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Forecasting the equity premium using weighted regressions: Does the jump variation help?

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Abstract

Growing literature documents that jump variations are important for comprehending the evolution of asset prices. In this paper, we provide a novel insight on the jump components. Specifically, we forecast the equity premium using the weighted least squares (WLS) approach that assigns the inverse of variance weight to observations, and detect the role of jump contributions in it. The results indicate that the WLS models with jump-robust variance weights generate superior out-of-sample performance both statistically and economically relative to that with the jump-involved weights, suggesting that eliminating the jump variation in the variance weight helps to predict the stock returns. The predictive source of the jump-robust variance stems from its efficient measure of the continuous price process and forecast error variance reduced. Furthermore, we demonstrate that the jump component in the variance weight should rather be dumped than collected in terms of minimizing the forecast losses.

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Notes

  1. According to Barndorff-Nielsen and Shephard (2004a, b) and Andersen et al. (2012), \(d_{m,q} = u_{q/m}^{ - m}\), where \(u_{p} = 2^{\frac{p}{2}} \frac{{\Gamma \left( {{{\left( {p + 1} \right)} \mathord{\left/ {\vphantom {{\left( {p + 1} \right)} 2}} \right. \kern-0pt} 2}} \right)}}{{\Gamma \left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}}\). We also can obtain BPV = MPV(2;2).

  2. Additionally, using daily data enables us to forecast the excess return over a long sample because the intraday data availability is limited.

  3. The alternate combination methods are also employed for examining the robustness.

  4. The homepage of Amit Goyal is at https://sites.google.com/view/agoyal145/.

  5. For briefness, we only report the forecasting results of the mean combination for each model because the combination approach is a more reliable and efficient strategy to be applied in practice.

  6. The construction of the quarterly macroeconomic variables is similar to that of the monthly ones, the data of which is also available at the homepage of Amit Goyal.

  7. We thank the anonymous referee for this valuable suggestion.

  8. For more details about the threshold functions in TMPV, please see Sect. 2.2 and Appendix B in Corsi et al. (2010).

  9. When comparing different WLS-JRV models with the WLS-RV, we use the corresponding JRV estimator to detect the jump diffusion. More specifically, we calculate jump component using BPV, TPV, and QPV when evaluating the performance of WLS-BPV, WLS-TPV, and WLS-QPV, respectively.

  10. The results are similar when using the R2OS values and CW tests to investigate the conditional forecasting performance.

  11. For the jump detection based on BPV, TPV, and QPV, the discrete jumps occur 376, 446, and 456 times, respectively, in the out-of-sample periods within 900 observations in total.

  12. In this subsection, we report the results for the jump detection based on BPV, and those based on other jump-robust variances like TPV and QPV are robust.

  13. We follow Yu et al. (2021) to set the validation sample as long as 12 months. The results are robust when changing it to 24 months. For the sake of brevity, we do not again report these similar results for alternate specifications.

  14. The results for using different window lengths to produce volatility forecasts are quantitatively similar, therefore, these results are not reported to save space.

  15. The results for individual predictors under alternative conditions are not reported, but they are available upon request.

References

  • Andersen T (2001) The distribution of realized stock return volatility. J Financ Econ 61:43–76

    Article  Google Scholar 

  • Andersen TG, Bollerslev T, Diebold FX, Labys P (2003) Modeling and forecasting realized volatility. Econometrica 71:579–625

    Article  Google Scholar 

  • Andersen TG, Bollerslev T, Diebold FX (2007) Roughing it up: including jump components in the measurement, modeling, and forecasting of return volatility. Rev Econ Stat 89:701–720

    Article  Google Scholar 

  • Andersen TG, Dobrev D, Schaumburg E (2012) Jump-robust volatility estimation using nearest neighbor truncation. J Econom 169:75–93

    Article  Google Scholar 

  • Ang A, Bekaert G (2007) Stock return predictability: Is it there? Rev Financ Stud 20:651–707

    Article  Google Scholar 

  • Avramov D (2002) Stock return predictability and model uncertainty. J Financ Econ 64:423–458

    Article  Google Scholar 

  • Bandi FM, Russel JR (2008) Microstructure noise, realized variance, and optimal sampling. Rev Econ Stud 75:339–369

    Article  Google Scholar 

  • Barndorff-Nielsen OE, Shephard N (2002) Estimating quadratic variation using realized variance. J Appl Econom 17:457–477

