Persistence in volatility, conditional kurtosis, and the Taylor property in absolute value GARCH processes

Many authors have observed what is known as the Taylor property, namely that the time series dependencies of financial volatility as measured by the autocorrelation function of power-transformed absolute returns are stronger for absolute stock returns than for the squares. In this note, we devise a simple method for detecting the Taylor property at any lag in a class of GARCH(1, 1) models and fully characterize the relevant parameter space for several popular conditional distributions. It turns out that (i) very generally a first-order Taylor property implies the Taylor property at any lag, and (ii) the degree of conditional kurtosis is crucial for the appearance of the effect. This generalizes earlier findings in He and Teräsvirta [He, C., Teräsvirta, T., 1999. Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173–192] and Gonçalves et al. [Gonçalves, E., Leite, J., Mendes-Lopes, N., 2009. A mathematical approach to detect the Taylor property in TARCH processes. Statistics and Probability Letters 79, 602–610] which focus on first-order autocorrelations and/or pure ARCH processes only. An application to the S&P500 index illustrates the results.