Results 1  10
of
20
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 ..."
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Cited by 46 (34 self)
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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 ..."
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Cited by 18 (13 self)
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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 ..."
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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.
QuasiLikelihood Models and Optimal Inference
 Ann. Statist
"... Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasilikelihood model. The customary estimator for the parameter is the maximum quasilikelihood estimator. It is not ef ..."
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Cited by 16 (5 self)
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Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasilikelihood model. The customary estimator for the parameter is the maximum quasilikelihood estimator. It is not efficient, but as good as the best estimator that ignores the parametric model for the conditional variance. We construct two efficient estimators. One is a convex combination of solutions of two estimating equations, the other a weighted nonlinear onestep least squares estimator, with weights involving predictors for the third and fourth centered conditional moments of the transition distribution. Additional restrictions on the model can lead to further improvement. We illustrate this with an autoregressive model whose error variance is related to the autoregression parameter. 1 Introduction According to Wedderburn (1974), a quasilikelihood model is defined by a relation between mean and v...
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 ..."
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Cited by 13 (2 self)
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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.
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...
Adaptive estimators for parameters of the autoregression function of a Markov chain
"... Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one specifies, in addition, a parametric model for the conditional variance, one can define a simple estimator for the pa ..."
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Cited by 11 (8 self)
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Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one specifies, in addition, a parametric model for the conditional variance, one can define a simple estimator for the parameter, the maximum quasilikelihood estimator. It is robust against misspecification of the conditional variance, but not efficient. We construct an estimator which is adaptive in the sense that it is efficient if the conditional variance is misspecified, and asymptotically as good as the maximum quasilikelihood estimator if the conditional variance is correctly specified. The adaptive estimator is a weighted nonlinear least squares estimator, with weights given by predictors for the conditional variance.
Optimal testing for semiparametric AR models  From Gaussian Lagrange multipliers to autoregressive rank scores and adaptive tests
, 2006
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Efficient Estimation in Invertible Linear Processes
"... An invertible causal linear process is a process which has infinite order moving average and autoregressive representations. We assume that the coefficients in these representations depend on a Euclidean parameter, while the corresponding innovations have an unknown centered distribution with some m ..."
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Cited by 7 (7 self)
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An invertible causal linear process is a process which has infinite order moving average and autoregressive representations. We assume that the coefficients in these representations depend on a Euclidean parameter, while the corresponding innovations have an unknown centered distribution with some moment restrictions. We discuss efficient estimation of differentiable functionals in such a semiparametric model. For this we first obtain a suitable semiparametric version of local asymptotic normality and then use Hajek's convolution theorem to characterize efficient estimators. Then we apply this result to construct efficient estimators of the Euclidean parameter and of linear functionals of the innovation distribution.
An Alternative Asymptotic Analysis of ResidualBased Statistics,” mimeo
, 2005
"... This paper offers an alternative technique to derive the limiting distribution of residualbased statistics or, more general, the limiting distribution of statistics with estimated nuisance parameters. This technique allows us to unify many known results on twostage estimators and tests and we als ..."
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Cited by 5 (1 self)
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This paper offers an alternative technique to derive the limiting distribution of residualbased statistics or, more general, the limiting distribution of statistics with estimated nuisance parameters. This technique allows us to unify many known results on twostage estimators and tests and we also derive new results. The technique is especially useful in situations where smoothness of the statistic of interest with respect to the parameters to be estimated does not hold or is difficult to establish, e.g., rankbased statistics. We essentially replace this differentiability condition with a distributional invariance property that is often satisfied in specification tests. Our results on statistics that have not been considered before all use nonparametric statistics. On the technical side, we provide a novel approach to the preestimation problem using Le Cam’s third lemma. The resulting formula for the correction in the limiting variance as a result of preestimation some parameters is a simple expression involving some appropriate covariances. The regularity conditions required fairly minimal. Numerous examples show the strength and wide applicability of our approach.