    Article  Google Scholar 

  • Barndorff-Nielsen OE, Shephard N (2004a) Power and bipower variation with stochastic volatility and jumps. J Financ Econom 2:1–37

    Google Scholar 

  • Barndorff-Nielsen OE, Shephard N (2004b) Econometric analysis of realized covariation: high frequency based covariance, regression, and correlation in financial economics. Econometrica 72:885–925

    Article  Google Scholar 

  • Barndorff-Nielsen OE, Shephard N, Winkel M (2006) Limit theorems for multipower variation in the presence of jumps. Stoch Process Their Appl 116:796–806

    Article  Google Scholar 

  • Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31:307–327

    Article  Google Scholar 

  • Boudoukh J, Israel R, Richardson M (2021) Biases in long-horizon predictive regressions. J Financ Econ. https://doi.org/10.1016/j.jfineco.2021.09.013

    Article  Google Scholar 

  • Buncic D, Gisler KI (2017) The role of jumps and leverage in forecasting volatility in international equity markets. J Int Money Financ 79:1–19

    Article  Google Scholar 

  • Campbell JY, Shiller RJ (1988) Stock prices, earnings, and expected dividends. J Financ 43:661–676

    Article  Google Scholar 

  • Campbell JY, Thompson SB (2008) Predicting excess stock returns out of sample: Can anything beat the historical average? Rev Financ Stud 21:1509–1531

    Article  Google Scholar 

  • Chan F, Pauwels LL (2018) Some theoretical results on forecast combinations. Int J Forecast 34:64–74

    Article  Google Scholar 

  • Claeskens G, Magnus JR, Vasnev AL, Wang W (2016) The forecast combination puzzle: a simple theoretical explanation. Int J Forecast 32:754–762

    Article  Google Scholar 

  • Clark TE, West KD (2007) Approximately normal tests for equal predictive accuracy in nested models. J Econom 138:291–311

    Article  Google Scholar 

  • Clements A, Liao Y (2017) Forecasting the variance of stock index returns using jumps and cojumps. Int J Forecast 33:729–742

    Article  Google Scholar 

  • Corsi F (2009) A simple approximate long-memory model of realized volatility. J Financ Econom 7:174–196

    Google Scholar 

  • Corsi F, Pirino D, Reno R (2010) Threshold bipower variation and the impact of jumps on volatility forecasting. J Econom 159:276–288

    Article  Google Scholar 

  • Dai Z, Zhu H (2020) Stock return predictability from a mixed model perspective. Pac Basin Financ J 60:101267

    Article  Google Scholar 

  • Dai Z, Kang J, Wen F (2021) Predicting stock returns: a risk measurement perspective. Int Rev Financ Anal 74:101676

    Article  Google Scholar 

  • Diebold FX, Mariano RS (1995) Comparing predictive accuracy. J Bus Econ Stat 13:253–263

    Article  Google Scholar 

  • Duong D, Swanson NR (2015) Empirical evidence on the importance of aggregation, asymmetry, and jumps for volatility prediction. J Econom 187:606–621

    Article  Google Scholar 

  • Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econom J Econom Soc 50:987–1007

    Google Scholar 

  • Engle RF, Ghysels E, Sohn B (2013) Stock market volatility and macroeconomic fundamentals. Rev Econ Stat 95:776–797

    Article  Google Scholar 

  • Fama EF, French KR (1988) Dividend yields and expected stock returns. J Financ Econ 22:3–25

    Article  Google Scholar 

  • Fleming J, Kirby C, Ostdiek B (2003) The economic value of volatility timing using “realized” volatility. J Financ Econ 67:473–509

    Article  Google Scholar 

  • Gong X, Lin B (2018) Structural breaks and volatility forecasting in the copper futures market. J Futures Mark 38:290–339

    Article  Google Scholar 

  • Gong X, Zhang W, Xu W, Li Z (2022) Uncertainty index and stock volatility prediction: evidence from international markets. Financ Innov 8:57

    Article  Google Scholar 

  • Hansen PR, Lunde A, Nason JM (2011) The model confidence set. Econometrica 79:453–497

    Article  Google Scholar 

  • Hansen PR, Huang Z, Shek HH (2012) Realized GARCH: a joint model for returns and realized measures of volatility. J Appl Econom 27:877–906

