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International Review of Financial Analysis journal homepage: www.elsevier.com/locate/irfa

Economic constraints and stock return predictability: A new approach a

b,⁎

c

Yaojie Zhang , Yu Wei , Feng Ma , Yongsheng Yi a b c

c

T

School of Economics and Management, Nanjing University of Science and Technology, Nanjing, China School of Finance, Yunnan University of Finance and Economics, Kunming, China School of Economics and Management, Southwest Jiaotong University, Chengdu, China

ARTICLE INFO

ABSTRACT

JEL classifications: C32 C53 C58 G11 G17

In this paper, we propose a new approach to impose economic constraints on the time-series forecasts of stock return. It is unlikely or risky for a rational investor to rely on forecast outliers to trade stocks. Given this, our new constraint approach truncates the stock return forecasts at the extremely positive and negative values. The empirical results suggest that the new economic constraint approach generate more accurate and reliable return forecasts than the unconstrained method for both univariate regression models and multivariate models. Furthermore, our new constraint approach also outperforms two prevailing constraint approaches of Campbell and Thompson (2008) and Pettenuzzo, Timmermann, and Valkanov (2014). In addition, a mean-variance investor can realize sizeable economic gains by using our new constraint approach to allocate asset relative to using unconstrained counterpart or other popular constrained models.

Keywords: Stock return predictability Economic constraints Forecast outlier Asset allocation

1. Introduction Stock return forecasts play a central role in many financial areas such as asset pricing, asset allocation, and risk management. However, the seminal work by Welch and Goyal (2008) argue that it remains difficult to find reliable models that can accurately predict stock returns out-of-sample. To this end, a growing number of studies have constructed or used many powerful predictors and variables, which include the variance risk premium (Bollerslev, Marrone, Xu, & Zhou, 2014; Bollerslev, Tauchen, & Zhou, 2009), technical indicators (Gao, Han, Li, & Zhou, 2018; Lin, 2018; Neely, Rapach, Tu, & Zhou, 2014; Zhang, Ma, & Zhu, 2019), downside variance risk (Feunou, Jahan-Parvar, & Okou, 2015; Kilic & Shaliastovich, 2018), oil-related variables (Chiang & Hughen, 2017; Liu, Ma, & Wang, 2015; Nonejad, 2018), short interest index (Rapach, Ringgenberg, & Zhou, 2016), investor sentiment (Huang, Jiang, Tu, & Zhou, 2015), news-implied volatility (Manela & Moreira, 2017), economic policy uncertainty (Chen, Jiang, & Tong, 2017) and manager sentiment (Jiang, Lee, Martin, & Zhou, 2017), among others. On the other hand, a few articles attempt to improve return predictability from an economic constraint perspective. Campbell and Thompson (2008) (CT hereafter) truncate the stock return forecasts at zero and also constrain the sign of the regression slopes in predictive regression models. Another influential paper by Pettenuzzo et al.

⁎

(2014) (PTV hereafter) imposes reasonable bounds on the conditional Sharpe ratio, which depends on both the conditional mean and volatility of the return distribution, to lie between zero and one. Along the same lines, we propose a new approach to impose economic constraints on the time-series forecasts of stock return. A rational investor is unlikely to rely on extremely positive or negative stock return forecasts to trade stocks. There are two economic driving forces behind it. First, the possibility is too low when an extremely large or small forecast occurs in the future. Therefore, investors will not trust the extreme values. Second, investors are typically riskaverse, while the extremely positive or negative return forecasts make their portfolio allocation too aggressive and thus amplify risk. To some extent, our motivation is similar to that of Pettenuzzo et al. (2014). They present that the conditional Sharpe ratio should be restricted to lie between a lower and upper bound, which makes the portfolio performance more reasonable. Pettenuzzo et al. (2014) propose a Bayesian approach to realize the bounds on the conditional Sharpe ratio, while we use a straightforward but efficient method to smooth forecast outliers. More specifically, we use the well-known three-sigma rule to recognize the extremely positive or negative stock return forecasts (i.e., forecast outliers) and smooth the return forecasts by truncating them at the extreme values. We first investigate the return predictability of our new economic constraint based on univariate regression models, in which 14 popular

Corresponding author at: School of Finance, Yunnan University of Finance and Economics, 237 Longquan Road, Kunming, Yunnan, China. E-mail addresses: [email protected] (Y. Wei), [email protected] (F. Ma).

https://doi.org/10.1016/j.irfa.2019.02.007 Received 17 November 2018; Received in revised form 17 January 2019; Accepted 22 February 2019 Available online 26 February 2019 1057-5219/ © 2019 Elsevier Inc. All rights reserved.

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economic variables recommended by Welch and Goyal (2008) are used. The empirical results show that our new constraint approach generates larger out-of-sample R-squares (ROS2s) than the unconstrained counterpart for all the 14 univariate regression models. Furthermore, we make a comparison of our new constraint model and the popular CT and PTV constraint models and find empirical evidence that our new constraint model can beat the popular CT and PTV constraint models for most of the univariate regression models. Next, we further consider a few multivariate models, including a diffusion index-based regression model and five widely used combination approaches from Rapach, Strauss, and Zhou (2010). The corresponding results suggest that the multivariate models exhibit overall stronger forecasting performance than the univariate regression models. More importantly, the multivariate models subject to our new constraint yield larger ROS2s than not only the unconstrained counterparts but also the CT and PTV constrained counterparts. Finally, we measure the economic value of our new economic constraint's predictive ability from an asset allocation perspective. Specifically, we calculate the certainty equivalent return (CER) for a mean-variance investor who allocates between stocks and risk-free bills using various forecasts of stock return. The CER gain is calculated for each model relative to the simple mean benchmark. We find that a mean-variance investor can realize the largest CER gains when she uses the return forecasts based on our new constraint approach to guide her portfolio allocation. Furthermore, our empirical results are robust to alternative evaluation techniques, business cycles, various risk aversion coefficients, and the consideration of transaction cost. In summary, we propose a new economic constraint approach, which is simple but efficient. The new constraint approach is documented to outperform not only the unconstrained approach but also the prevailing CT and PTV constraint approaches via a comprehensive investigation. Apparently, this paper is related to the works of Campbell and Thompson (2008) and Pettenuzzo et al. (2014). A common feature of the CT and PTV constraints is that they both rule out negative return forecasts. However, the short-selling constraint on stocks is fading away. A stock investor can easily take a short position when she predicts a negative stock return in the future. For this consideration, our new economic constraint does not arbitrarily rule out the negative stock return forecasts. We just do not trust extremely negative return forecasts (as well as extremely positive forecasts) because an extreme forecast will hardly occur in the future and the corresponding portfolio will be extremely risky. Due to this potential economic source, we find that the new constraint approach outperforms CT and PTV constraint approaches from both statistical and economic perspectives. Therefore, this paper contributes to the literature on economic constraints and stock return predictability. The remainder of the paper is organized as follows. Section 2 describes our data. Section 3 introduces the forecasting strategies of various economic constraints. Section 4 reports the empirical results from both statistical and economic perspectives. Section 5 makes some robustness checks. Section 6 concludes.

