In the news: Last week, American Lars Peter Hansen was awarded the Nobel Prize in Economics, with the committee citing his work on the generalized method of moments.
The generalized method of moments is an econometric estimation method, the aim of which is to estimate the parameters of a relationship between several variables. Here, we aim to provide an intuitive understanding of this method through its link with the best-known estimation method: ordinary least squares (OLS).
The simplest example often used to illustrate the generalized method of moments is as follows. Suppose we want to estimate the mean of a variable in a population: we denote this parameter (the mean) by w. In general, we will not observe this variable for the entire population of interest, but only for a subset (our sample). How, then, can we estimate this parameter? The generalized method of moments tells us that the best estimator of the parameter of interest here will correspond to the estimator m, which is such that:
E(w – m) = 0
And we will say that this estimator satisfies the above moment condition (the « moment » being nothing more than the expectation in this context). We can show very simply that under simple assumptions[1], the best estimator of the parameter of interest (the mean) is the empirical mean of our sample. To do this, we simply replace the theoretical moments with the empirical moments, based on the law of large numbers[2].
What, then, is the link with OLS?
The framework of the OLS estimation method is different. In this framework, we have a variable Y (GDP, for example) that we assume is related to explanatory variables x1 (interest rate) and x2 (business competitiveness). Let’s consider the simple model:
Y = x a + e
Where a is the parameter to be estimated and e is the error term. For the OLS estimator (â) to give us an average value close to the true parameter linking y to x (a), certain assumptions must be made. A sufficient assumption in this context[3]is that the correlation between the explanatory variables and the error term must be zero, which can be written as:
E(x e) = 0 for each observation
What we will call the exogeneity assumption in the OLS model will be the condition relating to the moment of (x e) in the generalized method of moments. This condition will suffice to obtain exactly the same estimator as the OLS estimator.
Indeed, knowing that we can rewrite e = y – xb, we can rewrite the above expression E(x (y – ax) ) = 0, which can also be written as E(xy) – E(a(x2)) = 0 or a = E(xy)/E(x2). Replacing the theoretical moments with the empirical moments, as suggested by the method of moments, we then obtain as an estimator of a, â = cov(y,x) / var(x). Compare this estimator to the expression that anyone starting out in econometrics learns in this context: it is exactly the same!
Thus, in the context of the generalized method of moments, we will use the conditions that our model must satisfy to estimate the parameter of interest. Here we have presented a case where this method gives us exactly the same result as the OLS method. However, the results differ when the estimation becomes more complex (robust estimates for heteroscedasticity, instrumental variables, etc.) because the method can no longer be summarized in this way. Hansen did not strictly speaking « invent » this method, but developed its application in much more complex settings (notably by applying it to the famous Euler equation in macro/microeconomics).
Julien P.
[1] For a sufficiently large number of observations, and assuming that the observations are independent and identically distributed.
[2] Whose validity—in its strong form—is based on the assumptions made in the previous footnote.
[3] Implicitly, no constant is assumed here; we assume that the DGPs (data generating processes) are those cited here and that the x’s are independent random variables.
