GARCH Models for Daily Stock Returns: Impact of Estimation Frequency on Value-at-Risk and Expected Shortfall Forecasts

Abstract

We analyze the impact of the estimation frequency - updating parameter estimates on a daily, weekly, monthly or quarterly basis - for commonly used GARCH models in a large-scale study, using more than twelve years (2000-2012) of daily returns for constituents of the S&P 500 index. We assess the implication for one-day ahead 95% and 99% Value-at-Risk (VaR) forecasts with the test for correct conditional coverage of Christoffersen (1998) and for Expected Shortfall (ES) forecasts with the block-bootstrap test of ES violations of Jalal and Rockinger (2008). Using the false discovery rate methodology of Storey (2002) to estimate the percentage of stocks for which the model yields correct VaR and ES forecasts, we reach the following conclusions. First, updating the parameter estimates of the GARCH equation on a daily frequency improves only marginally the performance of the model, compared with weekly, monthly or even quarterly updates. The 90% confidence bands overlap, reflecting that the performance is not significantly different. Second, the asymmetric GARCH model with non-parametric kernel density estimate performs well; it yields correct VaR and ES forecasts for an estimated 90% to 95% of the S&P 500 constituents. Third, specifying a Student-t (or Gaussian) innovations' density yields substantially and significantly worse forecasts, especially for ES. In sum, the somewhat more advanced model with infrequently updated parameter estimates yields much better VaR and ES forecasts than simpler models with daily updated parameter estimates.

Keywords: GARCH, Value-at-Risk, Expected Shortfall, equity, frequency, false discovery rate

JEL Classification: C12, C22, C58, G17, G32