whole sample period is from 1927:01 to 2017:12. The first 20 years are used as the initial estimation period, so that the out-of-sample evaluation period is from 1947:01 to 2017:12. The excess stock return is computed as the monthly return on the S& P 500 index (including dividends) minus the risk-free rate. In addition, the 14 predictors are monthly economic fundamentals and their descriptions are as follows. 1. Log dividend-price ratio (DP): log of a 12-month moving sum of dividends paid on the S&P 500 index minus the log of stock prices (S&P 500 index). 2. Log dividend yield (DY): log of a 12-month moving sum of dividends minus the log of lagged stock prices. 3. Log earnings-price ratio (EP): log of a 12-month moving sum of earnings on the S&P 500 index minus the log of stock prices. 4. Log dividend-payout ratio (DE): log of a 12-month moving sum of dividends minus the log of a 12-month moving sum of earnings. 5. Stock return variance (SVAR): sum of squared daily returns on the S &P 500 index. 6. Book-to-market ratio (BM): book-to-market value ratio for the Dow Jones Industrial Average. 7. Net equity expansion (NTIS): ratio of a 12-month moving sum of net equity issues by NYSE-listed stocks to the total end-of-year market capitalization of NYSE stocks. 8. Treasury bill rate (TBL): interest rate on a three-month Treasury bill (secondary market). 9. Long-term yield (LTY): long-term government bond yield. 10. Long-term return (LTR): return on long-term government bonds. 11. Term spread (TMS): long-term yield minus the Treasury bill rate. 12. Default yield spread (DFY): difference between Moody's BAA- and AAA-rated corporate bond yields. 13. Default return spread (DFR): long-term corporate bond return minus the long-term government bond return. 14. Inflation (INFL): calculated from the Consumer Price Index (CPI) for all urban consumers.3 Table 1 provides the summary statistics for stock return and the 14 predictors. On average, the monthly excess stock return reaches 0.509%, which is quite appealing. Also, the monthly excess stock return has a standard deviation of 5.423%, resulting in a monthly Sharpe ratio of 0.094. In addition, the first-order autocorrelation of the excess stock return is as low as 0.086, suggesting that stock return is very difficult to be explained or predicted by its past. While the excess stock return has little autocorrelation, most of the 14 predictors are highly persistent. Overall, the summary statistics are generally consistent with the related literature on stock return predictability (see, e.g., Huang et al., 2015). 3. Forecasting strategies 3.1. Univariate predictive regression model We follow the convention in the literature on stock return predictability and begin with a univariate predictive regression model as follows.

2. Data

rt + 1 =

In this paper, we investigate the monthly stock return predictability based on 14 prevailing predictors originally analyzed in Welch and Goyal (2008).1 The data sample is extended to 2017.2 Therefore, the

i

+

i x i, t

+

i, t + 1,

(1)

where rt+1 denotes the excess stock return for month t + 1, xi,t is the ith predictor available at month t, and εi, t+1 is an error term with mean equal to zero. To generate out-of-sample forecasts of excess stock return, we use a

1 Numerous related studies also rely on these predictors to explore the predictability of stock returns. See Rapach et al. (2010), Zhu and Zhu (2013), Neely et al. (2014), Pettenuzzo et al. (2014), Li and Tsiakas (2017), and Wang et al. (2018a) for example. 2 Updated data for the variables in Welch and Goyal (2008) are available from Amit Goyal's webpage at http://www.hec.unil.ch/agoyal/. We are grateful to

(footnote continued) Amit Goyal for providing the data. 3 We lag inflation for an extra month to account for the delay in releases of the CPI. 2

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Table 1 Summary statistics. Variable

Mean

Std. Dev.

Min.

Median

Max.

Skewness

Kurtosis

ρ(1)

Ret (%) DP DY EP DE SVAR (%) BM NTIS TBL (%) LTY (%) LTR (%) TMS (%) DFY (%) DFR (%) INFL (%)

0.509 −3.373 −3.368 −2.738 −0.635 0.286 0.568 0.017 3.404 5.124 0.477 1.719 1.126 0.037 0.243

5.423 0.462 0.459 0.417 0.329 0.575 0.266 0.026 3.099 2.792 2.438 1.304 0.693 1.360 0.533

−33.927 −4.524 −4.531 −4.836 −1.244 0.007 0.121 −0.058 0.010 1.750 −11.240 −3.650 0.320 −9.750 −2.055

0.953 −3.348 −3.341 −2.790 −0.627 0.126 0.542 0.017 2.960 4.225 0.313 1.770 0.900 0.050 0.242

34.553 −1.873 −1.913 −1.775 1.380 7.095 2.028 0.177 16.300 14.820 15.230 4.550 5.640 7.370 5.882

−0.433 −0.218 −0.246 −0.602 1.516 5.794 0.779 1.651 1.078 1.085 0.589 −0.286 2.481 −0.387 1.078

7.961 −0.343 −0.363 2.620 6.031 43.652 1.464 8.247 1.282 0.603 4.689 0.165 8.870 7.790 13.818

0.086 0.992 0.992 0.986 0.991 0.633 0.985 0.979 0.993 0.996 0.043 0.961 0.975 −0.120 0.481

This table provides the summary statistics for the excess stock return (Ret), the log dividend-price ratio (DP), log dividend yield (DY), log earnings-price ratio (EP), log dividend payout ratio (DE), stock return variance (SVAR), book-to-market ratio (BM), net equity expansion (NTIS), Treasury bill rate (TBL), long-term bond yield (LTY), long-term bond return (LTR), term spread (TMS), default yield spread (DFY), default return spread (DFR), and inflation rate (INFL). ρ(1) refers to the firstorder autocorrelation. The sample period is from 1927:01 to 2017:12.

recursive (expanding) estimation window, as in Rapach et al. (2010), Neely et al. (2014), and Rapach et al. (2016), among others. Specifically, we divide the entire sample consisting of T observations into an in-sample portion consisting of the first m observations and an out-ofsample portion consisting of the last q observations. Then, the first outof-sample forecast based on the ith predictor can be obtained by

ri, m + 1 =

i, m

+

i, m + 1

+

r iPTV ,t+1 = where dictor.

