关键词: 股市波动预测, 稀疏特征, 稀疏因子, 滚动窗口回归, 规则化回归

Abstract: Financial volatility is one of the most fundamental issues in both academic research and market practice. It provides valuable reference for participants in investing, hedging and arbitraging. With its great importance, understanding its dynamics becomes challenging, especially after the financial crisis in 2008, which changes the public's perception and expectation about financial market substantially, and which still has an influence on today's global economy.
The development of data science in recent years can serve methods of analyzing asset volatility in complex situations. In this paper, we investigate the classic volatility forecasting problem in a data-rich environment, focusing on the roles of sparse components in macro fundamentals on determining future stock market volatility. The analysis can not only show us the relative performance of predictors in different sparse forms, but also provide us with access to learning the dynamics of stock volatility with respect to the macroeconomic environment.
Formally, the “sparse components” in this paper refer to the subsets of predictors extracted from a large set of macro variables. There are two kinds of sparse components under consideration depending on the extracting methods: The “sparse characteristics” are the predictors selected through linear shrinkage techniques, whereas the “sparse factors” are latent factors extracted from the macro variables using principal component analysis. To select the sparse characteristics, regularized regressions with the smoothly clipped absolute deviation (SCAD)penalization are employed. Robustness checks based on the least absolute shrinkage selection operator (LASSO) are also presented. These L1 regularization terms can reserve only the most relevant predictors in predictive regressions while eliminating irrelevant variables from the predicting process. Though the latent factors are linear combinations of all the macro variables, they are able to summarize a sufficient large proportion of variation in these variables and thus appear in the regressions as sparse predictors. These sparse components reflect different types of dimension reduction methods, and the related results can give us access to understanding how macro fundamentals affect stock market volatility.
After previous research articles, the problem in linear predictive regressions is studied. With the sparse components being predictors, realized volatility as the proxy for market volatility is used. Lags of volatility are also included in the regressions as predictors. The sparse components consider not only the first but the second moment of the variables. With a rolling window scheme, the regression and forecasting are estimated and then pushed forward. This dynamic forecasting process allows us to observe the time-varying impact of macro fundamentals on stock market volatility. The relative performance of the two kinds of sparse components according to the mean squared prediction error (MSE) is compared.
The macro variables in this paper can reflect different aspects of China's economic environment. These variables are financial market variables such as the average price earnings ratio of stocks traded on the Shanghai Exchange (PE) and the Fama-French factors, macro-economic variables such as the growth rate of the consumer price index (CPI) and the growth rate of industrial production (IP), the global market variables such as the growth rate of the real effective exchange rate of CNY (FXI) and the growth rate of the commodity prices of crude oil (OIL), and policy uncertainty indexes of China mainland (CMPU), of the United States (USPU) and the global index (GPU). The monthly realized volatility is calculated using daily returns of the Shanghai Securities Composite Index. The data are from the CSMAR database, the RESSET database, the CEInet Statistics database, the database of the People's Bank of China, Yahoo Finance, the database of the International Monetary Fund (IMF) and the economic policy uncertainty index website.
The result shows that the forecasting accuracy of sparse factors is superior across various predicting horizons. There are more predictors in average in the sparse characteristic equations, which has increased the forecasting variance and thus leads to the inferior performance. The accuracy of the forecasts is strongly time-varying and is almost negatively correlated with volatility, indicating that the cause of turmoil of China's stock market is independent of information in the fundamentals to some degree. It appears that the mode of forecast is different for characteristics and factors. While two kinds of predictive regressions both include lags of volatility, its relative importance to the sparse components is different. In the characteristic equations, macro variables especially the price earnings ratio and the housing selling area have shown influential predicting powers. But in the factor equations, auto-regressive terms of volatility turn out to be prominent predictors, with the factors mainly serving as supplements. These outcomes reveal that single macro variables are more closely related to volatility in local time periods, but this relationship is strongly time-varying and not stable. The overall movements of macrofundamentals are more relevant to stock market volatility than single variables in the sense of prediction, since they can help produce better forecasting performance. The conclusion of the study can provide reference for investing strategy making and financial risk management.

Key words: stock market volatility forecast, sparse characteristic, sparse factor, rolling window regression, regularized regression