DEPARTMENT OF ECONOMICS

ISSN 1441-5429

DISCUSSION PAPER 06/15

The Financial Econometrics of Price Discovery and Predictability

Seema Narayan and Russell Smyth*

Abstract: This paper reviews recent econometric developments in the literature on price discovery and

price predictability. For both areas, we discuss traditional approaches to econometric

modeling, limitations to these approaches, and recent developments designed to overcome

them. We also discuss the state-of-the-art and suggest future research. Three main

conclusions are drawn. First, while many recent empirical applications in price discovery and

price predictability are on the frontier of econometric methods, further developments are

needed to increase relaxation of relevant assumptions and push the boundaries of

applications. Second, future research in econometric modelling needs to combine/synthesize

recent developments across multiple econometric issues rather than proceeding in a

piecemeal manner, for instance, by integrating developments in the time series literature into

panel-based frameworks. Third, recent econometric literature is generating findings that

challenge long-held beliefs about apparent empirical regularities in price discovery and price

predictability, thus presenting opportunities to develop relevant theory.

JEL Classification Numbers: C53, C58, G12, G13, G14

We thank Paresh Narayan for helpful suggestions on an earlier version of this paper.

© 2015 Seema Narayan and Russell Smyth

All rights reserved. No part of this paper may be reproduced in any form, or stored in a retrieval system, without the prior

written permission of the author

monash.edu/ business-economics

ABN 12 377 614 012 CRICOS Provider No. 00008C

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1. Introduction

The primary motive for investing in domestic or international stocks is to maximize

returns. This, however, encompasses risk, as both return and risk are conceptually tied

to the price of the stock. Given this, an understanding of price formation is

fundamental to achieving higher returns and lowering risks. It is therefore not

surprising that scholars and practitioners in the field of financial economics are

engaged in research that deals with the ‘what’ and ‘how’ of current and future price

formation. The following survey examines the role of econometrics in this venture,

and broadly covers financial econometric developments in the areas of price

discovery and price predictability.

The main reason we focus on price discovery and price predictability together is that

conceptually these two areas of financial economics are related. Dolatabadi et al.

(2014) show that if one can ascertain that in a two-market (say X and Y) setup,

market X dominates market Y in the price discovery process, the implication is that

one can use market X to predict (and also forecast) market Y. To put it differently, the

market with more information can be used to forecast the market with less

information. Price discovery is a first step because it tells us that market X has more

information content than market Y because the former dominates the price discovery

process.

A further reason for considering price discovery and price predictability in the same

review is that not only are these fields of financial economics conceptually related,

but in many respects both areas are informed by similar time series econometric

developments, particularly with respect to integration and cointegration. For example,

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both areas have been informed by developments in unit root and cointegration testing

designed to derive added power, such as the use of panel-based models. Similarly,

both areas have been advanced by ongoing econometric developments designed to

address non-linearities and structural change in data.

The catalyst for many recent econometric developments in both areas is the

theoretical insights of Figuerola-Ferretti and Gonzala (2010). Building on the

components share (CS) methods and a simultaneous price dynamics model proposed

by Garbade and Silber (1983), Figuerola-Ferretti and Gonzala (2010) propose a

theoretical model of price discovery characterized by finite elasticity of supply of

arbitrage services and endogenous convenience yields. Their model demonstrates a

perfect link between an extended Garbade and Silber (1983) framework and the CS

approach, suggesting that the cointegrated vector autoregressive (CVAR) model

(Johansen, 1995) is the natural empirical model to test for price discovery.

This insight is subsequently extended by Dolatabadi et al. (2014a, 2015) to develop a

fractionally cointegrated vector autoregressive (FCVAR) model to examine price

discovery, with the authors showing how CVAR and FCVAR links price discovery

and price predictability. Thus, econometrics developments in both areas are starting to

come together, and it is likely that this convergence will continue in the future.

Consequently, given that there is a theoretical basis to using CVAR/FCVAR to test

for price discovery and price predictability based on Figuerola-Ferretti and Gonzala

(2010) and extensions of their theoretical model, it makes sense to focus on the

evolution of unit root/cointegration testing in both literatures and how these are

linked.

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It should be noted that in chronicling the development in financial econometrics in the

period 1980-2000, Bollerslev (2001) argues that time-varying volatility models (such

as ARCH/GARCH and stochastic volatility formulations) and robust methods-of-

moments-based estimations procedures (such as Generalized Methods of Moments)

stand out as milestones. We do not review these important developments, however, as

there are already several broad textbook treatments (see e.g. Campbell et al., 1997) as

well as surveys on ARCH/GARCH and stochastic volatility/realized volatility (see

Bollerslev et. al., 1992, 1994; Engle, 1995; Ghysels et al., 1996; Shephard, 1996;

Barndorff-Nielsen and Shephard, 2014; Bollerslev, 2010; see also Bollerslev and

Hodrick, 1995; Pagan, 1996; Bollerslev, 2001).

The structure of the survey is as follows. For both topics the main areas covered will

be symmetric, namely: motivation and implications; overview of existing studies;

state of the art econometric techniques; and suggestions for future research.

2. Price Discovery

2.1 Motivation and implications

Price discovery refers to the means through which information is transformed into

prices. The price discovery literature centers on the speed at which related markets

reach the equilibrium asset price, with the market in which new information is

reflected most rapidly considered to be the one with the price discovery function.

There are multiple factors motivating studies of price discovery in financial markets.

Improved understanding of price discovery not only assists investors to make better-

informed decisions, but because it involves knowing how information translates into

price across closely related markets, it facilitates risk management (Westerlund et al.,

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2014). Improved understanding of price discovery also facilitates more efficient asset

evaluation. Evidence from experiments suggests that trading the same asset across

two markets simultaneously increases the speed at which efficient prices are reached

(Forsythe et al., 1982, 1984). Given this, better understanding of the transmission

mechanism of information across markets, and the rate at which information is

reflected in prices, can provide evidence on the relative efficiency of markets.

Studying price discovery also provides insights into how markets incorporate new

information about asset values, as well as the relative importance of fundamentals and

speculation in this process. For example, the lead-lag relationship between spot and

future markets has implications for evaluating the role of speculation in driving

commodity prices. If price discovery occurs in the futures market, this suggests that

speculation is driving commodity prices; however, if price discovery occurs in the

spot market, the implication is that changes in market fundamentals are driving

demand and supply (hence prices) for the commodity (Kaufmann and Ullman, 2009).

Price discovery also has implications for the preferred markets in which informed

traders invest. If price discovery occurs in futures markets, this implies that informed

traders might prefer to trade in futures markets rather than spot markets.

2.2 Types of price discovery in financial markets

There are at least three related sets of studies in the price discovery literature. One set

of studies estimates the information share of each market for fragmented securities

traded across multiple markets (see e.g. Eun and Subherwal, 2003; Harris et al., 1995,

2002; Hasbrouck, 1995). There has been a surge in the number of studies of this sort,

which at least in part reflects growing market fragmentation (De Jong and Schotman,

2010; Westerlund et al., 2014). The principal finding from these studies is that price

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discovery predominantly occurs in the home market, and that prices in the foreign

market adjust to the home market. Notwithstanding, evidence exists that the relative

importance of home and foreign market may change over time, particularly if there

have been financial crises (Caporale et al., 2014; Fernandes and Scherrer, 2014).

A second set of studies estimates the information share of each market across

interconnected markets such as spot and futures markets (see e.g. Stein, 1961;

Hasbrouck, 1995; Gonzalo and Granger, 1995). While findings vary across

commodities, most studies find that price discovery occurs in the futures market.

There is, however, slightly more evidence of price discovery in the spot market using

a fractionally cointegrated model (Dolatabadi et al., 2014, 2015).

A third set of studies estimates the information share across markets for a particular

product. An example is Westerlund et al.’s (2014) study that investigates which

country leads the price discovery process for crude oil. These authors examine oil

prices in the United Kingdom, United States and Oman and find that price discovery

occurs in Oman, reflecting the fact that it is the dominant oil producing market.