where i, m + 1 and i, m + 1 are estimated by regressing {rt}t=2 on a constant and {xi, t}t=1m. Proceeding in this manner through the end of the out-of-sample period, we can obtain a series of q out-of-sample forecasts of stock return, { ri, t }Tt = m + 1. 3.2. Existing economic constraint models

+

i , t x i, t ) ,

1,

= 1, …, t ,

(5)

PTV i,

PTV i, t

r iPTV ,t+1

+

PTV x i, t , i, t

(6)

is the PTV forecast at month t + 1 based on the ith pre-

ri, t + 3 t , if ri, t + 1 > ri, t + 3 r iNew , t + 1 = ri, t

In this paper, we consider two widely used economic constraints models as competing strategies relative to our new economic constraint. The first is form Campbell and Thompson (2008). They propose an economic constraint that a rational investor will rule out a negative stock return forecast and therefore set the forecast to zero whenever it is negative.4 Statistically, the CT forecasts are given by i, t

)

In this paper, we propose a new economic constraint model, in which the forecast outliers are smoothed. This is because a rational investor is unlikely to trust an extremely large or small forecast. Similar to the PTV constraint, our constraint approach bounds the return forecasts within a rational range. Specifically, we employ the wellknown three-sigma rule5 and smooth the return forecasts as

m+1

r iCT , t + 1 = max (0,

PTV x i, i,

3.3. A new economic constraint model

(3)

i, m + 1 x i, m + 1,

+

where and are the coefficient estimates imposed by the PTV constraint, and is the empirical estimate of stock volatility at month τ. Accordingly, the PTV forecasts are computed as

where ri, m + 1 is the stock return forecast at month m + 1 based on the ith predictor, i, m and i, m are the ordinary least squares (OLS) estimates of regression slopes in (1) generated by regressing {rt}t=2m on a constant and {xi, t}t=1m−1. The next (second) out-of-sample forecast is calculated as

ri, m + 2 =

PTV i,

PTV i,

(2)

i, m x i, m ,

12 (

0

ri, t + 1,

3 t , if ri, t + 1 < ri, t otherwise,

3

t t

(7)

where is our new constraint forecast at month t + 1 based on the ith predictor, ri, t + 1 is the original forecast generated by the regression model in (1), and σt is the standard deviation of excess stock returns for month t.6 It is important to note that our new economic constraint approach relies on the three-sigma deviation and does not consider other deviations. On one hand, the two-sigma deviation is not extremely large and is thus likely to happen in the future. Therefore, it is not rational for an investor to rule out the two-sigma deviation. On the other hand, the four-sigma (or even larger) deviation is too extreme to happen in the future, which leads our economic constraint to have no binding effect. Considering these important facts, we only rely on the prevailing threesigma rule to perform our new economic constraint.

r iNew ,t+1

(4)

where is the CT forecast at month t + 1 based on the ith predictor. The second is from Pettenuzzo et al. (2014). They propose a Sharpe ratio constraint approach that bounds the in-sample annualized Sharpe ratios between zero and one. More specifically, in the PTV constraint approach, we estimate the regression slopes in Eq. (1) via constrained least squares,

r iCT ,t+1

4 In addition to the non-negative forecast constraint, Campbell and Thompson (2008) also constrain the sign of the regression slopes in predictive regression models. In this study, we do not consider the constraint on the sign of the regression slopes, so that our focus is all on the constraint of return forecasts. Of course, the results are qualitatively similar when we further include the slope constraint.

5

Note that we rely on the last stock return other than the simple mean in the three-sigma rule. 6 We use a recursive estimation window to calculate the standard deviation of excess stock returns. The results are qualitatively similar when we rely on a five-year rolling window. 3

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4. Empirical results

Table 2 Out-of-sample R-squares.

4.1. Out-of-sample forecasting performance Following the convention in return forecasting (see, e.g., Rapach et al., 2010; Neely et al., 2014; Huang et al., 2015; Rapach et al., 2016; Jiang et al., 2017; Lin, Wu, & Zhou, 2018; Zhang, Ma, & Zhu, 2019; Zhang, Zeng, Ma, & Shi, 2019), we use the out-of-sample R2 statistic advocated by Campbell and Thompson (2008) to evaluate the out-ofsample predictive accuracy of the model relative to the popular mean benchmark. The out-of-sample R2 statistic is defined as q k=1 q k=1

2 ROS =1

(rm + k

rm + k ) 2

(rm + k

rm + k )2

,

(8)

where rm+k, rm + k , and rm + k are the actual stock return, historical average, and return forecast, respectively, at month m + k, and m and q are the length of initial estimation period and forecast evaluation period, respectively. In particular, the historical average is computed as

rt =

1 t

ri.

(9)

Fig. 1 plot the time series of the actual stock return and historical average benchmark over the out-of-sample period. The historical average is always positive and exhibits a stable trend during the out-ofsample period, while the actual stock return fluctuates greatly. Although the simple mean cannot match the actual stock return, this benchmark is found to be difficult to be outperformed (see, e.g., Campbell & Thompson, 2008; Welch & Goyal, 2008). Hence, it is very necessary to use a more reliable forecasting model to predict future stock returns. The ROS2 statistic measures the reduction in mean squared forecast error (MSFE) for the forecasting model of interest relative to the prevailing historical average. To further ascertain whether the forecasting model yields a statistically significant improvement in MSFE, the Clark and West (2007) statistic is employed. More specifically, the Clark and West (2007) statistic tests the null hypothesis that the MSFE of the historical average benchmark is smaller than or equal to the MSFE of the forecasting model of interest against the alternative hypothesis that the MSFE of the historical average benchmark is larger than the MSFE of the forecasting model of interest. Mathematically, the Clark and West (2007) statistic is computed by first defining

ft = (rt

rt )2

(rt

rt ) 2 + (rt

rt )2 ,

Original

CT

PTV

New

DP DY EP DE SVAR BM NTIS TBL LTY LTR TMS DFY DFR INFL

−0.117* −0.445* −1.526* −1.466 0.157 −1.551 −0.526 0.085* −0.665* −0.778 0.089 −0.173 −0.241 −0.057

0.054* 0.039** −0.628** −1.213 0.077 −1.075 −0.525 0.271* 0.294** −0.651 0.085 −0.173 −0.423 −0.036

−0.123 −0.120 −0.001 0.113 −0.215 −0.202 0.211* 0.304* 0.069* 0.127 0.411** 0.052 0.002 −0.004

