Predicting stock returns: A momentum and lasso approach for stock markets of the G7.

MSc Thesis

Alfons Wolthuis

Faculty of Economics and Business

University of Groningen

June 26, 2020

Supervisor:

Ioannis Souropanis

Abstract

This research investigates the improvement of the out-sample predictability of the stock returns of the

G7 countries, by implementing two novel constraints on the predictive regression of the stock return.

Based on the sign momentum these constraints truncate the predictive regression and either replacing

it with a no change forecast or a forecast based on the predictive lasso regression. The empirical

findings demonstrate mixed result on our G7 countries. Two countries outperform the average

historical benchmark for the momentum constraint. Compared to the unconstraint both constraint

models show an improved change of predictive performance.

Student number: s2056992

Name: Alfons R.F. Wolthuis

Study Programme: MSc International Financial Management

Field Key Words: Stock return; Out-of-sample prediction; G7; Shrinkage constraint; Momentum constraint; Lasso; International.

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Acknowledgement

I would like to contribute this thesis to my new born daughter Isabelle.

And I would like to extend my deepest gratitude to my partner, family, friends, as well as my

supervisor and the program coordinator. For supporting me to complete my thesis, despite

of the circumstances.

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1 Introduction

Many researchers have been interested in the stock return forecasting for a long period,

especially since reliable forecasts are considered of utmost importance in enhanced and

realistic asset pricing and risk assessment models (Cochrane, 2009). In comparison to the

straightforward in-sample stock return forecasting, the seminal research of Welch and Goyal

(2008) demonstrate difficulty of out-of-sample forecasting in beating the average historical

benchmark.

Despite of this, researchers took up the challenge to beat the average or the historical

benchmark. One avenue of research started constructing new robust predictor variables to

better predict the stock return and challenge the benchmark. Another avenue of research

aims on using technical analysis to improve the estimation performance. For example, by

introducing economic constraints such as in Campbell and Thompson (2008). This results in

favourable out-of-sample results with positive 𝑅2 statistics. Other positive results are

obtained by combining different predictive variables (Rapach et al. 2010; Neely et al. 2014),

or by making use of the momentum of predictors of stock returns (Wang et al., 2018).

Most research on stock return takes place on the United States (U.S.) stock market, due to

the fact that this is the best documented capital market (Dimson et al. 2011). However,

there is also research emphasizing on the importance of doing international stock return

predictability. These are for example the researches from Hjalmarsson (2010), Henkel et al.

(2010), and the research of Jordan et al. (2014). These researches all demonstrate a

successful method for international stock return forecasting.

This study aims on researching the out-of-sample stock return forecast of the different

countries of the G7. In this way the aim of study twofold. First, it aims to research the out-of-

sample performance by introducing two new constraints, by introducing a momentum

constraint truncating the forecast based in sign momentum. Next to that implementing a

lasso restriction, which instead of truncating it as the momentum approaches replaces it by a

forecast based on a lasso regression. Secondly, this study aims to contribute to international

research on stock return forecasting by applying our constraints to the countries of the G7.

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The contribution of our research is on existing stock return forecasting, by introducing the

two constraints to challenge the average of the historical benchmark (Welch and Goyal,

2008). Furthermore, this research adds a contribution to the existing international stock

return forecasting research by employing an international sample. This sample consists out

of the data rich G7 countries, similar to the research of Jordan et al. (2014).

This study is structured in 7 sections. Firstly, we present the theoretical background by

means of a literary review. Afterwards, we will focus on the data description in section 3 and

methodology in section 4. In the fifth section the empirical findings will be central, while

section 6 discusses the results. Finally, the seventh section marks the conclusion of this study

and addresses probable further research possibilities.

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2 Literature Review

2.1 Forecasting Stock Returns

A long history of literature on stock market predictability in finance consists. Especially

considering that reliable forecasts are imperative for the creation of more enhanced and

realistic capital asset pricing models explaining financial metrics(Sharpe, 1964; Cochrane,

2009). The debate on stock return predictability originates from the efficient market

hypothesis, which is stating that prices of securities are highly efficient in reflecting both

individual and stock market information (Malkiel, 2003). This is associated to the random walk

theory (Fama, 1970) which assumes that that stock returns in times-series are following a

random walk.

In the field of stock return predictability there have emerged two parts of asserted predictor

variables. Various researchers have argued that interest rate variables could predict stock

market returns. For example, the predictability of stock return with the short rate (Fama and

Schwert, 1977; Fama, 1981), or the dividend yield (Schiller, 1981), followed by the term

premium (Fama, 1984; Campbell, 1987), and the default premium (Chen et al., 1986).

Other researchers provided evidence that various economic variables predicted stock returns,

when regressing the U.S. stock return. The economic variables used where, for example,

nominal interest rates (Fama and Schwert, 1977; Ang and Bekaert, 2007), the dividend-price

ratio (Fama and French, 1988; Cambell and Schiller, 1988; Cochrane, 2009; Pastor and

Stambaugh, 2009), the earnings price-ratio (Campbell and Schiller, 1988), and inflation

(Nelson, 1976; Campbell and Vuolteenaho, 2004). Finally, this resulted in a debate on the

predictability of stock returns, since several researchers observed a lack of robust out-of-

sample result such as Bossaerts and Hillion (1999), Ang and Bekeart (2007) and Goyal and

Welch (2008).

The researchers Welch and Goyal (2008) for example discuss in their seminal work that

forecasting stock market returns is extremely difficult. Demonstrating that stock market

returns can be forecasted in-sample with prominent predictors from the literature, such as

the dividend yield and the dividend-price ratio. They emphasize that out-of-sample forecasts

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based on single predictors are being outperformed by the historical average benchmark,

showing a lack of robust results.

Despite of it being notoriously difficult, several scholars have taken up the challenge to beat

the historical average benchmark. In the financial literature two main avenues of research

have become eminent, namely testing novel predictors and technical analyses of existing

prediction models.

This first avenue of research aims on the construction of novel and robust predictors able to

challenge the benchmark. Examples of these new predictors are the short interest index

(Rapach et al., 2016), the variance risk premium (Bollerslev et al. 2009), technical indicators

(Ludvigson and Ng 2009; Neely et al. 2014; Jordan et al. 2014), news-implied return (Manela

and Moreira, 2017), and autocorrelations for stock return (Xue and Zhang 2017).

The second avenue of research aims on the use of technical analysis to improve the

estimation performance by addressing the model uncertainty and the instability of its

constraints. Examples of these methods are the economic constraint method, the forecast

combination method, a method making use of regime shifts, the momentum method, the

Sum-of-parts method, and the use statistical constraints. Underneath these methods will be

discussed in more detail.

Several researches have introduced economic constraints to improve the stock return

predictions. For example, the influential article of Campbell and Thompson (2008) truncates

the stock return predictions at zero and constrains the sign of the slope coefficient in the

prediction model. Resulting in favourable out-of-sample results with positive 𝑅2 statistics.

Another research of Pettenuzzo et al. (2014) added that the conditional Sharpe ratio could

be constrained between zero and one, showing an improved predictive performance.

Additionally, Zhang et al. (2019) argue that rational investors are unlikely to trade forecast

outlier stocks and introducing a new constraint to truncate both the extreme positive as

negative stock return forecasts.

The forecast combinations methods are model proposed to combine the information from

different predictor variables, which improves the forecast performance. For example,

Rapach et al. (2010) combine different predictive regressions to deliver an improved

forecast. Furthermore, Neely at al. (2014) improve the forecast by making use of the

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principal components, which are obtained by standardizing the predictors. On top of that,

Zhang et al. (2019) identify low correlating predictors as having complementary information

and regresses them in a multivariate regression.

Another technical method used is making use of regime shifts. Henkel et al. (2011), for

instance, demonstrate that the stock return predictors become non-existent during time-

varying regimes shift of the business cycle from an expansion to a contraction. Similar to

this, Zhu and Zhu (2013) apply a regime switching method to the combination method of

Rapach et al. (2010) we mentioned earlier.

Other researches work with the momentum method, which uses the direction of a variable

to make a prediction. For example, Wang et al. (2018) makes use of the momentum of the

predicted variables, where the momentum of the past predictions can be utilized to

successfully make a prediction. In addition, Zhang, Ma, and Zhu (2019) employed a short-

term intra-day momentum for the stock return of China.

The research of Ferreira and Santa-Clara (2011) and Faria and Verona (2018) makes use of

the sum-of-parts method. In this method the stock returns are first decomposed into

different parts and then forecasts are made separately to get an estimated stock return

after.

The last method is the use of an statistical constraint. For example, the research of Li and

Tsiakas (2017) imposes a statistical constraint by making use of shrinkage estimator which

reduces the effect of less informative predictor variables in stock return forecasting and

improves the performance. In previous research, Li et al. (2015) use this shrinkage method

to forecast exchange rates with an improved performance.

2.2 International stock returns

The main literature on stock market return predictability focuses on the United States

returns, which is according to Dimson et al. (2011) since the United States is the best

documented capital market. However, the general conclusion from countries predicting

stock returns worldwide, is that stock returns are actually predictable worldwide. Despite of

this there is still a limited number of studies also researching out-of-sample stock return

forecasting for an international sample.

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The research of Engsted and Pedersen (2010) demonstrates long-term international

evidence on the predictability of stock returns and the growth of dividend of the United

Kingdom, Denmark, and Sweden. Showing that three European stock markets are different

in predictability patterns than the United States are.

