Small sample properties of GARCH estimates and persistence

Abstract

It is shown that the ML estimates of the popular GARCH(1,1) model are significantly negatively biased in small samples and that in many cases converged estimates are not possible with Bollerslev's non-negativity conditions. Results also indicate that a high level of persistence in GARCH(1,1) models obtained using a large number of observations has autocorrelations lower than these ML estimates suggest in small samples. Considering the size of biases and convergence errors, it is proposed that at least 250 observations are needed for ARCH(1) models and 500 observations for GARCH(1,1) models. A simple measure of how much GARCH conditional volatility explains squared returns is proposed. The measure indicates that for a typical index return volatility whose ARCH parameter is very small, the conditional volatility hardly explains squared returns.Small sample, volatility, GARCH, persistence,

Small sample properties of copula-GARCH modelling: a Monte Carlo study

“…Instead, if we consider median estimates, biases are much smaller and α is negatively biased, thus highlighting a strong asymmetric distribution for the variance parameters. These results point out the difficulties of estimating GARCH models with small samples, thus confirming previous simulation evidence in Hwang and Valls Pereira (2006) who, however, considered only univariate models with normally distributed errors and did not examine the effect of different joint distributions. Moreover, strong biases were expected, since it has been shown by Newey and Steigerwald (1997) that a QML estimator can be biased when data are not symmetric.…”

“…Interestingly, no qualitative differences are found across different copula dimensions as well as across different correlation levels. In general, these results point out the difficulties of estimating GARCH models with small samples, thus extending previous simulation evidence in Hwang and Valls Pereira (2006) who, however, considered only univariate models with normally distributed errors and did not examine the effect of different joint distributions. As for the dependence parameters, when there is skewness in the data and symmetric marginals are used, the estimated correlations are negatively biased, and the bias increases when moving from the Student's t to the normal (marginal) distribution, reaching values as high as 25% of the true values.…”

Conditional Variances in UK Regional House Prices

“…Selection of the Nationwide index gives a sample size of 137 quarterly observations for each of the 13 UK regions: not only is GARCH more applicable to high frequency data (monthly regional series exist but for fewer observations), but also it is best suited to data sets of at least 250 observations. Hwang and Pereira (2006) suggest that with small samples there is a tendency for negative parameter bias, so any conclusions should be tempered with caution as positive but non-significant parameters may suffer from this reduction and could in fact be significant. Similarly, small negative parameters may have been made significant due to the bias.…”

UK Housing Market: Time Series Processes with Independent and Identically Distributed Residuals

“…Thus, one or more of the rejected GARCH-based processes may explain the data as well as or better than that of the identified EGARCH in mean. Secondly the concerns of Hwang and Pereira (2006) relate to small samples, which may result in the parameters in equation 13 having a negative bias. Their study was on GARCH and not the Exponential version, and not in mean, so it is possibly not applicable to the explaining format identified here.…”

“…If the identified process has GARCH for the conditional variance, then Hwang and Pereira (2006) suggest that 150 data points is too small and could result in bias in the estimated parameters. In addition, combining a GARCH process with use of the BDS test has a second factor to contend with, in that Caporale et al (2005) discuss how small sample sizes may distort the BDS test.…”