.S. Dollar exchange rate from 1997 to 2003 and posits this estimator could be

well-suited to applications to emerging markets which can sometimes lack in

financial infrastructure and/or availability of high-frequency intraday data. Molnar

(2012) introduces his model as the Range GARCH (RGARCH). The variance

specification of the RGARCH is a straightforward modification of the GARCH, and

ease of estimation is emphasized. Molnar shows that the RGARCH(1,1)

outperforms the standard GARCH model, both in- and out-of-sample, in empirical

tests involving 30 stocks, 6 stock indices, as well as simulated volatility processes.

Another particularly relevant range-based adaptation of GARCH factors

into the component models of volatility and correlation proposed in Section 3 of

this paper, and will be employed in the empirical experiments that follow, so it is

certainly worth discussing here. The Conditional Autoregressive Range (CARR)

model, proposed by Chou (2005), specifies the conditional range of an asset’s price,

within a fixed time interval, to evolve according to GARCH-like dynamics. In his

paper, Chou emphasizes ease of estimation and demonstrates that the CARR

outperforms the conventional GARCH in predictive accuracy for out-of-sample

weekly volatility forecasts of the S&P500 index.

Suppose that 𝑃𝑖,𝜏 represents the logarithmic price of asset i, which could be

governed by geometric Brownian motion with stochastic volatility. The observed

range ℜ𝑖,𝑡 measured at discrete intervals (e.g. daily), for asset price 𝑃𝑖,𝜏 with a

discrete-path sampled at finer intervals (e.g, every 5 minutes), can be defined as:

19

ℜ𝑖,𝑡 = 𝑀𝑅𝑒𝑃𝑖,𝜏 − 𝑀𝑖𝑅𝑃𝑖,𝜏 (11)

𝜏 = 𝑡 − 1, 𝑡 − 1 +1𝑅

, 𝑡 − 1 +2𝑅

, … , 𝑡.

where n represents the number of partitions or equally-spaced intervals at which

𝑃𝑖,𝜏 is sampled along its price path, within each range-measured interval. The finer

the sampling interval of the price path is (i.e. every 5 minutes versus hourly), the

more accurate the measured range will be. Chou (2005) specifies the CARR(p,q)

model as:

ℜ𝑖,𝑡 = 𝜆𝑖,𝑡𝜀𝑖,𝑡 (12)

𝜆𝑖,𝑡 = 𝜔𝑖 + ∑ 𝛼𝑖,𝑛ℜ𝑖,𝑡−𝑛2𝑞

𝑛=1 + ∑ 𝛽𝑖,𝑚𝜆𝑖,𝑡−𝑚𝑝𝑚=1 (13)

where 𝜔𝑖 > 0

𝛼𝑖,𝑛 ≥ 0 for n = 1,2,…,q

𝛽𝑖,𝑚 ≥ 0 for m = 1,2,…,p.

∑ 𝛼𝑖,𝑛𝑞𝑛=1 + ∑ 𝛽𝑖,𝑚

𝑝𝑚=1 < 1

and 𝜀𝑖,𝑡|𝐼𝑡−1 ~ 𝑅𝑒𝑒 (1) (14)

where 𝜆𝑖,𝑡 represents the conditional mean of the range based on the information

set up to time t, and closely resembles the conditional variance specification of the

GARCH(p,q) model shown in (7). Impact parameters 𝛼𝑖,𝑛 and 𝛽𝑖,𝑚 respectively

characterize the short- and long-term effects of shocks to the range, and their sum

∑ 𝛼𝑖,𝑛𝑞𝑛=1 + ∑ 𝛽𝑖,𝑚

𝑝𝑚=1 regulates the persistence of shocks to conditional

autoregressive range measures. The disturbance term, or normalized range

20 𝜀𝑖,𝑡 = ℜ𝑖,𝑡

𝜆𝑖,𝑡, is assumed to follow a probability density function with a unit mean -

and it can be specified to fit either an exponential or Weibull distribution.

Parameters 𝜔𝑖,𝛼𝑖,𝑛,𝛽𝑖,𝑚 are restricted so that 𝜆𝑖,𝑡 > 0 for all values of t and to

ensure stationarity. The unconditional mean of range, denoted as 𝜔𝚤, can be

calculated as 𝜔𝑖/(1 − (∑ 𝛼𝑖,𝑛𝑞𝑛=1 + ∑ 𝛽𝑖,𝑚

𝑝𝑚=1 )).

The CARR model, like the REGARCH, is not a conditional variance

specification, it is a dynamic model of the conditional mean of the range; it is

estimated by fitting the conditional distribution of range, rather than the

conditional variance of returns (as is the case with GARCH, GARCH-PARK-R,

and RGARCH). It can still, however, be successfully utilized as a range-based

estimator for conditional volatility based on the well-established statistical property

that the range is an estimator of the standard deviation for any random variable.

The results of Parkinson (1980) and Lo (1991) support this concept by showing the

range of the distribution for any random variable to be proportional to its standard

deviation.

A particularly convenient characteristic of the CARR model is in its ease of

estimation. The likelihood function of a CARR model with an exponential density

function is identical to that of the GARCH model with a normal density function,

but with a simple adjustment to the conditional mean specification. Engle and

Russell (1998) show that the CARR model can be consistently estimated by QMLE

with a unit mean exponential density function for the residual term 𝜀𝑖,𝑡.

Furthermore, a proof is provided to suggest that the QMLE estimation of an

exponential CARR can be attained by estimating a GARCH model but for the

square root of range and without a constant term in the mean equation. As such,

21 any programming environment or statistical software that is capable of estimating

the GARCH model can also be used to easily estimate the CARR.

2.2.3 Multivariate Conditional Volatility Models

While the empirical properties and stylized facts of individual asset returns are

easily recognized and widely accepted, so too is observation that these volatilities

tend to move together over time - across assets and markets. Furthermore,

estimates of covariance or correlations are required inputs for numerous decision

tools which are regularly applied by individuals and institutions operating in the

financial markets alike. Better estimates of these inputs allow for better decisions

to be made across many areas of finance including portfolio selection, hedging and

risk management, asset pricing and option pricing. Co-volatilities can be

accommodated for, and estimates of conditional correlations can be obtained, by

employing a Multivariate GARCH (MGARCH) modelling framework.

Various MGARCH models have been proposed which differ in how they trade

off flexibility and parsimony in their specifications for the time-varying conditional

covariance matrix 𝐻𝑡. Increased flexibility allows for more complex conditional

covariance processes to be captured, while increased parsimony restricts parameter

proliferation, making estimation feasible for higher-dimensional systems. The

discussion in this subsection will focus on conditional correlation MGARCH models

and the widely applied DCC model in particular.4

The Conditional correlation (CC) family of models use nonlinear combinations

of univariate GARCH models to represent the conditional covariances. In these

models, the conditional covariance matrix 𝐻𝑡 can be specified as:

4 See Engle (2009) and Bauwems et al. (2006) for a thorough introduction and review of MGARCH models.

22

𝐻𝑡 = 𝐷𝑡12𝑅𝑡𝐷𝑡

12

where ℎ𝑖𝑖 = 𝜌𝑖𝑖,𝑡𝜎𝑖,𝑡𝜎𝑖,𝑡 (15)

and 𝐷𝑡 =

⎝

⎜⎛𝜎1,𝑡2 0 ⋯ 00 𝜎2,𝑡

2 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜎𝑚,𝑡

2⎠

⎟⎞

where 𝐻𝑡 is decomposed into a matrix of conditional correlations 𝑅𝑡 and a diagonal

matrix of conditional variances 𝐷𝑡. Each diagonal element of 𝐷𝑡 represents the

conditional variance of the ith asset 𝜎𝑖,𝑡2 . Each of these values are specified by, and

evolve according to, the univariate GARCH(p,q) specification shown in Equation

(7). The parameter 𝜌𝑖𝑖,𝑡 characterizes the extent to which error processes 𝜀𝑖 and 𝜀𝑖

- from the univariate GARCH conditional variance estimation of assets i and j -

move together. The various CC MGARCH specifications differ only in how they

parameterize the conditional correlations 𝑅𝑡.

Bollerslev (1990) proposes the Constant Conditional Correlation (CCC)

MGARCH model wherein the correlation matrix is time invariant.5 The CCC

represents a near complete trade-off of flexibility for parsimony and is widely

viewed as being too restrictive for most applications. Tse and Tsui (2002) propose

a more flexible specification - the Varying Conditional Correlation (VCC)

MGARCH model - in which each period’s conditional correlations are computed as

a weighted sum of a time invariant component, a measure of the recent correlations

5 For the CCC, the time-invariant correlation matrix is specified as: 𝑅 =

1 𝜌12 ⋯ 𝜌1𝑚𝜌12 1 ⋯ 𝜌2𝑚⋮ ⋮ ⋱ ⋮

𝜌1𝑚 𝜌2𝑚 ⋯ 1

23 among the residuals, and the estimate of conditional correlation from the previous

period. To preserve parsimony, all conditional correlations are restricted to follow

the same dynamics.

A closely related specification and perhaps the most widely applied

multivariate volatility modelling framework in research and in practice is the DCC.

The Dynamic Conditional Correlation (DCC) MGARCH model was first

introduced in Engle (2002). Like the VCC MGARCH model, the DCC employs a

time-varying parameterization of the conditional correlation matrix 𝑅𝑡 and thus

offers significantly more flexibility compared to the CCC model. Following the CC

framework in (15) the conditional correlation matrix, 𝑅𝑡, of the DCC model is

parameterized as:

𝑅𝑡 = 𝑑𝑖𝑅𝑅(𝑄𝑡)−1/2𝑄𝑡𝑑𝑖𝑅𝑅(𝑄𝑡)−1/2

=

1 𝜌12,𝑡 ⋯ 𝜌1𝑚,𝑡𝜌12,𝑡 1 ⋯ 𝜌2𝑚,𝑡⋮ ⋮ ⋱ ⋮

𝜌1𝑚,𝑡 𝜌2𝑚,𝑡 ⋯ 1

(16)

and 𝑞𝑖𝑖,𝑡 = (1 − 𝛼 − 𝛽)𝑟𝑖𝑖 + 𝛼𝜀,𝑡−1𝜀,𝑡−1 + 𝛽𝑞𝑖𝑖,𝑡−1

where 𝛼 ≥ 0, 𝛽 ≥ 0, and 0 ≤ 𝛼 + 𝛽 ≤ 1

(17)

where 𝜀 represents a vector of m standardized residuals from conditional variance

estimation.6 As such, each element 𝜌𝑖𝑖,𝑡 of the conditional correlation matrix

represents a standardization of the quasicorrelation estimate 𝑞𝑖𝑖,𝑡 - which is

computed upon the cross-product of standardized residuals from the univariate

conditional variance estimation of assets i and j and evolves according to the

GARCH(1,1)-like process shown in Equation (17). Elements of 𝑄 are not

6 Disturbance terms from univariate conditional variance estimation are standardized as: = 𝜀

𝑑𝑖𝑑𝑑(𝐷𝑡)

24 standardized to be correlation estimates and so are referred to as ‘quasicorrelations’.

Parameter 𝑅 represents the unconditional correlation matrix, with element 𝑟𝑖𝑖

denoting the observed sample correlation between assets i and j. Parameters 𝛼 and

𝛽 govern the GARCH-like dynamics of conditional quasicorrelations - which are

restricted to ensure positive definiteness of the conditional covariance matrix 𝐻𝑡,𝐷𝐷𝐷.

For demonstrative purposes, the bivariate-DCC quasicorrelation matrix can be

constructed by:

𝑞1,𝑡 𝑞12,𝑡𝑞12,𝑡 𝑞2,𝑡

= 1 𝑞12𝑞12 1 o 1 − 𝑅1 − 𝑏1 1 − 𝑅3 − 𝑏3

1 − 𝑅3 − 𝑏3 1 − 𝑅2 − 𝑏2

+ 𝑅1 𝑅3𝑅3 𝑅2 o

𝜀1,𝑡−12 𝜀1,𝑡−1

2 𝜀2,𝑡−12

𝜀1,𝑡−12 𝜀2,𝑡−1

2 𝜀2,𝑡−12 + 𝑏1 𝑏3

𝑏3 𝑏2 o

𝑞1,𝑡−1 𝑞12,𝑡−1𝑞12,𝑡−1 𝑞2,𝑡−1

where 𝑞12 = 1𝑇∑ 𝜀1,𝑡𝜀2,𝑡𝑇𝑡=1

(18)

A particularly convenient feature of the DCC-GARCH is that the

conditional covariance matrix can be decomposed into a matrix of conditional

correlations 𝑅𝑡 and a diagonal matrix of conditional variances 𝐷𝑡. And so, these

matrices can be estimated separately in a two-stage estimation process. Taking

advantage of this characteristic as well as the statistical and asymptotic similarities

between the CARR and GARCH models, Chou et al. (2009) propose the range-

based DCC-CARR model wherein the CARR(p,q) model is employed in first-stage

conditional volatility estimation, in place of the conventional GARCH(p,q).

Compared to its return-based counterpart and a number of other benchmark

estimators, Chou et al. showed that the bivariate DCC-CARR is more accurate and

efficient in estimating and forecasting covariance for empirical applications

involving the S&P500, NASDAQ, and 10-year Treasury bond yield. Chou and Liu

25 (2010) further demonstrate the economic value of the DCC-CARR versus the DCC-

GARCH in a low-dimensional mean-variance portfolio allocation experiment.

2.3 Mixed Data Sampling Methods (MIDAS)

2.3.1 GARCH-MIDAS

This class of model is called GARCH-MIDAS because it uses a mean-reverting

unit daily GARCH process, inspired by the Spline-GARCH model of Engle and

Rangel (2008), and a Mixed-Data Sampling (MIDAS) polynomial applied to low-

frequency (weekly, monthly, quarterly, etc.) macroeconomic or financial variables.

Following Engle, Ghysels, and Sohn (2013) the basic GARCH-MIDAS model can

be specified as:

𝑟𝑖,𝑡 = µ + 𝜏𝑡𝑅𝑖,𝑡𝑅𝑖,𝑡 where 𝑅𝑖,𝑡|𝐼𝑖−1,𝑡 ~ 𝑁(0,1) (19)

git = (1 − α − β) + 𝛼𝑟𝑖−1,𝑡 − µ

2

𝜏𝑡+ 𝛽𝑅𝑖−1,𝑡

(20)

𝜏𝑡 = 𝑚 + 𝜃𝜓𝑘(𝑤)𝑉𝑡−𝑘

𝐾

𝑘=1

(21)

where 𝑉𝑡 = ∑ 𝑟𝑖,𝑡2𝑁𝑖=1

and 𝜓𝑘(𝑤) ∝ 1 − 𝑘𝐾𝑤−1

or 𝜓𝑘(𝑤) ∝ 1 − 𝑘𝐾𝑤1−1

𝑘𝐾𝑤2−1

where 𝑟𝑖𝑡 represents an asset return observed at the ith high-frequency interval of

low-frequency period t, and N is a scalar integer that specifies the aggregation

periodicity between our high- and low-frequency data series. For instance, 𝑟𝑖𝑡 could

26 represent an asset’s return on the ith day of week t in the case of ‘daily to weekly’

temporal aggregation (where N=5, to approximate ~5 trading days per week).

Other possible parameterizations could include those in which N equals 22, 63, 126

or 252 to approximate daily to monthly, quarterly, semiannually or yearly

aggregation, respectively.

The location-parameter, or unconditional mean of asset returns in this case,

is denoted by µ. The conditional variance is decomposed into two-parts: a short-

run ‘transient’ component of volatility 𝑅𝑖,𝑡 that follows the GARCH(1,1)-like

recursion shown in Equation (20), and a long-run ‘secular’ component of volatility

𝜏𝑡 which is based on a lagged-history 𝑉𝑡−1,𝑉𝑡−2, … ,𝑉𝑡−𝑘 of the asset’s realized

volatility (defined as the sum of daily squared returns) over each of the past K low-

frequency periods 𝑡1, 𝑡2, … , 𝑡𝑘.7 The lagged terms in the secular component can be

filtered by MIDAS weights as governed by either the single- or double-parameter

beta polynomial equations shown by 𝜓𝑘(𝑤).8 An appropriate number of lags K*

can be chosen, for instance, according to Bayesian Information Criterion (BIC),

Akaike Information Criterion (AIC), or by ‘profiling’ the likelihood function for

various levels of K - as described by Engle, Ghysels, and Sohn. Parameter 𝑅𝑖,𝑡

represents the independent standardized residual on the ith day of low-frequency

period t.

7 Alternatively, the long-run component can be, and frequently is, determined by the low-frequency average value of some exogenous variable or macroeconomic series 𝑉𝑡 = 1

𝑁∑ 𝑒𝑖,𝑡𝑁𝑖=1 where 𝑒𝑖,𝑡 is fixed for 𝑖 = 1, … , N

8 In addition to the simple beta polynomial specifications that are shown here and employed later in the experiments that follow, various other parameterizations for the polynomial lag structure exist which could be used. Refer to Ghysels, Sinko, Valkanov (2006) and Foroni, Marcellino, and Schumacher (2015) for example.

27

2.3.2 DCC-MIDAS

The DCC-MIDAS model of Colacito, Engle and Ghysels (2011) is a

multivariate extension to the GARCH-MIDAS for 𝑚-dimensional time series data.

Through utilization of mixed-data sampling methods, and inspired by its univariate

counterpart presented in the preceding chapter, the DCC-MIDAS builds upon the

multivariate DCC-GARCH by further decomposing the conditional correlation

specification into two-parts. Accordingly, DCC-MIDAS correlation estimates are

obtained through combined estimation of short-run ‘transient’ components, as well

as long-term ‘secular’ components, of conditional correlations. Like the DCC-

GARCH, the DCC-MIDAS employs a two-step specification and estimation

strategy wherein the conditional covariance matrix is decomposed into a diagonal

matrix of 𝑚 conditional variances 𝐷𝑡 - and an 𝑚 𝑒 𝑚 conditional correlation matrix

𝑅𝑡 that evolves over time but with GARCH-MIDAS-like dynamics:

𝐻𝑡,𝐷𝐷𝐷−𝑀 = 𝐷𝑡12𝑅𝑡𝐷𝑡

12

where ℎ𝑖𝑖 = 𝜌𝑖𝑖,𝑡𝜎𝑖,𝑡𝜎𝑖,𝑡 (22)

and 𝐷𝑡 =

⎝

⎜⎛𝜎1,𝑡2 0 ⋯ 00 𝜎2,𝑡

2 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜎𝑚,𝑡

2⎠

⎟⎞

In the first-stage of estimation, a matrix of conditional variances 𝐷𝑡 is constructed

such that each diagonal element 𝜎𝑖,𝑡2 is specified to follow a GARCH(p,q) or

GARCH-MIDAS process - as described in Sections 2.2.1 and 2.3.1 respectively.

Following Colacito, Engle and Ghysels - the second-stage of the DCC-MIDAS can

be formally specified as:

28

𝑅𝑡 = 𝑑𝑖𝑅𝑅(𝑄𝑡)−1/2𝑄𝑡𝑑𝑖𝑅𝑅(𝑄𝑡)−1/2

=

1 𝜌12,𝑡 ⋯ 𝜌1𝑚,𝑡𝜌12,𝑡 1 ⋯ 𝜌2𝑚,𝑡⋮ ⋮ ⋱ ⋮

𝜌1𝑚,𝑡 𝜌2𝑚,𝑡 ⋯ 1

(23)

𝑞𝑖𝑖,𝑡 = (1 − 𝑅 − 𝑏)𝑒𝑖𝑖,𝑡 + 𝑅𝜀𝑖,𝑡−1𝜀𝑖,𝑡−1 + 𝛽𝑞𝑖𝑖,𝑡−1

(24)

where 𝑒𝑡 = ∑ 𝜓𝑘(𝑤)𝑐𝑡−𝑘𝐾𝑘=1

and 𝜓𝑘(𝑤) ∝ 1 − 𝑘𝐾𝑤−1

(25)

where 𝑞𝑖𝑖,𝑡 represents element (i,j) in the quasi-correlation matrix 𝑄𝑡 - computed in

part upon the one-period lagged cross-products of standardized residuals between

the ith and jth series, 𝜀𝑖,𝑡−1𝜀𝑖,𝑡−1, from first-stage estimation. In the secular

component, 𝑒𝑖𝑖,𝑡 represents element (i,j) in the long-run quasi-correlation matrix 𝑃𝑡

- which also factors into quasi-correlation specification. This latter component, as

expressed by Equation (25), is calculated as the MIDAS-weighted sum of the

observed unconditional correlation matrices sampled at frequency t. Parameter K

specifies the discrete number of lags used in filtering the long-run component, and

thus governs the cyclical pattern in the low-frequency trend of correlations.

Appropriate parameterizations of K can be selected using the same approaches

alluded to in the previous chapter. Quasi-correlation estimates are also governed

by a GARCH-type beta coefficient 𝛽𝑖,𝑚 that characterizes the degree to which the

influence of conditional correlation shocks observed in past periods decay over time.

Holistically, the system formulated by Equations (23)-(25) exhibits GARCH-

MIDAS-like dynamics. 𝑄𝑡 is referred to as the ‘quasi-correlation’ matrix because,

as is the case with other DCC-type estimation methods, its parameters are not

29 standardized to be correlations. Quasi-correlation matrices are therefore rescaled so

that the diagonal elements are unity, as shown in Equation (23), so as to arrive at

an estimate of the conditional correlation matrix 𝑅𝑡.

In a minimum-variance portfolio allocation experiment involving various

combinations of international stock indices - Colacito, Engle and Ghysels (2011)

employ the GARCH(1,1) and DCC-MIDAS estimators in the first- and second-

stages of conditional covariance matrix estimation, respectively, and compare the

accuracy of these estimates against those of the DCC-GARCH. Their results

demonstrate that use of the DCC-MIDAS in second-stage estimation yields

improved predictive accuracy and economic efficiency gains, both in- and out-of-

sample, versus the standard DCC-GARCH. Furthermore, these efficiency gains are

shown to increase with portfolio size - providing a compelling indication that the

DCC-MIDAS could be better suited to estimating and forecasting higher-

dimensional covariance matrices versus conventional DCC-type estimation methods.

3 Range-based Component Models for

Conditional Volatility and Dynamic

Correlations

In this section a pair of novel range-based component models are proposed.

The inspiration behind these models is straightforward and follows naturally from

the progression of return- and range-based models of conditional volatility and

dynamic correlations presented throughout this paper. Essentially, the estimation

methods proposed here amount to nothing more than an amalgamation of existing

30 volatility modelling techniques, which are well-established in the literature or

widely applied in practice.

There are low costs associated with implementing the estimation strategies

proposed in this section; not necessarily with regard to considerations of model

parsimony or computational requirements, but rather concerning data and

programming requirements. Daily high- and low-prices are typically published for

all publicly traded stocks and made freely available alongside more conventionally

utilized daily closing prices. Furthermore, these models represent simple

adaptations of well-established volatility and correlation estimation methods such

that model estimations can be attained by making only minor modifications to

existing statistical software. Thus, ease of estimation is emphasized.

These range-based models of volatility and correlation are not the first of their

kind. Instead, they represent unique contributions to the ongoing development of

advanced volatility and correlation estimators which integrate various combinations

of daily price information into their estimation. The practical relevance of these

alternative range-based proxies rely upon the well-established statistical property

that the range is an estimator of the standard deviation of a random variable. This

concept is supported by both Parkinson (1980) and Lo (1991) whose results

demonstrate the range of any distribution to be proportional to its standard

deviation.

3.1 Range-based DCC-MIDAS

The multivariate component model presented in this subsection, like many

existing DCC-type estimation methods, employs a two-stage specification and

estimation strategy. However, this model differentiates itself by employing range-

31 based conditional volatility models in the first-stage of estimation and decomposing

conditional correlations into short- and long-run components. Following the

nomenclature established in the literature, this model is proposed as the

‘Range-based DCC-MIDAS’ (or the ‘DCC-CARR-MIDAS’):

𝑅𝑡 = 𝑑𝑖𝑅𝑅(𝑄𝑡)−1/2𝑄𝑡𝑑𝑖𝑅𝑅(𝑄𝑡)−1/2

=

1 𝜌12,𝑡 ⋯ 𝜌1𝑚,𝑡𝜌12,𝑡 1 ⋯ 𝜌2𝑚,𝑡⋮ ⋮ ⋱ ⋮

𝜌1𝑚,𝑡 𝜌2𝑚,𝑡 ⋯ 1

(27)

where 𝑞𝑖𝑖,𝑡 = (1 − 𝑅 − 𝛽)𝑒𝑖𝑖,𝑡 + 𝑅𝜀,𝑡−1𝜀,𝑡−1 + 𝛽𝑞𝑖𝑖,𝑡−1 (28)

and 𝑒𝑡 = ∑ 𝜓𝑘(𝑤)𝑐𝑡−𝑘𝐾𝑘=1

𝜓𝑘(𝑤) ∝ 1 − 𝑘𝐾𝑤−1

(29)

and 𝐷𝑡 =

𝜆1,𝑡 0 ⋯ 00 𝜆2,𝑡 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ 𝜆𝑚,𝑡

(30)

Specification and estimation of the Range-based DCC-MIDAS is nearly

identical, in form and function, to that of the conventional DCC-MIDAS

conditional correlation estimator but with the key distinguishing characteristic

being that this model is range-based. Each diagonal element 𝜆𝑖,𝑡 of conditional

volatility matrix 𝐷𝑡 represents an estimate of the conditional autoregressive price

range of the ith asset - as specified by Equation (13). As such, this model could be

perceived simply as a range-based analog of the traditionally return-based DCC-

MIDAS model of Colacito, Engle and Ghysels (2011), wherein CARR(p,q) replaces

32 the return-based conditional variance estimation methods traditionally employed in

the first-stage of conditional covariance matrix estimation. One-period lagged

cross-products of the standardized residuals from first-stage conditional range

estimation 𝜀,𝑡−1𝜀,𝑡−1 are then passed through to the specification presented in

Equation (27). A DCC-MIDAS-type estimation of the conditional quasi-correlation

matrix is then performed. Finally, quasi-correlation matrix estimates are rescaled,

according to Equation (26), to arrive at an estimate of the conditional correlation

matrix 𝑅𝑡. This subtle transformation reflects a variation on the same theme

which motivates the range-based adaptation of the DCC-GARCH. Since the use of

range-based volatility proxies in first-stage of estimation could translate to less-

noisy error processes - so to could these modifications result in improved second-

stage estimation and forecasting of conditional correlations which have been

estimated on the basis of these residuals.

Alternatively, this model could instead be perceived as an extension of the

DCC-CARR model of Chou et al. (2009) wherein the DCC-MIDAS replaces the

single-component conditional correlation specification of 𝑅𝑡 in the second-stage of

conditional covariance matrix estimation. And so, the estimator proposed through

Equations (26)-(30) represents a simple amalgamation of existing dynamic

conditional correlation modelling techniques. With the antecedent models having

been described in full, and their dynamics being well-understood, the remainder of

this subsection will focus on the motivations behind the proposed model, as well as

a practical discussion of how to approach its estimation.

This model was developed, at least in part, to address some of the

anticipated shortcomings of the DCC-CARR. While Chou et al. (2009) postulate

that their DCC-CARR could be readily applied to modelling large systems of assets

33 - the empirical applications presented in that seminal paper, as well as the

applications which have followed, have been limited in scale and scope. The

superiority of the DCC-CARR in modelling bivariate and trivariate systems of

assets vis-à-vis its return-based counterpart has been well-established in the

literature - but the question of how this range-based adaptation performs when

modelling correlation matrices of larger systems of assets has remained unexplored

until this study. The shortcomings of the DCC-CARR are formally established in

the empirical application presented in Section 4 of this paper. The experimental

model presented in this chapter is formulated with the aim of, in theory, combining

aspects of accuracy and efficiency that can be recognized through use of a range-

based estimator like the DCC-CARR, with the robustness of the DCC-MIDAS in

modelling the conditional correlation matrices of larger asset systems.

Any statistical software package that is capable of estimating the DCC-

MIDAS can also estimate the Range-based DCC-MIDAS, with only a few

modifications needing to be made. For instance, version 2.2 of the ‘MIDAS

MATLAB Toolbox’ includes a DCC-MIDAS function which can be adapted to

estimate this proposed model. Quasi-maximum likelihood estimation of the Range-

based DCC-MIDAS, as proposed by Equations (26)-(29), can be attained by

overriding first-stage GARCH-MIDAS conditional variance estimation within the

‘DccMidas.m’ function. Native GARCH estimation functions can then be utilized

to estimate an exponential CARR model - as proposed in Engle and Russell (1998)

and as described in Subsection 2.2.2 of this paper - as a proxy for first-stage

conditional volatility estimation. Standardized residuals from first-stage of

estimation must then be constructed and input into the modified ‘DccMidas.m’

function, ahead of second-stage estimation. And so, the resulting correlation

matrices will be estimated on the basis of error processes that result from range-

34 based conditional volatility estimation, and exhibiting GARCH-MIDAS-like

dynamics.

An alternative univariate range-based component model is proposed in the

subsection that follows; one which could conceivably be employed in first-stage

estimation of the Range-based DCC-MIDAS, and to serve as an alternative to the

CARR(p,q).

3.2 CARR-MIDAS

The univariate component model that follows can be thought of as a range-

based analog to the GARCH-MIDAS model of Engle, Ghysels, and Sohn (2013)

wherein we seek to replace each of the mechanisms through which the individual

components of volatility are estimated in the conventional GARCH-MIDAS with

their range-based counterparts. Simply put - the short-run component of volatility

is governed by the CARR(1,1) of Chou (2005) in place of the GARCH(1,1), and

the long-run component of volatility is estimated according to temporally

aggregated measures of the scaled Realized Range, as proposed by Martens and

Van Dijk (2006), in place of the realized variance. These range-based estimators

have been shown in the literature, as highlighted throughout this paper, to provide

better estimates of conditional or historical volatility respectively vis-à-vis the

return-based mechanisms they replace.

Following the established nomenclature, this range-based analog of the

GARCH-MIDAS is proposed as the ‘CARR-MIDAS’ model, which can be formally

expressed as:

35

ℜ𝑖,𝑡 = 𝜏𝑡𝑅𝑖,𝑡𝜀𝑖,𝑡 where 𝜀𝑖,𝑡|𝐼𝑖−1,𝑡 ~ 𝑅𝑒𝑒(1) (31)

𝑅𝑖,𝑡 = (1 − 𝛼 − 𝛽) + 𝛼ℜ𝑖−1,𝑡2

𝜏𝑡+ 𝛽𝑅𝑖−1,𝑡 (32)

𝜏𝑡 = 𝑚 + 𝜃𝜓𝑘(𝑤)𝑅𝑅𝑡−𝑘𝑆𝐾

𝑘=1

(33)

𝜓𝑘(𝑤) ∝ 1 −𝑘𝐾𝑤−1

𝑅𝑅𝑡𝑠 = 𝑅𝑅𝑖𝑠𝑁

𝑖=1

(34)

where 𝑅𝑅𝑖𝑠 = ∑ (log𝐻𝑖−𝑞−log𝐿𝑖−𝑞)𝑄∗𝑞=1

∑ 𝑅𝑅𝑖𝑄∗𝑞=1

𝑅𝑅𝑖 (35)

and 𝑅𝑅𝑖 = 14 𝑙𝑙𝑑 2

∑ 𝑙𝑙𝑅𝐻𝑖,𝑡 − 𝑙𝑙𝑅 𝐿𝑖,𝑡2𝑁

𝑛=1 (36)

where ℜ𝑖,𝑡 represents the log price range of a particular asset observed at the ith

high-frequency interval of low-frequency period t - ℜ𝑖,𝑡 = (log𝐻𝑖,𝑡 − log 𝐿𝑖,𝑡).

Parameter N represents the scalar integer which specifies the aggregation

periodicity between the high- and low-frequency data series. Many of the

parameters and much of the dynamics governing the CARR-MIDAS are shared

with the GARCH-MIDAS. Since the return-based model is well understood and

has previously been described in full, the remaining discussion will focus towards

those characteristics that distinguish the CARR-MIDAS from the GARCH-MIDAS

and the intuition behind these changes. The conditional specification of the

CARR-MIDAS, like its return-based counterpart, is decomposed into two-parts.

Taking advantage of the statistical and asymptotic similarities between the CARR

and GARCH models - the short-term, high-frequency component of volatility 𝑅𝑖,𝑡 is

36 specified to exhibit an exponential CARR(1,1)-like recursion shown in Equation

(31). Daily measures of Realized Range are first computed as proposed by

Parkinson (1980), as shown in Equation (35). These measures are then scaled

based on the past 𝑄∗ trading days, as proposed by Martens and Van Dijk (2006),

and as demonstrated in Equation (34). Finally, the scaled Realized Range

estimates are temporally aggregated using the familiar MIDAS approach shown in

Equation (33). The long-term secular component of volatility 𝜏𝑡 is then computed

on the lagged history 𝑅𝑅𝑡−1,𝑅𝑅𝑡−2, … ,𝑅𝑅𝑡−𝑘 of an asset’s temporally aggregated

measures of scaled Realized Range over each of the past K low-frequency time

periods 𝑡1, 𝑡2, … , 𝑡𝑘 , as per Equation (32). The estimated conditional price range

of the asset, observed at the ith high-frequency interval of low-frequency period t, is

represented by 𝜆𝑖,𝑡 = 𝜏𝑖,𝑡𝑅𝑖,𝑡 .

As is also the case with the traditional GARCH-MIDAS - selection of an

appropriate number of lags 𝐾∗ is left to the discretion of the investor, and so too is

selection of an appropriate number of trading days 𝑄∗ to use in computing a

scaling factor for the realized range series. Appropriate values of these parameters

will likely differ on an application-by-application basis. The considerations for

setting 𝑄∗ are exactly as described in the Scaled Realized Range model of Martens

and Van Dijk (2006) and thus depend on the dynamics of microstructure frictions

for the price range data series in question. If trading intensity and bid-ask spread

were to remain constant, accuracy could be gained by setting 𝑄∗ as high as possible.

However, since the magnitude of these microstructure frictions tend to vary over

time, only a recent price history should be used and so 𝑄∗ should not be set too

large. The issue of appropriate 𝑄∗ parameterization for the CARR-MIDAS model

will be partially explored in the empirical experiment that follows.

37

Any statistical software package that is capable of estimating the GARCH-

MIDAS can also estimate the CARR-MIDAS model, with only a few minor

modifications needing to be made. For instance, version 2.2 of the ‘MIDAS

MATLAB Toolbox’ includes a GARCH-MIDAS function which can be adapted to

estimate this proposed model. Quasi-maximum likelihood estimation of the

exponential CARR-MIDAS model, as proposed by Equations (30)-(35), can be

attained by estimating the GARCH-MIDAS but for the square root of range,

without a constant term µ in the mean equation, and with the low-frequency scaled

realized range series (separately computed as shown above) input as an exogenous

variable for the purpose of long-term volatility component estimation.

