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Maximum Likelihood Estimation of Misspecified Dynamic Models

In this paper we present a number of results which describe the behavior of the maximum likelihood estimator of the parameters of a dynamic model which is incorrectly or incompletely specified. We provide conditions which ensure the existence of the Quasi-Maximum Likelihood Estimator (QMLE) and its consistency for the parameters of an approximation to the unknown true probability density which has optimal information theoretic properties. The ability of the QMLE to consistently estimate certain parameters of interest despite misspecification is investigated. We give conditions ensuring the asymptotic normality of the QMLE together with conditions under which its asymptotic covariance matrix may be consistently estimated. Two model specification tests are briefly discussed.
Halbert White

The Geometry of Model Selection in Regression

This paper deals with the selection of the linear model, i.e. the design matrix X, In linear regression model Ey=Xß. Special cases of this model selection problem are: the selection of predictor variables, selection of transformations, and selection of interactions to be included in the model. The combined process of model selection and parameter estimation is sketched in the classical Neyman-Pearson-Wald framework of mathematical statistics.
Albert Verbeek

Misspecification and the Choice of Estimators, A Heuristic Approach

We study a classical simultaneous system which is possibly misspecified in the sense that too many variables may have been deleted from the structural equations. Suppose the system is to be used for predictive purposes. We compare unrestricted least squares, full-information and pre-test estimators based on the outcome of a chi-square test-statistic. A plea is made to take two times the degrees of freedom of the hypothesized model as critical value for the chi-square test.
Theo K. Dijkstra

Discrete Normal Linear Regression Models

In this paper we continue our study of the Pearsonian approach to discrete multivariate analysis, in which structural properties of the multivariate normal distribution are combined with the essential discreteness of the data into a single comprehensive model. In an earlier publication we studied these ‘block-multinormal’ methods for covariance models. Here we propose a similar approach for the regression model with fixed regressors. Likelihood methods are derived and applied to some examples. We review the related literature and point out some interesting possible generalizations. The effect of continuous misspecification of a discrete model is studied in some detail. Relationships with the optimal scaling approach to multivariate analysis are also investigated.
Jan de Leeuw

Specification in Simultaneous Linear Equations Models: The Relation Between A Priori Specifications and Resulting Estimators

The estimation of coefficients of a linear model is discussed, where we do not assume that the random vector X satisfies the model, but that the model is only a model of best fit. We consider both the situation without additional constraints on the coefficients and the situation with differentiable constraints. In both cases, the vector of parameters is a differentiable function of a covariance matrix, making the δ-method applicable. It is shown that the asymptotic distribution of the estimators is normal. Closed formulas for their asymptotic covariance matrices are derived.
Bernard M. S. van Praag, Jan T. A. Koster

Measurement Error and Endogeneity in Regression: Bounds for ML and 2SLS Estimates

We consider the single equation errors in variables model and assume that a researcher is willing to specify upper bounds on the possible measurement errors in the exogenous variables. We prove that as a result the set of possible ML estimates is bounded by an ellipsoid. The result is generalized to IV estimation of a structural equation of a simultaneous system, which has only endogenous variables on the right hand side.
Paul Bekker, Arie Kapteyn, Tom Wansbeek

Testing Parameter Constancy of Linear Regressions

In this paper we propose a new test for parameter constancy of linear regressions. The alternative hypothesis is that the data set can be split up in a finite number of subsets such that at least two of the OLS estimators corresponding to these subsets have different probability limits. Conditions are set forth such that under the null hypothesis the test statistic involved is asymptotically χ2 distributed, whereas under the alternative hypothesis the test statistic converges in probability to infinity. These conditions allow for stochastic regressors and time series applications.
Herman J. Bierens

Prediction Performance and the Number of Variables in Multivariate Linear Regression

The multivariate linear regression model is considered for prediction purposes. It will be assumed that there are in principle infinitelymany regressors ordered according to decreasing importance. The conditional variance of the regressand given the first p regressors will be denoted by ω p 2 . A natural measure of the performance of regression-predictors is mean squared prediction error MSEP(n,p), n being the sample size and p the number of variables. If {ω p 2 } is known, then p will be chosen such that MSEP(n,p) is minimized. Typically, this may give an optimal p* much smaller than n; in other words it pays to delete variables when coefficients have to be estimated. In practice the quantity MSEP(n,p) is unknown but it can be unbiasedly estimated by msep(n,p), the so-called Sp-criterion. It is frequently suggested to choose the number of variables p̂ such that msep(n,p) is minimized. This rule is asymptotically optimal in the sense that msep(n, p̂)/MSEP(n, p*) → 1 in probability as n → ∞. It will be shown that there are a lot of other selection-of-variables procedures, which share this property. So, asymptotic optimality is by itself not very compelling, and minimization of the Sp-criterion needs more justification.
Ton Steerneman
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