Showing papers on "Asymptotic distribution published in 1999"

Pooled Mean Group Estimation of Dynamic Heterogeneous Panels

Abstract: It is now quite common to have panels in which both T, the number of time series observations, and N, the number of groups, are quite large and of the same order of magnitude. The usual practice is either to estimate N separate regressions and calculate the coefficient means, which we call the mean group (MG) estimator, or to pool the data and assume that the slope coefficients and error variances are identical. In this article we propose an intermediate procedure, the pooled mean group (PMG) estimator, which constrains long-run coefficients to be identical but allows short-run coefficients and error variances to differ across groups. We consider both the case where the regressors are stationary and the case where they follow unit root processes, and for both cases derive the asymptotic distribution of the PMG estimators as T tends to infinity. We also provide two empirical applications: Aggregate consumption functions for 24 Organization for Economic Cooperation and Development economies over th...

Numerical distribution functions of likelihood ratio tests for cointegration

Abstract: This paper employs response surface regressions based on simulation experiments to calculate asymptotic distribution functions for the Johansen-type likelihood ratio tests for cointegration. These are carried out in the context of the models recently proposed by Pesaran, Shin, and Smith (1997) that allow for the possibility of exogenous variables integrated of order one. The paper calculates critical values that are very much more accurate than those available previously. The principal contributions of the paper are a set of data files that contain estimated asymptotic quantiles obtained from response surface estimation and a computer program for utilizing them. This program, which is freely available via the Internet, can be used to calculate both asymptotic critical values and P-values. Copyright © 1999 John Wiley & Sons, Ltd.

Some tests for parameter constancy in cointegrated VAR-models

Abstract: Some methods for the evaluation of parameter constancy in cointegrated vectorautoregressive (VAR) models are discussed. Two different ways of re-estimating the VAR-model are proposed; one in which all parameters are estimated recursively based upon the likelihood function for the first observations, and another in which the cointegrating relations are estimated recursively from a likelihood function, where the short-run parameters have been concentrated out. We suggest graphical procedures based on recursively estimated eigenvalues to evaluate the constancy of the long-run parameters in the model. Specifically,we look at the time paths of the eigenvalues using a new result on the asymptotic distribution of the estimated eigenvalues. Furthermore, we show that the fluctuation test by Ploberger et al. (1989) and the Lagrange multiplier (LM) type test for constancy of parameters by Nyblom (1989) can be applied to test the constancy of the long-run parameters in the cointegrated VAR-model. All results are illustrated using a model for the term structure of interest rates on US Treasury securities.

Nonparametric estimation of triangular simultaneous equations models

Abstract: This paper presents a simple two-step nonparametric estimator for a triangular simultaneous equation model. Our approach employs series approximations that exploit the additive structure of the model. The first step comprises the nonparametric estimation of the reduced form and the corresponding residuals. The second step is the estimation of the primary equation via nonparametric regression with the reduced form residuals included as a regressor. We derive consistency and asymptotic normality results for our estimator, including optimal convergence rates. Finally we present an empirical example, based on the relationship between the hourly wage rate and annual hours worked, which illustrates the utility of our approach.

Testing for Linearity

Abstract: The problem of testing for linearity and the number of regimes in the context of self-exciting threshold autoregressive (SETAR) models is reviewed. We describe least-squares methods of estimation and inference. The primary complication is that the testing problem is non-standard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is non-standard. Simulation methods to calculate asymptotic and bootstrap distributions are presented. As the sampling distributions are quite sensitive to conditional heteroskedasticity in the error, careful modeling of the conditional variance is necessary for accurate inference on the conditional mean. We illustrate these methods with two applications--annual sunspot means and monthly U.S. industrial production. We find that annual sunspots and monthly industrial production are SETAR(2) processes. Copyright 1999 by Blackwell Publishers Ltd

The existence and asymptotic properties of a backfitting projection algorithm under weak conditions

Abstract: We derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function. The procedure employs a recent projection interpretation of popular kernel estimators provided by Mammen, Marron, Turlach and Wand and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in Bickel, Klaassen, Ritov and Wellner. Our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components, and in this sense improves on the method analyzed in Opsomer and Ruppert. We provide ‘‘high level’’ conditions independent of the sampling scheme. We then verify that these conditions are satisfied in a regression and a time series autoregression under weak conditions.

Estimation When a Parameter is on a Boundary

Abstract: This paper establishes the asymptotic distribution of an extremum estimator when the true parameter lies on the boundary of the parameter space. The boundary may be linear, curved, and/or kinked. Typically the asymptotic distribution is a function of a multivariate normal distribution in models without stochastic trends and a function of a multivariate Brownian motion in models with stochastic trends. The results apply to a wide variety of estimators and models. Examples treated in the paper are: (i) quasi-ML estimation of a random coefficients regression model with some coefficient variances equal to zero and (ii) LS estimation of an augmented Dickey-Fuller regression with unit root and time trend parameters on the boundary of the parameter space.

