# Solution of overdetermined system of moment conditions

In Hayashi Econometrics, page 207-8, ex3 (see hint), he says that even if a population moment conditions (GMM setting) system is overidentified it still has a solution, while the sample moment conditions system cannot.

I'm having a hard time understanding this.

Any help would be appreciated.

By the way, I've also posted this question on CrossValidated. I hope there's no problem.

Edit: Check Jayk answer in CV. ;)

• Well the conditions of Weirestrass theorem are usually used to guarantee a solution. So informally I would say that the sample analogue could fail to have a solution because of lack of boundedness or closedness of the parameter space. Normally, this can be forced, what is problematic is uniqueness and thus identification. The population case is simple because is a quadratic objective minimized over a well behaved set of parameters (at least closed and bounded and usually compact). – user157623 Jan 15 '15 at 22:50
• I've posted an answer to this on cross validated: stats.stackexchange.com/a/134056/52022 – jayk Jan 19 '15 at 16:34
• Please don't cross-post – 410 gone Jan 19 '15 at 17:12
• Did you also try to solve Exercises 1 and 2 in the same section? I am looking for potential ways of solving them. For 2) I would suggest 0 as the solution. – Bonsaibubble Nov 25 '16 at 12:47

Now, you'll notice the population version having a solution implies that $$\text{rank}\left( \Sigma_{xz} \right ) = \text{rank}\left(\left [\Sigma_{xz} \mid \sigma_{xy} \right ]\right ),$$ as discussed in exercise 5 of section 3.3. In the hint they note that the population version has a solution when $$\left [\Sigma_{xz} \mid \sigma_{xy} \right ]$$ is of rank $L$. If it has rank $L+1$, it will not have a solution. In the hint, they say that this is a set of "equality conditions." This is referring to the fact that one of the columns must be a linear combination of the remaining columns. This is a very hard condition to satisfy. In many models, this condition will hold with probability zero. This is because the data has to come out so that the equality condition holds perfectly.
On the other hand, the condition that the sample version $S_{xy}$ of $\sum_{xz}$ be of full column rank is a set of "inequality" conditions on the matrix. Because they are inequality conditions on the matrix, they are much easier to satisfy by chance.
In conclusion, $S_{xy}$ will have full column rank for sufficiently large $n$ because the conditions are inequality conditions which occur with probability larger than zero. However, to have a solution we need $\left [S_{xz} \mid s_{xy} \right ]$ to have rank $L$. But because this is an equality condition on the matrix, it only occurs with probability zero.