This paper investigates the problem of assessing local influence of small perturbations in GARCH processes. First, we examine the local influence on the Lagrange multiplier (LM) statistic. Second, we assess the local influence on the pseudo-likelihood of the GARCH model. We find that short patches of high volatility observations that have a strong influence on the LM statistic may not necessarily be influential on the pseudo-likelihood. This is mainly due to the fact that the effects of high volatility could be incorporated through GARCH modeling. An empirical example is presented to illustrate the effectiveness of the proposed methods. It is interesting to note that observations which have a very strong influence on the LM statistic are far less influential on the GARCH pseudo-likelihood, suggesting that under the GARCH model they should not be regarded as outliers