Testing the order of a model using locally conic parametrization : population mixtures and stationary ARMA processes

Abstract: In this paper, we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models, with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model. We give two major applications: the case where the number of populations has to be tested in a mixture and the case of stationary ARMA$(p, q)$ processes where the order $(p, q)$ has to be tested. We give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model. In the case of order selection for ARMAs, the asymptotic distribution is invariant with respect to the parameters generating the process. A locally conic parametrization is a key tool in deriving the limiting distributions; it allows one to discover the deep similarity between the two problems.

Estimating equations based on eigenfunctions for a discretely observed diffusion process

Abstract: A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discrete-time observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study

Improved rates and asymptotic normality for nonparametric neural network estimators

Abstract: We obtain an improved approximation rate (in Sobolev norm) of r/sup -1/2-/spl alpha//(d+1)/ for a large class of single hidden layer feedforward artificial neural networks (ANN) with r hidden units and possibly nonsigmoid activation functions when the target function satisfies certain smoothness conditions. Here, d is the dimension of the domain of the target function, and /spl alpha//spl isin/(0, 1) is related to the smoothness of the activation function. When applying this class of ANNs to nonparametrically estimate (train) a general target function using the method of sieves, we obtain new root-mean-square convergence rates of Op([n/log(n)]/sup -/(1+2/spl alpha//(d+1))/[4(1+/spl alpha//(d+1))])=op(n/sup -1/4/) by letting the number of hidden units /spl tau//sub n/, increase appropriately with the sample size (number of training examples) n. These rates are valid for i.i.d. data as well as for uniform mixing and absolutely regular (/spl beta/-mixing) stationary time series data. In addition, the rates are fast enough to deliver root-n asymptotic normality for plug-in estimates of smooth functionals using general ANN sieve estimators. As interesting applications to nonlinear time series, we establish rates for ANN sieve estimators of four different multivariate target functions: a conditional mean, a conditional quantile, a joint density, and a conditional density. We also obtain root-n asymptotic normality results for semiparametric model coefficient and average derivative estimators.

On Single-Index Coefficient Regression Models

Abstract: In this article we investigate a class of single-index coefficient regression models under dependence. This includes many existing models, such as the smooth transition threshold autoregressive (STAR) model of Chan and Tong, the functional-coefficient autoregressive (FAR) model of Chen and Tsay, and the single-index model of Ichimura. Compared to the varying-coefficient model of Hastie and Tibshirani, our model can avoid the curse of dimensionality in multivariate nonparametric estimations. Another advantage of this model is that a threshold variable is chosen automatically. An estimation method is proposed, and the corresponding estimators are shown to be consistent and asymptotically normal. Some simulations and applications are also reported.

Efficient estimation of the partly linear additive Cox model

Abstract: The partly linear additive Cox model is an extension of the (linear) Cox model and allows flexible modeling of covariate effects semiparametrically. We study asymptotic properties of the maximum partial likelihood estimator of this model with right-censored data using polynomial splines. We show that, with a range of choices of the smoothing parameter (the number of spline basis functions) required for estimation of the nonparametric components, the estimator of the finite-dimensional regression parameter is root-$n$ consistent, asymptotically normal and achieves the semiparametric information bound. Rates of convergence for the estimators of the nonparametric components are obtained. They are comparable to the rates in nonparametric regression. Implementation of the estimation approach can be done easily and is illustrated by using a simulated example.

Censored Median Regression Using Weighted Empirical Survival and Hazard Functions

Abstract: For median regression models that regress the median of the survival time or a transform thereof on the covariates, some semi-parametric estimators that include the intercept component are introduced when the survival time may be censored. These new median regression estimators do not require estimating the censoring distributions. They can be viewed as an extension of the sample median to the censored regression model. These estimators are based on some weighted empirical survival and hazard functions and are shown to be consistent and asymptotically normal. They performed very well in various numerical studies. The proposed procedures are illustrated in some real data examples.

On goodness-of-fit tests for weakly dependent processes using kernel method

Abstract: In this paper, we extend some existing goodness-of-fit tests for independent observations using kernel method to tests for weakly dependent processes. The tests considered include: (i) a two sample goodness-of-fit test; (ii) a symmetry test; and (iii) a test for the goodness-of-fit of a parametric density function. We also develop a center-free test for the goodness-of-fit of a parametric density function. We establish the asymptotic normality of the tests under the corresponding null hypotheses and verify their consistency.

