Stock return predictability : a study for the United States

Stock return predictability

A study for the United States

Thesis by:

M.F.A. Mijlof

10666532

June 2018

Master of Science (MSc) in Finance

Thesis supervisor:

Dr. E. Eiling

Thesis coordinator:

Dr. J.E. Ligterink

University of Amsterdam: Amsterdam Business School

Faculty of Economics and Business

Specialization: Quantitative Finance

List of contents

2.1.1. Valuation variables: dividend and earnings ratios ... 6

2.1.2. Macro-economic variables: inflation, interest rates, spreads and labor market ... 7

2.1.3. Stock-related variables: stock volatility and short interest index. ... 8

2.2.COMBINING PREDICTORS ... 9

2.3.LIMITATIONS FROM EXISTING LITERATURE ... 10

2.4.LINK BETWEEN LITERATURE AND RESEARCH ... 11

3. METHODOLOGY ... 11

3.1.TYPE OF DATA ... 11

3.2.ECONOMETRIC MODELS ... 12

3.2.1. In-sample predictive regression ... 12

3.2.2. Out-of-sample predictive regression ... 12

3.2.3. Combination predictive variable ... 13

3.2.4. ‘Kitchen-sink’ regression ... 14

3.3.FORECAST EVALUATION ... 15

3.4.HYPOTHESES ... 15

3.4.1. In-sample regressions... 15

3.4.2. Out-of-sample predictions ... 16

4. DATA AND DESCRIPTIVE STATISTICS ... 16

4.1.MAIN DEPENDENT VARIABLE: MARKET RETURN OF S&P500 ... 16

4.2.FORECASTING VARIABLES ... 17 4.2.1. Stock volatility ... 17 4.2.2. Valuation ratios ... 18 4.2.3. Inflation rate ... 18 4.2.4. Term spread ... 18 4.2.5. Default spread ... 18 4.2.6. Employment growth ... 19

4.2.7. Short Interest Index ... 19

4.3.SUMMARY STATISTICS ... 20

5. RESULTS ... 22

5.1.IN-SAMPLE REGRESSIONS ... 22

5.2.OUT-OF-SAMPLE PREDICTIONS ... 25

5.3.‘KITCHEN-SINK’ METHOD ... 28

5.4.WEIGHTS OF DMSPE METHOD ... 30

6. ROBUSTNESS CHECKS ... 34

6.1.IN-SAMPLE REGRESSIONS ... 34

6.2.OUT-OF-SAMPLE PREDICTIONS ... 37

6.3.ALTERNATIVE DETRENDING METHODS ... 39

7. CONCLUSION ... 40

8. REFERENCE LIST ... 43

9. APPENDIX ... 45

9.1.AUTOCORRELATION FUNCTIONS ... 45

9.2.PROGRAMMING CODE:STATASE15 ... 46

Abstract

Stock return predictability is a subject which is not only very interesting due to the fact that large profits could be made if it is proven to be true, but also very debatable. Many financial economists state that stock returns are not predictable and just follow a random walk. However, many researchers have proven that there is some predictability within stock returns, especially when using the correct predictors. Variables such as the price-dividend ratio, inflation rate and term spread are popular predictors and have proven to be able show some predictability. Many of these individual predictors focus on in-sample. Out-of-sample is clearly more like a real-world situation and as a result, out-of-sample predictability is more important than in-sample. Many of the predictors that were used in previous literature show bad out-of-sample predictability. Besides using a simple individual predictor, it could be better to use a combined forecast variable which will lead to better out-of-sample predictability. Therefore, in this research, both individual predictors as combined forecasting variables will be used to look for out-of-sample predictability. The combination variables are based on simple methods such as the mean but also on more complex methods such as the discounted mean square prediction error. It can be concluded that the individual predictor, short interest index, is the best among other individual predictors, this especially holds for horizons less than a year. For horizons of one year, it is better to use a combination predictor whereas the short interest index should be included.

Statement of originality

Hereby I, M.F.A. Mijlof, declare that this thesis is written solely by myself. I am taking the full responsibility of the full content of this thesis. Also, I am declaring that the text and work are written by myself and everything is original and authentic. Except the sources that are mentioned in the reference list, no other sources have been used to conduct the research for this thesis and to write it. Finally, the Faculty of Economics and Business at the University of Amsterdam is solely responsible for the supervision and completion of this thesis. However, the content of this thesis is excluded from the scope of the responsibilities of the above-mentioned faculty.

1. Introduction

Stock return predictability is a subject which is widely debated. However, it is very interesting and important for both the academic world as for the financial world. For investors, it is important due to the fact that when there is evidence for return predictability, investors should make their strategic asset allocation choices based on this predictability. If they do not base their decisions on this predictability, they could suffer losses (Campbell & Viceira, 2002). For academics, it is important due to the fact that understanding the dynamics of future risks and/or returns for which investors demand compensation in equilibrium, could lead to other insights about risk premia (Schrimpf, 2010). Lots of variables are proposed as forecasting variables of stock returns. Examples of such variables are valuation rations such as dividend-price ratio, earnings-price ratio and book-to-market ratio, interest rates, inflation rate, term and other spreads, consumption ratios and stock market volatility. Most of the studies simply focus on the in-sample forecasts and they conclude that there is a significant evidence of return predictability (Rapach, Strauss & Zhou, 2010).

However, recent studies show that stock return predictability is not that significant anymore when looking out-of-sample. For instance, the article of Welch and Goyal (2008) shows that just one model is doing well in predicting equity premium out-of-sample and some other models need further research. All the remaining models they studied based on earlier academic research does not provide good results in predicting stock returns out-of-sample. The absence of good out-of-sample predictions in various studies indicates that there is a need of improved forecasting methods and therefore, to be able to better explain the reliability of stock return predictability.

One improvement for the forecasting methods is to combine the predictors stated in earlier academic research. This is also investigated by Rapach et al. (2010), where they propose a combination of predicting variables to predict the out-of-sample stock returns. The intuition behind their research, and finally this research, is that variables such as dividend yield or stock volatility could detect changes in the economy and financial markets that, in turn, indicate changes in the equity premium. However, these variables alone could identify different parts of the financial economic conditions and some other individual macro-economic variables could give a number of wrong signals and therefore, imply an incorrect forecast for equity premiums during certain periods. If the individual predictors are correlated in some way, the forecast of the two combined should be less volatile and more reliable in predicting stock returns.