We give a small selection of international out-of-sample researches. The research of

Hjalmarsson (2010) focuses on the stock return predictability in an enormous data set of

20000 monthly observations covering 40 international markets. Concluding that the

traditional valuation measures, such as earnings-price ratio and dividend-price ratio, have a

limited forecasting performance in an international data sample. Another out-of-sample

research is the research from Henkel et al. (2010), which finds mixed results a sample of G7

countries concerning the countercyclical risk premiums and the time-variation dynamics of

predictor variables. It concludes that return predictors such as the dividend yield appear to

be non-existent during business cycles expansions. Rapach et al. (2013) argues that the

United States has a leading role in stock returns, since its lagged returns have a predictive

performance for other countries. With using a relatively big sample of 14 countries Jordan

et al. (2014) researched both in-sample predictability as well as out-of-sample predictability

of stock returns. Using three types of predictor variables to forecast stock returns, namely

fundamental, macro variables and technical variables. Lawrenz and Zorn (2017) conducted

an out-of-sample test on 27 equity indices to research international asset allocation. The test

provided both a strong in-sample as out-of-sample evidence for stock return forecasting, by

conditioning a predictive regression on time-series and cross-sectional information.

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3 Data Description

In this section, we discuss the data sources and the variables used in this research. This

research, focuses on the monthly stock predictability in the seven economies of the G7.

These leading economies have longitudinal time series available. Therefore, the sample

covers monthly data for the countries of the G7 over an approximate 50-year period of

January 1970 to January 2020. The sample comprises the G7 countries: Canada (CAN),

France (FR), Germany (DEU), Italy (ITA), Japan (JPN), the United Kingdom (GBR), and the

United States (USA). Below, the variables we use for the sample countries will be explained.

3.1 Variables description

Our dependent variable is the stock return, which is the continuously compounded return of

the biggest stock indices of each country of the G7. For Canada we used the TSX, for France

the CAC40, for Germany DAX, for Italy the FTSE MIB, Japan the NIKKEI 225, the United

Kingdom the FTSE 100, and finally for United States the S&P500.

For the stock return predictors, we collected variables available for each country on basis of

variables employed by the research of Christiansen et al. (2012). Table 1 displays the

selected predictor variables accompanied by a short description.

Table 1: Variables Description

Nr. Predictors Abbr. Description

Interest Rates and Spreads

1 T-Bill rate TBL Three-month Treasury Bill rate; Risk-free rate

2 Long Term Bond Yield LTY Yield on long term government bonds over 10 years

3 Term Spread TMS Difference of long-term bond yield and three-month T-Bill rate

4 Libor rate LIBOR London Inter-bank Offered Rate, Bank rate

Liquidity

5 TED Spread TED Measure of funding Illiquidity, difference of 3 Month Libor rate minus 3-month T-Bill rate

Macro-economic variables

6 Money Supply MSM Monthly growth rate of aggregate money supply countries. Using the M1, or M3, or M0.

7 Unemployment UNEMP Monthly growth of unemployment

8 Industrial Production IPM Growth rate of the industrial production; Industrial Production Index

9 Inflation rate INFM Monthly growth inflation; Consumer Price Index

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First of all, we include a set of bond market variables including the Treasury bill rate (TBL),

which can be seen as the risk free rate and is according to Ang and Bekaert (2007) and Goyal

and Welch (2008) a valuable predictor of the stock return. We use the long term

government bond yield and the calculated term spread, both identified by Cambell and

Schiller (1991) being useful for forecasting stock returns. Where the latter is showing the

spread between long-term and short-term interest rates. The last variables we add the three

month LIBOR, which is the London Interbank Offered Rate and is reflecting the riskiness as it

is measuring the premium demanded by banks for lending an unsecured loan to another

bank and used to calculate the following variable.

This variables called the TED spread is a measure of liquidity, and is the difference between

the three-month Libor and the T-bill rate. This measure displays the funding (il)liquidity of

the interbank market, which can be used as a predictor for stock return according to

Brunnermeier et al. (2016) and Buncic and Piras (2016).

Finally, we select a number of macro-economic predictors available for each country, we use

the industrial production growth (Engle et al. 2008), the inflation rate (Fama and Schwerts,

1977; Fama, 1981), the unemployment rate (Chen and Zhang, 2009), and the growth in

money supply (Fama, 1981).

3.2 Data set and sources

The data for this research we gather from different available data sources. For example,

stock market data from Yahoo Finance, Thomson Reuters Eikon, and Stooq.com. The data for

our other variables is coming from the databases of the Organisation for Economic Co-

operation and Development (OECD), the International Monetary Fund (IMF), the Federal

Reserve Economic Data (FRED), and from Thomson Reuters Eikon. For a detailed description

of the used data sources see Table 2.

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Table 2: Data set and sources

Country Stock return Consumer Price Index Treasury Bill

Canada Stooq.com, ^TSX OECD, CANCPIALLMINMEI IMF, INTGSTCAM193N

Yahoo!. ^GSPTSE Eikon, CA3MT=RR

France Stooq.com, ^CAC40 OECD, FRACPIALLMINMEI IMF, INTGSTFRM193N

Eikon, FR3MT=RR

Germany Stooq.com, ^DAX OECD, DEUCPIALLMINMEI IMF, INTGSTDEM193N

Eikon, DE3MT=RR

Italy Eikon, MIB Storico OECD, ITACPIALLMINMEI IMF, INTGSTITM193N

Eikon, FTMIB

Japan Yahoo!, Nikkei 225 (^N225) OECD, JPNCPIALLMINMEI IMF, INTGSTJPM193N

Eikon, JP3MT=RR

United Kingdom Eikon, FT30 (.FTII) OECD, GBRCPIALLMINMEI IMF, INTGSTGBM193N

Eikon, FTSE 100 (.FTSE)

United States Yahoo!, S&P 500 (^GSPC) OECD, CPALTT01USQ657N FRED, DTB3

Long Term Bond Yield Unemployment rate Money Supply

Canada OECD, IRLTLT01CAM156N OECD, LRUNTTTTCAM156S OECD, MANMM101CAM189S

France OECD, IRLTLT01FRM156N Eikon, aFRCUNPQ/A Eikon, aFRM1

Germany OECD, IRLTLT01DEM156N OECD, LMUNRRTTDEM156S Eikon, aDECMS3B/A

Italy OECD, IRLTLT01ITM156N OECD, LRHUTTTTITM156S Eikon, aITCHBPM1

Japan Eikon, JP10YT=RR OECD, LRHUTTTTJPM156S OECD, MANMM101JPM189S

United Kingdom OECD, RLTLT01GBM156N IMF, LUR_PT OECD, MANMM101GBM18

FRED, MBM0UKM

United States OECD, IRLTLT01USM156N FRED, UNRATE OECD, MANMM101USM189S

Libor Industrial Production Index

Canada

FRED, LIOR3MUKM FRED, GBP3MTD156N

OECD, MEI_ARCHIVE

France OECD, MEI_ARCHIVE

Germany OECD, MEI_ARCHIVE

Italy OECD, MEI_ARCHIVE

Japan OECD, MEI_ARCHIVE

United Kingdom OECD, MEI_ARCHIVE

United States OECD, MEI_ARCHIVE

Notes: For the variable stock return we use for each of the G7 economies the benchmark indices. However not all current

stock price indices are covering the sample period 1970 till 2020. Therefore, we combine the following index data. For

Canada we use a combination of the current S&P/TSX Composite Index (starting in 2001) and the predecessor the TSE 300

index and a calculation. For France, we use a combination of the CAC40 index founded in December 1987 and a

calculation of the ‘indice Insee de la Bourse de Paris' by Le Bris & Hautcœur (2010). For Germany, the DAX (Deutscher

Aktienindex) is founded in 1988 and of the period up to 1970 we use a calculation. For Italy we use the FTSE MIB index of

the Borsa Italiana starting in 1997, and before this the available MIB Storico dataset up to 1975. For the United Kingdom

we combine the FTSE 100 starting in 1984 and the Financial Times Ordinary Index (FTOI) or also known as the FT 30 to

reach 1970. For the Treasury bill rate, we combine monthly data from the IMF and Eikon to compose a full timeseries

from 1970 – 2020. However, the timeseries of the German equivalent Bubill start at 1974 and the Italian starts at 1977.

For the data on the money supply we use time series on the M1 of most countries, but two exceptions apply. First for

Germany we use the M3 instead. And secondly for the United Kingdom we extend the M1 time series with the M0 time

series due to data availability. Lastly we construct a complete monthly timeseries of the 3-month Libor rate, we use a

combination of ICE Benchmark Administration data and data by the Bank of England obtained at the FRED. Note that the

official data has been used, even though the LIBOR scandal showed that the rates were at times manipulated.

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3.3 Summary Statistics

The summary statistics of our data are presented in Table 3. The stock return and the 8

country predictor variables are presented per G7 country. The LIBOR rate is presented

separately since it is the same for each of the countries of our sample.

For most countries, the time series of each variables are balanced and range over the sample

period of 1970 till 2020 containing 600 observations. However, this is not the case for the

countries France, Germany, and Italy. There is an incomplete time series for the variable

unemployment growth (UNEMP) of France starting at 1975. And for Germany the treasury

bill rate (TBL) is not complete and as such results in an incomplete TED spread and Term

spread. These variables range from the 1974 till 2020. Finally, for Italy several variables are

not available in time series starting in 1970, for example the monthly long government bond

yield of 10 years is only available until 1991. Therefore, a balanced data panel for Italy starts

from 1991.

Additionally, we have calculated the first-order autocorrelation of the of our predictor

variables. We can observe that in general for most countries the autocorrelation of stock

return is quiet low, from which we can conclude that the stock returns are very difficult to

forecast on basis of their past values. For the predictor variables we observe mixed result,

however most predictor variables show an high auto-correlation making them useful for

predicting the stock return.