Cointegration rank inference with stationary regressors in VAR models

Abstract: The issue of including stationary explanatory variables is addressed in the vector autoregressive (VAR) model, when testing for cointegration rank. It is shown that simply including stationary explanatory variables as extra regressors will lead to nuisance parameters in the asymptotic distribution of the trace statistic for cointegration rank. The nuisance parameters are characterized as canonical correlations between the common trends (which in this case also involve the accumulated stationary explanatory process) and the accumulated innovations. Thus, in particular, the trace test is not similar, even asymptotically, and an alternative model is discussed, which will lead to nuisance-free rank determination. The alternative model is the extended VAR model where the cumulated explanatory variables enter the system in the error correction term. Possible loss in power due to overparametrization is briefly addressed and the proposed analysis is illustrated by an empirical example.

Symmetrized random permutations

Abstract: Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed involutions. We are interested in the statistics of the length of the longest up/right path of random points selections in each symmetry type as the number of points increases to infinity. The limiting distribution functions are expressed in terms of Painleve II equation. Some of them are Tracy-Widom distributions in random matrix theory, while there are two new classes of distribution functions interpolating GOE and GSE, and GUE and GOE^2 Tracy-Widom distribution functions. Also some applications and related topics are discussed.

Asymptotic normality of the maximum likelihood estimator in state space models

Abstract: State space models is a very general class of time series models capable of modelling dependent observations in a natural and interpretable way. Inference in such models has been studied by Bickel, Ritov and Ryden, who consider hidden Markov models, which are special kinds of state space models, and prove that the maximum likelihood estimator is asymptotically normal under mild regularity conditions. In this paper we generalize the results of Bickel, Ritov and Ryden to state space models, where the latent process is a continuous state Markov chain satisfying regularity conditions, which are fulfilled if the latent process takes values in a compact space.

On asymptotics of estimating functions

Abstract: The asymptotic theory of estimators obtained from estimating functions is reviewed and some new results on the multivariate parameter case are presented Specifically, results about existence of consistent estimators and about asymptotic normality of these are given First a very general stochastic process setting is considered Then it is demonstrated how more specific conditions for existence of √ n-consistent and asymptotically normal estimators can be given for martingale estimating functions in the case of observations of a Markov process

Bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters. I: A use of bootstrap standard error.

Abstract: In population pharmacokinetic studies, one of the main objectives is to estimate population pharmacokinetic parameters specifying the population distributions of pharmacokinetic parameters. Confidence intervals for population pharmacokinetic parameters are generally estimated by assuming the asymptotic normality, which is a large-sample property, that is, a property which holds for the cases where sample sizes are large enough. In actual clinical trials, however, sample sizes are limited and not so large in general. Likelihood functions in population pharmacokinetic modelling include a multiple integral and are quite complicated. We hence suspect that the sample sizes of actual trials are often not large enough for assuming the asymptotic normality and that the asymptotic confidence intervals underestimate the uncertainties of the estimates of population pharmacokinetic parameters. As an alternative to the asymptotic normality approach, we can employ a bootstrap approach. This paper proposes a bootstrap standard error approach for constructing confidence intervals for population pharmacokinetic parameters. Comparisons between the asymptotic and bootstrap confidence intervals are made through applications to a simulated data set and an actual phase I trial.

Semiparametric Inference in the Proportional Odds Regression Model

Abstract: For fitting the proportional odds regression model with right-censored survival times, we introduce some weighted empirical odds functions. These functions are solutions of some self-consistency equations and have a nice martingale representation. From these functions, several classes of new regression estimators, such as the pseudo–maximum likelihood estimator, martingale residual-based estimators, and minimum distance estimators, are derived. These estimators have desirable properties such as easy computation, asymptotic normality via a martingale analysis, and reliable asymptotic covariance estimation in closed form. Extensive numerical studies show that the minimum L 2 distance estimators have very good finite-sample behaviors compared to existing methods. Results of some simulation studies and applications to a real dataset are given. The weighted odds function–based approach also provides inference on the baseline odds function and some measures for lack-of-fit analysis.

Subsampling intervals in autoregressive models with linear time trend

Abstract: A new method is proposed for constructing confidence intervals in autoregressive models with linear time trend. Interest focuses on the sum of the autoregressive coefficients because this parameter provides a useful scalar measure of the long-run persistence properties of an economic time series. Since the type of the limiting distribution of the corresponding OLS estimator, as well as the rate of its convergence, depend in a discontinuous fashion upon whether the true parameter is less than one or equal to one (that is, trend-stationary case or unit root case), the construction of confidence intervals is notoriously difficult. The crux of our method is to recompute the OLS estimator on smaller blocks of the observed data, according to the general subsampling idea of Politis and Romano (1994a), although some extensions of the standard theory are needed. The method is more general than previous approaches in that it works for arbitrary parameter values, but also because it allows the innovations to be a martingale difference sequence rather than i.i.d. Some simulation studies examine the finite sample performance.