0.619** 0.204** −0.848** −0.691 0.847 −0.875 0.433* 0.873** 0.091** 0.077 0.992** 0.639 0.504 0.741

Average

−0.515

−0.279

0.045

0.257

This table reports the out-of-sample R-square (R2OS) for univariate predictive regressions based on 14 popular predictors. Original refers to the original forecasts without economic constraints, while CT, PTV, and New correspond to the Campbell and Thompson (2008) constraint approach, the Sharpe ratio constraint approach of Pettenuzzo et al. (2014), and our new economic constraint approach, respectively. Also reported are averages across all predictor variables. Bold figures highlight instances in which the constrained R2OS is higher than its unconstrained counterpart. Statistical significance for the R2OS statistic is derived by using the Clark and West (2007) test. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. The initial estimation period is 1927:01-1946:12, while the out-of-sample period is 1947:01-2017:12.

t i=1

Predictor

on the forecasting model of interest, respectively. By regressing {fs}s=m+1T on a constant, we can conveniently derive the Clark and West (2007) statistic, which is just the t-statistic of the constant. Moreover, a p-value for the one-sided (upper-tail) test is derived with the standard normal distribution. Table 2 reports the out-of-sample R2s for univariate predictive regressions subject to different economic constraints. When we use the CT and PTV constraints, 10 and 12, respectively, out of the 14 predictors generate larger ROS2s than the unconstrained counterparts. Surprisingly, our new constraint approach yields larger ROS2s than the unconstrained counterparts for all of the 14 predictors. Moreover, 11 of them are positive. On average, the CT and PTV constrained forecasts generate larger ROS2s than the original unconstrained forecasts, while our new approach yields the largest value of average ROS2 at 0.257%. Therefore, we can conclude that our new constraint approach of smoothing forecast outliers shows better out-of-sample forecasting performance than not only the original models without economic

(10)

where rt, rt , and rt are the actual stock return, the simple mean benchmark forecast of stock return, and the stock return forecast based

Fig. 1. Actual stock returns and historical average over the out-of-sample period. The out-of-sample period is from 1947:01 to 2017:12. Vertical bars depict NBERdated recessions. 4

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constraints but also the famous CT and PTV constraint approaches.

Table 3 Multivariate results.

4.2. Multivariate results

Forecasting model

Original

CT

PTV

New

So far, we follow much of the literature on stock return predictability and focus on univariate regression models. We next extend the analysis to a multivariate setting. Specifically, we follow Pettenuzzo et al. (2014) and provide two strands of multivariate forecasting models. The first strand is using principal component analysis (PCA) to construct a common factor, which is the so-called diffusion index. We then regress stock returns on a constant and the diffusion index as follows.

Diffusion index Mean Median Trimmed mean DMSPE(1) DMSPE(0.9)

0.250** 0.513*** 0.401*** 0.472*** 0.500*** 0.539***

0.291** 0.424** 0.401** 0.432** 0.412** 0.441**

0.217** 0.117 0.145 0.127 0.117 0.120

1.136** 1.283** 1.157* 1.268** 1.270** 1.306**

rt + 1 =

+

DI

DI FDI , t

+

i, t + 1,

This table reports the out-of-sample R-square (R2OS) for multivariate models. The diffusion index constructed by the principal component analysis based on the 14 predictors. Five popular combination approaches include the mean, median, trimmed mean, DMSPE(1), and DMSPE(0.9), which combine the 14 individual forecasts. Original refers to the original forecasts without economic constraints, while CT, PTV, and New correspond to the Campbell and Thompson (2008) constraint approach, the Sharpe ratio constraint approach of Pettenuzzo et al. (2014), and our new economic constraint approach, respectively. Bold figures highlight instances in which the constrained R2OS is higher than its unconstrained counterpart. Statistical significance for the R2OS statistic is derived by using the Clark and West (2007) test. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. The initial estimation period is 1927:01-1946:12, while the out-of-sample period is 1947:01-2017:12.

(11)

where FDI, t is the diffusion index that is computed as the first principal component of all the 14 predictors, which are standardized to have zero mean and unit variance. To avoid the look-ahead bias, for each onestep-ahead forecast, we only use the data available up to t to generate the diffusion index for forecasting stock return at t + 1, and the procedure is repeated throughout the entire out-of-sample forecasting period. Note that we follow Rapach et al. (2010) and Pettenuzzo et al. (2014), and restrict our analysis to considering a single factor (i.e., the first principal component). This is because the out-of-sample forecasting performance does not appear to improve if we include two or more factors in Eq. (11). The second strand is forecast combination. Rapach et al. (2010) document that forecast combination can reduce the forecast variance and thus generate more reliable stock return forecasts. Statistically, combination forecasts are computed as weighted averages of the N (N = 14) individual forecasts, which is given by

among the different constraint approaches. In other words, our new economic constraint of smoothing forecast outliers exhibits better forecasting performance in a multivariate setting. In particular, forecast combination can substantially reduce forecast variance, thus generating relatively reliable forecasts (see, e.g., Rapach et al., 2010). However, our results suggest that the combination forecasts based on the original individual forecasts still have extreme value. Furthermore, the superior out-of-sample performance indicates that our new economic constraint can avoid forecast outliers when using combination strategy.

N

rc, t + 1 =

i, t ri, t + 1,

(12)

i=1

where rc, t + 1 is the combination forecast at month t + 1, ri, t + 1 is the ith individual forecast, and ωi, t represents the combining weight of the ith individual forecast calculated at month t. Following Pettenuzzo et al. (2014), we consider five popular combination approaches recommended by Rapach et al. (2010): mean, median, trimmed mean, DMSPE(1), and DMSPE(0.9). The mean combination forecast takes the mean of the N individual forecasts, { ri, t + 1 }iN= 1. The median combination forecast takes the median of { ri, t + 1 }iN= 1. The trimmed mean combination forecast discards the smallest and largest individual forecasts in { ri, t + 1 }iN= 1 and sets ωi, t = 1/(N − 2) for the remainder of the individual forecasts. In the discount mean squared prediction error (DMSPE) combining method, the combining weights of the ith individual forecast at month t are expressed as i, t

