Consistent Sets of Estimates for Regressions with Correlated or Uncorrelated Measurement Errors in Arbitrary Subsets of all Variables

Abstract

OVER THE LAST DECADE the problem of measurement errors in the independent variables of a regression equation has attracted renewed interest among econometricians. In the fifties and sixties, the problem was considered to be more or less hopeless due to its inherent underidentification (e.g., Theil (1971)). Apart from instrumental variables, the most frequently cited textbook solution was Wald's method of grouping (Wald (1940)). Recent insight into the properties of the method of grouping can be interpreted as making this method worthless in most practical cases (Pakes (1982)). Since about 1970, new approaches to the problem have been explored, basically along three lines, viz. embedding the error-ridden equation into a set of multiple equations (e.g., Zellner (1970), Goldberger (1972)), into a set of simultaneous equations (e.g., Hsiao (1976), Geraci (1976)), and using the dynamics of the equation, if present (e.g., Maravall and Aigner (1977)). In view of the underidentification of the basic model, it is clear that all these methods invoke additional information of some kind. If this information takes the form of exact or stochastic knowledge of certain parameters in the model, the construction of consistent estimators is fairly straightforward (e.g. Fuller (1980), Kapteyn and Wansbeek (1984)). For an overview of the state of the art, see Aigner et al. (1984). An approach somewhat orthogonal to the ones described above has been to take the model as it is and to use prior ideas about the size of the measurement errors to diagnose how serious the probem is. Examples are Blomqvist (1972), Hodges and Moore (1972), and Davies and Hutton (1975). Leamer (1983) starts from the opposite direction by asking how serious the measurement error problem has to be in order to render the data useless for inference, that is to say, when measurement error is large enough to make it impossible to put bounds on regression parameters. In an empirical example, he shows that even very small measurement errors in some explanatory variables would open up the possibility of perfectly collinear explanatory variables and hence make the data useless for statistical inference (at least without additional prior information). The most systematic analysis of the information loss caused by measurement error is due to Klepper and Leamer (1984). They start out by invoking a minimal amount of prior information and then ask the question under what conditions it is still possible to make some inferences regarding the vector of unknown regression parameters p. In the special case where the measurement errors are assumed uncorrelated and the k + 1 estimates of ,3, obtained by regressing each of the k +1 variables involved (i.e. the one dependent variable and the k independent variables) on the remaining k variables, are all in the same orthant, one can bound the ML estimates of p. In that case, the convex hull of the k + 1 regressions contains all possible ML estimates and any point in the hull is a possible ML estimate. If the k + 1 regressions are not all in the same orthant then the set of ML estimates is unbounded. In that case Klepper and Leamer (1984) introduce extra prior information which allows them to bound the set of maximum likelihood estimates. The prior information comes in two forms. Firstly, a researcher is supposed to be able to specify a maximum value of R2 if all exogenous variables were measured accurately. It is shown that if this maximum is low enough, one can again bound the set of ML estimates by a convex hull. Secondly, if