The Estimation of Misspecied Long Memory Models

We consider time series that, possibly after integer di¤erencing or integrating or other detrending, are covariance stationary with spectral density that is regularly varying near zero frequency, and unspeci…ed elsewhere. This semiparametric framework includes series with short, long and negative memory. We establish consistency of the popular log-periodogram memory estimate that, conventionally but wrongly, assumes the spectral density obeys a pure power law. The local-to zero misspeci…cation leads, however, to increased bias, which is liable to prevent the usual central limit theorem from holding. The order of the bias is calculated for several slowly-varying factors, and some discussion of mean squared error and bandwidth choice is included. JEL classi…cations: C14; C22