Moreover, a great amount of predicting variables are already tested, both in-sample as out-of-sample. Also, Rapach et al. (2010), test the most common variables in both in-sample as

out-of-sample by using them in a combining variable. This research will follow up on the existing research by using a new predicting variable which has proven to be the strongest predictor for equity premium nowadays, namely the short interest index. By comparing this new strong variable to the existing individual variables and by comparing it to the method of combining the variables, it will, to my best knowledge, contribute to the existing literature by showing which variables, either individual or combining, are better in predicting stock returns, especially for the out-of-sample returns. Concluding, the main aim of the research is to establish a variable, either individual or combined and which is financial and/or (macro)economic, that provides an improvement for predicting excess stock returns.

As a result, the main research question will be:

‘Is it better to use the short interest index or a combined variable to predict stock returns out-of-sample?’

In order to get an answer on the main research question, this research is outlined in the following way. First, an extensive literature review will be conducted to point out all the important variables, the importance of using a combination variable and to explain all the factors that are involved in this research. Second, the proposed methodology for the research will be described with a focus on the most important parts in the econometric analysis. Third, the data that is used in the research will be portrayed by showing where all data comes from, by showing descriptive statistics and some visualizations of the data. Fourth, the results of the analysis will be shown and explained extensively. Fifth, robustness checks will be conducted and also explained to show that the research could be generalized to other samples. Sixth and last, a conclusion will be discussed and answer on the research question will be provided as well as some potential limitations of the research and suggestions for future research.

To conclude this section, we will summarize the main findings and the conclusion of the research. In this research, the most important main finding is that the short interest index is the strongest individual predictor among other popular predictors that were used in previous literature. Another main finding is that, for long horizons such as one year, a combined variable has a strong predictive ability. This holds for many kinds of combination methods but especially for the combination based on a more complex method. As a conclusion, it could be stated that the short interest index should be taken into account when predicting stock market returns. A combination variable, including the short interest index, should also be considered, especially for longer horizons such as one year.

2. Literature review

In this section, an extensive literature review will be conducted to shed a light on the existing literature about stock return predictability. Specifically, the variables that are used in the research will be motivated and also, the possible surplus of using a combination predicting variable will be explained. Also, some limitations from current research will be elaborated. To end the section, the literature will be linked to the research that is going to be conducted.

2.1. Predicting variables

Attempts to predict stock market returns or equity premia are a long tradition in finance. A typical regression used is an independent lagged predictor on the stock market of return. The most prominent variables explored in the existing literature are various financial variables such as valuation ratios but also a lot of macro-economic variables, for instance, the inflation rate. Moreover, also variables that are more directly related to the stock market such as volatility or short interest have been researched (Welch & Goyal, 2008).

2.1.1. Valuation variables: dividend and earnings ratios

Existing research shows that the price-dividend ratio is one of the most informative predictive variables. For the United States stock markets, there is evidence that stock returns and cash flows are predictable. Fama and French (1988) use price-dividend ratio to forecast stock returns on the portfolio of the New York Stock Exchange (NYSE) for a period from one to four years. Their tests verify existing proof of the predictable component which is a small fraction of variances of the short-term returns. Normally, regressions of returns on yields explain less than five percent of monthly variances. However, regression of returns on price-dividend ratios often explain more than 25 percent of the variances (Fama & French, 1988). The intuition behind using the price-dividend ratio as a predicting variable is due to the fact that it is supposed as reflecting the discount rate for today’s stock price of future dividends. This means that when discount rates are high, the dividend-price ratio is high too (Campbell & Shiller, 1988).

To be more general, dividends do have a great impact on stock returns. One of the reasons is that a lot of valuation models use the present value of future dividends to calculate the current value of the stock. The paper of Kothari and Shanken (1992) show that almost 90% of the variation in the return of stocks is explained by dividends and dividends ratios.

As a result, various valuation ratios such as price-dividend ratio, dividend-payout ratio, dividend yield and price-earnings ratio will be used as a predicting variable due to the fact that it

2.1.2. Macro-economic variables: inflation, interest rates, spreads and labor market

Other extensively used variables for forecasting stock returns are nominal interest rates, spreads and inflation rates. Fama and Schwert (1977) show the relationship between various assets and inflation rate during the period between 1953 and 1971. The most important result from their research is that stock returns were negatively correlated to the inflation rate. It is widely known that stock returns tend to be low when short-term nominal interest rates are high. This is for the reason that investors do not want to take the risk of equities while the return on safe investments is also relatively high due to the high nominal interest rates.

Campbell (1987) shows more generally that term spread of (nominal) interest rates predicts stock returns. He states that stock premiums seem to move closely together with risk premiums on 20-year Treasury bonds. Campbell (1987) and also, Fama and French (1988) show that macro-variables such as short-term (nominal) interest rates, inflation rate and the term premium forecast excess stock market returns.

Another widely used interest rate spread to predict stock returns is the default spread. As explained by Chiang, Li and Yang (2015), the default spread is a measure of risk for bonds. It measures the premium investors get for the risk they are bearing in the bond market. Evidence shows that the S&P 500 index is negatively correlated with the default spread. This is due to the fact that a lower default spread leads to a better market sentiment and therefore results in higher stock prices (Chiang, et al., 2015). Also, the exposure of spreads of portfolios to economic shocks are very important to finding predictability in stock market returns (Kelly & Pruitt, 2013). Furthermore, there is also a relationship between the stock market and the labor market besides the fact that it is maybe not the most feasible relationship. The research of Chen and Zhang (2011) uses labor market variables to predict excess stock market returns. For instance, they show that employment growth predicts excess stock market returns, especially when focusing on business cycles. There is a low predictive power for short-term horizons, high predictive power for medium-term horizons and again low predictive power for long-term horizons. This is in line with the fact that stock market returns are moving inversely to the economic cycle whereas employment growth is moving with the economic cycle (Chen & Zhang, 2011).

As a result, macro-economic variables such as inflation rate, term spread, default spread and employment growth will be used as a predicting variable for the reasons that they are interconnected with the stock market.

2.1.3. Stock-related variables: stock volatility and short interest index.

According to the research of Day and Lewis (1992), the volatility of return included in the price of an option can be understood as a post-forecast of the mean of the volatility of the underlying asset over the duration of the option. The ability of implied volatilities to predict the future volatility of an underlying asset is considered as a measure of the information content of call option prices. Their study suggests that (implied) volatilities explain more of the variation for future standard deviations of individual returns of stocks compared to historical standard deviations of stock returns. As a result, it could be stated that volatility has a relationship with stock market returns. (Day & Lewis, 1992).

