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Stock return forecasting: U.S. S&P500For hundreds of years, investors have been fascinated by the variability of speculative prices. Investment practitioners have developed many a tool to forecast the future path of the prices in the hope that they can make great fortunes with good forecasts. One of the most commonly used forecasting tools is technical analysis. Despite its popularity among technicians, the value of technical analysis is still controversial due to its subjective nature. In contrast to fundamental analysis, which was quick to be adopted by the scholars of modern quantitative finance, academic scrutiny of technical analysis is still in its infancy. It has been argued that the difference between fundamental analysis and technical analysis is not unlike the difference between astronomy and astrology. Among some circles, technical analysis is even known as “voodoo finance”. However, several academic studies suggest that technical analysis may well be an effective means for extracting useful information from market prices. For example, Lo and MacKinlay (1988, 1999) have shown that past prices may be used to forecast future returns to some degree, a fact that all technical analysts take for granted. Studies by Tabell and Tabell (1964), Treynor and Ferguson (1985), Brown and Jennings (1989), Jegadeesh and Titman (1993), Blume et al. (1994), Chan et al. (1996), Lo and MacKinlay (1997), Grundy and Martin (1998), and Rouwenhorst (1998) have also provided indirect support for technical analysis, and more direct support has been given by Pruitt and White (1988), Neftci (1991), Brock et al. (1992), Neely et al. (1997), Neely and Weller (1998), Chang and Osier (1994), Osier and Chang (1995), and Allen and Karjalainen (1999). Lo et al. (2000) found that over the 31year sample period, several technical indicators do provide incremental information and may have some practical value. Recent academic results show that high low price extremes are valuable for forecasting speculative prices (Xie et al., 2012; Xie et al., 2012; Xie and Wang, 2013; Xie et al., 2013; Xie et al., 2014; Xie et al., 2015; Xie and Wang, 2018). Despite the growing evidence supporting the practical value of technical analysis, it is still frequently criticized due to its highly subjective nature and its lack of theoretical underpinnings. In this chapter, we are going to scrutinize the forecasting power of the candlestick using the DVAR model proposed in Chapter 5. The main purpose of this chapter is to see if the statistical properties of the candlestick can be used to improve stock return forecasting. 10.1 Introduction The predictability of stock market returns is of great interest to both academic researchers and investment practitioners, and numerous economic and financial variables have been identified as predictors of stock returns in academic literature. Examples include valuation ratios, such as the dividendprice (Dow, 1920; Fama and French, 1988, 1989), earningsprice (Campbell and Shiller, 1988, 1989), and booktomarket (Kothari and Shanken, 1997; Pontiff and Schall, 1998), as well as nominal interest rates (Fama and Schwert, 1977; Campbell, 1987; Breen et al., 1989; Ang and Bekaert, 2007), the inflation rate (Nelson, 1976; Fama and Schwert, 1977; Campbell and Vuolteenaho, 2004), term and default spreads (Campbell, 1987; Fama and French, 1989), corporate issuing activity (Baker and Wurgler, 2000; Boudoukh et al., 2007), consumption wealth ratio (Lettau and Ludvigson, 2001), stock market volatility (Guo, 2006).^{1} Almost all of these existing studies focus on the insample tests and conclude significant evidence of insample return predictability. Despite the consistent agreement on the insample predictability of stock returns, evidence of outofsample predictability remains controversial. Bossaerts and Hillion (1999), Ang and Bekaert (2007), and Goyal and Welch (2003) casted doubt on the insample evidence documented by the early authors by showing that these variables have negligible outofsample predictive power. Among these studies, Goyal and Welch (2008) take a comprehensive look at the empirical performance of stock returns, and show that a long list of predictors from the literature is unable to deliver consistently superior outofsample forecasts of the U.S. stock returns relative to a simple historical mean forecast. In contrast, recent empirical studies confirm new predietor variables and econometric methods that can improve the outofsample predictability of stock returns. New predictor variables include technical indicators (Neely et al., 2014; Han et al., 2013; Goh et al., 2013; Huang and Zhou, 2013), sentiment index (Baker