2.
The equation of interest is
$y_t = m(X_t, \beta) + e_t$
with $E[Z_t e_t] = 0$, where $m(x, \beta)$ is a known nonlinear function, $\beta^T = (\beta_1^T, \beta_2^T)$ is
a $k \times 1$ vector of unknown parameters and $Z$ is an $l \times 1$ vector of instrumental
variables.
(a) Show all steps on how to construct the GMM estimator for $\beta$, denoted
by $\hat{\beta}_{gmm}$.
(b) Show that $\hat{\beta}_{gmm}$ is efficient in the GMM sense.
(c) By considering testing $H_0 : \beta_2 = 0$, where $\beta_2$ is a vector of $k_2 \times 1$
parameters, please give all steps on using Langrange Multiplier (LM)
or Rao test to this testing problem.
