The bootstrap of the maximum likelihood estimator of the mean of a sample of iid normal random variables with mean µ and variance one is not asymptotically correct to first order when the mean is restricted to be nonnegative. The problem occurs when the true value of the mean µ equals zero. This counterexample to the bootstrap generalizes to a wide variety of estimation problems in which the true parameter may be on the boundary of the parameter space. We provide some alternatives to the bootstrap that are asymptotically correct to first order.
We consider two types of bootstrap percentile confidence intervals in the above example. We find that they both have asymptotic coverage probability that exceeds the nominal asymptotic level when the true value of the mean it equals zero.
This paper establishes the asymptotic distribution of extremum estimators when the true parameter lies on the boundary of the parameter space. The boundary may be linear, curved, and/or kinked. 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 explicitly in the paper are: (1) quasi-ML estimation of a random coefficients regression model with some coefficient variances equal to zero, (2) LS estimation of a regression model with nonlinear equality and/or inequality restrictions on the parameters and iid regressors, (3) LS estimation of an augmented Dickey-Fuller Fuller regression with unit root and time trend parameters on the boundary of the parameter space, (4) method of simulated moments estimation of a multinomial discrete response model with some random coefficient variances equal to zero, some random effect variances equal to zero, or some measurement error variances equal to zero, (5) quasi-ML estimation of a GARCH(1,q*) or IGARCH(1,q*) model with some GARCH MA parameters equal to zero, (6) semiparametric LS estimation of a partially linear regression model with nonlinear equality and/or inequality restrictions on the parameters, and (7) LS estimation of a regression model with nonlinear equality and/or inequality restrictions on the parameters and integrated regressors.
This paper considers a generalized method of moments (GMM) estimation problem in which one has a vector of moment conditions, some of which are correct and some incorrect. The paper introduces several procedures for consistently selecting the correct moment conditions. The procedures also can consistently determine whether there is a sufficient number of correct moment conditions to identify the unknown parameters of interest.
The paper specifies moment selection criteria that are GMM analogues of the widely used BIC and AIC model selection criteria. (The latter is not consistent.) The paper also considers downward and upward testing procedures.
All of the moment selection procedures discussed in the paper are based on the minimized values of the GMM criterion function for different vectors of moment conditions. The procedures are applicable in time series and cross-sectional contexts.
Application of the results of the paper to instrumental variables estimation problems yields consistent procedures for selecting instrumental variables.
Keywords: Akaike information criterion, Bayesian information criterion, consistent selection procedure, downward testing procedure, generalized method of moments estimator, instrumental variables estimator, model selection, moment selection, test of over-identifying restrictions, upward testing procedure
This paper considers the problem of choosing the number of bootstrap repetitions B for bootstrap standard errors, confidence intervals, and tests. For each of these problems, the paper provides a three-step method for choosing B to achieve a desired level of accuracy. Accuracy is measured by the percentage deviation of the bootstrap standard error estimate, confidence interval endpoint(s), test’s critical value, or test’s p-value based on B bootstrap simulations from the corresponding ideal bootstrap quantities for which B = ∞. Monte Carlo simulations show that the proposed methods work quite well.
The results apply quite generally to parametric, semiparametric, and nonparametric models with independent and dependent data. The results apply to the standard nonparametric iid bootstrap, moving block bootstraps for time series data, parametric and semiparametric bootstraps, and bootstraps for regression models based on bootstrapping residuals
This paper considers tests for seasonal and non-seasonal serial correlation in time series and in the errors of regression models. The problem of testing for white noise against multiplicative seasonal ARMA(l,l)-ARMA(l,l) alternatives is investigated. This testing problem is non-standard due to nuisance parameters that appear under the alternative but not under the null hypothesis. The likelihood ratio (LR), sup Lagrange multiplier (LM), and exponential average LM and LR tests are considered and are shown to be asymptotically admissible for multiplicative seasonal ARMA(l,l)-ARMA(l,l) alternatives. In addition, they are shown to be consistent against all (weakly stationary strong mixing) non-white noise alternatives. Simulation results compare the tests to several tests in the literature. The exponential average test, Exp-LRinfinity, is found to be the best test overall. It performs substantially better than the Box-Pierce, Durbin-Watson, and Wallis tests.
To obtain consistency and asymptotic normality, a generalized method of moments (GMM) estimator typically is defined to be an approximate global minimizer of a GMM criterion function. To compute such an estimator, however, can be problematic because of the difficulty of global optimization. In consequence, practitioners usually ignore the problem and take the GMM estimator to be the result of a local optimization algorithm. This yields an estimator that is not necessarily consistent and asymptotically normal. The use of a local optimization algorithm also can run into the problem of instability due to flats or ridges in the criterion function, which makes it difficult to know when to stop the algorithm.
