We present a general theory of consistent estimation for possibly misspecified parametric models based on recent results of Domowitz and White. This theory extends the unification of Burguete, Gallant, and Souza by allowing for heterogeneous, time-dependent data and dynamic models. The theory is applied to yield consistency results for quasi-maximum-likelihood and method of moments estimators. Of particular interest is a new generalized rank condition for identifiability.

This paper describes some techniques for proving asymptotic normality of statistics defined by maximization of random criterion function. The techniques are based on a combination of recent results from the theory of empirical processes and a method of Huber for the study of maximum likelihood estimators under nonstandard conditions.

This paper is concerned with the estimation of a nonlinear regression function which is not assumed to belong to a prespecified parametric family of functions. An orthogonal series estimator is proposed, and Hilbert space methods are used in the derivation of its properties and the proof of several convergence theorems. One of the main objectives of the paper is to provide the theoretical basis for a practical stopping rule which can be used for determining the number of Fourier coefficients to be estimated from a given sample.

In this paper the testing and estimation problems are discussed in the case of serial correlation. Various models are particular cases of the general framework considered: the nonlinear simultaneous equations models, the probit models, the tobit models, the disequilibrium models, the frontier models, etc. In this context, it is shown that the score test can be written explicitly and that the statistic obtained is a generalization of that of Durbin and Watson; moreover, the maximum likelihood estimation procedure is shown to be robust with respect to serial correlation.

Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, . We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and . We obtain the first and second derivatives of λ(X) and the first derivative of u(X). Two alternative expressions for the first derivative of λ(X) are also presented.

Methods of maximum-likelihood search (eg., Hildreth-Lu) for single equations models with autocorrelated disturbances are not easily extended to the multi-equation systems encountered in econometric studies of consumer demand. The obstacle is the large dimension of the space of independent serial correlation coefficients to be partitioned and searched. This paper treats stochastic disturbance vectors in demand/expenditure systems as functions of random disturbances in consumer utility functions. It specifies marginal utilities to be products of a systematic function and a nonnegative random disturbance. Under this specification, three hypotheses that readily come to mind in the context of consumer theory drastically reduce the number of independent elements of serial covariance matrices. This in turn makes practicable the extension of maximum-likelihood search methods in the estimation of parameters of large demand systems.

Linear rational expectations models generally have a large number of solutions. It is thus important to describe them exhaustively in order to study their properties and subsequently estimate which solution best fits the data. In this paper, a global approach is suggested allowing a simultaneous treatment of all possible cases. The fundamental concepts are the revision processes appearing in the procedure of updating expectations. It isfound that the set of solutions is completely described by using a limitednumber of these processes. We show how the method may be applied to determine the set of stationary solutions admitting an infinite moving-average representation. We give a natural parametrization of this set and discuss the exact number of independent parameters.

In this paper we develop nonparametric estimators of the joint time series data generating process (DGP) of (xt, yt) at different t-values, of conditional DGP, of the conditional mean of xt given the past values of x and y, and, more generally, the conditional mean of (xt, yt) given their past values (vector autoregression). We establish, among other results, the central limit theorems for these estimators under far weaker mixing conditions than those used in Robinson [23], where only the xt series is considered. Uniform consistency and rate results for the consistencies of various estimators are also obtained. The results of the paper are useful in light of the fact that often the functional form of the dynamic regression is not known and also the assumption of the Gaussian process is not true.

This paper is concerned with derivation of a new efficient algorithm for computing the exact Gaussian likelihood for structural parameters in nonstationary higher-order continuous-time dynamic models and with its application in the estimation of these parameters. The algorithm completely avoids the computation of the covariance matrix of the observations and is applicable to a system of any order with mixed stock and flow data. It is used as the basis for an iterative procedure in which the structural parameters and the initial state vector are estimated alternately.