Concerning the skewness and kurtosis it can be observed that our data is more or less evenly

skewed, but displays leptokurtic values for all variables. These high leptokurtic values

indicate that the variables show unsuspected peaks of outliers. For all countries we observe

high leptokurtic values, however the UK shows several predictor variables with extreme high

kurtosis values. Concerning single predictor variables, we see some extreme leptokurtic

results, such as the money supply (MSM) of Italy displaying a high kurtosis value of 202.70

and for the unemployment (UNEMP) of the United Kingdom with a very high value of

176.00. This is often combined with a skewed sample, for example the money supply of Italy

is skewed to the left (-11.105) and the unemployment is skewed to the left (9.715). This can

indicate unexpected shocks in, respectively, the money supply of Italy at the beginning of the

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sample and for the unemployment of the United Kingdom in the tail of the sample. These

unexpected shocks can bias the estimation based on these predictor variables.

To get a better understanding on our independent variable the change in stock return over

time, we plot the stock price and the stock returns in Appendix 1. As can be observed in both

Table 3 and in the Appendix 1, the range of the stock return3.3 lies general between 0.2 and

-0.2. However, for the stock return of the UK we can see a high maximum value of 0.468

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Table 3: Summary Statistics

Country Variable Mean Std. Dev. Min. Median Max. Skew. Kurt. 𝛒(1) Obs Begin

Canada Return 0.006 0.044 -0.226 0.009 0.160 -0.677 5.90 0.10 600 1970 TBL 0.057 0.042 0.002 0.048 0.208 0.763 3.14 0.99 600 1970 LTY 0.068 0.035 0.010 0.072 0.170 0.266 2.46 0.99 600 1970 IPM 0.001 0.005 -0.017 0.001 0.015 -0.265 3.69 -0.02 600 1970 MSM 0.007 0.016 -0.053 0.008 0.060 -0.205 3.58 0.15 600 1970 UNEMP 0.001 0.027 -0.076 0.000 0.171 0.011 7.17 0.06 600 1970 TED 0.013 0.020 -0.066 0.012 0.097 0.301 5.52 0.94 600 1970 TMS 0.011 0.015 -0.044 0.012 0.042 -0.867 4.49 0.96 600 1970 INFM 0.001 0.002 -0.005 0.001 0.011 0.535 4.53 0.34 600 1970

France Return 0.006 0.056 -0.229 0.010 0.245 -0.132 4.20 0.08 600 1970 TBL 0.058 0.045 -0.009 0.056 0.189 0.278 2.12 0.99 600 1970 LTY 0.070 0.042 -0.003 0.070 0.173 0.303 2.37 1.00 600 1970 IPM 0.000 0.006 -0.022 0.000 0.024 -0.060 3.97 -0.33 600 1970 MSM 0.006 0.014 -0.023 0.005 0.066 0.011 5.56 -0.17 599 1970 UNEMP 0.002 0.010 -0.023 0.000 0.036 0.438 4.02 0.84 538 1975 TED 0.013 0.020 -0.063 0.012 0.064 -0.449 3.48 0.95 600 1970 TMS 0.012 0.013 -0.040 0.014 0.038 -0.968 4.16 0.95 600 1970 INFM 0.001 0.002 -0.020 0.001 0.008 0.018 2.61 0.49 600 1970

Germany Return 0.007 0.056 -0.254 0.007 0.214 -0.383 4.98 0.05 600 1970 TBL 0.036 0.028 -0.009 0.036 0.121 0.309 2.62 0.99 546 1974 LTY 0.055 0.029 -0.007 0.061 0.108 -0.405 2.22 1.00 600 1970 IPM 0.000 0.007 -0.043 0.001 0.050 -0.340 9.92 -0.28 600 1970 MSM 0.005 0.029 -0.101 0.005 0.134 0.085 4.35 0.04 600 1970 UNEMP 0.006 0.087 -0.350 0.000 0.651 1.980 16.00 0.33 600 1970 TED 0.033 0.025 -0.002 0.025 0.107 0.825 2.80 0.33 546 1974 TMS 0.016 0.014 -0.043 0.016 0.054 -0.345 3.72 0.97 546 1974 INFM 0.001 0.002 -0.025 0.001 0.008 -4.677 71.98 0.09 600 1970

Italy Return 0.008 0.008 -0.201 0.005 0.317 0.485 4.76 0.09 538 1975 TBL 0.073 0.061 -0.004 0.050 0.216 0.412 1.90 0.99 514 1977 LTY 0.056 0.034 0.009 0.045 0.144 1.142 3.31 0.99 346 1991 IPM 0.000 0.009 -0.069 0.000 0.055 -0.293 11.58 0.00 599 1970 MSM 0.005 0.057 -1.000 0.003 0.158 -11.105 202.70 0.28 478 1980 UNEMP 0.004 0.036 -0.167 0.000 0.333 2.902 26.78 -0.23 599 1983 TED -0.007 0.030 -0.098 0.003 0.036 -1.008 3.12 0.98 514 1977 TMS -0.036 0.077 -0.216 0.008 0.043 -0.865 2.08 0.90 514 1977 INFM 0.002 0.002 -0.003 0.001 0.014 1.594 6.12 0.34 599 1970

Table3 continues on next page

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Table 3: Summary Statistics (cont.)

Country Variable Mean Std. Dev.

Min. Median Max. Skew. Kurt. 𝛒(1) Obs Range

Japan Return 0.005 0.055 -0.238 0.010 0.201 -0.379 4.16 0.04 600 1970 TBL 0.023 0.024 -0.004 0.016 0.068 0.423 1.55 0.99 600 1970 LTY 0.040 0.031 -0.003 0.035 0.103 0.265 1.60 0.99 600 1970 IPM 0.001 0.007 -0.075 0.001 0.028 -2.545 2.40 0.00 600 1970 MSM 0.006 0.009 -0.030 0.005 0.090 2.078 17.54 0.28 600 1970 UNEMP 0.002 0.039 -0.154 0.000 0.200 0.349 4.94 .023 600 1970 TED 0.047 0.030 -0.006 0.049 0.124 0.192 2.30 0.98 600 1970 TMS 0.017 0.009 0.000 0.016 0.064 0.324 3.36 0.90 600 1970 INFM 0.001 0.003 -0.005 0.000 0.018 1.888 9.31 0.34 600 1970

UK Return 0.006 0.055 -0.260 0.009 0.468 0.741 12.86 0.05 600 1970 TBL 0.065 0.044 0.002 0.058 0.162 0.168 2.08 099. 600 1970 LTY 0.075 0.041 0.006 0.077 0.163 0.114 1.91 0.99 600 1970 IPM 0.000 0.006 -0.036 0.000 0.041 -0.179 13.90 -0.17 599 1970 MSM 0.008 0.012 -0.029 0.007 0.144 3.968 38.04 0.04 599 1970 UNEMP 0.002 0.042 -0.159 0.000 0.765 9.715 175.99 0.28 600 1970 TED 0.006 0.005 -0.005 0.005 0.037 2.424 11.42 0.88 600 1970 TMS 0.011 0.016 -0.040 0.012 0.058 -0.143 3.27 0.79 600 1970 INFM 0.002 0.003 -0.003 0.001 0.018 2.261 11.98 0.05 600 1970

US Return 0.005 0.055 -0.238 0.010 0.201 -0.379 4.16 0.03 600 1970 TBL 0.046 0.034 0.000 0.049 0.155 0.579 3.23 0.99 600 1970 LTY 0.063 0.031 0.015 0.063 0.153 0.538 2.92 0.99 600 1970 IPM 0.001 0.003 -0.019 0.001 0.010 -1.143 8.44 0.34 600 1970 MSM 0.003 0.042 -1.000 0.005 0.058 -23.283 5.62 0.15 600 1970 UNEMP 0.000 0.028 -0.085 0.000 0.125 0.546 4.02 0.10 600 1970 TED 0.024 0.024 -0.027 0.019 0.103 0.702 2.98 0.96 600 1970 TMS 0.017 0.012 -0.020 0.018 0.047 -0.422 2.69 0.94 600 1970 INFM 0.001 0.002 -0.008 0.001 0.008 -0.076 6.06 0.03 600 1970

All LIBOR 0.070 0.046 0.003 0.064 0.181 0.184 2.09 0.99 600 1970

Notes: This table reports the summary statistics of the stock return and the predictors employed for each of the seven countries, where UK stands for the United Kingdom and US for the United States. It includes the mean, standard deviation (Std.), Minimum (Min.), Median, Maximum (Max.), Skewness (Skew.), Kurtosis (Kurt.), Observations (Obs.), as well as the available range of the time series in first year and last year since the data set is not balanced for each country and variable. The variable Return is the independent variable return on the stock price from each benchmark stock index. Next the predictor variables are TBL is the treasury bill rate, LTY is the long term government bond yield, IPM is the monthly log change of the industrial price index, the MSM is the monthly rate of change in the money supply, UNEMP is the rate of change of the unemployment rate, INFM is the monthly inflation rate. For all countries the used LIBOR rate is the same. We make use of two indicators of spread; The monthly TED spread between the 3 month Libor rate and the 3 month treasury bill rate. And the monthly term spread TMS between the long-term yield and 3 month T-bill rate.

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4 Methodology

In this section, we discuss the predictive models employed. How we construct the forecasts

and how we evaluate these forecasts.

4.1 Forecasting model To analyse the predictive power of our predictor variables on stock return we start by

considering a conventional bivariate regression model. This bivariate predictive model is

defined by

𝑟𝑡+1 = 𝛼 + 𝛽1𝑥𝑖,𝑡 + 𝜀𝑖,𝑡+1 (1)

where the independent variable 𝑟𝑡+1 is the return on the stock market indices of each

country from period 𝑡 until 𝑡 + 1. The candidate predictor variable 𝑖, with 𝑖 = 1, … ,8 for the

predictor variables, at time 𝑡 is set out by 𝑥𝑖,𝑡. The error term is illustrated by 𝜀𝑖,𝑡+1 from

which the mean is equal to zero. In the situation of the null hypothesis being zero 𝐻0: 𝛽1 =

0 there is no predictability, downgrading it to a model of constant returns. This goal is for

the ordinary least squares (OLS) to reduce the residual sum of squares (RSS) to a minimum,

which is defined by

where 𝑁 the is the predictor variable taken; and 𝑇 the point in time in the sample.