The paper of Paye (2012) tests whether it is possible to improve volatility predictions by conditioning on additional macroeconomic variables. He finds that several variables related to macroeconomic uncertainty and time-varying expected stock returns Granger cause volatility. Unfortunately, it is more difficult to find the same causality for out-of-sample. Also, he states that the best predictions could be done when individual forecasts are combined. It is also really important to understand the links between macro-economic variables and volatility of the financial world. For example, characterizing the pattern of time series volatility could be important for improving asset pricing models and, as a result, improving stock return predictability (Paye, 2012). This again shows that volatility has a strong connection with the economy and the stock market.

Guo (2006) shows that the stock market volatility has a strong predictive power for stock market returns. He also suggests that the out-of-sample forecasting results for stock returns is statistically and economically significant. However, most papers fail to find strong predictive power of stock volatility for stock market returns. For example, Guo and Qiu (2014) show that there is a negative relationship between options-implied variance and future stock market returns.

The relationship between stock returns and volatility is still an ongoing debate within the existing literature. Several prominent researchers such as Sharpe (1964) and Linter (1965) show that there is a positive relationship between volatility and expected return and this positive relationship is also widely incorporated in various asset-pricing models. However, some more recent papers such as Bakaert and Wu (2000) and Whitelaw (2000) confirm that there is a negative relationship between volatility and expected returns which could lead to a negative sign between them. Summarizing, we can again state that, in general, there is a relationship between the volatility and the stock market returns.

The research of Rapach et al. (2016) shows that the short interest index is one of the strongest predictors nowadays. This index could be seen as a measure of market pessimism. As a

result, if the short interest index contains information about future stock market returns, we would assume higher values for the short interest index to predict lower future stock market returns. One of the reasons for using a measure based on short selling is due to the fact that short sellers are declared to be informed traders. Boehmer, Jones and Zhang (2008) show that investors who go short are most of the time informed traders. Stocks that are heavily shorted underperform compared to stocks that are lightly shorted or not shorted. Investors that go short, in particular, institutional investors, are important for the contribution of efficiency in the stock markets (Boehmer, et al., 2008).

As a result, stock volatility and short interest index will be used as a predicting variable in this research on the grounds that it has a major effect for predicting excess stock returns.

2.2. Combining predictors

Rapach et al. (2010) show that numerous combinations of forecasts from individual predictive regression models, each of them based on economic variables from existing literature, generate consistent and significant out-of-sample predictability compared to the simple historical average of stock returns. They argue that modelling uncertainty and instability of a single variable seriously weakens the forecasting ability of predictive regression models. This could probably be due to omitted variable bias in the predictive regression model. They provide two empirical explanations for the benefits of forecast combination. The first one is that combining forecasts incorporates information from numerous variables while substantially reducing forecasting errors. Second, combination forecasts are more related to the real economy. In general, various factors such as potential predictive economic variables, economic policy changes, developments in information technology increase the difficulty to predict stock returns with a single predictive regression model. By combining individual predictive regression models, they find that economic variables jointly are valuable and also consistently outperform their historical average of the stock returns (Rapach, et al., 2010).

Schrimpf (2010) examines stock return predictability when the investor is uncertain about the right state variables. He also states a regression model which is really suitable for this research namely a predictive regression model with a vector of predictors that contain financial and (macro-)economic variables. Investors should make their decisions about asset allocation based on the future return prediction. Also, the question whether equity premiums are predictable is important to know to determine the compensation investors should get for their investment.

The model that Kelly and Pruitt (2013) use is based on an idea that dynamic state variables, which drives aggregate expectations, also affect the dynamic variables of the whole

panel of asset-specific valuation ratios. They present a model which links the total centralized market expectations to decentralized valuation ratios in a system of factors.

In the paper of Hjalmarsson (2010), a large worldwide data set is used among several countries both developed as emerging. It is well-known that combining the data could lead to more powerful forecasting methods, meaning that there will be more statistical evidence of stock return predictability. This is obviously relevant when studying stock return predictability because, as mentioned before, any predictable component of a variable will always be small compared to the overall variation in the stock returns (Hjalmarsson, 2010).

2.3. Limitations from existing literature

The research of Welch and Goyal (2008) reexamines the performance of some predicting variables such as price-dividend ratio and they conclude that most predicting variables perform poorly and seem to be volatile, especially out-of-sample. Schrimpf (2010) focuses on more international stock markets instead of just focusing on the stock market in the United States which is already extensively researched when it comes to stock return predictability. An interesting conclusion from that paper is that the interest-rate related variables are one of the best predictors whereas the valuation ratios such as the dividend yield generally perform poorly. This is contrary to paper of Welch and Goyal (2008).

The research of Ferson, Sarkissian and Simin (2003) even goes one step further and discover spurious regressions in the existence of serially correlated independent variables. They suggest that many of the predictive regressions in the existing literature, especially when based on an individual predictor variable, may be spurious. The lack of consistent out-of-sample evidence in the researches stated above implies that there is a demand for improved forecasting methods to determine the empirical consistency of stock return predictability (Rapach, et al., 2010). It is very important to get significant out-of-sample results instead of just in-sample significant results due to the fact that out-of-sample results are more reliable for a real-world application.

The paper of Inoue and Kilian (2005) stresses out the importance between in-sample and out-of-sample predictions. If both in-sample as out-of-sample tests seem to give the same result, it would not matter which one to use. Though, in practice, in-sample tests tend to reject the null hypothesis of no predictability more frequently compared to out-of-sample tests. One possible explanation could be that in-sample forecasting tests have a trend to detect spuriously the existence of predictability when there is actually no predictability. This could be due to the fact that there is look-ahead bias within in-sample regressions. (Inoue & Kilian, 2005).

2.4. Link between literature and research

According to the above stated literature review, we are able to conclude in the first place that there are possibilities to predict stock returns despite the fact that there is a great part of the excess stock market returns that remains unpredictable. However, at the end, this could probably lead to some doubts about the economic significance of the predictability. Also, important for the predicting stock returns are the results that are obtained out-of-sample and not just in-sample. Rapach et al. (2010) also emphasizes that out-of-sample stock return prediction is more similar to a real-world situation.

Nevertheless, we see that there are a lot of variables which have a connection with the stock market. Some of them, such as stock volatility of short interest have a direct connection with stock returns whereas others such as inflation rate or labor market variables have an indirect connection through the economy to the stock market. In the recent literature, short interest index showed to be the strongest predictor among other frequently used predictors. Existing literature also illustrates that combining a variable could lead to significant results in terms of predictability. As a result, we are able to conclude at this point that we need to answer the question whether it is better to use an individual predictor such as the short interest index or a combined one including the short interest index, that gives economic significant results out-of-sample.

