پیش بینی چند متغیره تحقق یافته نوسانات بازار سهام

We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of lagged volatility, lagged returns and other forecasting variables. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite estimated covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics

The variances and covariances of stock returns vary over time (e.g. Andersen et al., 2005). As a result, many important financial applications require a model of the conditional covariance matrix. Three distinct categories of methods for estimating a latent conditional covariance matrix have evolved in the literature. In the first category are the various forms of the multivariate GARCH model where forecasts of future volatility depend on past volatility and shocks (e.g. Bauwens et al., 2006). In the second category, authors have modeled asset return variances and covariances as functions of a number of predetermined variables (e.g. Ferson, 1995). The third category includes multivariate stochastic volatility models (e.g. Asai et al., 2006). In this paper, we introduce a new model of the realized covariance matrix. 1 We use high-frequency data to construct estimates of the daily realized variances and covariances of five size-sorted stock portfolios. By using high-frequency data we obtain an estimate of the matrix of ‘quadratic variations and covariations’ that differs from the true conditional covariance matrix by mean zero errors (e.g. Andersen et al. (2003) and Barndorff-Nielsen and Shephard (2004a)). This provides greater power in determining the effects of alternative forecasting variables on equity market volatility when compared to efforts based on latent volatility models. We transform the realized covariance matrix using the matrix logarithm function to yield a series of transformed volatilities which we term the log-space volatilities. The matrix logarithm is a non-linear function of all of the elements of the covariance matrix and thus the log-space volatilities do not correspond one-to-one with their counterparts in the realized covariance matrix. 2 However, modeling the time variation of the log-space volatilities is straightforward and avoids the problems that plague existing estimators of the latent volatility matrix. In particular, we do not have to impose any constraints on our estimates of the log-space volatilities. We then model the dynamics of the log-space volatility matrix using a latent factor model. The factors consist of both past volatilities and other variables that can help forecast future volatility. We thus are able to model the conditional covariance matrix by combining a large number of forecasting variables into a relatively small number of factors. Indeed we show that two factors can capture the volatility dynamics of the size-sorted stock portfolios. The factor model is estimated by GMM yielding a series of filtered estimates. We then transform these fitted values, using the matrix exponential function, back into forecasts of the realized covariance matrix. Our estimated matrix is positive definite by construction and does not require any parameter restrictions to be imposed. The approach can thus be viewed as a multivariate version of standard stochastic volatility models, where the variance is an exponential function of the factors and the associated parameters. In addition to introducing our new realized covariance matrix we also test the forecasting ability of alternative variables for time-varying equity market covariances. Our motivation is that researchers have examined a number of variables for forecasting returns but there is much less evidence that the variables forecast risks. The cross-section of small- and large-firm volatility has been examined in a number of earlier papers (e.g., Kroner and Ng (1998), Chan et al. (1999), and Moskowitz (2003)). However, these papers used models of latent volatility to capture the variation in the covariances. In contrast, we construct daily measures of the realized covariance matrix of small and large firms over the 1988 to 2002 period. Our precise measures of volatility allow a more detailed examination of the drivers of conditional covariances than prior work. Naturally all of these advantages come at a cost. The main cost is that by performing our analysis on the log-space volatilities and then using the (non-linear) matrix exponential function, the estimated volatilities are not unbiased. However, as we show below, a simple bias correction is available that greatly reduces the problem. Another cost is that direct interpretation of the effects of an instrument on expected volatility is difficult due to the non-linear nature of the model. However, using our factor model estimates, we can obtain the derivatives of the realized covariance matrix with respect to the forecasting variables. We are able to calculate the derivatives at each point in our sample, yielding a series of conditional volatility elasticities that are functions of both the level of the volatility and the factors driving the volatility. The time series allows us to determine which variables have a large impact on time-varying expected volatility. The paper is organized as follows. In Section 2, we present our model of matrix logarithmic realized volatility. In Section 3, we outline our method for constructing the realized volatility matrices and give the sources of the data. In Section 4, we give our results. In Section 5, we conclude.

This paper has introduced a new model for the realized covariance matrix of returns. The model is parsimonious, guarantees a ter constraints to be imposed. The model allows a number of variables to forecast the covariance matrix, yet restricts the number of factors in the estimation process. In addition, time-varying elasticities can be calculated that show the extent to which a per cent shock to the forecasting variable influences any particular element of the realized covariance matrix. The model is applied to the covariance matrix of daily realized stock returns over the 1988 to 2002 period. Four alternative sets of forecasting variables are tested. Lagged principal components of realized weekly and monthly bi-power covariation have a strong predictive power. Some variables (e.g., the dividend yield) that forecast stock market returns also forecast the cross section of volatility. The estimated elasticities show that the influence of some of these variables changes over the sample period while others are more stable.