2.3 Econometric issues in modeling price discovery

The econometric approach to modeling price discovery involves three steps. The first

is to ascertain the order of integration of asset prices as a precursor to examining

whether there is a long-run relationship between the markets. If two markets are

theoretically linked, it follows that any shock in the form of new information will

affect both markets. The second step is to establish if prices across the markets are

cointegrated. If prices are cointegrated, the question becomes which of the

cointegrated markets is the first to absorb the new information. The third step is to

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determine in which market price discovery occurs. This is done either through

examining the error correction coefficient to ascertain the relative speed of adjustment

to the long-run equilibrium price (Gonzalo and Granger, 1995), or estimating the

information shares from the error correction model (Hasbrouck, 1995).

2.3.1 Static measures of price discovery

The two main static approaches to measuring price discovery are the information

share (IS) method (Hasbrouck 1995) and the component share (CS) method that relies

on a permanent-transitory decomposition of the cointegrated system (Gonzalo and

Granger, 1995; Harris et al., 1995, 2002). Both approaches are reliant on the vector

error correction model (VECM) proposed by Engle and Granger (1987). The

approaches differ, however, in that Gonzalo and Granger (1995), and Harris et al.

(1995, 2002) focus on the proportion of each market’s innovation that contributes to

the common efficient price, while Hasbrouck (1995) focuses on the proportion of the

variance in the common efficient price that can be attributed to individual markets.

An alternative way of seeing the difference in the approaches is that Gonzalo and

Granger (1995), and Harris et al. (1995, 2002) define price discovery in terms of the

speed of adjustment to the long-run equilibrium based on the error correction

mechanism, while Hasbrouck (1995) defines price discovery in terms of the

information share.

2.3.2 Limitations of the CS and IS methods

The main limitation of both the CS and IS methods is the very fact that they are static

approaches to price discovery; as such, they are based on market innovations in their

reduced form (Scherrer, 2012). However, as price discovery is inherently a dynamic

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process in which shocks to prices are likely to be correlated, neither of these static

methodologies is able to address the price dynamics involved (Lehmann, 2012). The

only way to properly address the price dynamics involved in price discovery is to

adopt a structural measure of price discovery in which changes in structural

innovations can be appropriately attributed across markets (Scherrer, 2012; Yan and

Zivot, 2010).

A second problem is that the CS approach to price discovery is premised on

adjustment to the long-run equilibrium price being continuous and linear. For

example, this is implicit in the assumption made by Harris et al. (1995, 2002).

However, there are several reasons to doubt that adjustment is continuous and linear

due to market frictions in the presence of policy interventions and transaction costs.

Clearly, market frictions can be explicit or implicit (Gagnon and Karolyi, 2010).

Examples of potential explicit market frictions impeding linear adjustment include

holding costs (e.g. lending fees, unrealized interest on short-sales proceeds and

opportunity cost of capital), short-sale restrictions, taxation and transaction costs (e.g.

bid-ask spreads on cross-listed pairs, and foreign exchange and commission) (Chen et

al., 2013). Implicit market frictions can be due to cross-border differential adverse

selection risks and agency walls between arbitrageur and client (Shleifer and Vishney,

1997). Further, if speculators are active in the spot market-futures market, their

trading may destabilize the market and generate a potential non-linear effect of

hedging relative to speculation in the futures market’s share of price discovery

(Chang et al., 2013).

A third limitation of the CS model relates to the time-varying nature of price

discovery. Several studies in related branches such as cointegration show that markets

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are characterized by time-varying variance and co-variance. Hence, this time-varying

characteristic should have implications for price discovery. What this means is that

there may be phases of time over which market X will dominate price discovery,

while in other phases it may well be that market Y will dominate price discovery.

This will particularly be the case for markets such as oil prices that are prone to

volatility (Bessembinder et al., 1995). This is problematic, for example, when

examining the price adjustment process in futures markets, in which switches under

different market conditions can be expected to induce shifts in the adjustment process

to restore the cost-of-carry relationship (Brenner and Kroner, 1995; Caporale et al.,

2014).

A fourth limitation with both the CS and IS approaches is that neither allows for the

possibility of fractional cointegration. In the context of spot and futures markets,

several studies find evidence of fractional integration in the forward premium (see

e.g. Baillie and Bollerslev, 1994; Maynard and Phillips, 2001). Hence, fractional

integration in the forward premium implies that spot and futures prices are

fractionally cointegrated. The reason for this is that although the spot and futures

prices themselves may be integrated of order 1 (I(1), the spread between spot and

futures prices will be fractionally integrated of a lower order (Doltabadi et al., 2014).

A fifth limitation of the original CS and IS frameworks is that the microstructure

models are limited to two markets. This means that the frameworks do not have the

ability to allow for price discovery across multiple markets simultaneously. It also

means that, as with any time series measure, that power of the unit root and

cointegration tests are limited by the time series and fail to exploit the additional

power in a panel.

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The final limitation is the econometric challenge in estimating price discovery. The

measures are not easy to estimate in practice, however, as they very technical, involve

several steps, and are econometrically burdensome. In other words, unlike tests for

cointegration which are built into standard software packages and for which many

codes are available, the same is not true for price discovery estimators. This thereby

limits the application of these tests despite their merits.

2.3.3. Extensions of the CS and IS methods

The CS and IS methods have been extended in various directions to address each of

these limitations. First, the fact that the CS and IS frameworks are unable to capture

price dynamics has led to the development of structural models of price discovery that

allow for changes in structural innovations to be correctly assigned to markets (see

e.g. Yan and Zivot, 2010, 2010a; Kim, 2010; Scherrer, 2012). Yan and Zivot (2010)

propose the first two-variable structural model of price discovery in which they

consider only one common factor. The authors apply their model to the Euro/Yen

exchange rate and find that price discovery in this market occurs through the US

dollar market. Kim (2010) expands the Yan and Zivot (2010) framework to a three-

variable model, sharing two common variables or one cointegrating vector.

Scherrer (2012) presents a structural model of price discovery for a seven-variable

model. She first uses the Gonzalo and Granger (1995), and Gonzalo and Ng (2001)

method to retrieve permanent and transitory innovations from market residuals in

their reduced form. The author then modifies Gonzalo and Ng’s (2001) method to

allow the variance of the structural innovations to differ from the identity matrix,

which works well with up to three common factors. One advantage of Scherrer’s

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(2012) approach compared with those of Kim (2010), and Yan and Zivot (2010) is

that the innovations attached to each of the three common factors are allowed to

contemporaneously correlate. Another advantage is that Scherrer (2012) decomposes

the covariance matrix of the reduced innovations in such a manner that common

factors may impact all prices in the long run. The author applies her method to

examine price discovery for Brazilian firms cross-listed in Brazil and the United

States using high frequency data. She finds that price discovery occurs on the

Brazilian stock exchange in the short-run, but that in the long run both exchanges are

equally important. Scherrer (2012) presents Monte Carlo evidence suggesting that her

structural model outperforms that proposed by Yan and Zivot (2010).

Second, the assumption of continuous and linear adjustment has been relaxed in

studies that develop threshold cointegration/error-correction models. Chen et al.

(2013) extends the CS method to consider the exchange-respective information shares

in Canadian firms cross-listed in Canada and the United States, while Chang et al.

(2013) builds on the IS method to examine the effect of non-linearities in the futures

market on price discovery in the foreign exchange futures market.

Chen et al. (2013) first use Balke and Fomby’s (1997) threshold cointegration model

and then, because dynamics may alternatively be gradual, generalize it to a smooth

transition version. Their information share measures are obtained from outer-regime

adjustment coefficients based on a two-regime threshold ECM and average

coefficient estimates based on a smooth transition ECM. Chen et al.’s (2013)

approach has the advantage in that it takes into account non-linearities associated with

both abrupt and smooth party changes due to market frictions. Their approach

captures the relative contributions to price discovery with a higher degree of precision

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compared with, for example, the ECM in Harris et al. (1995, 2002), which does not

address the time-and regime-contingent characteristics of information shocks.