In order to estimate the stock market returns the frequently used one-step-ahead

estimation approach is considered (Campbell and Thompson, 2008; Neely et al., 2014;

Rapach et al., 2016). This recursively generates a continuously expanding estimation window

out of the in-sample period until the end of the out-of-sample period is reached. The total

sample of 𝑇-observation compromises an in-sample period of 𝐴 and the out-of-sample

period 𝑃 = 𝑇 − 𝐴. With respect to the multi-country sample used each country sample is

composed as described by

𝑅𝑆𝑆 = ∑ (𝑟𝑖 − 𝛼 − ∑ 𝑥𝑖,𝑡𝛽1

𝑇

𝑡=1

)

2𝑁

𝑖=1

(2)

16

Table 4: Sample division per country

Country Range T A P

Canada 1971M1 – 2019M12 588 120 468 France 1976M1 – 2019M12 528 60 468 Germany 1976M1 – 2019M12 528 72 456 Italy 1992M1 – 2019M12 336 60 276 Japan 1971M1 – 2019M12 588 120 468 United Kingdom 1971M1 – 2019M12 588 120 468 United States 1971M1 – 2019M12 588 120 468

Following the one step ahead forecast approach the following out-of-sample estimation

model for return is defined by

where ��𝑖,𝑡+1 is the predicted stock return based on the 𝑖th predictor. The ordinary least

square estimates ��𝑖,𝑡 and ��𝑖,𝑡 are acquired from respectively 𝛼 and 𝛽 by doing the following

regressions. Namely, {𝑟𝑡+1,}𝑡=1

𝑃 on a constant and {𝑥𝑖,𝑡}

𝑡=1

𝑃−1. Recursively executing this for

each time 𝑡 in our sample, this results in a series of 𝑃 out-of-sample estimations for the

stock return {��𝑖,𝑡+1}𝑡=1

𝑃.

This forecast obtained by the out-of-sample estimation model for stock return is compared

to the benchmark of the historical average. This approach can be found in numerous

forecasting articles (e.g. Welch and Goyal, 2008; Campbell and Thompson, 2008) and it is

based on the assumption that the stock return is expected to be constant (𝑟𝑡+1 = 𝛼 + 𝜀𝑡+1).

Consequently, the historical average forecast is defined by

where �� is the historical average return. And 𝑇 is the total sample considered.

Demonstrating that this is a very strict benchmark. In general regression forecast based on

individual macroeconomic variables tend to fail to outperform this historical benchmark.

��𝑖,𝑡+1 = ��𝑖,𝑡 + ��𝑖,𝑡𝑥𝑖,𝑡 (3)

�� =1

𝑇∑ 𝑟𝑖

𝑇

𝑖=1

(4)

17

4.2 Principal component model

To incorporate the information of the various predictors we use the principal component

analysis earlier employed by Neely et al. (2014). We estimate a predictive regression based

on a small number of principal components of the entire data set. This summarizes and

extracts information out of a large group of variables and reduces the noise. By this method

we transform our 𝑁 = 8 predictor variables 𝑋𝑡 = (𝑋1,𝑡, . . . , 𝑋𝑁,𝑡)𝑡 to a novel uncorrelated

variable ��𝑡𝐸𝐶𝑂𝑁 = (��1,𝑡

𝐸𝐶𝑂𝑁 , . . . , ��𝐾,𝑡𝐸𝐶𝑂𝑁), which contains the first 𝐾 principal components

extracted from 𝑋𝑡. The predictive regression model of the principal component is defined by

𝑟𝑡+1 = 𝛼 + ∑ 𝛽𝑘��𝑘,𝑡

𝐸𝐶𝑂𝑁

𝐾

𝑘=1

+ 𝜀𝑡+1

(5)

where ��𝑘,𝑡𝐸𝐶𝑂𝑁 is the 𝐾th principal component of the 𝑁 group of predictors, which are

recursively estimated until time 𝑡. And 𝛼 and 𝛽𝑘 are constants calculated by the least

squares and 𝐾 is the number of principal components.

4.3 Combination forecast model

Next to the principal component model we employ another model to incorporate the

information of various predictors. This is a simple model of forecast combination, a popular

procedure to both decrease model uncertainty and effectively include the information of

substantial sets of potential predictors. Numerous related researches report a significant

better performance of combination forecast in comparison to individual forecast (Rapach et

al., 2010; Zhu and Zhu, 2013; Buncic and Piras, 2016; Ekaterini and Souropanis, 2019; Zhang

et al. 2019).

The combination forecast is computed as weighted averages of the 𝑁 predictor forecast

based on equation (1). Statistically the combination forecast is defined by

��𝑐,𝑡+1 = ∑ 𝜔𝑖,𝑡��𝑖,𝑡+1

𝑁

𝑖=1

(6)

where ��𝑐,𝑡+1 represents the combination forecast at month 𝑡 + 1. And rc,t+1 is the 𝑖th

individual forecast, and 𝜔𝑖,𝑡 represents the combining weight of the 𝑖th individual forecast

18

calculated at month 𝑡. For simplicity and comprehensibility, this research uses the equal-

weighted mean combination, which is 𝜔𝑖,𝑡 =1

𝑁=

1

8 for the 8 predictor variables.

4.4 Forecasting constraints

In this section, we discuss the two employed constraints in detail. First we will discuss the

momentum constraint and secondly the lasso constraint.

4.4.1 Momentum Constraint

The first restriction we employ to forecast stock return is an economic based constraint

using the momentum of stock returns to form a forecast. This is previously used by Wang et

al. (2015) on forecasting the real oil prices, where they anticipated that the signs of the

regression coefficients are consistent with economic theory. In contrast of using the signs of

the regression coefficient, we apply the economic constraint on the sign of the predicted

forecast.

When in case of our sample the stock return is showing an increase for a longer period, we

assume this increase of stock return remains the same for the next period. Though, when

suddenly a shock occurs the stock return will change sharply in a short period of time. From

an economic point of view forecasts based on this sudden change are misleading and could

lead to a larger loss compared to a no-change forecast. Consequently, it is more rational to

discard the forecast in case an abnormal in-sample prediction is found. Therefore, this

constraint can be defined by

where ��𝑖,𝑡+1 is the forecasted return at time 𝑡 + 1 and 𝑟𝑖,𝑡 is actual return at time 𝑡. In

summary, for the momentum constraint if the sign of the predicted return is a poorly

approximation of the sign of the actual return, the forecast zero change is used. Otherwise,

the predicted forecast is used.

��𝑖,𝑡+1𝑆𝑖𝑔𝑛

= {0, 𝑖𝑓 𝑠𝑖𝑔𝑛(��𝑖,𝑡+1)~𝑠𝑖𝑔𝑛(𝑟𝑖,𝑡)

��𝑖,𝑡+1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(7)

19

4.4.2 Lasso constraint

The second constraint is comparable to the previous constraint, however it offers a way to

minimize an abnormal prediction. If the sign of previous actual 𝑟𝑖,𝑡 at time 𝑡 is different

compared to the sign of the forecast ��𝑖,𝑡+1 at 𝑡 + 1, we apply a lasso regression. If the signs

are not different we apply a simple OLS regression equation, or in other words we keep the

prediction ��𝑖,𝑡+1. This constrain is defined by

where the 𝑙𝑎𝑠𝑠𝑜 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 expresses the case in which we will perform a Lasso regression

instead of an OLS. And otherwise when this condition is not met the predicted OLS forecast

��𝑖,𝑡+1 is used.

The Lasso regression is used as statistical constraint based on shrinkage estimation.

Shrinkage estimation entails that by shrinking the mean squared error (MSE) that the

performance of the out-of-sample estimation improves. It has been popularized by

Tibshirani (1996) and is an acronym for Least Absolute Shrinkage and Selection Operator.

This operator is designed to shrink the absolute value of the regression coefficient and

therefore performing variable selection and promoting model interpretation. The lasso

regression is previously been used in forecasting by Li et al. (2015) and Li and Tsiakas (2017)

as part of their kitchen sink regression to predict equity returns.

In a normal OLS regression the goal is to minimize the residual sum of squares, with the

Lasso regression the goal is to minimalize the Penalized Sum of Squares with respect to our

𝛼 and 𝛽. This PSS is defined by

��𝑖,𝑡+1𝑆𝑖𝑔𝑛

= {𝑙𝑎𝑠𝑠𝑜 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 , 𝑖𝑓 𝑠𝑖𝑔𝑛(��𝑖,𝑡+1)~𝑠𝑖𝑔𝑛(𝑟𝑖,𝑡)

��𝑖,𝑡+1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(8)

𝑃𝑆𝑆 = ∑ (𝑟𝑖 − 𝛼 − ∑ 𝑥𝑖,𝑡𝛽𝑡

𝑇

𝑡=1

)

2𝑁

𝑖=1

+ 𝜆 ∑|𝛽𝑡|

𝑇

𝑡=1

(9)

20

where 𝜆 ∑ |𝛽𝑡|𝑇𝑡=1 is the introduced penalty term compared to the OLS regression; with 𝜆

being the penalty term penalizing the measurement error. And |𝛽𝑡| is the absolute value of

the slope of the regression at time 𝑡, making it impossible to take a negative sign.

5 Empirical Findings

In this section, the empirical findings will be discussed. Beginning with discussing the out-of-

sample performance of our forecast, including statistical significance of the findings. Next the

economic significance of the result will be described. And finally, the economic performance

will be discussed, via the Sharpe ratio and the certainty equivalent return.