and Wurgler, 2006; Stambaugh et al., 2012; Huang et al., 2013). New econometric methods include support vector machine method (Huang et al., 2005), economically motivated model restrictions (Campbell and Thompson, 2008; Ferreira and SantaClara, 2011), combination forecast (Rapach et al., 2010), diffusion index (Ludvigson and Ng, 2007; Kelly and Pruitt, 2012; Neely et al., 2014), regime shifts (Guidolin and Timmermann, 2007; Henkel et al., 2011; Dangl and Hailing, 2012), sequential learning (Johannes et al., 2014). Rapach and Zhou (2013) made a more extensive survey of the vast literature on predicators and methodologies concerning stock return predictability. Different from the existing the methods, we use the DVAR model proposed in Chapter 5 to scrutinize the the outofsample predictability of the U.S. stock market and the economic value of candlestick forecasting. Outofsample predictability of the U.S. stock market is performed on the monthly stoek returns of the U.S. S&P500 index over 1995.012015.12. To mitigate the concern of datamining, we use the outofsample Яsquare of Campbell and Thompson (2008) as the statistical performance measure and the certainty equivalent return (CER) gain as the economic performance measure. We find the DVAR model reports significant outofsample predictability for the U.S. stock market. The rest of the chapter is organized as follows. Section 2 describes the econometric methodologies. Section 3 presents the empirical results on the predictability' of the U.S. stock market. Section 4 presents some details of U.S. stock market predictability. A summary' is presented in Section 5.
For univariate time series modeling, one of the most commonly used benchmark models is the ARMAGARCHinMean model (see Eq. (5.3), p.28). This model can simultaneously capture the linear autocorrelation in return series and the risk return tradeoff. In this chapter, only the following ARMAGARCHinMean of order (1, 1) is used as it has been well documented that the order (1, 1) is sufficient to capture the return dynamics.
where у is the coefficient of relative risk aversion, reflecting the riskreturn tradeoff. Another model used in this chapter is the DVAR model proposed in Chapter 6. A DVAR model of order p is given by
where T, = (AR_{r}, А И^{7},)^{7}, X_{t}_i is a vector of exogenous variables. The exogenous variables used in this chapter include the upper shadow, the lower shadow, and other variables. Upper shadow and lower shadow are used because we have shown in Chapter 6 that these two variables are informative for predicting both AR_{t} and AW,. For more about DVAR, readers can refer to Chapter 5. Note that the DVAR model does not directly predict the stock return. To obtain the return forecasts, we proceed in an indirect way. To be specific, the return forecasts are constructed through the following equation where r{ is the return forecast, ДЩ and A Wf are respectively the forecasts of А1Ц and Д W{ reported by the DVAR model. 10.2.2 Outofsample evaluation A potential problem with insample predictability is overfitting. In a comprehensive study, Goyal and Welch (2008) found that many macroeconomic variables, though they deliver significant insample forecasts, perform poorly out of sample. Following Goyal and Welch (2008), we also use the outofsample R^{2}, R^{2}aot statistic which is defined as where r{(m) is the return forecast given by model m, r, is the historical mean forecast given by The historical mean forecast is equivalent to an efficient market model, and thus serves as a natural benchmark. A positive R^{2}M indicates the better performance of model m forecast over the simple historical mean forecast, while a negative K]_{uu }indicates the opposite. The R^{2} statistic measures the reduction in mean squared forecast error (MSFE) for model m relative to the historical mean forecast. To see if the reduction is significant, we test the null hypothesis that Rr_{m} < 0 against the alternative hypothesis that Rr_{m} > 0. Following Rapach et al. (2010), we also test this hypothesis by using the Clark and West (2007) MSFEadjusted statistic. Define
