Results 1  10
of
36
On adaptive estimation in nonstationary ARMA models with GARCH errors
 Ann. Statist
, 2003
"... This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfying a generalised autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the loglikelihood ratio for the model is obtained. It is show ..."
Abstract

Cited by 46 (34 self)
 Add to MetaCart
This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfying a generalised autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the loglikelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. Adaptivity is discussed and it is found that the parameters in the model are generally not adaptively estimable if the density of the rescaled error is asymmetric. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined Mνestimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to be
Weighted ResidualBased Density Estimators For Nonlinear Autoregressive Models
"... This paper considers residualbased and randomly weighted kernel estimators for innovation densities of nonlinear autoregressive models. The weights are chosen to make use of the information that the innovations have mean zero. Rates of convergence are obtained in weighted L1norms. These estimators ..."
Abstract

Cited by 18 (13 self)
 Add to MetaCart
This paper considers residualbased and randomly weighted kernel estimators for innovation densities of nonlinear autoregressive models. The weights are chosen to make use of the information that the innovations have mean zero. Rates of convergence are obtained in weighted L1norms. These estimators give rise to smoothed and weighted empirical distribution functions and moments. It is shown that the latter are efficient if an efficient estimator for the autoregression parameter is used to construct the residuals.
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 ..."
Abstract

Cited by 17 (17 self)
 Add to MetaCart
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.
Semiparametric Duration Models
, 2001
"... In this paper we consider semiparametric duration models and efficient estimation of the parameters in a noni.i.d. environment. In contrast to classical time series models where innovations are assumed to be i.i.d., we show that, for example in the oftenused Autoregressive Conditional Duration (AC ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
In this paper we consider semiparametric duration models and efficient estimation of the parameters in a noni.i.d. environment. In contrast to classical time series models where innovations are assumed to be i.i.d., we show that, for example in the oftenused Autoregressive Conditional Duration (ACD) model, the assumption of independent innovations is too restrictive to describe financial durations accurately. Therefore, we consider semiparametric extensions of the standard specification that allow for arbitrary kinds of dependencies between the innovations. The exact nonparametric specification of these dependencies determines the flexibility of the semiparametric model. We calculate semiparametric efficiency bounds for the ACD parameters, discuss the construction of efficient estimators, and study the efficiency loss of the exponential pseudolikelihood procedure. This efficiency loss proves to be sizeable in applications. For durations observed on the Paris Bourse for the Alcatel stock in July and August 1996, the proposed semiparametric procedures clearly outperform pseudolikelihood procedures. We analyze these efficiency gains using a simulation study which confirms that, at least at the Paris Bourse, dependencies among rescaled durations can be exploited.
Efficiency improvements in inference on stationary and nonstationary fractional time series
 ANNALS OF STATISTICS
, 2005
"... We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
(Show Context)
We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing descriptions of trending behavior. The stationary input to the stochastic component has parametric autocorrelation, but innovation with distribution of unknown form. The model is thus semiparametric, and we develop estimates of the parametric component which are asymptotically normal and achieve an Mestimation efficiency bound, equal to that found in work using an adaptive LAM/LAN approach. A major technical feature which we treat is the effect of truncating the autoregressive representation in order to form innovation proxies. This is relevant also when the innovation density is parameterized, and we provide a result for that case also. Our semiparametric estimates employ nonparametric series estimation, which avoids some complications and conditions in kernel approaches featured in much work on adaptive estimation of time series models; our work thus also contributes to methods and theory for nonfractional time series models, such as autoregressive moving averages. A Monte Carlo study of finite sample performance of the semiparametric estimates is included.
Reversible Markov chains and optimality of symmetrized empirical estimators
 Bernoulli
, 1998
"... Suppose we want to estimate the expectation of a function of two arguments under the stationary distribution of two successive observations of a reversible Markov chain. Then the usual empirical estimator can be improved by symmetrizing. We show that the symmetrized estimator is efficient. We point ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
Suppose we want to estimate the expectation of a function of two arguments under the stationary distribution of two successive observations of a reversible Markov chain. Then the usual empirical estimator can be improved by symmetrizing. We show that the symmetrized estimator is efficient. We point out applications to discretely observed continuoustime processes. The proof is based on a result for general Markov chain models which can be used to characterize efficient estimators in any model defined by restrictions on the stationary distribution of a single or two successive observations. 1 Introduction Suppose we observe X 0 ; : : : ; X n from an ergodic Markov chain with unknown transition distribution Q(x; dy) and invariant distribution ß(dx). We want to estimate the expectation of a function f(x; y) under the joint stationary distribution (ß\Omega Q)(dx; dy) = ß(dx)Q(x; dy) of two successive observations. Greenwood and Wefelmeyer (1995) show that the empirical estimator E n f =...
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... ..."
Abstract