CARR-MIDAS performance can only be assessed indirectly, relative to the

established single-component CARR, in the multivariate empirical application that

follows. Each of these univariate models are separately employed in the first-stage

of Range-based DCC-MIDAS conditional correlation matrix estimation and

forecasting in this dynamic asset allocation exercise. By comparing the forecasting

performance of these two estimators, gains (or losses) to predictive accuracy which

can be attributed to use of each of these alternative univariate range-based

volatility estimation methods can be isolated. A more direct, univariate assessment

of this models performance will remain a topic for further research.

4 Empirical Application

In this empirical experiment, the forecasting accuracy of a select variety of

return- and range-based covariance matrix estimation techniques is assessed and

compared. The estimators tested in this dynamic asset allocation experiment are

38 tasked with modelling a vast array of simulated asset systems and optimally

allocating portfolio weights across low- and mid-to-high dimensional systems alike.

This study is analogous to an empirical application presented by Colacito,

Engle and Ghysels (2011) but with a few contrasting differences. First and

foremost, this study primarily follows a range-based framework. A set of novel

Range-based DCC-MIDAS conditional covariance matrix estimators, as proposed in

the preceding section, are employed. Furthermore, the DCC-CARR model serves

as the ‘benchmark’ estimator in this study - in place of the traditional DCC-

GARCH (though the latter return-based model is also tested and included for

comparison). Furthermore, the detailed analysis presented in this study is

extended from asset systems consisting of six major international equity market

indices to eight - and so higher-dimension portfolio allocation problems can be

considered in depth.

In a novel line of inquiry, the suitability of the DCC-CARR(1,1) in modelling

larger asset systems vis-à-vis its return-based counterpart, the DCC-GARCH(1,1),

is evaluated for the first time. Simultaneously, univariate range-based models of

conditional volatility - the CARR(1,1) and a variety of parameterizations for the

CARR-MIDAS model proposed in this paper - are employed in the first-stage of

DCC-MIDAS conditional covariance matrix estimation, as previously described.

The various CARR-MIDAS specifications employed in the first-stage of Range-

based DCC-MIDAS conditional covariance matrix estimation parameterize the

exogenous Realized Range scaling factor 𝑄∗, from the long-term component of

conditional volatility, at a select variety of levels.

All return- and range-based covariance matrix estimators are evaluated within

a minimum-variance portfolio risk management framework. This approach

overcomes some of the difficulties associated with directly evaluating forecasting

39 accuracy for latent processes such as covariance and represents a practical

application allowing for the economic significance of our models to be evaluated

across a range of portfolio sizes. While model performance could alternatively be

analyzed through a mean-variance portfolio optimization approach, a minimum-

variance portfolio risk management framework abstracts away from the difficult

task of simultaneously estimating expected returns. The estimators in this study

are compared using the economic loss function proposed by Engle and Colacito

(2006) - whose approaches are integrated into and described in the subsection that

follows.

4.1 Methodology and Data

Suppose that a risk-averse investor chooses optimal portfolio weights across M

securities in order to minimize the expected one-day-ahead portfolio variance. The

asset allocation problem can be formulated as:

minwt

wt′Htwt

s.t. wt′µ = µ0

(36)

where 𝑤𝑡 is the vector of portfolio weights for time t chosen at time t–1, 𝐻𝑡 is the

forecasted one-period ahead conditional covariance matrix of a vector of excess

returns, 𝜇 is the estimated vector of excess returns relative to the risk-free asset,

and 𝜇0 > 0 is the required return constraint. The variance minimizing solution to

the problem formulated in Equation (36) is:

𝑤𝑡 = 𝐻𝑡−1µ µ′𝐻𝑡−1µ

µ0 (37)

40 For a portfolio consisting of M risky securities and with 𝑤𝑖,𝑡 representing the

percentage of total portfolio value invested in risky security i for time t, ∑ 𝑤𝑖,𝑡𝑁𝑖=1

need not be equal to ‘1’. Investors are permitted to lend at the risk-free rate; they

can allocate their portfolio budget between the tangency portfolio and a risk-free

asset with zero standard deviation and zero correlation with other asset classes

(such as treasury bills or cash equivalents). The percentage of total portfolio value

invested in the risk-free asset as such is 1 − ∑ 𝑤𝑖,𝑡𝑁𝑖=1 .

Now suppose that Ω𝑡 represents the latent true conditional covariance

matrix. The portfolio of an investor choosing optimal portfolio weights on the

basis of the forecasted one-period ahead conditional covariance matrix 𝐻𝑡 would

end up with a standard deviation of:

σtµ0

= 𝐸𝑡−1[𝑤𝑡′(𝑟𝑡 − 𝐸𝑡−1𝑟𝑡)]2

µ0

= 𝑤𝑡′Ω𝑡𝑤𝑡

µ0

= µ′𝐻𝑡−1Ω𝑡𝐻𝑡−1µ

µ′𝐻𝑡−1µ (38)

If an investor had knowledge of the true conditional covariance matrix Ω𝑡, then the

optimal vector of portfolio weights would be 𝑤𝑡∗ = Ω𝑡−1µ

µ′Ω𝑡−1µµ0 with a resulting

portfolio standard deviation of:

σt∗

µ0 =

𝑤𝑡∗′Ω𝑡𝑤𝑡∗ µ0

41

= 1

µ′Ω𝑡−1µ (39)

It stands to reason that the variance of portfolios optimized on the basis of

incorrect, forecasted estimates of the conditional covariance matrix will always be

greater than what could be achieved with knowledge of the true covariance matrix.

In fact, Engle and Colacito (2006) show that 𝐸 1𝑇∑ (σt∗)2𝑇𝑡=1 ≤ 𝐸 1

𝑇∑ (σt)2𝑇𝑡=1 for

any suboptimal estimator of the conditional covariance matrix, and they provide

strategies for testing the accuracy and equivalency of covariance estimation

methods.

The strategy employed to test the ‘accuracy’ of any given covariance matrix

estimation method is to test whether the actual realized portfolio variance divided

by the forecasted portfolio variance has a conditional mean equal to ‘1’. Engle and

Colacito propose that this can be achieved simply by estimating 𝛽 in the regression:

(𝑤𝑡′𝑟𝑡)2

𝑤𝑡′𝐻𝑡𝑤𝑡− 1 = 𝑋𝑡𝛽 + 𝜀𝑡

(40)

where (𝑤𝑡′𝑟𝑡)2 represents the vector of the actual, realized variance of portfolio

returns on day t and 𝑤𝑡′𝐻𝑡𝑤𝑡 is the series of minimized one-day-ahead forecasted

portfolio variances. Optimal daily portfolio weights wt′ are established upon

forecasts generated by the conditional covariance estimator in question 𝐻𝑡. The

null hypothesis of our accuracy test is that 𝐻0: 𝛽 = 0, in which case the estimator

is deemed to be ‘accurate’ at a given level of significance.

Gains attributable to superior covariance information can be interpreted as

the percentage reduction in portfolio investment that could be achievable with

42 knowledge of the true conditional covariance matrix, and can be quantified by the

ratio:

𝐸 1𝑇∑ (σt)𝑇

𝑡=1 − 𝐸 1𝑇∑ (σt∗)𝑇𝑡=1

𝐸 1𝑇∑ (σt∗)𝑇𝑡=1

(41)

This represents a simple measure of the percent reduction in portfolio standard

deviation Δσt% that could be achieved by employing a better estimator of H𝑡. An

efficiency loss implies that the benchmark estimator H𝑡∗ provides forecasts that

more closely approximate the true conditional covariance matrix vis-à-vis the

alternative model under consideration. An efficiency gain would suggest that

improvements to forecasting accuracy could be recognized by employing an

alternative estimator in place of the benchmark.

Engle and Colacito also propose the following methodology to test for

‘equality’ in predictive accuracy between two alternative covariance matrix

estimation methods. Suppose we have a set of two different time series of one-day-

ahead conditional covariance matrix forecasts. For instance, suppose we have one

series produced by a DCC type estimator, 𝐻𝑡𝐷𝐷𝐷, and another produced by a DCC-

MIDAS type estimator, 𝐻𝑡𝐷𝐷𝐷−𝑀. Following the framework outlined by Equations

(36)-(37), minimum-variance portfolio weights are selected using the covariance

forecasts produced by each estimator. The returns to each portfolio can be denoted

as:

𝜋𝑡𝑖 = 𝑤𝑡

𝑖′(𝑟𝑡), ∀𝑖 ∈ 𝐷𝐷𝐷,𝐷𝐷𝐷-𝑀 (42)

where 𝑟𝑡 is the demeaned vector of asset returns. The difference between the

squared returns to each portfolio, in this case, can be expressed as:

43

𝑢𝑡 = (𝜋𝑡𝐷𝐷𝐷)2 − (𝜋𝑡𝐷𝐷𝐷-𝑀)2 (43)

and the forecasting accuracy of the two estimators can be said to be ‘equal’ if 𝑢 has

an expected value of zero for all t. This can be tested following the Diebold and

Mariano (1995) procedure, wherein the null hypothesis of equal predictive accuracy

is simply a test that 𝑢, regressed on a constant and using heteroscedasticity-

corrected covariance estimates, has a mean value of zero.

This dynamic asset allocation experiment involves an exercise wherein risk is

minimized for various portfolios consisting of major international equity indices

from the stock markets of each G8 constituent country: United States, Canada,

Germany, United Kingdom, France, Italy, Japan, and Russia.9 Daily high-, low-,

and closing-prices are collected for each of these indices at a daily frequency over a

15-year sample period - spanning from January 4 2001 to December 30 2015. All

series are synchronized with each other such that, in total, a sample of 3672

observations are utilized from each.

For the purpose of this experiment, the first 14-years of data (3428

observations) represents the ‘in-sample’ period - which ends on December 31 2014;

data from this time period is utilized for model estimation and to evaluate the

performance of one-day-ahead in-sample forecasts produced by the return- and

range-based estimators tested in this study.10 The final years’ worth of collected

data is partitioned for ‘out-of-sample’ forecasting and evaluation. This pseudo out-

of-sample period runs 244 trading days spanning from January 2 2015 to December

30 2015.

9 The major stock indices used are: S&P500 (US), S&P/TSX (CAN), DAX30 (GER), FTSE100 (UK), CAC40 (FRA), FTSEMIB (ITA), NIKKEI225 (JPN), RTSI (RUS). Data was retrieved from Bloomberg, and all prices are expressed in US dollars. 10 Parameter estimates are not reported due to space constraints.

44

Simulated portfolios of different sizes are constructed by selecting all

possible combinations of international stock indices ranging in size from 2 to 8

assets11. Portfolios are formed from each simulated asset system, and allocated by

following the minimum-variance framework outlined by Equations (36)-(37), and

according to the in- and out-of-sample one-period-ahead conditional covariance

matrix forecasts produced by each estimator under consideration.

Practically speaking - univariate GARCH and exponential CARR estimates

and forecasts are attained through GARCH functions native to the MATLAB

Financial Econometrics Toolbox. Return- and Range-based DCC conditional

correlation matrix estimates and forecasts are attained by using functions contained

within the Oxford MFE Toolbox. Quasi-maximum likelihood estimation and

forecasting of return- and range-based DCC-MIDAS models are attained using

functions from version 2.2 of the MIDAS MATLAB Toolbox published by Hang

Qian. That package also contains a GARCH-MIDAS function which has been

modified slightly to estimate and forecast the CARR-MIDAS - as discussed in

Section 3 of this paper. The temporal data aggregation periodicities used in our

univariate and multivariate component models are both ‘daily-to-weekly’ such that

𝑁 = 5 trading days. The number of lagged variance and correlation terms, 𝐾,

included in the long-run specifications of our univariate and multivariate

component models are arbitrarily set to 100 weeks and 150 weeks, respectively.

Ahead of CARR-MIDAS estimation - daily Realized Range measures are calculated

for each of the 8 international stock indices, scaled at a variety of

11 For 2, 3, 4, 5, 6 and 7 dimensional asset systems there are respectively 28. 56, 70, 56, 28, and 8 possible combinations of international stock indices. Including a single 8 asset portfolio - this makes for a total of 247 possible asset systems with which to test estimators

45 Figures 1-3: Log-Return and Log-Range Time-Series of Major International Stock Indices (In-Sample)

R

ET

UR

N

R

AN

GE

S&P 500 (US)

S&P/TSX (CAN)

DAX 30 (GER)

46 Figures 4-6: Log-Return and Log-Range Time-Series of Major International Stock Indices (In-Sample)

R

ET

UR

N

R

AN

GE

FTSE 100 (UK)

CAC 40 (FRA)

FTSE M IB (ITA)

47 Figures 7-8: Log-Return and Log-Range Time-Series of Major International Stock Indices (In-Sample)

R

ET

UR

N

R

AN

GE

N IKKEI 225 (JPN)

RTSI (RUS)

48 parameterizations of 𝑄∗, and then temporally aggregated to weekly values.12

Appendix D includes much of the remaining MATLAB code produced for and

employed in this study.

4.2 Results and Discussion

Before discussing and interpreting the results of the dynamic asset allocation

experiment, a cursory analysis of the sample data is conducted. Log daily return

and price range series are utilized in the quasi-maximum likelihood estimation of

the return- and range-based estimation methods employed in this study,

respectively. These time series are portrayed for each international stock index in

Figures 1-8. Some basic statistics pertaining to each time series are also

extracted and included for reference. A collection of absolute-return series (which

are non-negative like price range values) does not factor into our experiment, but

are also analyzed and included to facilitate some comparison between our daily

return- and range-based statistics. Tables 1, 2, and 3 present summary statistics

for all log- daily return, absolute-return, and price range series. A variety of

commonly understood statistics are provided in each table, pertaining to each

international stock index, have been calculated on data from the in-sample period.

Jarque-Bera tests of normality are included for each series under

consideration, with the null hypothesis being that the data of any given sample is

expected to have originated from a normal distribution. The first number reported

is the Jarque-Bera test statistic, and the parenthesized number below reports the p-

value of the test. The null hypothesis is rejected for each series across all 3

12 For CARR-MIDAS estimation - Q* is parameterized as 63, 126, 252, 756 or 1260 such that realized range scaling factors are computed on the past 3-months, 6-months, 1-year, 3-years or 5-years of trading history

49

Table 1: Summary Statistics for Log-Return Time-Series of Major International Stock Indices (In-Sample) (USA) (CAN) (GER) (UK) (FRA) (ITA) (JPN) (RUS)

S&P 500

S&P /TSX

DAX30 FTSE 100

CAC40 FTSE MIB

NIKKEI 225

RTSI

Mean 0.013 0.015 0.012 0.002 -0.009 -0.023 0.007 0.049 Median 0.068 0.061 0.075 0.048 0.033 0.059 0.036 0.153

Maximum 10.424 9.370 13.463 11.112 13.305 14.470 13.235 20.204 Minimum -9.470 -9.788 -7.433 -9.266 -9.472 -10.136 -12.111 -21.199 Std.Dev. 1.273 1.129 1.572 1.261 1.542 1.577 1.568 2.180 Skewness -0.189 -0.656 0.153 -0.048 0.160 0.054 -0.404 -0.432 Kurtosis 11.640 13.335 8.369 11.333 9.247 9.681 9.366 15.168 Jarque- 10682.77

(0) 15502.35

(0) 4131.11

(0) 9919.84

(0) 5589.46

(0) 6377.08

(0) 5882.47

(0) 21254.62

(0) Bera ACF(1) -0.073 -0.024 -0.004 -0.060 -0.039 -0.019 -0.040 0.121 ACF(2) -0.034 -0.036 -0.026 -0.035 -0.050 -0.036 -0.008 0.013 ACF(3) 0.005 0.023 -0.018 -0.026 -0.022 0.008 -0.005 -0.012 ACF(4) -0.003 0.008 0.021 0.019 0.002 0.020 -0.038 -0.020 ACF(5) -0.066 -0.117 -0.070 -0.095 -0.099 -0.080 -0.009 -0.030 ACF(6) -0.007 0.055 -0.020 -0.001 0.005 0.020 0.002 0.023 ACF(7) 0.010 0.012 0.035 0.071 0.044 0.036 0.022 0.037 ACF(8) 0.045 0.008 0.003 -0.005 -0.011 0.014 -0.002 -0.027 ACF(9) -0.029 -0.020 0.026 -0.003 0.005 -0.010 -0.019 -0.019 ACF(10) 0.047 0.052 -0.001 -0.017 -0.015 -0.012 0.016 -0.035 ACF(11) 0.005 -0.043 0.034 0.036 0.044 0.040 0.016 0.023 ACF(12) -0.004 -0.023 -0.004 -0.048 -0.027 -0.015 -0.001 0.011 ACF(13) -0.009 0.012 -0.011 0.012 0.025 0.015 -0.002 0.115 ACF(14) -0.040 -0.020 -0.002 -0.021 0.003 -0.002 0.008 0.032 ACF(15) 0.037 0.046 -0.004 0.009 0.010 0.027 0.006 -0.018 ACF(16) 0.002 0.013 0.008 -0.009 -0.010 0.010 -0.043 0.007 ACF(17) 0.015 0.028 -0.009 0.008 0.001 -0.007 0.027 -0.024 ACF(18) -0.022 -0.038 -0.030 -0.034 -0.029 -0.012 0.027 -0.008 ACF(19) 0.018 -0.010 -0.006 0.015 0.004 -0.009 -0.014 0.031 ACF(20) -0.042 -0.006 -0.021 -0.029 -0.022 -0.051 0.009 0.011 ACF(21) 0.011 0.009 0.022 -0.013 0.000 -0.014 -0.052 -0.008 ACF(22) -0.039 -0.075 -0.009 -0.016 -0.015 0.007 -0.014 -0.044 LBQ(22) 81.436 123.235 41.813 93.348 74.204 57.685 38.299 136.445

50

Table 2: Summary Statistics for Absolute Log-Return Time-Series of Major International Stock Indices (In-Sample) (USA) (CAN) (GER) (UK) (FRA) (ITA) (JPN) (RUS) S&P

500 S&P /TSX

DAX30 FTSE 100

CAC40 FTSE MIB

NIKKEI 225

RTSI

Mean 0.840 0.764 1.087 0.845 1.060 1.084 1.121 1.463 Median 0.552 0.534 0.746 0.576 0.739 0.750 0.835 1.037

Maximum 10.424 9.788 13.463 11.112 13.305 14.470 13.235 21.199 Minimum 0.001 0.001 0.000 0.002 0.001 0.000 0.001 0.001 Std.Dev. 0.957 0.843 1.137 0.936 1.119 1.147 1.098 1.632 Skewness 3.170 3.520 2.564 3.182 2.797 2.836 2.877 3.863 Kurtosis 20.104 24.522 14.496 20.264 16.491 17.684 19.825 30.037 Jarque- 47525.2 73240.4 22635.0 48355.1 30464.2 35393.0 45160.7 112938.4

Bera (0) (0) (0) (0) (0) (0) (0) (0) ACF(1) 0.296 0.345 0.244 0.299 0.223 0.217 0.192 0.293 ACF(2) 0.344 0.319 0.310 0.346 0.295 0.278 0.265 0.260 ACF(3) 0.322 0.334 0.301 0.341 0.273 0.259 0.277 0.260 ACF(4) 0.338 0.323 0.290 0.290 0.258 0.239 0.219 0.234 ACF(5) 0.371 0.392 0.280 0.321 0.270 0.255 0.255 0.237 ACF(6) 0.364 0.396 0.310 0.276 0.271 0.276 0.225 0.195 ACF(7) 0.308 0.291 0.273 0.251 0.228 0.215 0.230 0.203 ACF(8) 0.349 0.326 0.276 0.292 0.258 0.244 0.199 0.218 ACF(9) 0.295 0.297 0.266 0.268 0.248 0.234 0.221 0.189 ACF(10) 0.360 0.358 0.276 0.266 0.250 0.239 0.189 0.221 ACF(11) 0.330 0.350 0.242 0.259 0.210 0.196 0.185 0.195 ACF(12) 0.298 0.282 0.276 0.282 0.268 0.249 0.209 0.206 ACF(13) 0.280 0.281 0.265 0.256 0.221 0.225 0.179 0.231 ACF(14) 0.264 0.272 0.226 0.240 0.222 0.241 0.168 0.208 ACF(15) 0.300 0.305 0.253 0.240 0.227 0.209 0.202 0.198 ACF(16) 0.307 0.320 0.259 0.257 0.234 0.205 0.156 0.194 ACF(17) 0.273 0.276 0.216 0.226 0.221 0.218 0.153 0.243 ACF(18) 0.254 0.261 0.250 0.237 0.206 0.201 0.148 0.192 ACF(19) 0.267 0.284 0.255 0.228 0.223 0.207 0.182 0.179 ACF(20) 0.258 0.280 0.239 0.200 0.195 0.198 0.136 0.154 ACF(21) 0.249 0.307 0.195 0.244 0.187 0.170 0.143 0.152 ACF(22) 0.257 0.292 0.229 0.205 0.196 0.190 0.124 0.181 LBQ(22) 7095.126 7527.82 5200.366 5408.755 4266.593 3909.393 2960.683 3456.383

51

Table 3: Summary Statistics for Log-Range Time-Series of Major International Stock Indices (In-Sample) (USA) (CAN) (GER) (UK) (FRA) (ITA) (JPN) (RUS) S&P

500 S&P /TSX

DAX30 FTSE 100

CAC40 FTSE MIB

NIKKEI 225

RTSI

Mean 0.112 0.104 0.128 0.116 0.123 0.126 0.115 0.134 Median 0.105 0.097 0.121 0.109 0.116 0.120 0.110 0.127

Maximum 0.330 0.415 0.334 0.328 0.304 0.314 0.371 0.513 Minimum 0.045 0.043 0.048 0.048 0.054 0.048 0.049 0.030 Std.Dev. 0.037 0.034 0.042 0.036 0.038 0.040 0.033 0.047 Skewness 1.418 1.940 1.083 1.280 1.067 0.884 1.427 1.540 Kurtosis 6.609 10.024 4.535 5.575 4.591 3.977 7.969 8.333 Jarque- 3008.3 9198.0 1007.2 1882.7 1011.7 583.3 4691.5 5416.1

Bera (0) (0) (0) (0) (0) (0) (0) (0) ACF(1) 0.664 0.666 0.699 0.683 0.662 0.678 0.542 0.568 ACF(2) 0.670 0.624 0.695 0.668 0.636 0.655 0.524 0.520 ACF(3) 0.651 0.600 0.682 0.642 0.614 0.648 0.484 0.485 ACF(4) 0.637 0.597 0.661 0.630 0.607 0.618 0.460 0.472 ACF(5) 0.630 0.608 0.658 0.631 0.603 0.627 0.443 0.453 ACF(6) 0.615 0.585 0.649 0.615 0.582 0.594 0.443 0.432 ACF(7) 0.597 0.561 0.636 0.607 0.570 0.586 0.439 0.429 ACF(8) 0.601 0.556 0.623 0.601 0.556 0.570 0.423 0.411 ACF(9) 0.587 0.540 0.619 0.584 0.550 0.567 0.443 0.391 ACF(10) 0.573 0.551 0.610 0.571 0.547 0.566 0.426 0.395 ACF(11) 0.566 0.530 0.590 0.549 0.515 0.540 0.404 0.382 ACF(12) 0.545 0.512 0.585 0.553 0.521 0.546 0.401 0.386 ACF(13) 0.540 0.508 0.588 0.541 0.520 0.542 0.400 0.385 ACF(14) 0.543 0.503 0.587 0.528 0.515 0.534 0.381 0.387 ACF(15) 0.530 0.505 0.568 0.532 0.508 0.524 0.388 0.373 ACF(16) 0.527 0.492 0.569 0.530 0.504 0.519 0.367 0.347 ACF(17) 0.510 0.483 0.547 0.516 0.485 0.514 0.387 0.347 ACF(18) 0.504 0.490 0.548 0.510 0.491 0.507 0.362 0.336 ACF(19) 0.505 0.481 0.552 0.505 0.489 0.507 0.361 0.336 ACF(20) 0.500 0.472 0.541 0.494 0.475 0.494 0.344 0.335 ACF(21) 0.484 0.471 0.530 0.493 0.465 0.484 0.341 0.324 ACF(22) 0.483 0.465 0.528 0.487 0.466 0.493 0.315 0.326 LBQ(22) 24551.2 21998.6 27723.3 24577.9 22317.4 23938.5 13129.1 12477.1

52 information sets - indicating that all data series analyzed can be said to have

originated from non-normal distributions. The Autocorrelation Function (ACF)

checks for correlation between observations separated by a fixed number of time-

lags and is employed to investigate underlying patterns in each series. Tests are

employed for time-lags of up to 22 trading days. A data series can be said to be

stationary if the ACF values exhibit exponential decay in residuals as the number

of lags are increased. The Ljung-Box Q (LBQ) statistic is a method used to test

for autocorrelations in time series data, with the null hypothesis being that no

autocorrelation is exhibited for a series of ACF residuals, up to a fixed number of

lags. Higher LBQ values indicate a greater degree of persistence for the series

under consideration. Every data series in this study was found to exhibit temporal

dependence. In comparing the ACF and LBQ(22) test statistics of the absolute-

return and range time series - it is evident that the range time series exhibits a

relatively high degree of persistence, which we seek to capture through conditional

range-based modelling.

With respect to the allocation experiment itself, we first make reference to

Appendix A - which details the resulting variances of portfolios allocated upon

one-day-ahead forecasts of conditional covariance, as produced by the estimation

methods specified in each column header. ‘IS’ denotes results from throughout the

‘in-sample’ forecasting period, while ‘OOS’ denotes statistics pertaining to ‘out-of-

sample’ forecasts. These values represent raw-output from our in- and out-of-

sample forecasting exercises. Following the methodology outlined throughout

Section 4.1 - the estimators which produce the most accurate covariance matrix

forecasts should, in theory, yield the lowest resulting portfolio variance, on average,

across the simulated asset systems they’ve been tasked with modelling.

53

P-values pertaining to tests of predictive accuracy, as presented in Equation

(40), are reported in Appendix B, for each estimator employed in this study. P-

values are reported for tests pertaining to both in- and out-of-sample forecasts, and

for each of the 247 possible portfolios. Recall that each covariance matrix

estimator is tasked with modelling the risk-characteristics of available investment

opportunities, and allocations have been made to minimize one-day-ahead portfolio

variance based on these estimates. A failure to reject the null hypothesis

that 𝐻0: 𝛽 = 0, in this context, indicates that one-day-ahead variance forecasts

accurately represent realized portfolio variances that are subsequently observed,

when optimally allocated. As such, a failure to reject the null indicates that the

estimator in question provides accurate conditional variance forecasts, at a certain

degree of significance. Tests are run at significance levels 𝛼 such that the

probability of making a type I error are small - 1%, 5% and 10%, respectively. The

null hypothesis is rejected in favor of the alternative hypothesis for each estimator,

portfolio, and sampling period for which 𝑃 ≤ α. To assist in interpreting this vast

array of p-values, a summary is provided in the table that follows.

The results in Table 4 indicate the average percentage of in- and out-of-

sample accuracy tests for which we fail to reject the null hypothesis 𝐻0: 𝛽 = 0,

across portfolio dimensions 𝑚 = 2, 3, 4, 5, 6 and 7, and for each estimation

method. These average percentages are reported for p-Tests at varying degrees of

significance α = 0.10, 0.05, 0.01. Each figure in this table summarizes the

percentage of portfolios of size ‘𝑚’ for which the estimator identified in the column

heading was deemed ‘accurate’ in identifying the one-day-ahead realized variance of

its implied allocations. Each figure in this table summarizes the percentage of

portfolios of size ‘𝑚’ for which the estimator identified in the column heading is

54 Table 4: Summary of Accuracy Test Results from Appendix B, by Portfolio Size m Stage 1

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

m = 2 : α = .01 0.500 1 0.571 0.893 0.429 0.964 0.893 0.929 0.786 1 0.750 0.929 0.786 0.893 0.750 0.821

.05 0.321 0.357 0.214 0.321 0.429 0.786 0.786 0.607 0.750 0.714 0.679 0.643 0.571 0.536 0.679 0.393

.10 0.179 0.214 0.179 0.321 0.393 0.357 0.714 0.429 0.679 0.571 0.607 0.536 0.536 0.393 0.536 0.250 m = 3 : α = .01 0.696 0.946 0.643 0.804 0.518 0.804 0.643 0.768 0.625 0.964 0.554 0.839 0.643 0.696 0.554 0.518

.05 0.429 0.536 0.446 0.393 0.429 0.643 0.482 0.571 0.518 0.714 0.375 0.643 0.554 0.429 0.464 0.250

.10 0.321 0.304 0.304 0.304 0.411 0.446 0.357 0.357 0.446 0.446 0.321 0.446 0.464 0.196 0.321 0.107 m = 4 : α = .01 0.729 0.886 0.714 0.814 0.557 0.871 0.457 0.714 0.500 0.929 0.386 0.800 0.500 0.586 0.400 0.286

.05 0.486 0.629 0.443 0.443 0.543 0.600 0.286 0.414 0.300 0.529 0.243 0.586 0.357 0.257 0.300 0.100

.10 0.343 0.429 0.371 0.357 0.514 0.500 0.214 0.271 0.243 0.314 0.129 0.357 0.257 0.057 0.214 0.029 m = 5 : α = .01 0.696 0.911 0.786 0.875 0.643 0.946 0.375 0.571 0.304 0.857 0.143 0.821 0.339 0.500 0.268 0.161

.05 0.482 0.714 0.536 0.518 0.643 0.661 0.125 0.339 0.179 0.464 0.071 0.500 0.179 0.125 0.161 0.036

.10 0.411 0.589 0.446 0.321 0.625 0.625 0.054 0.268 0.125 0.250 0.036 0.357 0.107 0 0.125 0

55 Table 4: Summary of Accuracy Test Results from Appendix B, by Portfolio Size m Stage 1

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

m = 6 : α = .01 0.679 0.964 0.821 0.964 0.750 1 0.179 0.500 0.179 0.821 0.036 0.821 0.214 0.393 0.214 0.143

.05 0.429 0.786 0.714 0.571 0.750 0.786 0.036 0.321 0.036 0.393 0.036 0.393 0.036 0.071 0.071 0

.10 0.357 0.679 0.607 0.286 0.750 0.750 0 0.179 0 0.357 0 0.214 0 0 0 0 m = 7 : α = .01 0.500 1 0.875 1 0.875 1 0 0.625 0 0.875 0 0.875 0 0.250 0.125 0.125

.05 0.250 1 0.875 0.875 0.875 0.875 0 0.125 0 0.625 0 0.500 0 0 0 0

.10 0.250 0.750 0.750 0.375 0.875 0.875 0 0.125 0 0.250 0 0.250 0 0 0 0

56 deemed capable of accurately estimating its own one-day-ahead realized variance.

Since these statistics are reported as the percentage of portfolios for which we fail

to reject the null, it’s understandable why these values decrease as levels of

significance increase. Increased significance levels in these tests indicate a higher

probability of incorrectly rejecting a true null hypothesis; confidence intervals

become narrower, resulting in more false positives. The results of these tests

demonstrate the difficulty in accurately forecasting the one-day-ahead realized

variance for a portfolio of assets. The results are mixed, but some broad

generalizations can be made. First, some comparisons are drawn regarding the

accuracy test results of the return- and range-based DCC estimators - as presented

in first two columns in Table 4. In-sample, the ‘CARR/DCC’ (or the ‘DCC-

CARR’) typically exhibits higher percentages of failed rejections in the results of

these tests compared to the more traditional ‘GARCH/DCC’ (DCC-GARCH)

model - particularly as portfolio size 𝑚 and significance levels of the test are

increased.13 The opposite holds true for the out-of-sample reference period -

wherein the return-based DCC is deemed to accurately predict one-day-ahead

portfolio variance for a greater percentage of portfolios than the range-based DCC

does. The results in the third column of Table 4 pertain to the ‘CARR/DCC-

MIDAS’, which indicate that this multivariate range-based component model is

comparatively accurate versus the well-established return- and range-based DCC

models, for both in- and out-of-sample forecasts, and across higher degrees of

significance. Furthermore, the Range-based DCC-MIDAS seems to provide the

best in- and out-of-sample accuracy test results at higher dimensions, compared to

13 For much of the remainder of this discussion, covariance matrix estimators will be referenced in the format 'S1/S2' - with 'S1' and 'S2' identifying the conditional volatility and dynamic correlation specifications employed in the first- and second-stages of estimation, respectively.

57 all models tested in this study. For instance, where 𝑚 ≥ 4 and at a significance

level of 10%, the ‘CARR/DCC-MIDAS’ produces favorable test results for 51.4 to

87.5% of portfolios throughout the in-sample period. Contrast these figures to the

performance of the ‘GARCH/DCC’ and ‘CARR/DCC’ which span, respectively,

from 25.0 to 41.1% and 37.1 to 75.0% over these same ranges.

Some generalizations can also be drawn from results of Table 4 pertaining to

those estimates which utilize various CARR-MIDAS parameterizations in first-

stage Range-based DCC-MIDAS estimation - as presented in columns 4 through 7.

These models are demonstrated to be quite accurate, particularly in forecasting the

variance of low-dimensional portfolios, but the results also indicate a high degree of

sensitivity to appropriate parameterization of Q* (the number of trading days used

to scale realized range in the long-run component of volatility). Across lower

dimension portfolios (where 𝑚 ≤ 3), The ‘CARR-MIDAS(Q*=63)/DCC-MIDAS’

specification, wherein Q* = 63 trading days to approximate ‘quarterly’ scaling of

the exogenous realized range series, exhibits the best in-sample p-Test results of all

parameterizations presented in this study. This model yields favorable results in

accuracy tests at the 10% significance level for 71.4% and 35.7% of portfolios

consisting of 𝑚 equals 2 and 3 assets, respectively. However, it seems these in-

sample percentages begin to deteriorate as portfolio size increases. Of all models

tested, the ‘CARR-MIDAS(Q*=126)/DCC-MIDAS’ approximates ‘semi-annual’

scaling of the low-frequency realized range series and exhibits the most consistent

out-of-sample predictive accuracy for one-day-ahead realized variance of portfolios

across lower dimensions (where 𝑚 ≤ 3). In tests at the 10% significance level, this

model provides accurate out-of-sample realized variance forecasts for 57.1% and

44.6% of portfolios where 𝑚 equals 2 and 3, respectively. The out-of-sample results

58 of this particular parameterization also seem to deteriorate as the portfolio size

increases. In fact, it seems that p-Test results deteriorate across all CARR-MIDAS

parameterizations as portfolio size increases, but to a lesser-degree for those

parameterizations where ‘Q*’ approaches what could be described as an

‘appropriate’ level of trading history on which to scale RR. Based on this

particular set of tests - specifications where Q* = 126 and 252 - which indicate

‘semi-annual’ and ‘annual’ scaling, respectively - seem to demonstrate more

accurate out-of-sample realized portfolio variance forecasts compared to the other

CARR-MIDAS parameterizations tested.

It is worth noting that these p-Tests only assess the degree to which

predicted one-day-ahead portfolio variances are realized. These tests do not assess

the ability of a given estimator 𝐻𝑡 to approximate the true conditional covariance

matrix and to allocate optimally within this minimum-variance framework. Better

estimators of 𝐻𝑡 should be better able to identify variance minimizing investment

opportunities - such that the variance in daily returns to portfolios allocated on the

basis of these estimates should be lower than for portfolios allocated on the basis of

forecasts produced by less-optimal alternatives. This represents the most

meaningful approach towards evaluating and comparing the forecasting accuracy of

covariance matrix estimators in this study. These ideas are explored further in the

analysis that follows.