5.1 Performance Evaluation

To evaluate the predictive accuracy of the our-of-sample return forecast, we use the

conventional 𝑅2 statistics. This statistic was endorsed by Campbell and Thompson (2008) and

measures the out-of-sample predictive accuracy of the model relative to the benchmark of

historical average. This out-of-sample 𝑅2 is statistically defined by

𝑅𝑂𝑆

2 = 1 −∑ (𝑟𝐴+𝑘 − 𝑟 𝐴+𝑘)𝑃

𝑘=12

∑ (𝑟 𝐴+𝑘 − 𝑟 𝐴+𝑘)𝑃𝑘=1

2 (10)

where 𝑟 𝐴+𝑘 is the actual stock return, 𝑟 𝐴+𝑘 the historical benchmark average, and 𝑟 𝐴+𝑘 the

predicted return at the month 𝐴 + 𝑘. And 𝐴 indicates the length of the initial estimation

period and 𝑃 the forecast evaluation period.

The 𝑅𝑂𝑆2 statistic proportionally measures the reduction in the mean squared forecast error

(MFSE) for the return forecast relative to the historical average benchmark. In case the 𝑅𝑂𝑆2

value is positive the MFSE of the historical mean is outperformed by the forecast of our

predictive regression. And when negative, the predictive forecast is outperformed. To further

ascertain whether a forecasting model yields a statistically significant improvement of the

MFSE, the Clark and West (2007) statistic is relevant.

21

5.2 Statistical significance

For testing the statistical significance of our estimation results we employ the Mean Squared

Forecast Error (MFSE) test statistic of Clark and West (2007). This test statistic is useful for

our research, since it considers estimation bias introduced by larger nested models. The

Clark and West (2007) statistic tests the null hypothesis 𝐻0 whether the MSFE of the

historical benchmark average is smaller than or equal to the MFSE of a particular forecasted

model. Or the alternative hypothesis 𝐻1 is that the MSFE of the historical benchmark

average is larger than that of the forecasted model.

𝑓𝑡 = (𝑟𝑡 − 𝑟 𝑡)2 − (𝑟𝑡 − 𝑟 𝑡)2 + (𝑟�� − 𝑟 𝑡)2 (11)

where 𝑟𝑡 is the stock return, 𝑟 𝑡 is the forecast of the stock return, and 𝑟 𝑡 is historical mean

of stock return. Next we derive the Clark and West (2007) statistic by regressing {𝑓𝑠}𝑠=𝑚+1𝑇

on a constant, which is actually the t-statistic of the constant. And the p-value for the one-

sided upper-tail test, for using a standard normal distribution can be obtained accordingly.

In Table 5 the out-of-sample performance is reported for the unconstrained model, the

momentum constraint, and the Lasso constrained model. The out-of-sample performance is

measured using the 𝑅𝑂𝑆2 statistic, a positive value for this statistic corresponds to a predictor

variable outperforming the historical average benchmark. Firstly, we observe positive and

significant 𝑅2-values for the countries Canada and France in the momentum constraint

model. Therefore, we can see that they are out-performing the historical benchmark.

However, the other countries are outperformed by the benchmark showing negative 𝑅2-

values.

Secondly, we note that for the lasso constraint only some predictors of Canada showing

positive 𝑅2-values and statically significant results. And all other countries and predictors

being again appear to be outperformed by historical average benchmark.

However, we do see a positive change in the 𝑅2-values compared to the unconstraint model

to the momentum constraint model for Japan. And an overall positive change between the

unconstraint model and the Lasso constraint model for all countries. Therefore, we can

assume that applying the Lasso constraint shows positive results compared to the

unconstrainted model.

22

Table 5: Out-of-sample Predictability of Stock Returns

Unconstraint Momentum Constraint Lasso Constraint CAN FRA DEU ITA JPN GBR USA CAN FRA DEU ITA JPN GBR USA CAN FRA DEU ITA JPN GBR USA

INFM -0.65 -0.66 -0.46 -0.85 -0.56 -1.90 -0.17 2.02*** 0.81** -0.90 -0.90 -0.48 -1.84 -1.28 0.30* 0.17 -0.35 -0.64 -0.27 -0.96 0.07

TBL -0.58 -0.59 -0.61 -1.51 -0.76 -0.64 -0.46 2.22*** 0.39* -0.82 -2.27 -0.72 -2.24 -1.78 0.73** -0.25 -0.19 -1.92 -0.58 -0.30 -0.33

LTY -1.13 -0.62 -0.90 -1.75 -0.79 -0.38 -1.03 2.26*** 0.34* -0.84 -2.50 -0.60 -2.10 -1.58 0.71* -0.21 -0.25 -2.16 -0.25 0.00 -0.36

IPM -0.75 -0.47 -0.24 -1.61 -0.63 0.06 -0.03 1.32** 0.48* -1.22 -1.33 -0.52 -2.13 -0.93 -0.38 -0.30 -0.55 -0.77 -0.36 -0.03 0.51*

MSM -0.44 -0.33 -0.54 -0.31 0.18 -13.74 -0.98 1.78*** 0.62* -1.02 -0.39 0.01 -12.25 -2.46 -0.06 0.15 -0.30 -0.05 0.18* -10.95 -1.03

LIBOR -0.37 -0.61 -0.66 -1.85 -1.01 -1.19 -0.40 2.05*** 0.23 -1.03 -1.36 -0.81 -2.81 -1.62 0.21 -0.32 -0.43 -0.74 -0.93 -1.45 -0.33

UNEMP -0.95 -0.52 -0.58 0.55** -0.26 -9.12 -2.62 1.66*** 0.24 -0.90 -0.53 -0.40 -4.64 -2.67 -0.11 -0.33 -0.25 -0.59 -0.44 -3.25 -1.28

TMS 0.07 -0.37 -0.31 -0.72 -0.52 -0.80 -0.98 2.01*** 0.58* -0.72 -0.85 -0.45 -2.63 -1.58 0.17 -0.23 -0.12 -0.35 -0.26 -0.58 -0.31

TED -0.31 -0.39 -0.38 -2.09 -0.22 -1.80 -0.39 1.70*** 0.37* -1.02 -2.68 -0.43 -4.13 -1.63 0.08 -0.27 -0.43 -2.42 -0.36 -1.88 -0.41

POOL -0.16 -0.25 -0.35 -0.27 -0.11 -1.10 -0.19 2.11*** 0.62* -0.85 -0.94 -0.27 -2.88 -1.40 0.43** -0.01 -0.22 -0.56 -0.14 -1.15 -0.03

PCA -1.12 -1.12 -1.03 -2.41 -1.63 -9.62 -1.30 2.33*** 0.29* -1.32 -2.87 -1.23 -6.94 -1.35 0.85** -0.38 -0.75 -1.96 -0.63 -4.71 -0.05*

Notes: This table reports the out-of-sample (𝑅2) from equation (10) for predictability regressions of stock returns for the unconstraint model, the momentum constraint model, and the Lasso constraint model. The unconstraint model describes the initial forecast without any imposed constraints as described in equation (3). Next the momentum constraint is described in equation (7). The Lasso constraint describes the results of the sign restriction with the Lasso regression see equation (8).

The abbreviations for the predictors INFM, TBL, LTY, IPM, MSM, LIBOR, UNEMP, TMS, TED are described in Table 1. Furthermore, we demonstrate the OOS predictability

of the combined forecast POOL and the principal component PCA forecast. CAN, FRA, DEU, ITA, JPN, GBR and USA are the ISO abbreviation we use for respectively Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States. INFM is the growth rate of Inflation. The statistical significance of the is measured measure statistical significance relative to the prevailing mean model using the Clark and West (2007) test statistic. * significance at 10% level; ** significance at 5% level; *** significance at 1% level.

23

5.3 Economic evaluation

5.3.1 Economic performance

Next, we measure the economic performance of the three out-of-sample prediction models.

We do this by computing, the out-of-sample Sharpe ratio (SR) and the certainty equivalent

rates of return (CER) we incorporate risk aversion into an asset allocation decision. This in

line with previous researches such as Campbell and Thomson (2008) and Neely et al. (2014).

5.3.2 Sharpe Ratio A commonly used measure of the economic value of stock return predictability is the Sharpe

Ratio (SR), for each of the predictive models we compute this ratio. This ratio was introduced

by Sharpe (1964) and is the average return per unit of volatility, where volatility is a measure

of the fluctuation. Statistically the Sharpe ratio is defined by

𝑆𝑅 = 𝑟𝑝 − 𝑖𝑡

𝜎𝑝

(12)

where 𝑟𝑝 − 𝑖𝑡 is the average return and 𝜎𝑝 is the standard deviation of the equivalent

returns. For each predictive model we test the significance based on difference between the

Sharpe ratios of our predictive models and the benchmark, testing whether the benchmark

Sharpe ratio is equivalent to the ratio of the prediction.

5.3.3 Certainty Equivalent Return

In addition, the next economic performance measure is the certainty equivalent return (CER)

and is used to judge the relative performance of our models. This is an asset allocation

perspective and incorporates individual investor risk tolerance. To be more precise, the CER

can be interpreted as the willingness of investors to accept a certain risk-free return instead

of investing in a risky return (Kandel and Stambaugh, 1996; Pastor and Stambaugh, 2000).

The 𝐶𝐸𝑅 is computed as the relative performance between the 𝐶𝐸𝑅 of the return forecasts,

and the 𝐶𝐸𝑅 of the historical benchmark. According to Zhang et al. (2019) the CER gain can

be examined as the fee a investor is willing to pay to have access to the predictive forecast

next to forecasts from the historical benchmark. Statistically the CER is divined by

24

𝐶𝐸𝑅𝑗 = 𝜇𝑗 +𝛾

2 𝜎𝑗

2 (13)

where 𝜇𝑗 and 𝜎𝑗2 are respectively the mean and variance of out-of-sample excess returns for

the unconstraint, the momentum constraint, and Lasso constraint model. And 𝛾 is the risk

aversion coefficient, for which we will take two levels of risk-aversion 𝛾 = 5 and the more

risk averse 𝛾 = 10. This is base on the idea that risk-averse investor is more likely to pay a

performance fee to shift from a more risky portfolio based on a random walk model, to a

portfolio based on the certainty of the historical benchmark (Corte et al. 2009).

