An exact form of the local Whittle likelihood is studied with the intent of developing a general purpose estimation procedure for the memory parameter (d) that does not rely on tapering or differencing prefilters. The resulting exact local Whittle estimator is shown to be consistent and to have the same N(0,1/4) limit distribution for all values of d if the optimization covers an interval of width less than 9/2 and the initial value of the process is known.
Asymptotic properties of the local Whittle estimator in the nonstationary case (d > 1/2) are explored. For 1/2 < d < 1, the estimator is shown to be consistent, and its limit distribution and the rate of convergence depend on the value of d. For d = 1, the limit distribution is mixed normal. For d > 1 and when the process has a linear trend, the estimator is shown to be inconsistent and to converge in probability to unity.
JEL Classification: C22
Keywords: Discrete Fourier transform, fractional Brownian motion, fractional integration, long memory, nonstationarity, semiparametric estimation, trend, Whittle likelihood, unit root
Semiparametric estimation of the memory parameter is studied in models of fractional integration in the nonstationary case, and some new representation theory for the discrete Fourier transform of a fractional process is used to assist in the analysis. A limit theory is developed for an estimator of the memory parameter that covers a range of values of d commonly encountered in applied work with economic data. The new estimator is called the modified local Whittle estimator and employs a version of the Whittle likelihood based on frequencies adjacent to the origin and modified to take into account the form of the data generating mechanism in the frequency domain. The modified local Whittle estimator is shown to be consistent for 0 < d < 2 and is asymptotically normally distributed with variance 1/4 for 1/2 < d < 7/4. The approach allows for likelihood-based inference about d in a context that includes nonstationary data, is agnostic about short memory components and permits the construction of valid confidence regions for d that extend into the nonstationary region.
Estimation of the memory parameter in time series with long range dependence is considered. A pooled log periodogram regression estimator is proposed that utilizes a set of mL periodogram ordinates with L approaching infinity rather than m ordinates used in the conventional log periodogram estimator. Consistency and asymptotic normality of the pooled regression estimator are established. The pooled estimator is shown to have smaller variance but larger bias than the conventional log periodogram estimator. Finite sample performance is assessed in simulations, and the methods are illustrated in an empirical application with inflation and stock returns.
Keywords: Discrete Fourier transform, log periodogram regression, long memory parameter, pooling frequency bands, semiparametric estimation