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. 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 establishes the asymptotic normality of series estimators for nonparametric regression models. Gallant’s Fourier flexible form estimators, trigonometric series estimators, and polynomial series estimators are prime examples of the estimators covered by the results. The results apply to a wide variety of estimands in the regression model under consideration, including derivatives and integrals of the regression function. The errors in the model may be homoskedastic or heteroskeclastic. The paper also considers series estimators for additive interactive regression (AIR), seimparametric regression, and semiparametric index regression models and shows them to be consistent and asymptotically normal. All of the consistency and asymptotic normality results in the paper follow from one set of general results for series estimators.
This paper extends the classical Chow (1960) test for structural change in linear regression models to a wide variety of nonlinear models, estimated by a variety of different procedures. Wald, Lagrange multiplier-like, and likelihood ratio-like test statistics are introduced. The results allow for heterogeneity and temporal dependence of general unifying results for estimation and testing in nonlinear parametric econometric models.
JEL Classification: 211, 212
Keywords: Chow test, dynamic model, econometric model, Lagrange multiplier test, likelihood ratio test, structural change, Wald test, nonlinear model
This paper is concerned with the use of power properties of tests in econometric applications. Power radius and inverse power functions are defined. These functions are designed to yield summary measures of power that facilitate the interpretation of test results in practice. Simple approximations are introduced for the power radius and inverse power functions of Wald, likelihood ration, Lagrange multiplier, and Hausman tests. These approximations readily convey the general qualitative features of the power of a test. Examples are provided to illustrate their usefulness in interpreting test results.
JEL Classification: 211
Keywords: Power function, Hypothesis tests, Inverse power function
A basic tool of modern econometrics is a uniform law of large numbers (LLN). It is a primary ingredient used in proving consistency and asymptotic normality of parametric and nonparametric estimators in nonlinear econometric models. Thus, in a well-known review article, Burguete, Gallant, and Sousa [8, p. 162] introduce a uniform LLN with the statement: “The following theorem is the result upon which the asymptotic theory of nonlinear econometrics rests.” So pervasive is the use of uniform LLNs, that numerous authors appeal to an unspecified generic uniform LLN. Others appeal to some specific result. The purpose of this paper is to provide a generic uniform LLN that is sufficiently general to incorporate most applications of uniform LLNs in the nonlinear econometrics literature. In summary, the paper presents a result that can be used to turn state of the art pointwise LLNs into uniform LLNs over compact sets, with the addition of a single smoothness condition — either a Lipschitz condition or a derivative condition. The latter is particularly easy to verify, and is implied by common assumptions used to prove asymptotic normality of estimators. Thus, the additional condition is not particularly restrictive. In contrast to other uniform LLNs that appear in the literature, the one given here allows the full range of heterogeneity of summands (i.e., non-identical distributions), and temporal dependence, that is available with pointwise LLNs.
JEL Classification: 211
Keywords: Uniform law of large numbers, Consistency, Nonlinear econometric models
This paper and its sequel, Andrews [4], extend the Pearson chi-square testing method to non-dynamic parametric econometric models, in particular, models with covariates. The present paper introduced the test and discusses a wide variety of applications. Andrews [4] establishes the asymptotic properties of the test, by extending recent probabilistic results for the weak convergence of empirical processes indexed by sets. The chi-square test that is introduced can be used to test goodness-of-fit of a parametric model, as well as to test particular aspects of the parametric model that are of interest. In the event of rejection of the null hypothesis of correct specification, the test provides information concerning the direction of departure from the null. The results allow for estimation of the parameters of the model by quite general methods. The cells used to construct the test statistic my be random and can be specified in a general form.
This paper extends the Pearson chi-square testing method to non-dynamic parametric econometric models, in particular, to models with covariates. The paper establishes the asymptotic distribution of the test statistic under the null and local alternatives, when the test statistic is based on data-dependent random cells of a general form, and on an arbitrary asymptotically normal estimator. These results are attained by extending recent probabilistic results for the weak convergence of empirical processes indexed by sets. The chi-square test that is introduced can be used to test goodness-of-fit of a parametric model, as well as to test particular aspects of the parametric model that are of interest. In the event of rejection of the null hypothesis, the test provides information concerning the direction of departure from the null. The diagnostics provided by the test are intuitive and particularly easy to interpret.
This paper considers the linear regression model with multiple stochastic regressors, intercept, and errors that have undefined means. This model is of interest from a robustness perspective as a polar case. Generally, least squares estimators are inconsistent in this context. It is shown, however, that this inconsistency is restricted to the estimation of the intercept, if the regressors are highly variable. Rates of convergence of the least squares slope estimators are determined, and are shown to exceed the standard rate, n-1/2, in certain contexts. The results place no restrictions on the temporal dependence of the errors, and require an unusually weak exogeneity condition between the regressors and errors. Implications of the results for robustness theory are discussed.