Starting from the conditional density of the instrumental variable (IV) estimator given the right-hand-side endogenous variables, we provide an alternative derivation of Phillips' result on the joint density of the IV estimator for the endogenous coefficients, and derive an expression for the marginal density of a linear combination of these coefficients. In addition, we extend Phillips' approximation to the joint density to 0(T−2,) and show how this result can be used to improve the approximation to the marginal density. Explicit formulae are given for the special case of no simultaneity, and the case of an equation with just three endogenous variables. The classical assumptions of independent normal reduced-form errors are employed throughout.

This paper reconsiders King's [12] locally optimal test procedure for first-order moving average disturbances in the linear regression model. It recommends two tests, one for problems involving positively correlated disturbances and one for negatively correlated disturbances. Both tests are most powerful invariant at a point in the alternative hypothesis parameter space that is determined by a function involving the sample size and the number of regressors. Selected bounds for the tests' significance points are tabulated and an empirical comparison of powers demonstrates the overall superiority of the new test for positively correlated moving average disturbances.

Forecasts from a univariate autoregressive model estimated by OLS are unbiased, irrespective of whether the model fitted has the correct order; this property only requires symmetry of the distribution of the innovations. In this paper, this result is generalized to vector autoregressions and a wide class of multivariate stochastic processes (which include Gaussian stationary multivariate stochastic processes) is described for which unbiasedness of predictions holds: specifically, if a vector autoregression of arbitrary finite order is fitted to a sample from any process in this class, the fitted model will produce unbiased forecasts, in the sense that the prediction errors have distributions symmetric about zero. Different numbers of lags may be used for each variable in each autoregression and variables may even be missing, without unbiasedness being affected. This property is exact in finite samples. Similarly, the residuals from the same autoregressions have distributions symmetric about zero.

In this paper we consider the situation in which ordinary least squares (OLS) is used to estimate an ARMA (1,1) model with one exogenous variable. Applying Edgeworth expansion techniques, we examine the misspecification errors and the approximate distributions of the OLS estimator. Extensive numerical studies were performed and selected results are shown graphically. In addition, a technical device is developed to calculate the Edgeworth coefficients.

In this paper, we shall derive the asymptotic expansion for the distribution of the likelihood ratio criterion for a Gaussian autoregressive moving average process under a sequence of local alternative hypotheses converging to the null hypothesis with rate of convergence where n is the sample size. Explicit algebraic formulae are presented for certain special cases, including the ARMA(1,1).

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. In the latter case, it either converges to a point not equal to the true parameter with probability one, or it diverges with probability one. These results are shown to hold 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 randomvariables to strong mixing random variables.

A size correction to adjust the critical values of asymptotic χ2 tests in a SUR context has been developed by Rothenberg and verified by Phillips. The adjustment centers the distribution of the observed test statistic so that it coincides more closely with its hypothetical distribution. The correction itself is an approximation to O(T−1), and while the adjustment should be superior to the asymptotic test (no adjustment) in small-sample situations, its performance can be expected to deteriorate as the sample size decreases. The Edgeworth corrections, using formulas provided by Phillips, are calculated for the Wald, likelihood ratio, and Lagrange multiplier tests in the context of symmetry testing in demand analysis using the Laitinen–Meisner simulated data set. The sample consists of 31 observations and the corrections were “adequate to useful” in the context of 5, 8, and 11 equations. However, in the extreme case of 14 equations the corrections typically only took up 60% of the adjustment required. The computational cost of calculating the Edgeworth correction for large SUR systems with a large number of restrictions also turns out to be quite heavy. The conclusion reached is that while Edgeworth corrections are not totally satisfactory, they are easy to include in a program and provide a useful critical value correction.

This paper is concerned with Cornish–Fisher corrections of some instrumental variables test statistics. The tests based on the corrected statistics have size with error of a smaller order of magnitude than the original tests. Symmetric Edgeworth-corrected confidence regions are also defined for the structural parameters. All these corrections are given as analytic formulas that require only limited information, so their implementation is a relatively easy task.

The estimation of K (K ≥ 3) location parameters is considered under quadratic loss when the coordinates of the best invariant estimators are spherically symmetrically distributed. Under these stochastic mechanisms traditional Stein estimators are evaluated for finite samples and shown to have a risk performance superior to some conventional rules.