JOURNAL OF APPLIED ECONOMETRICS, VOL. 10,477-479 (1995)

Estirnation, Inference and Specification Analysis, by H. WHITE. Econometric Society Monographs No. 22, Cambridge University Press, 1994, pp. x+380. ISBN 0-521-25280-6; Price E30 (hardback).

The econometrics profession has found it notoriously difficult to get to grips with the subject of this monograph: the interpretation of estimates and test results when the model may not be a correctly specified representation of the data generation process (DGP). It is hard even to agree on what it means to say a model is correct, given that a model simplifies reality by definition. David Hendry and his various co-workers have been pre-eminent (not to say largely alone) in making a serious assault on these problems. Halbert White’s research programme has clearly been inspired by their work, and his ambition has been to put what are often intuitive insights into the modelling problem onto a mathematically rigorous footing.

If the length of time it has taken to get the work into print is any measure, the task has been a testing one. A manuscript was rumoured as long ago as 1988. There has clearly been some revision in the interim, for chapter and theorem references which appear in his (1990) contribution to Modelling Economic Series (wherein the book is referenced as ‘in press’), do not appear applicable to the present version. Some ideas on model selection put forward in that earlier reference seem to have fallen by the wayside, or at any rate are not expanded here. But whatever the history, the text finally appearing has much to commend it, and I believe is substantially succeeds in its aim, while throwing into relief some major outstanding difficulties.

Estimation almost always involves optimising a criterion function, typically a quadratic form in the model residuals, or the explicit likelihood function for the assumed distribution of the data. As a rule, the procedure consistently estimates something, although that something may not be the theoretical constant of economic behaviour which the investigator has in mind. If it is possible to treat the optimand as a ‘quasi-likelihood’, one can interpret the optimisation estimator as consistent for the parameters which minimise the average Kullback-Leibler information criterion (KLIC), such that the quasi-likelihood lies as close as possible to the true density of the data. As such, the estimate is an artefact of the chosen specification as well as reflecting the characteristics of the DGP. The criterion for a specification to be ‘correct in its entirety’, according to White, is that there exists a model element (corresponding to a particular parameter value 0,) at which the quasi-likelihood equals the true joint density of the observations. But while this gives O,, a probabilistically valid interpretation, as a parameterization of the DGP, it does not guarantee a valid economic or behavioural interpretation. It is probabilistic validity alone which henceforth defines White’s notion of correct specification.

These notions can be successfully generalized to cases in which the quasi-likelihood is not a true joint density but a product of conditional densities, following the separation of variables into the categories of ‘jointly dependent’ and ‘explanatory’, the latter being unmodelled. As White emphasizes, this categorization may be entirely model based and independent of the DGP, but correct specification is a well-defined concept in terms of the parameterization of the conditional densities. The analysis can also be extended to models in which the only aspect of the distribution represented is the conditional expectation of the dependent variables, by embedding the specification in a quasi-likelihood belonging to the linear exponential family. This allows standard cases such as least squares regression to be covered by the theory. The analysis is further extended from continuous distributions to arbitrary distributions by characterizing the likelihood function explicitly as the Radon-Nikodym derivative of an arbitrary probability measure with respect to a suitable dominating measure, so that, in particular, mixed continuous-discrete cases such as the Tobit model can be accommodated. For some readers, this feature of the analysis will represent a major technical

0 1995 by John Wiley & Sons, Ltd.

478 BOOK REVIEW

overhead and will give some difficulty. While the gain in generality is worth while, the author could perhaps have been freer in giving familiar examples, to make the mathematics less forbidding.