5.3.4 Economic performance findings

For the economic evaluation we report two tables, Table 6 and Table 7 respectively

reporting the Sharpe ratio (SR) and certainty equivalent return (CER) for the unconstraint

model, the momentum constraint model and the Lasso constraint model.

Regarding the Sharpe ratio we observe that there are no ratios higher than the value 1.

However, we do observe a gain in the Sharpe ratios when implementing the Momentum

constraint and the Lasso constraint. Especially for the latter we observe gain for almost

predictive variables of all the sampled G7 countries, except for most predictor variables of

Italy. For the momentum constraint we observe lower values for all or most predictor

variables of the countries Germany, Italy, the United Kingdom, and the United States.

Regarding the CER in Table 7 we see specifically that for both the risk aversion of 𝛾 = 5 as

𝛾 = 10 a relative gain is caused by the lasso constraint compared to the unconstraint model

for almost all the predictor variables including POOL and PCA. However, for the momentum

constraint we see this gain partly. We observe a loss in the CER for the countries Germany,

United Kingdom, and the Unites States. Additionally, in the momentum constraint we see a

gain for the POOL model and a loss for the PCA model.

Noteworthy is that although most CER values of the forecast are negative, we do observe for

some positive values. For both the risk aversion of 𝛾 = 5 as 𝛾 = 10 positive values for

Canada, France, and Italy are observed when using the momentum constraint and for

Canada even when using the Lasso constraint.

25

Table 6: Economic Evaluation with the Sharpe Ratio

Unconstrained Momentum Constrained Lasso Constrained CAN FRA DEU ITA JPN GBR USA CAN FRA DEU ITA JPN GBR USA CAN FRA DEU ITA JPN GBR USA

INFM 0.34 0.39 0.45 0.04 0.42 0.22 0.52 0.63 0.49 0.39 0.02 0.37 0.09 0.50 0.43 0.44 0.49 -0.02 0.43 0.30 0.58 TBL 0.38 0.31 0.47 0.17 0.38 0.29 0.50 0.61 0.45 0.40 0.13 0.32 0.08 0.42 0.46 0.35 0.51 0.10 0.39 0.31 0.52

LTY 0.34 0.30 0.44 0.20 0.42 0.31 0.44 0.58 0.45 0.41 0.15 0.37 0.11 0.43 0.46 0.35 0.50 0.12 0.44 0.33 0.50

IPM 0.30 0.33 0.49 -0.05 0.36 0.34 0.50 0.58 0.45 0.35 0.00 0.29 0.08 0.49 0.31 0.35 0.45 0.01 0.36 0.32 0.56

MSM 0.33 0.37 0.43 0.00 0.40 0.22 0.46 0.62 0.49 0.36 0.09 0.38 -0.03 0.34 0.35 0.40 0.47 0.03 0.42 0.15 0.46

LIBOR 0.39 0.32 0.49 -0.09 0.44 0.29 0.49 0.61 0.44 0.38 -0.03 0.35 0.07 0.41 0.43 0.36 0.48 -0.01 0.38 0.28 0.50

UNEMP 0.25 0.36 0.46 0.39 0.42 -0.04 0.36 0.61 0.43 0.38 0.32 0.32 -0.02 0.30 0.35 0.37 0.50 0.22 0.36 0.16 0.43

TMS 0.38 0.34 0.49 -0.03 0.37 0.27 0.46 0.64 0.48 0.41 0.02 0.32 0.03 0.44 0.38 0.37 0.51 0.01 0.39 0.29 0.53

TED 0.30 0.35 0.49 0.11 0.45 0.34 0.48 0.59 0.46 0.37 0.09 0.34 0.06 0.41 0.34 0.36 0.47 0.05 0.37 0.28 0.49

POOL 0.35 0.35 0.48 0.12 0.43 0.26 0.50 0.62 0.48 0.39 0.12 0.36 0.05 0.45 0.41 0.38 0.49 0.07 0.41 0.27 0.53 PCA 0.33 0.33 0.44 0.20 0.41 0.10 0.44 0.62 0.45 0.35 0.12 0.30 -0.09 0.47 0.47 0.38 0.45 0.12 0.41 0.23 0.56

Notes: This table reports the Sharpe ratio (SR) for the unconstraint model and the momentum constraint model and the Lasso constraint model. The unconstraint model

describes the initial forecast without any imposed constraints as described in equation (3). Next the momentum constraint is described in equation (7). The Lasso constraint

describes the results of the sign restriction with the Lasso regression see equation (8). And the Sharpe ratio is calculated using Sharpe (1994) methodology, this can be

found in equation (12). For estimation period range see Table 4. The abbreviations for the predictors INFM, TBL, LTY, IPM, MSM, LIBOR, UNEMP, TMS, TED are described in

Table 1. Furthermore, we demonstrate the OOS predictability of the combined forecast POOL and the principal component PCA forecast. CAN, FRA, DEU, ITA, JPN, GBR and

USA are the ISO abbreviation we use for respectively Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States. INFM is the growth rate of

Inflation.

26

Table 7: Economic evaluation with Certainty Equivalent Return

Unconstrained Momentum Constrained Lasso Constrained CAN FRA DEU ITA JPN GBR USA CAN FRA DEU ITA JPN GBR USA CAN FRA DEU ITA JPN GBR USA

Panel A: Certainty Equivalent Return γ = 5

INFM -1.66 -3.64 -1.19 -1.92 -1.97 -0.20 -1.70 3.16 0.36 -1.25 0.07 0.04 -0.81 -1.13 0.43 -1.46 -0.40 -2.57 -1.03 0.23 0.10

TBL -2.04 -0.67 -1.96 -3.01 -3.46 -0.60 -0.61 2.96 1.60 -1.78 -0.95 -1.30 -2.06 -1.51 0.42 -0.23 -0.75 -3.44 -2.52 -0.26 -0.22

LTY -3.92 -0.44 -3.10 -4.63 -6.72 -0.51 -0.97 2.58 1.69 -1.94 -2.45 -2.87 -2.10 -1.25 0.04 -0.05 -1.02 -4.94 -3.84 -0.24 -0.24

IPM -0.59 -0.81 0.20 -1.13 -1.20 0.26 -0.39 2.47 1.51 -1.37 1.92 -0.11 -1.79 -0.88 -0.29 -0.52 -0.56 -0.07 -1.23 0.07 0.41

MSM -0.39 -0.36 -1.65 -0.54 0.55 -1.53 -0.68 2.84 1.94 -1.71 2.43 1.78 -3.14 -1.94 0.00 0.22 -0.74 -0.05 0.66 -2.49 -0.76

LIBOR -1.06 -0.74 -1.12 -1.28 0.45 -1.17 -0.44 3.01 1.56 -1.77 1.81 0.42 -2.59 -1.40 0.16 -0.18 -0.86 -0.34 -1.46 -0.95 -0.30

UNEMP -1.67 -1.42 -0.60 2.29 1.50 -16.73 -3.44 2.82 0.90 -1.21 1.97 1.46 -4.60 -2.86 0.06 -0.75 -0.16 -0.55 0.02 -3.22 -1.86

TMS 0.15 -0.75 -0.64 -0.81 -3.72 -0.44 -2.02 3.01 1.81 -1.27 1.99 -0.78 -2.04 -1.41 0.17 -0.19 -0.34 -0.30 -1.94 -0.23 -0.29

TED -0.62 -0.01 -0.47 -3.99 1.73 -1.33 -0.60 2.63 1.85 -1.59 -1.44 1.07 -3.74 -1.42 -0.04 0.01 -0.73 -4.03 -0.44 -1.34 -0.39

POOL -0.80 -0.61 -0.80 -0.44 -0.13 -0.73 -0.56 3.03 1.70 -1.34 1.47 0.66 -2.01 -1.22 0.43 -0.12 -0.45 -0.95 -0.64 -0.51 -0.09

PCA -3.20 -1.88 -3.56 -3.13 -4.22 -1.66 -3.92 3.09 1.23 -2.99 -1.67 -3.13 -2.27 -1.63 0.56 -0.72 -2.14 -3.31 -2.47 -0.76 -0.57

Panel B: Certainty Equivalent Return γ = 10

INFM -1.00 -2.33 -0.61 -0.92 -1.21 -0.15 -1.06 1.69 -0.26 -0.70 0.25 -0.27 -0.54 -0.99 0.13 -1.19 -0.19 -1.22 -0.71 0.12 -0.13

TBL -1.24 -0.38 -1.04 -1.26 -1.83 -0.37 -0.30 1.53 0.84 -1.06 -0.25 -0.88 -1.20 -1.00 0.09 -0.09 -0.44 -1.62 -1.43 -0.14 -0.14

LTY -2.26 -0.26 -1.64 -2.03 -3.59 -0.35 -0.49 1.35 0.91 -1.13 -0.99 -1.85 -1.24 -0.80 -0.10 0.02 -0.56 -2.37 -2.29 -0.16 -0.08

IPM -0.28 -0.44 0.41 -0.48 -0.53 0.16 -0.26 1.47 0.75 -0.64 1.29 -0.17 -1.04 -0.66 -0.12 -0.31 -0.15 0.17 -0.63 0.04 0.21

MSM -0.21 -0.14 -0.86 -0.26 0.29 -0.49 -0.26 1.63 0.98 -0.93 1.37 0.84 -1.51 -1.14 0.01 0.08 -0.32 -0.01 0.37 -1.03 -0.37