3. Methodology

In this section, the proposed method to conduct the research will be explained. To be more specific, we will explain what kind of data we use. We will also explain which model we are going to use, and how we are going to use these mathematical/econometric models. Additionally, (statistical) hypothesis will be stated so we can actually test our findings at a later stage.

3.1. Type of data

The data that we are using in this research can be categorized as time-series data. We have one particular entity namely the market return which is collected over multiple time periods. An important statistical feature that we have to take into account when using time-series data is autocorrelation or serial correlation. No autocorrelation means that there is independence between different data points over time which is an important requisition for conducting time series ordinary least squares regressions. We will show an autocorrelation function in section 4.1.

3.2. Econometric models

In this research, we are going to predict excess stock returns for different horizons namely, one month, three months, six months and one year. We are going to do this by performing a predictive regression for both in-sample as out-of-sample. The regression technique that is used is the ordinary least squares (OLS) regression. Time series ordinary least squares regression is based on a couple assumptions namely: 1) conditional mean zero assumption; 2) dependent and independent variable are jointly stationary and are weakly dependent; 3) no outliers; 4) no perfect multicollinearity. In the next section, we will give a glance of the data to show that these assumptions are true.

3.2.1. In-sample predictive regression

To perform in-sample predictive regression, we are going to use equation (1):

(1)

In this equation, rt:t+h stands for the excess market return. Xt is a vector of different predictive

variables which are in this research: stock volatility, price-dividend ratio, dividend payout ratio, dividend yield, price-earnings ratio, inflation rate, term spread, default spread, employment growth and short interest index. ϵt:t+h is the usual error term in a regression equation. To obtain

the in-sample predictions for the excess market returns, we run this regression over the full sample period for each predicting variable for each horizon. Important to mention is that the compounding of the excess market return for different horizons is based on a rolling window.

To account for potential autocorrelation and heteroskedasticity in the error term and still get reliable results, meaning that standard errors should not be too high, which could lead that we too often reject our null-hypothesis, we have to use heteroskedasticity and autocorrelation consistent (HAC) standard errors. These are often referred as Newey-West standard errors (Newey & West, 1987). When we are performing regression over longer horizons, say more than one month, we need to use Hodrick standard errors. This is due to the fact that excess stock market returns for periods of three months, six months or longer are partially overlapping and as a result, R2 _{will increase for longer horizons (Hodrick, 1992).}

3.2.2. Out-of-sample predictive regression

To execute the out-of-sample predictive regression, we need to perform a slightly more complex method namely by using an expanding estimation window. This expanding window means the starting date is always the same namely January 1973, but the end date is always increasing by one month. We follow the methodology of Rapach et al. (2010). First, we are going to run the

regression for each predicting variable of equation (1) for the first 20 years to obtain an estimated α and β which equals 240 observations in the case of the one-month horizon. Then, we are going to run the regression again for the next period to obtain another estimated α and β which equals 241 observations in the case of the one-month horizon and so forth.

(2)

Second, we need this estimated α and β to calculate rt:t+h by using equation (2). Proceeding in this

way for each t will generate a time series of out-of-sample predictions of the excess market return based on the 20 years before for each single variable for each horizon.

Following several researchers1_{, we use the historical average of the excess market return}

as a natural benchmark forecasting model which is similar to a constant expected excess market return.

3.2.3. Combination predictive variable

As pointed out before in the literature review, Rapach et al (2010) state that combinations of individual predictive regressions could outperform the individual predictions themselves. This is due to the fact that each individual forecast could suffer from omitted variable bias. To solve this omitted variable bias problem and exploit the different parts of information across the individual predictions, we use combining methods to compute the combining variables.

(3)

The combination predicting variables are weighted averages of the N individual predicting regressions that are based on equation (2). The simple combination methods that we are going to use are the arithmetic mean, median and trimmed mean. In the case of the trimmed mean method, we exclude the smallest and largest value and take the mean of the N-2 remaining variables.

We also use a more complex combination method namely the ‘discount mean square

prediction error’ (DMSPE). This method is based on Stock and Watson (2004) where the

combining weights are based on the historical performance of the individual models. The weights are computed by:

(4) where

Important to mention is that we need a holdout period to calculate the combining forecast based on the discounted mean squared prediction error method. This holdout period is equal to q0

which is equal to the first ten years of the out-of-sample period. This holdout period is needed to test the out-of-sample performance of the combination variables. As a result, it means that the combination forecasts have an out-of-sample period equal to the period from January 2003 till December 2014. All the other forecasts, the individual forecasts, have an out-of-sample period equal to the period of January 1993 till December 2004.

In equation (5),  is the discount factor. This method assigns weights depending on the value of the mean square prediction error (MSPE). Greater weights are assigned to individual predicting regressions that have a lower value of MSPE. Lower value of MSPE means that it has a better predicting performance. If  is equal to one, there is no discounting. In this latter case, the weights and, in turn, the combination forecast yields an optimal combination that was derived by Bates & Granger (1969). They derived this for a special case namely where the individual forecasts are not correlated. If  is smaller than one, a greater weight is assigned to the more recent precision of the individual predictive regression model. In this research, we are going to consider three values of  namely, 1.00, 0.90 and 0.75.

3.2.4. ‘Kitchen-sink’ regression

To predict excess market returns, we could also use a multiple regression where all the predictive variables are included to predict excess stock market return in one regression. This is often called the ‘kitchen-sink’ regression. For instance, the article of Li, Tsiakas and Wang (2015) shows that economic fundamentals could lead to significant out-of-sample forecasts for exchange rates when the predictive regression is based on a ‘kitchen-sink’ regression which includes multiple forecast variables. The key to establishing strong forecasting of returns is estimating the kitchen-sink regression that improves performance both in-sample as out-of-sample (Li, et al., 2015). However, this method has a poor out-of-sample performance due to overfitting of the model.

In section 5.3, the results based on a ‘kitchen-sink’ regression will be shown and discussed to proof the poor out-of-sample performance. To perform a multiple regression with all the predicting variables included, we are going to use equation (6) for the in-sample predictions.

(6) For the out-of-sample predictions, we are going to conduct the exact same method as pointed out in section 3.2.2. However, in this case, we are using equation (7) to calculate rt:t+h.

