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Consider the following linear model

$$y_t = x_t' \beta +u_t$$

where $t =1,...,T$ and $x_t = (x_{1t} x_{2t} ... x_{kt})'$ , $ \beta$ is $k \times 1$ vector of unknown coefficients, $u_t$ is an iid disturbance term with the variance $\sigma^2$ and $E(x_tu_t)=0$ for all t.

Find the consistent but inefficient GMM estimator.


My solution:

I know that $E(x_tu_t)= \frac{1}{T} \sum^T_{t=1} [x_t(y_t-x_t'\beta)]=0$

Define the Jacobian matrix

$$J(B)= g(B)' W g(B)$$

where $g(B)=\frac{1}{T} \sum^T_{t=1} [x_t(y_t-x_t'\beta)]$ and $W=I_k$

Here, I define W as an identity matrix, because efficiency depends on W matrix and when W=I, I guess that this estimator become inefficient. (Maybe wrong, I don't know exactly)

Then, the Jacobean matrix $J(B)$ in the matrix form is written as

$$J(B)=[\frac{1}{T} X'(y-X\beta)]' I_k [\frac{1}{T} X'(y-X\beta)]$$

Let's minimize J(B) w.r.t $\beta$

That's, $\partial J(B) / \partial \beta =0 $

Then, $$\hat{\beta} = (X'XX'X)^{-1} X'XX'y$$

As it is seen in the answer, I obtained OLS estimator $\hat{\beta} = (X'X)^{-1} X’y $. But OLS estimator is efficient, not inefficient. Thus my answer is wrong.

What should I make an extra assumption to obtain inefficient but consistent estimator? Assuming Weight matrix as I is wrong.

How do you solve for this question ? Where I'm wrong? Please share your ideas with me.

Note: the question is as follows. I am asking the part a.

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New Update

The efficient GMM estimator depends on the weight matrix.

I firstly define the population moment condition as $g(B)=\frac{1}{T} \sum^T_{t=1} [x_t(y_t-x_t'\beta)]$

Based on the Central limit theorem,$\sqrt{T} g(\beta) \to N(0, P)$ where $P=var(\sqrt{T}g(\beta)=\frac{\sigma^2}{T^2}(X'X)$

Also, Based on Central limit theorem,

$\sqrt{T} (\hat{\beta}_{GMM} - \beta) \to N(0, V)$

where $V=(D'WD)^{-1}D'WPW'D(D'WD)^{-1}$ for the expected value of the matrix go the first derivates of the moment $D$.

I I choose the weight matrix $W=P^{-1}= (\frac{\sigma^2}{T^2}(X'X))^{-1}$ instead of choosing any other positive definite matrix $S$ for $W$, then I can obtain the minimum variance convince matrix.

Well, what is this positive definite matrix $S$ which makes the GMM estimator is inefficient.

Can you give me an example for $S$ matrix?

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1 Answer 1

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You basically, and correctly, rederived OLS, as these are the moment conditions exploited by your GMM estimator (namely, that regressors and errors are orthogonal, so that no external instruments are needed (equivalently, that regressors serve as their own instruments).

Notice that, assuming no multicollinearity, $X'X$ has full rank and hence is invertible. Hence, $$(X'XX'X)^{-1}= (X'X)^{-1}(X'X)^{-1}$$ so that $$\hat{\beta} = (X'X)^{-1}\underbrace{(X'X)^{-1}X'X}_{=I}X'y=(X'X)^{-1}X'y.$$

I am not sure if that is the actual answer to your question, as we seek an inefficient estimator, but OLS is, under the present assumptions, efficient, by the Gauß-Markov theorem.

P.S. Not that it is not correct to equate expected values and sample averages. Rather, the sample moment condition is the analogon exploited by the population moment condition.

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