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Return direction forecasting: a conditional autoregressive shape model with beta density
Financial Innovation volume 9, Article number: 82 (2023)
Abstract
This paper derives a new decomposition of stock returns using price extremes and proposes a conditional autoregressive shape (CARS) model with beta density to predict the direction of stock returns. The CARS model is continuously valued, which makes it different from binary classification models. An empirical study is performed on the US stock market, and the results show that the predicting power of the CARS model is not only statistically significant but also economically valuable. We also compare the CARS model with the probit model, and the results demonstrate that the proposed CARS model outperforms the probit model for return direction forecasting. The CARS model provides a new framework for return direction forecasting.
Introduction
The predictability of the stock market has long been an interesting and important topic in both financial economics and econometrics. This topic is interesting from different perspectives. For practitioners, better forecasts mean better investment returns. For theoretical researchers, market predictability is highly related to asset pricing theories. There is a vast volume of literature that focuses on the prediction of stock returns using public information. For a comprehensive and uptodate overview of the literature on stock return predictability, we refer to Rapach and Zhou (2013).
It is wellknown that the overall level, or the conditional mean of the excess return, is notoriously difficult to forecast (Welch and Goyal 2008); however, the direction of stock returns is much more predictable. For example, Christoffersen and Diebold (2006) and Christoffersen et al. (2007) established the theoretical connection between asset return volatility and return direction predictability and demonstrated that return direction is predictable. Many studies have documented that only the direction of asset returns is predictable (see, among others, Breen et al. 1988; Hong and Chung 2006; Chung and Hong 2007).
Directional predictability is important for market timing, which is crucial for asset allocation decisions between stock and riskfree interest rate investments. Leitch and Tanner (1991, 1995) show that the directionofchange criterion may be better able to capture a utilitybased measure of forecasting performance such as economic profits (see Granger and Pesaran 2000; Pesaran and Skouras 2001 for further discussion). Various classificationbased qualitative models, such as traditional static logit and probit models were considered by Leung et al. (2000), whereas Hong and Chung (2006), Rydberg and Shephard (2003), and Anatolyev and Gospodinov (2007) used the socalled autologistic model, while Nyberg (2011) used the socalled dynamic probit model to predict the return direction. Nyberg and Ponka (2016) extended the dynamic probit model to a bivariate case and include interaction effects with the United States for return predictions in other financial markets. The recent academic literature shows a rising interest in using machine learning to forecast return direction (Kara et al. 2011; Qiu and Song 2016; Zhong and Enke 2019).
This study attempts to provide a new approach to forecast the direction of stock return. Traditionally, return direction forecasting models are based on the sign of return, which totally ignores the possible value of price extremes in return direction forecasting. The recent academic literature shows that price extremes are informative for return forecasting (see Xie and Wang 2013, 2018). Thus, the question of “how to use price extremes to forecast the direction of stock returns” is of great interest and also the focus of this paper.
With price extremes, asset return can be decomposed into a linear combination of two parts, and the direction of the stock return is determined by the relative strength of these two parts. A simple transformation shows that direction prediction is equivalent to a ratio prediction. To model the dynamics of this ratio series, a conditional autoregressive shape model with beta (henceforth BCARS) density is proposed. The specification of the BCARS model is much like the GARCH model and continuously valued, which makes it totally different from classificationbased qualitative models.
An empirical study is performed on the monthly US stock index to evaluate the forecasting power of the BCARS model in return direction forecasting. The results show that the return direction is significantly predictable in both statistical and economic senses, judging by either the Sharpe ratio or utility gain. According to the utility gain, investors would like to pay an annual return of 3.84% to access the forecasts relative to the simple market portfolio, and such payment is even as much as 21.70% in market recession.
We also compare the BCARS model with the commonly used classification probit model. The results show that the BCARS model significantly outperforms the probit model in an economic sense. Another interesting finding is that the macroeconomic and financial variables selected by the BCARS model are different from those by the probit model, which confirms that the BCARS model is different from the probit model in a new perspective.
