Well-known and recent long-memory GARCH models and their semiparametric extensions

This paper considers two tractable special cases of the fractionally integrated asymmetric power ARCH (FIAPARCH) model, called FIGJR-GARCH and FITGARCH, which exhibit improved numerical stability relative to the general FIAPARCH specification. Under a restriction on the leverage parameter, almost sure positivity of the conditional variance process is ensured by the Conrad and Haag (2006) conditions. Building on these parametric specifications, we develop semiparametric extensions. In this framework, we first estimate the time-varying long-run component for unconditional variance by a local linear estimator, and then estimate the time-invariant parameters in GARCH-type short-run component by a quasi maximum likelihood estimator based on descaled returns. Next, we construct pointwise confidence bands for inference on the long-run component. An application to equity returns suggests that part of the persistence attributed to fractional integration in parametric long-memory GARCH models may instead reflect long-run variation in the unconditional variance. The empirical evidence also suggests that individual stocks exhibit more pronounced long-run variation in volatility than aggregate indices.