Appendix C processes the output presented in Appendix A and Appendix

B. Some comparisons are drawn between the estimators employed in this study,

such that the economic relevance of each can begin to be assessed. For each of the

247 simulated portfolios, the efficiency gain (or loss) of estimators employed in this

study - as computed in Equation (41) - is reported relative to the benchmark DCC-

59 CARR model. A reported efficiency gain implies that the estimator designated in

the column heading, when tasked with modelling a particular asset system,

generates covariance matrix forecasts which result in a lower resulting portfolio

variance than does those estimates produced by the benchmark. Also embedded

within Appendix C are results from the proposed tests of equivalency, as presented

through Equations (42)-(43). The parenthesized values represent t-stats from

Diebold-Mariano tests of equivalence between the squared daily returns to

portfolios, allocated on the basis of forecasts produced by the methods indicated in

the column header, versus those of the benchmark. The asterisk(s) appended to

the end of each reported efficiency gain (or loss) provides an indication that the

null hypothesis of the equivalence test has been rejected - implying that portfolio

returns yielded through use of the model in question are significantly different from

what results from use of the DCC-CARR. One, two, or three asterisks signify that

tests of equivalency versus the benchmark have been rejected for the portfolio, and

sample period under consideration, at a significance level of 10%, 5% or 1%,

respectively. Though Appendix C is rich with information and the results of this

experiment, it is vast and admittedly difficult to parse. The remainder of this

discussion presents these results in an economically relevant and more easily

interpretable fashion.

Table 5 summarizes and extracts some interesting insights from the vast

array of efficiency gain information presented throughout Appendix C, and is

perhaps the most illuminating set of results presented in this study. The first

statistic identifies, for each estimator under consideration, the proportion of

portfolios (out of a possible 247) for which the reported efficiency gains - as

reported in Appendix C - were found to be non-negative. This can be interpreted

60 Table 5: Average and Maximum Efficiency Gains across Simulated Portfolios (vs. DCC-CARR benchmark), by Portfolio Size

Stage 1 𝐷𝑡 = GARCH CARR

CARR-MIDAS

Q = 63 (3-months)

CARR-MIDAS

Q = 126 (6-months)

CARR-MIDAS

Q = 252 (1-year)

CARR-MIDAS

Q = 756 (3-years)

CARR-MIDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-MIDAS DCC-MIDAS DCC-MIDAS DCC-MIDAS DCC-MIDAS DCC-MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

% lower variance

0.5425 0.6275 0.9838 0.8988 0.9312 0.9474 0.9636 0.9636 0.9514 0.9474 0.9676 0.8421 0.9150 0.8623

Average Efficiency Gain (Loss) % : m 2 -0.0067 -0.0008 0.0644 0.0537 0.0480 0.0449 0.0632 0.0572 0.0605 0.0647 0.0556 0.0538 0.0398 0.0371 3 0.0006 0.0104 0.0952 0.0778 0.0689 0.0649 0.0939 0.0846 0.0931 0.0947 0.0830 0.0749 0.0593 0.0495 4 0.0096 0.0242 0.1206 0.1025 0.0822 0.0820 0.1173 0.1089 0.1188 0.1205 0.1056 0.0938 0.0735 0.0603 5 0.0171 0.0380 0.1424 0.1268 0.0913 0.0982 0.1358 0.1314 0.1394 0.1442 0.1243 0.1121 0.0846 0.0714 6 0.0226 0.0517 0.1607 0.1510 0.0979 0.1140 0.1507 0.1526 0.1560 0.1669 0.1396 0.1305 0.0938 0.0834 7 0.0260 0.0659 0.1753 0.1751 0.1030 0.1297 0.1621 0.1727 0.1690 0.1891 0.1516 0.1498 0.1017 0.0959 Maximum Efficiency Gains % : m 2 0.0088 0.0112 0.2357 0.3655 0.2373 0.1818 0.2409 0.2483 0.2168 0.3376 0.2439 0.3480 0.1612 0.2307 3 0.0668 0.1148 0.2175 0.2827 0.2353 0.1535 0.2239 0.2243 0.2220 0.3008 0.2418 0.2959 0.1464 0.1664 4 0.0809 0.1215 0.2395 0.2471 0.2213 0.1661 0.2089 0.2138 0.2505 0.2710 0.2526 0.2496 0.1512 0.1671 5 0.0947 0.1126 0.2432 0.2384 0.1945 0.1565 0.1957 0.2044 0.2506 0.2665 0.2715 0.2349 0.1524 0.1469 6 0.0856 0.1075 0.2458 0.2298 0.1779 0.1559 0.2056 0.2034 0.2395 0.2472 0.2642 0.2196 0.1410 0.1401 7 0.0565 0.0921 0.2422 0.2096 0.1463 0.1462 0.2088 0.1982 0.2238 0.2325 0.1916 0.2018 0.1192 0.1308

61 as a measure of how consistently use of the estimation method indicated in the

column header outperforms, and thus provides efficiency gains (as opposed to

losses), relative to the benchmark across all portfolios under consideration. These

percentages are reported separately for both in-sample and out-of-sample

forecasting periods.

Relative to the benchmark, use of the traditional return-based DCC-

GARCH results in one-day-ahead covariance matrix forecasts which yield a lower

resulting portfolio variance for 54.25% of portfolios in-sample, and 62.75% of

portfolios out-of-sample. This provides an early indication that, for the majority of

the 247 portfolios tested, the DCC-CARR actually underperformed vis-à-vis its

traditional return-based analog, the DCC-GARCH. The suitability of the return-

and range-based DCC models to modelling asset systems of various sizes is a

crucial question that will be further investigated later in this discussion. It’s also

interesting to note that each of the estimators which employ the Range-based

DCC-MIDAS specification into second-stage conditional correlation estimation (as

reported in columns 2 through 7) outperform the benchmark DCC-CARR for the

vast majority of portfolios tested. In-sample proportions range from 91.50% to

98.38% of applications yielding favorable outcomes relative to the benchmark, and

86.23% to 96.36% out-of-sample. Across the in- and out-of-sample forecasting

periods, these values are maximized under the ‘CARR/DCC-MIDAS’ (‘IS’ =

98.38%) and the ‘CARR-MIDAS(Q*=126)/DCC-MIDAS’ (‘OOS’ = 96.36%),

respectively.

Another illuminating summarization of the efficiency gain (or loss) data from

Appendix C is presented in Table 5. The remainder of this table, and the figure

that follows, facilitate analysis that is central to the objective of this study - how

62 do the established covariance matrix estimation methods, as well as those methods

proposed in this paper, perform and compare in modelling asset systems of various

sizes? To this end, the average magnitude of efficiency gains (or losses) realized

across all possible portfolios of a given size ‘𝑚’ are reported separately for 𝑚 = 2,

3, 4, 5, 6 and 7 and for each estimator. In contrast with ‘% lower variance’

figures, the reported ‘average efficiency gain (loss)’ measures communicate the

expected magnitude of efficiency gains (or losses), rather than reporting on

directions alone. Furthermore, results presented in this fashion allow for a more

detailed analysis of how these values can be expected to change as additional assets

are added into consideration. From the results of this table, the suitability

estimators to the modelling of low- and higher-dimension systems of assets can be

evaluated and compared.

A novel comparison between the suitability of the DCC-CARR versus the

DCC-GARCH in modelling larger systems of assets can be drawn from data in the

first column of the ‘average efficiency gain’ results reported in Table 5. These

results confirm what has been demonstrated extensively in the DCC-CARR

literature - the range-based model seems to provide some improvements in

predictive accuracy over the DCC-GARCH when applied to smaller systems of

assets. When applied to the 28 bivariate portfolios simulated for the purpose of

this study, use of the DCC-GARCH in lieu of the benchmark DCC-CARR

produces modest average efficiency losses in the order of 67 basis points in-sample

and 8 basis points out-of-sample. Comparably modest efficiency gains are

recognized through use of the traditional return-based DCC model in applications

to trivariate systems. In fact, efficiency gains owing to use of the DCC-GARCH

over the DCC-CARR seem to increase with portfolio size - particularly in out-of-

63 sample forecasting of the conditional covariance matrix. At 𝑚 = 7, the reported

in- and out-of-sample efficiency gains of the DCC-GARCH are 2.6% and 6.59%

respectively. These results refute the postulation that the DCC-CARR could

provide improvements over its return-based counterpart in applications of

modelling small-scale and larger asset systems alike. The DCC-GARCH seems to

be comparatively better suited to modelling larger systems of assets in this

minimum-variance portfolio risk management application (where 𝑚 ≥ 4 ).

The remainder of this discussion pertains to the performance of the novel

range-based component models proposed in this paper, as reported in columns 2

through 7 of Table 5. All Range-based DCC-MIDAS models yield efficiency gains

both in- and out-of-sample on average, and across all portfolio sizes, compared to

the benchmark DCC-CARR (albeit to different degrees). Across all dimensions

tested, the ‘CARR/DCC-MIDAS’ conditional covariance matrix estimator yielded

the greatest average in-sample efficiency gains (ranging from 6.44% for the

bivariate case and up to an average of 17.53% across 7-asset portfolios). The

greatest average out-of-sample efficiency gains were obtained by the ‘CARR-

MIDAS(Q*=252)/DCC-MIDAS’ covariance matrix estimator across all portfolio

sizes. The average out-of-sample efficiency gains of this model ranged from 6.47%

and up to 18.91% for 2- and 7-asset portfolios, respectively. It could be that this

particular parameterization of the CARR-MIDAS provides efficiency gains to

volatility estimation that translates to less-noisy error processes and improved

second stage DCC-MIDAS estimation. Maximum efficiency gains are also reported

in Table 5. The maximum in-sample and out-of-sample gains across all portfolio

sizes are 27.15% and 34.80% respectively - both obtained by the ‘CARR-

64 MIDAS(Q*=756)/DCC-MIDAS’ specification. In general, the maximum attainable

efficiency gains realized across portfolios seem to decrease as system size increases.

Another key observation regarding the performance of the multivariate

range-based component estimation methods proposed in this paper is that, on

average, realizable efficiency gains seem to be increasing in portfolio size in

practically all cases - across the various specifications employed, and for both in-

and out-of-sample forecasting periods. With respect to the in-sample forecasting

period and use of the ‘CARR/DCC MIDAS’ estimator, for instance, there appears

to be an average efficiency improvement of about 1.33% for each asset that is being

added to the analysis. For the out-of-sample forecasting period, and using the

‘CARR-MIDAS(Q*=252) /DCC-MIDAS’ estimator, the average marginal efficiency

gain per asset under consideration is around 2.49%. Following the methodology of

Engle and Colacito (2006), these values can be practically interpreted as the

average percentage reduction in portfolio investment that could be achieved by

employing these novel covariance matrix estimators in lieu of the benchmark DCC-

CARR. For instance, an investor seeking to allocate $100 million who selects

optimal portfolio weights according forecasts produced by the ‘CARR-

MIDAS(Q*=252)/DCC-MIDAS’ rather than the DCC-CARR could save an

average of $2.49 million, per asset under consideration, in achieving the minimum-

variance portfolio. It should be noted however that if a model like the DCC-

GARCH, which is comparatively better suited to modelling larger asset systems

than the DCC-CARR, had served as benchmark in this study - the perceived

benefits of the novel estimation methods proposed in this paper would appear to be

more modest.

65 Figure 9: Average Efficiency Gains across Simulated Portfolios (vs. DCC-CARR benchmark), by Portfolio Size

IN-S

AM

PLE

OU

T-O

F-SA

MP

LE

66

Figure 9 is revealing of those most pertinent findings that emerge from the

results of this study - as presented by Table 5 and highlighted throughout the

preceding discussion. Each bar reports average efficiency gains, relative to the

benchmark DCC-CARR, for all permutations of G8 counties’ equity market returns

in a portfolio of the size displayed on the horizontal axis. The estimators presented

in this figure include the traditional return-based DCC-GARCH, as well as two

separate Range-based DCC-MIDAS specifications - one which employs CARR, and

another that employs the CARR-MIDAS (where Q*=252) in the first-stage of

conditional covariance matrix estimation.14 The relative performance of these

estimators in modelling asset systems of varying sizes is illustrated for both in-

sample and out-of-sample forecasting exercises. Visualizing the average efficiency

gains in this manner illuminates patterns that underlie the results of this study,

and proves particularly useful towards facilitating an understanding of how existing

covariance matrix estimation methods, and those proposed in this paper, compare

in modelling low- and mid-to-high dimensional asset systems alike.

The fact that gains are increasing in portfolio size could be related to the

findings of Engle and Sheppard (2001) who found that, with respect to return-

based estimation, the DCC works well in modelling smaller systems of assets but

correlation estimates appear to become excessively smooth around their

unconditional means as the number of dimensions becomes excessively large. In

any case, the results of this experiment replicate the findings of Colacito, Engle and

Ghysels (2011) but from a unique range-based perspective and framework. These

results imply that the Range-based DCC-MIDAS estimation methods proposed in

14 The CARR-MIDAS(Q*=252)/DCC-MIDAS was selected for representation of the CARR-MIDAS/DCC-MIDAS because this particular parameterization seemed to perform best amongst those considered in this study.

67 this paper could be capable of addressing some of the now formally established

shortcomings of the antecedent DCC-CARR model - particularly in applications

involving larger-system covariance matrix estimation, such as for mid- to high-

dimensional asset allocation. Our models also performed favorably relative to the

traditional DCC-GARCH.

The average efficiency gains reported in Table 5 also emphasize the

importance of setting the ‘Q*’ parameter at an appropriate level when utilizing

CARR-MIDAS specifications for first-stage conditional volatility estimation. This

demonstrates a well-understood trade-off that exists in setting this value too high

or too low - when it comes to realized range estimation. The parameterization of

CARR-MIDAS wherein Q* = 63, for instance, scales the exogenous Realized Range

series by the ratio between the average level of the daily range and the average

daily realized range estimates obtained over the past 63 trading days

(approximating ‘quarterly’ scaling, across ~3 months). A lack of trading history on

which to scale leaves our processed long-term measures of realized range especially

susceptible to short-term, transient microstructure effects. The model where Q* =

63 was the second worst performer, both in- and out-of-sample, amongst all

reported CARR-MIDAS parameterizations. ‘Monthly’ and ‘unscaled’

parameterizations of CARR-MIDAS (where Q*=22 and Q*=0, respectively) were

also employed in testing and model performance further deteriorated at these levels.

Use of an unscaled ‘CARR-MIDAS(Q*=0)/DCC-MIDAS’ estimator actually yielded

negative average efficiency gains across the vast majority of portfolios in this study

relative to the DCC-CARR. Due to space constraints, results pertaining to those

parameterizations have not been reported. The opposite extreme parameterization

of the CARR-MIDAS, where Q* = 1260, calculates a scaling factor in the same

68 fashion but based instead upon the previous 5 years of trading history. This

parameterization was the worst performer, both in- and out-of-sample, compared to

all other CARR-MIDAS parameterizations reported in this study.

The fact that setting ‘Q*’ at its highest tested value does not achieve the

highest predictive accuracy of those parameterizations under consideration implies

that the magnitude of trading volumes and microstructure effects are changing over

time for the assets included this study, and so only a recent history of daily price

range and realized range values should be utilized for microstructure bias

adjustments. The value of ‘Q*’ must be set at a level that is appropriate for the

application at hand. The results presented in Table 5 show clearly that the in- and

out-of-sample efficiency gains for CARR-MIDAS models are increasing with ‘Q*’

from the parameterizations where Q*=63 and up to the best-performing

specification wherein Q* = 252 trading days. Increasing ‘Q*’ beyond this point,

such as parameterizations where Realized Range is scaled on the basis of the past 3

or 5 years of trading history, yields marginal losses to the average efficiency gains

which can be realized through use of estimators which utilize CARR-MIDAS in

first-stage of estimation, versus the DCC-CARR.

Finally, it can be noted that the CARR-MIDAS specification which yielded

the greatest out-of-sample efficiency gains also performed best in-sample. And so,

for out-of-sample forecasting applications using the CARR-MIDAS in the first-stage

of Range-based DCC-MIDAS conditional covariance matrix estimation -

appropriate ‘Q*’ values could potentially be selected on the basis of in-sample fit, or

by profiling the log-likelihood function.

69

5 Conclusion

The results of this study confirm what has been extensively demonstrated in

the literature - the range-based DCC-CARR model of Chou et al. (2009) represents

a relevant alternative to its return-based analog, the DCC-GARCH of Engle (2002),

in the modelling of low-dimension systems of assets. Even still, efficiency gains

attributable to use of the range-based DCC estimator in lieu of its return-based

counterpart across low-dimensional systems appear to be modest at best. In

general, the results of this study confirm that improvements to forecasting accuracy

could be realized in applications of the DCC-CARR to the volatility modelling of

systems consisting of thee-assets or less.

However, analyzing the relative performance of these alternative models in

applications to larger systems clearly illustrates insights and patterns which

challenge the economic relevance and suitability of the DCC-CARR for certain

applications. As system size increases beyond four-assets, the DCC-CARR begins

to clearly underperform the DCC-GARCH for both in- and out-of-sample

covariance matrix forecasts. In fact, the results of this study indicate that a

decision to utilize the DCC-CARR in lieu of its return-based counterpart beyond a

three-asset portfolio would, on average, yield losses to efficiency that are increasing

with system size.

This result seem to refute the postulation that the DCC-CARR could

represent an attractive alternative to the DCC-GARCH for the modelling of low-

and high-dimensional systems alike, and challenges the economic value of the

range-based estimator for volatility timing strategies. In fact, the DCC-CARR

70 seems to be suitable only for minimum-variance hedging, and the most basic of

portfolio selection exercises.

The Range-based DCC-MIDAS conditional correlation specification, as

proposed in this paper, effectively addresses the shortcomings of its single-

component, range-based counterpart in the modelling of larger asset systems.

Compared with the DCC-CARR, use of the Range-based DCC-MIDAS yields gains

to efficiency that increase with each additional asset added to consideration.

Furthermore, use of the Range-based DCC-MIDAS yields gains to efficiency, across

all portfolio sizes tested, versus both return- and range-based DCC models. And so,

the results of this study establish the economic relevance of the Range-based DCC-

MIDAS in this context; this model could represent an attractive alternative for

practical applications of minimum-variance hedging, as well as the selection and

volatility timing of sufficiently diversified portfolios.

The CARR-MIDAS model proposed in this paper, when employed ahead of

Range-based DCC-MIDAS estimation, exhibited mixed results but demonstrated

some promise - particularly in out-of-sample forecasting applications. However, the

Range-based DCC-MIDAS specification that employs CARR in first-stage

conditional volatility estimation is more parsimonious and robust, with CARR-

MIDAS specifications being sensitive to appropriate parameterization of the

number of trading days ‘Q*’ that are used in scaling realized range measures, ahead

of long-run component of volatility estimation. To isolate for any potential

idiosyncratic benefits to using the CARR-MIDAS alone, further testing of this

model is warranted, within a univariate framework.

Finally, it should be emphasized that the range-based component models

proposed in this paper are not purported to be the ‘best’ estimators of their kind,

nor as occupying the forefront of contemporary volatility and correlation modelling

71 techniques. The models proposed here simply represent another contribution to the

ongoing development of advanced volatility and correlation estimation methods.

Ease of estimation is emphasized, with marginal data- or programming-

requirements being low, relative to other methods which are well-established in the

literature or widely applied in practice. Investors are encouraged to diligently

back-test strategies integrating the estimation methods proposed in this paper for

their unique applications. Doing so could potentially contribute to an improved

understanding of the risk-return characteristics of available investment

opportunities, and allow for better investment decisions to be made.

72

References

Alizadeh, Sassan, Michael W. Brandt, and Francis X. Diebold. 2002. "Range-Based Estimation of Stochastic Volatility Models". The Journal of Finance 57 (3): 1047-1091.

Bauwens, Luc, Sébastien Laurent, and Jeroen V. K. Rombouts. 2006. "Multivariate GARCH Models: A Survey". Journal of Applied Econometrics 21 (1): 79-109.

Bollerslev, Tim. 1986. "Generalized Autoregressive Conditional Heteroskedasticity". Journal of Econometrics 31 (3): 307-327.

Bollerslev, Tim. 1990. "Modelling The Coherence In Short-Run Nominal Exchange Rates: A Multivariate Generalized Arch Model". The Review of Economics and Statistics 72 (3): 498.

Brandt, Michael W and Christopher S Jones. 2006. "Volatility Forecasting With Range-Based EGARCH Models". Journal of Business & Economic Statistics 24 (4): 470-486.

Chou, Ray Yeu-Tien. 2005. "Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model". Journal of Money, Credit, And Banking 37 (3): 561-582.

Chou, Ray Yeutien, Chun-Chou Wu, and Nathan Liu. 2009. "Forecasting Time-Varying Covariance with A Range-Based Dynamic Conditional Correlation Model". Review of Quantitative Finance And Accounting 33 (4): 327-345.

Chou, Ray Yeutien and Nathan Liu. 2010. "The Economic Value Of Volatility Timing Using A Range-Based Volatility Model". Journal Of Economic Dynamics And Control 34 (11): 2288-2301.

Chou, Ray Yeutien, Heng-chih Chou, and Nathan Liu. 2010. "Range Volatility Models and Their Applications in Finance". Handbook of Quantitative Finance and Risk Management, 1273-1281.

Chou, Ray Yeutien and Nathan Liu. 2011. "Estimating Time-Varying Hedge Ratios With A Range-Based Multivariate Volatility Model". Working Paper, Academia Sinica.

Christensen, Kim and Mark Podolskij. 2007. "Realized Range-Based Estimation of Integrated Variance". Journal of Econometrics 141 (2): 323-349.

73 Colacito, Riccardo, Robert F. Engle, and Eric Ghysels. 2011. "A Component Model

for Dynamic Correlations". Journal of Econometrics 164 (1): 45-59.

Diebold, Francis X. and Roberto S. Mariano. 1995. "Comparing Predictive Accuracy". Journal of Business & Economic Statistics 13 (3): 253.

Engle, Robert F. 1982. "Autoregressive Conditional Heteroscedasticity With Estimates Of The Variance Of United Kingdom Inflation". Econometrica 50 (4): 987.

Engle, Robert F. 2002. "Dynamic Conditional Correlation". Journal of Business & Economic Statistics 20 (3): 339-350.

Engle, Robert F. 2009. Anticipating Correlations. 1st ed. Princeton, NJ: Princeton University Press.

Engle, Robert F. and Tim Bollerslev. 1986. "Modelling the Persistence Of Conditional Variances". Econometric Reviews 5 (1): 1-50.

Engle, Robert F., Eric Ghysels, and Bumjean Sohn. 2013. "Stock Market Volatility and Macroeconomic Fundamentals". Review of Economics and Statistics 95 (3): 776-797.

Engle, Robert F. and Riccardo Colacito. 2006. "Testing And Valuing Dynamic Correlations For Asset Allocation". Journal of Business & Economic Statistics 24 (2): 238-253.

Engle, Robert F. and Jose Gonzalo Rangel. 2008. "The Spline-GARCH Model For Low-Frequency Volatility And Its Global Macroeconomic Causes". Review of Financial Studies 21 (3): 1187-1222.

Foroni, Claudia, Massimiliano Marcellino, and Christian Schumacher. 2015. "Unrestricted Mixed Data Sampling (MIDAS): MIDAS Regressions with Unrestricted Lag Polynomials". Journal of The Royal Statistical Society: Series A (Statistics In Society) 178 (1): 57-82.

Ghysels, Eric, Arthur Sinko, and Rossen Valkanov. 2007. "MIDAS Regressions: Further Results and New Directions". Econometric Reviews 26 (1): 53-90.

Glosten, Lawrence R., Ravi Jagannathan, and David E. Runkle. 1993. "On the Relation Between The Expected Value And The Volatility Of The Nominal Excess Return On Stocks". The Journal of Finance 48 (5): 1779.

74 Huang, Chi-Fu and Robert H. Litzenberger. 1988. "Foundations For Financial

Economics.". The Journal of Finance 44 (2): 529.

Hansen, Peter R and Asger Lunde. 2006. "Realized Variance and Market Microstructure Noise". Journal of Business & Economic Statistics 24 (2): 127-161.

Hansen, Peter Reinhard and Asger Lunde. 2006. "Consistent Ranking of Volatility Models". Journal of Econometrics 131 (1-2): 97-121.

Hayashi, Fumio. 2000. Econometrics. 1st ed. Princeton: Princeton University Press.

Lahiani, Amine and Khaled Guesmi. 2014. "Commodity Price Correlation And Time Varying Hedge Ratios". Journal Of Applied Business Research (JABR) 30 (4): 1053.

Lo, Andrew W. 1991. "Long-Term Memory In Stock Market Prices". Econometrica 59 (5): 1279.

Maginn, John L, Donald L Tuttle, and Jerald E Pinto. 2007. Managing Investment Portfolios: A Dynamic Process. 3rd ed. Hoboken: John Wiley & Sons.

Mandelbrot, Benoit. 1963. "The Variation of Certain Speculative Prices". The Journal of Business 36 (4): 394.

Mandelbrot, Benoit and Howard M. Taylor. 1967. "On The Distribution Of Stock Price Differences". Operations Research 15 (6): 1057-1062.

Mapa, Dennis S. 2003. "A Range-Based GARCH Model For Forecasting Volatility". The Philippine Review of Economics XL (2): 73-90.

Martens, Martin and Dick van Dijk. 2007. "Measuring Volatility with The Realized Range". Journal of Econometrics 138 (1): 181-207.

Molnár, Peter. 2012. "Properties Of Range-Based Volatility Estimators". International Review of Financial Analysis 23: 20-29.

Nelson, Daniel B. 1991. "Conditional Heteroskedasticity In Asset Returns: A New Approach". Econometrica 59 (2): 347.

Parkinson, Michael. 1980. "The Extreme Value Method for Estimating The Variance Of The Rate Of Return". The Journal of Business 53 (1): 61.

75 Qian, Hang. MIDAS MATLAB Toolbox (version v2.2). MATLAB.

Rachev, Svetlozar Todorov, Stefan Mittnik, and Frank J Fabozzi. 2007. Financial Econometrics. 1st ed. Hoboken, New Jersey: John Wiley & Sons.

Rogers, L. C. G. and S. E. Satchell. 1991. "Estimating Variance from High, Low and Closing Prices". The Annals of Applied Probability 1 (4): 504-512.

Sheppard, Kevin. 2013. Oxford MFE Toolbox. Oxford.

Tse, Y. K and Albert K. C Tsui. 2002. "A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model with Time-Varying Correlations". Journal of Business & Economic Statistics 20 (3): 351-362.

76 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.008 0.008 0.007 0.008 0.007 0.008 CAN, JPN 0.010 0.011 0.010 0.011 0.008 0.010 0.009 0.010 0.009 0.010 0.008 0.010 0.008 0.010 0.008 0.010 UK, RUS 0.012 0.012 0.012 0.013 0.009 0.011 0.009 0.011 0.008 0.010 0.010 0.012 0.010 0.012 0.009 0.011 FRA, ITA 0.008 0.008 0.008 0.008 0.007 0.008 0.008 0.009 0.008 0.008 0.008 0.008 0.007 0.008 0.007 0.008 FRA, JPN 0.014 0.019 0.013 0.019 0.009 0.010 0.013 0.014 0.011 0.012 0.009 0.011 0.009 0.011 0.011 0.013 FRA, RUS 0.012 0.015 0.012 0.015 0.011 0.014 0.011 0.013 0.011 0.014 0.010 0.013 0.010 0.013 0.010 0.012 ITA, JPN 0.017 0.020 0.017 0.021 0.016 0.021 0.016 0.021 0.016 0.021 0.017 0.020 0.017 0.020 0.018 0.021 ITA, RUS 0.020 0.022 0.020 0.022 0.018 0.023 0.017 0.022 0.016 0.022 0.020 0.024 0.022 0.025 0.023 0.026 JPN, RUS 0.011 0.013 0.011 0.013 0.011 0.013 0.011 0.013 0.010 0.013 0.011 0.012 0.011 0.013 0.011 0.013 USA, GER 0.019 0.024 0.019 0.024 0.017 0.020 0.019 0.022 0.018 0.021 0.017 0.020 0.017 0.020 0.021 0.024 USA, UK 0.015 0.015 0.013 0.015 0.011 0.014 0.010 0.013 0.011 0.014 0.010 0.013 0.011 0.014 0.010 0.013 USA, FRA 0.011 0.012 0.011 0.012 0.011 0.011 0.011 0.011 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 USA, ITA 0.016 0.017 0.016 0.017 0.012 0.015 0.011 0.014 0.010 0.014 0.013 0.015 0.014 0.017 0.013 0.016 USA, JPN 0.009 0.011 0.009 0.011 0.009 0.011 0.009 0.011 0.009 0.011 0.009 0.011 0.009 0.011 0.009 0.011 USA, RUS 0.017 0.022 0.016 0.022 0.012 0.015 0.014 0.017 0.013 0.016 0.012 0.014 0.012 0.014 0.014 0.017 CAN, GER 0.022 0.022 0.022 0.022 0.020 0.022 0.019 0.021 0.019 0.021 0.022 0.023 0.022 0.024 0.022 0.024 CAN, UK 0.012 0.013 0.012 0.013 0.011 0.014 0.011 0.013 0.011 0.013 0.011 0.013 0.011 0.013 0.011 0.013 CAN, FRA 0.020 0.024 0.020 0.024 0.019 0.020 0.020 0.021 0.019 0.020 0.019 0.019 0.018 0.020 0.020 0.021 CAN, ITA 0.014 0.014 0.014 0.014 0.012 0.014 0.012 0.013 0.012 0.013 0.012 0.013 0.012 0.014 0.012 0.013 CAN, RUS 0.024 0.026 0.023 0.026 0.023 0.022 0.024 0.023 0.023 0.023 0.023 0.022 0.023 0.022 0.023 0.022

77 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK 0.017 0.019 0.017 0.019 0.014 0.015 0.014 0.016 0.014 0.016 0.014 0.014 0.014 0.015 0.014 0.016 GER, FRA 0.008 0.009 0.008 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 GER, ITA 0.012 0.011 0.012 0.011 0.012 0.011 0.012 0.010 0.011 0.010 0.011 0.010 0.012 0.011 0.012 0.011 GER, JPN 0.015 0.013 0.015 0.013 0.013 0.012 0.012 0.011 0.011 0.011 0.012 0.011 0.014 0.013 0.014 0.014 GER, RUS 0.009 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.009 0.010 UK, FRA 0.016 0.019 0.015 0.019 0.012 0.012 0.014 0.014 0.012 0.012 0.011 0.011 0.011 0.011 0.015 0.015 UK, ITA 0.009 0.011 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.007 0.010 0.007 0.009 0.007 0.009 UK, JPN 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 USA, CAN, GER 0.009 0.010 0.008 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.007 0.009 0.007 0.009 USA, UK, FRA 0.008 0.009 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.009 0.010 CAN, GER, FRA 0.014 0.017 0.015 0.017 0.012 0.012 0.014 0.014 0.012 0.012 0.011 0.011 0.012 0.011 0.015 0.015 CAN, ITA, JPN 0.011 0.010 0.011 0.011 0.009 0.010 0.009 0.010 0.009 0.010 0.009 0.010 0.009 0.010 0.009 0.010 GER, UK, ITA 0.012 0.011 0.013 0.012 0.010 0.011 0.009 0.010 0.009 0.010 0.010 0.011 0.011 0.012 0.011 0.012 FRA, ITA, JPN 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.008 0.009 0.008 0.010 FRA, ITA, RUS 0.013 0.016 0.014 0.017 0.011 0.011 0.013 0.013 0.011 0.012 0.010 0.010 0.010 0.011 0.013 0.014 FRA, JPN, RUS 0.014 0.013 0.014 0.012 0.013 0.013 0.012 0.011 0.011 0.011 0.012 0.011 0.014 0.013 0.014 0.013 ITA, JPN, RUS 0.009 0.009 0.009 0.011 0.009 0.011 0.009 0.010 0.008 0.010 0.008 0.009 0.008 0.010 0.009 0.011 USA, CAN, UK 0.015 0.017 0.016 0.017 0.013 0.012 0.014 0.014 0.013 0.012 0.011 0.011 0.013 0.012 0.015 0.014 USA, CAN, FRA 0.009 0.010 0.011 0.011 0.009 0.011 0.009 0.010 0.008 0.010 0.008 0.010 0.009 0.011 0.010 0.011 USA, CAN, ITA 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.006 0.008

78 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, JPN 0.016 0.018 0.016 0.017 0.013 0.013 0.014 0.014 0.013 0.013 0.012 0.012 0.014 0.014 0.016 0.016 USA, CAN, RUS 0.010 0.012 0.011 0.015 0.009 0.011 0.010 0.012 0.009 0.011 0.008 0.010 0.008 0.010 0.010 0.013 USA, GER, UK 0.008 0.010 0.008 0.010 0.007 0.009 0.008 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.007 0.009 USA, GER, FRA 0.010 0.011 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.010 USA, GER, ITA 0.011 0.012 0.011 0.012 0.009 0.010 0.009 0.010 0.009 0.010 0.010 0.011 0.010 0.012 0.009 0.012 USA, GER, JPN 0.008 0.009 0.007 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.007 0.009 0.007 0.008 USA, GER, RUS 0.012 0.017 0.013 0.016 0.009 0.011 0.012 0.013 0.011 0.012 0.009 0.011 0.008 0.010 0.011 0.013 USA, UK, ITA 0.009 0.010 0.009 0.010 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 USA, UK, JPN 0.010 0.011 0.011 0.011 0.008 0.010 0.007 0.010 0.007 0.009 0.009 0.010 0.009 0.011 0.008 0.010 USA, UK, RUS 0.007 0.008 0.007 0.008 0.007 0.008 0.008 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 USA, FRA, ITA 0.010 0.010 0.009 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 USA, FRA, JPN 0.011 0.016 0.011 0.016 0.008 0.010 0.011 0.012 0.010 0.011 0.008 0.010 0.008 0.010 0.010 0.012 USA, FRA, RUS 0.012 0.012 0.012 0.012 0.009 0.011 0.009 0.011 0.009 0.010 0.010 0.012 0.010 0.012 0.009 0.011 USA, ITA, JPN 0.008 0.009 0.008 0.010 0.007 0.009 0.008 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.007 0.009 USA, ITA, RUS 0.013 0.017 0.013 0.017 0.009 0.011 0.012 0.013 0.011 0.012 0.009 0.011 0.009 0.011 0.011 0.012 USA, JPN, RUS 0.009 0.009 0.009 0.010 0.008 0.009 0.009 0.009 0.008 0.009 0.008 0.010 0.008 0.010 0.008 0.010 CAN, GER, UK 0.014 0.018 0.014 0.018 0.010 0.012 0.013 0.013 0.011 0.012 0.011 0.012 0.011 0.013 0.011 0.013 CAN, GER, ITA 0.010 0.012 0.010 0.014 0.008 0.009 0.010 0.011 0.009 0.010 0.008 0.009 0.008 0.009 0.009 0.011 CAN, GER, JPN 0.013 0.015 0.013 0.016 0.011 0.013 0.011 0.014 0.011 0.013 0.011 0.012 0.011 0.013 0.012 0.014 CAN, GER, RUS 0.021 0.024 0.020 0.023 0.018 0.021 0.019 0.021 0.017 0.021 0.020 0.022 0.020 0.023 0.022 0.023

