Showing papers in "Econometric Theory in 1992"
TL;DR: In this paper, the estimation and testing of general cointegrated systems by using an autoregressive approximation is studied, and the asymptotic distributions of the estimators and test statistics are derived by assuming that the order of the auto-regressive approximation increases with the sample size at a suitable rate.
Abstract: This paper studies the estimation and testing of general cointegrated systems by using an autoregressive approximation. Simple estimators for both the cointegration vectors and their weight matrix in the autoregressive error correction model representation of the system are developed. Since these estimators assume that the number of cointegration vectors and their normalization are fixed in advance, convenient specification tests for checking the validity of these assumptions are also provided. The asymptotic distributions of the estimators and test statistics are derived by assuming that the order of the auto-regressive approximation increases with the sample size at a suitable rate. This generalizes some previous results derived for finite-order autoregressions as no assumption of a finite-parameter data-generating process is imposed. The estimators and tests of the paper are interpreted in terms of autoregressive spectral density estimators at the zero frequency and, in the special case of a finite-order Gaussian autoregression, their relation to maximum likelihood procedures is discussed. All estimators of the paper can be applied with simple least-squares techniques and used to construct conventional Wald tests with asymptotic chi-square distributions under the null hypothesis. The limit theory of the specification tests is nonstandard, similar to that in univariate unit root tests.
419 citations
TL;DR: In this paper, conditions for weak convergence of stochastic integrals are established under the assumption that the innovations are strong mixing with uniformly bounded 2-h moments, and several applications of the results are given, relevant for the theories of estimation with I(1) processes, I(2) processes with nonstationary variances, near-integrated processes, and continuous time approximations.
Abstract: This paper provides conditions to establish the weak convergence of stochastic integrals. The theorems are proved under the assumption that the innovations are strong mixing with uniformly bounded 2-h moments. Several applications of the results are given, relevant for the theories of estimation with I(1) processes, I(2) processes, processes with nonstationary variances, near-integrated processes, and continuous time approximations.
331 citations
TL;DR: In this paper, the authors provide conditions under which pointwise convergence almost surely or in probability can be strengthened to uniform convergence, and the results are useful for establishing asymptotic properties of estimators and test statistics.
Abstract: This paper presents several generic uniform convergence results that include generic uniform laws of large numbers. These results provide conditions under which pointwise convergence almost surely or in probability can be strengthened to uniform convergence. The results are useful for establishing asymptotic properties of estimators and test statistics. The results given here have the following attributes, (1) they extend results of Newey [15] to cover convergence almost surely as well as convergence in probability, (2) they apply to totally bounded parameter spaces (rather than just to compact parameter spaces), (3) they introduce a set of conditions for a generic uniform law of large numbers that has the attribute of giving the weakest conditions available for i.i.d. contexts, but which apply in some dependent nonidentically distributed contexts as well, and (4) they incorporate and extend the main results in the literature in a parsimonious fashion.
296 citations
TL;DR: In this paper, the authors investigate vector autoregressive processes and find the condition under which the processes are I(2) and prove a representation theorem for such processes and the interpretation of the AR model as an error correction model is discussed.
Abstract: We investigate vector autoregressive processes and find the condition under which the processes are I(2). A representation theorem forsuch processes is proved and the interpretation of the AR model as an error correction model is discussed.
234 citations
TL;DR: In this paper, the same set of random numbers are used to simulate the choice probabilities for each individual in the sample, and an asymptotic theory for such procedures is provided.
Abstract: This article considers methods of simulated moments for estimation of discrete response models. It is possible to use the same set of random numbers to simulate the choice probabilities for each individual in the sample. In addition to the method of simulated moments of McFadden, we have considered also maximum simulated likelihood estimation methods. An asymptotic theory for such procedures is provided. The estimators are shown to be consistent and asymptotically normal by the theory of generalized U-statistics. Asymptotic efficiency is discussed. Monte Carlo experiments on the finite sample performance of the estimators are reported.
170 citations
TL;DR: In this article, the authors propose tests on semiparametric models based on the sum of squared residuals from a least-squares procedure, which yields tests of specification, significance, smoothness and concavity and allows for heteroskedastic residuals.
Abstract: This paper proposes tests on semiparametric models based on the sum of squared residuals from a least-squares procedure. Smoothness conditions are imposed on the nonparametric portion of the model to obtain asymptotic normality of the sum of squared residuals. The approach yields tests of specification, significance, smoothness and concavity and allows for heteroskedastic residuals.