In a related contribution, Chang et al. (2013) examine the effect of the relative size of

hedger and speculator open interests on price discovery in the foreign exchange

futures market. They first use the IS approach of Hasbrouck (1995) to ascertain the

contribution of the EUR-USD and JPY-USD futures markets to the futures/spot price

discovery process. With the IS for the futures market, they then estimate a logistic

smooth transition model to ascertain how price discovery depends on the trading

positions held by hedgers and speculators in the futures markets. Chang et al. (2013)

find a non-linear relationship between speculative trading volume and the efficacy of

trading volume, in which the effect of speculative trading volume on market

efficiency depends on its relation to endogenously determined thresholds.

Third, to accommodate structural change, models have been developed that allow for

time varying price dynamics. For example, Caporale et al. (2014) build on the CS

approach to take into account parameter instability in the price discovery process.

Specifically, in order to detect structural changes in the adjustment coefficients, they

estimate time-varying parameter models for the loading weights using a Kalman filter

approach. Kalman filtering has the advantage that it does not impose a priori

restrictions on the timing of structural breaks, making it useful to ascertain and

interpret the effect of any financial turmoil on price discovery. Caporale et al. (2014)

apply their model to price discovery in crude oil spot and futures. One interesting

conclusion to emerge from their study is that while futures markets play a more

important role in price discovery than spot markets, their relative contributions exhibit

high instability. Another interesting finding is that spot markets become more

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important for price discovery in times of financial crisis such as the Global Financial

Crisis.

Fourth, the assumption of a I(0)/I(1) distinction has been relaxed in studies that

examine price discovery using FCVAR. Dolatabadi et al. (2014a, 2015) extend

Figuerola-Ferretti and Gonzalo’s (2010) model to develop a fractionally cointegrated

equilibrium model, with Dolatabadi et al. (2015) using this new model as the

theoretical basis on which to apply FCVAR to price discovery in spot and futures

markets.

As outlined in Dolatabadi et al. (2014a, 2015), the main advantage of the FCVAR

model lies with its flexibility. It permits one to ascertain the cointegrating rank and to

jointly estimate the adjustment coefficients and cointegrating relations while

accounting for the short-run dynamics. Dolatabadi et al. (2015) find that generally the

fractional cointegration parameter is significant, suggesting that a non-fractional

model is inappropriate. The authors also find that, when allowing for fractional

integration in the long-run equilibrium relations, fewer lags are needed in the

autoregressive augmentation of the model, which underpins the utility of the

fractional model. Their main substantive finding is that when applying the fractional

model, the spot market is slightly more important in the price discovery process

relative to the futures market compared with the non-fractional model.

Fifth, the power limitations associated with short time series have been addressed by

the development of panel data models of price discovery. De Jong and Schotman

(2010) generalize the IS framework to multiple markets. In contrast to the traditional

information share that is defined within a reduced-form time series model, De Jong

and Schotman (2010) define the information share directly from the unobserved

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components model. This approach avoids the wide information share bounds one

sometimes gets when using the Hasbrouck (1995) VECM (see e.g. Covrig and

Melvin, 2002), particularly in settings in which there are multiple dealers and

markets.

More recently, Narayan et al. (2014) propose a panel model in which, conditional on

finding panel cointegration, a panel VECM is specified. The panel VECM is used to

extract the price discovery coefficients, thus extending the CS and IS frameworks.

The conventional panel VECM, such as that proposed in Narayan et al. (2014), has

the limitation that if one considers a large number of exchanges (prices) then the

model becomes over-parameterized. This comes at the cost of precision in estimation

of price discovery. Narayan et al. (2014) only have two price variables and so escape

this criticism. However, in most other literature the subject of price discovery

involves multiple exchanges, which inhibits the utility of using this approach.

Building on the CS and IS frameworks as well as on De Jong and Schotman (2010),

Westerlund et al. (2014) propose a panel data model of prices that have a common

factor representation and in which prices are modelled as non-stationary and

cointegrated processes. The Westerlund et al. (2014) factor-based panel model is

motivated in part by the need to address the econometric complexities with measuring

price discovery. Their proposed test is simple and easy to estimate compared to the

Generalized Methods of Moments test of De Jong and Schotman (2010), and has two

advantages. First, it is free from the limitations of the panel VECM used in Narayan et

al. (2014) in that it neither restricts the time-series dimension nor the cross-sectional

dimension. Thus, the concern of over-parametrizing the model with the traditional

panel VECM is addressed. Second, with existing price discovery tests, there are

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restrictions on short-term (transitory) and long-term (permanent) shocks. However,

because the Westerlund et al. (2014) test is built on a factor model, it does not impose

these restrictions. With the Westerlund et al. (2014) test, all shocks are entertained in

their original form, that is, without any restriction.

2.4 State of the art

The above discussion suggests that by using the CS and IS frameworks as its starting

point the price discovery literature has followed two directions. One involves

developing panel models of price discovery, while the other focuses on relaxing

various assumptions of the standard IS and CS frameworks in a time series context.

Perhaps the nearest to the state-of-the-art in the traditional IS/CS framework is the

seminal ‘unifying’ contribution of Figuerola-Ferretti and Gonzalo (2010), who

demonstrate that the Garbade and Silber (1983) framework coincides exactly with the

permanent component in the Gonzalo and Granger (1995) permanent-transitory

decomposition. This has paved the way for treating the CVAR model as the natural

model to analyze price discovery, and provided a relatively simple way of

ascertaining which of two prices is dominant in terms of long-run price discovery.

Figuerola-Ferretti and Gonzalo’s (2010) framework, however, no longer represents

the state-of-the-art in price discovery per se. Other models have been developed to

address the limitations of the CS/IS framework, with Figuerola-Ferretti and Gonzalo’s

(2010) framework now serving as a platform for the FCVAR proposed by Dolatabadi

et al. (2015, in press) Further, the CS and IS frameworks have been extended in other

directions, including the development of structural models to better model price

dynamics, models that take into account non-linearities or structural change, as well

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as panel models to deal with multiple markets and/or small time series. Because these

contributions emanate from myriad directions, it is difficult at this point to pinpoint

one state-of-the-art method for determining price discovery. The preferred model

really comes down to the issue affecting price discovery that one wants to focus on.

Perhaps the most that can be said is that these developments are at the frontier and in

compact represent the state-of-the-art in modeling price discovery.

2.5. Suggestions for future research

Given that the time series literature has moved in two directions, namely, the

development of panel models and relaxing assumptions underpinning the CS and IS

frameworks, one avenue for future research could be to integrate some of the

developments of these two avenues of inquiry into a panel framework. Obvious

examples would be to incorporate non-linearities, structural change or fractionally

integrated processes into recently proposed panel models.

A second avenue for future research could be to further develop structural models of

price discovery. The most general model of this form in the literature appears to be

from Scherrer (2012), who presents a method to measure the instantaneous effects of

permanent shocks on prices. Her model allows the observed price to depend on three

common factors, which are allowed to be contemporaneously correlated, providing

the required conditions for linkages across common factors. One potential extension

to the Scherrer (2012) framework could be to extend the number of common factors.

A third direction for future research could be to develop regime switching price

discovery models, which may be sufficient to allow an examination of price discovery

during economic expansions and recessions about which nothing is known.

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A fourth direction for future research in this area could follow on from recent

developments in econometric modelling rather than extending its frontier. Recent

innovations in econometric modelling have challenged long-held beliefs about the

markets in which price discovery occurs. One example is that incorporating fractional

cointegration seems to provide more evidence that price discovery occurs in the spot

market. Another example is that allowing for the effects of financial crises suggests

that incorporating time-varying dynamics provides evidence that fundamental factors

may have a greater role in driving prices, particularly in periods of financial crises.

Future research could address why one finds more evidence of price discovery in

specific spot markets with fractional integration or time-varying dynamics given the

economic fundamentals in particular markets.

