This paper discusses some uses econometrics of functional limit theory for dependent random variables. Attention is focused on empirical process-type results rather than partial sum results that are prevalent in unit root econometrics. Examples considered include nonstandard parametric hypotheses tests and semiparametric estimation. The application of bracketing functional limit results is discussed in some detail.
This paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null. The results of the paper are of interest, because the testing problem considered in non-standard and the classical asymptotic optimality results for the Wald, Lagrange multiplier (LM), and likelihood ratio (LR) tests do not apply. In the non-standard cases of main interest, new optimal tests are obtained and the LR test is not found to be an optimal test.
JEL Classification: C12
Keywords: Asymptotics, changepoint, nonstandard testing problem
This paper determines a class of finite sample optimal tests for the existence of a changepoint at an unknown time in a normal linear multiple regression model with known variance. Optimal tests for multiple changepoints are also derived. Power comparisons of several tests are provided based on simulations.
JEL Classification: C12
Keywords: Optimal test, Multiple changepoints, Structural change test
This paper is concerned with the estimation of first-order autoregressive/unit root models with independent identically distributed normal errors. The models considered include those without an intercept, those with an intercept, and those with an intercept and time trend. The autoregressive (AR) parameter alpha is allowed to lie in the interval (-1,1], which includes the case of a unit root. Exactly median-unbiased estimators of the AR parameter alpha are proposed. Exact confidence intervals for this parameter are introduced. Corresponding exactly median-unbiased estimators and exact confidence intervals are also provided for the impulse response function and the cumulative impulse response. An unbiased model selection procedure is discussed. The procedures that are introduced are applied to several data series including real exchange rates, the velocity of money, and industrial production.
Keywords: Autoregressive process, confidence interval, time trend, model selection, unit roots
This paper shows how the modern machinery for generating abstract empirical central limit theorems can be applied to arrays of dependent variables. It develops a bracketing approximation based on a moment inequality for sums of strong mixing arrays, in an effort to illustrate the sorts of difficulty that need to be overcome when adapting the empirical process theory for independent variables. Some suggestions for further development are offered. The paper is largely self-contained.
Keywords: Strong mixing, functional central limit theorem, empirical process
This paper considers tests of parameter instability and structural change with unknown change point. The results apply to a wide class of parametric models including models that satisfy maximum likelihood type regularity conditions and models that are suitable for estimation by generalized method of moments procedures. The paper considers likelihood ratio and likelihood ratio like tests, as well as asymptotically equivalent Wald and Lagrange multiplier tests. Each test implicitly uses an estimate of change point. Tests of both “pure” and “partial” structural change are discussed.
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 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 iid contexts, but which apply in dependent non-identically distributed contexts as well, and (4) they incorporate and extend the main results in the literature in a parsimonious fashion.
Keywords: Consistency, law of large numbers, uniform convergence, asymptotic theory, test statistics, estimators
This paper considers a new class of heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators. The estimators considered are prewhitened kernel estimators with vetor autoregressions employed in the prewhitening stage. The paper establishes consistency, rate of convergence, and asymptotic truncated mean squared error (MSE) results for the estimators when a fixed or automatic bandwidth procedure is employed. Conditions are obtained under which prewhitening improves asymptotic truncated MSE. Monte Carlo results show that prewhitening is very effective in reducing bias, improving confidence interval coverage probabilities, and rescuing over-rejection of t-statistics constructed using kernel-HAC estimators. On the other hand, prewhitening is found to inflate variance and MSE of the kernel estimators. Since confidence interval coverage probabilities and over-rejection of t-statistics are usually of primary concern, prewhitened kernel estimators provide a significant improvement over the standard non-prewhitened kernel estimators.
This paper considers series estimators of additive interactive regression (AIR) models. AIR models are nonparametric regression models that generalize additive regression models by allowing interactions between different regressor variables. They place more restrictions on the regression function, however, than do fully nonparametric regression models. By doing so, they attempt to circumvent the curse of dimensionality that afflicts the estimation of fully nonparametric regression models.
In this paper, we present a finite sample bound and asymptotic rate of convergence results for the mean average squared error of series estimators that show the AIR models do circumvent the curse of dimensionality. The rate of convergency of these estimators is shown to depend on the order of the AIR model and the smoothness of the regression function, but not on the dimension of the regressor vector. Series estimators with fixed and data-dependent truncation parameters are considered.
Keywords: Additive interactive regression model, cross-validation, curse of dimensionality, generalized cross-validation, mean average squared error, nonparametric estimation, nonparametric regression, series estimator
This paper establishes a central limit theorem (CLT) for empirical processes indexed by smooth functions. The underlying random variables may be temporally dependent and non-identically distributed. In particular, the CLT holds for near epoch dependent (i.e., functions of mixing processes) triangular arrays, which include strong mixing arrays, among others. The results apply to classes of functions that have series expansions. The proof of the CLT is particularly simple; no chaining argument is required. The results can be used to establish the asymptotic normality of semiparametric estimators in time series contexts. An example is provided.