To alleviate these problems of global and local optimization, we propose a stopping-rule (SR) procedure for computing GMM estimators. The SR procedure eliminates the need for global search with high probability. And, it provides an explicit SR for problems of stability that may arise with local optimization problems.
This paper provides a consistent and asymptotically normal estimator for the intercept of a semiparametrically estimated sample selection model. The estimator uses a decreasingly small fraction of all observations as the sample size goes to infinity, as in Heckman (1990). In the semiparametrics literature, estimation of the intercept typically has been subsumed in the nonparametric sample selection bias correction term. The estimation of the intercept, however, is important from an economic perspective. For instance, it permits one to determine the “wage gap” between unionized and nonunionized workers, decompose the wage differential between different socioeconomic groups (e.g., male-female and black-white), and evaluate the net benefits of a social program.
This paper introduces a conditional Kolmogorov test of model specification for parametric models with covariates (regressors). The test is an extension of the Kolmogorov test of goodness-of-fit for distribution functions. The test is shown to have power against 1/root{n}-local alternatives and all fixed alternatives to the null hypothesis. A parametric bootstrap procedure is used to obtain critical values for the test.
This paper is concerned with tests for serial correlation in time series and in the errors of regression models. In particular, the nonstandard problem of testing for white noise against ARMA(1,1) alternatives is considered. Sup Lagrange multiplier (LM) and exponential average LM tests are introduced and are shown to be asymptotically admissible for ARMA(1,1) alternatives. In addition, they are shown to be consistent against all (weakly stationary strong mixing) non-white noise alternatives. Simulation results compare the tests to several tests in the literature. These results show that the Exp-LMinfinity test has very good all-around power.
This paper provides an introduction to the use of empirical process methods in econometrics. These methods can be used to establish the large sample properties of econometric estimators and test statistics. In the first part of the paper, key terminology and results are introduced and discussed heuristically. Applications in the econometrics literature are briefly reviewed. A select set of three classes of applications is discussed in more detail.
The second part of the paper shows how one can verify a key property called stochastic equicontinuity. The paper takes several stochastic equicontinuity results from the probability literature, which rely on entropy conditions of one sort or another, and provides primitive sufficient conditions under which the entropy conditions hold. This yields stochastic equicontinuity results that are readily applicable in a variety of contexts. Examples are provided.
This paper considers an alternative asymptotic framework to standard sequential asymptotics for nonlinear models with deterministically trending variables. The asymptotic distributions of generalized method of moments estimators and corresponding test statistics are derived using this framework. The asymptotic distributions are shown to be the same with deterministically trending variables as with non-trending variables. That is, the distributions are normal and chi-squared respectively. The asymptotic covariance matrices of the estimators, however, are found to depend on the form of the trends. These findings provide a justification for the use of standard asymptotic approximations in nonlinear models even when the variables have deterministic trends.
Keywords: Asymptotics, Deterministic trend, Generalized method of moments estimator, Hypothesis test, Nonlinear econometric model, Time Trend
This paper establishes the asymptotic admissibility of the likelihood ratio (LR) test for a general class of testing problems in which a nuisance parameter is present only under the alternative hypothesis. The paper also establishes the finite sample admissibility of the LR test for testing problems of this sort that arise in Gaussian linear regression models with known variance.
This paper considers hypothesis tests for nonlinear econometric models when the parameter space is restricted under the alternative hypothesis. Multivariate one-sided tests are a leading example. Optimal tests, called directed tests, are derived using a weighted average power criterion. The likelihood ratio test is shown to be admissible and to maximize power against alternatives that are arbitrarily distant from the null hypothesis. Exact results are established first for Gaussian linear regression models with known variance. Asymptotic analogues are then established for dynamic nonlinear models.
Simulation is used to compare the tests discussed in the paper. The D–W∞ directed test is found to perform best in an overall sense for multivariate one-sided alternatives. The he D–W∞ and LR tests are found to perform likewise for mixed one- and two-sided alternatives.
This paper establishes a correspondence in large samples between classical hypothesis tests and Bayesian posterior odds tests for models without trends. More specifically, tests of point null hypotheses and one- or two-sided alternatives are considered (where nuisance parameters may be present under both hypotheses). It is shown that for certain priors the Bayesian posterior odds test is equivalent in large samples to classical Wald, Lagrange multiplier, and likelihood ratio tests for some significance level and vice versa.
This paper introduces approximately median-unbiased estimators for univariate AR(p) models with time trends. Confidence intervals also are considered. The methods are applied to the Nelson–Plosser macroeconomic data series, the extended Nelson–Plosser macroeconomic data series, and some annual stock dividend and price series. The results show that most of the series exhibit substantially greater persistence than least squares estimates and some Bayesian estimates suggest. For example, for the extended Nelson–Plosser data set, eight of the fourteen series are estimated to have a unit root, while six are estimated to be trend stationary. In contrast, the least squares estimates indicate trend stationarity for all of the series.