3. Price Predictability

3.1. Motivation and implications

Two variables that commonly appear in the financial econometrics literature on

predictability are securities and exchange rates. Prior to the seminal review by Fama

(1970), empirical studies provided strong evidence that securities’ markets are

predictable using past returns. Fama (1970) defines this form of return predictability

as the weak form of the efficient market hypothesis (EMH). However, relying only on

past returns lacks power as past realised returns are noisy measures of expected

returns (Fama, 1991). To improve the power of return predictability tests, predictor

variables that are less noisy proxies for expected returns could be used (Fama, 1991).

Some early evidence that short-horizon returns can be predicted from past values of

other financial and macroeconomic variables include expected inflation (Bodie, 1976;

Fama, 1981), short-term interest rates (Fama and Schwert, 1977), dividend yields

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(Rozeff, 1984; Shiller, 1984; Fama and French, 1988), and earnings to price ratios

(Campbell and Shiller, 1988), all of which are able to explain a larger fraction of the

return variation for longer return horizons. In the past three decades an influx of other

predictors of stock markets has appeared in the literature, including the seminal work

of Campbell and Shiller (1988) that shows that market returns can be predicted to

some extent by using financial ratios such as dividend yields and price-earnings ratios

(see Fama, 1991, and Malkiel, 2004, for a discussion of the success of past studies in

predicting future equity returns).

The success of predictability studies in the 1980s and 1990s led many studies to

question the EMH, although Fama (1991) suggests that the results can be achieved

within the confinement of rationality, thus extending the definition of the weak-form

EMH to include past movements in variables other than past returns (see also Lim and

Brooks, 2009). However, many studies, particularly in the behavioural finance

literature, conclude that investors are prone to irrational behavior (or psychological

pitfalls); hence, this implies the potential to make abnormal profits (see Malkiel,

2003; Schwert, 2003; Malkiel et al., 2005; Lo, 2004; Yen and Lee, 2008). This

change in attitude towards stock market predictability has been fuelled by the growth

of computer power and has led to the development of a number of estimation and

predictability techniques.

Thus, while testing exchange rate predictability is strongly supported by economic

theory, the challenge since Meese and Rogoff (1983) has largely remained one of

finding predictive models for the exchange rate beyond the simple random walk

model.

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The rest of this part of the survey is concerned with financial econometric

developments in the predictability literature on securities and exchange rates – by far

the most common financial variables covered in this literature. Our focus is on the

predictability of these two variables using past values of financial and macroeconomic

variables. Although this literature is vast, our particular focus is on advances in

econometric techniques. We initially provide an overview of predictive modelling

using unrestricted and restricted vector autoregressions (VARs), including both panel

and time series analyses. This is followed by a review of recent developments in

financial econometrics in forecasting financial variables.

3.2. Predictive modelling using unrestricted and restricted VARs

Sims (1980) highlights the importance of the VAR formulation as a convenient

representation for forecasting and estimation, while Engle and Granger (1987)

establish that an error correction framework provides the means for testing, estimating

and forecasting a cointegrated system (see also Engle et al., 1989; Christoffersen and

Diebold, 1998). Engle and Yoo (1987) illustrate that when the unrestricted VAR (in

levels) is compared to cointegrated systems, it leads to suboptimal forecasts at the

longer horizon. Hence, they conclude that it is important to “build models designed

for cointegrated time series or vector autoregressive error-correction models

(VECMs)” (p. 159) (see also Phillips, 1998; Hoffman and Rasche, 1996; Clement and

Hendry, 1995).

The advancement of this technique as a forecasting tool has also gone hand-in-hand

with the quest to explain foreign exchange rate behavior within the confines of the

monetary model of exchange rate. Indeed, a long-run cointegrating relationship

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between the exchange rate and monetary fundamentals is a central tenet of the

monetary model of the exchange rate. Thus, if this model depicts the long-run

behavior of the exchange rate correctly, then the exchange rate only deviates from

monetary fundamentals in the short-run, making the vector error correction model

(VECM) in forecasting exchange rates a natural contender (see also Groen, 2005).1

While early studies such as Mark (1995), Chin and Meese (1995), and Groen (2000)

show the importance of first establishing the cointegration relationship between the

monetary fundamentals and exchange rate in the exchange rate predictability exercise,

predictability in these studies has been confined only to the long horizon. These long

horizon-to-weak-to-no predictability results, however, have fueled a search for a more

versatile, efficient and powerful error-correction framework predictability test.

In terms of versatility then, and unlike Engle and Granger’s (1987) test, Johansen

(1988, 1991) allows for more than one cointegrating relationship. The autoregressive

distributed lag (ARDL) - bounds testing methodology of Pesaran and Shin (1999),

and Pesaran et al. (2001) allows for a mixture of I(0) and I(1) variables, provides for

more variables to enter the long run model, and has been proven useful in predicting

long-run asset returns. The fractional cointegration test developed by Johansen (2008)

is suitable for several financial variables well described by long-memory fractionally

integrated processes, and has also been proven useful in explaining the return-

volatility relationship (see e.g. Bollerslev et al., 2013). With evidence of parameter

instability in the exchange rate as well as in the stock market, this time-varying

cointegration approach represents a logical tool. Further, Park and Park (2013) show

that the time-varying approach has strong predictive power for future changes in

1 Hence, predictability of exchange rates and econometric developments in the area of predictability

using the VECM framework have gone hand-in-hand.

20

exchange rates through in-sample and out-of-sample analysis.

In order to address the low power of time series, the Engle-Granger type tests have

also followed developments in testing for a unit root. The predictability literature has

also analyzed time series across similar cross-sections in panel datasets. As

Westerlund and Narayan (2014c, p. 2) point out, “…the use of panel rather than time

series data not only increases the total number of observations and their variation, but

also reduces the noise coming from the individual time series regressions. This is

reflected in the power of the resulting panel predictability test, which is increasing in

both N (number of cross-sections) and T (time span), as opposed to a time series/unit-

by-unit approach where power is only increasing in T. Thus, from a power/precision

point of view, a joint (panel) approach is always preferred. Furthermore, since power

is increasing in both N and T, this means that in panels one can effectively

compensate for a relatively small T by having a relatively large N, and vice versa”.

As indicated above, the panel version of the Engle and Granger (1987) test involves

estimating the panel long-run model with a panel unit root test (see e.g. Levin et al.,

2002; Im et al., 2003) being conducted on the residuals. This approach implies both

homogeneous long-run coefficients and homogenous adjustment parameters, with

only the serial correlation in the residuals panel unit root test assumed to be

heterogeneous (Groen, 2000). Error-correction based forecasting models from these

techniques have the disadvantage that the differences in the speed of adjustment and

dynamics across different countries are not taken into account (see e.g. Groen, 2000).

Commonly used variants of the panel-based Engle-Granger approach are the Pedroni

(1995), and Kao (1997) tests, which assume heterogeneity in the model parameter and

21

statistics. These tests are based on cross-sectional averages of the individual

parameters and statistics. However, because these parameters and statistics are

independent, Groen and Kleibergen (2003) argue that these tests do not capture the

panel dimension that is crucial to enhance the power of cointegration tests. Thus, to

allow for interdependence between panels, Groen and Kleibergen (2003) stack a fixed

number of VECM of different individuals into a joint panel VECM and develop a

likelihood-based framework for cointegration analysis, generalising Johansen (1988,

1991) (see also Larsson et al., 2001).

These aforementioned tests have become increasingly efficient over time as they

incorporate better estimating procedures for forecasting parameters. For example,

apart from using ordinary least squares (OLS) as the standard estimation method,

methods such as fully-modified OLS (FMOLS) and dynamic ordinary least squares

(DOLS) (see Saikkonen, 1991; Stock and Watson, 1993; Kao and Chiang, 2000) have

been developed for time series and panels, with Kao and Chiang’s (2000) examination

of the finite-sample properties of the DOLS estimator finding it to be superior to both

OLS and FMOLS in panel regressions.

In panel-based forecasting analysis, it has become common to pool the parameters of

the forecasting equation, as in Mark and Sul (2001). Using simulations, Westerlund

and Basher (2007) show that pooling individual prediction tests (such as the Theil U

statistic and the S prediction statistics of Diebold and Mariano, 1995) can lead to

substantial power gains. These authors further show that pooling only the parameters

of the forecasting equation does not seem to generate more powerful tests.