JEL Classification: 211
Keywords: Central limit theorem, empirical process, Fourier series, semiparametric estimator, time series
This paper considers tests of nonlinear parametric restrictions in semiparametric econometric models. To date, only Wald tests of such restrictions have been considered in the literature. Here, Wald, Lagrange multiplier, and likelihood ratio-like test statistics are considered and are shown to have asymptotic chi-square distributions under the null and local alternatives. The results hold for a wide variety of underlying estimation techniques and in a wide variety of model scenarios. A number of examples are given to illustrate the testing results of this paper and the estimation and stochastic equicontinuity results of the antecedents to this paper, viz. Andrews (1989b, c).
JEL Classification: 211
Keywords: Lagrange multiplier test, likelihood ratio test, semiparametric model, semiparametric tests, Wald test, asymptotic theory
This paper provides a general framework for proving the square root of T consistency and asymptotic normality of a wide variety of semiparametric estimators. The results apply in time series and cross-sectional modeling contexts. The class of estimators considered consists of estimators that can be defined as the solution to a minimization problem based on a criterion function that may depend on a preliminary infinite dimensional nuisance parameter estimator. The criterion function need not be differentiable. The method of proof exploits results concerning the stochastic equicontinuity or weak convergence of normalized sums of stochastic processes.
This paper also considers tests of nonlinear parametric restrictions in seimparametric econometric models. To date, only Wald tests of such restrictions have been considered in the literature. Here, Wald, Lagrange multiplier, and likelihood ratio-like tests statistics are considered. A general framework is provided for proving that these test statistics have asymptotic chi-square distributions under the null hypothesis and local alternatives. The results hold for a wide variety of underlying estimation techniques and in a wide variety of model scenarios.
The problem considered here is that of using a data-driven procedure to select a good estimate from a class of linear estimates indexed by a discrete parameter. In contrast to other papers on this subject, we consider models with heteroskedastic errors. The results apply to model selection problems in linear regression and to nonparametric regression estimation via series estimators, nearest neighbor estimators, and local regression estimators, among others. Generalized CL, cross-validation, and generalized cross-validation procedures are analyzed.
JEL Classification: 211
Keywords: Heteroskedasticity, linear regression, nonparametric regression, model selection, asymptotic theory, cross validation
This paper presents several stochastic equicontinuity results that are useful for establishing the asymptotic properties of estimators and tests in parametric, semiparametric, and nonparametric econometric models. In particular, they can be applied straightforwardly in the estimation and testing results of Andrews (1989b). The paper takes various stochastic equicontinuity results from the probability literature, which rely on entropy conditions of one sort or another, and provides primitive conditions under which the entropy conditions hold. This yields stochastic equicontinuity results that are readily applicable in a variety of contexts.
This paper also presents a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives. These results are particularly useful in semiparametric estimation and testing problems that rely on preliminary nonparametric estimators, as in Andrews (1989b). The results allow for near epoch dependent non-identically distributed random variables, data-dependent bandwidth sequences, preliminary estimation of parameters (e.g., regression based on residuals), and nonparametric regression on index functions. Some of the results make use of the stochastic equicontinuity results of the paper.
JEL Classification: 211
Keywords: nonparametric density estimator, nonparametric regression estimator, semiparametric estimator, semiparametric test, series expansion, Sobolev norm, stochastic equicontinuity
This paper is concerned with the estimation of covariance matrices in the presence of heteroskedasticity and autocorrelation of unknown forms. Currently available estimators that are designed for this context depend upon the choice of a lag truncation parameter and a weighting scheme. Results in the literature provide a condition on the growth rate of the lag truncation parameter as T → ∞ that is sufficient for consistency. No results are available, however, regarding the choice of a lag truncation parameter for a fixed sample size, regarding data-dependent automatic lag truncation parameters, or regarding the choice of weighing scheme. In consequence, available estimators are not entirely operational and the relative merits of the estimators are unknown.
This paper addresses these problems. Upper and lower bounds on the asymptotic mean squared error of each estimator in a given class are determined and compared. Asymptotically optimal kernel/weighting scheme and bandwidth/lag truncation parameters are obtained using a minimax asymptotic mean squared error criterion. Higher order asymptotically optimal corrections to the first order optimal bandwidth/lag truncation parameters are introduced. Using these results, data-dependent automatic bandwidth/lag truncation parameters are defined and are shown to possess certain asymptotic optimality properties. Finite sample properties of the estimators are analyzed via Monte Carlo simulation.