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
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Acknowledgement
I would like to contribute this thesis to my new born daughter Isabelle.
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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.
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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).
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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; Pástor 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).
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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.
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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
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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
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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
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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
10 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
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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
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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.
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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
14 Table 3: Summary Statistics (cont.)
Country Variable Mean Std. Dev.
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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
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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 𝑟̂𝑖,𝑡+1 is the predicted stock return based on the 𝑖th predictor. The ordinary least square estimates 𝛼̂𝑖,𝑡 and 𝛽̂𝑖,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 {𝑟̂𝑖,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 𝑟̅ 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.
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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 𝐹̂𝑡𝐸𝐶𝑂𝑁 = (𝐹̂1,𝑡𝐸𝐶𝑂𝑁, . . . , 𝐹̂𝐾,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 + ∑ 𝛽𝑘𝐹̂𝑘,𝑡𝐸𝐶𝑂𝑁
𝐾
𝑘=1
+ 𝜀𝑡+1
(5) where 𝐹̂𝑘,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
𝑟̂𝑐,𝑡+1 = ∑ 𝜔𝑖,𝑡𝑟̂𝑖,𝑡+1
𝑁
𝑖=1 (6)
where 𝑟̂𝑐,𝑡+1 represents the combination forecast at month 𝑡 + 1. And r̂c,t+1 is the 𝑖th
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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 𝑟̂𝑖,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.
𝑟̂𝑖,𝑡+1𝑆𝑖𝑔𝑛 = {
0, 𝑖𝑓 𝑠𝑖𝑔𝑛(𝑟̂𝑖,𝑡+1)~𝑠𝑖𝑔𝑛(𝑟𝑖,𝑡) 𝑟̂𝑖,𝑡+1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
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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 𝑟̂𝑖,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 𝑟̂𝑖,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 𝑟̂𝑖,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
20 where 𝜆 ∑𝑇 |𝛽𝑡|
𝑡=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
𝑅𝑂𝑆2 = 1 − ∑ (𝑟𝐴+𝑘− 𝑟 ̂𝐴+𝑘) 𝑃 𝑘=1 2 ∑𝑃 (𝑟 𝐴+𝑘− 𝑟 ̅𝐴+𝑘) 𝑘=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.
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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+ (𝑟̅ − 𝑟 𝑡 ̂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.
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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).
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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
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.
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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
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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
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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.
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.
29
Table 8: Extreme Values of Predictor
Unconstraint Momentum Constraint Lasso Constraint
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
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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.
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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.
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
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