We provide both a theoretical and empirical analysis of the relation between administrative and survey data. By distinguishing between different sources of deviations between survey and administrative data we are able to reproduce several stylized facts. We illustrate the implications of different error sources for estimation in (simple) econometric models and find potentially very substantial biases. This article shows the sensitivity of some findings in the literature for the assumption that administrative data represent the truth. In particular, the common finding of substantial mean reversion in survey data largely goes away once we allow for a richer error structure.

[A class of decompositions is derived for the variance-covariance matrix Ω of a generalized error components model, introduced in [18 and 19]. The spectral decomposition of Ω is a member of this class. For estimation purposes certain other members of the class are preferred, especially those that allow for simplifying transformations of the model not depending on unknown parameters. The transformations suggest simple and asymptotically efficient estimators of both the parameters in Ω and the parameters in the systematic part of the model.]

In this paper the authors estimate a model of the allocation of time within the household using data that allows them to d istinguish between a large number of time uses. The model is explicitly derived within a utility maximization framework and can be estimated by relatively simpl e two-step estimation procedures. The model providesa natural framework to test implications of the more restrictive models of Reuben Gronau_(1977, 1980) and John Graham and Carole Green_(1984). Copyright 1987 by University of Chicago Press.

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

T HE hypothesis that maximization underlies human behaviour is perhaps the most widely accepted paradigm among economists. Particularly in the study of consumer behaviour, numerous models have been built upon the hypothesis of maximization. Reviews of these models can inter alia be found in Houthakker (1961), Brown and Deaton (1972) and Barten (1977). The testing of the maximization hypothesis (HM) in real life situations appears to be a complicated affair. The main problem is that HM can only be tested conditional upon other assumptions. An individual's function' is commonly measured via the individual's observed behaviour. We call that indirect measurement. But the relationship between an individual' s and his behaviour is based on HM itself. Hence, having measured functions via HM it becomes difficult to use the measured to test HM. Therefore, testing HM mostly reduces to testing certain restrictions which have to be satisfied by parameters in a system of demand equations. However, testing these restrictions is not without problems, as testing a certain restriction has to take place conditional upon the validity of other restrictions.2 As far as testing has been carried out, results are not very encouraging (cf. Barten, 1977; Wales and Woodland, 1976). But, since many additional assumptions are involved,3 no firm conclusions can be drawn from these negative outcomes. Given these problems, several paths are open to the student of consumer behaviour. First he may want to dispense with the concept altogether and only hypothesize certain consistency properties of individual choices. This approach was taken by Samuelson (1938). If, however, the assumptions on individual preferences are made sufficiently strong, especially if one adopts the strong axiom of revealed preference, their implications for behaviour are equivalent to the restrictions derived from HM (cf. Houthakker, 1950; Stigum, 1973). Hence, testing the restrictions implied by the strong axiom of revealed preference is equivalent to testing HM. Empirical work in this area (cf. Koo, 1963; Mossin 1972) suggests that for everyday commodities (mainly food) most purchases of individual families are not inconsistent with the strong axiom of revealed preference theory. However, in many cases purchases are such that neither consistency nor inconsistency can be assessed (cf. Koo, 1963). Koo (1974) states that '6with few exceptions, almost all families made at least some inconsistent choices (p. 174). He finds that inconsistencies do not arise very often if purchases are in the neighbourhood of past experience. For less routine-like purchases inconsistencies are more likely to occur.4 Parenthetically, it may be mentioned that aggregate demand functions have a tendency to be in agreement with the strong axiom of reReceived for publication November 15, 1977. Revision accepted for publication November 16, 1978. * University of Southern California, Leyden University and Leyden University, respectively. The research reported in the paper is supported by grants from the Netherlands Organisation for the Advancement of Pure Research (ZWO) and the Dutch Ministry of Cultural Affairs, Recreation and Social Welfare. At the time of writing Arie Kapteyn was successively a fellow of the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS) and at Leyden University. The empirical results are based on a survey, designed by B. M. S. van Praag, of members of the Dutch Consumer Union. Their willingness to make available the data for the study is gratefully acknowledged. We thank Floor van Herwaarden, Bernard van Praag, Roberto Wessels, and the referees for their helpful suggestions with respect to an earlier version. The research reported is part of the Leyden Income Evaluation Project. I Throughout the paper the term utility function will be used to denote the general concept, whereas the term welfare function will be used for the more narrowly defined concept introduced in section II. 2 In any case one has to specify functional forms for the demand equations. Even when using flexible forms (e.g., Christensen, Jorgenson, and Lau, 1975) the specification may be expected to affect the result. 3 For example, it is usually assumed that individuals have identical functions, or that an individual's parameters are not affected by consumption patterns of other individuals, that functions do not shift over time, etc. Moreover, estimation is often based on aggregate data. 4 The empirical investigations in the present paper are concerned with durables. Extrapolating Koo's findings we would expect HM to be violated relatively often for these expenditure categories, since durables are bought infrequently.

Arie Kapteyn, Peter Kooreman, Arthur van Soest, Quantity Rationing and Concavity in a Flexible Household Labor Supply Model, The Review of Economics and Statistics, Vol. 72, No. 1 (Feb., 1990), pp. 55-62