We consider a volatility model, named ARCH-NNH model, that is specifically an ARCH process with a nonlinear function of a persistent, integrated or nearly integrated, explanatory variable.
We first establish the asymptotic theories showing that the time series properties of our model successfully describe stylized facts about volatility in financial time series. Due to persistent covariates, the model generates time series showing the long memory property in volatility and leptokurtosis which are commonly observed in speculative return series.
Next, we derive the asymptotic distribution theory of the maximum likelihood estimator in our model. In particular, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator in the model, which ensure the standard inference procedure is valid in our model. Additionally, we conduct misspecification analysis and provide an explanation of the commonly observed IGARCH behavior of financial time series. Our theory shows that the IGARCH behavior could be spurious and could be the result of ignoring persistent covariates in ARCH type models.
Finally, we present two empirical applications and a forecast evaluation of our model. Both empirical applications show that our model fits the data very well, and the estimation results confirm the findings of other literature. It is shown that the default premium (the yield spread between Baa and Aaa corporate bonds) predicts stock return volatility, and the interest rate differential between two countries accounts for exchange rate return volatility. The forecast evaluation shows that our model generally performs better than other standard volatility models at relatively lower frequencies.