=

i, t

1

/

N =1

1 ,t ,

4.3. Asset allocation Following a large body of related literature on stock return predictability,7 we further measure the economic value of various stock return forecasts from an asset allocation perspective. More specifically, we calculate the certainty equivalent return (CER) for a mean-variance investor who allocates between stocks and risk-free bills using various forecasts of stock returns. In order to achieve the maximum CER, the investor would allocate the weight of stocks during month t + 1 as

wt =

=

t s=m+1

t s (r s

(13)

ri, s )2 ,

2 t+1

,

(15)

where γ is the investor's risk aversion coefficient, rt + 1 denotes a return forecast, and t2+ 1 denotes a forecast of the stock return variance. As in Campbell and Thompson (2008), Rapach et al. (2010), Neely et al. (2014), and Jiang et al. (2017), among others, we estimate the variance forecasts using a five-year moving window of past stock returns.8 In addition, we restrict wt to the range between −0.5 and 1.5 to allow no more than 50% leverage.9

where i, t

1 rt + 1

(14)

m is the length of the initial training sample period and θ is a discount factor. Following Rapach et al. (2010), Zhu and Zhu (2013), and Zhang, Ma, Shi, and Huang (2018), we consider two values of θ, namely, 1 and 0.9. Consequently, two DMSPE methods, DMSPE(1) and DMSPE(0.9), are used in this study. Table 3 reports the ROS2s for the multivariate models. Two observations follow the table immediately. First, the multivariate forecasting models using all the information from the 14 predictors yield larger ROS2s than the univariate regression models. The better out-ofsample forecasting performance of the multivariate models is consistent with the previous literature on return predictability (see, e.g., Rapach et al., 2010; Zhu & Zhu, 2013; Neely et al., 2014). Second and more importantly, our new constraint approach yields the largest ROS2s

7 See, for example, Campbell and Thompson (2008), Rapach et al. (2010), Neely et al. (2014), Phan, Sharma, and Narayan (2015), Rapach et al. (2016), Jiang et al. (2017), and Wang, Qian, and Wang (2018). 8 Alternatively, Rapach et al. (2016) uses a ten-year moving window to estimate the return volatility. The economic value results are qualitatively similar for a ten-year moving window. To save space, we do not report these results, but they are available upon request. 9 More studies restrict the portfolio weight to lie between 0 and 1.5; see, e.g., Campbell and Thompson (2008), Rapach et al. (2010), Neely et al. (2014), Huang et al. (2015), and Jiang et al. (2017). However, we follow Rapach et al. (2016) and use the weight range between −0.5 and 1.5. This is because if we restrict the portfolio weight to the range between 0 and 1.5, the portfolio

5

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5. Robustness checks

Table 4 Portfolio performance.

5.1. Alternative evaluation techniques

Forecasting model

Original

CT

PTV

New

DP DY EP DE SVAR BM NTIS TBL LTY LTR TMS DFY DFR INFL Diffusion index Mean Median Trimmed mean DMSPE(1) DMSPE(0.9)

−1.829 −2.137 0.418 −0.274 −0.205 −2.407 0.538 1.518 0.630 0.435 1.570 −0.395 0.471 0.236 0.230 0.870 0.669 0.805 0.884 0.992

−1.236 −1.292 0.540 −0.210 −0.205 −1.642 0.538 1.500 0.972 0.434 1.578 −0.366 0.404 0.276 0.150 0.425 0.669 0.516 0.435 0.511

−1.067 −1.106 −0.485 −0.199 −0.802 −1.345 −0.366 0.287 −0.389 0.303 0.523 −0.931 −1.019 −0.456 −0.658 −0.562 −0.207 −0.340 −0.561 −0.543

−1.609 −2.080 0.548 0.090 0.266 −2.342 0.980 1.827 0.923 0.773 2.044 −0.036 0.760 0.577 0.699 1.260 1.007 1.163 1.275 1.383

Although the ROS2 statistic and Clark and West (2007) test are the conventions in examining the stock return predictability, a few articles rely on other statistical evaluation techniques. For example, Charles, Darné, and Kim (2017) and Nonejad (2018) both use the model confidence set (MCS) to assess the out-of-sample forecasting performance of stock returns. Given this, we also use the MCS test to check the economic constraint approaches. The MCS test is originally proposed by Hansen, Lunde, and Nason (2011). This test can ascertain whether the forecasting models have a statistically significant difference in the out-of-sample performance without specifying a benchmark model. A MCS is a subset of models that contains the best model with a given level of confidence. The interpretation of a MCS p-value is analogous to that of a classical p-value (Hansen et al., 2011). It is evident that a model with a larger MCS pvalue shows stronger predictive ability. Following Hansen et al. (2011) and Ma, Li, Liu, and Zhang (2018), among others, we consider the significance (confidence) level of 25% (75%). That is, a forecasting model will be included in the MCS when its MCS p-value is larger than 0.25. The excluded models are viewed as significantly inferior models. In particular, all of the MCS p-values reported in this paper is calculated based on the range statistic using the circular block bootstrap.10 In this study, the MCS test is based on two popular loss functions, namely, mean squared error (MSE) and mean absolute error (MAE).11 Table 5 provides the mean values of the two loss functions and the corresponding MCS p-values. In Panel A of Table 5, we find that our new constraint approach yields lower forecast error including both MSE and MAE than the unconstrained counterparts for all the used forecasting models. This is consistent with the evidence from the ROS2 statistic. We report the MCS p-values in Panel B of Table 5. While there is no statistically significant difference among the four economic constraint approaches for some forecasting models, the overall forecasting performance of our new economic constraint is the best. Specifically, when we depend on the loss function of MSE, only our new constraint approach can fall into the MCS with the 75% confidence level for all the 20 forecasting models (including 14 univariate models and 6 multivariate models). Furthermore, our new constraint approach yields the largest MCS p-value of 1 for most of the forecasting models. In terms of MAE, only one model based on our new economic constraint does not enter the MCS with the 75% confidence level. In contrast, the unconstrained approach as well as the CT and PTV constrained approaches have more forecasting models that cannot enter the MCS. In addition, our new constraint approach yields the largest MCS p-value of 1 for 17 out of the 20 forecasting models based on MAE, while the CT and PTV approaches only have 1 and 2, respectively, models that yield the largest MCS p-value of 1. In summary, compared to the ROS2 statistic and Clark and West (2007) test, the MCS test based on MSE and MAE produces robust results. That is, the new constraint approach outperforms not only the original version but also the prevailing CT and PTV constrained versions.