3.3. Forecast evaluation

To evaluate the predictive regression, we have to look at the coefficient of determination, denoted as R2_{. For the in-sample forecasts, we can simply use the R}2 _{given by the regression}

output. However, for the out-of-sample forecasts, we need to compute it manually. To do this, we follow the method of Campbell and Thompson (2008) to construct the R2_{OOS. In this}

evaluation, we compare the estimated equity premium with the historical average. The estimated equity premium in this case could be either the individual forecast or the forecast based on the combined variable.

(8)

The R2_{OOS statistic measures the reduction in the mean square prediction error (MSPE) for the}

individual forecast or combination forecast compared to the benchmark. This benchmark is the average excess stock market over the full sample period. This means that when R2_{OOS is greater}

than zero, the out-of-sample prediction outperforms the historical average prediction. Intuitively, if x_i,t contains information which could be meaningful for predicting excess market returns, then the out-of-sample predicted excess market return should perform better than just the historical average of the returns. This R2

OOS can also be seen as the proportional reduction in the mean

squared prediction error (MSPE) for the out-of-sample predictive regressions. For the significance of R2

OOS, we use the method of Clark and West (2007). We test the hypothesis that

the MSPE of the average excess stock market return is greater than the MSPE of the out-of-sample prediction. If the MSPE is greater, it means that the prediction error is also greater and thus, we could say that the prediction is less precise.

3.4. Hypotheses

According to the previous sections, we could state some hypotheses to test them later in the results section and to provide an answer for the research question. If the hypotheses are proven to be true, we are able to conclude that either an individual variable or a combined variable has predictive ability out-of-sample for stock market returns.

3.4.1. In-sample regressions

When considering the in-sample predictive regression, we can assess one hypothesis which will be based on the coefficients of β of each predictive variable for each horizon. The corresponding

greater than zero (H1: β > 0). If we are able to reject the null hypothesis, it means that the

predictive variable has some in-sample predictability for stock returns. We have to take into account the relationship of the predicting variable with the stock market, so we need to look at the sign. For some variables, we know what the sign should be. For instance, the short interest index and inflation have a negative relation with the stock market. Therefore, we need to take the negative of it. This should result in all coefficients of the predictors being greater than zero.

3.4.2. Out-of-sample predictions

When considering the out-of-sample results, we can assess two different hypotheses. The first will be based on the coefficient of R2_{OOS being larger than zero and the second will be based}

on the comparison between R2_{OOS for the individual predictors and the combination predictors.}

The corresponding alternative hypothesis for the first out-of-sample prediction will be that the R2

OOS is greater than zero (H1: R2OOS > 0). If the null hypothesis will be rejected, we could

state that the predictive variable in particular is an improvement relative to the historical average forecast and if we do not reject the null hypothesis, the adverse is true.

The corresponding alternative hypothesis for second the out-of-sample prediction will be that the R2_{OOS of the combined prediction is greater than the R}2

OOS of each of the individual

predictive regression. If we are able to reject the null hypothesis in this case, it shows that the combined predictive variable outperforms the individual predictive variable out-of-sample.

4. Data and descriptive statistics

In this section, all the data that is used in the research will be explained. More specifically, the origin of the data will be pointed out and some descriptive statistics and visualizations will be given to show a glance of the data.

For all data holds that it contains monthly data over 41 years starting from January 1973 till December 2014. Reason to use this time-frame is because a couple crises are incorporated such as the last financial crisis of 2008.

4.1. Main dependent variable: market return of S&P500

We define excess market return as the return measured in logarithms on the S&P500 index in excess of the return measured in logarithms on a one-month Treasury Bill rate. These excess market returns are similar to Rapach et al. (2016). The data is obtained from David Rapach’s website. Mathematically, we compute the excess return as the return minus the lagged risk-free rate. In the case of predicting stock returns, we are using market returns. More explicitly, we are

using the value-weighted diversified portfolio of the 500 greatest firms in the United States which is proxied by the S&P500 index.

To show some characteristics, we plot an autocorrelation function of the excess market returns to see if there is any autocorrelation, or also called serial correlation, in the data. We plot the data on 12 lags meaning that it is lagged for one year. As we can see from Figure 1, the data is not autocorrelated at any lag, which should be the case for stock market returns and it is also preferable for conducting ordinary least squares regression and hence, for this research.

4.2. Forecasting variables

We use ten different predicting variables which are based on previous research and are proven to be either a strong estimator and/or have a correlation with stock markets. These predicting variables are: stock volatility, price-dividend ratio, dividend payout ratio, dividend yield, price earnings ratio, inflation rate, term spread, default spread, employment growth and short interest index.

4.2.1. Stock volatility

To compute the stock volatility, daily returns are obtained from CRSP (The Center for Research in Security Prices) using the database of WRDS (Wharton Research Data Services) powered by

by the standard deviation of these daily returns over the past month and thereafter, it is also annualized by using 252 trading days.

4.2.2. Valuation ratios

To compute the valuation ratios, prices, earnings and dividends of the S&P 500 are obtained from Robert Shiller’s website. After obtaining the data, we convert prices, earnings and dividends to logarithms.

The price-dividend ratio is calculated as the difference between the log of dividends and the log of stock prices of the S&P 500 index (following Rapach, et al., 2016). The dividend payout ratio is calculated as the difference between the log of dividends and log of earnings. The dividend yield is calculated as the difference between the log of dividends and the log of a one-period lagged stock price. The price earnings ratio is then calculated as the difference between the log of earnings and the log of stock prices.

4.2.3. Inflation rate

To compute the inflation rate, the consumer price index (CPI) on the United States is obtained from the database of the Federal Reserve in St. Louis. After obtaining the data, the inflation rate is computed by the log growth rate of the consumer price index. Also, the inflation rate is lagged by one month to account for the data release of the inflation data.

4.2.4. Term spread

To compute the term spread, the yields on long-term bonds and short-term bonds are obtained from the database of the Federal Reserve in St. Louis. Specifically, the long-term bond is a 10-year U.S. Government Bond and the short-term bond is a three-month Treasury Bill. After obtaining the data, the term spread is calculated by the difference between the 10-year U.S. Government Bond and the three-month Treasury Bill.

4.2.5. Default spread

To compute the default spread, the yields on AAA and BAA corporate bonds are obtained from the database of the Federal Reserve in St. Louis. The ratings are based on the rating scheme of Moody’s. After obtaining the data, the default spread is calculated by the difference between BAA corporate bond and the AAA corporate bond.