the Clark and West (2007) MSFEadjusted statistic is the rstatistic from the regression of/) on a constant. 10.3 Statistical evidence This section describes the data and presents the insample and outofsample forecasting results. 10.3.1 The data We scrutinized the monthly return predictability of the U.S. stock market using the S&P500 index. The data spans from January 1950 through December 2015 with 792 observations and was downloaded from the website, www.finance.yalioo.com. For each month, the high, low, close and open prices were reported. The riskfree bill rates came from www.bus.emory.edu/AGoyal/Research.html. From these prices, we constructed the lower shadow (LS,) by Eq. (3.5)(3.6), the upper shadow (US,) by Eq. (3.3)(3.4), ДА, and Д W, by Eq. (5.6)(5.7), and die asset returns. Candlestick chart forecasting is often used with market states. We define the market is in uptrend state if the market index is above its 200day moving average and in downtrend state otherwise. The 200day moving average has been widely used by practitioners and is available in investment letters, trading software, and newspapers, which can thus mitigate the concerns of data mining and data snooping. The 200day moving average has also been used in Huang et al. (2013) to define market states. To be specific, the market state variable, ms, is constructed from the daily stock price, given by where P, is the daily price level of the market index.^{2} Table 10.1 reports the summary statistics of stock returns together with some other variables. We find no significant linear autocorrelation in the return series, which is consistent with our intuition that the U.S. stock market is quite efficient. For the other time series variables, the LjungBox Q statistics report significant linear autocorrelations. The ADF statistics show that all these time series are stationary. 10.3.2 Insample estimation Table 10.2 reports the estimates of the ARMAGARCHinMean model. Panel A reports the estimates of the mean equation in the ARMAGARCHinMean model and Panel В presents the estimates of the GARCH equation. The results indicate Table 10.1 Summary statistics of stock return and other variables
We use ***, **, and * to mean significance at the level of 1%, 5%, 10%. Table 10.2 Estimates of the ARMAGARCHinMean model
Table 10.3 Estimates of the DVAR model: S&P500
there is a significant volatility clustering effect in stock return. Also we can see from the estimates of the mean equation that there is a positive but insignificant riskreturn tradeoff. The low Лsquare statistics (A^{2} = 1.5%) demonstrate that the forecasts reported by the ARMAGARCHinMean model contain almost no information about the true return observations. We also present in Table 10.2 the estimates of the ARMAGARCHinMean model with the state variable, ms_{r}__{l}. The coefficient on state variable is reported to be 0.005 and statistically significant at the level of 10%. The Лsquare reported by ARMAX GARCHinMean model sharply increases to 2.4%. This result means that the state variable is very informative for forecasting stock return. Table 10.3 presents the estimates of the DVAR model. The lag k= 1 in the DVAR model is chosen by the SIC criteria. Consistent with the theoretical results given in Chapter 6, we find both upper shadow and lower shadow are informative for predicting All, and Д If,. Also we find the state variable is very important for forecasting ДR, and Д W_{t}. The high Rsquare statistics indicate the high predictability' of AR_{t} and AW_{r}. We also calculate the insample Rsquare, Rr_{in} using where r is the mean of r, over the whole sample, and is the forecast given by DVAR model (see Eq. (10.3)). The insample Rsquare reported by DVAR model is 2.98%. 10.3.3 Outofsample forecast For outofsample forecasting, the total T observations are divided into two portions. The first portion from 1 to M observations is used to estimate the coefficients, and the remaining portion from M + 1 to T is used for forecasting evaluation. A static forecasting procedure is used. To be specific, we first use the M observations to obtain the estimates of the parameters, and then make outofsample forecasts with these estimates being fixed. In other words, the estimates of the parameters in the DVAR model are not updated with new information
where C, A_{t} and T, are estimates of the parameters using the only the first M observations. We use the data over 1995.012015.12 as the outofsample time period. We compute the cumulative squared forecast error of each competing model
where S_{4} is the cumulative squared forecast error, r, and r{ (m) are respectively the return observation and the return forecast given by model m. Figure 10.1 presents the time series plots of the cumulative squared forecast error. We use CUM_M to mean the cumulative squared forecast error reported by model M. It is clear that the DVAR model reports the lowest cumulative squared forecast error and then comes the ARMAGARCHinMean model, and the historical mean (HM) model has the largest cumulative squared forecast error. Following Goyal and Welch (2008), we also compute the difference between the cumulative squared forecast error for the historical mean model and the Figure 10.1 Time series of cumulative squared forecast error: 1995.012015.12 cumulative squared forecast error