Abstract
We establish an Edgeworth expansion for the distribution of the Whittle maximum likelihood estimator of the parameter of a time series generated by a linear regression model with Gaussian, stationary, and long-memory residuals. This is done by imposing an extra condition on coefficients of the regression model in addition to the standard conditions imposed on the the spectral density function and the parameter values and making use of the results of Andrews et al. (2005), who provided an Edgeworth expansion for the residual component.
Highlights
We consider a linear regression model {Yt = Xtβ + t, t ≥ 1} where β is a p vector of deterministic but unknown real numbers, {Xt ∈ Rp, t ≥ 1} are non-stochastic regressors, and the error terms { t, t ≥ 1} are stationary, Gaussian, and strongly dependent discrete time series
We establish an Edgeworth expansion for the distribution of the Whittle maximum likelihood estimator of the parameter of a time series generated by a linear regression model with Gaussian, stationary, and long-memory residuals
This is done by imposing an extra condition on coefficients of the regression model in addition to the standard conditions imposed on the the spectral density function and the parameter values and making use of the results of Andrews et al (2005), who provided an Edgeworth expansion for the residual component
Summary
Let X denote the design matrix given by X = (xi j) for i = 1, ..., n and j = 1, ..., p of our regression model. The n × n (Toeplitz) covariance matrix of fθ(λ) is denoted by Tn( fθ) and has ( j, k) element defined by: The log-likelihood function is π. Fθ) as n (Beran, 1994), Ln(θ, μ) can be approximated by the Whittle log-likelihood function, Wn(θ, μ), as. We refer to Wn(θ, μ), where μis replaced for μ in (1.3) above, as the plug-in Whittle log-likelihood (PWLL) function. Likelihood estimators (WMLE), θn, of the true parameter θ solves the equation:. We shall make use of the results of Andrews et al (2005) and acquire an Edgeworth expansion of the PWMLE of our linear regression processes by imposing an extra condition on the regression coefficients.
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