79 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

CAN, UK, FRA 0.012 0.013 0.013 0.013 0.011 0.014 0.011 0.013 0.010 0.013 0.012 0.013 0.012 0.013 0.012 0.013 CAN, UK, ITA 0.012 0.011 0.011 0.011 0.008 0.010 0.008 0.009 0.008 0.009 0.009 0.010 0.009 0.011 0.009 0.011 CAN, UK, JPN 0.020 0.024 0.019 0.023 0.017 0.020 0.019 0.021 0.018 0.020 0.018 0.020 0.018 0.020 0.020 0.022 CAN, UK, RUS 0.011 0.013 0.012 0.013 0.011 0.013 0.010 0.012 0.010 0.012 0.011 0.013 0.011 0.013 0.011 0.013 CAN, FRA, ITA 0.020 0.022 0.020 0.022 0.017 0.022 0.016 0.021 0.016 0.021 0.020 0.023 0.021 0.024 0.021 0.024 CAN, FRA, JPN 0.017 0.022 0.017 0.020 0.012 0.015 0.014 0.016 0.014 0.016 0.012 0.014 0.012 0.014 0.014 0.017 CAN, FRA, RUS 0.010 0.012 0.009 0.012 0.009 0.012 0.009 0.011 0.009 0.011 0.009 0.011 0.009 0.011 0.009 0.011 CAN, ITA, RUS 0.016 0.017 0.016 0.017 0.012 0.015 0.011 0.014 0.011 0.014 0.013 0.016 0.014 0.017 0.013 0.016 CAN, JPN, RUS 0.014 0.015 0.013 0.015 0.010 0.014 0.010 0.013 0.010 0.014 0.010 0.013 0.010 0.014 0.010 0.014 GER, JPN, RUS 0.012 0.015 0.012 0.016 0.010 0.013 0.010 0.013 0.010 0.013 0.009 0.012 0.009 0.012 0.010 0.014 GER, ITA, RUS 0.020 0.023 0.018 0.021 0.012 0.016 0.014 0.017 0.013 0.016 0.014 0.016 0.014 0.017 0.015 0.018 GER, ITA, JPN 0.011 0.012 0.012 0.012 0.010 0.012 0.009 0.012 0.009 0.011 0.010 0.012 0.010 0.013 0.010 0.012 GER, FRA, RUS 0.008 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 GER, FRA, JPN 0.019 0.022 0.018 0.021 0.012 0.015 0.014 0.017 0.013 0.016 0.012 0.015 0.012 0.015 0.014 0.017 GER, FRA, ITA 0.011 0.012 0.010 0.012 0.009 0.012 0.009 0.012 0.009 0.012 0.009 0.012 0.009 0.012 0.009 0.012 GER, UK, RUS 0.017 0.017 0.017 0.017 0.012 0.015 0.011 0.014 0.011 0.014 0.013 0.016 0.014 0.017 0.013 0.016 GER, UK, JPN 0.013 0.014 0.013 0.013 0.012 0.014 0.011 0.013 0.011 0.013 0.012 0.013 0.012 0.013 0.012 0.013 GER, UK, FRA 0.022 0.024 0.022 0.024 0.019 0.021 0.020 0.021 0.019 0.021 0.020 0.022 0.021 0.022 0.021 0.022 UK, JPN, RUS 0.013 0.015 0.013 0.016 0.011 0.014 0.012 0.014 0.011 0.014 0.011 0.013 0.011 0.013 0.012 0.014 UK, ITA, RUS 0.014 0.015 0.014 0.017 0.012 0.014 0.012 0.014 0.012 0.014 0.012 0.013 0.012 0.013 0.012 0.014

80 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, ITA, JPN 0.013 0.016 0.012 0.016 0.008 0.010 0.011 0.012 0.010 0.011 0.009 0.010 0.008 0.010 0.010 0.012 UK, FRA, RUS 0.009 0.010 0.010 0.011 0.009 0.010 0.009 0.010 0.009 0.010 0.009 0.009 0.009 0.010 0.008 0.009 UK, FRA, JPN 0.012 0.012 0.011 0.012 0.011 0.011 0.011 0.011 0.010 0.010 0.010 0.010 0.011 0.011 0.011 0.011 UK, FRA, ITA 0.014 0.013 0.014 0.013 0.012 0.012 0.011 0.011 0.010 0.011 0.012 0.012 0.014 0.013 0.014 0.014 FRA, ITA, JPN, RUS

0.014 0.015 0.014 0.016 0.012 0.013 0.012 0.014 0.011 0.013 0.011 0.013 0.012 0.013 0.012 0.014

GER, UK, ITA, JPN

0.012 0.015 0.012 0.016 0.009 0.012 0.010 0.013 0.009 0.012 0.009 0.011 0.009 0.012 0.010 0.013

CAN, UK, FRA, RUS

0.020 0.023 0.018 0.021 0.012 0.016 0.014 0.017 0.013 0.016 0.014 0.016 0.015 0.018 0.016 0.018

CAN, GER, UK, ITA

0.011 0.012 0.012 0.013 0.010 0.012 0.009 0.011 0.009 0.011 0.010 0.012 0.010 0.012 0.010 0.012

USA, UK, FRA, ITA

0.018 0.022 0.017 0.021 0.012 0.015 0.014 0.016 0.013 0.016 0.012 0.014 0.012 0.015 0.014 0.018

USA, CAN, JPN, RUS

0.010 0.011 0.010 0.012 0.009 0.012 0.009 0.011 0.008 0.011 0.009 0.011 0.009 0.012 0.009 0.012

USA, CAN, GER, JPN

0.017 0.017 0.016 0.017 0.012 0.016 0.010 0.014 0.011 0.015 0.012 0.016 0.013 0.017 0.013 0.017

USA, CAN, GER, FRA

0.010 0.012 0.011 0.014 0.008 0.010 0.010 0.011 0.009 0.011 0.009 0.010 0.008 0.010 0.009 0.011

USA, CAN, GER, 0.009 0.012 0.010 0.014 0.008 0.010 0.010 0.011 0.009 0.010 0.008 0.010 0.008 0.010 0.009 0.011

81 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK UK, ITA, JPN, RUS

0.015 0.017 0.014 0.017 0.010 0.012 0.012 0.013 0.011 0.012 0.011 0.013 0.011 0.013 0.011 0.013

UK, FRA, JPN, RUS

0.009 0.009 0.009 0.010 0.008 0.009 0.009 0.010 0.008 0.009 0.008 0.010 0.008 0.010 0.008 0.009

UK, FRA, ITA, RUS

0.013 0.015 0.013 0.016 0.010 0.013 0.010 0.013 0.010 0.013 0.010 0.012 0.010 0.013 0.010 0.013

UK, FRA, ITA, JPN

0.009 0.011 0.009 0.013 0.007 0.009 0.009 0.011 0.008 0.010 0.008 0.009 0.007 0.009 0.008 0.010

GER, ITA, JPN, RUS

0.014 0.018 0.013 0.017 0.009 0.011 0.011 0.012 0.009 0.011 0.010 0.011 0.010 0.012 0.011 0.013

GER, FRA, JPN, RUS

0.009 0.009 0.009 0.010 0.007 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.010 0.008 0.009

GER, FRA, ITA, RUS

0.013 0.017 0.013 0.016 0.008 0.010 0.011 0.012 0.010 0.011 0.008 0.010 0.008 0.010 0.009 0.012

GER, FRA, ITA, JPN

0.008 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009

GER, UK, JPN, RUS

0.011 0.011 0.011 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.009 0.011 0.009 0.011 0.008 0.011

GER, UK, ITA, RUS

0.009 0.012 0.010 0.013 0.008 0.009 0.009 0.011 0.009 0.010 0.008 0.009 0.008 0.009 0.009 0.011

82 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, FRA, RUS

0.014 0.017 0.014 0.017 0.009 0.011 0.012 0.013 0.011 0.012 0.011 0.013 0.011 0.013 0.012 0.014

GER, UK, FRA, JPN

0.009 0.009 0.009 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.009 0.009 0.008 0.010 0.008 0.009

GER, UK, FRA, ITA

0.013 0.017 0.013 0.016 0.009 0.011 0.012 0.013 0.011 0.012 0.009 0.011 0.009 0.011 0.011 0.013

CAN, ITA, JPN, RUS

0.013 0.015 0.013 0.016 0.010 0.013 0.010 0.013 0.010 0.013 0.009 0.012 0.009 0.012 0.010 0.013

CAN, FRA, JPN, RUS

0.008 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.007 0.009 0.008 0.009

CAN, FRA, ITA, RUS

0.012 0.012 0.011 0.012 0.009 0.011 0.009 0.011 0.009 0.011 0.010 0.012 0.010 0.012 0.009 0.012

CAN, FRA, ITA, JPN

0.012 0.016 0.012 0.015 0.008 0.010 0.011 0.012 0.010 0.011 0.008 0.010 0.008 0.010 0.010 0.012

CAN, UK, JPN, RUS

0.007 0.009 0.007 0.009 0.007 0.009 0.008 0.009 0.007 0.009 0.007 0.008 0.007 0.008 0.007 0.008

CAN, UK, ITA, RUS

0.010 0.011 0.011 0.011 0.008 0.010 0.008 0.010 0.008 0.009 0.009 0.010 0.009 0.011 0.009 0.011

CAN, UK, ITA, JPN

0.009 0.010 0.008 0.010 0.007 0.009 0.007 0.009 0.007 0.010 0.007 0.009 0.007 0.009 0.007 0.009

CAN, UK, FRA, 0.010 0.012 0.012 0.015 0.010 0.011 0.010 0.012 0.009 0.011 0.008 0.010 0.009 0.011 0.010 0.013

83 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

JPN CAN, UK, FRA, ITA

0.010 0.012 0.012 0.015 0.010 0.011 0.010 0.012 0.009 0.011 0.008 0.010 0.009 0.011 0.010 0.012

CAN, GER, JPN, RUS

0.016 0.018 0.016 0.017 0.013 0.013 0.014 0.014 0.013 0.013 0.012 0.012 0.014 0.014 0.016 0.016

CAN, GER, ITA, RUS

0.010 0.010 0.011 0.011 0.009 0.011 0.009 0.010 0.008 0.010 0.008 0.010 0.009 0.011 0.010 0.011

CAN, GER, ITA, JPN

0.020 0.023 0.019 0.021 0.013 0.016 0.014 0.017 0.014 0.016 0.014 0.016 0.014 0.017 0.015 0.018

CAN, GER, FRA, RUS

0.010 0.012 0.011 0.015 0.009 0.011 0.009 0.012 0.008 0.011 0.008 0.010 0.008 0.010 0.010 0.012

CAN, GER, FRA, JPN

0.015 0.018 0.015 0.017 0.011 0.012 0.012 0.013 0.011 0.012 0.011 0.012 0.012 0.013 0.014 0.015

CAN, GER, FRA, ITA

0.009 0.010 0.010 0.011 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.010 0.009 0.011 0.009 0.011

CAN, GER, UK, RUS

0.015 0.017 0.015 0.017 0.011 0.012 0.012 0.013 0.011 0.012 0.010 0.011 0.010 0.011 0.013 0.014

CAN, GER, UK, JPN

0.008 0.009 0.009 0.011 0.008 0.011 0.008 0.010 0.008 0.010 0.007 0.009 0.008 0.010 0.008 0.011

CAN, GER, UK, FRA

0.012 0.011 0.013 0.012 0.010 0.012 0.009 0.010 0.009 0.010 0.010 0.011 0.012 0.012 0.011 0.012

84 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, ITA, JPN, RUS

0.010 0.012 0.011 0.015 0.009 0.011 0.010 0.012 0.009 0.011 0.008 0.009 0.009 0.010 0.010 0.013

USA, FRA, JPN, RUS

0.016 0.018 0.016 0.017 0.012 0.013 0.014 0.014 0.012 0.013 0.012 0.013 0.014 0.014 0.016 0.017

USA, FRA, ITA, RUS

0.009 0.010 0.011 0.012 0.009 0.011 0.009 0.010 0.008 0.010 0.008 0.010 0.009 0.011 0.010 0.011

USA, FRA, ITA, JPN

0.015 0.017 0.015 0.017 0.012 0.012 0.014 0.014 0.012 0.012 0.011 0.011 0.012 0.012 0.015 0.015

USA, UK, JPN, RUS

0.012 0.012 0.012 0.012 0.010 0.012 0.009 0.012 0.009 0.012 0.010 0.012 0.010 0.012 0.010 0.012

USA, UK, ITA, RUS

0.008 0.009 0.009 0.011 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.009 0.008 0.010 0.009 0.011

USA, UK, ITA, JPN

0.014 0.013 0.014 0.013 0.012 0.013 0.011 0.011 0.010 0.011 0.011 0.012 0.014 0.013 0.014 0.014

USA, UK, FRA, RUS

0.014 0.017 0.015 0.017 0.011 0.012 0.013 0.013 0.011 0.012 0.010 0.010 0.010 0.011 0.013 0.014

USA, UK, FRA, JPN

0.008 0.009 0.008 0.010 0.008 0.010 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.009 0.008 0.010

USA, GER, JPN, RUS

0.012 0.011 0.013 0.012 0.010 0.011 0.009 0.010 0.009 0.010 0.010 0.011 0.012 0.013 0.011 0.013

USA, GER, ITA, 0.010 0.011 0.010 0.011 0.009 0.010 0.008 0.010 0.009 0.010 0.009 0.009 0.009 0.010 0.009 0.010

85 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

RUS USA, GER, ITA, JPN

0.009 0.011 0.010 0.014 0.008 0.009 0.009 0.011 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.010

USA, GER, FRA, RUS

0.014 0.016 0.013 0.016 0.009 0.011 0.012 0.012 0.010 0.011 0.010 0.011 0.010 0.012 0.011 0.013

USA, GER, FRA, JPN

0.008 0.009 0.009 0.010 0.008 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.008 0.009

USA, GER, FRA, ITA

0.013 0.016 0.013 0.015 0.009 0.010 0.012 0.012 0.010 0.011 0.009 0.010 0.009 0.010 0.011 0.012

USA, GER, UK, RUS

0.013 0.015 0.014 0.016 0.011 0.013 0.011 0.013 0.011 0.013 0.011 0.013 0.012 0.013 0.012 0.014

USA, GER, UK, JPN

0.008 0.008 0.008 0.009 0.007 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009

USA, GER, UK, ITA

0.011 0.011 0.011 0.010 0.009 0.010 0.009 0.010 0.009 0.009 0.009 0.010 0.010 0.011 0.009 0.011

USA, GER, UK, FRA

0.012 0.015 0.012 0.015 0.008 0.010 0.010 0.011 0.009 0.010 0.008 0.009 0.007 0.010 0.009 0.011

USA, CAN, ITA, RUS

0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008

USA, CAN, ITA, JPN

0.010 0.010 0.010 0.010 0.008 0.009 0.007 0.009 0.007 0.009 0.009 0.010 0.009 0.010 0.008 0.010

86 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, FRA, RUS

0.009 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009

USA, CAN, FRA, JPN

0.012 0.016 0.012 0.015 0.009 0.010 0.011 0.012 0.010 0.011 0.009 0.010 0.008 0.010 0.010 0.012

USA, CAN, FRA, ITA

0.008 0.008 0.007 0.009 0.007 0.009 0.008 0.009 0.007 0.009 0.007 0.008 0.007 0.008 0.007 0.008

USA, CAN, UK, RUS

0.011 0.011 0.011 0.011 0.008 0.010 0.008 0.009 0.008 0.009 0.009 0.010 0.010 0.011 0.009 0.011

USA, CAN, UK, JPN

0.009 0.010 0.009 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009

USA, CAN, UK, ITA

0.013 0.015 0.013 0.016 0.011 0.013 0.011 0.013 0.011 0.013 0.011 0.013 0.011 0.013 0.012 0.015

USA, CAN, UK, FRA

0.008 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.007 0.009 0.006 0.008

USA, CAN, GER, RUS

0.022 0.024 0.020 0.024 0.017 0.020 0.018 0.021 0.017 0.020 0.019 0.021 0.020 0.022 0.021 0.022

USA, CAN, GER, ITA

0.012 0.013 0.013 0.013 0.011 0.013 0.011 0.012 0.010 0.012 0.011 0.013 0.012 0.013 0.012 0.013

UK, FRA, ITA, JPN, RUS

0.013 0.015 0.013 0.016 0.010 0.013 0.010 0.013 0.010 0.013 0.010 0.012 0.010 0.013 0.010 0.013

CAN, GER, ITA, 0.014 0.017 0.013 0.017 0.009 0.011 0.011 0.013 0.010 0.012 0.009 0.012 0.010 0.012 0.010 0.013

87 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

JPN, RUS USA, UK, ITA, JPN, RUS

0.009 0.009 0.009 0.010 0.007 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.010 0.007 0.009

USA, GER, UK, FRA, RUS

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.009 0.010 0.009 0.010 0.009 0.011 0.009 0.012

USA, CAN, UK, FRA, JPN

0.009 0.012 0.010 0.013 0.008 0.009 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.010 0.009 0.011

USA, CAN, GER, UK, RUS

0.014 0.017 0.014 0.017 0.009 0.012 0.012 0.013 0.011 0.012 0.011 0.013 0.011 0.013 0.012 0.014

USA, CAN, GER, UK, JPN

0.009 0.009 0.009 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.010 0.008 0.009

USA, CAN, GER, UK, ITA

0.009 0.012 0.010 0.013 0.007 0.009 0.009 0.011 0.008 0.010 0.008 0.009 0.007 0.009 0.008 0.011

USA, CAN, GER, UK, FRA

0.014 0.017 0.013 0.016 0.009 0.011 0.011 0.012 0.010 0.011 0.010 0.012 0.010 0.012 0.011 0.013

GER, FRA, ITA, JPN, RUS

0.009 0.009 0.009 0.010 0.007 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.008 0.009

GER, UK, ITA, JPN, RUS

0.013 0.017 0.012 0.016 0.008 0.010 0.011 0.012 0.009 0.011 0.008 0.010 0.008 0.010 0.010 0.012

GER, UK, FRA, JPN, RUS

0.014 0.015 0.014 0.016 0.011 0.013 0.011 0.013 0.011 0.013 0.011 0.013 0.012 0.013 0.012 0.014

88 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, FRA, ITA, RUS

0.008 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009

GER, UK, FRA, ITA, JPN

0.011 0.011 0.011 0.012 0.008 0.010 0.008 0.010 0.008 0.010 0.009 0.011 0.009 0.011 0.008 0.011

CAN, FRA, ITA, JPN, RUS

0.011 0.012 0.012 0.015 0.010 0.011 0.010 0.012 0.009 0.011 0.008 0.010 0.009 0.011 0.010 0.012

CAN, UK, ITA, JPN, RUS

0.011 0.012 0.012 0.015 0.009 0.011 0.009 0.012 0.008 0.011 0.008 0.010 0.009 0.011 0.010 0.012

CAN, UK, FRA, JPN, RUS

0.010 0.012 0.011 0.015 0.009 0.011 0.009 0.012 0.008 0.011 0.008 0.010 0.008 0.011 0.009 0.012

CAN, UK, FRA, ITA, RUS

0.016 0.018 0.015 0.017 0.011 0.013 0.012 0.013 0.011 0.012 0.011 0.012 0.012 0.013 0.014 0.015

CAN, UK, FRA, ITA, JPN

0.009 0.009 0.010 0.011 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.009 0.011 0.009 0.011

CAN, GER, FRA, JPN, RUS

0.011 0.012 0.012 0.015 0.009 0.011 0.010 0.012 0.009 0.011 0.008 0.010 0.009 0.011 0.011 0.013

CAN, GER, FRA, ITA, RUS

0.010 0.012 0.012 0.015 0.009 0.011 0.010 0.012 0.009 0.011 0.008 0.010 0.009 0.011 0.010 0.013

CAN, GER, FRA, ITA, JPN

0.016 0.018 0.016 0.017 0.012 0.013 0.014 0.014 0.012 0.012 0.012 0.013 0.014 0.014 0.016 0.016

CAN, GER, UK, 0.013 0.015 0.013 0.016 0.010 0.012 0.010 0.013 0.009 0.012 0.010 0.012 0.010 0.013 0.011 0.014

89 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

JPN, RUS CAN, GER, UK, ITA, RUS

0.009 0.010 0.011 0.011 0.009 0.011 0.009 0.010 0.008 0.009 0.008 0.009 0.009 0.011 0.010 0.011

CAN, GER, UK, ITA, JPN

0.010 0.012 0.011 0.015 0.009 0.011 0.009 0.012 0.008 0.011 0.008 0.009 0.008 0.010 0.009 0.012

CAN, GER, UK, FRA, RUS

0.016 0.017 0.015 0.017 0.011 0.013 0.012 0.013 0.011 0.012 0.011 0.012 0.012 0.014 0.014 0.015

CAN, GER, UK, FRA, JPN

0.009 0.010 0.010 0.011 0.008 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.009 0.011 0.009 0.011

CAN, GER, UK, FRA, ITA

0.014 0.017 0.015 0.017 0.010 0.012 0.012 0.013 0.011 0.012 0.010 0.011 0.010 0.012 0.013 0.014

USA, FRA, ITA, JPN, RUS

0.008 0.009 0.009 0.010 0.008 0.010 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.010 0.008 0.010

USA, UK, FRA, JPN, RUS

0.012 0.011 0.013 0.012 0.010 0.012 0.009 0.010 0.009 0.010 0.010 0.011 0.011 0.013 0.011 0.013

USA, UK, FRA, ITA, RUS

0.009 0.011 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.010 0.009 0.011

USA, UK, FRA, ITA, JPN

0.009 0.011 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.009 0.009 0.011

USA, GER, ITA, JPN, RUS

0.014 0.016 0.014 0.016 0.010 0.011 0.012 0.012 0.010 0.011 0.010 0.011 0.010 0.012 0.011 0.013

90 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, GER, FRA, JPN, RUS

0.012 0.015 0.013 0.016 0.009 0.012 0.010 0.013 0.009 0.012 0.009 0.012 0.009 0.012 0.010 0.014

USA, GER, FRA, ITA, RUS

0.009 0.009 0.009 0.010 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009 0.008 0.009

USA, GER, FRA, ITA, JPN

0.009 0.011 0.009 0.013 0.007 0.009 0.009 0.011 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.010

USA, GER, UK, JPN, RUS

0.014 0.016 0.013 0.015 0.008 0.010 0.010 0.012 0.009 0.011 0.009 0.011 0.009 0.011 0.010 0.012

USA, GER, UK, ITA, RUS

0.008 0.009 0.009 0.010 0.007 0.009 0.008 0.009 0.007 0.008 0.008 0.009 0.008 0.009 0.008 0.009

USA, GER, UK, ITA, JPN

0.013 0.016 0.013 0.015 0.008 0.010 0.010 0.012 0.009 0.011 0.008 0.010 0.008 0.010 0.009 0.012

USA, GER, UK, FRA, JPN

0.008 0.008 0.008 0.009 0.007 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009

USA, GER, UK, FRA, ITA

0.010 0.010 0.010 0.010 0.008 0.010 0.008 0.009 0.008 0.009 0.008 0.010 0.009 0.011 0.008 0.010

USA, CAN, ITA, JPN, RUS

0.009 0.011 0.010 0.013 0.008 0.009 0.009 0.011 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.011

USA, CAN, FRA, JPN, RUS

0.014 0.016 0.013 0.015 0.009 0.011 0.011 0.012 0.010 0.011 0.010 0.011 0.010 0.012 0.011 0.014

USA, CAN, FRA, 0.008 0.009 0.009 0.010 0.008 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.008 0.009

91 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

ITA, RUS USA, CAN, FRA, ITA, JPN

0.020 0.023 0.018 0.021 0.012 0.016 0.014 0.017 0.013 0.016 0.013 0.016 0.014 0.017 0.015 0.018

USA, CAN, UK, JPN, RUS

0.012 0.016 0.013 0.015 0.009 0.010 0.011 0.012 0.010 0.011 0.009 0.010 0.009 0.010 0.011 0.012

USA, CAN, UK, ITA, RUS

0.008 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.007 0.008 0.007 0.008 0.007 0.009 0.007 0.009

USA, CAN, UK, ITA, JPN

0.011 0.011 0.011 0.011 0.009 0.010 0.008 0.010 0.008 0.009 0.009 0.010 0.010 0.011 0.009 0.011

USA, CAN, UK, FRA, RUS

0.012 0.015 0.012 0.015 0.008 0.010 0.011 0.012 0.010 0.011 0.008 0.009 0.008 0.010 0.009 0.012

USA, CAN, UK, FRA, ITA

0.007 0.008 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.008 0.007 0.008 0.007 0.008 0.007 0.008

USA, CAN, GER, JPN, RUS

0.010 0.010 0.010 0.010 0.008 0.010 0.008 0.009 0.008 0.009 0.009 0.010 0.009 0.011 0.008 0.010

USA, CAN, GER, ITA, RUS

0.009 0.010 0.008 0.010 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009 0.007 0.009

USA, CAN, GER, ITA, JPN

0.011 0.012 0.012 0.012 0.009 0.012 0.009 0.011 0.009 0.011 0.010 0.011 0.010 0.012 0.010 0.012

USA, CAN, GER, FRA, RUS

0.010 0.012 0.011 0.014 0.008 0.010 0.010 0.011 0.009 0.010 0.009 0.010 0.008 0.010 0.009 0.011

92 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, FRA, JPN

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.011

USA, CAN, GER, FRA, ITA

0.009 0.012 0.010 0.014 0.007 0.010 0.009 0.011 0.008 0.010 0.008 0.009 0.007 0.010 0.008 0.011

GER, UK, FRA, ITA, JPN, RUS

0.013 0.015 0.013 0.016 0.009 0.012 0.010 0.012 0.009 0.012 0.010 0.012 0.010 0.013 0.010 0.014

USA, GER, UK, FRA, ITA, RUS

0.011 0.013 0.012 0.015 0.009 0.011 0.009 0.012 0.008 0.011 0.008 0.010 0.009 0.011 0.010 0.013

USA, CAN, GER, FRA, ITA, JPN

0.010 0.012 0.012 0.015 0.008 0.011 0.009 0.012 0.008 0.011 0.008 0.010 0.008 0.011 0.010 0.013

USA, CAN, GER, UK, JPN, RUS

0.016 0.018 0.015 0.017 0.011 0.013 0.012 0.013 0.011 0.012 0.011 0.012 0.012 0.013 0.014 0.015

USA, CAN, GER, UK, ITA, RUS

0.009 0.009 0.010 0.011 0.008 0.011 0.008 0.010 0.007 0.009 0.008 0.009 0.009 0.011 0.009 0.011

USA, CAN, GER, UK, ITA, JPN

0.010 0.011 0.011 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.010 0.009 0.011

USA, CAN, GER, UK, FRA, RUS

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.010 0.008 0.011

USA, CAN, GER, UK, FRA, JPN

0.009 0.011 0.010 0.014 0.007 0.010 0.009 0.011 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.011

USA, CAN, GER, 0.014 0.016 0.013 0.015 0.009 0.011 0.011 0.012 0.009 0.011 0.009 0.011 0.009 0.012 0.010 0.012

93 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, FRA, ITA CAN, UK, FRA, ITA, JPN, RUS

0.008 0.009 0.009 0.010 0.007 0.009 0.008 0.009 0.007 0.009 0.007 0.009 0.008 0.009 0.008 0.009

CAN, GER, FRA, ITA, JPN, RUS

0.010 0.011 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.010 0.009 0.011

CAN, GER, UK, ITA, JPN, RUS

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.011

CAN, GER, UK, FRA, JPN, RUS

0.009 0.011 0.010 0.013 0.008 0.009 0.009 0.010 0.008 0.010 0.007 0.009 0.007 0.009 0.009 0.011

CAN, GER, UK, FRA, ITA, RUS

0.014 0.016 0.013 0.015 0.009 0.011 0.011 0.012 0.010 0.011 0.010 0.011 0.010 0.012 0.011 0.013

CAN, GER, UK, FRA, ITA, JPN

0.009 0.009 0.009 0.010 0.008 0.009 0.008 0.009 0.007 0.008 0.008 0.009 0.008 0.009 0.008 0.009

USA, UK, FRA, ITA, JPN, RUS

0.009 0.011 0.010 0.013 0.007 0.009 0.009 0.011 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.011

USA, GER, FRA, ITA, JPN, RUS

0.014 0.016 0.013 0.015 0.008 0.011 0.010 0.012 0.009 0.011 0.009 0.011 0.009 0.012 0.011 0.013

USA, GER, UK, ITA, JPN, RUS

0.008 0.009 0.009 0.010 0.007 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.008 0.009

USA, GER, UK, FRA, JPN, RUS

0.012 0.016 0.012 0.015 0.008 0.010 0.010 0.011 0.009 0.010 0.008 0.010 0.008 0.010 0.009 0.012

94 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, GER, UK, FRA, ITA, JPN

0.008 0.008 0.008 0.009 0.007 0.009 0.007 0.009 0.007 0.008 0.007 0.008 0.007 0.009 0.007 0.009

USA, CAN, FRA, ITA, JPN, RUS

0.010 0.010 0.010 0.011 0.008 0.010 0.007 0.009 0.008 0.009 0.008 0.010 0.009 0.011 0.008 0.011

USA, CAN, UK, ITA, JPN, RUS

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.009 0.010 0.008 0.010 0.009 0.011

USA, CAN, UK, FRA, JPN, RUS

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.009 0.011

USA, CAN, UK, FRA, ITA, RUS

0.009 0.012 0.010 0.014 0.007 0.009 0.009 0.011 0.008 0.010 0.008 0.009 0.007 0.009 0.008 0.011

USA, CAN, UK, FRA, ITA, JPN

0.014 0.017 0.013 0.016 0.009 0.011 0.011 0.012 0.010 0.011 0.009 0.012 0.010 0.012 0.011 0.013

USA, CAN, GER, ITA, JPN, RUS

0.009 0.009 0.009 0.010 0.007 0.009 0.008 0.009 0.007 0.009 0.008 0.009 0.008 0.009 0.008 0.009

USA, CAN, GER, FRA, JPN, RUS

0.011 0.012 0.012 0.015 0.009 0.011 0.009 0.012 0.008 0.011 0.008 0.010 0.009 0.011 0.010 0.012

USA, CAN, GER, FRA, ITA, RUS

0.011 0.012 0.012 0.015 0.009 0.011 0.010 0.011 0.009 0.010 0.008 0.010 0.009 0.011 0.011 0.013

USA, CAN, GER, UK, FRA, ITA, JPN

0.008 0.009 0.009 0.010 0.007 0.009 0.007 0.009 0.007 0.008 0.007 0.009 0.008 0.009 0.008 0.009

95 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, UK, FRA, ITA, RUS

0.014 0.016 0.013 0.015 0.008 0.010 0.010 0.012 0.009 0.011 0.009 0.011 0.009 0.012 0.010 0.013

USA, CAN, GER, UK, FRA, JPN, RUS

0.009 0.011 0.010 0.013 0.007 0.009 0.009 0.010 0.008 0.010 0.007 0.009 0.007 0.009 0.008 0.011

USA, CAN, GER, UK, ITA, JPN, RUS

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.008 0.009 0.008 0.010 0.009 0.011

USA, CAN, GER, FRA, ITA, JPN, RUS

0.010 0.011 0.010 0.014 0.008 0.009 0.009 0.010 0.008 0.009 0.008 0.009 0.008 0.010 0.009 0.011

USA, CAN, UK, FRA, ITA, JPN, RUS

0.010 0.012 0.010 0.014 0.008 0.010 0.009 0.011 0.008 0.010 0.007 0.009 0.008 0.010 0.008 0.011

USA, GER, UK, FRA, ITA, JPN, RUS

0.011 0.012 0.012 0.015 0.009 0.011 0.009 0.011 0.008 0.011 0.008 0.010 0.009 0.011 0.010 0.013

CAN, GER, UK, FRA, ITA, JPN, RUS

0.010 0.012 0.010 0.014 0.007 0.009 0.009 0.011 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.011

96 Appendix A: Resulting Simulated Portfolio Variances (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR- MIDAS (Q = 63)

CARR- MIDAS (Q = 126)

CARR- MIDAS (Q = 252)

CARR- MIDAS (Q = 756)

CARR- MIDAS (Q = 1260)

Stage 2 𝑅𝑡 = DCC DCC DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS DCC-

MIDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, UK, FRA, ITA, JPN, RUS

0.010 0.012 0.010 0.014 0.007 0.009 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.010 0.008 0.011

97 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN 0.000 0.031 0.000 0.023 0.000 0.082 0.000 0.094 0.003 0.150 0.002 0.203 0.000 0.047 0.000 0.012 CAN, JPN 0.000 0.018 0.000 0.015 0.000 0.066 0.234 0.031 0.251 0.031 0.073 0.017 0.024 0.013 0.012 0.023 UK, RUS 0.002 0.199 0.018 0.172 0.002 0.095 0.148 0.121 0.235 0.139 0.371 0.049 0.161 0.039 0.270 0.064 FRA, ITA 0.217 0.013 0.561 0.016 0.169 0.122 0.217 0.257 0.304 0.168 0.357 0.189 0.489 0.107 0.456 0.150 FRA, JPN 0.038 0.032 0.027 0.040 0.536 0.480 0.559 0.008 0.855 0.042 0.539 0.185 0.858 0.132 0.937 0.013 FRA, RUS 0.000 0.024 0.000 0.023 0.000 0.095 0.072 0.028 0.029 0.028 0.028 0.021 0.010 0.033 0.004 0.032 ITA, JPN 0.000 0.029 0.000 0.022 0.001 0.002 0.001 0.171 0.000 0.168 0.031 0.006 0.039 0.001 0.079 0.000 ITA, RUS 0.000 0.134 0.017 0.117 0.001 0.020 0.026 0.772 0.004 0.981 0.003 0.028 0.004 0.029 0.000 0.016 JPN, RUS 0.099 0.013 0.164 0.006 0.326 0.054 0.427 0.087 0.881 0.048 0.355 0.114 0.359 0.110 0.243 0.140 USA, GER 0.116 0.058 0.123 0.162 0.515 0.933 0.700 0.015 0.846 0.120 0.703 0.260 0.870 0.172 0.392 0.000 USA, UK 0.014 0.026 0.000 0.024 0.000 0.025 0.000 0.029 0.001 0.021 0.000 0.021 0.000 0.020 0.000 0.028 USA, FRA 0.000 0.028 0.000 0.033 0.000 0.056 0.191 0.132 0.204 0.122 0.000 0.139 0.047 0.094 0.033 0.231 USA, ITA 0.000 0.153 0.020 0.214 0.001 0.011 0.012 0.132 0.007 0.114 0.223 0.033 0.250 0.008 0.206 0.025 USA, JPN 0.072 0.014 0.027 0.006 0.064 0.087 0.071 0.097 0.081 0.080 0.108 0.077 0.096 0.095 0.088 0.100 USA, RUS 0.069 0.045 0.027 0.021 0.195 0.858 0.398 0.032 0.797 0.120 0.746 0.324 0.986 0.284 0.350 0.022 CAN, GER 0.016 0.108 0.001 0.153 0.001 0.014 0.014 0.080 0.003 0.136 0.000 0.001 0.000 0.000 0.000 0.000 CAN, UK 0.007 0.016 0.006 0.007 0.129 0.055 0.199 0.083 0.357 0.063 0.233 0.069 0.255 0.059 0.229 0.081 CAN, FRA 0.309 0.081 0.349 0.134 0.975 0.945 0.336 0.005 0.618 0.028 0.887 0.088 0.650 0.083 0.275 0.003 CAN, ITA 0.096 0.095 0.045 0.193 0.258 0.139 0.346 0.300 0.418 0.270 0.856 0.166 0.681 0.109 0.842 0.166 CAN, RUS 0.235 0.067 0.604 0.362 0.325 0.376 0.685 0.031 0.977 0.093 0.177 0.282 0.125 0.288 0.705 0.097