The main contributions of this paper are presented as follows: first, return direction forecasting models are based on the sign of the return, which ignores the possible value of price extremes in return direction forecasting; thus, this paper proposes a method to predict the direction of stock return by using stock extreme value information. Second, we conduct empirical analysis on the monthly US stock index to evaluate the forecasting power of the BCARS model in return direction forecasting and compare the BCARS model with the commonly used classification probit model. The remainder of this paper is organized as follows: “Econometric methods” section introduces the BCARS model. “Empirical results on S &P500” section empirically investigates the forecasting power of the BCARS model on the US stock market index. “Further evidence” section compares the BCARS model with the classic classification probit model. We conclude in “Conclusion” section.
Econometric methods
In this section, we will first show that return can be decomposed into a multiplication of two components using price extremes and then propose a new econometric method for return direction forecasting.
Return decomposition
Let \(p_{t}\) be the logarithmic price of a speculative asset observed at time t. The high price \(h_t\) over time interval [t1, t] is defined as
The asset return \(r_{t}\) over [t1, t] can thus be expressed as
where \(u_t\) measures the quantity of maximum price increase, and \(d_{t}\) measures the quantity of maximum price decrease.
Let \(R_{t}\)=\(u_{t}\)+\(d_{t}\), which measures the total price variation over [t1, t]. We define
where \(ur_{t}\) (\(dr_{t}\)) gauges the contribution of maximum price increase (decrease) to the total price variation. We call \(ur_t\) the up ratio and \(dr_t\) the down ratio. It is evident that
From Eqs. (2)–(4), it follows that asset return can be rewritten as
which means that the return can be decomposed into a multiplication of a return variation and a ratio. Given that \(R_t\)>0, Eq. (5) implies that if \(ur_{t}\)>\(\frac{1}{2}\) then the return is positive. This fact hints that we can predict the direction of the return by predicting \(ur_t\): if \(ur_t\) is predicted to be larger than 1/2, then we should predict a positive return; otherwise, we should forecast a negative return.
The model
Given that \(ur_t\) \(\in\)[0, 1] and is continuously valued, a natural choice for its density is the beta distribution
where \(\Gamma (.)\) is a gamma function. The shape of the beta density is controlled by the ratio of \(\alpha\) over \(\beta\): if \(\frac{\alpha }{\beta }\)=1, the beta density is symmetric around 1/2; if \(\frac{\alpha }{\beta }\)>1, the density function is leftskewed about 1/2; otherwise, it is rightskewed about 1/2.
Given the fact that the expectation of \(ur_t\) is
we thus propose the following model to describe the dynamics of \(ur_t\)
where x is a variable in the probability density function, \(k_t\)=\({\alpha _t}/{(\alpha _t +\beta )}\), and
As \(k_t\) controls the shape, Eq. (8) is thus called the conditional autoregressive shape of order (P, Q) with beta density, or BCARS(P,Q) for short. Condition (9) is required to ensure that \(k_t\) \(\in\) [0, 1].
Let information filtration \(\Omega _{t}\)={\(k_t\), \(k_{t1}\),...}. The conditional expectation of \(ur_t\) is given by
The unconditional expectation of \(ur_t\) is
The BCARS model can be further extended by including exogenous variables,
where \(X_{t1}\) is an exogeneous variable. To ensure that \(k_t\) \(\in\) [0, 1], we also require that \(X_t\) \(\in\) [0, 1] and that
Model estimation
The maximum likelihood estimation (MLE) method can be used to estimate the unknown parameters in the BCARS model.
Conditional on information filtration \(\Omega _{T}\), the joint density of (\(ur_{T}, ur_{T1},\ldots , ur_{1}\)) is given by
where
The loglikelihood function of the joint density is given by
By maximizing the likelihood function, the unknown parameters in the BCARS model can be obtained.
Empirical results on S &P500
This section is used to evaluate the performance of the BCARS model for forecasting the direction of stock returns.
Data
We collect the monthly Standard & Poor’s 500 (S &P500) stock index data for the sample period from January 1928 to December 2019 with 1,104 data observations. For each month, four pieces of price information—open, high, low, and close—are reported.^{Footnote 1} The data set is downloaded from the finance subdirectory of the website “www.finance.yahoo.com”.
Table 1 presents the summary statistics of monthly return and up ratio. Consistent with the welldocumented facts, the return on S &P500 is far from being normal with high kurtosis and negative skewness. Both the mean and median of the up ratio are larger than 0.5, indicating that asset returns are more likely to be positive. Figure 1 presents the histogram and autocorrelation plots of the return and up ratio. The left panel shows the sample autocorrelations of return series and its histogram, and the right panel presents the sample autocorrelations of the up ratio series and its histogram. Two interesting findings emerge from Fig. 1. First, there is no autocorrelation in the return series, while the autocorrelation is significant in the up ratio series. The sample autocorrelation function (acf) of the up ratio at lag 5 is reported to be 0.123. Second, the distribution of the return is bellshaped while the distribution of the up ratio is Ulike.
We also collect macroeconomic and financial variables, including riskfree rate (rfree), dividendprice ratio (dp), dividend yield (dy), earningprice ratio (ep), dividendpayout ratio (de), stock variance (svar), booktomarket ratio (bm), net equity expansion (ntis), treasury bill rate (tbl), longterm return (ltr), term spread (tms), default yield spread (dfy), and inflation (infl).^{Footnote 2}