for the competing model, where I)S_{4} is a series of difference. Figure 10.2 presents the time series plots of the differences. We use HM_M to mean the difference between the cumulative squared forecast error for the historical mean model and the cumulative squared forecast error for the competing model M. This is an informative graphical device that provides a visual impression of the consistency of a competing model’s outofsample forecasting performance relative to the historical mean model over time. When the curve increases, the competing model outperforms the historical mean model, while the opposite holds when the curve decreases. The plots conveniently illustrate whether a competing model has a lower mean squared forecast error (MSFE) than the historical mean model for any particular outofsample period by redrawing the horizontal zero line to the start of the outofsample period. A competing model that always dominates the historical mean model for any outofsample period will have a curve with a slope that is always positive; the closer a competing mode is to this ideal, the greater its ability to consistently beat the historical mean model in terms of MSFE. Figure 10.2 Cumulative squared forecast error for the historical mean benchmark forecasting model minus the cumulative squared forecast error for the competing model: 1995.012015.12 Several findings emerge from Figure 10.2. First, both DVAR and ARMA GARCHinMean models outperform the historical mean benchmark model as the curves have ending points higher than starting points. Second, DVAR outperforms ARMAGARCHinMean since the ending point of the DVAR curve is higher than the ending point of the ARMAGARCHinMean curve. Third, the DVAR model is more robust than the ARMAGARCHinMean model for outofsample forecasting as the DVAR curve is less volatile than the ARMAGARCHinMean curve. The outofsample Rsquares reported by the DVAR model and the ARMA GARCHinMean are respectively 4.82% and 1.70%, indicating outperformance of DVAR and ARMAGARCHinMean over the historical mean. To see if the outperformance is statistically significant, we calculate the MSFEadjusted rstatistic by Eq. (10.6). The MSFEadjusted rstatistic for the DVAR is 2.633, which is significant at the level of 1%. The MSFEadjusted rstatistic for the ARMAGARCH inMean model is 1.612, which is not significant at the level of 10%. 10.4 Economic evidence A limitation to the K^{2}ml measure is that it does not explicitly account for the risk borne by an investor over the outofsample period. To address this, following Campbell and Thompson (2008), we also calculate realized utility gains for a meanvariance investor on a realtime basis. More specifically, we first compute the average utility for a meanvariance investor with relative risk aversion parameter у who allocates his or her portfolio monthly between stocks and riskfree bills using forecasts of the equity premium based on the historical sample mean. This exercise requires the investor to forecast the variance of stock returns, and similar to Campbell and Thompson (2008), we assume that the investor estimates the variance using a tenyear rolling window. A meanvariance investor who forecasts the equity premium using the historical average will decide at the end of period t to alloeate the following share of his or her portfolio to equities in period r+1: where tf_{t+} j and o^{2}l+l are respectively riskfree rate and the rollingwindow estimate of the variance of stoek returns.^{3} Over the outofsample period, the investor realizes an average utility level of where fi_{()} and a~_{a} are the sample mean and variance, respectively, over the outof sample period for the return on the benchmark portfolio formed using forecasts of the equity premium based on the historical sample mean. We then compute the average utility for the same investor when he or she forecasts the equity premium using the DVARor ARMAGARCHinMean model. He or she will choose an equity share of
and realizes an average utility level of