22

3.3.Developments in econometric modelling of forecasting financial variables

Beginning with Mankiw and Shapiro (1986), and Stambaugh (1986), innovation in

financial econometrics in the predictability literature over the past three decades has

primarily addressed various challenges which we characterize under three broad

topics. These include the salient features of the predictive models and its predictors,

predictability in the presence of model instability and uncertainty, and combination

forecasts. We focus on each of these three areas in turn.

3.3.1 Salient features of predictive models

(a) Persistent regressors

There is now strong evidence that widely accepted predictors of excess returns (such

as dividend-price ratio, earnings price ratio and measures of interest rates) are

persistent and behave like non-stationary variables, although they may not be fully

integrated (see Campbell and Yogo, 2006; Lewellen, 2004; Torous et al., 2004).

Meanwhile, the dependent variable (e.g. excess returns) may exhibit less persistence

than their predictors (Maynard et al., 2013). Cavanagh et al. (1995) show that

classifying the predictor variables as I(0) or I(1) leads to large size distortions when

regressors have local-to-unit roots. Predictability studies indicate that ignoring this

persistence in the regressors can lead to over-rejection of the null hypothesis using

conventional critical values (Campbell and Yogo, 2006; Torous et al., 2004).

Furthermore, Phillips and Magdalinos (2007a, b), and Magdalinos and Phillips

(2009a) recently analysed a wide range of close-to-unity processes and named those

in the vicinity of stationary, but not quite stationary, as mildly integrated processes.

Many variables, including Treasury Bill rates, show varying degrees of persistence

and seem to be mildly integrated processes.

23

While Phillip and Lees (2013) only recently accounted for the mildly integrated

process in predictive regression methodology, there are several studies in the

predictability literature that have paid attention to nearly integrated variables. Elliot

and Stock (1994), for instance, propose an asymptotic framework in which the

regressor is allowed to follow an autoregressive process with a root near to unity so

that it approximates a finite sample distribution of test statistics when the predictor is

persistent. However, allowing for persistence in the regressor leads to a nonstandard

limit distribution of the t-statistic that becomes dependent on a nuisance parameter,

and this cannot be consistently estimated (see Cavanagh et al., 1995; Torous et al.,

2004; Campbell and Yogo, 2006). Torous et al. (2004), and Campbell and Yogo

(2006) are other studies that develop asymptotics for inference using a local-to-unity

autoregressive specification for a single regressor. The Campbell and Yogo (2006)

method has been particularly popular among empirical researchers, is an industry

standard in finance, and provides a benchmark for competitors (Phillips, 2014, p.

1191).

These aforementioned studies follow Cavanagh et al. (1995), and use Bonferroni

intervals of the coefficient (of the predictor variables) in which the dependence from

the localising coefficient, c, is eliminated. Nonetheless, Bonferroni-type modelling

requires simulations to compute critical values to perform inference and confidence

interval construction, since the limit distribution employed in the calculations is

nonstandard (Phillips and Lee, 2013). These tests are also difficult to extend beyond

one regressor, which is not a desirable feature as there are typically several predictors

for each financial variable (Phillips and Lee, 2013). Furthermore, empirical size may

be substantially lower than nominal size and result in a conservative test. Cavanagh et

24

al. (1995) suggest aligning the nominal size to a desired level asymptotically (see

Torous et al., 2004), while Phillips and Less (2013) argue that the power of the

conservative test is often negligible in near local alternatives to the null of non-

predictability.

To address the uncertainty surrounding the root, these papers follow Stock (1991) to

construct confidence intervals for the largest autoregressive root of many explanatory

variables used in predictive regressions. However, Phillips (2014) shows that Stock’s

confidence intervals for the root are invalid and seriously bias asymptotically when c

→ −∞. This failure in the approach leads to poor performance in predictive regression

tests based on Bonferroni methods when the regressor is stationary or mildly

integrated (Phillips, 2014). Elliott et al. (2012) develop a ‘nearly optimal’ test so that

the procedure does not suffer from the problems inherent in the Campbell–Yogo test.

Furthermore, as Phillips (2014) argues, the computational and design considerations

make this method most useful in the single regressor case.

Finally, in these local-to-unity cases, the finite sample bias in estimation (discussed

below) is still present in the limit, and correcting the asymptotic bias is generally not

possible since the bias depends on the localizing coefficient, c, and this parameter is

not consistently estimable (Phillips and Lee, 2013). Lewellen (2004) extends the work

of Stambaugh (1999) to account for the finite sample bias and persistence in the

regressor. Lewellen uses conditional tests (which are estimated from OLS) to show

that predictability can be improved if information in the predictor autocorrelation,

especially when it is close to one, is utilized. Other approaches also provide solutions

to the nonstandard and non-pivotal limit distribution problem, including the

25

conditional likelihood approach with sufficient statistics developed by Jansson and

Moreira (2006), a control function approach by Elliott (2011), and the instrumental

variable (IVX) approach proposed by Magdalinos and Phillips (2009b) (see Phillips

and Lee, 2013, for a detailed discussion of each of these approaches).

In particular, Phillips and Lee (2013) extend Magdalinos and Phillips (2009b) IVX

approach to solve most existing problems in predictive regression methods. The IVX

approach generates a less persistent instrument for the regressor (X) than the regressor

itself, which uses no extraneous information so that the instrument relies only on the

regressor. Phillips and Lee (2013) add to this by allowing for multivariate regressors,

including predictive processes whose roots may lie in a wide vicinity of unity and

mildly integrated, local-to-unity and unit root regressors, all of which are examined

with standard chi-square tests and can be applied easily to long-horizon predictive

regressions. While the implementation of this approach is by straightforward linear

regression, the method requires user input on the parameter choice for instrument

construction (Phillips, 2014).

The above studies address size distortions in the context of the local-to-unity case.

However, persistence can also manifest itself in the form of long memory (Maynard

et al., 2013). Fractionally differenced processes can be observed in several predictors

such as forward premium (Baillie and Bollerslev, 1994; Maynard and Phillips, 2001),

volatility (Comte and Renault, 1998; Baillie and Bollerslev, 2000) and dividend yields

(Koustas and Serletis, 2005).

26

Maynard et al. (2013) develop a predictive test based on a two-stage rebalancing

procedure in which a semiparametric or parametric estimator may be used to estimate

the degree of long memory in the regressor in the first stage. In the second stage, the

regression is rebalanced by fractionally differencing the regressor. This also

rebalances the alternative hypothesis while leaving the null hypothesis unchanged and

thus allowing for a valid test of non-predictability. The authors’ Monte Carlo study

confirms that fractionally differencing the regressor removes the source of the size

distortion and yields a t statistic in the second-stage regression with correct size. They

also find that estimation and inference in the second stage are robust to estimation

error or even modest misspecification in the first stage.

The latter result has connections with recent studies that show that the functional form

may affect the power of predictive tests under non-stationarity with a parametric

approach (Kasparis et al., 2014). And this also may be true for some nonparametric

nonstationary specifications (Wang and Phillips, 2009). With this in mind and using

the recent work of Wang and Phillips (2009), Kasparis et al. (2014), in particular,

develop a unifying framework for inference in predictive regressions where the

predictor has unknown integration properties. They propose two easily implemented

nonparametric F-tests in which limit distributions is nuisance parameter free and

holds for a wide range of predictors, including stationary as well as non-stationary

fractional and near unit root processes. Kasparis et al.’s simulations suggest that this

test is more powerful than existing parametric tests (e.g. Jansson and Moreira, 2006)

when deviations from unity are large or the predictive regression is nonlinear.

Karemera and Kim (2006) suggest that using long memory models for forecasting

nominal exchange rates is a viable alternative to conventional models. They show that

27

an autoregressive fractionally integrated moving-average (ARFIMA) model is more

efficient than the random walk model in steps-ahead forecasts beyond one month. The

authors also find it to be more efficient than the random walk model in multi-step-

ahead forecasts.

