Two-step estimators, where the first step is the predicted value from a nonparametric regression, are useful in many contexts. Examples include a non-parametric residual variance, probit with nonparametric generated regressors, efficient GMM estimation with randomly missing data, heteroskedasticity corrected least squares, semiparametric regression, and efficient nonlinear instrumental variables estimators. The purpose of this paper is the development of consistency and asymptotic normality results when the first step is a series estimator. The paper presents the form of a correction term for the first step on the second-step asymptotic variance and gives a consistent variance estimator. Data-dependent numbers of terms are allowed for, and the regressor distribution can be discrete, continuous, or a mixture of the two. Results for several new estimators are given.

This paper investigates the sampling behavior of the quasi-maximum likelihood estimator of the Gaussian GARCH(1,1) model. The rescaled variable (the ratio of the disturbance to the conditional standard deviation) is not required to be Gaussian nor independent over time, in contrast to the current literature. The GARCH process may be integrated (α + β = 1), or even mildly explosive (α + β > 1). A bounded conditional fourth moment of the rescaled variable is sufficient for the results. Consistent estimation and asymptotic normality are demonstrated, as well as consistent estimation of the asymptotic covariance matrix.

By appropriately partitioning the joint hypothesis of weak exogeneity and the maintained overidentifying restrictions in the linear dynamic simultaneous equations model and showing that the component subhypotheses are separable, asymptotically optimal tests for the weak exogeneity hypothesis may be constructed using limited information statistics. A necessary and sufficient condition for the separability of parametric hypotheses of the mixed implicit function and constraint equation type is derived which generalizes conditions previously obtained in the literature. Consequently, limited and full information procedures for testing the weak exogeneity hypothesis are asymptotically equivalent. The impact of these results for testing strong exogeneity in the linear dynamic simultaneous equations model is also explored.

In this paper, a consistent model specification test is proposed. Some consistent model specification tests have been discussed in econometrics literature. Those tests are consistent by randomization, display a discontinuity in sample size, or have an asymptotic distribution that depends on the data-generating process and on the model, whereas our test does not have one of those disadvantages. Our test can be viewed upon as a conditional moment test as proposed by Newey but instead of a fixed number of conditional moments, an asymptotically infinite number of moment conditions is employed. The use of an asymptotically infinite number of conditional moments will make it possible to obtain a consistent test. Computation of the test statistic is particularly simple, since in finite samples our statistic is equivalent to a chi-square conditional moment test of a finite number of conditional moments.

This paper proposes a residual-based test of the null of cointegration using a structural single equation model. It is shown that the limiting distribution of the test statistic for cointegration can be made free of nuisance parameters when the cointegrating relation is efficiently estimated. The limiting distributions are given in terms of a mixture of a Brownian bridge and vector Brownian motion. It is also shown that this test is consistent. Critical values are given for standard, demeaned, and detrended cases. Combining results from our test for cointegration with results from the Phillips-Ouliaris test for no cointegration, we find that there is evidence of cointegration between real consumption and real disposable income over the postwar period.

When values of regressors are symmetrically disposed, many M-estimators in a wide class of models have a reflection property, namely, that as the signs of the coefficients on regressors are reversed, their estimators' sampling distribution is reflected about the origin. When the coefficients are zero, sign reversal can have no effect. So in this case, the sampling distribution of regression coefficient estimators is symmetric about zero, the estimators are median unbiased and, when moments exist, the estimators are exactly uncorrelated with estimators of other parameters. The result is unusual in that it does not require response variates to have symmetric conditional distributions. It demonstrates the potential importance of covariate design in determining the distributions of estimators, and it is useful in designing and interpreting Monte Carlo experiments. The result is illustrated by a Monte Carlo experiment in which maximum likelihood and symmetrically censored least-squares estimators are calculated for small samples from a censored normal linear regression, Tobit, model.

In this paper, we derive an instrumental variables estimator for the generalized Hausman and Taylor [4] model with a two-way error component. We also extend the work of Breusch, Mizon, and Schmidt [2] to this model and derive two additional instrumental variables estimators. Consistency and asymptotic normality of the parameter estimator is demonstrated, allowing both the number of time periods as well as the number of individuals to increase, their ratio bounded. Consistent estimates of the variance components are discussed.

This paper considers some finite sample properties of the partially restricted reduced form estimators in a general (n + 1) endogenous variable model. In particular, the characteristic functions, density functions, and moments are examined for both the vector of estimators and a linear combination. The approach utilizes both invariant polynomials of matrix argument (see Chikuse and Davis [4] and Davis [6]) and fractional calculus techniques (see Phillips [20,25,26]).

This paper explores the robustness of minimum distance (GMM) estimators focusing particularly on the effect of intermediate covariance matrix estimation on final estimator performance. Asymptotic expansions to order Op(n−3/2) are employed to construct O(n−2) expansions for the variance of estimators constructed from preliminary least-squares and general M-estimators. In the former case, there is a rather curious robustifying effect due to estimation of the Eicker-White covariance matrix for error distributions with sufficiently large kurtosis.

This paper treats the theory of identification presented in Haavelmo's classic work, The Probability Approach in Econometrics. This was the first identification theory for stochastic models to be developed in econometrics. The paper presents a detailed commentary on Haavelmo's analysis. It also examines the development of Haavelmo's theory from Frisch's analysis of multicollinearity and also the relationship between Haavelmo's analysis and later work on identification.