A Note on Whittle's Likelihood |
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Authors: | Alberto Contreras-Cristán Eduardo Gutiérrez-Peña Stephen G. Walker |
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Affiliation: | 1. Departamento de Probabilidad y Estadística , Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas , UNAM , Mexico D. F , Mexico alberto@sigma.iimas.unam.mx;3. Departamento de Probabilidad y Estadística , Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas , UNAM , Mexico D. F , Mexico;4. Institute of Mathematics Statistics and Actuarial Science, University of Kent , Canterbury , UK |
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Abstract: | The approximate likelihood function introduced by Whittle has been used to estimate the spectral density and certain parameters of a variety of time series models. In this note we attempt to empirically quantify the loss of efficiency of Whittle's method in nonstandard settings. A recently developed representation of some first-order non-Gaussian stationary autoregressive process allows a direct comparison of the true likelihood function with that of Whittle. The conclusion is that Whittle's likelihood can produce unreliable estimates in the non-Gaussian case, even for moderate sample sizes. Moreover, for small samples, and if the autocorrelation of the process is high, Whittle's approximation is not efficient even in the Gaussian case. While these facts are known to some extent, the present study sheds more light on the degree of efficiency loss incurred by using Whittle's likelihood, in both Gaussian and non-Gaussian cases. |
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Keywords: | ARCH process Autocorrelation function Gamma process Gaussian process Periodogram Spectral density Stationary time series |
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