This note presents (i) necessary and sufficient conditions for the consistency of estimators of Moore-Penrose inverted matrices, and (ii) sufficient conditions for convergence to a chi-square distribution of quadratic forms based on g-inverted weighting matrices. The latter results are needed to establish asymptotic significance levels and local power properties of generalized Wald tests (i.e., Wald tests with singular covariance matrices). Included in this class of tests are Hausman specification tests and various goodness of fit tests, among others. The results are relevant to procedures currently in the literature, since they illustrate that some results stated in the literature hold only under more restrictive assumptions than those given.
This note presents a set of conditions on the defining functions of regression parameter estimators of the linear model. These conditions guarantee that the estimators are symmetrically distributed about the true parameter value, and hence are median unbiased, provided the conditional distribution of the vector of errors is symmetric given the matrix of regressors. The symmetry result holds even if the regression parameters are subject to linear restrictions. If the estimators posses one or more moments, then the symmetry result also implies mean unbiasedness. Similar conditions are provided that establish the property of origin (or shift) equivariance for the estimators. Common feasible GLS, quasi-ML, robust, adaptive, and spectral estimators are seen easily to satisfy the requisite conditions.
JEL Classification: 211
Keywords: Unbiasedness, Linear model, Parameter estimators
A property of estimators called stability is investigated in this paper. The stability of an estimator is a measure of the magnitude of the affect of any single observation in the sample on the realized value of the estimator. High stability often is desirable for robustness against misspecification and against highly variable observations.
Stabilities are determined and compared for a wide variety of estimators and econometric models. Estimators considered include: least squares, maximum likelihood (including both LIML and FIML), instrumental variables, M-, and multi-stage estimators such as tow and three stage least squares, Zellner’s feasible Aikten estimator of the multivariate regression model, and Heckman’s estimator of censored regression and self-selection models. The general results of the paper apply to numerous additional estimators of various and sundry models.
The stability of an estimator is found to depend on the number of finite moments of its influence curve (evaluated at a random observation in the sample). An estimator’s stability increases strictly and continuously from zero to one as the number of finite moments of its influence curve increases from one to infinity. The more moments, the higher the stability. Since it often is possible to construct estimators with a specified influence function, estimators with different stabilities can be constructed. For example, one can attain the maximum stability possible by formulating a bounded influence estimator, since they have an infinite number of finite moments.
This paper extends the results of Andrews (1984) which considers the problem of robust estimation of location in a model with stationary strong mixing Gaussian parametric distributions. Three neighbourhood systems are considered, each of which contains the Hellinger neighbourhoods used in Andrews (1984). Optimal robust estimators for this dependent random variable model are found to be bounded influence estimators with optimal psi functions which are very nearly of Huber shape. These estimators are quite robust against different “amounts” of dependence, and against lack of dependence. To generate the optimal estimators a minimax asymptotic risk criterion is used, where minimaxing is done over neighbourhoods of the parametric Gaussian distributions. The neighbourhood systems include distributions of strong mixing processes. They allow for deviations from stationarity and from the Gaussian structure of dependence. In addition, deviations from the normal univariate parametric distributions are allowed within the neighbourhoods defined by (i) εn-contamination, (ii) variational metric distance, and (iii) Kolmogorov metric distance.
The least squares estimator for the linear regression model is shown to converge to the true parameter vector either with probability one or with probability zero under weak conditions on the dependent random variable and regressor variables. No additional conditions are placed on the errors. The dependent and regressor variables are assumed to be weakly dependent — in particular, to be strong mixing. The regressors may be fixed or random and must exhibit a certain degree of independent variability. No further assumptions are needed. The model considered allows the number of regressors to increase without bound as the sample size increases. The proof proceeds by extending Kolmogorov’s 0-1 law for independent random variables to strong mixing random variables.
A sufficient condition is given such that first-order autoregressive processes are stong mixing. The condition is specified in terms of the univariate distribution of the independent identically distributed innovation random variables. Normal, exponential, uniform, Cauchy, and many other continuous innovation random variables are shown to satisfy the condition. In addition, an example of a first-order autoregressive process which is not strong mixing is given. This process has Bernoulli (p) innovation random variables and any autoregressive parameter in (0, 1/2).
This paper considers the problem of robust estimation of location in a model with stationary strong mixing Gaussian parametric distributions. An estimator is found that is within epsilon of being asymptotically efficient at the Gaussian parametric distribution and is within epsilon of being optimally robust! For the robustness results a Huber-type minimax criterion is used, where minimaxing takes place over neighborhoods of the parametric Gaussian distributions. The neighborhood system considered includes distributions of strong mixing processes and allows for deviations from the normal univariate parametric distributions within a Hellinger metric neighborhood, as well as deviations from stationarity and from the Gaussian structure of independence.