98 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK 0.022 0.323 0.022 0.021 0.004 0.659 0.729 0.024 0.679 0.042 0.737 0.229 0.767 0.202 0.773 0.023 GER, FRA 0.000 0.034 0.000 0.031 0.000 0.082 0.178 0.193 0.138 0.191 0.001 0.163 0.038 0.124 0.060 0.169 GER, ITA 0.000 0.026 0.000 0.034 0.000 0.026 0.105 0.107 0.082 0.125 0.000 0.125 0.038 0.049 0.077 0.075 GER, JPN 0.000 0.230 0.012 0.347 0.000 0.059 0.808 0.281 0.206 0.349 0.122 0.123 0.956 0.027 0.631 0.045 GER, RUS 0.021 0.022 0.050 0.028 0.128 0.094 0.124 0.297 0.190 0.226 0.190 0.180 0.191 0.150 0.126 0.296 UK, FRA 0.133 0.036 0.020 0.024 0.804 0.630 0.534 0.028 0.929 0.292 0.302 0.733 0.937 0.515 0.363 0.009 UK, ITA 0.000 0.020 0.000 0.020 0.009 0.099 0.397 0.035 0.348 0.031 0.290 0.018 0.036 0.014 0.002 0.022 UK, JPN 0.000 0.015 0.000 0.017 0.007 0.110 0.164 0.115 0.130 0.093 0.050 0.045 0.004 0.028 0.008 0.044 USA, CAN, GER 0.000 0.021 0.000 0.045 0.000 0.078 0.000 0.084 0.005 0.106 0.000 0.136 0.000 0.024 0.000 0.015 USA, UK, FRA 0.061 0.023 0.173 0.007 0.153 0.031 0.170 0.125 0.283 0.099 0.007 0.161 0.104 0.118 0.079 0.178 CAN, GER, FRA 0.114 0.069 0.012 0.050 0.824 0.904 0.172 0.004 0.476 0.054 0.005 0.261 0.235 0.137 0.095 0.001 CAN, ITA, JPN 0.008 0.016 0.000 0.056 0.000 0.017 0.001 0.037 0.001 0.040 0.000 0.064 0.001 0.032 0.000 0.033 GER, UK, ITA 0.000 0.108 0.069 0.302 0.000 0.010 0.002 0.152 0.001 0.154 0.001 0.038 0.068 0.004 0.083 0.010 FRA, ITA, JPN 0.030 0.028 0.060 0.011 0.032 0.079 0.037 0.160 0.070 0.174 0.021 0.188 0.090 0.153 0.033 0.145 FRA, ITA, RUS 0.311 0.054 0.008 0.036 0.325 0.872 0.089 0.007 0.265 0.058 0.004 0.378 0.230 0.215 0.061 0.003 FRA, JPN, RUS 0.011 0.190 0.201 0.864 0.000 0.006 0.001 0.080 0.000 0.140 0.000 0.022 0.002 0.004 0.000 0.001 ITA, JPN, RUS 0.038 0.028 0.015 0.007 0.015 0.020 0.078 0.105 0.114 0.097 0.017 0.132 0.146 0.058 0.095 0.087 USA, CAN, UK 0.163 0.089 0.026 0.060 0.943 0.966 0.130 0.002 0.371 0.025 0.010 0.224 0.234 0.086 0.071 0.001 USA, CAN, FRA 0.037 0.068 0.029 0.209 0.001 0.052 0.107 0.260 0.131 0.267 0.217 0.139 0.391 0.037 0.140 0.084 USA, CAN, ITA 0.000 0.018 0.000 0.025 0.000 0.068 0.001 0.135 0.003 0.140 0.000 0.138 0.000 0.029 0.000 0.032

99 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, JPN 0.415 0.081 0.196 0.208 0.948 0.623 0.033 0.061 0.068 0.357 0.136 0.472 0.940 0.169 0.094 0.026 USA, CAN, RUS 0.102 0.808 0.004 0.008 0.155 0.951 0.482 0.010 0.761 0.039 0.235 0.219 0.352 0.130 0.436 0.007 USA, GER, UK 0.000 0.011 0.000 0.025 0.000 0.111 0.012 0.019 0.002 0.014 0.001 0.008 0.000 0.007 0.000 0.008 USA, GER, FRA 0.000 0.034 0.002 0.010 0.000 0.001 0.001 0.315 0.000 0.229 0.008 0.013 0.002 0.002 0.001 0.000 USA, GER, ITA 0.000 0.058 0.074 0.034 0.000 0.005 0.002 0.547 0.000 0.863 0.004 0.008 0.004 0.006 0.000 0.002 USA, GER, JPN 0.234 0.017 0.796 0.012 0.186 0.071 0.315 0.127 0.556 0.059 0.373 0.146 0.537 0.079 0.307 0.150 USA, GER, RUS 0.040 0.103 0.071 0.176 0.891 0.939 0.207 0.007 0.593 0.061 0.755 0.094 0.654 0.052 0.416 0.001 USA, UK, ITA 0.007 0.002 0.000 0.011 0.000 0.014 0.000 0.027 0.000 0.019 0.000 0.019 0.000 0.011 0.000 0.015 USA, UK, JPN 0.000 0.061 0.019 0.091 0.001 0.013 0.005 0.093 0.003 0.069 0.100 0.019 0.116 0.005 0.038 0.013 USA, UK, RUS 0.213 0.014 0.260 0.010 0.049 0.110 0.068 0.132 0.100 0.089 0.167 0.079 0.242 0.075 0.204 0.085 USA, FRA, ITA 0.000 0.021 0.000 0.022 0.000 0.022 0.005 0.065 0.011 0.129 0.000 0.175 0.000 0.018 0.000 0.011 USA, FRA, JPN 0.054 0.046 0.005 0.017 0.292 0.890 0.195 0.035 0.727 0.145 0.634 0.264 0.692 0.145 0.319 0.020 USA, FRA, RUS 0.013 0.075 0.026 0.036 0.001 0.003 0.003 0.007 0.000 0.014 0.000 0.000 0.000 0.000 0.000 0.000 USA, ITA, JPN 0.099 0.016 0.129 0.011 0.025 0.064 0.149 0.154 0.239 0.099 0.357 0.122 0.495 0.056 0.411 0.121 USA, ITA, RUS 0.077 0.116 0.092 0.139 0.811 0.781 0.107 0.003 0.447 0.021 0.903 0.053 0.570 0.038 0.268 0.004 USA, JPN, RUS 0.179 0.050 0.172 0.133 0.009 0.125 0.220 0.298 0.360 0.233 0.675 0.133 0.728 0.068 0.839 0.123 CAN, GER, UK 0.135 0.074 0.649 0.433 0.911 0.410 0.182 0.025 0.741 0.113 0.200 0.120 0.121 0.097 0.794 0.045 CAN, GER, ITA 0.026 0.910 0.002 0.014 0.155 0.859 0.406 0.032 0.694 0.064 0.517 0.250 0.711 0.171 0.923 0.029 CAN, GER, JPN 0.027 0.702 0.010 0.012 0.110 0.886 0.750 0.014 0.875 0.030 0.438 0.138 0.543 0.130 0.512 0.009 CAN, GER, RUS 0.945 0.019 0.050 0.177 0.332 0.641 0.013 0.006 0.007 0.178 0.024 0.002 0.013 0.003 0.006 0.000

100 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

CAN, UK, FRA 0.005 0.097 0.003 0.021 0.001 0.071 0.174 0.094 0.227 0.105 0.022 0.029 0.014 0.012 0.002 0.018 CAN, UK, ITA 0.000 0.202 0.005 0.368 0.000 0.124 0.505 0.147 0.175 0.267 0.048 0.146 0.070 0.019 0.001 0.013 CAN, UK, JPN 0.068 0.017 0.044 0.221 0.713 0.573 0.007 0.003 0.003 0.034 0.071 0.001 0.080 0.000 0.171 0.000 CAN, UK, RUS 0.234 0.004 0.001 0.007 0.005 0.043 0.085 0.185 0.078 0.140 0.048 0.055 0.047 0.010 0.025 0.004 CAN, FRA, ITA 0.021 0.161 0.001 0.027 0.000 0.009 0.001 0.131 0.000 0.203 0.000 0.003 0.000 0.003 0.000 0.001 CAN, FRA, JPN 0.400 0.077 0.139 0.047 0.268 0.780 0.074 0.004 0.107 0.021 0.453 0.046 0.271 0.048 0.202 0.001 CAN, FRA, RUS 0.110 0.013 0.131 0.002 0.120 0.032 0.066 0.039 0.115 0.020 0.042 0.049 0.055 0.064 0.009 0.078 CAN, ITA, RUS 0.000 0.088 0.247 0.172 0.000 0.008 0.001 0.139 0.000 0.143 0.004 0.004 0.005 0.006 0.003 0.006 CAN, JPN, RUS 0.015 0.032 0.000 0.005 0.000 0.001 0.000 0.102 0.000 0.074 0.000 0.057 0.000 0.009 0.000 0.001 GER, JPN, RUS 0.013 0.818 0.010 0.005 0.777 0.784 0.395 0.006 0.498 0.014 0.199 0.086 0.206 0.076 0.403 0.006 GER, ITA, RUS 0.542 0.224 0.603 0.253 0.144 0.982 0.013 0.027 0.026 0.090 0.371 0.092 0.291 0.013 0.038 0.005 GER, ITA, JPN 0.015 0.061 0.018 0.174 0.005 0.051 0.010 0.147 0.008 0.119 0.078 0.059 0.168 0.033 0.066 0.061 GER, FRA, RUS 0.180 0.023 0.440 0.014 0.106 0.094 0.085 0.264 0.095 0.187 0.037 0.197 0.160 0.086 0.113 0.111 GER, FRA, JPN 0.280 0.299 0.225 0.086 0.068 0.870 0.006 0.002 0.014 0.006 0.010 0.040 0.006 0.031 0.003 0.002 GER, FRA, ITA 0.039 0.008 0.064 0.002 0.007 0.020 0.014 0.065 0.022 0.047 0.030 0.064 0.040 0.050 0.012 0.073 GER, UK, RUS 0.027 0.148 0.055 0.610 0.000 0.005 0.002 0.066 0.001 0.056 0.000 0.012 0.000 0.005 0.000 0.003 GER, UK, JPN 0.008 0.272 0.005 0.015 0.006 0.054 0.033 0.085 0.019 0.063 0.005 0.009 0.003 0.002 0.001 0.005 GER, UK, FRA 0.084 0.028 0.181 0.170 0.489 0.875 0.023 0.000 0.017 0.008 0.001 0.000 0.000 0.000 0.001 0.000 UK, JPN, RUS 0.143 0.811 0.016 0.012 0.106 0.839 0.600 0.011 0.751 0.019 0.386 0.111 0.426 0.091 0.601 0.010 UK, ITA, RUS 0.149 0.719 0.196 0.027 0.033 0.296 0.853 0.065 0.957 0.102 0.875 0.228 0.772 0.131 0.747 0.058

101 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, ITA, JPN 0.030 0.032 0.001 0.009 0.409 0.504 0.123 0.010 0.426 0.180 0.037 0.744 0.014 0.211 0.008 0.003 UK, FRA, RUS 0.000 0.037 0.000 0.048 0.000 0.064 0.016 0.051 0.005 0.050 0.000 0.065 0.005 0.045 0.001 0.060 UK, FRA, JPN 0.000 0.050 0.000 0.016 0.000 0.001 0.000 0.255 0.000 0.150 0.000 0.112 0.002 0.013 0.008 0.000 UK, FRA, ITA 0.000 0.108 0.278 0.512 0.000 0.008 0.000 0.072 0.000 0.107 0.000 0.022 0.006 0.010 0.001 0.010 FRA, ITA, JPN, RUS

0.032 0.933 0.620 0.025 0.166 0.453 0.035 0.012 0.027 0.027 0.003 0.005 0.001 0.001 0.002 0.000

GER, UK, ITA, JPN

0.088 0.626 0.011 0.028 0.880 0.807 0.167 0.003 0.263 0.006 0.063 0.055 0.054 0.043 0.106 0.003

CAN, UK, FRA, RUS

0.512 0.073 0.189 0.117 0.452 0.599 0.001 0.003 0.000 0.028 0.010 0.001 0.007 0.001 0.002 0.000

CAN, GER, UK, ITA

0.003 0.046 0.008 0.024 0.000 0.039 0.010 0.050 0.008 0.030 0.003 0.009 0.002 0.005 0.000 0.011

USA, UK, FRA, ITA

0.904 0.050 0.082 0.240 0.329 0.375 0.000 0.012 0.000 0.059 0.000 0.036 0.001 0.003 0.006 0.000

USA, CAN, JPN, RUS

0.026 0.008 0.026 0.005 0.000 0.023 0.006 0.202 0.002 0.146 0.011 0.135 0.012 0.036 0.005 0.009

USA, CAN, GER, JPN

0.024 0.222 0.006 0.040 0.000 0.008 0.000 0.070 0.000 0.042 0.000 0.012 0.000 0.008 0.000 0.004

USA, CAN, GER, FRA

0.116 0.880 0.110 0.036 0.198 0.513 0.188 0.052 0.423 0.102 0.633 0.162 0.831 0.074 0.950 0.040

USA, CAN, GER, 0.052 0.804 0.001 0.013 0.320 0.917 0.183 0.017 0.396 0.034 0.462 0.178 0.685 0.116 0.672 0.018

102 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK UK, ITA, JPN, RUS

0.061 0.074 0.962 0.247 0.876 0.829 0.001 0.000 0.000 0.006 0.000 0.000 0.001 0.000 0.000 0.000

UK, FRA, JPN, RUS

0.003 0.065 0.009 0.026 0.000 0.059 0.009 0.135 0.002 0.084 0.007 0.017 0.010 0.005 0.004 0.007

UK, FRA, ITA, RUS

0.847 0.457 0.901 0.068 0.944 0.976 0.024 0.008 0.023 0.019 0.064 0.038 0.095 0.009 0.064 0.003

UK, FRA, ITA, JPN

0.024 0.820 0.001 0.006 0.828 0.777 0.138 0.011 0.275 0.027 0.241 0.137 0.364 0.109 0.413 0.009

GER, ITA, JPN, RUS

0.589 0.304 0.774 0.194 0.135 0.766 0.001 0.025 0.004 0.089 0.215 0.049 0.177 0.005 0.016 0.003

GER, FRA, JPN, RUS

0.013 0.012 0.030 0.106 0.001 0.050 0.010 0.171 0.010 0.110 0.119 0.054 0.284 0.023 0.122 0.042

GER, FRA, ITA, RUS

0.520 0.384 0.126 0.078 0.102 0.915 0.001 0.003 0.003 0.012 0.002 0.061 0.001 0.027 0.001 0.002

GER, FRA, ITA, JPN

0.071 0.001 0.254 0.004 0.003 0.028 0.020 0.138 0.033 0.087 0.064 0.116 0.102 0.068 0.047 0.106

GER, UK, JPN, RUS

0.017 0.015 0.060 0.140 0.000 0.002 0.001 0.027 0.000 0.020 0.000 0.005 0.000 0.002 0.000 0.001

GER, UK, ITA, RUS

0.015 0.820 0.001 0.014 0.361 0.980 0.233 0.016 0.594 0.033 0.425 0.166 0.645 0.116 0.555 0.011

103 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, FRA, RUS

0.827 0.047 0.167 0.150 0.946 0.817 0.000 0.007 0.000 0.154 0.016 0.001 0.010 0.001 0.004 0.000

GER, UK, FRA, JPN

0.003 0.047 0.023 0.021 0.000 0.064 0.128 0.292 0.110 0.194 0.028 0.052 0.052 0.018 0.005 0.018

GER, UK, FRA, ITA

0.045 0.035 0.064 0.161 0.823 0.620 0.003 0.035 0.001 0.231 0.052 0.027 0.018 0.003 0.117 0.000

CAN, ITA, JPN, RUS

0.863 0.506 0.022 0.054 0.828 0.744 0.063 0.003 0.077 0.006 0.036 0.054 0.034 0.027 0.060 0.003

CAN, FRA, JPN, RUS

0.423 0.003 0.143 0.015 0.001 0.061 0.113 0.486 0.124 0.367 0.127 0.214 0.134 0.057 0.078 0.015

CAN, FRA, ITA, RUS

0.006 0.151 0.034 0.001 0.000 0.002 0.001 0.149 0.000 0.184 0.000 0.001 0.000 0.001 0.000 0.000

CAN, FRA, ITA, JPN

0.355 0.161 0.175 0.041 0.277 0.961 0.020 0.005 0.024 0.027 0.185 0.059 0.035 0.025 0.224 0.001

CAN, UK, JPN, RUS

0.177 0.008 0.502 0.005 0.095 0.048 0.067 0.071 0.113 0.027 0.085 0.068 0.117 0.066 0.043 0.082

CAN, UK, ITA, RUS

0.000 0.022 0.048 0.036 0.000 0.004 0.000 0.076 0.000 0.071 0.001 0.001 0.001 0.002 0.001 0.002

CAN, UK, ITA, JPN

0.002 0.062 0.000 0.013 0.000 0.001 0.000 0.251 0.000 0.173 0.000 0.085 0.000 0.019 0.000 0.004

CAN, UK, FRA, 0.534 0.990 0.116 0.036 0.245 0.708 0.105 0.017 0.157 0.064 0.072 0.151 0.258 0.039 0.156 0.009

104 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

JPN CAN, UK, FRA, ITA

0.216 0.949 0.003 0.013 0.208 0.923 0.287 0.004 0.473 0.010 0.034 0.090 0.243 0.033 0.256 0.003

CAN, GER, JPN, RUS

0.060 0.044 0.620 0.490 0.838 0.837 0.000 0.001 0.000 0.026 0.000 0.009 0.001 0.001 0.000 0.000

CAN, GER, ITA, RUS

0.001 0.176 0.033 0.395 0.000 0.045 0.002 0.148 0.000 0.175 0.000 0.072 0.002 0.022 0.000 0.016

CAN, GER, ITA, JPN

0.065 0.131 0.415 0.237 0.204 0.806 0.002 0.002 0.001 0.011 0.000 0.008 0.000 0.002 0.000 0.000

CAN, GER, FRA, RUS

0.227 0.847 0.003 0.007 0.684 0.671 0.132 0.003 0.242 0.010 0.028 0.121 0.128 0.064 0.117 0.002

CAN, GER, FRA, JPN

0.577 0.254 0.764 0.253 0.144 0.938 0.000 0.011 0.001 0.073 0.000 0.104 0.028 0.009 0.001 0.001

CAN, GER, FRA, ITA

0.003 0.033 0.035 0.224 0.000 0.037 0.004 0.185 0.003 0.194 0.010 0.080 0.055 0.016 0.015 0.046

CAN, GER, UK, RUS

0.473 0.305 0.040 0.059 0.145 0.885 0.004 0.001 0.010 0.004 0.000 0.087 0.006 0.028 0.002 0.000

CAN, GER, UK, JPN

0.013 0.004 0.088 0.005 0.003 0.013 0.008 0.087 0.015 0.096 0.003 0.136 0.035 0.063 0.006 0.085

CAN, GER, UK, FRA

0.017 0.238 0.401 0.810 0.000 0.006 0.002 0.133 0.000 0.137 0.000 0.039 0.000 0.008 0.000 0.005

105 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, ITA, JPN, RUS

0.076 0.990 0.003 0.011 0.234 0.889 0.265 0.004 0.498 0.012 0.009 0.122 0.142 0.057 0.189 0.003

USA, FRA, JPN, RUS

0.588 0.044 0.335 0.184 0.941 0.472 0.000 0.000 0.000 0.013 0.000 0.006 0.003 0.001 0.000 0.000

USA, FRA, ITA, RUS

0.001 0.068 0.014 0.438 0.000 0.056 0.051 0.148 0.021 0.112 0.001 0.060 0.002 0.018 0.001 0.039

USA, FRA, ITA, JPN

0.141 0.027 0.012 0.069 0.783 0.553 0.000 0.006 0.000 0.080 0.000 0.096 0.006 0.006 0.025 0.000

USA, UK, JPN, RUS

0.008 0.275 0.014 0.111 0.000 0.034 0.002 0.073 0.001 0.048 0.000 0.027 0.000 0.010 0.000 0.013

USA, UK, ITA, RUS

0.047 0.009 0.004 0.008 0.002 0.022 0.033 0.260 0.012 0.207 0.002 0.193 0.022 0.037 0.011 0.008

USA, UK, ITA, JPN

0.008 0.461 0.375 0.194 0.000 0.011 0.000 0.026 0.000 0.021 0.000 0.018 0.000 0.007 0.000 0.002

USA, UK, FRA, RUS

0.550 0.111 0.025 0.037 0.450 0.773 0.017 0.001 0.020 0.012 0.000 0.079 0.061 0.040 0.063 0.000

USA, UK, FRA, JPN

0.021 0.015 0.128 0.005 0.070 0.030 0.042 0.091 0.065 0.063 0.002 0.126 0.039 0.094 0.005 0.124

USA, GER, JPN, RUS

0.000 0.075 0.263 0.652 0.000 0.005 0.000 0.045 0.000 0.046 0.000 0.008 0.001 0.002 0.001 0.007

USA, GER, ITA, 0.002 0.093 0.000 0.011 0.000 0.004 0.000 0.067 0.000 0.048 0.000 0.102 0.000 0.025 0.000 0.002

106 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

RUS USA, GER, ITA, JPN

0.033 0.976 0.000 0.009 0.111 0.966 0.218 0.011 0.392 0.039 0.053 0.223 0.313 0.083 0.368 0.007

USA, GER, FRA, RUS

0.210 0.063 0.047 0.107 0.760 0.511 0.007 0.038 0.032 0.338 0.056 0.569 0.114 0.095 0.007 0.012

USA, GER, FRA, JPN

0.082 0.070 0.017 0.126 0.000 0.050 0.099 0.242 0.116 0.233 0.046 0.158 0.335 0.032 0.115 0.050

USA, GER, FRA, ITA

0.043 0.073 0.005 0.024 0.685 0.716 0.015 0.003 0.082 0.042 0.001 0.366 0.001 0.053 0.001 0.001

USA, GER, UK, RUS

0.623 0.684 0.275 0.012 0.104 0.504 0.214 0.028 0.262 0.073 0.032 0.021 0.016 0.006 0.008 0.001

USA, GER, UK, JPN

0.175 0.029 0.024 0.006 0.007 0.022 0.069 0.111 0.087 0.121 0.004 0.168 0.129 0.043 0.086 0.053

USA, GER, UK, ITA

0.011 0.102 0.150 0.485 0.000 0.012 0.000 0.011 0.000 0.044 0.000 0.016 0.000 0.001 0.000 0.000

USA, GER, UK, FRA

0.066 0.039 0.001 0.011 0.730 0.878 0.005 0.015 0.025 0.121 0.000 0.585 0.000 0.167 0.000 0.006

USA, CAN, ITA, RUS

0.148 0.020 0.131 0.010 0.029 0.085 0.029 0.147 0.045 0.137 0.007 0.163 0.070 0.075 0.029 0.064

USA, CAN, ITA, JPN

0.000 0.075 0.020 0.214 0.000 0.026 0.001 0.083 0.000 0.101 0.000 0.037 0.002 0.003 0.000 0.003

107 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, FRA, RUS

0.007 0.005 0.000 0.031 0.000 0.010 0.000 0.024 0.000 0.032 0.000 0.057 0.000 0.011 0.000 0.006

USA, CAN, FRA, JPN

0.032 0.054 0.006 0.021 0.581 0.880 0.008 0.006 0.034 0.087 0.000 0.311 0.000 0.072 0.002 0.000

USA, CAN, FRA, ITA

0.220 0.026 0.379 0.009 0.122 0.033 0.110 0.103 0.155 0.073 0.002 0.159 0.073 0.067 0.082 0.097

USA, CAN, UK, RUS

0.000 0.071 0.069 0.636 0.000 0.011 0.000 0.038 0.000 0.123 0.000 0.006 0.000 0.001 0.000 0.001

USA, CAN, UK, JPN

0.000 0.059 0.000 0.014 0.000 0.001 0.000 0.463 0.000 0.333 0.000 0.139 0.000 0.010 0.000 0.000

USA, CAN, UK, ITA

0.012 0.724 0.017 0.009 0.123 0.939 0.278 0.029 0.280 0.063 0.105 0.046 0.101 0.010 0.204 0.001

USA, CAN, UK, FRA

0.000 0.019 0.000 0.054 0.000 0.062 0.000 0.024 0.000 0.029 0.000 0.036 0.000 0.007 0.000 0.006

USA, CAN, GER, RUS

0.060 0.008 0.298 0.307 0.280 0.829 0.001 0.000 0.001 0.010 0.000 0.000 0.000 0.000 0.002 0.000

USA, CAN, GER, ITA

0.001 0.114 0.008 0.022 0.000 0.054 0.022 0.116 0.013 0.086 0.001 0.013 0.000 0.003 0.000 0.002

UK, FRA, ITA, JPN, RUS

0.032 0.729 0.334 0.086 0.724 0.966 0.003 0.004 0.002 0.009 0.000 0.013 0.000 0.002 0.000 0.000

CAN, GER, ITA, 0.049 0.305 0.861 0.231 0.205 0.996 0.000 0.004 0.000 0.016 0.000 0.007 0.000 0.001 0.000 0.000

108 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

JPN, RUS USA, UK, ITA, JPN, RUS

0.005 0.023 0.024 0.101 0.000 0.040 0.004 0.132 0.002 0.076 0.000 0.039 0.000 0.011 0.000 0.015

USA, GER, UK, FRA, RUS

0.471 0.813 0.258 0.014 0.381 0.784 0.036 0.032 0.035 0.082 0.029 0.016 0.029 0.004 0.009 0.001

USA, CAN, UK, FRA, JPN

0.014 0.338 0.002 0.010 0.374 0.906 0.172 0.088 0.256 0.196 0.207 0.251 0.223 0.062 0.249 0.003

USA, CAN, GER, UK, RUS

0.034 0.028 0.709 0.367 0.785 0.765 0.000 0.007 0.000 0.096 0.000 0.001 0.000 0.001 0.000 0.000

USA, CAN, GER, UK, JPN

0.000 0.026 0.033 0.015 0.000 0.060 0.028 0.358 0.013 0.292 0.003 0.055 0.002 0.011 0.001 0.005

USA, CAN, GER, UK, ITA

0.109 0.315 0.002 0.034 0.888 0.773 0.070 0.005 0.158 0.010 0.096 0.102 0.122 0.066 0.175 0.005

USA, CAN, GER, UK, FRA

0.384 0.187 0.855 0.092 0.259 0.673 0.000 0.002 0.000 0.019 0.002 0.001 0.003 0.000 0.001 0.000

GER, FRA, ITA, JPN, RUS

0.002 0.009 0.013 0.015 0.000 0.036 0.012 0.103 0.005 0.052 0.006 0.013 0.004 0.005 0.002 0.012

GER, UK, ITA, JPN, RUS

0.992 0.072 0.219 0.137 0.277 0.508 0.000 0.053 0.000 0.199 0.000 0.189 0.000 0.028 0.003 0.001

GER, UK, FRA, JPN, RUS

0.009 0.799 0.589 0.036 0.120 0.564 0.025 0.024 0.014 0.063 0.001 0.006 0.000 0.001 0.001 0.000

109 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, FRA, ITA, RUS

0.126 0.003 0.331 0.013 0.000 0.033 0.014 0.451 0.007 0.355 0.049 0.299 0.053 0.099 0.025 0.025

GER, UK, FRA, ITA, JPN

0.009 0.116 0.034 0.006 0.000 0.003 0.000 0.145 0.000 0.105 0.000 0.014 0.000 0.005 0.000 0.003

CAN, FRA, ITA, JPN, RUS

0.009 0.672 0.319 0.046 0.464 0.830 0.002 0.005 0.001 0.022 0.000 0.016 0.001 0.002 0.000 0.000

CAN, UK, ITA, JPN, RUS

0.484 0.307 0.590 0.080 0.912 0.819 0.004 0.004 0.004 0.015 0.002 0.037 0.021 0.005 0.009 0.001

CAN, UK, FRA, JPN, RUS

0.891 0.345 0.007 0.047 0.938 0.617 0.036 0.001 0.063 0.004 0.004 0.068 0.040 0.019 0.028 0.001

CAN, UK, FRA, ITA, RUS

0.046 0.166 0.787 0.427 0.148 0.661 0.000 0.004 0.000 0.026 0.000 0.024 0.000 0.003 0.000 0.000

CAN, UK, FRA, ITA, JPN

0.003 0.134 0.053 0.638 0.000 0.034 0.001 0.188 0.000 0.211 0.000 0.112 0.000 0.032 0.000 0.028

CAN, GER, FRA, JPN, RUS

0.215 0.934 0.343 0.019 0.346 0.978 0.019 0.003 0.007 0.008 0.001 0.010 0.001 0.002 0.001 0.000

CAN, GER, FRA, ITA, RUS

0.163 0.590 0.003 0.009 0.204 0.813 0.128 0.018 0.119 0.051 0.006 0.121 0.054 0.015 0.073 0.000

CAN, GER, FRA, ITA, JPN

0.038 0.016 0.879 0.643 0.850 0.572 0.000 0.001 0.000 0.015 0.000 0.012 0.000 0.002 0.000 0.000

CAN, GER, UK, 0.410 0.576 0.660 0.057 0.965 0.954 0.009 0.002 0.004 0.004 0.002 0.004 0.001 0.001 0.001 0.000

110 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

JPN, RUS CAN, GER, UK, ITA, RUS

0.000 0.146 0.051 0.195 0.000 0.060 0.014 0.174 0.005 0.116 0.000 0.074 0.001 0.026 0.000 0.008

CAN, GER, UK, ITA, JPN

0.328 0.429 0.004 0.024 0.742 0.688 0.072 0.001 0.141 0.004 0.003 0.082 0.047 0.037 0.043 0.001

CAN, GER, UK, FRA, RUS

0.338 0.144 0.620 0.193 0.290 0.379 0.000 0.000 0.000 0.004 0.000 0.002 0.001 0.000 0.000 0.000

CAN, GER, UK, FRA, JPN

0.000 0.028 0.019 0.351 0.000 0.036 0.002 0.061 0.001 0.034 0.000 0.038 0.001 0.010 0.000 0.027

CAN, GER, UK, FRA, ITA

0.871 0.088 0.034 0.094 0.512 0.429 0.000 0.004 0.000 0.024 0.000 0.099 0.000 0.011 0.001 0.000

USA, FRA, ITA, JPN, RUS

0.009 0.007 0.035 0.006 0.000 0.019 0.005 0.220 0.001 0.162 0.001 0.255 0.009 0.071 0.003 0.015

USA, UK, FRA, JPN, RUS

0.011 0.607 0.201 0.193 0.000 0.015 0.000 0.045 0.000 0.025 0.000 0.030 0.000 0.012 0.000 0.007

USA, UK, FRA, ITA, RUS

0.363 0.902 0.059 0.038 0.190 0.732 0.034 0.014 0.063 0.050 0.024 0.139 0.245 0.021 0.132 0.005

USA, UK, FRA, ITA, JPN

0.067 0.819 0.000 0.015 0.178 0.969 0.104 0.004 0.209 0.013 0.007 0.120 0.209 0.031 0.245 0.003

USA, GER, ITA, JPN, RUS

0.066 0.074 0.967 0.240 0.853 1.000 0.000 0.000 0.000 0.013 0.000 0.006 0.000 0.000 0.000 0.000

111 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, GER, FRA, JPN, RUS

0.243 0.600 0.023 0.041 0.863 0.598 0.041 0.018 0.020 0.053 0.024 0.086 0.018 0.015 0.033 0.001

USA, GER, FRA, ITA, RUS

0.002 0.087 0.027 0.115 0.000 0.045 0.002 0.116 0.000 0.122 0.000 0.062 0.002 0.010 0.001 0.007

USA, GER, FRA, ITA, JPN

0.072 0.682 0.000 0.008 0.512 0.706 0.058 0.004 0.110 0.016 0.005 0.153 0.093 0.053 0.109 0.003

USA, GER, UK, JPN, RUS

0.752 0.281 0.334 0.115 0.331 0.869 0.000 0.013 0.000 0.097 0.000 0.137 0.000 0.005 0.000 0.001

USA, GER, UK, ITA, RUS

0.007 0.021 0.017 0.103 0.000 0.036 0.004 0.150 0.003 0.139 0.003 0.075 0.046 0.015 0.015 0.021

USA, GER, UK, ITA, JPN

0.739 0.302 0.011 0.023 0.378 0.873 0.000 0.001 0.000 0.009 0.000 0.143 0.000 0.021 0.000 0.001

USA, GER, UK, FRA, JPN

0.046 0.004 0.092 0.003 0.002 0.014 0.008 0.103 0.009 0.099 0.001 0.153 0.016 0.042 0.006 0.050

USA, GER, UK, FRA, ITA

0.017 0.044 0.284 0.877 0.000 0.015 0.001 0.043 0.000 0.052 0.000 0.038 0.000 0.003 0.000 0.001

USA, CAN, ITA, JPN, RUS

0.026 0.841 0.000 0.013 0.179 0.924 0.091 0.006 0.205 0.016 0.001 0.130 0.107 0.046 0.210 0.003

USA, CAN, FRA, JPN, RUS

0.729 0.051 0.165 0.085 0.770 0.667 0.000 0.000 0.000 0.008 0.000 0.001 0.000 0.000 0.000 0.000

USA, CAN, FRA, 0.002 0.052 0.010 0.055 0.000 0.041 0.041 0.181 0.010 0.137 0.000 0.048 0.002 0.008 0.000 0.009

112 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

ITA, RUS USA, CAN, FRA, ITA, JPN

0.050 0.026 0.801 0.626 0.587 0.460 0.000 0.007 0.000 0.034 0.000 0.009 0.000 0.002 0.000 0.000