155 citations
TL;DR: In this paper, a test for neglected nonlinearities in regression models is proposed, which is based on sieve estimation of the alternative model, and the test statistic is shown to be asymptotically standard normal under the null, while rejecting with probability going to one if the linear model is misspecified.
Abstract: A test for neglected nonlinearities in regression models is proposed. The test is of the Davidson-MacKinnon type against an increasingly rich set of non-nested alternatives, and is based on sieve estimation of the alternative model. For the case of a linear parametric model, the test statistic is shown to be asymptotically standard normal under the null, while rejecting with probability going to one if the linear model is misspecified. A small simulation study suggests that the test has adequate finite sample properties, but one must guard against over fitting the nonparametric alternative.
154 citations
TL;DR: The use of probit and logit models has become quite common whenever the dependent variable in a regression is qualitative as mentioned in this paper, and these models have been used to explain either/or choices and decisions involving multiple alternatives.
Abstract: The use of probit and logit models has become quite common whenever the dependent variable in a regression is qualitative. These models have been used to explain either/or choices and decisions involving multiple alternatives. A two-dimensional graphical interpretation of these different models has been provided by Johnson [3]. The purpose of this paper is to provide a three-dimensional graphical exposition of the ordered probit model, which was first estimated by McKelvey and Zavoina [4] and is now built into computer packages, such as LIMDEP [1].
100 citations
100 citations
TL;DR: In this article, it was shown that the set of instruments employed by Balestra and Varadharajan-Krishnakumar is a subset of those used in Baltagi.
Abstract: Baltagi [3] derived 2SLS and 3SLS analogues for a simultaneous equation model with error components. These were denoted by EC2SLS and EC3SLS. More recently, Balestra and Varadharajan-Krishnakumar [1] derived alternative 2SLS and 3SLS analogues; these were denoted by G2SLS and G3SLS. This note explains the relationship between these estimators and shows that the set of instruments employed by Balestra and Varadharajan-Krishnakumar is a subset of those used in Baltagi. In addition this paper shows that for the single equation case the extra set of instruments is redundant and both EC2SLS and G2SLS have the same asymptotic variance-covariance matrix. However, for the system of equations, it can be shown that EC3SLS is asymptotically more efficient than G3SLS.
64 citations
TL;DR: In this article, a central limit theorem for dependent stochastic processes is proved for the case of martingale differences due to McLeish and suitably defined Bernstein blocks satisfy the required conditions.
Abstract: A central limit theorem is proved for dependent stochastic processes. Global heterogeneity of the distribution of the terms is permitted, including asymptotically unbounded moments. The approach is to adapt a CLT for martingale differences due to McLeish and show that suitably defined Bernstein blocks satisfy the required conditions.
TL;DR: In this paper, a method for finding an appropriate pooled bandwidth is developed, which is shown that this method is much more reliable than single curve cross-validation, because of too much variability across curves.
Abstract: The size distributions of net income in Great Britain changed systematically in the 1970s. This can be shown by visual comparison of nonparametric density estimates. Typical bandwidth selection methods, such as least squares and biased cross-validation, tend to hinder comparison, because of too much variability across curves. Hence, a method for finding an appropriate pooled bandwidth is developed. It is seen that this method is much more reliable than single curve cross-validation. Normalized net income distributions are often described in the literature as quite constant over time if one is examining the income of the whole population, see Goseke and Bedau [3], Hartog and Venbergen [5], and Blinder [1]. The central example in this paper was motivated during the course of studying this issue for the Family Expenditure Survey of the United Kingdom. The data utilized in this article were made available by the ESCR Data Archive
TL;DR: In this paper, a generalized least square based estimator is proposed based on nonparametric nearest neighbor estimates of the conditional variance matrices for the multiple equations nonlinear regression model with heteroskedasticity of unknown form.
Abstract: Asymptotically efficient estimates for the multiple equations nonlinear regression model are obtained in the presence of heteroskedasticity of unknown form. The proposed estimator is a generalized least squares based on nonparametric nearest neighbor estimates of the conditional variance matrices. Some Monte Carlo experiments are reported.
TL;DR: In this paper, the distribution function of the first few terms in a stochastic expansion of an econometric estimator or test statistic provided an asymptotic approximation to the original estimator with an error of order less than that of the limiting normal or chi-square approximation.
