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

This result seems odd to me.

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

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

This result seems odd to me.

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

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, $y_t$ 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 consistency depends on W matrix and when W=I, I guess that this estimator become inconsistent. (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, $$\hat{\beta} = (X'XX'X)^{-1} X'XX'y$$

This result seems odd to me.

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

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

This result seems odd to me.

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