(b) Endogeneity

In financial predictive models where returns are regressed on a lagged predictor

variable (whether for exchange rates or stocks), the regression results will suffer from

small sample bias if the regression disturbances are correlated with the predictor’s

innovations. This creates the so-called embedded endogeneity problem that leads to

biased estimates. Stambaugh (1999) illustrates through the example of excess stock

returns and dividend yield that in the presence of this bias, “…finite-sample

estimation and inference become less straightforward, for at least two reasons. First,

the ordinary least squares (OLS) estimators, although consistent, are biased and have

sampling distributions that differ from those in the standard setting. Second,

differences between classical and Bayesian methods become more apparent in the

presence of this Stambaugh bias, whereas those approaches are distinguished less

often in the standard regression setting”. Stambaugh’s (1999) approach to accounting

for endogeneity applies in the case of a single predictor, while more recently, Amihud

and Hurvich (2004), and Amihud et al. (2009) account for endogeneity in the

presence of multiple predictors. Furthermore, Stambaugh accounts for endogeneity

assuming that the predictive regressor’s stochastic explanatory variable is stationary.

Lewellen (2004) allows for a persistent regressor and Stambaugh’s bias in his study,

and shows that without accounting for persistence the corrections for small sample

biases can substantially understate forecasting power.

28

Similarly, Cai and Wang (2014) are concerned with the presence of endogeneity in

the model and persistence of the regressor, although their approach is different from

Lewellen (2004) because they work with multiple predictors. Cai and Wang (2014)

address endogeneity by adopting the Amihud and Hurvich (2004), and Amihud et al.

(2009) linear projection method, and then using a two-step estimation procedure to

manage both highly persistent and non-stationary predictors.

(c) Heteroskedasticity

Another important feature of financial predictive regression is the presence of noise or

time-varying ARCH/GARCH effects in asset returns. The long horizon predictive

regressions as in Torous et al. (2004 in which the predictor variable may be monthly

but the returns variable can be measured over one, two, or five year/s, have long been

used as a way to reduce noise in stock returns (Campbell and Yogo, 2006). This is

because, under the alternative hypothesis that returns are predictable, the variance of

the return increases less than proportionally with the investment horizon (Campbell

and Yogo, 2006). Compared to the long horizon tests, the procedures proposed in

Lewellen (2004), and Campbell and Yogo (2006) are even more powerful as they

reduce noise not only under the alternative hypothesis, but also under the null.

Westerlund and Narayan’s (2014b) study proposes a test that exploits the information

contained in the heteroskedasticity of returns, which is found to lead to higher power.

Similar to the above studies, their generalised least squares (GLS) test accounts for

persistence in regressors and endogenity issues. The asymptotic test’s dependence on

nuisance parameters reflecting persistence, endogeneity and heteroskedasticity in the

data, however, means that the test can suffer from size distortions. To account for this

29

lack of robustness, the authors propose a subsample feasible qausi-GLS (FQGLS) test

that has correct asymptotic size even in the presence of nuisance parameters. This

differs from many of the above tests, which use Bonferroni-type tests that are known

to be conservative.

3.3.2 Predictability in the presence of model instability and uncertainty

This literature is vast and complex. In this section, we focus only on areas that have

seen recent econometric developments or are seen as emerging areas. We begin by

looking at model instability, focusing on studies that have developed the predictability

framework under the assumption of structural break uncertainty (see e.g. Pesaran and

Timmermann, 1995, 2002). We also review another group of studies that treat

structural breaks as being deterministic in nature and, hence, address issues

surrounding the number and duration of breaks. Related to this is the idea that there is

a stochastic process underlying the structural breaks that is used to predict future

structural breaks. The time-varying literature takes the issue of model instability to the

extreme, and is premised on the notion that there are not just one, two, or five

structural breaks, but many structural breaks across each period that render parameter

uncertainty. To this end, this literature argues for time-varying approaches for

estimating parameters, and developing many tests to both detect the phenomenon and

to develop such approaches. Furthermore, choosing the right set of variables for

forecasting has led to some studies trying to deal with model uncertainty. Finally,

there are those we refer to as ‘hybrid studies’ within the literature on model instability

and uncertainty that address aspects of instability and uncertainty within the same

predictability framework.

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3.3.3 Model instability

(a) Structural break uncertainty

Structural breaks refer to both permanent shifts and ‘breaks’ (temporary jumps) in the

parameters of the return generating process. There could be various reasons for

permanent shifts, including legislative, institutional or technological changes, shifts in

economic policy, or even large macroeconomic shocks such as the doubling or

quadrupling of oil prices that have been experienced over recent decades (Pesaran et

al., 2006). Breaks (or jumps) in the parameters may also arise due to a number of

factors, including major changes in market sentiment, the creation or bursting of

speculative bubbles, or regime switches that affect monetary and debt management

policies. The latter, for example, might include switches from money supply targeting

to inflation targeting, or from short-term to long-term debt instruments (Pesaran and

Timmermann, 2002).

To address parameter instability, many studies are concerned with the problem of

determining in real time how much historical information to use when estimating a

forecasting model. Studies use a range of methods, such as a rolling window of

observations with a fixed size to generate the forecast, discounted least squares that

assigns smaller weights to observations further away from the point of the prediction,

or recursive least squares that allows for expansion of the observation window to

allow for new information (see Pesaran and Timmermann, 2002). While Pesaran and

Timmermann (1995) allow the forecasting model to change over time, they do

mention in their 2002 paper that this may be inappropriate in the presence of breaks in

the parameters of the forecasting model. The choice of estimation window of a

forecasting model is endogenised in Cooper and Gulen (1999), but that study is

31

constrained to either rolling windows of fixed length or an expanding window.

However, none of these methods explicitly detect, and condition on, the occurrence of

one or several breaks. This is a concern of Pesaran and Timmermann (2002), who

develop a conditionally time-varying window size using a two-stage approach. A

reversed ordered Cusum (ROC) test is developed to detect the most recent breakpoint.

This test is related to the standard CUSUM test developed by Brown et al. (1975) (see

Tian and Anderson, 2014, for a detailed discussion). When adopted recursively

through time, the reversed ordered Cusum procedure yields a sequence of estimation

windows whose lengths effectively indicate the ‘memory’ of the return model under

consideration. The resulting sequence of ROC tests statistics is used to choose and

forecast from a single post-break estimation window.

(b) Deterministic structural breaks

There are studies that treat structural breaks as being deterministic in nature. The aim

of these studies is to explicitly identify the number and duration of break dates, and to

examine the influence of the structural breaks in the forecasting exercise. As implied

above, the identification of the number and duration of the structural breaks or regime

shifts is critical here. Early studies determined the number and duration of structural

breaks exogenously. However, given the uncertainty with this approach, various

studies have attempted to develop techniques that allow determination of the breaks

endogenously (see e.g. Rapach and Wohar, 2006).

The Chow (1960) procedure is a familiar test with the null hypothesis of no structural

change, but one may use the standard F-test to test this against the alternative only if

the breakpoint is known. Quandt (1960) suggests using the maximum F-statistic over

all values of break dates as a means to test parameter stability. This search over a set

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of dependent F-statistics affects the asymptotic distribution of the test, which ceases

to be χ2 (Elliott & Müller, 2003). Expanding on Quandt (1960), Andrews (1993)

derives the limiting distribution of the supremum of the Fk statistics (SupF statistic),

which is nonstandard and dependent on a trimming parameter. At any given trimming

parameter, the null hypothesis of no structural break can be tested using the

asymptotic critical values in Andrews (1993) and, if the null is rejected, the break date

can be estimated (see also Rapach and Wohar, 2006). Hansen (2000) develops a

heteroskedastic fixed-regressor bootstrap procedure that delivers the correct

asymptotic distribution for the SupF statistic in the presence of general

nonstationarities in the regressors, including mean and variance breaks and unit roots,

which Andrew’s asymptotic distribution fails to account for. Bai (1997), and Bai and

Perron (1998, 2003a, 2004) explicitly test for multiple structural breaks at unknown

dates in the bivariate predictive regression models. Apart from allowing for multiple

breaks, the Bai and Perron methodology accounts for autocorrelation in the regression

model residuals, heteroskedasticity in the residuals, and different moment matrices for

the regressors in the different regimes when computing test statistics and confidence

intervals for the break dates and regression coefficients.