4.2.6. Employment growth

To compute the employment growth, (seasonally adjusted total non-farm) payrolls of all employees in the United States are obtained from the database of the United States Bureau of Labor Statistics. After obtaining the data, the payrolls are converted to logarithms and to calculate the employment growth, the log growth rate is used.

4.2.7. Short Interest Index

To compute the short interest index, the data from David Rapach’s website is used. The data we use is the equally-weighted short interest. Before using the data in the research, it has to be normalized/standardized first. This is for the reason that we want to have a robust proof for the trend in short interest over time. Therefore, we have to eliminate the secular increase in aggregate short interest. When eliminating this, we can just focus on the relevant economic information in short selling which mirrors the views of the investors that go short in a better way.

Due to the fact that the short interest index is one of the most important and interesting variables in this research, we will elaborate more on this variable.

In Figure 2A, we see the series of the total of short selling across all publicly listed stocks in the United States. From the figure, we see that there is an upward trend. One reason for this upward trend could be due to the improvement in equity lending which made it easier to short sell over time. Another reason for the upward trend could be found in the fact that the amount of hedge funds has increased in the past decades which led to a rise in the amount of capital reserved to short arbitrage (Rapach, et al., 2016).

In Figure 2B, we see the short interest index which equals the difference between the solid and dashed line from part A. It is also normalized to a standard deviation equal to one.

To perform this standardization, we first regress the logarithm of the equally weighted short interest on time over the full sample period, as pointed out by equation (8):

(9) This is also called a time trend model and in this particular case, it is referred to as a linear time trend model. After this regression, we are going to store the fitted values of the logarithm of the equally weighted short interest. Finally, to get the short interest index, we use these fitted values to detrend the logarithm of the equally weighted short interest. Then, we standardize this detrended measure to get a standard deviation of one and a mean of zero for the short interest index.

At last, it is also worth mentioning why we use the equally-weighted short interest while we are using the value-weighted market return of the S&P500 because it seems obviously to also use the value-weighted short interest. One reason to use equally-weighted short interest is because short-selling is less important for large-capitalization stocks, as stated by Asquith, Pathak and Ritter (2005). Additionally, we want to investigate the aggregate effect of short-selling, so it is better to use the equally-weighted short interest.

4.3. Summary statistics

Before we are going to discuss the results of the main analysis, we are going to look at the data by discussing the summary statistics and the correlation matrix.

From Table 1, we can see indeed see that the short interest index is normalized as mentioned before. We also see that almost all variables have strong first order autocorrelation (AR(1)). This means that the data at time t+1 has a strong correlation with the data at time t. However, for these variables, it is common to have serial correlation, especially for the short run.

Table 1) Summary statistics

Note: this table contains the summary statistics for all the predictive variables that are used in this research. It contains 504 observations due to monthly data running from January 1973 till December 2014. It reports the mean, standard deviation, median, minimum value, maximum value and the first-order autocorrelation coefficient (AR(1)).

Mean Std. Dev. Median Min Max AR(1)

Volatility (annualized) 0.1509 0.0863 0.1304 0.0408 0.9100 0.6498

Dividend price ratio -3.6148 0.4449 -3.5689 -4.5021 -2.7747 0.9958 Dividend payout ratio -0.8001 0.3453 -0.8637 -1.2443 1.3797 0.9848

Dividend yield -3.6091 0.4451 -3.5668 -4.5093 -2.7727 0.9958

Price earnings ratio -2.8147 0.4908 -2.8294 -4.8181 -1.9152 0.9914

Inflation rate 0.0034 0.0034 0.0029 -0.0179 0.0179 0.6398

Term spread 0.0174 0.0131 0.0194 -0.0265 0.0442 0.9537

Default spread 0.0110 0.0047 0.0096 0.0055 0.0338 0.9623

Employment growth 0.0012 0.0021 0.0015 -0.0077 0.0123 0.6775

Short interest index 0 1 -0.0891 -2.2761 2.9358 0.9497

Number of observations (N) 504 504 504 504 504 504

Table 2) Correlation matrix

Note: this table contains the correlation matrix. It reports the Pearson correlation coefficient for both the excess stock market return and the predicting variables that used in this research. Significance levels: *p<0.05. SII is abbreviation for short interest index.

Market return Volatility price ratio Dividend payout ratio Dividend Dividend yield

Price earnings

ratio Inflation rate spread Term Default spread Employment growth SII Market return 1.00

Volatility -0.33* 1.00

Dividend price ratio -0.02 -0.10* 1.00

Dividend payout ratio -0.00 0.28* 0.25* 1.00

Dividend yield 0.03 -0.14* 1.00* 0.25* 1.00

Price earnings ratio -0.02 -0.29* 0.73* -0.48* 0.73* 1.00

Inflation rate -0.06 -0.14* 0.45* -0.15* 0.44* 0.52* 1.00

Term spread 0.09 0.03 -0.03 0.35* -0.03 -0.28* -0.39* 1.00

Default spread 0.02 0.36* 0.49* 0.45* 0.49* 0.12* 0.01 0.15* 1.00

Employment growth -0.00 -0.34* 0.05 -0.35* 0.05 0.29* 0.09* -0.06 -0.46* 1.00

Short interest index -0.10* 0.13* -0.13* 0.16* -0.14* -0.23* -0.02 -0.08 -0.07 -0.12* 1.00

From Table 2, we see in the first column that most of the variables have a negative contemporaneous correlation with the stock market return. All of these negative correlations make sense. For example, as mentioned before, higher values for the short interest index could predict lower future market returns and this relation is also powered by the negative correlation coefficient. Also, as mentioned before, we see a negative correlation between the inflation rate and market return that was also shown by Fama and Schwert (1977). Additionally, we see that volatility has the greatest negative correlation coefficient and this could be due to the fact that if

volatility is high, investors tend to sell more stocks which leads to a decrease in stock prices and in turn, a decrease in market returns. Last, we see that the short interest index does not have a strong correlation with the other variables. The strongest correlation is the one between price-earnings ratio and the short interest index and it equals 0.23. Most of the other variables have a stronger correlation with each other. However, despite these low correlations between the short interest index and other variables, most of the pairwise correlations are significant at 5% level. This means that the short interest index contains substantial different information about stock market returns compared to the other variables.

5. Results

In this section, the results will be discussed. All the outcomes will be explained and also the hypotheses and economic causality will be addressed. First, the in-sample regressions will be discussed and second, the out-of-sample results will be discussed. Third, an alternative method namely a kitchen-sink method will be discussed and last, the weights calculated by the discounted mean square prediction error will be highlighted.

