Modeling and forecasting realized covariance matrices with accounting for leverage

ABSTRACT

The existing dynamic models for realized covariance matrices do not account for an asymmetry with respect to price directions. We modify the recently proposed conditional autoregressive Wishart (CAW) model to allow for the leverage effect. In the conditional threshold autoregressive Wishart (CTAW) model and its variations the parameters governing each asset's volatility and covolatility dynamics are subject to switches that depend on signs of previous asset returns or previous market returns. We evaluate the predictive ability of the CTAW model and its restricted and extended specifications from both statistical and economic points of view. We find strong evidence that many CTAW specifications have a better in-sample fit and tend to have a better out-of-sample predictive ability than the original CAW model and its modifications.

KEYWORDS:

• CAW
• leverage
• MIDAS
• realized volatility
• rolatility forecasting

JEL CLASSIFICATION:

• C32
• C58

Acknowledgements

Our thanks go to the Editor and two anonymous referees for numerous useful suggestions that significantly improved the article.

Notes

From time to time, we will use the terms regime, switch, and the like adapted in the literature on threshold autoregressions (Lanne and Saikkonen, 2002).

This is a set of convenient restrictions. In fact, it is sufficient to fix a sign of any element of each of the matrices A_i and B_i, not necessarily the first diagonal one and not necessarily at “positive,” see Engle and Kroner (1995). It is, of course, natural to fix the sign to be positive.

In Section 4, we consider a model with another indicator variable—a market-wide directional indicator.

For comparison, in Golosnoy et al. (2012) the Schwarz criterion preferred specification is the unrestricted CAW(2,2) model having 116 parameters; the leading one-step-ahead predicting model in the subprime crisis is the unrestricted CAW(3,2) model having 141 parameters.

Covariances with the first asset are represented by the elements s_k1, k = 1, 2, 3 and s_1m, m = 1, 2, 3.

Indeed, in our dataset the correlation between the signs of the market return and individual returns varies from 0.46 to 0.55; the correlation between the sign of the market return and the indicator that most of individual returns have the same sign is 0.69.

Each observation is a 5 × 5 realized covariance matrix.

Golosnoy et al. (2012) instead report p-values for the Ljung–Box test; these values, in contrast to ours, are very close to zero.

This proves to be crucial for relatively lower order models because some too parsimonious models like those of orders (0,1) or (1,1) are usually unable to capture all richness of the dynamics thus having their optimal solutions quite far from optimal solutions of higher order models. As a result, the simple bottom-up strategy may lead to suboptimal estimates of the whole category of models. Indeed, estimates that started from diagonal parameter matrices are chosen quite often.

A visual inspection of LM test p-values in “problematic” models indicates that errors in both variance and covariance equations may be equally subject to residual serial correlation, and that such autocorrelation tends to be present in the same equations for different “problematic” models. At the same time, given the autoregressive orders, the CTAW structure by itself does not necessarily correct autocorrelation compared to CAW; it is an increase in orders that helps whiten the errors.

Similarly, one can explain strongly persistent behavior of stationary series using threshold autoregressions (Lanne and Saikkonen, 2002).

Recall that the earliest 240 observations are used only in MIDAS models for estimating long-run components, hence no predictions at the beginning of the sample.

We use Kevin Sheppard's MFE Toolbox for MATLAB; please find more details at http://www. kevinsheppard.com/MFE_Toolbox

We conjecture that relative tightness of Stein MCS is due to the coherency of the Stein loss to the Wishart likelihood.