USA, CAN, UK, JPN, RUS

0.040 0.022 0.007 0.028 0.569 0.764 0.000 0.024 0.000 0.202 0.000 0.161 0.000 0.009 0.003 0.000

USA, CAN, UK, ITA, RUS

0.238 0.010 0.032 0.011 0.001 0.024 0.034 0.422 0.014 0.385 0.000 0.289 0.014 0.051 0.013 0.006

USA, CAN, UK, ITA, JPN

0.006 0.304 0.238 0.050 0.000 0.007 0.000 0.043 0.000 0.051 0.000 0.011 0.000 0.002 0.000 0.000

USA, CAN, UK, FRA, RUS

0.275 0.108 0.018 0.014 0.714 0.987 0.001 0.002 0.001 0.021 0.000 0.115 0.000 0.026 0.000 0.000

USA, CAN, UK, FRA, ITA

0.106 0.013 0.208 0.005 0.060 0.031 0.027 0.068 0.039 0.042 0.001 0.105 0.021 0.045 0.006 0.055

USA, CAN, GER, JPN, RUS

0.000 0.036 0.051 0.395 0.000 0.011 0.000 0.017 0.000 0.024 0.000 0.002 0.000 0.001 0.000 0.001

USA, CAN, GER, ITA, RUS

0.002 0.085 0.000 0.020 0.000 0.001 0.000 0.193 0.000 0.129 0.000 0.117 0.000 0.023 0.000 0.002

USA, CAN, GER, ITA, JPN

0.002 0.170 0.023 0.064 0.000 0.044 0.004 0.126 0.002 0.081 0.000 0.039 0.000 0.014 0.000 0.009

USA, CAN, GER, FRA, RUS

0.015 0.468 0.340 0.025 0.440 0.682 0.003 0.018 0.002 0.046 0.006 0.011 0.009 0.002 0.003 0.000

113 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, FRA, JPN

0.765 0.208 0.911 0.089 0.851 0.893 0.009 0.011 0.013 0.026 0.105 0.039 0.209 0.007 0.089 0.002

USA, CAN, GER, FRA, ITA

0.563 0.241 0.004 0.055 0.739 0.733 0.031 0.007 0.066 0.014 0.075 0.118 0.099 0.052 0.115 0.007

GER, UK, FRA, ITA, JPN, RUS

0.010 0.833 0.369 0.114 0.903 0.853 0.004 0.015 0.002 0.045 0.000 0.022 0.000 0.003 0.000 0.000

USA, GER, UK, FRA, ITA, RUS

0.147 0.322 0.599 0.060 0.936 0.673 0.003 0.001 0.001 0.002 0.000 0.005 0.000 0.001 0.000 0.000

USA, CAN, GER, FRA, ITA, JPN

0.535 0.398 0.007 0.030 0.923 0.506 0.021 0.008 0.013 0.031 0.001 0.144 0.012 0.019 0.015 0.000

USA, CAN, GER, UK, JPN, RUS

0.034 0.048 0.830 0.681 0.387 0.380 0.000 0.005 0.000 0.026 0.000 0.031 0.000 0.005 0.000 0.000

USA, CAN, GER, UK, ITA, RUS

0.001 0.179 0.069 0.249 0.000 0.055 0.003 0.170 0.002 0.119 0.000 0.116 0.000 0.043 0.000 0.020

USA, CAN, GER, UK, ITA, JPN

0.009 0.426 0.542 0.039 0.366 0.861 0.001 0.004 0.000 0.017 0.000 0.012 0.001 0.001 0.000 0.000

USA, CAN, GER, UK, FRA, RUS

0.519 0.226 0.429 0.086 0.893 0.819 0.002 0.004 0.002 0.014 0.001 0.039 0.025 0.003 0.011 0.001

USA, CAN, GER, UK, FRA, JPN

0.639 0.280 0.001 0.053 0.878 0.667 0.014 0.002 0.024 0.006 0.000 0.091 0.019 0.019 0.026 0.002

USA, CAN, GER, 0.052 0.246 0.744 0.197 0.329 0.851 0.000 0.003 0.000 0.021 0.000 0.021 0.000 0.001 0.000 0.000

114 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, FRA, ITA CAN, UK, FRA, ITA, JPN, RUS

0.003 0.041 0.045 0.238 0.000 0.036 0.002 0.141 0.000 0.138 0.000 0.098 0.000 0.015 0.000 0.012

CAN, GER, FRA, ITA, JPN, RUS

0.256 0.720 0.187 0.020 0.270 0.986 0.006 0.004 0.002 0.010 0.000 0.005 0.002 0.001 0.001 0.000

CAN, GER, UK, ITA, JPN, RUS

0.021 0.238 0.277 0.078 0.662 0.897 0.001 0.009 0.001 0.023 0.000 0.021 0.000 0.003 0.001 0.001

CAN, GER, UK, FRA, JPN, RUS

0.047 0.482 0.001 0.010 0.159 0.793 0.064 0.036 0.049 0.112 0.001 0.189 0.030 0.025 0.079 0.000

CAN, GER, UK, FRA, ITA, RUS

0.041 0.025 0.988 0.369 0.750 0.660 0.000 0.002 0.000 0.033 0.000 0.005 0.000 0.000 0.000 0.000

CAN, GER, UK, FRA, ITA, JPN

0.000 0.065 0.057 0.050 0.000 0.045 0.010 0.284 0.002 0.227 0.000 0.082 0.001 0.016 0.000 0.004

USA, UK, FRA, ITA, JPN, RUS

0.161 0.349 0.001 0.031 0.536 0.717 0.029 0.002 0.054 0.006 0.000 0.088 0.022 0.028 0.043 0.001

USA, GER, FRA, ITA, JPN, RUS

0.414 0.179 0.464 0.086 0.462 0.557 0.000 0.000 0.000 0.004 0.000 0.001 0.000 0.000 0.000 0.000

USA, GER, UK, ITA, JPN, RUS

0.001 0.017 0.010 0.036 0.000 0.031 0.004 0.084 0.001 0.043 0.000 0.026 0.001 0.005 0.000 0.008

USA, GER, UK, FRA, JPN, RUS

0.820 0.076 0.032 0.037 0.677 0.646 0.000 0.022 0.000 0.120 0.000 0.222 0.000 0.026 0.000 0.000

115 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, GER, UK, FRA, ITA, JPN

0.064 0.009 0.129 0.011 0.000 0.018 0.004 0.369 0.001 0.330 0.000 0.339 0.005 0.082 0.004 0.012

USA, CAN, FRA, ITA, JPN, RUS

0.010 0.271 0.118 0.081 0.000 0.011 0.000 0.082 0.000 0.057 0.000 0.028 0.000 0.005 0.000 0.002

USA, CAN, UK, ITA, JPN, RUS

0.004 0.247 0.308 0.040 0.417 0.809 0.008 0.080 0.006 0.226 0.002 0.037 0.001 0.004 0.001 0.001

USA, CAN, UK, FRA, JPN, RUS

0.293 0.228 0.736 0.059 0.840 0.781 0.004 0.004 0.002 0.008 0.003 0.005 0.003 0.001 0.002 0.000

USA, CAN, UK, FRA, ITA, RUS

0.173 0.247 0.005 0.039 0.773 0.569 0.032 0.054 0.026 0.165 0.068 0.306 0.061 0.076 0.072 0.002

USA, CAN, UK, FRA, ITA, JPN

0.032 0.077 0.444 0.484 0.412 0.556 0.000 0.034 0.000 0.143 0.000 0.027 0.000 0.004 0.000 0.000

USA, CAN, GER, ITA, JPN, RUS

0.001 0.039 0.069 0.046 0.000 0.048 0.009 0.318 0.005 0.242 0.000 0.095 0.000 0.028 0.000 0.015

USA, CAN, GER, FRA, JPN, RUS

0.013 0.432 0.378 0.100 0.738 0.766 0.001 0.004 0.000 0.015 0.000 0.029 0.000 0.004 0.000 0.000

USA, CAN, GER, FRA, ITA, RUS

0.003 0.583 0.424 0.065 0.362 0.889 0.005 0.014 0.002 0.045 0.000 0.030 0.000 0.004 0.000 0.000

USA, CAN, GER, UK, FRA, ITA, JPN

0.001 0.068 0.083 0.090 0.000 0.040 0.003 0.271 0.001 0.216 0.000 0.131 0.000 0.029 0.000 0.010

116 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, UK, FRA, ITA, RUS

0.038 0.072 0.916 0.381 0.515 0.517 0.000 0.012 0.000 0.066 0.000 0.027 0.000 0.002 0.000 0.000

USA, CAN, GER, UK, FRA, JPN, RUS

0.268 0.340 0.002 0.033 0.767 0.528 0.010 0.024 0.004 0.096 0.000 0.240 0.005 0.035 0.017 0.001

USA, CAN, GER, UK, ITA, JPN, RUS

0.173 0.226 0.424 0.063 0.879 0.690 0.001 0.001 0.000 0.003 0.000 0.003 0.000 0.000 0.000 0.000

USA, CAN, GER, FRA, ITA, JPN, RUS

0.004 0.370 0.515 0.053 0.318 0.988 0.002 0.024 0.001 0.079 0.000 0.024 0.000 0.002 0.000 0.000

USA, CAN, UK, FRA, ITA, JPN, RUS

0.015 0.257 0.613 0.086 0.890 0.785 0.000 0.003 0.000 0.013 0.000 0.021 0.000 0.002 0.000 0.000

USA, GER, UK, FRA, ITA, JPN, RUS

0.006 0.554 0.459 0.122 0.867 0.675 0.001 0.009 0.001 0.034 0.000 0.054 0.000 0.007 0.000 0.000

CAN, GER, UK, FRA, ITA, JPN, RUS

0.007 0.274 0.297 0.103 0.762 0.768 0.003 0.050 0.003 0.164 0.000 0.077 0.000 0.010 0.000 0.001

117 Appendix B: p-Values Pertaining to Tests of Predictive Accuracy (raw output), by Portfolio Stage 1 𝐷𝑡 =

GARCH CARR CARR CARR-M

Q = 63

CARR-M

Q = 126

CARR-M

Q = 252

CARR-M

Q = 756

CARR-M

Q = 1260

Stage 2 𝑅𝑡 = DCC DCC DCC-M DCC-M DCC-M DCC-M DCC-M DCC-M

(IS) (IS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, UK, FRA, ITA, JPN, RUS

0.008 0.438 0.567 0.100 0.979 0.663 0.001 0.018 0.000 0.078 0.000 0.034 0.000 0.005 0.000 0.000

118

Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN -0.01** -0.00 0.020 -0.01 0.007 -0.00 -0.00 -0.01 -0.03 -0.02 0.018** -0.01 0.043** 0.000

(-2.55) (-0.94) (1.14) (-0.27) (1.26) (0.239) (0.293) (-0.33) (-1.18) (-1.12) (2.16) (-0.81) (2.29) (0.143)

CAN, JPN -0.01** -0.00 0.086*** 0.058** 0.051*** 0.037** 0.053*** 0.039** 0.086*** 0.049** 0.087*** 0.053** 0.114*** 0.067**

(-2.09) (-1.49) (2.9) (2.19) (2.98) (2.31) (2.95) (2.24) (3.22) (2.29) (3.12) (2.26) (3.27) (2.33)

UK, RUS -0.00 0.010 0.202*** 0.099** 0.185*** 0.098** 0.206*** 0.113** 0.12*** 0.046** 0.113*** 0.027** 0.161*** 0.059**

(-0.81) (0.769) (3.27) (2.44) (3.81) (2.37) (3.73) (2.4) (3.68) (2.43) (3.2) (2.15) (3.38) (2.41)

FRA, ITA -0.01* -0.00 0.036 0.015 -0.04 -0.01 -0.00 0.007 -0.01 0.000 0.020 0.010 0.016 0.006

(-1.92) (-1.07) (0.794) (0.04) (-0.2) (-0.74) (0.44) (-0.25) (0.311) (-0.38) (0.747) (-0.12) (0.656) (-0.21)

FRA, JPN -0.01 0.001 0.236*** 0.365** 0.022 0.182* 0.096 0.248** 0.217*** 0.338** 0.244*** 0.348** 0.113 0.231**

(-1.21) (-0.76) (2.99) (2.52) (-0.11) (1.69) (1.48) (2.14) (3.32) (2.54) (3.41) (2.56) (1.57) (1.99)

FRA, RUS -0.00** -0.00 0.057*** 0.049 0.060*** 0.066 0.047*** 0.050 0.087*** 0.087** 0.089*** 0.082** 0.112*** 0.109**

(-2.13) (-1.37) (3.17) (0.809) (3.4) (1.5) (3) (0.96) (3.72) (1.97) (3.81) (1.97) (3.92) (2.08)

119 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

ITA, JPN -0.00 0.005 0.006 -0.00 0.016 0.003 0.019 -0.00 -0.00 0.007 -0.01 0.005 -0.03 -0.00

(-1.19) (0.826) (0.381) (-0.05) (0.647) (0.467) (0.687) (-0.17) (0.018) (0.226) (-0.30) (0.032) (-0.82) (-0.39)

ITA, RUS 0.001 -0.00 0.056** -0.00 0.091** 0.012 0.098** 0.015* -0.01 -0.03* -0.04 -0.05** -0.06** -0.07**

(1.21) (-0.96) (2.11) (-0.53) (2.29) (1.36) (2.23) (1.7) (-0.75) (-1.84) (-1.46) (-2.1) (-2.03) (-2.25)

JPN, RUS -0.00 -0.00 0.012 -0.01 0.023 0.009 0.040 0.005 0.001 0.016 0.012 0.011 -0.01 0.008

(-1.1) (-1.14) (1.28) (0.002) (1.39) (0.902) (1.55) (0.61) (1.04) (0.999) (1.23) (1.02) (0.711) (0.749)

USA, GER -0.00 -0.00 0.057 0.077** 0.008 0.040 0.044 0.063 0.053 0.077* 0.057 0.074 -0.04 -0.01

(-1.34) (0.124) (0.718) (2) (-0.15) (0.045) (0.447) (1.04) (0.766) (1.82) (0.864) (1.54) (-1.62) (-1.11)

USA, UK -0.05 -0.01** 0.119 0.034 0.144 0.051** 0.13 0.043* 0.146 0.052** 0.12 0.034* 0.153 0.052**

(-1.2) (-2.14) (1.45) (1.07) (1.52) (2.17) (1.49) (1.78) (1.52) (2.25) (1.46) (1.67) (1.54) (2.21)

USA, FRA -0.00 -0.00 0.012 0.024 0.018 0.050** 0.027 0.061** 0.036 0.099*** 0.021 0.081** 0.045 0.086***

(1.26) (-1.01) (0.044) (-0.25) (0.438) (2.4) (0.758) (2.53) (0.2) (2.86) (-0.75) (2.42) (1.11) (3.4)

USA, ITA 0.003 -0.00 0.185*** 0.055 0.237*** 0.096** 0.241*** 0.099** 0.121*** 0.040 0.083** -0.00 0.127*** 0.025

(0.996) (-1.47) (2.71) (1.64) (2.89) (2.12) (2.89) (2.13) (2.75) (1.42) (2.4) (-0.15) (2.66) (1.01)

120 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, JPN -0.01 0.001 0.016 -0.04 0.010 -0.04 0.026 -0.03 0.000 -0.03 0.011 -0.03 0.005 -0.04

(-1.47) (-0.42) (1.1) (-0.96) (1.08) (-1.38) (1.25) (-0.97) (0.907) (-0.77) (1.07) (-0.76) (1) (-1.18)

USA, RUS -0.01* 0.001 0.17 0.217* 0.061 0.142 0.104 0.177 0.176** 0.245** 0.18** 0.228** 0.06 0.133

(-1.66) (0.449) (1.63) (1.8) (0.141) (0.667) (0.63) (1.13) (2.16) (2.22) (2.34) (2.02) (0.115) (0.567)

CAN, GER 0.008 0.011 0.054** -0.00 0.078** 0.018** 0.082** 0.023** 0.009 -0.02** 0.000 -0.03** 0.006 -0.03**

(1.33) (1.29) (2.08) (0.425) (2.08) (2.03) (2.03) (2.07) (-0.64) (-2.41) (-0.76) (-2.55) (-0.63) (-2.54)

CAN, UK -0.00 -0.00 0.028* -0.02 0.033* -0.00 0.045** -0.00 0.026* -0.00 0.037* -0.00 0.028* -0.00

(-1.44) (-1.03) (1.81) (0.405) (1.93) (0.79) (2.04) (0.747) (1.79) (0.813) (1.9) (0.798) (1.85) (0.823)

CAN, FRA -0.00 7.19e 0.043 0.099 0.006 0.060 0.028 0.083 0.047 0.105 0.053 0.104 0.002 0.054

(-0.28) (-0.00) (0.702) (1.4) (-0.10) (-0.08) (0.305) (0.406) (1.02) (1.59) (1.17) (1.51) (-0.17) (-0.15)

CAN, ITA 0.006 0.002 0.060** -0.00 0.061** 0.023 0.065** 0.023 0.070** 0.015 0.066** -0.00 0.069** 0.011

(1.34) (-0.55) (2.38) (0.792) (2.31) (1.6) (2.42) (1.6) (2.47) (1.23) (2.48) (0.748) (2.5) (1.21)

CAN, RUS -0.00 -0.00 0.001 0.078 -0.01 0.048 -0.00 0.062 0.006 0.074 0.000 0.07 0.002 0.074

(-0.67) (-1.05) (-0.13) (0.851) (-0.64) (0.145) (-0.42) (0.341) (0.246) (0.866) (0.21) (0.81) (-0.04) (0.673)

121 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK -0.00 0.007 0.086*** 0.126* 0.087*** 0.107 0.090*** 0.116 0.097*** 0.158* 0.098*** 0.155* 0.089*** 0.105

(-1.2) (0.3) (2.81) (1.66) (2.87) (1.23) (2.92) (1.36) (2.91) (1.87) (2.94) (1.83) (2.87) (1.01)

GER, FRA 0.000 0.005 -0.03 -0.01 -0.01 -0.00 -0.02 -0.00 -0.04 -0.00 -0.02 -0.01 -0.01 -0.00

(-0.51) (1.05) (-0.83) (-1.35) (0.474) (-1.12) (-0.31) (-0.75) (-1.57) (-0.71) (-0.50) (-0.95) (0.551) (-1.41)

GER, ITA -0.00 -0.01 0.000 0.002 0.017 0.036*** 0.031 0.046*** 0.046 0.059** 0.001 0.024 0.006 0.025*

(0.428) (-0.85) (-0.99) (0.054) (-0.77) (2.72) (-0.43) (2.75) (-0.14) (2.35) (-1.48) (1.05) (-0.95) (1.95)

GER, JPN 0.002 -0.00 0.096 0.021 0.131 0.076** 0.154 0.101** 0.131 0.066 0.035 0.001 0.024 -0.02

(1.01) (0.765) (1.07) (0.563) (1.21) (2.52) (1.29) (2.34) (1.34) (1.37) (-0.17) (-1.12) (-0.47) (-1.48)

GER, RUS -0.01* -0.00 0.013 -0.00 0.004 -0.02 0.047 0.006 0.054 -0.00 0.028 -0.02 -0.01 -0.04

(-1.72) (-0.41) (0.925) (-0.77) (0.891) (-1.18) (1.26) (-0.61) (1.24) (-0.73) (1.04) (-1.42) (0.614) (-1.5)

UK, FRA -0.00 -0.00 0.126 0.243** 0.037 0.148 0.113 0.232 0.185** 0.322** 0.153 0.292** 0.011 0.114

(-1.32) (0.217) (1.4) (2.09) (-0.02) (0.78) (0.888) (1.64) (2.21) (2.43) (1.15) (2.27) (-0.34) (0.43)

UK, ITA -0.00 0.003 0.067** 0.073** 0.037** 0.048** 0.030** 0.043** 0.089*** 0.084*** 0.099*** 0.094*** 0.123*** 0.117**

(-0.08) (0.216) (2.5) (2.36) (2.25) (2.47) (2.06) (2.35) (3.46) (2.66) (3.6) (2.62) (3.84) (2.49)

122 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, JPN 0.002 0.001 -0.01 -0.00 -0.02 -0.01 -0.01 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00

(1.38) (-0.65) (0.296) (-0.44) (0.094) (-1.43) (0.226) (-1.1) (0.577) (-0.89) (0.682) (-0.15) (0.615) (-0.41)

USA, CAN, GER

-0.01* -0.01* 0.056 0.028 0.035 0.031* 0.031 0.03 0.028 0.054* 0.073* 0.054* 0.113*** 0.074*

(-1.69) (-1.73) (1.55) (1.13) (0.975) (1.76) (1.12) (1.64) (0.476) (1.91) (1.91) (1.85) (3.46) (1.82)

USA, UK, FRA

0.017 0.072** 0.017 0.009 0.020 0.028 0.060 0.055 0.073 0.095* 0.044 0.060 -0.00 0.035

(0.465) (2.54) (1.1) (0.080) (1.2) (0.758) (1.59) (1.49) (1.44) (1.94) (1.26) (1.17) (0.819) (0.63)

CAN, GER, FRA

0.025 -0.00 0.114 0.175** 0.041 0.116 0.112 0.181* 0.181** 0.262*** 0.141 0.227** 0.017 0.069

(1.18) (0.222) (1.06) (2.03) (-0.04) (0.732) (0.885) (1.79) (1.99) (2.76) (0.7) (2.44) (-0.47) (-0.00)

CAN, ITA, JPN

-0.01 0.018* 0.055* 0.009 0.097** 0.040* 0.089* 0.043* 0.087* 0.052* 0.067 0.016 0.095** 0.032

(-1.09) (1.67) (1.69) (0.149) (2) (1.67) (1.88) (1.84) (1.77) (1.88) (1.51) (0.713) (2.02) (1.16)

GER, UK, ITA

0.039 0.030** 0.126** 0.028 0.194*** 0.084** 0.199*** 0.102** 0.117** 0.054 0.057 -0.00 0.074** -0.01

(1.39) (2.21) (2.46) (1.16) (2.99) (2.08) (2.89) (2.16) (2.06) (1.3) (1.36) (-1.07) (2.28) (-1.13)

FRA, ITA, JPN

-0.00 0.023 -0.00 -0.01 -0.00 -0.01 0.024 0.015 0.019 0.014 0.001 -0.01 -0.02 -0.02

(-0.64) (0.538) (0.683) (-1.15) (0.662) (-1) (1.04) (-0.48) (0.842) (-0.62) (0.693) (-1.24) (0.456) (-1.19)

123 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

FRA, ITA, RUS

0.016 0.026 0.141 0.22* 0.048 0.14 0.11 0.202 0.172* 0.284** 0.173 0.255** 0.031 0.116

(-0.34) (1.58) (0.975) (1.84) (-0.19) (0.594) (0.449) (1.4) (1.83) (2.29) (1.34) (2.08) (-0.34) (0.288)

FRA, JPN, RUS

0.010 -0.00 0.072 -0.00 0.114 0.054** 0.134 0.075** 0.1 0.043 0.008 -0.01 0.010 -0.03

(-0.27) (0.587) (0.532) (0.115) (0.766) (2.36) (0.899) (2.3) (0.463) (1.42) (-1.39) (-1.38) (-0.92) (-1.64)

ITA, JPN, RUS

0.029 0.077** 0.025 -0.00 0.027 0.018 0.065** 0.048 0.091** 0.067 0.044 0.017 0.008 0.001

(0.756) (2.38) (1.51) (-0.01) (1.64) (0.661) (2.08) (1.23) (2.07) (1.35) (1.59) (0.272) (1.32) (0.289)

USA, CAN, UK

0.024 0.004 0.099 0.181** 0.039 0.128 0.108 0.191* 0.177* 0.256*** 0.118 0.217** 0.013 0.099

(0.954) (0.744) (0.864) (2.34) (-0.11) (0.801) (0.768) (1.95) (1.72) (2.95) (0.605) (2.53) (-0.53) (0.344)

USA, CAN, FRA

0.066** 0.087** 0.082** 0.016 0.101** 0.058** 0.147*** 0.091** 0.152*** 0.082* 0.084** 0.019 0.056** 0.000

(2.14) (2.14) (2.35) (1.29) (2.5) (1.97) (2.87) (2.16) (2.79) (1.83) (2.38) (0.364) (2.22) (0.303)

USA, CAN, ITA

-0.01*** -0.00 -0.00 -0.02 -0.00 -0.01* -0.02 -0.01 -0.05 -0.02 0.002 -0.02 0.020* -0.01

(-2.7) (-0.66) (0.89) (-1.18) (0.947) (-1.81) (0.21) (-1.4) (-1.52) (-1.48) (1.11) (-1.4) (1.74) (-1.28)

USA, CAN, JPN

-0.00 -0.01 0.103 0.156** 0.050 0.112 0.124 0.177** 0.156 0.195*** 0.062 0.138 -0.00 0.059

(-1.03) (-0.53) (0.211) (2.48) (-0.36) (0.786) (0.292) (2.13) (0.89) (2.58) (-0.13) (1.36) (-1.02) (-0.30)

124 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, RUS

0.061** 0.115*** 0.112*** 0.186** 0.076*** 0.122 0.14*** 0.186 0.204*** 0.259** 0.163*** 0.223** 0.058** 0.097

(2.04) (3.2) (2.93) (1.98) (2.58) (0.835) (3.12) (1.52) (3.49) (2.26) (3.28) (1.98) (2.38) (0.372)

USA, GER, UK

0.017*** 0.008* 0.054** 0.058* 0.042** 0.052* 0.030* 0.045* 0.090*** 0.073* 0.097*** 0.080** 0.126*** 0.096*

(3.29) (1.86) (2.02) (1.77) (2.04) (1.78) (1.73) (1.65) (2.98) (1.91) (3.19) (2.02) (3.59) (1.84)

USA, GER, FRA

-0.01 0.002 0.074*** 0.071*** 0.054*** 0.052*** 0.052*** 0.048*** 0.083*** 0.071*** 0.078*** 0.073** 0.091*** 0.077**

(-1.41) (-0.33) (2.96) (2.82) (2.94) (2.94) (2.79) (2.87) (3.61) (2.63) (3.53) (2.47) (3.67) (2.08)

USA, GER, ITA

-0.00 0.004 0.15*** 0.090*** 0.15** 0.084*** 0.152** 0.09*** 0.079* 0.036* 0.061** 0.008 0.093*** 0.029

(-1.5) (-0.27) (2.69) (2.67) (2.52) (2.59) (2.3) (2.62) (1.77) (1.72) (2.08) (0.12) (2.7) (1.07)

USA, GER, JPN

-0.01 0.031 0.021 0.041 -0.03 0.010 -0.00 0.014 -0.01 0.042 0.014 0.052 0.013 0.065

(-1.47) (0.967) (0.911) (1.5) (-0.44) (0.128) (0.356) (0.527) (0.034) (1.15) (0.726) (1.57) (0.504) (1.45)

USA, GER, RUS

0.006 -0.02* 0.179** 0.213*** 0.015 0.101 0.077 0.15*** 0.191*** 0.223*** 0.219*** 0.23*** 0.093 0.118

(-0.05) (-1.91) (2.5) (3.25) (-0.53) (1.52) (0.822) (2.92) (2.88) (3.43) (3.06) (3.42) (0.866) (1.44)

USA, UK, ITA

-0.03 -0.00 0.097*** 0.042 0.097*** 0.035 0.089*** 0.034 0.118*** 0.043 0.101*** 0.041 0.133*** 0.055

(-1.14) (-0.47) (2.63) (1.29) (2.92) (1.32) (2.8) (1.28) (2.99) (1.34) (2.8) (1.42) (3.05) (1.61)

125 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, UK, JPN

0.022 0.003 0.169*** 0.082** 0.197*** 0.092** 0.209*** 0.103** 0.107*** 0.044 0.084*** 0.015 0.135*** 0.049*

(1.37) (0.047) (3.53) (2.07) (3.86) (2.1) (3.9) (2.16) (3.35) (1.64) (2.85) (0.948) (3.3) (1.69)

USA, UK, RUS

-0.01 -0.01 0.009 -0.00 -0.04 -0.02 -0.01 -0.00 -0.03 -0.01 -0.00 -0.00 -0.00 -0.00

(-1.44) (-1.45) (0.662) (-0.97) (-0.72) (-1.37) (0.019) (-1.08) (-0.14) (-1.13) (0.408) (-0.91) (0.27) (-1)

USA, FRA, ITA

-0.02** -0.02 0.068* 0.009 0.050** 0.020* 0.051* 0.022* 0.049 0.025 0.073** 0.016 0.094*** 0.03

(-2.3) (-1.39) (1.79) (1.03) (2.03) (1.83) (1.83) (1.78) (1.42) (1.63) (2.18) (1.15) (2.73) (1.55)

USA, FRA, JPN

-0.00 -0.00 0.181** 0.259*** 0.017 0.139 0.083 0.191* 0.184*** 0.258*** 0.203*** 0.26*** 0.077 0.162

(-0.97) (-1.09) (2.03) (2.62) (-0.75) (1.01) (0.346) (1.89) (2.67) (2.7) (2.73) (2.74) (0.228) (1.41)

USA, FRA, RUS

-0.00 0.002 0.148** 0.067** 0.141** 0.067** 0.147** 0.077** 0.097*** 0.020** 0.085*** -0.00 0.132*** 0.031**

(-1.32) (-0.07) (2.52) (2.33) (2.57) (2.27) (2.54) (2.31) (3.19) (2.09) (2.85) (-0.12) (3.04) (2.28)

USA, ITA, JPN

-0.00 0.032 0.040* 0.029 -0.02 -0.00 0.003 0.011 0.002 0.019 0.027 0.024 0.034 0.034

(-0.97) (1.07) (1.79) (0.977) (-0.12) (-0.19) (0.888) (0.296) (0.86) (0.544) (1.61) (0.6) (1.57) (1.03)

USA, ITA, RUS

0.009 -0.00 0.185** 0.226*** 0.027 0.122* 0.086 0.171*** 0.179*** 0.228*** 0.199*** 0.235*** 0.112 0.154**

(-0.11) (-1.14) (2.46) (3.46) (-0.32) (1.84) (1.03) (2.95) (2.95) (3.54) (3.03) (3.55) (1.22) (2.52)

126 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, JPN, RUS

0.027 0.041* 0.105** 0.054** 0.050* 0.036 0.082** 0.055** 0.055** 0.025 0.067** 0.009 0.086** 0.030

(1.07) (1.79) (2.41) (1.96) (1.77) (1.58) (2.33) (2.05) (2.18) (0.895) (2.37) (0.175) (2.43) (1.39)

CAN, GER, UK

-0.00 -0.00 0.209** 0.24*** 0.051 0.15** 0.117 0.2*** 0.139** 0.2*** 0.144** 0.19*** 0.109 0.163**

(-0.60) (0.3) (2.12) (3.05) (0.019) (2.14) (1.13) (2.64) (2.47) (2.89) (2.51) (2.79) (1.07) (2.49)

CAN, GER, ITA

0.031 0.087*** 0.141*** 0.242** 0.018 0.122 0.066*** 0.168* 0.106*** 0.229** 0.145*** 0.238** 0.083*** 0.154

(0.787) (3.34) (3.59) (2.58) (1.56) (1.1) (2.7) (1.77) (3.27) (2.44) (3.7) (2.48) (2.99) (1.29)

CAN, GER, JPN

0.014 0.042 0.071*** 0.11* 0.064*** 0.090 0.090*** 0.11 0.082*** 0.137** 0.092*** 0.132** 0.032** 0.055

(0.474) (1.63) (3.27) (1.93) (3.09) (1.14) (3.52) (1.55) (3.27) (2.37) (3.43) (2.31) (2.37) (0.3)

CAN, GER, RUS

-0.02 -0.01 0.067 0.061* 0.037 0.045 0.072 0.063 0.009 0.035 -0.00 0.017 -0.05 0.005

(-1.48) (-1.09) (0.695) (1.71) (0.001) (0.331) (0.379) (1.26) (0.053) (-0.17) (-0.16) (-0.61) (-1.35) (-0.95)

CAN, UK, FRA

0.024 0.005 0.064** -0.00 0.096** 0.030* 0.113** 0.029 0.053** 0.021 0.044** -0.00 0.034** 0.002

(1.03) (-0.87) (2.29) (0.442) (2.38) (1.76) (2.54) (1.57) (2.37) (1.38) (2.29) (0.366) (2.16) (0.634)

CAN, UK, ITA

-0.00 -0.02 0.164* 0.042* 0.174** 0.063** 0.18** 0.076** 0.133* 0.033 0.11** -0.00 0.131*** 0.005

(-1.22) (-1.36) (1.87) (1.76) (2.42) (2.3) (2.04) (2.22) (1.96) (1.59) (2.35) (-0.55) (2.81) (0.4)

127 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

CAN, UK, JPN

-0.00 -0.00 0.058 0.089** 0.017 0.055 0.049 0.076 0.048 0.087 0.048 0.083 -0.02 0.027

(-1.01) (-0.34) (0.4) (2.04) (-0.21) (0.284) (0.231) (1.36) (0.48) (1.32) (0.522) (0.958) (-0.96) (-0.71)

CAN, UK, RUS

0.018 -0.00 0.039* -0.01 0.053** 0.018 0.073** 0.013 0.029* 0.009 0.038* 0.002 0.015 -0.01

(1.53) (-1.22) (1.91) (0.284) (2.08) (1.07) (2.25) (0.862) (1.74) (0.9) (1.83) (0.798) (1.61) (0.344)

CAN, FRA, ITA

-0.00 0.011 0.062 -0.00 0.101* 0.019 0.106 0.019 0.001 -0.02 -0.02 -0.04* -0.03 -0.04*

(-1.05) (0.956) (1.33) (0.596) (1.7) (1.47) (1.59) (1.63) (-0.46) (-1.42) (-0.88) (-1.88) (-1.18) (-1.69)

CAN, FRA, JPN

-0.01* -0.03** 0.168* 0.162** 0.073 0.107 0.109 0.134 0.19*** 0.205*** 0.196*** 0.191*** 0.087 0.089

(-1.89) (-2.26) (1.72) (2.28) (0.338) (0.465) (0.808) (1.23) (2.6) (2.92) (2.89) (2.7) (0.647) (0.094)

CAN, FRA, RUS

-0.01 0.007 0.036 0.005 0.033 0.022 0.046* 0.016 0.024 0.033 0.034 0.029 0.028 0.03

(-1.51) (-0.52) (1.55) (-0.05) (1.55) (0.469) (1.76) (0.209) (1.31) (0.665) (1.45) (0.68) (1.34) (0.56)

CAN, ITA, RUS

-0.00 -0.00 0.157*** 0.054 0.208*** 0.099** 0.199*** 0.094** 0.102** 0.039 0.061* -0.01 0.095*** 0.010

(0.609) (-1.47) (3.32) (1.55) (3.48) (2.21) (3.37) (2.05) (2.24) (1.08) (1.88) (-1.24) (2.59) (-0.09)

CAN, JPN, RUS

-0.04 -0.00 0.102** 0.039 0.131** 0.062* 0.116** 0.047 0.132** 0.068* 0.105** 0.048 0.126** 0.058

