Lecture 9 (Generalized Method of Moments (GMM))

Question 1

Explain the concept of MM. Derive OLS and IV using MM.

Answer 1

Moment = expectations. with (G)MM we use assumptions in terms of expectations (moment conditions) to find estimators. E(something) = something (usually zero).

See notion for derivations.

Question 2

Explain the difference between MM and GMM.

What is the general GMM estimator?

Answer 2

MM is applied to exactly identified problems where the estimator is a unique solution to sample a set of moment restrictions. With GMM we can try to improve efficiency by adding restrictions (hence, making more assumptions).

when we have more equations than unknowns, we have no unique solutions. Instead, we will seek to simultaneously get as close as possible to solving all equations. We then minimize the weighted quadratic loss function:

$$
\hat\theta_{GMM}=\argmin_{\theta}\Bigg[ \frac1N\sum_{i=1}^Nh(w_i,\theta)\Bigg]’W_N \Bigg[ \frac1N\sum_{i=1}^Nh(w_i,\theta)\Bigg]=0
$$

where $W_N$ is a $r\times r$ symmetric positive definite matrix. Different choices of $W_N$ lead to different GMM estimators. For some choices of $W_N$ the GMM is more efficient than the OLS.

Question 3

Derive the GMM 2SLS estimator for the exact and over-identified case.

Answer 3

See notion.

Question 4

What effect do the weighting matrix have in GMM?

Answer 4

While the GMM is consistent for any weighting matrix $W_N$ , the choice of this matrix will affect the variance of the estimator, $\hat \theta_{GMM}$. If every weighting matrix yields a GMM estimator that is consistent, then we would ideally like to use the weighting matrix that minimizes the variance of the estimator.

Question 5

How do we optimally choose the weighting matrix in GMM?

Answer 5

Hence, when estimating we are using the minimized variance matrix. We, therefore, use a repeated two-step procedure when estimating the GMM:

1. Perform GMM with any arbitrary weighting matrix and
   estimate the variance of the moments using this GMM estimator.
2. Perform GMM again, but now using minimized variance matrix.

This is basically = “iterating” GMM. We keep on repeating this process until the parameter of interest stop changing significantly between iterations. Since GMM is consistent in large samples for any weight matrix, this is something that we usually just need to consider in small samples.

Question 6

GMM vs 2SLS

Answer 6

In the over-identified cases with heteroskedastic errors GMM has better precision than 2SLS in large samples but might be biased in small samples. 2SLS is also faster to run in the software.

In the just identified case, they are equivalent.

Question 7

What can we use GMM IV for?

Answer 7

• IV overidentification tests
• Bartik instruments