5.1. In-sample regressions

In Table 3, the results of the in-sample predictive regressions are reported. We will discuss the outcomes of each of the individual predictive variables. Remind that when performing the in-sample regression, we use equation (1). The dependent variable in this equation is the market return of the S&P500 minus the one-month Treasury Bill rate and the independent variables in this equation are the individual predictors.

From column (1) in Table 3, we see that only the short interest index shows predictive ability. Remember that we take the negative of the short interest index so in this case, the sign is correct which is also seen in Table 2 where the correlation between short interest index and market return is negative. Short interest has a coefficient equal to 0.00501 and it is significant at a 5% level. Therefore, we could state that we reject the null hypothesis for short interest as having no predictive power for a horizon of one month. We also see that the R2 _{in column (2) is 1.24%.}

Obviously, this value of R2 _{is relatively low but this is due to the fact that the monthly excess}

stock market return has a great component which is simply not predictable. According to Huang and Zhou (2017), which investigate the upper bounds on return predictability, a R2 _{between 0.5%}

and 1.5% already has an economically significant effect in terms of one period ahead prediction. In that case, the R2 _{of the short interest index is sufficient to posit that it has an economically}

Interesting to mention is the sign of volatility, which is wrong. As discussed before, there is an ongoing debate about finding the correct method to proof the relationship between risk and return of stocks in the data. We also see from Table 2 that the correlation between volatility and stock market return is around -0.32 which is the strongest among all predicting variables, so this could explain the fact that the in-sample regression leads to a negative relation.

From column (3) in Table 3, we see again that only short interest index shows predictive ability. Short interest index has a coefficient equal to 0.0168 and it is significant at a 5% level. Again, the sign here is correct because we take the negative of the short interest index. Therefore, we are again rejecting the null hypothesis of short interest index having no predictive power. This means that the short interest index has some ability to predict the excess stock market return for a horizon of three months. We also see in column (4) that the R2 _{is around 4.37%. From column}

(5) in Table 3, we once more see that only short interest index shows predictive ability. Short interest index has a coefficient equal to 0.0339 and it is significant at a 5% level. As a result, we can again reject the null hypothesis and posit that the short interest index has some ability to predict the excess stock market return for a horizon of six months. We also see in column (6) that the R2 _{is around 8.07%. The R}2 _{for a horizon of three months and six months are both}

economically significant due to the fact that they are higher than upper bound of 1.5 percent pointed out by Huang and Zhou (2017).

From column (7) in Table 3, we see that not only short interest index is significant but also inflation rate and term spread. Short interest index has a coefficient equal to 0.0631 and it is significant at a 1% level. Therefore, we are able to reject that null hypothesis of short interest having no predictive power. As a result, we are able to state that short interest index is a very strong predictor for excess market return due to its significance at every horizon, including a horizon of one year. We also see from column (8) that the R2 _{is around 12.89% which is high in}

terms of predictability and so, it has not only a statistical significant effect, but we are able to conclude that it also has economic significance. Furthermore, inflation rate has a coefficient equal to 9.375 and it is significant at a 5% level. Therefore, we are able to reject the null hypothesis of inflation rate having no predictive power at a horizon of one year. Term spread has a coefficient equal to 3.459 and it is also significant at a 5% level. Therefore, we are also able to reject the null hypothesis of term spread having no predictive power at a horizon of one year. We are able to see from column (8) that the R2 _{of inflation rate equals 3.64% and that the R}2 _{of term spread}

equals 7.30%. Especially the latter one has not only an econometric meaning but also an economic meaning.

It is interesting to elaborate a bit more on term spread and inflation rate. It is straightforward that these macroeconomic variables have a connection with the stock market and therefore are able to partly predict excess stock market returns. This is also seen in Table 2, where especially term spread shows a quite strong, contemporaneous correlation with the stock market. However, at short horizons, for instance one month or three months, they do not show any predictive ability. A reason for this could be that due to the fact that they are macroeconomic variables measured by consumer price index and bonds, which do not adapt that quick to shocks so they are not able to predict stock returns for short horizons, but they are for the long horizons. Besides, we see in general that the R2 _{is getting higher for each variable when we are}

looking at longer horizon (h>1). This is due to the fact that returns are partially overlapping.

Table 3) In-sample predictive regression results

Note: this table contains the results of the in-sample regression based on equation (1) where the dependent variable is the excess market return of the S&P500 and the independent variables are the single individual predictive variables.

Heteroscedasticity- and autocorrelation-robust t-statistics are (Newey-West (1987) and Hodrick (1992) for h>1) are in parentheses: *p<0.10, **p<0.05, ***p<0.01. (1) (2) (3) (4) (5) (6) (7) (8) h = 1 h = 3 h = 6 h = 12 Predicting variable β R2 _(%) β _R2 _(%) β _R2 _(%) β _R2 _(%) Volatility -0.0450 0.75 -0.0288 0.10 0.0202 0.02 0.0301 0.02 (-1.50) (-0.33) (0.20) (0.19)

Dividend price ratio 0.00323 0.10 0.0107 0.35 0.0250 0.92 0.0544 2.09

(0.68) (0.77) (0.93) (1.13)

Dividend payout ratio 0.00185 0.02 0.0108 0.22 0.0277 0.67 0.0492 1.03

(0.27) (0.58) (0.89) (1.21)

Dividend yield 0.00367 0.13 0.0116 0.42 0.0261 0.99 0.0559 2.21

(0.77) (0.83) (0.97) (1.16)

Price earnings ratio 0.00173 0.04 0.00338 0.04 0.00679 0.08 0.0200 0.35

(0.32) (0.21) (0.23) (0.44) Inflation rate (-) 0.152 0.01 1.487 0.40 4.596 1.80 9.375** 3.64 (0.19) (0.94) (1.61) (2.04) Term spread 0.258 0.57 0.714 1.38 1.344 2.27 3.459** 7.30 (1.59) (1.52) (1.56) (2.37) Default spread 0.322 0.11 1.031 0.37 3.052 1.52 4.740 1.75 (0.53) (0.64) (1.19) (1.18) Employment growth -0.180 0.01 -1.913 0.25 -3.591 0.42 -8.160 1.05 (-0.13) (-0.57) (-0.66) (-1.04)

Short interest index (-) 0.00501** 1.24 0.0168** 4.37 0.0339** 8.07 0.0631*** 12.89

(2.41) (2.50) (2.45) (2.69)

5.2. Out-of-sample predictions

In Table 4, the out-of-sample results are reported. More specifically, the R2

OOS is reported which

is defined in section 3.3, following the method of Campbell and Thompson (2008). We will discuss each of the R2

OOS for the variables, both individual and combined. We are going to test

the hypotheses that were stated in section 3.4. In addition, we will also state some remarks about the economic significance of the values of the R2

OOS.

