Indirect inference is, I think, a really important methodological advance, one which opens the door to doing a lot of useful statistics on models of complex systems. However, Gouriéroux and Monfort write for a reader who is very familiar with theoretical statistics, in particular with concepts such as the likelihood and maximum likelihood estimation, Fisher information, the score, consistency and efficiency, and so forth, though no measure theory. (Say, Wasserman's All of Statistics.) No special knowledge of econometrics is really needed, though the last three chapters may seem under-motivated to those not committed to standard econometric models. All this being the case, in the rest of my review I will presume the reader has at least some recollection of the basic ideas of probability, expectation, etc.
Let me start by giving some concrete examples of what I mean by "what the
model predicts for different parameters". Typically, predictions will depend
not just on the parameters,
In the "generalized method of moments", one picks a number of functions of
the data y and the exogenous variables, say
The method of least squares works similarly. We assume that
Finally, the method of maximum likelihood asks "how often should we expect
to see data like this, under this model?", and tries to maximize that probability:
Originally, all of these methods of estimation were practical only if one could derive a simple formula for the best-fitting parameter values as a function of the data. Latter, with the rise of numerical optimization on cheap, fast computers, one could get away from needing an exact formula, provided it was possible to say precisely what the model predicted --- most often, what the likelihood function was.
This sounds like it ought to be easy, but there are many models which are very natural from a scientific view-point (because they nicely represent mechanisms we guess are at work) for which exact expressions for the likelihood, or indeed for other predictions, just are not available. In modeling dynamics, for example, if what we observe is not the full state of the system, but rather only part of it (and generally a part distorted by noise and nonlinearity at that), it becomes exceedingly difficult to calculate the probability of seeing a given sequence of observations. Or, again, if one's model is specified in terms of the behavior of large numbers of interacting entities (like molecules or economic agents), each possibly with an unobserved internal state, finding an exact likelihood function is pretty much hopeless. If we nonetheless want to connect our models to reality, and estimate parameters, what then should we do?
Gouriéroux and Monfort's answer turns on the fact that even though many interesting models can be simulated even when they can't be solved. That is, one can fairly quickly and cheaply "run them forward" to generate examples of the kind of behavior they say should happen, if necessary making many simulation runs to get many samples of the behavior they predict. One can then use those samples for estimation, and this in two ways, "direct" and "indirect".
The "direct" method of simulation-based inference is older and more
straightforward; just use the sample of simulation runs as an approximation to
the probability distribution generated by the model. In the formulas where one
would want to use the theoretical probabilities to calculate expectations,
likelihoods, etc., substitute the appropriate average over simulations. The
easiest way to see how this works is with the method of moments. The actual
expectations
The "principle of indirect inference" (ch. 4) is more subtle, and to me much
more exciting. In this approach, one introduces an "auxiliary" or
"instrumental" model, which is not in general expected to be correct, but is
supposed to be something which is easy to fit to the data. One then fits the
auxiliary model both to the data, getting auxiliary parameter
values
For this to work, there are essentially two requirements. The first
requirement is that, if we feed in larger and larger samples from the primary
model, with its parameters held to
The first assumption, convergence of auxiliary parameter estimates, is very weak, though not altogether trivial. The second assumption basically demands that the auxiliary model be rich enough to distinguish between different versions of the primary model. Typically, but not necessarily always, this will entail their being at least as many auxiliary parameters as there are primary ones, though these needn't correspond in any useful or comprehensible way. The distributional and Cramér-Rao-style results are of the kind one would expect: the indirect estimates will be more precise when the auxiliary parameters can be precisely estimated from the data, and when small differences in the auxiliary parameters correspond to large differences in the primary parameters.
Chapters 5, 6 and 7 apply direct and indirect simulation inference to a range of popular models from econometrics, comparing the results to those of other estimation methods on both simulated and real-world data. Some of these are extremely impressive — in particular some of the results on complicated time-series models are simply astonishing — but these chapters will frankly be very hard going for anyone who has not seen these econometric models before. (Chapter 5, in particular, includes an awful lot on how to simulate discrete choice models.) Other applications will readily suggest themselves to any reader who has worked with simulation models.
x+174 pp., bibliography, line figures, index (spotty)
Economics / Probability and Statistics
In print as a hardback, ISBN 0-19-877475-3
Updated 16 March 2012: small typo fixes, switched to using MathJax