    Article  Google Scholar 

  • Hao X, Zhao Y, Wang Y (2020) Forecasting the real prices of crude oil using robust regression models with regularization constraints. Energy Econ 86:104683

    Article  Google Scholar 

  • Harvey DS, Leybourne SJ, Newbold P (1998) Tests for forecast encompassing. J Bus Econ Stat 16:254–259

    Article  Google Scholar 

  • He M, Hao X, Zhang Y, Meng F (2021a) Forecasting stock return volatility using a robust regression model. J Forecast 40:1463–1478

    Article  Google Scholar 

  • He M, Zhang Y, Wen D, Wang Y (2021b) Forecasting crude oil prices: a scaled PCA approach. Energy Econ 97:105189

    Article  Google Scholar 

  • Huang X, Tauchen G (2005) The relative contribution of jumps to total price variance. J Financ Econom 3:456–499

    Google Scholar 

  • Huang D, Jiang F, Tu J, Zhou G (2015) Investor sentiment aligned: a powerful predictor of stock returns. Rev Financ Stud 28:791–837

    Article  Google Scholar 

  • Jeon B, Seo SW, Kim JS (2020) Uncertainty and the volatility forecasting power of option-implied volatility. J Futures Mark 40:1109–1126

    Article  Google Scholar 

  • Jiang F, Lee J, Martin X, Zhou G (2019) Manager sentiment and stock returns. J Financ Econ 132:126–149

    Article  Google Scholar 

  • Johnson TL (2019) A fresh look at return predictability using a more efficient estimator. Rev Asset Pricing Stud 9:1–46

    Article  Google Scholar 

  • Johnson JA, Medeiros MC, Paye BS (2022) Jumps in stock prices: new insights from old data. J Financ Mark. https://doi.org/10.1016/j.finmar.2022.100708,100708

    Article  Google Scholar 

  • Lang Q, Wang J, Ma F, Huang D, Mohamed Ismail MW (2021) Is Baidu index really powerful to predict the Chinese stock market volatility? New evidence from the internet information. China Finance Rev Int ahead-of-print

  • Lee SS, Mykland PA (2008) Jumps in financial markets: a new nonparametric test and jump dynamics. Rev Financ Stud 21:2535–2563

    Article  Google Scholar 

  • Liang C, Li Y, Ma F, Wei Y (2021) Global equity market volatilities forecasting: a comparison of leverage effects, jumps, and overnight information. Int Rev Financ Anal 75:101750

    Article  Google Scholar 

  • Lin H, Wu C, Zhou G (2018) Forecasting corporate bond returns with a large set of predictors: an iterated combination approach. Manag Sci 64:4218–4238

    Article  Google Scholar 

  • Liu L, Ma F, Wang Y (2015) Forecasting excess stock returns with crude oil market data. Energy Econ 48:316–324

    Article  Google Scholar 

  • Ma F, Liao Y, Zhang Y, Cao Y (2019a) Harnessing jump component for crude oil volatility forecasting in the presence of extreme shocks. J Empir Finance 52:40–55

    Article  Google Scholar 

  • Ma F, Wahab MIM, Zhang Y (2019b) Forecasting the U.S. stock volatility: an aligned jump index from G7 stock markets. Pac Basin Financ J 54:132–146

    Article  Google Scholar 

  • Neely CJ, Rapach DE, Tu J, Zhou G (2014) Forecasting the equity risk premium: the role of technical indicators. Manag Sci 60:1772–1791

    Article  Google Scholar 

  • Newey WK, West KD (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55:703–708

    Article  Google Scholar 

  • Pan Z, Pettenuzzo D, Wang Y (2020) Forecasting stock returns: a predictor-constrained approach. J Empir Finance 55:200–217

    Article  Google Scholar 

  • Paye BS (2012) ‘Déjà vol’: predictive regressions for aggregate stock market volatility using macroeconomic variables. J Financ Econ 106:527–546

    Article  Google Scholar 

  • Rapach DE, Zhou G (2020) Time-series and cross-sectional stock return forecasting: new machine learning methods. In: Jurczenko E (ed) Machine learning for asset management. Wiley, Hoboken, pp 1–33

    Google Scholar 

  • Rapach D, Strauss JK, Zhou G (2010) Out-of-sample equity premium prediction: combination forecasts and links to the real economy. Rev Financ Stud 23:821–862

