The memory parameter is usually assumed to be constant in traditional long memory time series. We relax this restriction by considering the memory a time-varying function that depends on a finite number of parameters. A time-varying Local Whittle estimator of these parameters, and hence of the memory function, is proposed. Its consistency and asymptotic normality are shown for locally stationary and locally non-stationary long memory processes, where the spectral behaviour is restricted only at frequencies close to the origin. Its good finite sample performance is shown in a Monte Carlo exercise and in two empirical applications, highlighting its benefits over the fully parametric Whittle estimator proposed by Palma and Olea (2010). Standard inference techniques for the constancy of the memory are also proposed based on this estimator.

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Long memory,Local Whittle estimation,Local stationarity,Time-varying parameter

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