(c) Stochastic structural breaks in forecasting

Pesaran et al. (2006) develop a test to examine how future values of the variables of

interest might be affected by multiple breaks. As the authors explain, this question

cannot treat structural breaks as deterministic (as in the case of past structural breaks)

unless they are known; rather, it requires a modelling of the stochastic process

underlying the breaks. They propose a Bayesian estimation and prediction procedure

that allows for the possibility of new breaks occurring over the forecast horizon,

taking into account the size and duration of past breaks (if any) by means of a

33

hierarchical hidden Markov chain model. Here, predictions form by integrating

parameters from the meta-distribution that characterizes the stochastic break-point

process.

(d) Time-varying parameter - parameter uncertainty

In the stock market predictability literature, various studies, including Paye and

Timmermann (2006), and Goyal and Welch (2008), show that the magnitude of asset

return predictability is distinctly time-varying and unstable; however, this is not just

limited to stock returns but is true of a wide range of time series (see Stock and

Watson, 1996). Merton’s (1973) intertemporal CAPM also suggests that time-varying

risk aversion can lead to a time-varying relationship between stock returns and

predictive variables (see Zhu and Zhu, 2013). And if predictability of returns partly

reflects market inefficiencies and not just time-varying risk premia, then predictive

relationships should disappear once discovered, provided that sufficient capital is

allocated towards exploiting them (Pesaran and Timmermann, 2002). Several studies

show the importance of business cycles in dealing with predictability in returns (see

e.g. Fama and French, 1989; Lettau and Ludvigson, 2001; Cooper and Priestly, 2009;

Henkel et al., 2011; Nitschka, 2013, 2014). Collectively, these studies make the point

that parameter uncertainty or the time varying nature of parameters seem to be driven

by business cycles.

While the studies on structural breaks/jump work mainly with one to five breaks, the

time-varying parameter literature is concerned with large as well as small parameter

changes that occur now and then. Furthermore, because the breaking process can

occur in any number of ways – for instance, over the review period, breaks can occur

34

every year or every second year, in clusters, or randomly – several studies have

focused on different breaking processes. To give a flavour, studies such as Shively

(1988b) work with Gaussian breaks of constant variance every period, Andrea and

Ploberger (1994) use a finite number of breaks by employing an average asymptotic,

and Bai and Perron (1998, 2003) work with the maximum of a sequence of F-statistics

for their power criterion. To compare various breaking processes used in the model

instability and time-varying parameter literature, Elliott and Müller (2003), in

particular, show that leaving the exact breaking process unspecified (apart from a

scaling parameter) does not result in a loss of power, at least asymptotically.

Here, as explained by Hackl and Westlund (1989), time-varying studies are concerned

with tests to detect non-constant parameters in regressions and, if the null of constant

regression is rejected, implement time varying approaches. A number of studies

examine the hypothesis of constant regression against the alternative hypothesis that

the parameter is stochastic and follows a first order autoregressive process – in this

case, the non-constant parameter is modelled to deviate only temporarily from zero,

so that the ’long-run’ value remains the estimated parameter (see Shively, 1988a).

Most Markov switching models with recurring states and threshold autoregressive

models are also closely related to the case of a stable autoregressive process for the

non-constant parameter (see also Elliott and Müller, 2003). Most studies in this area,

however, consider models in which deviations of the non-constant parameter from

zero are permanent. In these models the alternative hypothesis is that the non-constant

parameter follows a random walk. For the ‘unobserved components’ model, see

Chernoff and Zacks (1964), and Nyblom and Mäkeläinen (1983). For more general

35

models, see Garbade (1977), LaMotte and McWhorter (1978), Franzini and Harvey

(1983), Nabeya and Tanaka (1988), or Leybourne and McCabe (1989).

(e) Model uncertainty

Similarly, while some predictive variables such as financial ratios, including dividend

yields and price-earnings ratios, are popular in the predictability literature, many other

predictability variables are also covered.2 Given the high volume of predictors, it

might often seem unclear to an investor as to which set of variables comprises the

correct predictive variables (see also Schrimpf, 2010). Hence, an important question

that arises is how investors choose between so many competing models (Pesaran and

Timmermann, 2002). Moreover, choosing the best model or model thinning is

restrictive, as information from the discarded models is ignored in each period (Aiolfi

and Favero, 2005). Beginning with Cremers (2002), and Avramov (2002), studies

examine ways to efficiently deal with this model uncertainty. As Aiolri and Favero

(2005) explain, a natural way to interpret model uncertainty is to refrain from the

assumption of the existence of a ‘true’ model and instead attach probabilities to

different possible models – an approach known as ‘Bayesian model averaging’.

Cremers (2002), and Avramov (2002) use a pure Bayesian model averaging approach

that requires prior elicitation for the relevant parameters, conditional on the different

models. Avramov (2002) applies an empirical Bayesian approach for prior elicitation,

which uses data-information from the sample for prior elicitation. However, such an

2 Among others, predictors of equities in the literature include expected inflation (Bodie, 1976; Fama,

1981), short-term interest rates (Fama and Schwert, 1977; Hodrick, 1992), output gap (Cooper and

Priestly, 2009; Møller et al., 2014); the consumption wealth ratio (Lettau and Ludvigson, 2001);

consumer confidence (Møller et al., 2014); and investor sentiments/attention/anchoring behaviours (Li

and Yu, 2012). The fact that many variables have been found to be valuable predictors of returns has

raised concern that apparent predictability may be a result of data-snooping rather than of genuine

variation in economic risk-premia (see e.g. Goyal and Welch, 2008). Authors such as Schrimpf (2010),

Cremers (2002), and Avramov (2002) regard the long list of apparent predictive variables as a sign of

model uncertainty.

36

approach has been criticized for using information about the dependent variable,

which violates the rules of probability necessary for conditioning (Fernández et al.,

2001). In the Bayesian tradition, Cremers (2002) specifies subjective prior

distributions that are based on different beliefs about predictability. The specification

of prior beliefs, however, can be a problematic task when the set of models becomes

very large (Schrimpf, 2010). To avoid the drawback of the dependence on prior

distribution, studies such as Schrimpf (2010) adopt the Salai-Martin et al. (2004)

approach, that is, the Bayesian averaging of classical estimates. Furthermore, a pure

Bayesian model averaging approach takes all predictive variables as exogenous. To

remedy this, Schrimpf (2010) combines the Bayesian feature of model averaging with

coefficients estimated by standard OLS, which are adjusted for endogeneity bias

following Amihud and Hurvich (2004), and Amihud et al. (2009).

3.3.4 Combination forecasts

The pioneering work of Bates and Granger (1969) suggests that if a number of

unbiased forecasts of the same future variable are available, then, rather than using

one, the forecasts can always be combined in such a way that the composite forecast

has variance less than or equal to any of the competing forecasts. Since Bates and

Granger’s (1969) work, various ‘model averaging’ studies have tested the efficacy of

combination forecasts from different models (similar to Bates and Granger, 1969).

While there is clear support for combination forecasts in this literature, it is not

obvious which combination scheme is the best. Bates and Granger (1969) find that

combination forecasts that allow the weights to change can lead to better forecasts

than those that result from the application of a constant weight determined after

noting all individual forecast errors. Makridakis and Winkler (1983) show that

37

differential weights that are proportional to the reciprocal of a sum of squared errors

work better than a simple average.