    Article  Google Scholar 

  • Rapach D, Ringgenberg MC, Zhou G (2016) Short interest and aggregate stock returns. J Financ Econ 121:46–65

    Article  Google Scholar 

  • Shen D, Zhang Y, Xiong X, Zhang W (2017) Baidu index and predictability of Chinese stock returns. Financ Innov 3:4

    Article  Google Scholar 

  • Smith SC (2021) International stock return predictability. Int Rev Financ Anal 78:101963

    Article  Google Scholar 

  • Smith J, Wallis KF (2009) A simple explanation of the forecast combination puzzle. Oxf Bull Econ Stat 71:331–355

    Article  Google Scholar 

  • Stock JH, Watson MW (2004) Combination forecasts of output growth in a seven-country data set. J Forecast 23:405–430

    Article  Google Scholar 

  • Wang Y, Pan Z, Wu C (2017) Time-varying parameter realized volatility models. J Forecast 36:566–580

    Article  Google Scholar 

  • Wang Y, Pan Z, Liu L, Wu C (2019) Oil price increases and the predictability of equity premium. J Bank Finance 102:43–58

    Article  Google Scholar 

  • Wang Y, Pan Z, Wu C, Wu W (2020) Industry equi-correlation: a powerful predictor of stock returns. J Empir Finance 59:1–24

    Article  Google Scholar 

  • Wang Y, Hao X, Wu C (2021) Forecasting stock returns: a time-dependent weighted least squares approach. J Financ Mark 53:100568

    Article  Google Scholar 

  • Welch I, Goyal A (2008) A comprehensive look at the empirical performance of equity premium prediction. Rev Financ Stud 21:1455–1508

    Article  Google Scholar 

  • Xiu D, Kelly B, Gu S, Karolyi A (2020) Empirical asset pricing via machine learning. Rev Financ Stud 33:2223–2273

    Article  Google Scholar 

  • Yi Y, Ma F, Zhang Y, Huang D (2019) Forecasting stock returns with cycle-decomposed predictors. Int Rev Financ Anal 64:250–261

    Article  Google Scholar 

  • Yu H, Hao X, Wang Y (2021) Good volatility, bad volatility, and time series return predictability. Eur J Finance 28:1–25

    Google Scholar 

  • Zhang Y, Wang Y (2022) Forecasting crude oil futures market returns: a principal component analysis combination approach. Int J Forecast. https://doi.org/10.1016/j.ijforecast.2022.01.010

    Article  Google Scholar 

  • Zhang Y, Ma F, Wang Y (2019a) Forecasting crude oil prices with a large set of predictors: Can LASSO select powerful predictors? J Empir Financ 54:97–117

    Article  Google Scholar 

  • Zhang Y, Wei Y, Ma F, Yi Y (2019b) Economic constraints and stock return predictability: a new approach. Int Rev Financ Anal 63:1–9

    Article  Google Scholar 

  • Zhang Y, Ma F, Liang C, Zhang Y (2020) Good variance, bad variance, and stock return predictability. Int J Financ Econ 26:4410–4423

    Article  Google Scholar 

  • Zhang W, Yan K, Shen D (2021a) Can the Baidu Index predict realized volatility in the Chinese stock market? Financ Innov 7:7

    Article  Google Scholar 

  • Zhang Z, He M, Zhang Y, Wang Y (2021b) Realized skewness and the short-term predictability for aggregate stock market volatility. Econ Model 103:105614

    Article  Google Scholar 

  • Zhang Z, He M, Zhang Y, Wang Y (2022) Geopolitical risk trends and crude oil price predictability. Energy 258:124824

    Article  Google Scholar 

  • Zhu X, Zhu J (2013) Predicting stock returns: a regime-switching combination approach and economic links. J Bank Finance 37:4120–4133

    Article  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China [72001110; 72371131; 72071114].

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Authors and Affiliations

  1. College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Zhikai Zhang

  2. School of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210094, China

    Yaojie Zhang & Yudong Wang

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  1. Zhikai Zhang
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Zhang, Z., Zhang, Y. & Wang, Y. Forecasting the equity premium using weighted regressions: Does the jump variation help?. Empir Econ 66, 2049–2082 (2024). https://doi.org/10.1007/s00181-023-02521-8

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  • DOI: https://doi.org/10.1007/s00181-023-02521-8

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