This table reports the certainty equivalent return (CER) gains. Original refers to the original forecasts without economic constraints, while CT, PTV, and New correspond to the Campbell and Thompson (2008) constraint approach, the Sharpe ratio constraint approach of Pettenuzzo et al. (2014), and our new economic constraint approach, respectively. The annualized CER is calculated based on a mean-variance investor with relative risk aversion coefficient of three who allocates between stocks and risk-free bills using the return forecasts from various forecasting models. The CER gain is calculated as the difference between the CER for the investor when she uses return forecasts and the CER when she uses the prevailing mean benchmark. Bold figures highlight instances in which the constrained CER gain is higher than its unconstrained counterpart. The initial estimation period is 1927:01-1946:12, while the out-of-sample period is 1947:01-2017:12.

For a portfolio constructed by Eq. (15), the investor can realize an average CER as

CER = Rp

0.5

2 p,

(16)

where Rp and σp2 denote the mean and variance, respectively, of the realized portfolio returns during the out-of-sample evaluation period. The CER gain is calculated as the difference between the CER for the investor when she uses return forecasts and the CER when she uses the prevailing mean benchmark. Accordingly, the CER gain can be regarded as the portfolio management fee that a mean-variance investor would be willing to pay to have access to the return forecasts in instead of the historical average forecasts. Table 4 reports the CER gains for all the forecasting models subject to different economic constraints. Compared with the original forecasting models without economic constraints, the forecasting models subject to the CT and PTV constraints do not enhance the CER gains obviously. Fortunately, all of the forecasting models based on our new constraint approach yield larger CER gains than the original ones and most of the ones based on the CT and PTV constraints. Furthermore, all the multivariate forecasting models subject to our new constraint generate considerably positive CER gains, all of which are larger than not only the CT and PTV constrained counterparts but also the unconstrained counterparts. In conclusion, a mean-variance investor can realize larger economic gains using our new constraint approach to allocate her portfolio.

5.2. Business cycles While the overall ROS2 is interesting, it is also important to analyze the return predictability during business cycles. Following related studies (see, e.g., Rapach et al., 2010; Neely et al., 2014; Huang et al., 10

We obtain qualitatively similar results of MCS p-values when using the semi-quadratic statistic or the stationary bootstrap. To save space, we do not report these results, but they are available upon request. 11 Note that the loss function of MSE is consistent with the out-of-sample R2, which is calculated based on MSE.

(footnote continued) performance of the CT constraint method will reduce to that of the original forecasting models without economic constraints. 6

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Table 5 Out-of-sample performance using the MCS test. Forecasting model

MSE Original

MAE CT

PTV

New

Original

CT

PTV

New

Panel A: The mean value of loss functions DP 0.171 DY 0.172 EP 0.174 DE 0.173 SVAR 0.171 BM 0.174 NTIS 0.172 TBL 0.171 LTY 0.172 LTR 0.172 TMS 0.171 DFY 0.171 DFR 0.171 INFL 0.171 Diffusion index 0.171 Mean 0.170 Median 0.170 Trimmed mean 0.170 DMSPE(1) 0.170 DMSPE(0.9) 0.170

0.171 0.171 0.172 0.173 0.171 0.173 0.172 0.170 0.170 0.172 0.171 0.171 0.172 0.171 0.170 0.170 0.170 0.170 0.170 0.170

0.171 0.171 0.171 0.171 0.171 0.171 0.171 0.170 0.171 0.171 0.170 0.171 0.171 0.171 0.171 0.171 0.171 0.171 0.171 0.171

0.170 0.171 0.172 0.172 0.169 0.172 0.170 0.169 0.171 0.171 0.169 0.170 0.170 0.170 0.169 0.169 0.169 0.169 0.169 0.169

3.180 3.197 3.200 3.151 3.140 3.213 3.133 3.134 3.154 3.170 3.136 3.147 3.150 3.139 3.174 3.143 3.140 3.145 3.144 3.143

3.173 3.175 3.177 3.148 3.140 3.195 3.133 3.136 3.147 3.165 3.135 3.147 3.150 3.139 3.170 3.141 3.140 3.143 3.142 3.141

3.170 3.176 3.169 3.155 3.160 3.175 3.154 3.156 3.165 3.160 3.148 3.151 3.160 3.159 3.169 3.160 3.157 3.160 3.160 3.160

3.169 3.188 3.190 3.141 3.130 3.203 3.120 3.123 3.143 3.158 3.124 3.135 3.139 3.128 3.162 3.133 3.129 3.133 3.133 3.132

Panel B: MCS p-values DP DY EP DE SVAR BM NTIS TBL LTY LTR TMS DFY DFR INFL Diffusion index Mean Median Trimmed mean DMSPE(1) DMSPE(0.9)

0.508* 0.871* 0.737* 0.276* 0.427* 0.359* 0.447* 0.722* 1.000* 0.367* 0.509* 0.351* 0.462* 0.426* 0.440* 0.292* 0.239 0.300* 0.301* 0.294*

0.508* 0.871* 1.000* 1.000* 0.427* 1.000* 0.836* 0.722* 0.750* 1.000* 0.509* 0.383* 0.555* 0.426* 0.440* 0.203 0.239 0.212 0.206 0.181

1.000* 1.000* 0.737* 0.460* 1.000* 0.485* 1.000* 1.000* 0.833* 0.952* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000*

0.007 0.001 0.101 0.540* 0.402* 0.002 0.299* 0.357* 0.358* 0.111 0.349* 0.242 0.349* 0.345* 0.132 0.352* 0.121 0.282* 0.345* 0.353*

0.875* 1.000* 0.641* 0.704* 0.402* 0.127 0.299* 0.357* 0.840* 0.763* 0.349* 0.242 0.349* 0.345* 0.716* 0.363* 0.121 0.284* 0.364* 0.353*

0.943* 0.884* 1.000* 0.704* 0.058 1.000* 0.246 0.034 0.032 0.889* 0.237 0.242 0.318* 0.042 0.716* 0.028 0.041 0.042 0.028 0.025

1.000* 0.400* 0.569* 1.000* 1.000* 0.127 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000*

0.319* 0.263* 0.485* 0.265* 0.427* 0.103 0.447* 0.501* 0.473* 0.367* 0.509* 0.351* 0.462* 0.342* 0.440* 0.292* 0.239 0.300* 0.301* 0.294*

This table reports the mean values of the two loss functions of MSE and MAE in Panel A and the corresponding MCS p-values in Panel B. Original refers to the original forecasts without economic constraints, while CT, PTV, and New correspond to the Campbell and Thompson (2008) constraint approach, the Sharpe ratio constraint approach of Pettenuzzo et al. (2014), and our new economic constraint approach, respectively. Bold figures highlight instances in which the constrained value of MSE or MAE is smaller than its unconstrained counterpart. *Highlights instances in which the MCS p-value is larger than 0.25. The initial estimation period is 1927:011946:12, while the out-of-sample period is 1947:01-2017:12.