Cited by 11 (10 self)
 Add to MetaCart
Suppose we observe an invertible linear process with independent mean zero innovations, and with coefficients depending on a finitedimensional...
Efficiency Comparisons Of Maximum LikelihoodBased Estimators In Garch Models And Testing For Density Functional Form
"... . In this paper we investigate the loss of efficiency of semiparametric (SP) and quasimaximum likelihood (QMLE) estimators relative to maximum likelihood (MLE) estimators in models with Generalized Autoregressive Conditional Heteroscedasticity (GARCH). We demonstrate that the factors which contribu ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
. In this paper we investigate the loss of efficiency of semiparametric (SP) and quasimaximum likelihood (QMLE) estimators relative to maximum likelihood (MLE) estimators in models with Generalized Autoregressive Conditional Heteroscedasticity (GARCH). We demonstrate that the factors which contribute to differences in efficiency are the Fisher information for scale and the coefficient of kurtosis of the conditional density when this density is symmetric, and in addition the coefficient of skewness when it is asymmetric. We provide a necessary and sufficient condition for equal efficiency of the three estimators if the density is symmetric. If asymmetry is present, we show that there is no density for which equal efficiency holds, even though the SP estimator may be as efficient as the MLE. Based on the Fisher information for scale we propose both parametric and nonparametric tests for density functional form. We apply our proposed tests in the context of testing the null of a Student...
Sample Splitting With Markov Chains
 Bernoulli
, 2000
"... Sample splitting techniques play an important role in constructing estimates with prescribed influence functions in semiparametric and nonparametric models when the observations are independent and identically distributed. This paper shows how a contiguity result can be used to modify these techniqu ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
Sample splitting techniques play an important role in constructing estimates with prescribed influence functions in semiparametric and nonparametric models when the observations are independent and identically distributed. This paper shows how a contiguity result can be used to modify these techniques to the case when the observations come from a stationary and ergodic Markov chain. As a consequence, sufficient conditions for the construction of efficient estimates in semiparametric Markov chain models are obtained. The applicability of the resulting theory is demonstrated by constructing an estimate of the innovation variance in a nonparametric autoregression model, by constructing a weighted least squares estimate with estimated weights in an autoregressive model with martingale innovations, and by constructing an efficient and adaptive estimate of the autoregression parameter in a heteroscedastic autoregressive model with symmetric errors.
Testing a linear MA model against threshold MA models
 Ann. Statistics
, 2005
"... This paper investigates the (conditional) quasilikelihood ratio test for the threshold in MA models. Under the hypothesis of no threshold, it is shown that the test statistic converges weakly to a function of the centred Gaussian process. Under local alternatives, it is shown that this test has non ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
This paper investigates the (conditional) quasilikelihood ratio test for the threshold in MA models. Under the hypothesis of no threshold, it is shown that the test statistic converges weakly to a function of the centred Gaussian process. Under local alternatives, it is shown that this test has nontrivial asymptotic power. The results are based on a new weak convergence of a linear marked empirical process, which is independently of interest. This paper also gives an invertible expansion of the threshold MA models. 1. Introduction. Since Tong [30]