where (i_{p} and a_{f} are the sample mean and variance, respectively, over the outof sample period for the return on the portfolio formed using forecasts of the equity premium based on DVAR or ARMAGARCHinMean model. We measure the utility gain as the difference between Eq. (10.15) and Eq. (10.13), and we multiply this difference by 1200 for monthly observations to express it in an average annualized percentage return. The utility gain (or certainty equivalent return, CER) can be interpreted as the portfolio management fee that an investor would be willing to pay to have access to the additional information available in a predictive regression model relative to the information in the historical sample mean. We report results for у = 3; the results are qualitatively similar for other reasonable у = values. Table 10.4 presents the realized utilities of the historical mean model, DVAR model and ARMAGARCHinMean model and the CER gains relative to the historical mean (HM). The realized utility of DVAR model is 0.618 which is almost twice as much as that (0.313) of the ARMAGARCHinMean model. Of course, both earn a higher realized utility' than the HM model has. The annual CER gain of the DVAR (ARMAGARCHinMean) model in 5.098 (1.443). We also calculate the sharp ratio of each trading strategy, and the results are presented in Table 10.4. We find clear dominance of the DVAR model over the other models. For comparison, we also calculate the realized utility' of CER gain of the simple buyandhold (BH) trading strategy. We find a trading strategy based on the DVAR model even outperforms the BH method. Figure 10.3 presents the dynamic weights allocated on equities by different models. Figure 10.4 presents the outofsample cumulative returns given by Table 10.4 Realized utilities and CER gains
Figure 10.3 Dynamic weights allocated on equities over time: 1995.012015.12 Figure 10.4 Dynamic cumulative portfolio returns formed by different trading strategies over time: 1995.012015.12 different trading strategies. Figure 10.4 shows clear dominance of the trading strategy based on the DVAR forecast over the other trading strategies. 10.5 More details Rapach et al. (2010) found the predictability of the U.S. stock market depends highly on business cycle: stock returns are more predictable in recession than in expansion. To see how the predicting power of the DVAR model is related to the business cycle. We also divide the outofsample forecasts by the NBERdated business cycle phases.^{4} For economic expansion there are 226 months, and for economic recession there are 26 months. Figure 10.5 presents the cumulative squared forecast error of the DVAR model over expansion in left panel and over recession in right panel. It is clear that the DVAR model has lower forecast error than the historical mean model in both economic expansion and recession. Figure 10.6 presents the difference between the cumulative squared forecast error for the historical mean benchmark forecasting model and the cumulative squared forecast error for the DVAR model over expansion in the left panel and over recession in the right panel. It seems that the outperformance of the DVAR model over the historical mean Figure 10.5 Time series of cumulative squared forecast error over business cycle Figure 10.6 Cumulative squared forecast error for the historical mean benchmark forecasting model minus the cumulative squared forecast error for the DVAR model over business cycle model is quite robust, regardless of the economic cycle since the curve increases in a quite steady way. We also calculate the MSFEadjusted rstatistics by Eq. (10.6) over business cycle. The MSFEadjusted ^statistic over expansion is 2.293, which is significant Figure 10.7 Dynamic cumulative portfolio return formed by different trading strategies over business cycle at the level of 5%. The MSFEadjusted rstatistic over recession is 1.781, which is significant at the level of 10%. This result confirms that the forecasts given by the DVAR model significantly dominate those reported by the historical mean model over both economic expansion and recession. To see if there is any difference in economic value of the DVAR model relative to the historical mean model over the business cycle we compute the CER gains. The CER gain over expansion is 2.654 and 25.638 over recession. This result indicates that the forecasts reported by the DVAR model are more valuable in economic recession than in economic expansion. Figure 10.7 presents the out ofsample cumulative returns over expansion in the left panel and over recession in the right panel. 10.6 Summary This chapter scrutinizes the performance of the candlestick in return forecasting using the DVAR model. The empirical study is performed on the monthly S&P500 index. The results show that the DVAR model outperforms both the historical mean model and the ARMAGARCHinMean model. Moreover, we find the outperformance is not only statistically significant but also economically valuable. Further evidence indicates that the dominance of the DVAR model is robust to the business cycle. The results obtained in this chapter provide statistical evidence that the candlestick chart is valuable for predicting stock returns. We believe that the evidence presented in this chapter confirms, more or less, that candlestick chart forecasting is not a “voodoo” finance. Notes

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