In his treatment of specification testing, White shows that the Newey-Tauchen in-tests provide a unifying principle embracing most varieties of specification test, including Lagrange Multiplier tests. Hausman tests, Cox tests, and encompassing tests. But apart from pointing out the possibility of using the in-test framework to set up joint tests of specification, he does not have a great deal to say about model-building strategies and, curiously, does not enlarge on the ideas in White (1990) in any detail. He does have some interesting observations on the various concepts of exogeneity, and suggests an alternative to the well-known treatment of Engle, Hendry, and Richard (1983). These authors implicitly assume a correspondence between the estimated model and the DGP, notwithstanding that weak exogeneity is a property of the parameterization of interest, rather than of data. White finds it helpful to distinguish the notions of explanatory variables (as noted above), informational exogeneity, and conditional invariance. The first of these concepts relate entirely to the specified model, not to the DGP, although when the specification is correct something is naturally implied about the DGP, that a particular conditioning is relevant to probabilistically meaningful parameters. Informational exogeneity, on the other hand, is a property of the DGP in relation to a particular correctly specified conditional model. By definition, omitting to model informationally exogenous variables imposes no cost in terms of asymptotic efficiency of the estimates of the conditional submodel parameters. The most important difference from weak exogeneity, with which it will coincide in many practical cases, is that no explicit parameterization of the marginal model enters the definition. Conditional invariance is White’s proposed alternative to Engle et al’s super-exogeneity concept, and is wholly a property of the DGP, requiring that the conditional distribution be invariant to changes in the marginal processes. There will doubtless be an ongoing debate on the merits of these alternatives, and suffice it to say here that I find White’s discussion convincing and valuable.

However, although the book develops an analytical framework of undoubted elegance, I am not convinced by the attempt to define a single unifying estimation principle. The idea that all estimation methods should be viewed as analogues of maximum likelihood is a tempting and popular one, but the truth is that the method of moments, to cite just one example, does not fit happily into this framework. It is, of course, an optimization estimator (OE), but except with the eye of faith, it is very difficult to view the optimand as in any sense a quasi-likelihood. For that matter, models of the conditional mean, even in the least squares context, have an essential affinity with the method of moments, and to endow them with a spurious likelihood interpretation seems to me to be stretching a point too far. The motivation for these methods is not that the investigator is imposing simple assumptions about the complete density, but that he or she is eschewing any assumption about it at all. The asymptotic theory developed here is, almost in its entirety, a theory for OEs, and the QMLE derives its mathematical properties solely from its characterization as such. It is only the information matrix equality which is a special property of the MLE.

So why not develop a theory of misspecification based on OEs? The chief cost would seem to be that one would lose the information-based interpretation of the estimators, in terms of the KLIC; but how much would this really matter? That interpretation may still be available, as one among several, and there is still a full-dress asymptotic theory, of which the most important feature remains the full efficiency of the correctly specified MLE. The interpretation of the estimator as consistent for the optimizer of the expected optimand provides, in many cases, a satisfactory heuristic interpretation of the results under misspecification. By contrast, the QMLE framework gives birth to some odd justifications. For example (page 67). a result due to Gourieroux, Monfort, and Trognon (1984) shows that constructing the quasi-likelihood function appropriate to the linear exponential family yields a consistent QMLE for the model of conditional expectations, regardless of the true distribution of the data. What useful insight does this result really convey, beyond the fact that the assumption ensures that the O E problem has been correctly formulated? This is just one of several points at which the QMLE tail seems to be wagging the estimation dog.

Such reservations aside, this book is to be welcomed as setting new standards of rigour and generality for the econometric modelling literature. The style is austere by the standards of this literature, and more discussion, and especially more examples, would often have been welcome. Nonetheless, it can be highly recommended as an original and stimulating contribution to the debate.

JAMES DAVIDSON Departtnerit of Economics

Utiiversity of Wales Llatidiriatn Building

Petiglais, Aherysrwytfi, Dyfed SY23 3DB

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REFERENCES

Engle, R. F., D. F. Hendry and J. F. Richard (1983), ‘Exogeneity’, Econometricu, 51, 277-304. Gourieroux, C., A. Monfort and A. Trogon (1984), ‘Pseudo-maximum likelihood methods: theory’,

White, H. (1990). ‘A consistent model selection procedure based on In-testing’, Chapter 16 in Granger, C. Econornetrica, 52, 681-700.

W. J. (ed,), ModeZZirzg Economic Series, Oxford University Press, Oxford.