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21
Testing Distributional Assumptions: A GMM Approach. Universite de Montreal
, 2005
"... In this paper, we consider testing distributional assumptions. Special cases that we consider are the Pearson’s family like the normal, Student, gamma, beta and uniform distributions. The test statistics we consider are based on a set of moment conditions. This set coincides with the first moment co ..."
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Cited by 22 (1 self)
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In this paper, we consider testing distributional assumptions. Special cases that we consider are the Pearson’s family like the normal, Student, gamma, beta and uniform distributions. The test statistics we consider are based on a set of moment conditions. This set coincides with the first moment conditions derived by Hansen and Scheinkman (1995) when one considers a continuous time model. By testing moment conditions, we treat in detail the parameter uncertainty problem when the considered variable is not observed but depends on estimators of unknown parameters. In particular, we derive moment tests that are robust against parameter uncertainty. We also consider the case where the variable of interest is serially correlated with unknown dependence by adopting a HAC approach for this purpose. This paper extends Bontemps and Meddahi (2005) who considered this approach for the normal case. Finite sample properties of our tests when the variable of interest is a Student are derived through a comprehensive Monte Carlo study. An empirical application to StudentGARCH model is presented. Keywords: Pearson’s distributions; HansenScheinkman moment conditions; parameter uncertainty; serial correlation; HAC.
Estimating Linear Functionals of the Error Distribution in Nonparametric Regression
"... This paper addresses estimation of linear functionals of the error distribution in nonparametric regression models. It derives an i.i.d. representation for the empirical estimator based on residuals, using undersmoothed estimators for the regression curve. Asymptotic eciency of the estimator is p ..."
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Cited by 17 (9 self)
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This paper addresses estimation of linear functionals of the error distribution in nonparametric regression models. It derives an i.i.d. representation for the empirical estimator based on residuals, using undersmoothed estimators for the regression curve. Asymptotic eciency of the estimator is proved. Estimation of the error variance is discussed in detail.
Estimating the Innovation Distribution in Nonlinear Autoregressive Models
 Department of Mathematics, University of Siegen. http://www.math.unisiegen.de/statistik/wefelmeyer.html
"... The usual estimator for the expectation of a function under the innovation distribution of a nonlinear autoregressive model is the empirical estimator based on estimated innovations. It can be improved by exploiting that the innovation distribution has mean zero. We show that the resulting estimator ..."
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Cited by 17 (17 self)
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The usual estimator for the expectation of a function under the innovation distribution of a nonlinear autoregressive model is the empirical estimator based on estimated innovations. It can be improved by exploiting that the innovation distribution has mean zero. We show that the resulting estimator is efficient if the innovations are estimated with an efficient estimator for the autoregression parameter. Efficiency of this estimator is necessary except when the expectation of the function can be estimated adaptively. Analogous results hold for heteroscedastic models.
HIGH MOMENT PARTIAL SUM PROCESSES OF RESIDUALS IN GARCH MODELS AND THEIR APPLICATIONS 1
, 2006
"... In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and selfnormalized partial sum processes. The kth power partial sum process converges to a Brownian p ..."
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Cited by 14 (0 self)
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In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and selfnormalized partial sum processes. The kth power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the kth moment µk of the innovation sequence. If µk = 0, then the correction term is 0 and, thus, the kth power partial sum process converges weakly to the same Gaussian process as does the kth power partial sum of the i.i.d. innovations sequence. In particular, since µ1 = 0, this holds for the first moment partial sum process, but fails for the second moment partial sum process. We also consider the CUSUM and the selfnormalized processes, that is, standardized by the residual sample variance. These behave as if the residuals were asymptotically i.i.d. We also study the joint distribution of the kth and (k + 1)st selfnormalized partial sum processes. Applications to changepoint problems and goodnessoffit are considered, in particular, CUSUM statistics for testing GARCH model structure change and the Jarque– Bera omnibus statistic for testing normality of the unobservable innovation distribution of a GARCH model. The use of residuals for constructing a kernel density function estimation of the innovation distribution is discussed.
Estimating invariant laws of linear processes by Ustatistics
"... Suppose we observe an invertible linear process with independent mean zero innovations, and with coefficients depending on a finitedimensional... ..."