dividendprice ratio, dp: difference between the log of dividends paid on the S &P500 index and log of stock prices (S &P500 index), where dividends are measured using a oneyear moving sum.

dividend yield, dy: difference between the log of dividends and log of lagged stock prices.

earningsprice ratio, ep: difference between the log of earnings on the S &P 500 index and log of stock prices, where earnings are measured using a oneyear moving sum.

dividend payout ratio, de: difference between the log of dividends and log of lagged stock prices.

stock variance, svar: sum of squared daily returns on the S &P 500 index.

booktomarket ratio, bm: ratio of book value to market value for the Dow Jones Industrial Average.

net equity expansion, ntis: ratio of twelvemonth moving sums of net issues by NYSElisted stocks to total endofyear market capitalization of NYSE stocks.

treasure bill rate, tbl: interest rate on a threemonth Treasury bill (secondary market).

longterm return, ltr: return on longterm government bonds.

term spread, tms: difference between the longterm yield and Treasury bill rate.

default yield spread, dfy: difference between BAA and AAArated corporate bond yields.

inflation, infl: inflation is the Consumer Price Index (All Urban Consumers).
Table 2 presents the summary statistics. All these variables are regularized using the following formula:
Regularization is employed to ensure that \(r(x_t)\) satisfies the condition of Eq. (12). The last column shows the correlation between the up ratio and these regularized variables.
Insample estimation
It has been well documented that order (1,1) can well capture the dynamics of financial time series; thus, we only consider the following BCARS model of order (1,1)^{Footnote 3}
The estimates of the BCARS(1,1) models are presented in Table 3. Column 1 presents the estimate of the benchmark BCARS(1,1) model. We find that the up ratio is indeed predictable as its conditional mean, while \(k_t\) is reported to be persistent although the \(R^2\) = 0.66% is small. Figure 2 presents the series \(ur_t\) and its forecast. It is clear from Fig. 2 that the conditional mean of \(ur_t\) is meanreverting and predictable to some extent.
As it has been presented in “The model” section that the shape of the beta density is controlled by the relative strength of \(\alpha _t\) to \(\beta\), Fig. 3 presents the plots of the \(\alpha _t\) and \(\beta\) estimates. Series \(\alpha _t\) is calculated using the following formula:
Two main findings emerge from Fig. 3. First, both \(\alpha _t\) and \(\beta\) are less than 1, indicating that the density of \(ur_t\) concentrates more on tail than on central. Second, \(\alpha _t\) is larger than \(\beta\) most of the time, indicating that the right tail in the density of \(ur_t\) is higher than the left tail. Summarizing these two facts, the density of \(ur_t\) is more likely to be Jshaped.
Notably, BCARS models with exogeneous variables are also estimated, and the results are shown in the remaining columns. Only two variables, ntis and ltr are found to be significantly important for forecasting \(ur_t\). Different from the findings of Christoffersen et al. (2007), market variance is found to be insignificant, and the contribution of svar to up ratio forecasting is very small as the \(R^2\) statistic only increases by 0.04% from 0.66% to 0.70%. The reason may be that volatility clustering effect has been removed from return since
which indicates that the return series \(r_t\) has been standardized by price variation \(R_t\).
Outofsample forecast
Insample fitting does not guarantee outofsample predictability. This section investigates the outofsample forecasting power of the BCARS model. For outofsample forecasting, an extending window forecasting procedure is used:
Step 1: The first q data observations are used to estimate the model;
Step 2: Using the estimates to forecast \(k_{q+1}\);
Step 3: Let q=q+1, and return to step 1. Repeat the abovestated procedures till the end.
Following Campbell and Thompson (2008), the following outofsample Rsquare statistic, \(R^{2}_{oos}\), is used to evaluate the empirical performance of the BCARS model relative to the simple historical mean model:
where \({\hat{ur}}_{t}\) are forecasts reported by BCARS and \({\bar{ur}}_t\) are historical mean forecasts defined as follows:
Historical mean model simulates efficient market forecasting. Furthermore, BCARS beats the historical mean if \(R^2_{oos}\) is positive, otherwise the historical mean model outperforms.
Herein, the first 500 observations are used as the initial window, and the remaining 603 observations are used as testing samples. Table 4 presents the outofsample Rsquare statistics. We only report variables that are found to be important in the insample estimation. Two findings can be obtained from Table 4. First, the positive \(R^2_{oos}\)=0.099% indicates that the benchmark BCARS(1,1) model outperforms the simple historical mean model, which is consistent with the insample results. Second, including exogeneous variables does not necessarily improve outofsample forecasting performance. For example, when ntis is included, the \(R^2_{oos}\) decreases from 1.05% to \(\)0.044%.
