This paper is the text of the 1994 Tjalling Koopmans Lecture of the Cowles Foundation. The aim of this lecture was to survey the roles of misspecified models in econometrics. Through 10 stories we show how the misspecifipation problems can be dealt with and how misspecified models can play a positive role in inference processes.

A model M is said to encompass another model N if the former can explain the results obtained by the latter. In this paper, we propose a general notion of encompassing that covers both classical and Bayesian viewpoints and essentially represents a concept of sufficiency among models. We introduce the parent notion of specificity that aims at measuring lack of encompassing. Tests for encompassing are discussed and the test statistics are compared to Bayesian posterior odds. Operational approximations are offered to cover situations where exact solutions cannot be obtained.

We describe an intuitive, simple, and systematic approach to generating moment conditions for generalized method of moments (GMM) estimation of the parameters of a structural model. The idea is to use the score of a density that has an analytic expression to define the GMM criterion. The auxiliary model that generates the score should closely approximate the distribution' of the observed data but is not required to nest it. If the auxiliary model nests the structural model then the estimator is as efficient as maximum likelihood. The estimator is advantageous when expectations under a structural model can be computed by simulation, by quadrature, or by analytic expressions but the likelihood cannot be computed easily.

Let (X1) be a discrete multivariate Gaussian autoregressive process of order 1. The paper derives the exact finite-sample joint moment generating function (m.g.f.) of the three quadratic forms constituting the sufficient statistic of the process. The formula is then specialized to some cases of interest, including the m.g.f. of functional of multivariate Ornstein-Uhlenbeck processes that arise asymptotically from more general (X1) processes as well.

We derive the limit distribution of the number of crossings of a level by a random walk with continuously distributed increments, using a Brownian motion local time approximation. This complements the well-known result for the random walk on the integers. Use of the frequency of level crossings to test for a unit root is examined.

Said and Dickey (1984, Biometrika 71, 599–608) and Phillips and Perron (1988, Biometrika 75, 335–346) have derived unit root tests that have asymptotic distributions free of nuisance parameters under very general maintained models. Under models as general as those assumed by these authors, the size of the unit root test procedures will converge to one, not the size under the asymptotic distribution. Solving this problem requires restricting attention to a model that is small, in a topological sense, relative to the original. Sufficient conditions for solving the asymptotic size problem yield some suggestions for improving finite-sample size performance of standard tests.

This paper establishes conditions for the nonparametric identifiability of the mixed proportional hazards model with time-varying coefficients. Unlike the mixed proportional hazards model, a regressor with two distinct values is not sufficient to identify this model. An unbounded regressor, however, is sufficient for identification.

In “Modeling Stock Prices without Knowing How to Induce Stationarity” (1994, Econometric Theory 10, 701–719), we used posterior-odds calculations to evaluate restrictions imposed by a present-value model of stock prices across the equations of a VAR representation of stock prices and dividends. The results we reported are tainted by the omission of two factors: the Jacobians induced by the mapping of our priors over VAR parameters β into the restricted sample spaces relevant under hypotheses H2-H4 (hence, tainting our calculations of p(Hi|y,X) in (22) for i = 2–4), and an integrating constant needed in calculating the unrestricted probability p(Hi|y,X) in (22). Table 1 reports our revised calculations, which differ substantively from those reported previously.

The editor is grateful to Francisco Goerlich for pointing out that there was a misprint in the criterion defined in Problem 94.1.4. It should read as

instead of

as it was originally published.