LIBOR -0.68 -0.34 -0.57 -0.66 0.13 -0.68 -0.17 1.59 0.84 -1.00 1.09 -0.10 -1.48 -0.89 -0.04 -0.05 -0.44 -0.13 -1.17 -0.49 -0.12

UNEMP -0.85 -0.82 -0.22 1.18 1.12 -9.15 -1.90 1.64 0.33 -0.59 0.60 0.81 -2.50 -1.68 0.06 -0.49 -0.02 -0.60 0.15 -1.61 -1.09

TMS 0.04 -0.45 -0.33 -0.40 -1.85 -0.22 -1.30 1.66 0.90 -0.77 1.18 -0.51 -1.15 -1.00 0.03 -0.14 -0.20 -0.11 -1.03 -0.09 -0.23

TED -0.33 0.07 -0.20 -1.70 0.87 -0.48 -0.34 1.55 1.02 -0.89 -0.40 0.33 -2.04 -0.91 0.02 0.08 -0.36 -1.84 -0.32 -0.56 -0.18

POOL -0.49 -0.38 -0.38 -0.15 -0.09 -0.45 -0.36 1.67 0.82 -0.75 0.90 0.15 -1.14 -0.85 0.18 -0.12 -0.22 -0.44 -0.45 -0.22 -0.10

PCA -1.91 -1.09 -1.81 -1.19 -2.17 -0.80 -2.53 1.60 0.53 -1.66 -0.59 -2.06 -1.12 -1.28 0.15 -0.48 -1.14 -1.50 -1.30 -0.23 -0.58

Notes: This table reports the certainty equivalent return (CER) for the unconstrained and the momentum constraint and lasso constraint forecast. The unconstraint model describes the initial forecast without any imposed constraints as described in equation (3). Next the momentum constraint is described in equation (7). The Lasso constraint describes the results of the sign restriction with the Lasso regression see equation (8).The CER in panel A and panel B can be calculate by using equation (13) when respectively 𝛾 = 5 and 𝛾 = 10. For estimation period range see Table 4. The abbreviations for the predictors INFM, TBL, LTY, IPM, MSM, LIBOR, UNEMP, TMS, TED are described in Table 1. Furthermore, we demonstrate the OOS predictability of the combined forecast POOL and the principal component PCA forecast. CAN, FRA, DEU, ITA, JPN, GBR and USA are the ISO abbreviation we use for respectively Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States. INFM is the growth rate of Inflation.

27

5.4 Economic Significance

Now we have observed the statistical significance and the economic performance of the

stock return forecasts of our unconstrained and the constrained models, we will evaluate

the economic significance of the out-of-sample stock return forecast. This is done by using a

predictive density method used by Pettenuzo et al. (2014), where the candidate predictor

variables are replaced by their own density. In other words we conditionally change the

candidate predictor variables in the extreme values of the predictor by using 𝑥𝑖,𝑡 = 𝑥 𝑖,𝑇 as

well as 𝑥𝑖,𝑡 = 𝑥 𝑖,𝑇 ± 2 × 𝑆𝐸(𝑥). Here 𝑥 𝑖,𝑇 is the full-sample 𝑇 average and 𝑆𝐸(𝑥) standard

deviation of the predictors 𝑥𝑖,𝑡. The candidate predictor variable 𝑖, with 𝑖 = 1, … ,8 for the

predictor variables, at time 𝑡 is set out by 𝑥 𝑖,𝑇. This evaluation approach provides the full

predictive density and accounts for estimation errors in the predictors.

For all the extreme values of the predictor replacing the predictor variables we evaluate the

performance of the out-of-sample predictive accuracy using 𝑅𝑂𝑆2 statistic from equation (14).

Additionally, we determine the Clark and West (2007) test statistic from equation (11) to

test for the statistical significance.

Considering the results of the economic significance test reported in Table 8, we observe the

following in relation to the historical benchmark. On the one hand, the out-of-sample

momentum constraint model demonstrates that the density of predictor variables of Canada

and France are outperforming the historical benchmark with positive and significant 𝑅𝑂𝑆2

values. However, for the remaining five countries the results appear to be negative and

insignificant. On the other hand, the Lasso constraint appears to be outperformed by the

average historical benchmark for almost all the predictor variables in the countries. Except

for some minor results overall and interestingly for the combined (POOL) and the principal

component (PCA) predictive density forecasts of France.

Although, the overall results of the predictive density of this economic significance model fail

to outperform the historical benchmark, we do observe a positive effect of the constraints.

Compared to the unconstraint model, the momentum and Lasso constraint model show a

slight increase to a more positive out-of-sample 𝑅𝑂𝑆2 . The momentum constraint model

28

appears to show a positive change for the countries Canada, France, and Japan. However,

for the countries Germany, Italy, the UK, and the US this change is negative.

In general, these results are in line with the previous out of sample predictability

performance we observed in Table 5.

29

Table 8: Extreme Values of Predictor

Unconstraint Momentum Constraint Lasso Constraint Mean-2σ Mean Mean+2σ Mean-2σ Mean Mean+2σ Mean-2σ Mean Mean+2σ

CAN INFM -0.88 -0.38 -0.20 .. 1.41** 1.49** 1.60*** .. -0.43 -0.35 -0.24 TBL -1.29 -1.80 -1.97 0.97** 1.18** 1.11** 0.02 -0.42 -0.51 LTY -2.03 -2.04 -2.28 0.56* 1.06** 1.06** -0.57 -0.49 -0.46 IPM -0.42 -0.64 -0.30 1.62*** 2.14*** 1.70*** -0.21 0.30* -0.14 MSM -0.48 -0.17 -1.15 1.71*** 1.88*** 1.56** -0.13 0.04 -0.28 LIBOR -0.57 -0.58 -0.27 1.40** 1.43** 1.53** -0.44 -0.30 -0.19 UNEMP -0.80 0.10 -0.43 1.83*** 1.41** 1.33** 0.00 -0.43 -0.51 TMS -1.72 -1.62 -0.76 1.13** 1.30** 1.52** -0.55 -0.42 0.56 TED -1.26 -1.37 -0.23 1.44** 1.98*** 1.67*** -0.18 0.15 -0.20 POOL -0.61 -0.79 -0.65 1.58** 1.62*** 1.55** 0.03 -0.13 -0.10 PCA -3.57 -2.27 -2.45 0.43* 1.00** 1.12** -0.91 -0.73 -0.30

FRA INFM -1.64 -0.98 0.44* 0.44** -0.05 1.00** -0.20 -0.91 0.76** TBL -1.09 -0.02** 0.05 0.11* 1.00** 1.21** -0.42 -0.03 0.57** LTY -0.92 -0.23** -0.26 0.33* 0.74** 0.95** 0.21* -0.13 0.40* IPM -1.43 -0.52 -0.58 0.00 1.02** 0.75* -0.47 0.38* 0.11 MSM -0.35 -0.81 -0.55 0.64* 0.92** 0.80** 0.00 0.28* 0.17 LIBOR -1.33 -1.16 0.30** 0.03* 0.18* 0.91** -0.72 -0.17 0.27** UNEMP -0.50 -0.59 -0.62 0.21 0.69** 0.88** -0.28 0.05 0.24 TMS -0.63 -0.86 -0.84 0.64* 0.73** 0.64* 0.00 0.10 0.09 TED 0.87** 0.26** 0.11 1.90*** 1.52*** 0.91** 1.26*** 0.89** 0.53** POOL -0.14 0.25 0.02 0.91** 1.18** 1.05** 0.37* 0.55* 0.51* PCA -2.29 -1.73 -0.68 -0.06* 0.33** 0.85** -0.41 0.48** 0.26*

DEU INFM -0.86 -0.94 -0.83 -1.19 -1.23 -1.14 -0.62 -0.58 -0.58 TBL -0.92 -0.33 -0.40 -1.02 -0.81 -0.76 -0.46 -0.28 -0.15 LTY -1.02 -0.96 -0.86 -1.27 -1.20 -1.21 -1.02 -0.44 -0.66 IPM -0.52 -0.22 -0.92 -0.85 -0.76 -1.16 -0.27 -0.14 -0.43 MSM -1.01 -0.30 -0.59 -1.13 -0.76 -0.96 -0.51 -0.21 -0.36 LIBOR -1.17 -1.16 -0.16 -1.09 -1.43 -0.68 -0.66 -1.03 -0.09 UNEMP -0.36 -0.67 -0.87 -0.55 -0.99 -1.08 0.14 -0.43 -0.50 TMS -0.50 -0.66 -0.48 -0.82 -0.91 -0.90 -0.26 -0.35 -0.31 TED -0.90 -1.43 -0.36 -0.89 -1.14 -0.75 -0.28 -0.70 -0.16 POOL -0.54 -0.53 -0.47 -0.83 -0.91 -0.89 -0.27 -0.33 -0.29 PCA -1.75 -1.42 -1.46 -1.49 -1.53 -1.41 -1.05 -1.18 -1.09

ITA INFM -1.52 -1.76 0.02* -2.25 -2.60 -0.80 -1.99 -1.74 -0.06 TBL -1.20 -2.17 -1.40 -1.83 -2.05 -0.41 -1.61 -2.47 -0.01 LTY -1.24 -2.02 -1.01 -1.81 -2.31 -0.98 -1.38 -2.56 -0.58 IPM -4.12 -1.25 -0.87 -1.32 -0.91 -0.28 -0.44 -0.68 0.37 MSM -1.45 -0.27 -1.05 -1.17 -0.46 -0.71 -1.30 -0.12 -1.01 LIBOR -0.54 -0.73 -1.44 -0.75 -1.01 -1.36 -0.33 -0.53 -1.01 UNEMP -0.70 0.81* -0.77 -0.35 -0.40 -0.59 -0.03 -0.30 0.13 TMS -0.48 -1.77 -0.71 -0.49 -1.07 -0.94 -0.80 -1.06 -0.51 TED -0.39 -0.17 -2.83 -0.46 -0.53 -3.64 0.14 0.00 -3.25 POOL -0.63 -0.32 -0.34 -0.79 -0.88 -0.67 -0.38 -0.59 -0.12 PCA -3.19 -4.76 -3.22 -2.63 -4.35 -3.53 -2.12 -4.22 -3.14

30

Table 8: Extreme Values of Predictor (cont.)