(-1.23) (-0.40) (1.97) (0.824) (2.13) (1.8) (2.05) (1.22) (2.18) (1.68) (2.03) (1.12) (2.17) (1.01)

128 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, JPN, RUS

0.014 0.062*** 0.117*** 0.141* 0.082*** 0.104 0.109*** 0.128 0.122*** 0.173** 0.131*** 0.161* 0.080*** 0.099

(-0.15) (2.9) (3.35) (1.79) (2.96) (0.716) (3.31) (1.11) (3.33) (2.05) (3.46) (1.93) (2.89) (0.426)

GER, ITA, RUS

-0.04** -0.04** 0.201** 0.163** 0.129 0.123 0.169* 0.15 0.143** 0.148** 0.115** 0.111** 0.079 0.098

(-2.16) (-2.54) (2.39) (2.41) (1.45) (1.07) (1.88) (1.62) (2.44) (2.5) (2.31) (2) (0.947) (0.924)

GER, ITA, JPN

0.014 0.006 0.106*** 0.010 0.129*** 0.033 0.145*** 0.038 0.099*** 0.023 0.089*** -0.00 0.102*** 0.010

(1.19) (0.031) (2.86) (0.958) (3.04) (1.46) (3.18) (1.55) (2.84) (1.01) (2.83) (0.242) (2.89) (0.826)

GER, FRA, RUS

-0.03** -0.02** 0.018 -0.01 -0.03 -0.02 0.002 -0.00 -0.00 -0.02 0.017 -0.01 -0.45 -0.01

(-2.52) (-2.03) (0.93) (-1.01) (-0.15) (-1.54) (0.505) (-1.2) (0.27) (-1.44) (0.822) (-1.45) (0.631) (-1.29)

GER, FRA, JPN

-0.03** -0.03*** 0.201** 0.173** 0.116 0.121 0.149* 0.149 0.222*** 0.202*** 0.208*** 0.183*** 0.119 0.113

(-2.4) (-2.61) (2.23) (2.17) (1.49) (0.589) (1.8) (1.2) (2.62) (2.94) (2.62) (2.6) (1.51) (0.456)

GER, FRA, ITA

-0.03* 0.013 0.060** -0.00 0.061** 0.005 0.071** 0.006 0.052* 0.012 0.055* 0.001 0.060** 0.007

(-1.71) (-0.46) (2.03) (0.096) (2.04) (0.44) (2.16) (0.476) (1.86) (0.495) (1.95) (0.365) (1.97) (0.446)

GER, UK, RUS

-0.01 0.001 0.179* 0.042* 0.235* 0.084** 0.224* 0.082** 0.143 0.033 0.101 -0.01 0.146* 0.018

(-0.51) (0.672) (1.85) (1.68) (1.94) (2.23) (1.91) (2.13) (1.64) (1.53) (1.49) (-1.31) (1.7) (0.867)

129 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, JPN

0.015 -0.01 0.066** -0.01 0.088*** 0.018 0.099*** 0.015 0.076*** 0.013 0.072*** -0.00 0.075*** 0.008

(0.52) (-1.42) (2.58) (0.53) (2.79) (1.37) (2.92) (1.32) (2.73) (1.11) (2.71) (0.566) (2.75) (0.986)

GER, UK, FRA

-0.01 -0.00 0.065 0.077 0.042 0.062 0.065 0.081 0.032 0.056 0.025 0.042 0.011 0.048

(-0.02) (-0.34) (0.913) (1.22) (0.495) (0.458) (0.797) (0.923) (0.263) (0.375) (0.25) (0.060) (0.002) (0.065)

UK, JPN, RUS

0.012 0.039 0.079*** 0.105** 0.071*** 0.086 0.088*** 0.104 0.093*** 0.129** 0.104*** 0.128** 0.068*** 0.081

(0.419) (1.64) (3.47) (1.98) (3.33) (1.05) (3.57) (1.4) (3.48) (2.28) (3.65) (2.23) (3.27) (0.67)

UK, ITA, RUS

-0.00 0.042** 0.072*** 0.112** 0.063*** 0.087 0.069*** 0.102 0.088*** 0.131** 0.083*** 0.121* 0.074*** 0.094

(0.14) (2.01) (3.04) (2.06) (2.84) (1.31) (2.91) (1.5) (3.33) (2.18) (3.38) (1.92) (3.24) (1.01)

UK, ITA, JPN

-0.01 -0.01 0.217*** 0.283** 0.043 0.154 0.124 0.224** 0.197** 0.301*** 0.242*** 0.296*** 0.105 0.166

(-1.06) (-0.93) (2.79) (2.56) (-0.03) (1.28) (1.19) (2.07) (2.48) (2.68) (3.2) (2.64) (1.29) (1.43)

UK, FRA, RUS

0.024*** 0.023** 0.020 0.030 0.037 0.057 0.029 0.057 0.044 0.085** 0.055 0.068* 0.084** 0.086**

(2.63) (2.22) (1.08) (-0.32) (1.42) (1.24) (1.06) (1.25) (1.18) (2.41) (1.2) (1.76) (2.31) (2.12)

UK, FRA, JPN

-0.00 -0.00 0.021 0.014 0.038 0.051** 0.055 0.063** 0.062 0.086*** 0.014 0.049 0.011 0.029

(-1.02) (-1.16) (0.512) (-0.22) (0.965) (2.04) (1.43) (2.55) (1.35) (2.68) (-1.01) (1.05) (0.055) (-0.20)

130 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, FRA, ITA

0.007 0.006 0.085 0.027 0.13 0.082*** 0.153 0.103** 0.094 0.057 -0.00 -0.01* -0.00 -0.04**

(0.282) (0.709) (0.944) (1.17) (1.23) (2.58) (1.08) (2.5) (0.399) (0.85) (-1.52) (-1.67) (-1.04) (-1.97)

FRA, ITA, JPN, RUS

-0.00 0.031 0.090*** 0.106** 0.089*** 0.098 0.105*** 0.112* 0.102*** 0.126* 0.099*** 0.11 0.082*** 0.091

(0.341) (1.23) (3.29) (2.1) (3.11) (1.41) (3.29) (1.71) (3.45) (1.92) (3.5) (1.55) (3.24) (0.737)

GER, UK, ITA, JPN

0.012 0.035 0.129*** 0.141** 0.098*** 0.109 0.126*** 0.131 0.134*** 0.177** 0.144*** 0.163** 0.092*** 0.086

(-0.16) (1.32) (3.95) (2.01) (3.58) (1.12) (3.99) (1.5) (3.83) (2.45) (4.02) (2.34) (3.38) (0.55)

CAN, UK, FRA, RUS

-0.04** -0.04** 0.2** 0.145** 0.139 0.115 0.177 0.139* 0.14** 0.129** 0.109** 0.087 0.070 0.068

(-2.12) (-2.44) (2.19) (2.44) (1.31) (1.02) (1.54) (1.7) (2.31) (2.11) (2.2) (1.22) (1.08) (0.105)

CAN, GER, UK, ITA

0.018 0.014 0.109*** 0.016 0.132*** 0.056 0.151*** 0.053 0.085*** 0.041 0.073*** 0.010 0.083*** 0.024

(1.06) (-0.29) (2.9) (0.592) (3.04) (1.64) (3.18) (1.46) (2.87) (1.35) (2.78) (0.355) (2.82) (0.932)

USA, UK, FRA, ITA

-0.02** -0.04** 0.204** 0.166*** 0.119 0.116 0.159 0.145** 0.217*** 0.19*** 0.199*** 0.163** 0.095 0.078

(-2.46) (-2.4) (2.37) (2.66) (1.23) (0.983) (1.62) (2.04) (3.02) (2.93) (3.1) (2.32) (1.33) (-0.31)

USA, CAN, JPN, RUS

-0.01* 0.011 0.077** 0.003 0.081** 0.028 0.099** 0.024 0.060* 0.023 0.061** 0.007 0.055* -0.00

(-1.65) (-0.52) (2.23) (0.020) (2.25) (0.71) (2.43) (0.483) (1.96) (0.535) (2.01) (0.255) (1.94) (-0.16)

131 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, JPN

-0.02 0.005 0.164** 0.045 0.221*** 0.092* 0.209*** 0.083 0.126** 0.037 0.077** -0.01 0.112** 0.013

(-1.53) (0.734) (2.49) (1.25) (2.69) (1.68) (2.6) (1.6) (2.32) (1.01) (1.98) (-0.36) (2.3) (0.412)

USA, CAN, GER, FRA

0.028 0.087*** 0.143*** 0.201*** 0.047** 0.116 0.089*** 0.156** 0.103*** 0.176*** 0.12*** 0.17** 0.095*** 0.135

(0.982) (3.21) (2.98) (2.96) (2.19) (1.49) (2.86) (2.18) (3.17) (2.58) (3.24) (2.44) (2.85) (1.49)

USA, CAN, GER, UK

0.034 0.081*** 0.122*** 0.185*** 0.022* 0.106 0.063*** 0.144* 0.098*** 0.182*** 0.13*** 0.189*** 0.082*** 0.131

(0.663) (2.75) (3.42) (2.78) (1.81) (1.11) (3.06) (1.87) (3.38) (2.61) (3.69) (2.66) (3.24) (1.25)

UK, ITA, JPN, RUS

-0.01 -0.00 0.203** 0.216*** 0.064 0.138** 0.124 0.185** 0.146** 0.172*** 0.147** 0.156** 0.112 0.133**

(-0.79) (-0.04) (2) (2.85) (0.088) (1.98) (1.05) (2.49) (2.35) (2.75) (2.43) (2.57) (1.03) (2.21)

UK, FRA, JPN, RUS

0.025 0.031 0.094** 0.034* 0.051 0.023 0.080** 0.035* 0.058** 0.022 0.07** 0.005 0.089** 0.029

(0.437) (1.35) (2.4) (1.66) (1.59) (1.35) (2.15) (1.73) (2.19) (0.913) (2.43) (-0.17) (2.53) (1.41)

UK, FRA, ITA, RUS

-0.00 0.033 0.151*** 0.142** 0.136*** 0.112 0.162*** 0.133* 0.15*** 0.156** 0.142*** 0.133** 0.12*** 0.105

(0.103) (1.16) (3.85) (2.54) (3.65) (1.47) (3.85) (1.91) (3.88) (2.44) (3.94) (2.06) (3.7) (0.847)

UK, FRA, ITA, JPN

0.027 0.085*** 0.127*** 0.209** 0.023 0.116 0.070*** 0.159 0.103*** 0.21** 0.137*** 0.213** 0.072*** 0.138

(0.092) (2.88) (3.41) (2.32) (1.26) (0.592) (2.63) (1.36) (3.08) (2.23) (3.53) (2.26) (2.72) (0.765)

132 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, ITA, JPN, RUS

-0.02* -0.01 0.239*** 0.247*** 0.107 0.166* 0.176* 0.214** 0.171*** 0.212*** 0.16*** 0.188** 0.116 0.158**

(-1.75) (-0.40) (2.9) (2.66) (0.874) (1.83) (1.95) (2.27) (2.98) (2.64) (2.86) (2.53) (1.2) (1.99)

GER, FRA, JPN, RUS

0.026 0.024 0.106*** 0.048 0.074* 0.038 0.106** 0.055 0.063** 0.030 0.070** 0.009 0.089** 0.029

(0.681) (0.793) (2.62) (1.42) (1.94) (1.06) (2.51) (1.41) (2.11) (0.624) (2.38) (-0.08) (2.48) (0.735)

GER, FRA, ITA, RUS

-0.01 -0.01 0.236*** 0.242*** 0.080 0.144* 0.14** 0.19*** 0.25*** 0.25*** 0.251*** 0.248*** 0.151* 0.167**

(-1.63) (-1.52) (3.21) (3.28) (0.652) (1.82) (2.05) (2.78) (3.85) (3.35) (3.71) (3.35) (1.86) (2.31)

GER, FRA, ITA, JPN

-0.01* 0.035 0.051* 0.028 -0.00 0.010 0.022 0.020 0.017 0.028 0.040 0.029 0.048 0.039

(-1.81) (0.756) (1.78) (0.318) (0.363) (-0.28) (1.16) (0.029) (0.935) (0.153) (1.6) (0.302) (1.56) (0.483)

GER, UK, JPN, RUS

-0.00 0.009 0.16*** 0.064** 0.185*** 0.074** 0.181*** 0.080** 0.119*** 0.030 0.090*** 0.000 0.143*** 0.035*

(-1.21) (0.155) (3.41) (2.01) (3.71) (2.08) (3.61) (2.09) (3.6) (1.47) (3.04) (-0.22) (3.48) (1.67)

GER, UK, ITA, RUS

0.027 0.072** 0.114*** 0.193*** 0.011 0.103 0.057*** 0.143* 0.090*** 0.193*** 0.126*** 0.199*** 0.061*** 0.113

(0.339) (2.23) (3.75) (2.82) (1.28) (1.07) (2.96) (1.94) (3.43) (2.69) (3.91) (2.69) (2.94) (0.777)

GER, UK, FRA, RUS

-0.02 -0.01 0.201** 0.208*** 0.068 0.123** 0.133 0.174*** 0.125* 0.156*** 0.12** 0.134** 0.071 0.087

(-1.22) (-0.61) (2.06) (3.08) (0.005) (2.02) (0.939) (2.82) (1.74) (2.69) (1.99) (2.27) (0.517) (0.812)

133 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, FRA, JPN

0.026 0.036 0.101** 0.051** 0.059* 0.042** 0.096** 0.050** 0.041* 0.036 0.052* 0.011 0.062** 0.035

(0.843) (0.377) (2) (2.17) (1.67) (1.98) (2.01) (2.4) (1.79) (0.872) (1.89) (-0.48) (2) (0.589)

GER, UK, FRA, ITA

0.002 -0.02 0.186** 0.223*** 0.035 0.119*** 0.098 0.169*** 0.181*** 0.221*** 0.193*** 0.22*** 0.087 0.114

(-0.42) (-1.29) (2.51) (3.39) (-0.27) (2.6) (1.05) (3.45) (2.84) (3.35) (2.98) (3.35) (0.834) (1.29)

CAN, ITA, JPN, RUS

-0.00 0.037 0.148*** 0.132** 0.121*** 0.107 0.144*** 0.128 0.156*** 0.166** 0.161*** 0.152** 0.12*** 0.103

(-1.24) (1.22) (3.98) (2.03) (3.76) (1.17) (3.99) (1.54) (3.85) (2.43) (3.98) (2.3) (3.71) (0.819)

CAN, FRA, JPN, RUS

-0.00 0.026 0.049** 0.035 -0.00 0.011 0.024* 0.017 0.004 0.028 0.030* 0.030 0.025 0.026

(-0.33) (0.221) (2.05) (1.58) (0.735) (0.23) (1.73) (0.833) (0.96) (0.749) (1.66) (0.859) (1.39) (0.221)

CAN, FRA, ITA, RUS

-0.01 0.014 0.138** 0.081** 0.143** 0.082* 0.143** 0.086* 0.085* 0.035 0.067** 0.009 0.098** 0.032

(-1.44) (0.575) (2.37) (1.97) (2.23) (1.83) (2.06) (1.9) (1.74) (1.45) (2.17) (0.729) (2.44) (1.06)

CAN, FRA, ITA, JPN

0.005 -0.03* 0.205** 0.228*** 0.047 0.126 0.104 0.171*** 0.231*** 0.249*** 0.253*** 0.245*** 0.135* 0.139

(-0.54) (-1.82) (2.51) (3.21) (-0.18) (1.35) (1.04) (2.62) (3.63) (3.22) (3.69) (3.23) (1.69) (1.34)

CAN, UK, JPN, RUS

-0.01 0.026 0.032 0.040 -0.02 0.020 0.003 0.024 -0.00 0.045 0.025 0.055 0.028 0.064

(-1.26) (0.624) (1.26) (0.718) (-0.19) (-0.04) (0.576) (0.113) (0.273) (0.555) (0.926) (0.902) (0.799) (0.772)

134 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

CAN, UK, ITA, RUS

0.014 0.005 0.156*** 0.088** 0.178*** 0.098** 0.172*** 0.101** 0.101*** 0.049 0.073*** 0.013 0.118*** 0.039

(-0.32) (-0.17) (3.24) (2.17) (3.65) (2.19) (3.37) (2.21) (3.01) (1.51) (2.71) (0.491) (3.75) (1.04)

CAN, UK, ITA, JPN

-0.02 0.002 0.085*** 0.057** 0.093*** 0.053** 0.079*** 0.047** 0.114*** 0.062* 0.095*** 0.058 0.119*** 0.064

(-1.14) (-0.21) (3.12) (1.97) (3.54) (2) (3.23) (2) (3.83) (1.7) (3.6) (1.59) (3.8) (1.14)

CAN, UK, FRA, JPN

0.069** 0.112*** 0.114*** 0.155** 0.101*** 0.119 0.161*** 0.171** 0.203*** 0.207*** 0.136*** 0.157* 0.067*** 0.091

(2.1) (3.3) (3.3) (2.55) (3.24) (1.47) (3.69) (2.2) (3.8) (2.69) (3.51) (1.89) (2.93) (0.629)

CAN, UK, FRA, ITA

0.080** 0.118*** 0.104*** 0.145** 0.084*** 0.113 0.139*** 0.166* 0.202*** 0.226** 0.149*** 0.181** 0.065*** 0.095

(2.32) (2.96) (3.29) (2.02) (3.14) (1.17) (3.61) (1.81) (3.82) (2.49) (3.49) (2.11) (2.83) (0.714)

CAN, GER, JPN, RUS

-0.01 -0.01 0.1 0.135** 0.053 0.105 0.123 0.165** 0.146 0.173** 0.054 0.116 -0.01 0.047

(-1.37) (-0.69) (0.189) (2.16) (-0.44) (0.807) (0.2) (2.08) (0.637) (2.24) (-0.59) (1.08) (-1.2) (-0.34)

CAN, GER, ITA, RUS

0.067* 0.077** 0.073** 0.000 0.097** 0.040* 0.14*** 0.072** 0.15*** 0.080* 0.082** 0.021 0.056** 0.000

(1.84) (2.11) (2.2) (0.7) (2.45) (1.77) (2.82) (2.05) (2.96) (1.93) (2.4) (0.379) (2.16) (0.356)

CAN, GER, ITA, JPN

-0.04 -0.04 0.201** 0.15** 0.138 0.119 0.169* 0.142* 0.172** 0.142** 0.139** 0.105* 0.106 0.093

(-0.59) (-1.64) (2.24) (2.49) (1.61) (1.12) (1.87) (1.68) (2.41) (2.52) (2.3) (1.8) (1.46) (0.991)

135 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

CAN, GER, FRA, RUS

0.059 0.122*** 0.117*** 0.17 0.081*** 0.115 0.137*** 0.169 0.183*** 0.235** 0.155*** 0.197* 0.066** 0.097

(1.13) (3.18) (3.17) (1.59) (2.77) (0.613) (3.33) (1.22) (3.54) (1.98) (3.4) (1.67) (2.55) (0.253)

CAN, GER, FRA, JPN

-0.00 -0.01 0.174 0.171** 0.113 0.131 0.182 0.185** 0.178* 0.196** 0.112 0.138 0.037 0.075

(-1.28) (-0.70) (1.62) (2.52) (0.63) (0.872) (1.29) (2.08) (1.74) (2.46) (0.947) (1.38) (-0.16) (-0.28)

CAN, GER, FRA, ITA

0.066** 0.077* 0.093** 0.021 0.12*** 0.057 0.161*** 0.086* 0.134*** 0.077 0.082** 0.016 0.072** 0.006

(2.12) (1.94) (2.46) (1.02) (2.7) (1.6) (3.04) (1.76) (2.69) (1.51) (2.36) (0.155) (2.37) (0.36)

CAN, GER, UK, RUS

0.013 0.005 0.174* 0.194** 0.098 0.14 0.159 0.196* 0.234*** 0.266*** 0.2** 0.222** 0.081 0.116

(-1.15) (0.255) (1.89) (2.23) (0.797) (0.828) (1.58) (1.92) (2.88) (2.96) (2.03) (2.54) (0.572) (0.369)

CAN, GER, UK, JPN

0.018 0.085** 0.039 -0.00 0.042* 0.013 0.075** 0.040 0.085* 0.055 0.048 0.005 0.027 -0.00

(-0.73) (2.3) (1.62) (-0.52) (1.71) (0.134) (2.04) (0.678) (1.85) (0.715) (1.58) (-0.42) (1.47) (-0.17)

CAN, GER, UK, FRA

0.02 0.030** 0.122** 0.013 0.188*** 0.068** 0.186*** 0.080** 0.133** 0.043 0.057 -0.01 0.080** -0.01

(-0.72) (2.09) (2.5) (0.587) (3) (1.99) (2.84) (2.03) (2.37) (1.36) (1.25) (-1.25) (2.4) (-1.38)

USA, ITA, JPN, RUS

0.077** 0.114** 0.107*** 0.16** 0.081*** 0.116 0.139*** 0.171* 0.198*** 0.247** 0.159*** 0.205** 0.059*** 0.084

(2.41) (2.49) (3.28) (1.99) (3.04) (1.16) (3.58) (1.76) (3.84) (2.46) (3.66) (2.19) (2.68) (0.498)

136 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, FRA, JPN, RUS

-0.01 -0.01 0.121 0.144** 0.068 0.105 0.146 0.171** 0.144 0.162* 0.051 0.099 -0.01 0.017

(-1.14) (-0.35) (0.689) (2.13) (-0.07) (0.781) (0.639) (2.07) (0.916) (1.73) (-0.42) (0.547) (-1.12) (-0.87)

USA, FRA, ITA, RUS

0.069 0.095*** 0.083** 0.020 0.109** 0.067** 0.162** 0.097** 0.137** 0.098** 0.07** 0.036 0.042* 0.007

(1.54) (2.85) (2.08) (1.04) (2.28) (2.21) (2.56) (2.46) (2.55) (2.23) (2.09) (0.508) (1.85) (-0.03)

USA, FRA, ITA, JPN

0.024 -0.00 0.118 0.179** 0.047 0.122 0.126 0.194** 0.19* 0.253*** 0.127 0.198** 0.007 0.060

(0.301) (0.36) (0.877) (2.24) (-0.25) (1) (0.645) (2.31) (1.88) (2.8) (0.4) (2.17) (-0.68) (-0.41)

USA, UK, JPN, RUS

0.005 0.008 0.108*** 0.001 0.13*** 0.032 0.14*** 0.032 0.109*** 0.032 0.095*** 0.002 0.111*** 0.020

(0.208) (-0.34) (2.71) (0.601) (2.8) (1.27) (2.9) (1.26) (2.74) (1.1) (2.68) (0.312) (2.76) (0.866)

USA, UK, ITA, RUS

0.048 0.067*** 0.043* 0.004 0.049** 0.033 0.097** 0.063* 0.102** 0.081 0.054* 0.029 0.007 -0.01

(1.64) (2.89) (1.85) (0.055) (1.97) (0.958) (2.43) (1.76) (2.19) (1.64) (1.79) (0.31) (1.29) (-0.60)

USA, UK, ITA, JPN

0.003 0.010 0.090 0.010 0.137 0.073** 0.161 0.093*** 0.108 0.056 0.013 -0.00 0.002 -0.03*

(-0.42) (0.137) (0.786) (0.025) (1.05) (2.53) (1.1) (2.65) (0.566) (0.783) (-1.22) (-1.47) (-0.84) (-1.69)

USA, UK, FRA, RUS

0.031 -0.00 0.161* 0.195* 0.070 0.13 0.13 0.188 0.212*** 0.271*** 0.203** 0.237** 0.080 0.097

(0.017) (0.018) (1.65) (1.96) (0.122) (0.603) (0.888) (1.6) (3.08) (2.68) (2.01) (2.39) (0.342) (0.102)

137 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, UK, FRA, JPN

0.021 0.071** 0.025 0.015 0.022 0.033 0.056 0.053 0.063 0.082 0.041 0.049 0.012 0.037

(0.225) (2) (1.19) (-0.16) (1.22) (0.369) (1.57) (0.704) (1.33) (1.07) (1.22) (0.45) (0.957) (0.313)

USA, GER, JPN, RUS

0.037 0.040*** 0.126*** 0.040 0.184*** 0.096** 0.188*** 0.11** 0.112* 0.060 0.042 -0.00 0.060** -0.01**

(1.35) (2.6) (2.69) (1.23) (3.13) (2.36) (2.95) (2.32) (1.9) (1.26) (0.64) (-1.64) (2.07) (-1.96)

USA, GER, ITA, RUS

-0.00 0.016 0.062** 0.022 0.1*** 0.058 0.093*** 0.061* 0.091*** 0.076** 0.070** 0.04 0.093*** 0.037

(-1.04) (-0.04) (2.5) (-0.21) (2.86) (1.46) (2.66) (1.71) (2.68) (2.11) (2.02) (0.531) (3.02) (-0.29)

USA, GER, ITA, JPN

0.035 0.102*** 0.141*** 0.226** 0.037** 0.125 0.1*** 0.185* 0.153*** 0.249** 0.17*** 0.24** 0.080*** 0.138

(1.02) (3) (3.34) (2.46) (1.96) (0.903) (2.95) (1.69) (3.46) (2.41) (3.65) (2.35) (2.86) (0.972)

USA, GER, FRA, RUS

-0.00 -0.02 0.207 0.206*** 0.075 0.134* 0.155 0.192*** 0.188 0.198*** 0.165** 0.167** 0.092 0.113

(-0.95) (-1.32) (1.5) (2.96) (0.029) (1.73) (0.782) (2.74) (1.54) (2.78) (1.97) (2.22) (0.641) (1.12)

USA, GER, FRA, JPN

0.043 0.056** 0.1** 0.047* 0.069** 0.052* 0.113** 0.077** 0.097** 0.056 0.083** 0.022 0.080** 0.024

(1.21) (2) (2.32) (1.77) (1.97) (1.79) (2.49) (2.02) (2.23) (1.49) (2.35) (0.473) (2.37) (1.05)

USA, GER, FRA, ITA

0.012 -0.01 0.195** 0.21*** 0.047 0.129 0.122 0.186** 0.2** 0.241*** 0.219*** 0.228*** 0.107 0.137

(-0.06) (-0.81) (2.14) (2.84) (-0.18) (1.49) (0.903) (2.49) (2.03) (3.04) (2.6) (2.93) (0.996) (1.63)

138 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, GER, UK, RUS

0.003 0.042* 0.087*** 0.111** 0.087*** 0.104 0.113*** 0.12* 0.086*** 0.119* 0.078*** 0.099 0.049*** 0.059

(0.838) (1.87) (3.19) (2.12) (3.11) (1.47) (3.3) (1.83) (3.32) (1.87) (3.28) (1.31) (2.73) (0.032)

USA, GER, UK, JPN

0.002 0.053** 0.037 0.023 -0.01 0.014 0.024 0.038 0.035 0.049 0.040 0.032 0.029 0.032

(-0.83) (2.41) (1.61) (0.774) (0.149) (0.196) (1.13) (0.876) (1.14) (0.968) (1.58) (0.639) (1.46) (1.01)

USA, GER, UK, ITA

-0.00* -0.02 0.129 0.019 0.14 0.043** 0.145 0.053** 0.104 0.013 0.083 -0.02 0.108** -0.00

(-1.95) (-1.22) (1.27) (1.47) (1.44) (2.25) (1.19) (2.21) (1.12) (1.45) (1.61) (-1.11) (2.27) (0.152)

USA, GER, UK, FRA

-0.00 0.006 0.211** 0.241** 0.047 0.138 0.118 0.194* 0.181** 0.257** 0.239*** 0.25** 0.106 0.147

(-0.88) (-0.51) (2.46) (2.32) (-0.29) (0.917) (0.751) (1.74) (2.31) (2.46) (3.23) (2.4) (1.02) (1.06)

USA, CAN, ITA, RUS

-0.02** -0.00 0.004 -0.01 -0.04 -0.01 -0.00 0.004 -0.02 -0.00 0.003 -0.00 -0.00 -0.00

(-2.39) (-1.18) (0.639) (-1.12) (-0.64) (-1.23) (0.085) (-0.94) (-0.21) (-1.2) (0.472) (-1.31) (0.344) (-0.91)

USA, CAN, ITA, JPN

0.023 -0.00 0.153*** 0.038 0.19*** 0.061** 0.19*** 0.070** 0.104** 0.024 0.093** -0.01 0.128*** 0.005

(-0.18) (-0.38) (2.69) (1.55) (3.39) (2.04) (3.13) (2.11) (2.09) (0.942) (2.48) (-0.80) (3.12) (0.022)

USA, CAN, FRA, RUS

-0.03 -0.00 0.079** 0.009 0.087*** 0.022 0.075*** 0.021 0.059** 0.022 0.087** 0.008 0.121*** 0.028

(-1.18) (-0.09) (2.35) (0.485) (2.95) (1.07) (2.67) (1.15) (2.12) (1.12) (2.47) (0.472) (3.08) (0.975)

139 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, FRA, JPN

0.007 -0.02 0.199** 0.208*** 0.040 0.115 0.116 0.172** 0.194** 0.242*** 0.235*** 0.234*** 0.107 0.112

(-0.27) (-1.53) (2.43) (2.7) (-0.35) (1.11) (0.88) (2.19) (2.01) (2.99) (2.81) (2.93) (0.977) (0.961)

USA, CAN, FRA, ITA

-0.01 0.048* 0.026 0.038 -0.02 0.024 0.013 0.041 0.014 0.070 0.033 0.064 0.021 0.067

(-1.58) (1.93) (1.15) (1.3) (-0.07) (0.403) (0.639) (0.982) (0.247) (1.36) (1) (1.54) (0.904) (1.49)

USA, CAN, UK, RUS

-0.00 -0.01 0.135 0.048** 0.149 0.070*** 0.154 0.082*** 0.091 0.030 0.069 -0.00 0.085** -0.00

(-1.56) (-1.03) (1.32) (2.11) (1.32) (2.75) (0.968) (2.73) (0.818) (1.17) (1.35) (-1.24) (2.22) (-1.4)

USA, CAN, UK, JPN

-0.02* -0.02 0.060* 0.025 0.051 0.035** 0.054 0.036** 0.044 0.046** 0.063* 0.032 0.081*** 0.034

(-1.74) (-1.46) (1.66) (1.6) (1.46) (2.54) (1.52) (2.35) (1.04) (1.97) (1.79) (1.33) (3.07) (0.962)

USA, CAN, UK, ITA

0.029 0.043 0.094*** 0.119** 0.085*** 0.107 0.113*** 0.125** 0.096*** 0.135** 0.103*** 0.125** 0.053*** 0.057

(1.42) (1.61) (3.67) (2.26) (3.55) (1.64) (3.91) (2.02) (3.54) (2.48) (3.68) (2.28) (2.96) (0.082)

USA, CAN, UK, FRA

0.008 0.004 0.051 0.027 0.042 0.035 0.029 0.030 0.030 0.047 0.083** 0.045 0.127*** 0.063

(-0.26) (1.01) (1.52) (0.69) (1.38) (1.04) (1.1) (0.977) (0.725) (1.42) (2.04) (1.37) (3.37) (1.2)

USA, CAN, GER, RUS

-0.02 -0.01 0.077 0.075** 0.053 0.063 0.086 0.081** 0.027 0.051 0.010 0.034 -0.02 0.029

(-1.32) (-0.14) (0.52) (2) (0.241) (1.17) (0.521) (2.03) (0.151) (0.438) (0.006) (-0.07) (-0.71) (-0.32)

140 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, ITA

0.024 -0.00 0.076** -0.01 0.107*** 0.039* 0.127*** 0.032 0.069*** 0.019 0.061** -0.00 0.051** -0.00

(1.1) (-1.43) (2.5) (0.193) (2.72) (1.85) (2.85) (1.62) (2.59) (1.17) (2.51) (0.144) (2.39) (0.382)

UK, FRA, ITA, JPN, RUS

-0.01 0.025 0.154*** 0.133** 0.138*** 0.114 0.159*** 0.132* 0.159*** 0.158** 0.149*** 0.132* 0.13*** 0.11

(-0.43) (0.296) (3.75) (2.32) (3.5) (1.48) (3.66) (1.83) (3.83) (2.32) (3.83) (1.89) (3.64) (0.936)

CAN, GER, ITA, JPN, RUS

-0.03* -0.02 0.239*** 0.231*** 0.107 0.156** 0.164* 0.201** 0.192*** 0.196*** 0.176*** 0.171*** 0.135 0.145**

(-1.69) (-0.67) (2.8) (2.82) (0.784) (1.96) (1.78) (2.41) (3.29) (2.9) (3.07) (2.74) (1.47) (2.26)

USA, UK, ITA, JPN, RUS

0.017 0.033 0.105** 0.039 0.074* 0.035 0.101** 0.046 0.072** 0.036 0.078** 0.014 0.1*** 0.037

(-0.15) (0.796) (2.56) (1.24) (1.85) (0.981) (2.39) (1.29) (2.25) (0.792) (2.48) (-0.04) (2.61) (0.965)

USA, GER, UK, FRA, RUS

0.023 0.078** 0.147*** 0.195*** 0.059** 0.129* 0.11*** 0.168*** 0.096*** 0.162** 0.108*** 0.145* 0.075*** 0.098

(0.918) (2.57) (3.03) (2.97) (2.25) (1.83) (2.8) (2.58) (3.36) (2.23) (3.4) (1.76) (2.87) (0.372)

USA, CAN, UK, FRA, JPN

0.038 0.075** 0.137*** 0.2*** 0.035** 0.12* 0.084*** 0.158*** 0.103*** 0.196*** 0.133*** 0.19*** 0.069*** 0.102

(0.85) (2.06) (3.67) (3.03) (2.39) (1.82) (3.83) (2.6) (3.65) (2.9) (3.94) (2.8) (3.27) (0.534)

USA, CAN, -0.02 -0.01 0.199** 0.213*** 0.071 0.135** 0.133 0.185*** 0.137* 0.164** 0.131** 0.143** 0.081 0.1

141 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, RUS (-1.45) (-0.03) (1.97) (2.8) (0.080) (2.14) (0.973) (2.62) (1.89) (2.51) (2.16) (2.21) (0.539) (1.24)

USA, CAN, GER, UK, JPN

0.022 0.032 0.102** 0.044** 0.065* 0.040** 0.101** 0.047** 0.050** 0.033 0.058** 0.014 0.068** 0.024

(0.772) (0.393) (2.16) (2.07) (1.77) (2.07) (2.14) (2.45) (2.1) (0.947) (2.2) (-0.04) (2.17) (0.245)

USA, CAN, GER, UK, ITA

0.026 0.064* 0.142*** 0.202*** 0.039* 0.115 0.084*** 0.153* 0.119*** 0.212** 0.153*** 0.208** 0.092*** 0.123

(-0.08) (1.89) (3.93) (2.64) (1.92) (1.07) (3.33) (1.85) (3.57) (2.56) (4.06) (2.51) (3.39) (0.811)

USA, CAN, GER, UK, FRA

-0.02* -0.02 0.242*** 0.231*** 0.117 0.153* 0.178* 0.201** 0.175*** 0.185** 0.158*** 0.154** 0.112 0.11