Welch and Goyal (2008) present comprehensive evidence that few models significantly beat the historical mean forecast; therefore, a null hypothesis of interest to investors is \(R^{2}_{oos}\le\)0 against the alternative hypothesis that \(R^{2}_{oos}>\)0. We test this hypothesis by using the mean square forecast error (MSFE)adjusted statistic of Clark and West (2007). Define
Then, the Clark and West (2007) MSFEadjusted statistic is the tstatistic from the regression of \(f_{t}\) on a constant. The results are also presented in Table 4. It is found only the BCARS model with ltr significantly outperforms the historical mean, indicating that variable ltr is informative to return direction forecasting both insample and outofsample.
It has been documented the predictability of stock returns is related to business cycle (see Rapach et al. 2010); thus, we also calculate the outofsample MSFEadjusted statistics over business expansion and business recession. The expansion and recession time periods are dated by the NBER. The results show that the BCARS model with ltr outperforms the historical mean model both in economic recession and expansion. The benchmark BCARS model outperforms only in economic recession, while the BCARS model with ntis dominates only in economic expansion.
Economic value
The results in “Insample estimation” and “Outofsample forecast” sections confirm that the up ratio is predictable both insample and outofsample and that this predictability is significant in a statistical sense. This section investigates if this predictability can be translated into economic value. This is important, as for financial analysts and practitioners, the most important model evaluation criterion is the return on their investment.
In this paper, a switching trading strategy is employed. At the beginning of each month, the investors make an asset allocation decision. They can shift their assets either into stocks or the riskfree Treasury bills and hold either of these alternatives till the next decision date. The switching trading strategy is presented as
where \(r_t\) is the return on stock index, \(r^f_t\) is the riskfree return, \(r^s_t\) is the return on the switching strategy, and
At time \(t1\), if the forecast of the BCARS model \({\hat{ur}}_{t}\) is larger than historical mean forecast \({\bar{ur}}_t\), investors hold the stock index, otherwise they hold the riskfree asset at time t.
Table 5 reports the summary statistics of the returns on different modelbased switching strategies together with the commonly used Sharpe ratio (SR)
where \(\mu _x\) is the mean excess return based on model x and \(\sigma _x\) is the sample standard deviation of the mean excess return. Compared with the market portfolio, all the switching trading strategies deliver higher mean returns and smaller standard deviations. The Sharpe ratio statistics show that our switching trading strategies outperform the market portfolio. The Sharpe ratio of the ltr_based trading strategy is even two times larger than that of the market portfolio.
We also report in Table 5, the summary statistics of returns on different trading strategies over a business cycle. Consistent with the outofsample \(R^2\) statistics, the Sharpe ratio statistics show that the ltr_based trading strategy dominates the market portfolio in both expansion and recession, and the benchmark BCARSbased trading strategy outperforms only in market recession. Interestingly, we find that the ntis_based trading strategy outperforms the market portfolio in both recession and expansion, which is different from the results of the outofsample \(R^2\) statistics.
Figure 4 presents the cumulative return difference between the ltr_based trading strategy and the market portfolio. It can be observed from Fig. 4 that the outperformance of the ltr_based trading strategy mainly occurs when the market is in recession.
Following Campbell and Thompson (2008) and Rapach et al. (2010), we further calculate the utility gain of the switching strategy relative to the simple market portfolio. Over the outofsample period, the market portfolio investors realize an average utility level of
where \({\hat{u}}_{0}\) and \({\hat{\sigma }}^2_{0}\) are the sample mean and variance, respectively, over the outofsample period for the return on the market portfolio. Letter \(\gamma\) is the relative risk aversion parameter. If the same investor uses the switching strategy, then he/she realizes an average utility level of
where \({\hat{u}}_{s}\) and \({\hat{\sigma }}^2_{s}\) are, respectively, the sample mean and variance of the return on the switching strategy over the outofsample period. The utility gain is measured as the difference between (25) and (24). We multiply this difference by 1200 to express it in average annualized percentage return. The utility gain can be interpreted as the portfolio management fee that an investor would be willing to pay to have access to the switching strategy.