2015; Jiang et al., 2017; Ma, Liu, Wahab, & Zhang, 2018; Wang, Qian, & Wang, 2018), we compute the ROS2 statistic separately for expansions (ROS, EXP2) and recessions (ROS, REC2), 2 ROS ,c = 1

q k=1 q k=1

Imc + k (rm + k

rm + k ) 2

Imc + k (rm + k

rm + k ) 2

for c = EXP, REC,

new constraint approach is more effective in the forecasting of stock returns than the CT and PTV models for both expansions and recessions and our approach itself shows stronger forecasting performance during the recession period than during the expansion period. 5.3. Alternative risk aversion coefficients

(17)

where Im+kEXP (Im+kREC) is an indicator that takes a value of one when month m + k is in an NBER expansion (recession) period and zero otherwise. Table 6 reports the out-of-sample forecasting performance over business cycles. Two important findings emerge. First, consistent with the related literature on return predictability (see, e.g., Rapach et al., 2010; Neely et al., 2014; Huang et al., 2015; Jiang et al., 2017), the return predictability is concentrated over recessions for all the forecasting models. Second and more importantly, for both the expansion and recession periods, most of the forecasting models subject to our new constraint generate larger ROS2s than the ones subject to the CT and PTV constraints as well as the unconstrained counterparts. In summary, our

In the asset allocation exercise above, we assume that a mean-variance investor has a risk aversion coefficient of three. However, the optimal portfolio weight given in Eq. (15) changes with the value of risk aversion coefficient. For this consideration, we further consider other different risk aversion coefficients and investigate the impact of risk aversion coefficient on the portfolio performance. Table 7 reports the CER gains for alternative risk aversion coefficients, including two, four, and six. Although the CER gains vary with different risk aversion coefficients, all of the forecasting models based on our new constraint approach yield larger CER gains than the original ones. Furthermore, all the multivariate forecasting models subject to our new constraint still produce considerably positive CER gains, which 7

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Table 6 Out-of-sample performance over business cycles. Forecasting model

DP DY EP DE SVAR BM NTIS TBL LTY LTR TMS DFY DFR INFL Diffusion index Mean Median Trimmed mean DMSPE(1) DMSPE(0.9)

Expansions

Recessions

Original

CT

PTV

New

Original

CT

PTV

New

−0.800 −1.578 −1.283* −1.042 −0.025 −1.975 0.624** −0.348 −1.117 −1.374 −0.199 −0.174 −0.117 0.074 −0.808 0.378** 0.324** 0.321* 0.362** 0.363**

−0.562 −0.886 −0.508* −0.989 −0.042 −1.321 0.625** 0.130 0.080* −1.200 −0.224 −0.174 −0.198 0.082 −0.615 0.392** 0.324** 0.328** 0.377** 0.375**

−0.762 −0.847 −0.448 −0.273 −0.709 −0.764 −0.175 −0.276 −0.628 −0.510 0.001 −0.134 −0.453 −0.474 −0.555 −0.384 −0.330 −0.374 −0.385 −0.383

−0.057 −0.925 −0.524** −0.227 0.837 −1.270 1.483** 0.409* −0.372* −0.434 0.693 0.627 0.646 0.861 0.076 1.168* 1.098 1.131* 1.153* 1.153*

1.779** 2.696*** −2.199 −2.641 0.662 −0.374 −3.717 1.289 0.587 0.877 0.888 −0.172 −0.587 −0.419 3.183*** 0.889** 0.615** 0.888** 0.883** 1.027**

1.764** 2.608** −0.960 −1.835 0.407 −0.394 −3.717 0.663 0.889 0.873 0.941 −0.172 −1.048 −0.362 2.804** 0.511 0.615** 0.720** 0.508* 0.625*

1.650*** 1.898*** 1.241* 1.185** 1.157** 1.355** 1.282** 1.915** 2.000** 1.896*** 1.550*** 0.568* 1.265** 1.302** 2.361*** 1.507*** 1.463*** 1.515*** 1.507*** 1.514***

2.493** 3.335*** −1.747 −1.979 0.872 0.221 −2.478 2.160* 1.374 1.495* 1.822** 0.673 0.110 0.408 4.077*** 1.601** 1.320* 1.648** 1.595** 1.729**

This table reports the out-of-sample R-square (R2OS) over business cycles. Original refers to the original forecasts without economic constraints, while CT, PTV, and New correspond to the Campbell and Thompson (2008) constraint approach, the Sharpe ratio constraint approach of Pettenuzzo et al. (2014), and our new economic constraint approach, respectively. Bold figures highlight instances in which the constrained R2OS is higher than its unconstrained counterpart. Statistical significance for the R2OS statistic is derived by using the Clark and West (2007) test. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. The initial estimation period is 1927:01-1946:12, while the out-of-sample period is 1947:01-2017:12.

are larger than the other constrained and unconstrained counterparts. The CER results are thus robust to alternative risk aversion coefficients.

this, we follow Neely et al. (2014) and assume a proportional transaction cost equal to 50 basis points per transaction. Table 8 reports the CER gains net of transaction cost. Two observations follow the table. First, as expected, all the CER gains are lower than the ones without transaction cost. This is intuitive because the economic value is inversely related to the average monthly turnover and the historical average benchmark always yields a very low turnover. As evidenced in

5.4. Transaction cost In the practical application, we need to pay transaction cost when trading assets, which will influence the portfolio performance. Given Table 7 Portfolio performance for alternative risk aversion coefficients. Forecasting model

DP DY EP DE SVAR BM NTIS TBL LTY LTR TMS DFY DFR INFL Diffusion index Mean Median Trimmed mean DMSPE(1) DMSPE(0.9)

Risk aversion coefficient is 2

Risk aversion coefficient is 4

Risk aversion coefficient is 6

Original

CT

PTV

New

Original

CT

PTV

New

Original

CT

PTV

New

−2.957 −3.641 0.243 −0.342 −0.078 −3.080 0.298 0.761 −0.045 0.355 1.464 −0.800 0.215 −0.085 −0.434 0.223 0.225 0.351 0.227 0.264

−2.403 −2.732 0.445 −0.277 −0.078 −2.315 0.298 0.740 0.277 0.367 1.434 −0.757 0.127 −0.027 −0.623 0.181 0.225 0.242 0.178 0.158

−1.925 −1.988 −1.098 −0.696 −1.182 −2.149 −0.619 −0.030 −0.848 0.297 0.396 −0.939 −1.312 −1.030 −1.270 −0.725 −0.565 −0.607 −0.725 −0.715