Recent studies have resorted to combining models using either the same or different

estimation procedure but different predictors. Examples include shrinkage methods

such as ridge regression (Hoerl and Kennard, 1970), bootstrap aggregation (or

bagging) (Breiman, 1996), and the least absolute shrinkage and selection operator

(Lasso) (Tibshirani, 1996). These studies apply various flexible weighting to

individual predictors and include: bagging (Breiman, 1996), which applies differential

shrinkage weights to each coefficient; adaptive Lasso (Zou, 2006), which applies

variable-specific weights to the individual predictors in a data-dependent adaptive

manner; the elastic net (Zou and Hastie, 2005; Zou and Zhang, 2009), which

introduces extra parameters to control for the penalty of including additional

variables; and Bayesian methods such as adaptive Monte Carlo (Lamnisos et al.,

2012; Tian and Anderson, 2014), which assign heavier weights to forecasts that use

more recent information on different estimation windows. Elliot et al. (2013) show

that combinations of subset regressions can produce more accurate forecasts than

conventional approaches based on equal-weighted forecasts, combinations of

univariate forecasts, or forecasts generated by methods such as bagging, ridge

regression or Bayesian model averaging.

While combination forecasting is becoming popular in the economics literature,

predictability literature in finance is only just beginning to catch up (see Rapach et al.,

2010). Only a few methods have been developed/applied in the finance literature. For

instance, a fixed number of predictors of stock returns are combined using an equal-

38

weighted scheme on the basis of complete subset regressions (Elliot et al., 2013),

averaging of OLS and Kalman filter of Mamaysky et al. (2008) models of mutual

funds (Mamaysky et al., 2007), equal-weighted methods to combine many univariate

equity premium models (Rapach et al., 2010), and equal-weighted combinations of

forecasts of stock returns from all possible two variable models (Aiolfi and Favero,

2003).

Overall, the strong message in recent studies is that combination forecasting is better

than standalone forecasting. And while it is unclear which combination method is the

best, a scheme that applies differential weights seems to work well in this literature.

Moreover, it appears using a few schemes is the best approach.

3.4 State of the art

3.4.1 Persistence

At present, Bonferroni-type procedures are regarded as the state-of-the-art in this

literature, but they do have some properties that limit their adoption in empirical work

(Phillip and Lee, 2013). Other alternatives are available, including Phillip and Lee’s

(2013) parametric test and Kasparis et al.’s (2014) non-parametric test. Both tests can

be easily applied and are versatile enough to accommodate a wide range of predictors,

including stationary as well as non-stationary fractional and near unit root processes.

3.4.2 Persistence, endogeneity and heterogeneity

Modeling approaches that take account of endogeneity, heterogeneity and persistence

and can be implemented easily are state-of-the-art. Westerlund and Narayan (2014b)

(one predictor) and Lewellen (2004) (multiple predictors) are two approaches for time

series analysis, while Westerlund and Narayan (2014a) present a panel-based test.

39

3.4.3 Model instability and uncertainty

There are several standalone developments in this literature that have made significant

contributions. Methods that will make the most difference include those that

incorporate several features of model instability and uncertainty. For instance, using

Bayesian model averaging techniques, Pettenuzzo and Timmermann (2011) account

for model uncertainty and instability; that is, investors are not assumed to know the

true model or its parameter values, nor are they assumed to know the number, timing

or magnitude of past or future breaks. Instead, they come with prior beliefs about the

meta-distribution from which current and future values of the parameters of the return

model are drawn and update these beliefs efficiently as new data are observed.

Zhu and Zhu (2013), on the other hand, develop a Bayesian regime-switching

combination (BRSC) approach that explicitly incorporates model, regime and

parameter uncertainty to examine the out-of-sample return forecasting problem. Here

they incorporate model uncertainty, as in Barberis (2000), and Cremers (2002). The

uncertainty about parameters is summarized by the posterior distribution of the

parameters given by the data. Instead of constructing the distribution of expected

returns conditional on fixed parameter estimates, the Bayesian method integrates over

the uncertainty in the parameters captured by the posterior distribution. To account for

time varying predictability and address the regime uncertainty issue, the regime-

switching combination approach is used (see Rapach et al., 2010) to account for

model uncertainty and efficiently use the information contained in a large collection

of candidate predictive variables.

40

3.4.4 Combination forecasts

With many models and predictors of securities and exchange rates, this is one

important area of research. However, it is difficult to suggest state-of the art

techniques here, as many developments in this area have occurred outside the

financial econometrics literature that currently relies heavily on equal-weighted

forecasts from two to many univariate models of returns or equity premium or model

averaging schemes. A recent paper by Elliot et al., (2013) shows that combinations of

subset regressions can produce more accurate forecasts of stock returns than

conventional approaches based on equal-weighted forecasts, combinations of

univariate forecasts, or forecasts generated by methods such as bagging, ridge

regression or Bayesian model averaging.

3.5 Suggestions for future research

This survey highlights that econometric developments in predictability mainly address

challenges in modeling predictability one or two at a time. It is likely that future

research will provide more hybrid approaches that address several if not all

forecasting challenges simultaneously, and which can be easily implemented

empirically. There is a glimpse of this gradually starting to occur in some of the most

recent literature. For example, as mentioned above, Pettenuzzo and Timmermann

(2011) address model uncertainty and instability, Zhu and Zhu (2013), address model,

regime and parameter uncertainty, and Westerlund and Narayan, (2014a, 2014b)

address persistence, endogeneity and heterskedasticity. However, with the exception

of a few models, the performance of hybrid models set against each other as well as

against standard approaches is mostly unknown. Future research into robustly

evaluating such models is necessary to determine which model provides more

41

accurate results. Focusing on just the three studies mentioned above, Westerland and

Narayan (2014a, 2014b) models might be further extended to cater for time-varying

parameters and model uncertainty as in Pettenuzzo and Timmerman (2011) to derive

more fine-tuned results. Hence, a mixture of strategies that include data and model

features, and model and parameter uncertainty may possibly lead to more superior

models. However, for these innovations to occur, it is likely that financial

econometric research will need to embrace more of the literature on combination

methods, which is a large and growing area of research on its own.

4. Conclusion

The purpose of this paper has been to review econometric developments in the areas

of price discovery and price predictability. We have focused on these two topics

because they are conceptually linked. Both areas have benefited from similar

developments in econometric modelling and, in contrast with areas such as

ARCH/GARCH and stochastic volatility, have not been reviewed before.

In terms of the current status of the two literatures, both areas are informed by

developments in modelling of unit roots, cointegration and VAR/VECM processes.

While this is most evident in the price discovery literature, it is also reflected in the

price predictability literature in terms of, for example, addressing persistence and

modelling predictability on the presence of instability and uncertainty. Both literatures

essentially follow two directions. One is to develop new methods based on relaxing

the assumptions of time series models. Examples include allowing for fractionally

integrated/cointegrated processes, addressing non-linearities, and accommodating

structural breaks and parameter instability. The other is to develop panel-based

models to harness the extra power of cross-sectional and time series information. In

both literatures, it is difficult to pinpoint the exact state-of-the-art. It very much

42

depends on the extension/feature one wants to focus on, which, in turn, dictates the

assumption(s) one wants to relax. There are few methods dealing with either topic

that simultaneously address several issues at once.

In terms of looking forward, this survey suggests several directions. The first flows

from evidence that the financial econometric applications in price discovery and price

predictability, as well as in other areas of finance, are being fuelled by developments

within econometrics itself. Indeed, many of the recent empirical applications in price

discovery and price predictability are on the frontier of econometric methods. Given

this, further developments in econometric modelling are needed to relax the

assumptions and push the boundaries of applications to price discovery and price

predictability.

Second, a limitation of both the price discovery and price predictability literatures is

that there has largely been a piecemeal approach in which econometric issues have

been dealt with on a one-by-one basis. This reflects the manner in which the

econometrics literature has evolved. An area for future research, then, is to develop

new modelling methods that combine/synthesise recent developments across multiple

econometric issues and address more than one issue at once. One obvious way to

begin this is through integrating developments in the time series literature into panel-

based frameworks.

A third direction is to take a step back and reconsider what insights from recent

developments in econometric modelling that challenge stylised facts tell us about

long-held beliefs on the apparent empirical regularities in price discovery and price

predictability. For example, in the price discovery literature, allowing for fractional

43

integration and time-varying dynamics is challenging previous findings about whether

price discovery occurs in spot or futures markets, as well as the role of fundamental

factors in driving prices in financial crises. This opens up the opportunity to develop

theory to explain these empirical results.

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