Estimating Without a Full Model: Hansen's GMM

Maximum likelihood is the classical way to estimate a model, and it is powerful. But it demands an enormous price up front: you must specify the complete probability distribution of your data. Not just the average behaviour. The whole thing. Every moment, every tail, every dependency.

In finance, this is a disaster. Do you know the exact joint distribution of asset returns? Nobody does. Returns are fat-tailed, their volatility clusters, their correlations shift in crises, and their higher moments are unstable. If you assume normality because it is convenient, and it is wrong, your estimates and everything built on them are contaminated by an assumption you never believed in the first place.

In 1982, Lars Peter Hansen showed that you can throw almost all of that away. His framework, the Generalized Method of Moments, asks only for a handful of statements you actually believe. It won him a share of the Nobel Prize and it is the reason modern empirical asset pricing exists in the form it does.

The problem: theory gives you conditions, not distributions

Here is the crucial observation that motivates everything.

When economic theory speaks, it almost never says "returns are normally distributed." What it says is things like: "in equilibrium, no investor should be able to systematically improve their welfare by trading a bit more of asset X." That translates into a statement of the form: a certain quantity, on average, should be zero.

Or in asset pricing: "any priced asset's return, multiplied by the right discount factor, should average to one." Again: something, on average, hits a specific number.

Or in a simple regression: "the errors should be uncorrelated with the regressors." Again: something averages to zero.

These are called moment conditions. And theory produces them in abundance. What theory does not produce is a full probability distribution. So maximum likelihood forces you to invent one, and everything you build then inherits the fiction.

Hansen's question was: can we estimate using only the moment conditions, which is all we actually believe?

The key idea via analogy: the balancing act with too many scales

Imagine you are trying to figure out the weight of a single unknown object. You have one scale. You put the object on it, read the number, done. One equation, one unknown, exact answer.

Now imagine you have five different scales, all slightly inaccurate in different ways. Each one gives you a different reading. You now have five equations and one unknown. There is no single weight that makes all five scales read correctly. The system is overdetermined.

What do you do? You do not throw away four scales. You pick the weight that makes all five scales as nearly correct as possible, all at once. And if you know that some scales are more reliable than others, you weight them more heavily in the compromise.

That is GMM in one paragraph.

You have a set of moment conditions that theory says should hold exactly. In your finite sample, they will not hold exactly, because of noise. So you choose the parameters that make all the moment conditions come as close to zero as possible, simultaneously, weighting each condition by how reliable it is.

The word "generalized" in the name refers precisely to this weighting. The old "method of moments" (which goes back to Karl Pearson) required exactly as many conditions as parameters, so everything could be solved exactly. Hansen generalised it to the far more useful case where you have more conditions than parameters, and worked out the whole statistical theory: how to weight them optimally, what the resulting estimates converge to, how uncertain they are, and, crucially, what to do with the leftover disagreement.

The gift of the leftovers: a free specification test

That last part deserves its own attention, because it is where GMM does something the other methods cannot.

If you have five moment conditions and only one parameter, then after choosing the parameter that best satisfies all of them, four dimensions of disagreement remain. Those leftovers are information.

If your theory is correct, those leftovers should be small: just noise. If your theory is wrong, the leftovers will be large, because no choice of parameter can reconcile conditions generated by a false model.

Hansen formalised this into a test statistic, the J-test (also called the test of overidentifying restrictions). It asks: given how much noise there is, is the residual disagreement among my moment conditions bigger than luck can explain? If yes, your model is rejected.

This is genuinely remarkable. You get, for free, a test of whether the theory you are estimating is even compatible with the data, out of the same machinery that estimated it. No extra assumptions required.

Why it mattered

• It became the language of empirical asset pricing. The whole modern framework, where an asset pricing model is written as "there exists a stochastic discount factor such that expected discounted returns equal one," is a set of moment conditions, and GMM is the natural way to estimate and test it. Hansen and Singleton's tests of consumption-based asset pricing, and the entire literature that followed, run on this machinery.
• It freed estimation from distributional fiction. You no longer have to pretend returns are normal in order to estimate anything. You state what you believe, which is a set of averages, and estimate from exactly that.
• It unified a zoo of methods. Ordinary least squares, instrumental variables, two-stage least squares, and many other familiar estimators all turn out to be special cases of GMM with particular moment conditions. That kind of unification is what makes a framework rather than a technique.
• It handles serially correlated, heteroskedastic data natively. GMM's standard errors are computed using exactly the robust machinery of White and Newey-West. Time-series financial data was the intended use case, not an afterthought.

The honest limitations

GMM's flexibility is real, and so is its capacity to disappoint.

• It can behave badly in the sample sizes you actually have. This is the big one. GMM's beautiful properties are asymptotic, meaning they hold as your sample grows toward infinity. In finite samples, especially with many moment conditions, GMM estimates can be substantially biased and the J-test can reject far too often or far too rarely. The two-step version, which is standard, is particularly vulnerable because the weighting matrix estimated in the first step is itself noisy and that noise contaminates the second step.
• Too many moments is a trap, not a bonus. It is tempting to throw in more moment conditions, since more information sounds better. In practice, piling on moments makes the weighting matrix harder to estimate and the small-sample bias worse. Restraint is required, and there is no clean rule for how much.
• Weak identification quietly destroys it. If your moment conditions only weakly pin down the parameter (analogous to weak instruments), the estimates can be wildly unreliable and the standard errors will not warn you. This is a serious problem in asset pricing, where many moment conditions carry very little information about the parameter of interest.
• The J-test has low power and is easy to misread. Failing to reject your model does not mean your model is right. It very often means your moment conditions were too noisy to detect anything. And when the J-test does reject, it tells you something is wrong without telling you what.
• You must still get the moment conditions right. GMM frees you from specifying the distribution. It does not free you from specifying the economics. If the moment conditions come from a false theory, GMM will estimate the parameters of a false theory, efficiently.

The one-line takeaway

Hansen showed you can estimate and test economic models using only the handful of things you actually believe should average out to a known number, discarding the fiction of a fully specified distribution, and the leftover disagreement among those conditions gives you a free test of whether your theory was ever plausible in the first place.