−2.672 −3.556 0.409 0.080 0.373 −3.012 0.750 1.092 0.262 0.709 1.916 −0.394 0.586 0.294 0.015 0.675 0.600 0.754 0.680 0.721

−1.064 −1.368 0.272 −0.270 −0.337 −2.161 −0.033 1.270 0.733 0.102 1.202 −0.712 0.237 0.182 0.168 0.829 0.606 0.763 0.824 0.956

−0.429 −0.579 0.384 −0.221 −0.337 −1.429 −0.033 1.258 1.144 0.092 1.225 −0.690 0.185 0.216 0.247 0.541 0.606 0.591 0.535 0.603

−0.556 −0.586 −0.119 0.101 −0.739 −0.743 −0.094 0.518 −0.047 0.493 0.648 −0.970 −0.724 −0.082 −0.233 −0.176 0.107 −0.007 −0.175 −0.160

−0.886 −1.325 0.383 0.018 0.154 −2.100 0.369 1.572 0.986 0.436 1.651 −0.374 0.468 0.508 0.618 1.154 0.882 1.058 1.150 1.271

0.004 −0.304 0.177 −0.587 −0.489 −1.816 −0.223 1.020 0.642 −0.294 0.488 −0.838 0.141 0.083 0.396 1.014 0.708 1.028 1.008 1.150

0.618 0.540 0.335 −0.554 −0.489 −1.118 −0.223 1.069 1.129 −0.309 0.510 −0.823 0.102 0.106 0.671 0.757 0.708 0.913 0.752 0.860

0.234 0.214 0.527 0.676 −0.421 0.111 0.545 0.952 0.574 0.934 1.031 −0.282 −0.003 0.552 0.448 0.489 0.678 0.602 0.490 0.499

0.139 −0.260 0.268 −0.377 0.000 −1.759 0.149 1.263 0.827 −0.055 0.866 −0.513 0.312 0.322 0.766 1.247 0.908 1.241 1.241 1.376

This table reports the certainty equivalent return (CER) gains for alternative risk aversion coefficients, including 2, 4, and 6. Original refers to the original forecasts without economic constraints, while CT, PTV, and New correspond to the Campbell and Thompson (2008) constraint approach, the Sharpe ratio constraint approach of Pettenuzzo et al. (2014), and our new economic constraint approach, respectively. The annualized CER is calculated based on a mean-variance investor who allocates between stocks and risk-free bills using the return forecasts from various forecasting models. The CER gain is calculated as the difference between the CER for the investor when she uses return forecasts and the CER when she uses the prevailing mean benchmark. Bold figures highlight instances in which the constrained CER gain is higher than its unconstrained counterpart. The initial estimation period is 1927:01-1946:12, while the out-of-sample period is 1947:01-2017:12. 8

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acknowledges the support from the National Natural Science Foundation of China [71371157, 71671145], the Humanities and Social Science Fund of Ministry of Education of China [17YJA790015, 17XJA790002, 18YJC790132, 18XJA790002] and Key Laboratory of Service Computing and Safety Management of Yunnan Provincial Universities. Feng Ma acknowledges the support from the Natural Science Foundation of China [71701170], the Humanities and Social Science Fund of the Ministry of Education [17YJC790105, 17XJCZH002], and Fundamental research funds for the central universities [682017WCX01, 2682018WXTD05].

Table 8 Portfolio performance including transaction cost. Forecasting model

Original

CT

PTV

New

DP DY EP DE SVAR BM NTIS TBL LTY LTR TMS DFY DFR INFL Diffusion index Mean Median Trimmed mean DMSPE(1) DMSPE(0.9)

−2.066 −2.542 0.132 −0.308 −0.286 −2.676 0.373 1.349 0.439 −1.243 1.319 −0.501 −0.186 −0.474 −0.104 0.733 0.519 0.669 0.744 0.832

−1.413 −1.463 0.321 −0.235 −0.286 −1.810 0.373 1.404 0.882 −1.162 1.358 −0.465 −0.226 −0.412 −0.037 0.329 0.519 0.399 0.338 0.405

−1.092 −1.173 −0.527 −0.193 −1.100 −1.380 −0.353 0.254 −0.445 −0.539 0.401 −1.010 −1.416 −0.484 −0.716 −0.583 −0.259 −0.356 −0.582 −0.564

−1.867 −2.492 0.255 0.015 0.150 −2.614 0.762 1.618 0.694 −0.913 1.735 −0.188 0.103 −0.177 0.351 1.082 0.815 0.984 1.095 1.180

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This table reports the certainty equivalent return (CER) gains when we assume a proportional transaction cost of 50 basis points per transaction. Original refers to the original forecasts without economic constraints, while CT, PTV, and New correspond to the Campbell and Thompson (2008) constraint approach, the Sharpe ratio constraint approach of Pettenuzzo et al. (2014), and our new economic constraint approach, respectively. The annualized CER is calculated based on a mean-variance investor with relative risk aversion coefficient of three who allocates between stocks and risk-free bills using the return forecasts from various forecasting models. The CER gain is calculated as the difference between the CER for the investor when she uses return forecasts and the CER when she uses the prevailing mean benchmark. Bold figures highlight instances in which the constrained CER gain is higher than its unconstrained counterpart. The initial estimation period is 1927:01-1946:12, while the out-of-sample period is 1947:01-2017:12.

Fig. 1, the historical average is very stable. However, the return forecasts generated by the forecasting models are relatively fluctuant, thus yielding a higher turnover. The higher turnover leads to higher transaction cost. The second observation is that the forecasting models based on our new constraint approach typically yield larger CER gains than the other constrained and unconstrained models. The CER results are thus robust to the consideration of transaction cost. 6. Conclusion In this paper, we propose a new economic constraint approach to predict stock returns. Since a rational investor does not trust extremely large and small return forecasts, we truncate the forecasts at the extreme values. We compare the new constraint approach to two previous economic constraint approaches proposed by Campbell and Thompson (2008) and Pettenuzzo et al. (2014). The out-of-sample results suggest that our new constraint approach outperforms the prevailing CT and PTV constraints as well as the original models without economic constraints from both statistical and economic perspectives. The superiority of our new constraint is supported by not only the univariate regression models but also more reliable multivariate models. Acknowledgments The authors are grateful to the editor and three anonymous referees for insightful comments that significantly improved the paper. This work is supported by Doctoral Innovation Fund Program of Southwest Jiaotong University [D-CX201724] and Service Science and Innovation Key Laboratory of Sichuan Province [KL1704]. In particular, Yu Wei

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