Arch with Persistent Covariate

The paper considers a volatility model which introduces a persistent, integrated or nearly integrated, covariate to the standard ARCH(1) model. For such a model, we derive asymptotic theory of quasi-maximum likelihood estimator. In particular, we establish consistency and obtain limit distribution. The limit distribution is generally non-Gaussian and represented as a functional of Brownian motions. However, it becomes Gaussian if the covariate is strictly exogenous or the volatility function is linear in parameter. We also analyze the effect of omitting the persistent covariate. Our analysis shows that, if the relevant covariate is omitted and the usual GARCH(1,1) model is fitted, then the model would be estimated approximately as IGARCH. This may well explain the ubiquitous evidence of IGARCH in empirical volatility analysis