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Cited by 11 (10 self)
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Suppose we observe an invertible linear process with independent mean zero innovations, and with coefficients depending on a finitedimensional...
A note on the BickelRosenblatt test in autoregressive time series
"... Abstract In a recent paper Lee and Na (2001) introduced a test for a parametric form of the distribution of the innovations in autoregressive models, which is based on the integrated squared error of the nonparametric density estimate from the residuals and a smoothed version of the parametric fit ..."
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Cited by 10 (2 self)
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Abstract In a recent paper Lee and Na (2001) introduced a test for a parametric form of the distribution of the innovations in autoregressive models, which is based on the integrated squared error of the nonparametric density estimate from the residuals and a smoothed version of the parametric fit of the density. They derived the asymptotic distribution under the nullhypothesis, which is the same as for the classical BickelRosenblatt (1973) test for the distribution of i.i.d. observations. In this note we first extend the results of Bickel and Rosenblatt to the case of fixed alternatives, for which asymptotic normality is still true but with a different rate of convergence. As a byproduct we also provide an alternative proof of the Bickel and Rosenblatt result under substantially weaker assumptions on the kernel density estimate. As a further application we derive the asymptotic behaviour of Lee and Na's statistic in autoregressive models under fixed alternatives. The results can be used for the calculation of the probability of the type II error of the BickelRosenblatt test for the parametric form of the error distribution and for testing interval hypotheses in this context.
Fitting an error distribution in some heteroscedastic time series models
 Ann. Statist
, 2006
"... 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 a ..."
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Cited by 8 (2 self)
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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. 1. Introduction. Let {yi
Asymptotic theory for ARCHSM models: LAN and residual empirical processes
 Statist. Sinica
, 2005
"... In this paper, we study the two asymptotic objectives: the LAN and the residual empirical process in a class of ARCH(∞)SM (stochastic mean) models, which covers finiteorder ARCH and GARCH models. First, we establish the LAN for the ARCH(∞)SM model, and based on it, construct an asymptotically opt ..."
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Cited by 8 (3 self)
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In this paper, we study the two asymptotic objectives: the LAN and the residual empirical process in a class of ARCH(∞)SM (stochastic mean) models, which covers finiteorder ARCH and GARCH models. First, we establish the LAN for the ARCH(∞)SM model, and based on it, construct an asymptotically optimal test when the parameter vector contains a nuisance parameter. Also, we discuss asymptotically efficient estimators for unknown parameters under the circumstances: (i) the innovation density is known, and (ii) it is unknown. For the residual empirical process, we investigate its asymptotic behavior in ARCH(q)SM models. We show that unlike the usual autoregressive model, the limiting distribution in this case depends upon the estimator of the regression parameter as well as those of the ARCH parameters.
Estimating the innovation distribution in nonparametric autoregression
, 2008
"... We prove a Bahadur representation for a residualbased estimator of the innovation distribution function in a nonparametric autoregressive model. The residuals are based on a local linear smoother for the autoregression function. Our result implies a functional central limit theorem for the residua ..."
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Cited by 5 (1 self)
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We prove a Bahadur representation for a residualbased estimator of the innovation distribution function in a nonparametric autoregressive model. The residuals are based on a local linear smoother for the autoregression function. Our result implies a functional central limit theorem for the residualbased estimator. 1. Introduction. Regression
Asymptotic distributions of error density estimators in firstorder autoregressive models
 Sankhya
"... This paper considers the asymptotic distributions of the error density estimators in firstorder autoregressive models. At a fixed point, the distribution of the error density estimator is shown to be normal. Globally, the asymptotic distribution of the maximum of a suitably normalized deviation of ..."
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Cited by 4 (1 self)
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This paper considers the asymptotic distributions of the error density estimators in firstorder autoregressive models. At a fixed point, the distribution of the error density estimator is shown to be normal. Globally, the asymptotic distribution of the maximum of a suitably normalized deviation of the density estimator from the expectation of the kernel error density (based on the true error) is the same as in the case of the one sample set up, which is given in Bickel and Rosenblatt (1973). AMS (2000) subject classification. Primary 62G07; secondary 62M10.