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. ;)
Here's a little discussion that might help.

The population version has a solution because it is assumed to have a solution. You can see that this is the case from assumptions 3.1 (linearity), 3.3 (orthogonality conditions), and 3.4 (rank condition for identification). You can see the derivation of this fact from equations (3.3.3) and (3.3.4). 

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.

Search Keywords: Solutions, Exercise 3, section 3.4 Hayashi Econometrics