Higher Order Asymptotic Theory for Minimum Contrast Estimators of Spectral Parameters of Stationary Processes

Abstract

Let g~l! be the spectral density of a stationary process and let f u~l !, u ʦ Q, be a fitted spectral model for g~l!+ A minimum contrast estimator Z u n of u is defined that minimizes a distance D~f u , [ g n ! between f u and [ g n , where [ g n is a nonparametric spectral density estimator based on n observations+ It is known that Z u n is asymptotically Gaussian efficient if g~l! ϭ f u~l !+ Because there are infinitely many candidates for the distance function D~f u , [ g n !, this paper discusses higher order asymptotic theory for Z u n in relation to the choice of D+ First, the second-order Edgeworth expansion for Z u n is derived+ Then it is shown that the bias-adjusted version of Z u n is not second-order asymptotically efficient in general+ This is in sharp contrast with regular parametric estimation, where it is known that if an estimator is first-order asymptotically efficient, then it is automatically secondorder asymptotically efficient after a suitable bias adjustment~e+g+, Ghosh, 1994, Higher Order Asymptotics, p+ 57!+ The paper establishes therefore that for semiparametric estimation it does not hold in general that "first-order efficiency implies second-order efficiency+" The paper develops verifiable conditions on D that imply second-order efficiency+ This paper was written while the first author was visiting the University of Bristol as a Benjamin Meaker Professor+ The second author was previously at Bristol and is now supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences+ We are grateful to the co-editor Pentti Saikkonen and two anonymous referees for their valuable comments, which significantly

Second order optimality for estimators in time series regression models

“…This implies that the Hannan estimator has the same rate of convergence as in regular parametric estimation. This is a sharp contrast with the general semiparametric estimation theory, where it is known that the second order asymptotic properties are strongly influenced by the bandwidth (e.g., Taniguchi et al [12]). …”

“…In the context of semiparametric estimation, it is known that root-N asymptotics in general do not hold (e.g., [12]). However, our results claim that, in a linear regression model, standard root-N asymptotics hold up to second order.…”

“…Velasco and Robinson [15] derived Edgeworth expansions for the distribution of nonparametric estimates. Taniguchi et al [12] discussed higher order asymptotic theory for minimum contrast estimators of spectral parameters. They established that for semiparametric estimation it does not hold in general that first order efficiency implies second order efficiency.…”

Moderate deviations for quadratic forms in Gaussian stationary processes

“…Bentkus and Rudzkis [8] and Janas [26] [14]. Taniguchi et al [37] developed the higher-order asymptotic theory of minimum contrast estimators based on f T (·).…”

Edgeworth expansions for a class of spectral density estimators and their applications to interval estimation

“…See Lahiri (2007Lahiri ( , 2010 for some examples. In the frequency domain, Janas (1994) and Taniguchi et al (2003) derived EEs for estimators of weighted integrals of the spectral density and their smooth functions. For the spectral density itself and also for studentized sample mean, Velasco and Robinson (2001) constructed valid EEs for a univariate Gaussian stationary time series.…”