On the transmission of memory in GARCH-in-mean models

Abstract

In this article, we show that in times series models with in-mean and level effects, persistence will be transmitted from the conditional variance to the conditional mean and vice versa. Hence, by studying the conditional mean/variance independently, one will obtain a biased estimate of the true degree of persistence. For the specific example of an AR(1)-APARCH(1,1)-in-mean-level process, we derive the autocorrelation function, the impulse response function and the optimal predictor. Under reasonable assumptions, the AR(1)-APARCH(1,1)-in-mean-level process will be observationally equivalent to an autoregressive moving average (ARMA)(2,1) process with the largest autoregressive root being close to one. We illustrate the empirical relevance of our results with applications to S&P 500 return and US inflation data.