Generalized dynamic factor models and volatilities: Consistency, rates, and prediction intervals

Abstract

Volatilities, in high-dimensional panels of economic time series with a dynamic factor structure on the levels or returns, typically also admit a dynamic factor decomposition. We consider a two-stage dynamic factor model method recovering the common and idiosyncratic components of both levels and log-volatilities. Specifically, in a first estimation step, we extract the common and idiosyncratic shocks for the levels, from which a log-volatility proxy is computed. In a second step, we estimate a dynamic factor model, which is equivalent to a multiplicative factor structure for volatilities, for the log-volatility panel. By exploiting this two-stage factor approach, we build one-step-ahead conditional prediction intervals for large n × T panels of returns. Those intervals are based on empirical quantiles, not on conditional variances; they can be either equal-or unequaltailed. We provide uniform consistency and consistency rates results for the proposed estimators as both n and T tend to infinity. We study the finite-sample properties of our estimators by means of Monte Carlo simulations. Finally, we apply our methodology to a panel of asset returns belonging to the S&P100 index in order to compute one-step-ahead conditional prediction intervals for the period 2006-2013. A comparison with the componentwise GARCH benchmark (which does not take advantage of cross-sectional information) demonstrates the superiority of our approach, which is genuinely multivariate (and high-dimensional), nonparametric, and model-free.JEL Classification: C32, C38, C58.

Forecasting Conditional Covariance Matrices in High-Dimensional Time Series: A General Dynamic Factor Approach

“…In this section, we briefly describe the GDFM to be considered throughout, which basically contains as particular cases all other factor models proposed in the econometric and time series literature, along with the regularity assumptions we need for consistency, which are borrowed, essentially, from Barigozzi and Hallin (2018).…”

“…Assumptions (i)-(ix) are the main assumptions in Barigozzi and Hallin (2018); on top of these, they also require two less important and more technical ones (Assumptions (L4) and (L5), respectively), which we do not reproduce here. Under those assumptions, Barigozzi and Hallin (2018) show that a consistent reconstruction, based on X t , X t−1 , . .…”

“…(2017) derive estimators for (1)-( 2) and provide a complete asymptotic analysis for the same. On the other hand, Barigozzi and Hallin (2018) do not require i.i.d.ness and, under assumptions that include (i)-(ix), provide consistency and consistency rates for the Forni et al (2017) estimators. Finally, we assume the following.…”

“…Steps 1-7 are those described in Forni et al (2015Forni et al ( , 2017 and Barigozzi and Hallin (2018), where we refer to for details. The resulting estimator χ t , however, depends on the ordering of the panel obtained at Step 2: that ordering indeed determines which elements of Σ χ N T (θ) are kept in Σ χ N * T (θ) and belong to the diagonal blocks of Σ χ N * T (θ).…”

“…The resulting estimator χ t , however, depends on the ordering of the panel obtained at Step 2: that ordering indeed determines which elements of Σ χ N T (θ) are kept in Σ χ N * T (θ) and belong to the diagonal blocks of Σ χ N * T (θ). Forni et al (2017) and Barigozzi and Hallin (2018) explain how to deal with this by iterating Steps 2-7 (going back to Step 2, choosing a new random permutation, hence a new N * -dimensional subpanel, etc.) until numerical stabilization of the averaged (over the permutations) χ t values; this typically takes place after few iterations 13 .…”