Edgeworth expansions for semiparametric Whittle estimation of long memory.

The semiparametric local Whittle or Gaussian estimate of the long memory parameter is known to have especially nice limiting distributional properties, being asymptotically normal with a limiting variance that is completely known. However in moderate samples the normal approximation may not be very good, so we consider a refined, Edgeworth, approximation, for both a tapered estimate, and the original untapered one. For the tapered estimate, our higher-order correction involves two terms, one of order m-1/2 (where m is the bandwidth number in the estimation), the other a bias term, which increases in m; depending on the relative magnitude of the terms, one or the other may dominate, or they may balance. For the untapered estimate we obtain an expansion in which, for m increasing fast enough, the correction consists only of a bias term. We discuss applications of our expansions to improved statistical inference and bandwidth choice. We assume Gaussianity, but in other respects our assumptions seem mild.

On asymptotic distributions of weighted sums of periodograms

We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions of quadratic forms involving integrals of weighted periodograms. Conditions for asymptotic normality of these weighted sums are simple, minimal, and resemble Lindeberg-Feller condition for weighted sums of independent and identically distributed random variables. Our results are applicable to a large class of short, long or negative memory processes. The proof is based on sharp bounds derived for Bartlett type approximation of these sums by the corresponding sums of weighted periodograms of independent and identically distributed random variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ456 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotic normality for weighted sums of linear processes

We establish asymptotic normality of weighted sums of linear processes with general triangular array weights and when the innovations in the linear process are martingale differences. The results are obtained under minimal conditions on the weights and innovations. We also obtain weak convergence of weighted partial sum processes. The results are applicable to linear processes that have short or long memory or exhibit seasonal long memory behavior. In particular, they are applicable to GARCH and ARCH(∞) models and to their squares. They are also useful in deriving asymptotic normality of kernel-type estimators of a nonparametric regression function with short or long memory moving average errors

Estimation pitfalls when the noise is not i.i.d.

This paper extends Whittle estimation to linear processes with a general stationary ergodic martingale difference noise. We show that such estimation is valid for standard parametric time series models with smooth bounded spectral densities, e.g., ARMA models. Furthermore, we clarify the impact of the hidden dependence in the noise on such estimation. We show that although the asymptotic normality of the Whittle estimates may still hold, the presence of dependence in the noise impacts the limit variance. Hence, the standard errors and confidence intervals valid under i.i.d. noise may not be applicable and thus require correction. The goal of this paper is to raise awareness to the impact of a non-i.i.d. noise in applied work

Approximations and limit theory for quadratic forms of linear processes

AbstractThe paper develops a limit theory for the quadratic form Qn,X in linear random variables X1,…,Xn which can be used to derive the asymptotic normality of various semiparametric, kernel, window and other estimators converging at a rate which is not necessarily n1/2. The theory covers practically all forms of linear serial dependence including long, short and negative memory, and provides conditions which can be readily verified thus eliminating the need to develop technical arguments for special cases. This is accomplished by establishing a general CLT for Qn,X with normalization (Var[Qn,X])1/2 assuming only 2+δ finite moments. Previous results for forms in dependent variables allowed only normalization with n1/2 and required at least four finite moments. Our technique uses approximations of Qn,X by a form Qn,Z in i.i.d. errors Z1,…,Zn. We develop sharp bounds for these approximations which in some cases are faster by the factor n1/2 compared to the existing results

A time varying DSGE model with Financial frictions

We build a time varying DSGE model with financial frictions in order to evaluate changes in the responses of the macroeconomy to financial friction shocks. Using U.S. data, we find that the transmission of the financial friction shock to economic variables, such as output growth, has not changed in the last 30 years. The volatility of the financial friction shock, however, has changed, so that output responses to a one-standard deviation of the shock increase twofold in the 2007–2011 period in comparison with the 1985–2006 period. The time varying DSGE model with financial frictions improves the accuracy of forecasts of output growth and inflation during the tranquil period of 2000–2006, while delivering similar performance to the fixed coefficient DSGE model for the 2007–2012 period

Reduction principles for quantile and Bahadur-Kiefer processes of long-range dependent linear sequences

In this paper we consider quantile and Bahadur-Kiefer processes for long range dependent linear sequences. These processes, unlike in previous studies, are considered on the whole interval $(0,1)$. As it is well-known, quantile processes can have very erratic behavior on the tails. We overcome this problem by considering these processes with appropriate weight functions. In this way we conclude strong approximations that yield some remarkable phenomena that are not shared with i.i.d. sequences, including weak convergence of the Bahadur-Kiefer processes, a different pointwise behavior of the general and uniform Bahadur-Kiefer processes, and a somewhat "strange" behavior of the general quantile process.Comment: Preprint. The final version will appear in Probability Theory and Related Field