Unconstraint Momentum Constraint Lasso Constraint

Mean-2σ Mean Mean+2σ Mean-2σ Mean Mean+2σ Mean-2σ Mean Mean+2σ

JPN INFM -0.06 -1.11 -0.79 -0.12 -0.90 -0.70 0.04 -0.74 -0.60 TBL -1.08* -1.08* 0.83** -0.25* -0.17* 0.32* -0.64 -0.58 0.37* LTY -1.49* -0.87 0.44* -0.67* -0.11* 0.08 -1.42 -0.15 0.41* IPM -0.46 -0.37 -1.66 -0.41 -0.45 -0.91 -0.19 -0.28 -0.62 MSM 0.29 0.03 -0.86 0.22 0.01 -0.77 0.39** 0.21 -0.60 LIBOR -1.09 -0.51 -0.01 -0.89 -0.14 0.08 -0.50 0.15* 0.35** UNEMP -0.02 -0.76* -0.38 0.09 -0.56* 0.25* 0.95** -0.74 0.63** TMS -1.14 -0.36 0.10 -0.54 -0.21 0.07 0.10 -0.05 0.23* TED -0.49 0.36* 0.14 -0.68 0.29* 0.25* -0.46 0.42** 0.38** POOL 0.14 0.35 0.21 -0.04 0.12 0.07 0.31 0.36* 0.34** PCA -1.81 -0.77 -1.30 -0.58* -0.52 -0.94 -0.27 -0.15 -0.49

GBR INFM -1.39 -0.04* 0.64** -3.74 -1.86 -1.43 -1.64 0.08 0.15 TBL -0.44 -0.19 -0.29 -1.66 -2.46 -2.76 -0.89 -0.40 -0.66 LTY -0.51 0.37** -0.07 -1.82 -1.78 -2.63 0.23 -0.69 -0.52 IPM -1.25 -0.73 -1.90 -3.09 -3.17 -3.69 -0.99 -1.06 -1.58 MSM -0.17 0.10 -0.28 -2.02 -2.11 -2.05 0.09 0.00 0.06 LIBOR -0.63 -0.04 -0.12 -2.00 -2.20 -2.51 -1.42 -0.39 -0.41 UNEMP -0.76 0.01** -1.03 -1.86 -1.85 -2.86 0.41 0.55* -0.75 TMS -0.93 -0.42 0.08 -2.93 -2.32 -2.05 -0.83 -0.21 0.05 TED -0.23 -0.01 -0.18 -2.30 -1.93 -2.22 -0.19 0.17 -0.11 POOL -0.10 0.53* 0.10 -2.07 -1.83 -2.20 -0.22 0.22 -0.16 PCA -1.83 -1.43 -1.69 -2.99 -4.22 -3.45 -0.73 -2.65 -1.34

USA INFM -0.86 -0.66 0.00 -1.77 -1.47 -1.20 -0.31 -0.28 0.09 TBL -0.90 -1.11 -0.59 -2.11 -1.65 -1.42 -0.80 -0.52 -0.18 LTY -1.34 -1.19 -1.23 -2.32 -1.68 -1.64 -0.93 -0.44 -0.40 IPM -1.64 0.32* -0.84 -2.01 -1.17 -1.79 -0.58 0.12 -0.50 MSM -1.33 -0.69 -0.97 -2.21 -1.80 -2.11 -0.96 -0.43 -0.66 LIBOR -0.77 -0.43 0.11 -2.04 -1.55 -1.26 -0.19 -0.26 0.03 UNEMP -0.07 -0.35 -0.81 -1.35 -1.52 -1.81 -0.06 -0.31 -0.48 TMS -0.15 -0.32 0.01 -1.30 -1.49 -1.45 -0.01 -0.20 -0.16 TED -0.24 -0.12 0.00 -1.57 -1.36 -1.31 -0.31 -0.07 -0.02 POOL -0.40 -0.16 -0.23 -1.61 -1.33 -1.42 -0.21 -0.07 -0.11 PCA -1.85 -0.93 -0.86 -2.70 -1.79 -1.78 -1.14 -0.40 -0.55

Notes: This table reports the out-of-sample 𝑅2 base on equation (10) for predictability regressions of stock returns for the unconstraint model, the momentum constraint model, and the Lasso constraint model. The Unconstrained describes the initial forecast without any imposed constraints as described in equation (3). Next the momentum constraint is described in equation (7). The Lasso constraint describes the results of the sign restriction with the Lasso regression see equation (8). The abbreviations for the predictors INFM, TBL, LTY, IPM, MSM, LIBOR, UNEMP, TMS, TED are described in Table 1. Furthermore, we demonstrate the OOS predictability of the combined forecast POOL and the principal component PCA forecast. CAN, FRA, DEU, ITA, JPN, GBR and USA are the ISO abbreviation we use for respectively Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States. INFM is the growth rate of Inflation. The statistical significance of the is measured measure statistical significance relative to the prevailing mean model using the Clark and West (2007) test statistic. * significance at 10% level; ** significance at 5% level; *** significance at 1% level.

31

6 Discussion

The aim of our research on out-of-sample stock return predictability is twofold. Firstly, we

research the predictive power of the introduced momentum constraint and Lasso

constraint. Secondly, we research the differences between the predictability of stock

returns in various countries of the G7.

In our empirical findings we have observed the following results concerning the

implementation of the constraint models to stock return predictability. We evaluate the

unconstraint model, the momentum constraint and the Lasso constraint using three

different performance measures.

We start by considering the statistical performance of the three models. Here we can

observe that only the out-of-sample predictions of momentum constraint is outperforming

the historical benchmark with a positive and significant 𝑅2 for the countries Canada and

France. The predictions of the constraint models for all the other countries are

outperformed by the historical benchmark. Nevertheless, we do see more negative 𝑅2

values for the Japan using the momentum constraint and for the countries using the lasso

constrain, compared to the unconstraint model.

These improvements can also be observed in both the economic performance and

significance evaluation. When combining the results of the different methods of assessing

our forecasting models, we observe an overall increase of the result for the lasso regression

at the economic evaluation. This happens for the Sharpe and both levels of risk aversion of

the Certainty equivalent. For the momentum approach we again only see a part of the

countries showing positive results.

For the evaluation of the economical significance with the extreme values of predictors we

see similar improvement within the same countries. This are the significant effects of the

momentum for Canada and France, which is in line with the results found in the statistical

performance of the three models.

To summarize, we observe clear differences between the performances of our predictors in

the various G7 countries. Although the good significant results are observed in the three

evaluation methods for Canada and Italy, we see mixed results in the other countries.

32

Luckily, we do observe an improvement for the Lasso constraint model and partly for the

momentum constraint model when compared to its unconstraint variations.

These differences between countries could be explained by for example by country

characteristics which are not accounted for by our predictor variables. Bearing into mind the

result of Hjallmarsson et al. (2010), in which is stated that not all predictors can have a

predictive value in different countries. And the research of Henkel et al. 2011 showing

mixed results throughout the G7 countries, concluding that there are differences between

business cycles in the various countries.

For the research we see that implementing constraint forecasting models results in an

improvement of out-of-sample performance. Especially for the Lasso regression we can see

the positive effects. However, for the principal component models and the combined model

we see no clear improvement in out-of-sample performance. Nonetheless, by using these

constraint models on a data set with more predictor variables, such as the U.S. stock

market, could lead to different results.

Concerning our collected datasets, we used the longest possible historical sample period. To

be able to compare we choose variables available for all countries, not considering possible

differences in importance of these variables for the individual countries. Furthermore, one

could argue that taking shorter time period would offer more availability in different

variables.

Another limitation mentioned by Jordan et al. (2014) is that forecasting gains of European

countries outside the G7 countries are higher than those of the G7. However, we have

chosen for the G7 countries due to their importance in the world and data availability.

This research demonstrates that constraints, in general, offer positive change in the

predictability of out-of-sample stock returns. This is in line with previous research, such as

done in Campbell and Thomson (2008). However, when implementing constraint models,

we do observe clear differences on the predictive out-of-sample performance between

various countries. For future research it could be interesting to research the origins of these

differences.

33

7 Conclusion

This study researched the out-of-sample stock return predictability of the different

economies of the G7. First, this study aimed to research the out-of-sample performance

under the implementation of two novel restrictions. A momentum restriction that truncates

the forecasting model based on the sign momentum. And the Lasso restriction that,

similarly, replaces a prediction based on a lasso regression instead of truncating it. Secondly,

we applied these two constraints on an international sample of the G7 economies.

Accordingly, our research contributed to existing research on forecasting stock returns by

implementing two constraints to improve predictability performance as described by

Campbell and Thomson (2008). Additionally, we contributed to research on international

stock return predictability by observing the differences between countries as described by

Hjallmarsson et al. (2010).

Our results demonstrated that predictor variables are presented mixed results for the

countries of the G7. Some countries showed positive significant results under the

momentum constraint, whilst other countries ended in negative results. Overall, we can see

that in both constraints show an improved performance of predictability in most countries.

This could lead to better forecast performance for stock returns in the future. And can we

only propose to do more research with applying more technical analysis on stock return

predictability, such as the momentum constraint and lasso constraint we have used.

34

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Appendix

Appendix 1: Graphs Stock Price Index and Stock Return

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