Edgeworth Expansions for Spectral Density Estimates and Studentized Sample Mean

Abstract

We establish valid Edgeworth expansions for the distribution of smoothed nonparametric spectral estimates, and of studentized versions of linear statistics such as the same mean, where the studentization employs such a nonparametric spectral estimate. Particular attention is paid to the spectral estimate at zero frequency and, correspondingly, the studentized sample mean, to reflect econometric interest in autocorrelation-consistent or long-run variance estimation. Our main focus is on stationary Gaussian series, though we discuss relaxation of the Gaussianity assumption. Only smoothness conditions on the spectral density that are local to the frequency of interest are imposed. We deduce empirical expansions from our Edgeworth expansions designed to improve on the normal approximation in practice, and also a feasible rule of bandwidth choice.

Multivariate semi-nonparametric distributions with dynamic conditional correlations

“…Since then, they have been investigated from many perspectives and disciplines, Sargan (1975) being who brought them into econometrics. More recently, that literature has attracted a renewed interest and so we …nd contributions, in the theoretical …eld -see, e.g., Nishiyama & Robinson (2000) and Velasco & Robinson (2001)-and in the applied …eld, with particular emphasis in …nance, to deal with the unsolved problem of …tting the heavy-tailed distribution of high-frequency asset returns -see, e.g., Corrado & Su (1996), Mauleón & Perote (2000) and Verhoeven & McAleer (2004). The latter articles provide empirical evidence on the good in-sample performance of univariate seminonparametric (SNP hereafter) distributions.…”

Block Bootstrap Hac Robust Tests: The Sophistication of the Naive Bootstrap

“…bootstrap have ERPs that converge to zero at rates faster than the standard normal approximation. In Section 6 we discuss heuristic comparisons between fixed-b asymptotic approximations and the Edgeworth approximations derived by Velasco and Robinson (2001) in an effort to shed some light on the relative performance of Edgeworth approximations and the naive bootstrap/fixed-b asymptotics in the simple location model. Proofs are given in two mathematical appendices.…”

Positive Definiteness of Multivariate Densities Based on Hermite Polynomials

“…Therefore, all the moments of the distribution depend on the correlation coefficients and d si ∀s = 1, 2, ..., q and ∀i = 1, 2, ..., n. For the sake of clarity the bivariate case of this distribution (for the variables x t and y t ) is given in equation (30). Without loss of generality, we expand every marginal Gaussian density to the same order q.…”

“…Since then, these expansions have been used in many fields from mathematics or statistics to physics, but it was Sargan in the seventies who brought these expansions into econometrics - Sargan (1975) and Sargan (1976). More recently the densities based on these expansions have also been investigated - Nishiyama and Robinson (2000), Velasco and Robinson (2001) and Nabeya (2001) -and have been introduced in finance to capture the asymmetric and leptokurtic behaviour of high frequency financial data -see Mauleón (1997), Mauleón and Perote (2000) or Verhoeven and McAller (2004) as recent examples. Even more, these densities have also been used to price options -Jarrow and Rudd (1982), Corrado and Su (1996) and Rubinstein (1998) All these articles reveal densities based on Hermite polynomials as accurate and general specifications that capture skewness and kurtosis of most high frequency data -in particular, Mauleón and Perote (2000) proved that these densities are capable of capturing financial data behavior even better than other non-normal distributions such as the Student's t.…”