Asymptotic inference in some heteroscedastic regression models with long memory design and errors

This paper discusses asymptotic distributions of various estimators of the underlying parameters in some regression models with long memory (LM) Gaussian design and nonparametric heteroscedastic LM moving average errors. In the simple linear regression model, the first-order asymptotic distribution of the least square estimator of the slope parameter is observed to be degenerate. However, in the second order, this estimator is $n^{1/2}$-consistent and asymptotically normal for $h+H<3/2$; nonnormal otherwise, where $h$ and $H$ are LM parameters of design and error processes, respectively. The finite-dimensional asymptotic distributions of a class of kernel type estimators of the conditional variance function $\sigma^2(x)$ in a more general heteroscedastic regression model are found to be normal whenever $H<(1+h)/2$, and non-normal otherwise. In addition, in this general model, $\log(n)$-consistency of the local Whittle estimator of $H$ based on pseudo residuals and consistency of a cross validation type estimator of $\sigma^2(x)$ are established. All of these findings are then used to propose a lack-of-fit test of a parametric regression model, with an application to some currency exchange rate data which exhibit LM.Comment: Published in at http://dx.doi.org/10.1214/009053607000000686 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Fitting an error distribution in some heteroscedastic time series models

This paper addresses the problem of fitting a known distribution to the innovation distribution in a class of stationary and ergodic time series models. The asymptotic null distribution of the usual Kolmogorov--Smirnov test based on the residuals generally depends on the underlying model parameters and the error distribution. To overcome the dependence on the underlying model parameters, we propose that tests be based on a vector of certain weighted residual empirical processes. Under the null hypothesis and under minimal moment conditions, this vector of processes is shown to converge weakly to a vector of independent copies of a Gaussian process whose covariance function depends only on the fitted distribution and not on the model. Under certain local alternatives, the proposed test is shown to have nontrivial asymptotic power. The Monte Carlo critical values of this test are tabulated when fitting standard normal and double exponential distributions. The results obtained are shown to be applicable to GARCH and ARMA--GARCH models, the often used models in econometrics and finance. A simulation study shows that the test has satisfactory size and power for finite samples at these models. The paper also contains an asymptotic uniform expansion result for a general weighted residual empirical process useful in heteroscedastic models under minimal moment conditions, a result of independent interest.Comment: Published at http://dx.doi.org/10.1214/009053606000000191 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach

This paper discusses asymptotically distribution free tests for the classical goodness-of-fit hypothesis of an error distribution in nonparametric regression models. These tests are based on the same martingale transform of the residual empirical process as used in the one sample location model. This transformation eliminates extra randomization due to covariates but not due the errors, which is intrinsically present in the estimators of the regression function. Thus, tests based on the transformed process have, generally, better power. The results of this paper are applicable as soon as asymptotic uniform linearity of nonparametric residual empirical process is available. In particular they are applicable under the conditions stipulated in recent papers of Akritas and Van Keilegom and M\"uller, Schick and Wefelmeyer.Comment: Published in at http://dx.doi.org/10.1214/08-AOS680 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Minimum distance regression model checking with Berkson measurement errors

Lack-of-fit testing of a regression model with Berkson measurement error has not been discussed in the literature to date. To fill this void, we propose a class of tests based on minimized integrated square distances between a nonparametric regression function estimator and the parametric model being fitted. We prove asymptotic normality of these test statistics under the null hypothesis and that of the corresponding minimum distance estimators under minimal conditions on the model being fitted. We also prove consistency of the proposed tests against a class of fixed alternatives and obtain their asymptotic power against a class of local alternatives orthogonal to the null hypothesis. These latter results are new even when there is no measurement error. A simulation that is included shows very desirable finite sample behavior of the proposed inference procedures.Comment: Published in at http://dx.doi.org/10.1214/07-AOS565 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The transfer principle: A tool for complete case analysis

This paper gives a general method for deriving limiting distributions of complete case statistics for missing data models from corresponding results for the model where all data are observed. This provides a convenient tool for obtaining the asymptotic behavior of complete case versions of established full data methods without lengthy proofs. The methodology is illustrated by analyzing three inference procedures for partially linear regression models with responses missing at random. We first show that complete case versions of asymptotically efficient estimators of the slope parameter for the full model are efficient, thereby solving the problem of constructing efficient estimators of the slope parameter for this model. Second, we derive an asymptotically distribution free test for fitting a normal distribution to the errors. Finally, we obtain an asymptotically distribution free test for linearity, that is, for testing that the nonparametric component of these models is a constant. This test is new both when data are fully observed and when data are missing at random.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1061 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org