Recall that we are comparing our estimates against the average excess stock market return benchmark. To be able to conclude that our estimate is better than the natural benchmark, we need a R2

OOS greater than zero.

From column (1) in Table 4, we see that only the short interest index has a positive R2_OOS

which equals 1.68% and it is also significant at a 5% level. As discussed before, a R2

OOS greater

than 1.5% already means that it has an economic meaning. Consequently, we are able to reject the null hypothesis that the forecast based on the historical average better predicts one-month ahead stock returns. This is also in line with the in-sample results. The short interest index had some predictive ability when considering in-sample, but additionally we see that the short interest index also has predictive ability out-of-sample.

From column (2) in Table 4, we see again that only the short interest index has a positive

R2

OOS which equals 5.84% and that it is also significant at a 5% level. Accordingly, we are able to

state that, similar to the one-month horizon, the short interest index better predicts quarterly stock market returns than the historical average forecast. This is again in line with the in-sample results. As discussed before, the short interest index is a measure of market pessimism. Investors who go short are most of time the investors or traders with the most information about the stock market. On the grounds that they use the most information when trading, we see that it is a good predictor for future stock market returns, especially for short horizons.

From column (3) in Table 4, we see that both the inflation rate and short interest index have a positive R2

OOS which equals 1.59% and 9.41% respectively. They are both significant at 1%

level. Thus, we are able to posit that both the inflation rate and short interest index better predict semi-annually stock market returns in favor of the historical average forecast. Noteworthy to state is that the R2

OOS of the inflation rate is smaller than the one of short interest at a horizon of

one month. This means that the short interest index is a stronger predictor compared to the inflation rate, even at horizons longer than three months. Moreover, in line with the results in-sample, inflation rate has predictive ability for longer horizons. The difference between out-of-sample and in-out-of-sample is that inflation rate already has a significant effect at a horizon of six months instead of one year.

From column (4) in Table 4, we see that inflation rate, term spread and short interest index all have a positive R2

OOS which equal 2.93%, 1.82% and 10.76% respectively. They are all

significant at a 1% level. Therefore, we are able to conclude that inflation rate, term spread, and short interest have predictive ability for annual returns. Additionally, we are also able to state that they better predict annual returns compared to the historical average forecast. Remarkable to state is that the out-of-sample results are in line with the in-sample results. In addition to the discussion in section 5.1. about the macroeconomic variables having predictive ability in-sample, we could clarify more about the economic significant effect of the inflation rate, term spread and short interest.

An explanation for the term spread, which is defined as the difference between the yields of long-term government bonds and short-term government bonds, could be that it is a good prediction for annual returns but not for horizons shorter than one year due to the fact that it depends on the yields of the bonds. In general, bonds are considered less risky assets in comparison to stocks. Therefore, yields of bonds have a lower volatility than stock returns, especially for short horizons. Campbell (1987) confirms this by stating that there is a common movement between the risk premia of bonds and stocks. This common movement leads to predictability over time in excess stock market returns. However, there are some differences namely that the standard deviation, also known as the volatility, is higher for stocks returns compared to the standard deviation of bonds yields. As a result, the term spread could be a good predictor for the stock market on the long-run because it is a good reflection of the current state of the financial markets. The similar explanation holds for the inflation rate. The inflation rate is a good measure of the economic welfare because it is the rate in which prices are increasing compared to the previous period. However, the prices in the economy will not change very hard in the short run. This is due to the fact that, for instance, stores need to adapt their prices and they will not this on daily or weekly basis. As a result, it is able to predict stock prices in the long run but not in the short run.

From column (5) in Table 4, we see the combination variables pointed out in section 3.2.3. None of the combination variables are significant in the case of a horizon of one month. This seems surprising due to the fact that the value of R2_{OOS of the combination forecast based on}

the DMSPE method with  equal to 0.90 and 0.75 is greater than the R2_{OOS of the short interest}

index for the same horizon. However, it is important to recall that the period to calculate the

R2

OOS is not similar. For the combination forecasts, we use the post-holdout period where we use

the initial out-of-sample period for the individual predictors. This is the reason that it is not significant in the case of the combination forecasts, but it is in the case of the individual

forecasts. The reason to still use the different periods is to use the most data available to conduct the main research.

From column (6) in Table 4, we see that all the forecasts have a positive R2

OOS. However,

only two of them are significant at 10% level namely the combination forecast based on the median and based on the DMSPE method with  equal to 0.75. The reason for the weak significance is similar to the one stated above, namely that the period for the R2

OOS differs.

Furthermore, the outcome of the combination forecast based on the DMSPE method with  equal to 0.75 is in line with the theory, namely that a greater weight is assigned to an individual forecast that has a low mean square prediction error. The lower the discounting value, the more weight is assigned to this better forecast. As a result, it is in line that the combination forecast based on the DMSPE method with the lowest discounting value has the highest R2

OOS.

Concluding, we could state that the combination variable based on the DMSPE method and on the median has some predictive ability for quarterly stock market returns. Surprisingly, we see that the values of R2

OOS are relatively small compared to the R2OOS of short interest index in

column (3). An explanation is that nine out of ten variables are not individually significant and hence, when combining all the variables through various methods will not give substantial better results when these nine insignificant forecasts are also taken into account in the combination forecast.

From column (7) in Table 4, we see that all variables are positive and this time, they are also all significant. The combination based on the trimmed mean has the weakest significant namely at a 10% level whereas the combination based on the mean, median and DMSPE method with  equal to 1.00 are significant at 5% level. The combination forecasts based on the DMSPE method with  equal to 0.90 and 0.75 are very significant, namely at a 1% level. Consequently, we are able to suggest that all the combination variable have predictive ability for a horizon of six months. We again see that when the discounting value gets lower, the R2

OOS is getting higher. We

also see again that the combination based on the median is one of the best predictors among the combination variables.

From column (8) in Table 4, we see that all the variables are positive and that they are all very significant namely at a 1% level. Also, interesting to mention is that in this case, namely a horizon of one year, all the R2

OOS are at least as great as the R2OOS of the strongest individual

predictor which is the short interest index. We can therefore reject the second hypothesis that the individual predictors better predict excess stock return for a horizon of one year compared to combined variables. A clarification is that in the case of the one year ahead horizon, multiple