The utility gains are reported in the last column in Table 5. We report results for \(\gamma\) = 3; the results are qualitatively similar for other reasonable \(\gamma\) values. In general, the results confirm that modelbased trading strategies can improve investors’ utility, and this utility improvement is more dominant in recession than in expansion. Taking the ltr_based trading strategy as an example, overall investors would like to pay an annual return of 3.84% to access the forecasts relative to the simple market portfolio; however, when the market is in recession they would like to pay as much as an annual return of 21.12%, which is much higher than 2.99% in expansion.
Further evidence
Another interesting question is that if the BCARS model outperforms the other classification models. This section compares the BCARS model with the commonly used probit model. We define the direction of stock return as follows:
The probit model is presented as
where \(\Phi\)(.) is the cumulative distribution function of the standard normal and \(Z_{t}\)=c+\(\alpha\) \(r_{t1}\)+\(\beta\) \(X_{t1}\).
Table 6 reports the insample regression results of the probit model. The last column reports the pseudo \(R^2\). The lagged return is found almost unanimously to be informationless for predicting return direction, which is consistent with the weak efficient market hypothesis that the direction of stock return can be hardly predicted by its historical price information. Interestingly, we find that the most significant variable for predicting the direction of stock return in the probit model is inflation, which significantly differs from the BCARS model. The other two variables that may be useful for return direction forecasting are tbl and svar. These findings indicate from a new perspective that the BCARS model is indeed different from the probit model when predicting the direction of stock return.
The switching trading strategy under the probit model is presented as follows:
where \({\bar{s}}_t\) is the historical mean defined as
At time t1, if the probability of a positive return is predicted to be larger than its historical mean, investors hold the stock, otherwise they hold the riskfree asset at time t.
Table 7 presents the Sharpe ratio and utility gain of the probit model relative to the market portfolio. Three findings emerge in Table 7: (1) Consistent with the insample estimation results in Table 6, infl has the best outofsample trading performance judging by both the Sharpe ratio and utility gain. (2) The utility gain statistics show that the outperformance of the probit trading strategy only occurs in market recession. (3) Compared with the BCARS trading strategy, the probit trading strategy generally underperforms the BCARSbased trading strategies judging by either the Sharpe ratio or utility gain.
Conclusion
In this paper, we propose a BCARS model to predict the direction of stock returns. An empirical study is performed on the monthly US stock returns to evaluate the BCARS model, and the results show that the forecasting power of the BCARS model is significant both statistically and economically.
The BCARS model proposed in this paper provides a new perspective for return direction forecasting. For future studies, we suggest the following directions: first, it could be interesting to scrutinize the power of the BCARS model in forecasting the direction of other asset returns in different countries. Second, more predictive variables can be empirically investigated. In this paper, we only check the macroeconomic and financial variables. A considerable amount of literature has shown that technical indicators and market sentiment are valuable when forecasting the level of stock returns. However, how these variables contribute to return direction forecasting remains unknown. Third, extensions to the BCARS model are also interesting. For example, there is empirical evidence indicating that the directionofchange is businesscycle related; thus, it would be more practical to use a Markov regimeswitching model to describe the conditional mean of the up ratio. A beta model with both timevarying alpha and beta would be also of interest.
Notes
As the monthly high price collected from the website does not consider the last monthly close price, the following adjustment \(h^a_{t}\)=max{\(h_{t}\), \(c_{t1}\)} is used as the true high price in month t.
These macro and financial variables are downloaded from the homepage of Amit Goyal at www.hec.unil.ch/agoyal/.
We also estimate CARS(1,2) and CARS(2,1), and the results show that CARS(1,1) is sufficient to capture the up ratio dynamics. We do not present the estimation results in order to save space.
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Funding
Funding was provided by National Social Science Fund of China (Grant No. 22BJY259), National Natural Science Foundation of China (Grant Nos. 71971004, 72271055), Research on Modeling of Return Rate Based on Mixed Distribution and Its Application in Risk Management (Grant No. 19YB26).
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Xie, H., Sun, Y. & Fan, P. Return direction forecasting: a conditional autoregressive shape model with beta density. Financ Innov 9, 82 (2023). https://doi.org/10.1186/s4085402300489z
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DOI: https://doi.org/10.1186/s4085402300489z
Keywords
 Return direction forecasting
 Price extremes
 CARS
 Beta distribution