15.5 Applications and practical issues
Euler equations
Hansen and Singleton (1982) estimated a representative consumer’s preferences from the first-order condition for saving. With utility \(c^{1-\gamma }/(1 - \gamma )\), discount factor \(\delta \) and gross asset return \(R_{t+1}\), optimal saving requires \[ E_t\Bigl [\delta \Bigl (\frac {c_{t+1}}{c_t}\Bigr )^{-\gamma }R_{t+1} - 1\Bigr ] = 0, \] where \(E_t\) is the expectation given information at time \(t\). The expression in brackets is a forecast error, so it is uncorrelated with every variable \(z_t\) known at time \(t\), which gives the moment conditions \[ E\Bigl \{z_t\Bigl [\delta \Bigl (\frac {c_{t+1}}{c_t}\Bigr )^{-\gamma }R_{t+1} - 1\Bigr ]\Bigr \} = 0 . \] With a constant and lagged consumption growth and returns in \(z_t\), there are more moments than the two parameters \((\delta , \gamma )\), the model is overidentified, and the \(J\) test is a test of the theory itself. No assumption about the distribution of consumption or returns is needed, and no model of how income or returns are generated: this is the great attraction of GMM for testing economic theory. The equity premium puzzle of Mehra and Prescott (1985) appears in this framework as a rejection of the overidentifying restrictions, or an implausibly large \(\gamma \), when stock and bond returns are used together.
Finite-sample problems
The asymptotic theory can be a poor guide. The two-step standard errors are often much too small in moderate samples, because they ignore the estimation error in \(\hat S\); Windmeijer (2005) gives a correction. With many moment conditions, two-step GMM is biased in the direction of least squares, for the same reason as 2SLS with many instruments (Section 13.3), and the \(J\) test loses power, since \(\hat S\) becomes a poor estimate of a large matrix. Practical advice is to use moderately many, well-chosen moments, report the first-step as well as the efficient estimates, and treat a large discrepancy between them as a warning.
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