Hansen’s work is the most technical and most difficult to explain to a layperson. The brief version is that in 1982 Hansen developed the Generalized Method of Moments a new and elegant way to estimate many economic models that requires fewer assumptions and is often more powerful than other methods.
Here is the basic idea in a nutshell. The method of moments is an old and intuitive technique for estimating the parameters of a data generating process. A moment is an expectation of the form E(X^r), where r can be an integer. For example if r=1 then the first moment, E(X), is just the mean (you may also know that together the first and second moments, E(X^2), define the variance). If the true mean in the data generating process is M then we can write a moment condition, E(X)-M=0. Now the method of moments says to estimate M we should solve that condition by replacing, E(X), with the sample mean. In other words, a good estimator for the unknown population mean is the sample mean, i.e. the mean in the data that you have. Pretty obvious so far.
Now let’s start to generalize. First, there are many moments other than the mean and variance. Indeed economic theory often provides moment conditions that may be written E(f(X,M))=0 where M now stands for a moment not necessarily the mean and f can be a non-linear function. For example, rational expectation models often provide conditions that E(f(X))-M=0, i.e. that forecasts should equal true values, or macro models imply that various differences, such as in consumption levels should not be correlated and so forth. Indeed, finance and macroeconomic theory provided a surplus of moment conditions and many of these different conditions imply something about the same parameter. Now, and this is key, when we have more moment conditions than parameters we can’t choose the parameters to make all the moment conditions true, i.e. we can’t make all those moment conditions equal to zero. So what to do?
What Hansen did with the generalized method of moments is show that when we have more moment conditions than parameters we can best estimate those parameters by giving more weight to the conditions that we have better information about. In other words, if we have two conditions and we can’t force both of them to zero by a choice of parameter then choose the parameter such that the moment condition we know the most about (least variance) is closer to zero than the one we know less about. Again, the idea is intuitive, but Hansen showed how to make these choices and then he proved that when the parameters are chosen in this way they have good statistical properties such as consistency (they get closer to the true values as the sample size increases). Importantly, estimating a model using these moment conditions does not require untenable assumptions on the entire distribution of returns. Hansen then also showed, such as with Hansen and Hodrick (1980) and Hansen and Singleton (1982, 1983) how these methods could be applied to a large class of macro models and finance models including asset pricing, the latter of which links Hansen with the work of Fama and Shiller as does the important bound discovered by Hansen and Jaganatthan (1991).