Influence diagnostics in generalized autoregressive conditional heteroscedasticity processes

Influence diagnostics have become important tools for statistical analysis since the seminal work by Cook. In this article we present a curvature-based directional diagnostic, set up based on the slope-based diagnostic to assess the local influence of minor perturbations on influence graph in a regression model. Using both slope- and curvature-based diagnostics, we examine local influence in the generalized autoregressive conditional heteroscedasticity (LARCH) model under two perturbation schemes that involve model perturbation and data perturbation. We present a Monte Carlo study to obtain the approximate benchmark for determining the significance of a directional diagnostic, as well as the threshold for locating influential observations. An empirical study involving LARCH modeling of the continuously compounded daily return of the New York Stock Exchange composite index illustrates the effectiveness of the proposed diagnostics. The empirical study also shows that the curvature-based diagnostic can find a cluster of additive shocks that cannot be discovered by the slope-based diagnostic. Because observations may have different effects on the influence graph under different perturbation schemes, and both the slope-based and the curvature-based diagnostics are useful for assessing local influence (especially in LARCH models), it is advisable to assess local influence under different perturbation schemes through both diagnostics.