Abstract: Under general conditions the distribution function of the first few terms in a stochastic expansion of an econometric estimator or test statistic provides an asymptotic approximation to the distribution function of the original estimator or test statistic with an error of order less than that of the limiting normal or chi-square approximation. This can be used to establish the validity of several refined asymptotic methods, including the comparison of Nagar-type moments and the use of formal Edgeworth or Edgeworth-type approximations.
TL;DR: In this paper, the exact form of the orthogonalizing matrix R such that R'R = E-1, where -= E(uu') is the covariance matrix of u, generalizing the known formulae for AR(p) processes.
Abstract: For a general stationary ARMA(p, q) process u we derive the exact form of the orthogonalizing matrix R such that R'R = E-1, where -= E(uu') is the covariance matrix of u, generalizing the known formulae for AR(p) processes. In a linear regression model with an ARMA(p, q) error process, transforming the data by R yields a regression model with white-noise errors. We also consider an application to semi-recursive (being recursive for the model parameters, but not for the parameters of the error process) estimation. There have been many contributions to the literature concerned with estimation of the linear regression model with some autocorrelated process in the errors, surveys of which are provided by, for example, Judge et al. [15]. The main technical difficulties arise in inverting the error covariance matrix, finding the orthogonalizing transformation for the errors, or evaluating the relevant quadratic forms in the inverse matrix by other means. Among results applicable to a general ARMA(p, q) process, we may distinguish several approaches or classes of result. The numerical approach, exemplified by Harvey and Phillips [12], seeks maximum likelihood estimates for the full model through numerical optimization of the likelihood; in [12], this is achieved by expressing quadratic forms (e.g., in the likelihood function) through recursive residuals which can be computed by the Kalman filter. One difficulty in doing so lies in the necessity of initializing the recursive filtering algorithm. The other main approach to the problem uses the inverse of the error covariance matrix to obtain generalized least-squares estimates, following estimation of the parameters of the ARMA process in the errors. Within this
TL;DR: In this paper, a semiparametric method for the estimation of truncated regression models where the disturbances are independent of the regressors before truncation is presented, which provides useful information on model identification and estimation.
Abstract: This article provides a semiparametric method for the estimation of truncated regression models where the disturbances are independent of the regressors before truncation. This independence property provides useful information on model identification and estimation. Our estimate is shown to be -consistent and asymptotically normal. A consistent estimate of the asymptotic covariance matrix of the estimator is provided. Monte Carlo experiments are performed to investigate some finite sample properties of the estimator.
TL;DR: In this article, the authors show that the bias of both estimators is downward and that the bootstrap estimator exhibits a smaller bias than the conventional estimator, and they also show that neither estimator uniformly dominates the other.
Abstract: When estimating the seemingly unrelated regression (SUR) model in small samples, the bootstrap feasible generalized least-squares (FGLS) covariance estimator has been widely advocated as less biased than the conventional FGLS covariance estimator obtained by evaluating the asymptotic covariance matrix. Assuming multivariate normal errors and an unbiased estimator of the error covariance, Eaton proves that the conventional estimator is biased downward for a general SUR model. Ignoring terms O(T–2) for this model, we prove that the bootstrap estimator is also biased downward. However, from these results, the relative magnitude of these two biases is indeterminant in general. By ignoring terms O(T–2) for Zellner's two-equation, orthogonal regressor model with bivariate normal errors, we show that the bias of both estimators is downward and that the bootstrap estimator exhibits a smaller bias than the conventional estimator. Monte Carlo simulation results indicate that, in general, neither estimator uniformly dominates the other.
TL;DR: In this paper, the locally best invariant statistic to test for the constancy of regression coefficients under a random walk alternative is shown to be the same as a Bayesian-type statistic derived under a change-point alternative.
Abstract: The locally best invariant statistic to test for the constancy of regression coefficients under a random walk alternative is shown to be the same as a Bayesian-type statistic derived under a change-point alternative. Asymptotic theory for this and more general statistics is discussed.
TL;DR: In this article, a semiparametric estimator for the censored linear regression model is introduced, which is based on the regression version of Huber's [6] M-estimator.
Abstract: We introduce a semiparametric estimator for the censored linear regression model. It is based on the regression version of Huber's [6] M-estimator. It includes Powell's [19] censored least absolute deviations estimator as a special case and is related to Powell's [20] symmetrically censored least-squares estimator. We prove strong consistency and derive its asymptotic distribution which is √n-consistent with an easily computable covariance matrix. A small-scale simulation study shows that it works quite well in various cases.