(-1.83) (-0.86) (2.77) (2.72) (0.834) (1.81) (1.8) (2.38) (2.81) (2.51) (2.77) (2.23) (1.4) (1.13)

GER, FRA, ITA, JPN, RUS

0.022 0.031 0.11** 0.058 0.081** 0.055 0.116** 0.063* 0.054** 0.051 0.062** 0.027 0.080** 0.042

(0.672) (0.177) (2.47) (1.6) (1.98) (1.36) (2.42) (1.66) (2.02) (0.935) (2.25) (0.155) (2.34) (0.548)

GER, UK, ITA, JPN, RUS

-0.01 -0.03* 0.237*** 0.238*** 0.085 0.14** 0.147** 0.189*** 0.251*** 0.245*** 0.248*** 0.233*** 0.133** 0.126

(-1.58) (-1.79) (3.2) (3.29) (0.613) (2.4) (1.98) (3.26) (4.02) (3.19) (3.92) (3.18) (2.07) (1.13)

GER, UK, FRA, JPN, RUS

0.006 0.037 0.102*** 0.117** 0.103*** 0.116* 0.129*** 0.132** 0.098*** 0.128** 0.090*** 0.106 0.061*** 0.068

(0.865) (1.3) (3.41) (2.27) (3.36) (1.9) (3.54) (2.24) (3.56) (2.12) (3.51) (1.59) (3.02) (0.216)

142 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, UK, FRA, ITA, RUS

-0.01 0.028 0.065** 0.037 0.015 0.023 0.045* 0.028 0.021 0.035 0.042* 0.034 0.044 0.032

(-1.41) (0.19) (2.14) (0.754) (1.07) (0.066) (1.88) (0.352) (1.11) (0.328) (1.68) (0.339) (1.55) (-0.02)

GER, UK, FRA, ITA, JPN

-0.00 0.018 0.153*** 0.078* 0.181*** 0.093* 0.171*** 0.094* 0.113*** 0.044 0.079*** 0.010 0.122*** 0.037

(-1.26) (0.648) (3.39) (1.7) (3.76) (1.68) (3.42) (1.71) (3.75) (1.28) (3.27) (0.65) (3.91) (0.948)

CAN, FRA, ITA, JPN, RUS

0.056* 0.101*** 0.109*** 0.137** 0.098*** 0.116 0.154*** 0.165** 0.198*** 0.201** 0.128*** 0.147 0.066*** 0.090

(1.86) (3.23) (3.16) (2.25) (2.98) (1.47) (3.43) (2.12) (3.84) (2.41) (3.41) (1.61) (2.77) (0.615)

CAN, UK, ITA, JPN, RUS

0.055 0.095*** 0.151*** 0.152** 0.136*** 0.122 0.195*** 0.168** 0.214*** 0.204*** 0.159*** 0.152* 0.096*** 0.096

(1.47) (2.93) (3.7) (2.42) (3.54) (1.45) (3.94) (2.07) (4.02) (2.6) (3.83) (1.84) (3.31) (0.577)

CAN, UK, FRA, JPN, RUS

0.070 0.113*** 0.147*** 0.147* 0.116*** 0.117 0.17*** 0.164* 0.226*** 0.223** 0.182*** 0.173** 0.101*** 0.101

(0.972) (2.87) (3.78) (1.87) (3.55) (1.09) (3.99) (1.69) (4.06) (2.4) (3.85) (1.96) (3.33) (0.663)

CAN, UK, FRA, ITA, RUS

-0.01 -0.02 0.167 0.157** 0.109 0.125 0.169 0.176** 0.193* 0.187** 0.117 0.129 0.048 0.073

(-1.45) (-1.08) (1.38) (2.42) (0.47) (0.961) (1.08) (2.11) (1.95) (2.48) (0.963) (1.47) (-0.08) (-0.04)

CAN, UK, 0.063* 0.080** 0.094** 0.006 0.119*** 0.042 0.154*** 0.067 0.151*** 0.073 0.092** 0.013 0.079** 0.007

143 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

FRA, ITA, JPN (1.67) (2) (2.42) (0.505) (2.62) (1.33) (2.88) (1.55) (2.87) (1.49) (2.51) (0.024) (2.42) (0.228)

CAN, GER, FRA, JPN, RUS

0.063** 0.11*** 0.122*** 0.149** 0.109*** 0.124 0.173*** 0.176** 0.188*** 0.205** 0.119*** 0.145 0.056*** 0.071

(2.01) (3.08) (3.33) (2.17) (3.18) (1.53) (3.56) (2.15) (3.93) (2.23) (3.49) (1.41) (2.72) (0.114)

CAN, GER, FRA, ITA, RUS

0.094*** 0.112*** 0.128*** 0.163** 0.099*** 0.134 0.165*** 0.19** 0.214*** 0.24** 0.158*** 0.179** 0.061*** 0.071

(2.62) (2.61) (3.65) (2.16) (3.34) (1.56) (3.93) (2.19) (4.05) (2.56) (3.71) (2.1) (2.7) (0.18)

CAN, GER, FRA, ITA, JPN

-0.01 -0.01 0.122 0.138** 0.069 0.11 0.148 0.176** 0.155 0.172** 0.059 0.107 -0.01 0.025

(-1.5) (-0.80) (0.39) (2.03) (-0.28) (1.18) (0.387) (2.45) (0.878) (1.98) (-0.55) (0.839) (-1.19) (-0.80)

CAN, GER, UK, JPN, RUS

-0.00 0.031 0.154*** 0.142** 0.143*** 0.122 0.174*** 0.143** 0.137*** 0.155** 0.132*** 0.121 0.108*** 0.088

(0.239) (1.2) (3.97) (2.38) (3.75) (1.64) (3.94) (2.04) (4.02) (2.17) (4.01) (1.62) (3.77) (0.406)

CAN, GER, UK, ITA, RUS

0.072* 0.072*** 0.090** 0.010 0.116** 0.064** 0.169*** 0.091** 0.15*** 0.089** 0.079** 0.027 0.044* -0.00

(1.8) (2.72) (2.26) (0.495) (2.47) (2.29) (2.77) (2.48) (2.88) (2.03) (2.34) (0.005) (1.94) (-0.58)

CAN, GER, UK, ITA, JPN

0.074 0.105** 0.139*** 0.167* 0.101*** 0.123 0.157*** 0.171* 0.21*** 0.245** 0.178*** 0.201** 0.091*** 0.094

(1.34) (2.41) (3.67) (1.93) (3.26) (1.1) (3.81) (1.66) (3.97) (2.37) (3.85) (2.12) (3.06) (0.526)

144 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

CAN, GER, UK, FRA, RUS

-0.00 -0.01 0.186* 0.162** 0.126 0.128 0.194 0.185** 0.182** 0.181** 0.112 0.119 0.044 0.051

(-1.14) (-0.7) (1.67) (2.02) (0.744) (0.91) (1.44) (2.09) (2.08) (2.03) (1.18) (0.987) (0.125) (-0.54)

CAN, GER, UK, FRA, JPN

0.069* 0.092** 0.103** 0.029 0.128*** 0.067* 0.172*** 0.092** 0.133*** 0.095* 0.076** 0.034 0.066** 0.018

(1.88) (2.47) (2.46) (0.901) (2.62) (1.78) (2.9) (1.97) (2.74) (1.87) (2.35) (0.384) (2.24) (0.28)

CAN, GER, UK, FRA, ITA

0.018 -0.00 0.194** 0.199** 0.108 0.142 0.177 0.204** 0.246*** 0.267*** 0.209** 0.211** 0.080 0.082

(-1) (-0.11) (2.22) (2.27) (0.727) (1.17) (1.58) (2.32) (3.5) (2.82) (2.23) (2.29) (0.685) (-0.33)

USA, FRA, ITA, JPN, RUS

0.035 0.071** 0.063** 0.006 0.064** 0.035 0.106** 0.058 0.098** 0.069 0.060* 0.018 0.031 -0.00

(1.02) (1.99) (1.99) (-0.34) (2.03) (0.449) (2.39) (0.911) (2.01) (0.836) (1.79) (-0.38) (1.56) (-0.74)

USA, UK, FRA, JPN, RUS

0.017 0.037 0.137*** 0.024 0.195*** 0.085** 0.196*** 0.096** 0.135*** 0.055 0.057 -0.00 0.071** -0.01*

(-0.56) (1.58) (2.75) (0.236) (3.24) (2.53) (3.06) (2.49) (2.67) (0.867) (1.34) (-1.61) (2.53) (-1.8)

USA, UK, FRA, ITA, RUS

0.048 0.104*** 0.148*** 0.199*** 0.069** 0.13 0.13*** 0.182** 0.163*** 0.211*** 0.146*** 0.182** 0.089*** 0.129

(1.35) (3.5) (3.16) (2.92) (2.56) (1.63) (3.23) (2.34) (3.31) (2.74) (3.41) (2.43) (2.91) (1.35)

USA, UK, 0.054 0.105*** 0.133*** 0.186** 0.042** 0.119 0.099*** 0.171** 0.157*** 0.224*** 0.158*** 0.206** 0.084*** 0.133

145 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

FRA, ITA, JPN (1.4) (3) (3.37) (2.55) (2.23) (1.25) (3.26) (1.99) (3.66) (2.6) (3.69) (2.5) (3.09) (1.31)

USA, GER, ITA, JPN, RUS

-0.01 -0.02 0.193 0.186*** 0.071 0.127* 0.145 0.181*** 0.178 0.178** 0.156* 0.145* 0.09 0.099

(-1.45) (-1.37) (1.31) (2.75) (-0.17) (1.69) (0.561) (2.63) (1.35) (2.56) (1.72) (1.95) (0.457) (0.953)

USA, GER, FRA, JPN, RUS

0.019 0.04 0.167*** 0.149** 0.135*** 0.125 0.169*** 0.149** 0.161*** 0.173*** 0.164*** 0.152** 0.111*** 0.077

(0.153) (1.36) (4.24) (2.26) (4.03) (1.61) (4.33) (2.03) (4.06) (2.62) (4.15) (2.38) (3.71) (0.199)

USA, GER, FRA, ITA, RUS

0.038 0.051*** 0.090** 0.032 0.064* 0.037* 0.102** 0.059** 0.098** 0.054 0.079** 0.020 0.079** 0.027

(0.809) (2.65) (2.27) (1.54) (1.73) (1.65) (2.25) (1.98) (2.37) (1.53) (2.35) (0.306) (2.43) (1.12)

USA, GER, FRA, ITA, JPN

0.032 0.102*** 0.136*** 0.201** 0.038* 0.12 0.096*** 0.171 0.138*** 0.228** 0.161*** 0.216** 0.079*** 0.134

(0.193) (2.92) (3.28) (2.06) (1.76) (0.622) (2.89) (1.37) (3.33) (2.12) (3.6) (2.05) (2.83) (0.751)

USA, GER, UK, JPN, RUS

-0.02* -0.02 0.241** 0.212*** 0.119 0.145 0.194 0.195*** 0.195** 0.199*** 0.186** 0.166** 0.115 0.12

(-1.77) (-1.54) (2.28) (2.92) (0.654) (1.61) (1.47) (2.58) (1.96) (2.67) (2.49) (2.18) (1.03) (0.977)

USA, GER, UK, ITA, RUS

0.037 0.043 0.102** 0.043 0.082** 0.051 0.122*** 0.073 0.091** 0.054 0.083** 0.019 0.085** 0.029

(0.756) (1.36) (2.51) (1.28) (2.04) (1.31) (2.63) (1.56) (2.2) (1.03) (2.44) (0.12) (2.46) (0.738)

146 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, GER, UK, ITA, JPN

-0.00 -0.01 0.243*** 0.22*** 0.093 0.14 0.162* 0.191** 0.24*** 0.245*** 0.269*** 0.228*** 0.152* 0.147

(-1.5) (-0.98) (2.87) (2.8) (0.571) (1.44) (1.77) (2.39) (3.12) (2.96) (3.49) (2.85) (1.76) (1.57)

USA, GER, UK, FRA, JPN

-0.00* 0.059* 0.047* 0.021 0.003 0.020 0.036 0.039 0.038 0.045 0.048 0.026 0.042 0.037

(-1.78) (1.68) (1.67) (0.222) (0.456) (0.012) (1.27) (0.436) (0.983) (0.418) (1.62) (0.031) (1.54) (0.587)

USA, GER, UK, FRA, ITA

0.001 -0.00 0.149** 0.024 0.181*** 0.049** 0.173*** 0.053** 0.114** 0.014 0.094** -0.02 0.132*** 0.001

(-1.32) (-0.40) (2.47) (1.23) (3.13) (1.96) (2.72) (1.99) (2.22) (0.892) (2.42) (-1.24) (3.18) (-0.17)

USA, CAN, ITA, JPN, RUS

0.046 0.098** 0.135*** 0.2** 0.036* 0.121 0.095*** 0.172* 0.146*** 0.237*** 0.163*** 0.223** 0.078*** 0.121

(1.21) (2.49) (3.62) (2.57) (1.94) (1.26) (3.23) (1.95) (3.7) (2.58) (3.99) (2.52) (3.17) (0.97)

USA, CAN, FRA, JPN, RUS

-0.01 -0.02 0.202 0.194*** 0.081 0.126 0.161 0.187*** 0.168 0.173** 0.145* 0.134 0.071 0.066

(-1.41) (-1.5) (1.51) (2.76) (-0.04) (1.58) (0.744) (2.65) (1.26) (2.31) (1.66) (1.63) (0.31) (0.032)

USA, CAN, FRA, ITA, RUS

0.040 0.063** 0.099** 0.055** 0.073* 0.061** 0.12** 0.082** 0.082** 0.075 0.066** 0.039 0.064** 0.037

(1.02) (2) (2.01) (2.13) (1.79) (2.27) (2.14) (2.53) (1.98) (1.59) (2.03) (0.585) (2.12) (0.678)

USA, CAN, -0.04* -0.04 0.201** 0.147*** 0.143 0.119 0.179 0.144** 0.162*** 0.135** 0.124** 0.094 0.081 0.071

147 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

FRA, ITA, JPN (-1.82) (-1.41) (2.09) (2.6) (1.29) (1.53) (1.56) (2.22) (2.61) (2.37) (2.5) (1.57) (1.26) (0.358)

USA, CAN, UK, JPN, RUS

0.006 -0.02 0.199** 0.214*** 0.054 0.129* 0.134 0.19*** 0.199* 0.242*** 0.214** 0.217*** 0.092 0.099

(-0.59) (-1.24) (2.26) (2.85) (-0.25) (1.83) (0.957) (2.77) (1.89) (2.99) (2.54) (2.84) (0.773) (0.657)

USA, CAN, UK, ITA, RUS

0.010 0.044 0.050* 0.033 0.005 0.028 0.046* 0.046 0.039 0.060 0.043* 0.039 0.027 0.026

(-0.13) (1.43) (1.92) (1.37) (0.926) (0.632) (1.9) (1.44) (1.22) (1.2) (1.67) (0.818) (1.47) (0.319)

USA, CAN, UK, ITA, JPN

-0.01 -0.01 0.127 0.040** 0.143 0.066*** 0.15 0.075*** 0.094 0.028 0.068 -0.01 0.085* -0.00

(-1.61) (-0.93) (1.27) (2.13) (1.24) (2.89) (1) (2.79) (0.694) (1.37) (1.06) (-1.21) (1.81) (-1.38)

USA, CAN, UK, FRA, RUS

0.009 -0.02 0.224** 0.218*** 0.064 0.128 0.13 0.179** 0.215*** 0.248*** 0.272*** 0.235*** 0.149* 0.125

(-0.73) (-1.57) (2.52) (2.65) (-0.13) (1) (1) (2.06) (2.65) (2.88) (3.59) (2.81) (1.91) (0.881)

USA, CAN, UK, FRA, ITA

-0.00 0.047 0.034 0.039 -0.01 0.031 0.018 0.043 0.016 0.066 0.037 0.058 0.030 0.066

(-1.51) (1.34) (1.27) (0.718) (-0.09) (0.168) (0.634) (0.463) (0.192) (0.752) (0.993) (0.811) (0.972) (0.941)

USA, CAN, GER, JPN, RUS

0.019 0.002 0.146** 0.049* 0.176*** 0.075*** 0.173** 0.080** 0.094 0.032 0.081** -0.00 0.114*** 0.002

(-0.46) (-0.39) (2.03) (1.93) (2.84) (2.59) (2.46) (2.55) (1.37) (0.875) (2.19) (-1.35) (3.49) (-1.16)

148 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, ITA, RUS

-0.02 -0.00 0.073** 0.024 0.085*** 0.038 0.074** 0.034 0.058** 0.041 0.083** 0.024 0.115*** 0.031

(-1.19) (-0.60) (2.26) (0.864) (2.59) (1.43) (2.4) (1.49) (2.03) (1.25) (2.43) (0.521) (3.45) (0.107)

USA, CAN, GER, ITA, JPN

0.014 0.013 0.124*** 0.009 0.147*** 0.055 0.166*** 0.051 0.104*** 0.037 0.088*** 0.004 0.095*** 0.013

(0.819) (-0.40) (2.75) (0.271) (2.86) (1.63) (2.98) (1.38) (2.72) (1.13) (2.63) (-0.19) (2.64) (0.291)

USA, CAN, GER, FRA, RUS

0.023 0.080*** 0.141*** 0.188*** 0.055** 0.124* 0.097*** 0.161** 0.111*** 0.171** 0.126*** 0.158** 0.102*** 0.132

(0.814) (3.09) (3.15) (2.9) (2.05) (1.72) (2.69) (2.31) (3.31) (2.45) (3.46) (2.19) (3.02) (1.4)

USA, CAN, GER, FRA, JPN

0.015 0.069** 0.169*** 0.203*** 0.087*** 0.127 0.135*** 0.167** 0.132*** 0.188*** 0.145*** 0.172** 0.116*** 0.137

(0.269) (2.48) (3.51) (2.87) (2.63) (1.51) (3.21) (2.16) (3.44) (2.69) (3.61) (2.49) (3.21) (1.36)

USA, CAN, GER, FRA, ITA

0.023 0.074** 0.158*** 0.195*** 0.057*** 0.117 0.101*** 0.156* 0.137*** 0.203** 0.167*** 0.201** 0.113*** 0.141

(-0.52) (2.32) (3.81) (2.58) (2.68) (1.14) (3.65) (1.85) (3.64) (2.56) (3.93) (2.53) (3.63) (1.31)

GER, UK, FRA, ITA, JPN, RUS

-0.00 0.029 0.168*** 0.146** 0.15*** 0.133* 0.181*** 0.155** 0.155*** 0.162** 0.141*** 0.13* 0.114*** 0.087

(0.049) (0.628) (3.89) (2.4) (3.7) (1.88) (3.87) (2.24) (4.07) (2.42) (4.02) (1.91) (3.73) (0.413)

USA, GER, 0.058 0.096*** 0.162*** 0.159** 0.143*** 0.13 0.206*** 0.176** 0.208*** 0.208** 0.149*** 0.148 0.093*** 0.085

149 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, FRA, ITA, RUS (1.52) (2.75) (3.78) (2.24) (3.54) (1.56) (3.89) (2.11) (4.23) (2.28) (3.92) (1.57) (3.37) (0.305)

USA, CAN, GER, FRA, ITA, JPN

0.085 0.107** 0.174*** 0.168** 0.133*** 0.135 0.197*** 0.188** 0.237*** 0.238** 0.191*** 0.176** 0.101*** 0.079

(1.54) (2.47) (4.08) (2.1) (3.83) (1.47) (4.31) (2.11) (4.28) (2.52) (4.08) (2.05) (3.38) (0.17)

USA, CAN, GER, UK, JPN, RUS

-0.01 -0.02 0.19 0.162** 0.124 0.131 0.192 0.188** 0.201** 0.186** 0.121 0.122 0.047 0.050

(-1.54) (-1.33) (1.54) (2.22) (0.549) (1.37) (1.2) (2.48) (2.32) (2.22) (1.17) (1.21) (0.089) (-0.54)

USA, CAN, GER, UK, ITA, RUS

0.069* 0.077** 0.115** 0.015 0.137*** 0.064* 0.182*** 0.087* 0.152*** 0.082 0.091** 0.020 0.071** 0.003

(1.85) (2.24) (2.49) (0.3) (2.62) (1.7) (2.88) (1.9) (2.89) (1.56) (2.52) (-0.31) (2.3) (-0.59)

USA, CAN, GER, UK, ITA, JPN

0.035 0.096*** 0.141*** 0.183*** 0.068** 0.129* 0.125*** 0.177** 0.163*** 0.203** 0.142*** 0.171** 0.091*** 0.127

(1.1) (3.39) (3.11) (2.7) (2.25) (1.66) (2.91) (2.29) (3.48) (2.55) (3.48) (2.18) (2.94) (1.3)

USA, CAN, GER, UK, FRA, RUS

0.030 0.084*** 0.17*** 0.192*** 0.098*** 0.13 0.159*** 0.177** 0.173*** 0.207*** 0.164*** 0.176** 0.111*** 0.13

(0.508) (2.75) (3.46) (2.74) (2.84) (1.55) (3.42) (2.18) (3.57) (2.68) (3.66) (2.41) (3.17) (1.29)

USA, CAN, GER, UK, FRA, JPN

0.040 0.099*** 0.165*** 0.184** 0.070*** 0.122 0.127*** 0.169* 0.179*** 0.22** 0.188*** 0.199** 0.114*** 0.137

(-0.00) (2.62) (3.67) (2.39) (2.86) (1.18) (3.77) (1.87) (3.92) (2.47) (3.95) (2.36) (3.48) (1.28)

150 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, GER, UK, FRA, ITA

-0.02* -0.02 0.237** 0.198*** 0.113 0.138* 0.18 0.185*** 0.211** 0.186*** 0.192** 0.152** 0.127 0.114

(-1.85) (-1.58) (2.14) (2.86) (0.495) (1.7) (1.27) (2.58) (2.18) (2.64) (2.57) (2.12) (1.15) (1.11)

CAN, UK, FRA, ITA, JPN, RUS

0.032 0.053* 0.103** 0.031 0.081* 0.041 0.115** 0.059 0.103** 0.052 0.088** 0.016 0.093*** 0.032

(0.187) (1.69) (2.5) (1.05) (1.92) (1.16) (2.42) (1.44) (2.45) (1.08) (2.56) (-0.06) (2.62) (0.825)

CAN, GER, FRA, ITA, JPN, RUS

0.040 0.102*** 0.154*** 0.199*** 0.076** 0.139* 0.142*** 0.189** 0.15*** 0.207** 0.133*** 0.17* 0.078*** 0.106

(1.3) (3.01) (3.13) (2.71) (2.52) (1.76) (3.04) (2.39) (3.48) (2.36) (3.56) (1.89) (2.95) (0.617)

CAN, GER, UK, ITA, JPN, RUS

0.009 0.067** 0.169*** 0.196*** 0.087** 0.13 0.129*** 0.166** 0.142*** 0.19*** 0.152*** 0.169** 0.125*** 0.14

(-0.23) (2.16) (3.44) (2.78) (2.52) (1.61) (3.05) (2.19) (3.5) (2.58) (3.61) (2.32) (3.28) (1.46)

CAN, GER, UK, FRA, JPN, RUS

0.060 0.097** 0.151*** 0.204*** 0.055*** 0.134* 0.12*** 0.187** 0.159*** 0.235*** 0.162*** 0.206** 0.078*** 0.105

(1.52) (2.56) (3.57) (2.74) (2.7) (1.73) (3.9) (2.45) (3.91) (2.75) (3.95) (2.57) (3.2) (0.694)

CAN, GER, UK, FRA, ITA, RUS

-0.02* -0.02 0.2 0.195*** 0.082 0.135** 0.159 0.195*** 0.173 0.18** 0.146 0.139* 0.073 0.071

(-1.73) (-1.59) (1.44) (2.81) (-0.10) (2.15) (0.626) (2.93) (1.19) (2.44) (1.55) (1.81) (0.204) (0.223)

CAN, GER, 0.037 0.049* 0.101** 0.044** 0.078* 0.056** 0.123** 0.074** 0.092** 0.066 0.070** 0.028 0.065** 0.022

151 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

UK, FRA, ITA, JPN (1.05) (1.81) (2.14) (1.98) (1.88) (2.32) (2.26) (2.56) (2.32) (1.57) (2.22) (0.348) (2.2) (0.187)

USA, UK, FRA, ITA, JPN, RUS

0.042 0.089** 0.156*** 0.2** 0.054** 0.125 0.111*** 0.17* 0.158*** 0.235** 0.18*** 0.217** 0.101*** 0.126

(0.262) (2.19) (3.76) (2.48) (2.2) (1.19) (3.42) (1.86) (3.73) (2.47) (4.09) (2.41) (3.46) (0.931)

USA, GER, FRA, ITA, JPN, RUS

-0.02* -0.02 0.242** 0.209*** 0.124 0.142 0.195 0.195*** 0.192* 0.185** 0.179** 0.145* 0.113 0.087

(-1.71) (-1.6) (2.14) (2.78) (0.581) (1.61) (1.36) (2.61) (1.78) (2.39) (2.43) (1.82) (1.19) (0.267)

USA, GER, UK, ITA, JPN, RUS

0.038 0.056 0.11** 0.057 0.089** 0.064 0.133** 0.081* 0.084** 0.074 0.074** 0.037 0.078** 0.043

(0.915) (1.25) (2.42) (1.64) (2.05) (1.64) (2.5) (1.89) (2.15) (1.3) (2.35) (0.381) (2.4) (0.707)

USA, GER, UK, FRA, JPN, RUS

-0.00 -0.02 0.246*** 0.225*** 0.097 0.141* 0.17* 0.197*** 0.239*** 0.247*** 0.264*** 0.22*** 0.141** 0.11

(-1.54) (-1.47) (2.92) (2.83) (0.489) (1.77) (1.75) (2.69) (3.16) (2.91) (3.68) (2.77) (1.98) (0.611)

USA, GER, UK, FRA, ITA, JPN

0.001 0.048 0.064** 0.032 0.021 0.033 0.060* 0.047 0.043 0.055 0.053* 0.034 0.044* 0.030

(-1.37) (0.835) (2.02) (0.61) (1.13) (0.276) (1.96) (0.724) (1.09) (0.572) (1.72) (0.12) (1.65) (0.043)

USA, CAN, FRA, ITA, JPN, RUS

-0.00 0.006 0.146** 0.041* 0.178*** 0.069*** 0.171** 0.073*** 0.108** 0.027 0.083** -0.01 0.117*** -0.00

(-1.26) (-0.36) (2.2) (1.87) (2.91) (2.68) (2.43) (2.59) (2) (0.796) (2.33) (-1.52) (3.55) (-1.5)

152 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

USA, CAN, UK, ITA, JPN, RUS

0.018 0.077** 0.15*** 0.199*** 0.065** 0.138** 0.116*** 0.177*** 0.103*** 0.179*** 0.115*** 0.158** 0.079*** 0.105

(0.753) (2.4) (3.11) (3.04) (2.28) (2.36) (2.84) (2.93) (3.66) (2.71) (3.66) (2.32) (2.96) (0.692)

USA, CAN, UK, FRA, JPN, RUS

0.013 0.065** 0.176*** 0.206*** 0.095*** 0.134* 0.146*** 0.172** 0.125*** 0.189** 0.137*** 0.158** 0.107*** 0.114

(0.319) (2.04) (3.54) (2.84) (2.72) (1.75) (3.21) (2.41) (3.67) (2.44) (3.78) (1.98) (3.38) (0.658)

USA, CAN, UK, FRA, ITA, RUS

0.030 0.069* 0.176*** 0.21*** 0.070*** 0.13* 0.123*** 0.172** 0.142*** 0.213*** 0.169*** 0.203*** 0.104*** 0.112

(-0.05) (1.9) (4.06) (2.87) (3.24) (1.73) (4.26) (2.52) (3.94) (2.78) (4.19) (2.67) (3.79) (0.628)

USA, CAN, UK, FRA, ITA, JPN

-0.03** -0.02 0.239*** 0.23*** 0.115 0.156** 0.172* 0.203** 0.187*** 0.188*** 0.165*** 0.157** 0.113 0.112

(-1.99) (-0.53) (2.71) (2.72) (0.728) (2.05) (1.66) (2.49) (3.17) (2.59) (3.08) (2.34) (1.43) (1.31)

USA, CAN, GER, ITA, JPN, RUS

0.017 0.035 0.117** 0.050 0.089** 0.053 0.122** 0.060* 0.066** 0.045 0.069** 0.021 0.085** 0.034

(0.331) (0.37) (2.45) (1.58) (2.03) (1.46) (2.43) (1.77) (2.24) (0.919) (2.36) (0.019) (2.36) (0.316)

USA, CAN, GER, FRA, JPN, RUS

0.048 0.089*** 0.151*** 0.14** 0.132*** 0.119 0.185*** 0.161** 0.222*** 0.201** 0.161*** 0.146* 0.102*** 0.098

(1.02) (2.81) (3.59) (2.17) (3.36) (1.39) (3.72) (1.97) (4.07) (2.43) (3.82) (1.69) (3.32) (0.668)

USA, CAN, 0.061** 0.096*** 0.128*** 0.15** 0.112*** 0.137* 0.179*** 0.189** 0.2*** 0.208** 0.126*** 0.146 0.056*** 0.070

153 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

GER, FRA, ITA, RUS (1.99) (2.82) (3.4) (2.17) (3.2) (1.76) (3.63) (2.36) (4.2) (2.35) (3.69) (1.57) (2.63) (0.137)

USA, CAN, GER, UK, FRA, ITA, JPN

0.034 0.051 0.116** 0.043 0.096** 0.058 0.137** 0.073* 0.098** 0.062 0.080** 0.023 0.083** 0.027

(0.707) (1.06) (2.42) (1.43) (2.1) (1.6) (2.5) (1.89) (2.45) (1.17) (2.46) (-0.01) (2.44) (0.192)

USA, CAN, GER, UK, FRA, ITA, RUS

-0.03** -0.03* 0.242** 0.207*** 0.121 0.145** 0.19 0.198*** 0.206** 0.188** 0.182** 0.146** 0.113 0.085

(-1.99) (-1.85) (2.11) (2.83) (0.477) (2.12) (1.24) (2.87) (2.05) (2.51) (2.55) (1.96) (1.16) (0.377)

USA, CAN, GER, UK, FRA, JPN, RUS

0.049 0.091** 0.185*** 0.203*** 0.083*** 0.136* 0.148*** 0.186** 0.182*** 0.232*** 0.192*** 0.202** 0.111*** 0.111

(0.373) (2.19) (3.83) (2.66) (3.34) (1.65) (4.32) (2.37) (4.16) (2.65) (4.17) (2.47) (3.67) (0.699)

USA, CAN, GER, UK, ITA, JPN, RUS

0.031 0.085** 0.18*** 0.199*** 0.105*** 0.14* 0.169*** 0.185** 0.166*** 0.209** 0.155*** 0.17** 0.106*** 0.114

(0.668) (2.33) (3.5) (2.66) (2.88) (1.73) (3.36) (2.31) (3.81) (2.39) (3.86) (1.99) (3.37) (0.762)

USA, CAN, GER, FRA, ITA, JPN,

0.034 0.092*** 0.155*** 0.2*** 0.079** 0.146** 0.144*** 0.197*** 0.156*** 0.213*** 0.134*** 0.172** 0.077*** 0.103

(1.14) (2.77) (3.13) (2.73) (2.47) (2.06) (3.02) (2.64) (3.76) (2.59) (3.72) (2.16) (2.94) (0.701)

154 Appendix C: Efficiency Gains and Diebold Mariano Test Results (vs. DCC-CARR benchmark), by Portfolio

Stage 1 𝐷𝑡 = GARCH CARR

CARR-M IDAS

Q = 63 (3-months)

CARR-M IDAS

Q = 126 (6-months)

CARR-M IDAS

Q = 252 (1-year)

CARR-M IDAS

Q = 756 (3-years)

CARR-M IDAS

Q = 1260 (5-years)

Stage 2 𝑅𝑡 = DCC DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS DCC-M IDAS

(IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS) (IS) (OOS)

RUS

USA, CAN, UK, FRA, ITA, JPN, RUS

0.025 0.082*** 0.171*** 0.181*** 0.095*** 0.129 0.15*** 0.172** 0.185*** 0.202** 0.169*** 0.168** 0.119*** 0.131

(0.166) (2.58) (3.42) (2.58) (2.71) (1.57) (3.25) (2.14) (3.74) (2.52) (3.71) (2.21) (3.29) (1.35)

USA, GER, UK, FRA, ITA, JPN, RUS

0.056 0.087** 0.171*** 0.158** 0.146*** 0.137* 0.209*** 0.184** 0.224*** 0.21** 0.158*** 0.147* 0.095*** 0.080

(1.34) (2.54) (3.76) (2.2) (3.51) (1.7) (3.84) (2.27) (4.48) (2.4) (4.17) (1.67) (3.47) (0.227)

CAN, GER, UK, FRA, ITA, JPN, RUS

0.009 0.066* 0.182*** 0.21*** 0.097*** 0.146** 0.148*** 0.186*** 0.135*** 0.196*** 0.142*** 0.17** 0.108*** 0.115

(-0.04) (1.88) (3.47) (2.94) (2.72) (2.2) (3.19) (2.78) (3.92) (2.74) (3.89) (2.4) (3.44) (0.818)

USA, CAN, GER, UK, FRA, ITA, JPN, RUS

0.037 0.084*** 0.186*** 0.199*** 0.106*** 0.146** 0.17*** 0.192** 0.135*** 0.186** 0.16*** 0.169** 0.107*** 0.108

(0.477) (2.8) (3.42) (2.68) (2.87) (1.97) (3.32) (2.52) (3.9) (2.43) (3.96) (2.17) (3.43) (0.765)

155 Appendix D - MATLAB code

Note: Some of the scripts and functions presented here have been modified, after use, due to space constraints or for presentation. Furthermore, many commands herein reference files and directories on the author’s computer. The code provided in this appendix is included only to serve as a basic guide towards reproducing the results of this study. It has not been optimized for speed or memory efficiency, nor is it being presented as a ‘turn-key’ package for public use. The scripts and sub-functions presented in this appendix include: 1) script_1_G8_dataImportProcess.m (pg. 156)

sub-functions: -func_procLowFreq2 -func_procRealizedRange4

2) script_2_G8_enumerateAll.m (pg. 164)

sub-functions: -func_G8_reEnumerate

3) script_3_G8_summaryStats.m (pg. 167) 4) script_4_G8_mvpAllCombs.m (pg. 170)

sub-functions: - func_G8_mvpThisComb - func_S1_estForc - func_S2_estForc

5) script_5_G8_testTables (pg. 180)

6) script_6_G8_compareTables (pg. 183)

sub-functions: - func_G8_compareEstimators

Omitted functions include: - carrM idas - modification of ‘garchMidas.m’ function from ‘MIDAS MATLAB Toolbox’ v2.2 - dccM idas_S2 - modification of ‘dccMidas.m’ function from ‘MIDAS MATLAB Toolbox’ v 2.2 - dccM od - modification of ‘dcc.m’ function from ‘Oxford MFE Toolbox’

See paper for details. Omitted functions are available upon request. stephenswanson@trentu.ca

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