TL;DR: In this paper, the authors used the bootstrap method to simulate the distribution of the smallest eigenvalue of random matrices and to test their positive definiteness, which is a sufficient condition for the law of demand to hold.
Abstract: Positive definiteness of income effect matrices provides a sufficient condition for the law of demand to hold. Given cross section household expenditure data, empirical evidence for the law of demand can be obtained by estimating such matrices. Hardle, Hildenbrand, and Jerison [10] used the bootstrap method to simulate the distribution of the smallest eigenvalue of random matrices and to test their positive definiteness. Here, theoretical aspects of this bootstrap test of positive definiteness are considered. The asymptotic distribution of the smallest eigenvalue, lambda(1), of the matrix estimate is obtained. This theory applies generally to symmetric, asymptotically normal random matrices. A bootstrap approximation to the distribution of lambda(1) is shown to converge in probability to the asymptotic distribution of lambda(1). The bootstrap test is illustrated using British family expenditure survey data.
TL;DR: In this article, GLS estimates of the regression coefficients using kernel regression and spectral methods are shown to be adaptive, in the sense of having the same asymptotic distribution, to the first order, as GLS estimate based on knowledge of the actual heteroskedasticity and serial correlation.
Abstract: In a multiple time series regression model the residuals are heteroskedastic and serially correlated of unknown form. GLS estimates of the regression coefficients using kernel regression and spectral methods are shown to be adaptive, in the sense of having the same asymptotic distribution, to the first order, as GLS estimates based on knowledge of the actual heteroskedasticity and serial correlation. A Monte Carlo experiment about the performance of our estimator is described.
TL;DR: In contrast to the standard approach of driving N to infinity for a fixed sampling frequency, the current paper follows Phillips [35,36] and Perron [29] and examines the dual asymptotics implied by letting h tend to zero while the span N remains fixed as discussed by the authors.
Abstract: This paper considers estimation based on a set of T + 1 discrete observations, y (0), y ( h ), y (2 h ),…, y ( Th ) = y ( N ), where h is the sampling frequency and N is the span of the data. In contrast to the standard approach of driving N to infinity for a fixed sampling frequency, the current paper follows Phillips [35,36] and Perron [29] and examines the “dual” asymptotics implied by letting h tend to zero while the span N remains fixed. We suggest a way of explicitly embedding discrete processes into continuous-time processes, and using this approach we generalize the results of the above-mentioned authors and derive continuous record asymptotics for vector first-order processes with positive roots in a neighborhood of one and we also consider the case of a scalar second-order process. We illustrate the method by two examples. The first example is a near unit root model with drift and trend.
TL;DR: In this paper, an overview of the Cowles Commission effort in the area of econometric theory and a critical review of the Brookings Project are presented, but flaws are noted in both efforts.
Abstract: This paper presents an overview of the Cowles Commission effort in the area of econometric theory and a critical review of the Brookings Project, but flaws are noted in both efforts. The Brookings part reflects a view of the project as seen by one of the younger members of the coordinating team; it is a view shared to some extent by other members. The paper deals with the research work stimulated by both projects and contains a brief but frank discussion of the emergence of a commercial econometric services industry. Finally, promising areas for future research are noted.
TL;DR: A Sharpening of Non-uniform bounds of the Berry-Esseen type was proposed by Kolodjažnyĭ as mentioned in this paper, who also proved that they are, in some sense, optimal.
Abstract: A Sharpening Of Nonuniform bounds of the Berry-Esseen type initially obtained by Esseen and later generalized by Kolodjažnyĭ–who also proved that they are, in some sense, optimal–is proposed. Further, the corresponding inequalities are shown to provide uniformly improved Chebyshev bounds for the tail probabilities of the distribution functions to be approximated. In contrast with most results on Berry–Esseen bounds, which emphasize rates of convergence to normality, the bounds proposed are sufficiently explicit to allow the computation of numerical bounds on a distribution function. For example, they can be applied to the sum of a small number of independent random variables. The bounds are easy to compute and can be used in confidence estimation as well as in testing problems. Applications include signed-rank tests, permutation tests, and the chi-square approximation to Bartlett's test statistic for the homogeneity of several variances. © 1992, Cambridge University Press. All rights reserved.