pset 3.3 Consider the regression model$$y_{i}=\boldsymbol{x}_{i}^{\prime} \boldsymbol{\beta}+\varepsilon_{i}, \quad \text { with } \quad \varepsilon_{i} \mid \boldsymbol{x}_{i}, \boldsymbol{z}_{i} \sim N\left(0, \sigma^{2} \exp \left(\boldsymbol{z}_{i}^{\prime} \boldsymbol{\gamma}\right)\right)$$where $\boldsymbol{z}_{i}$ is a vector of regressors not including an intercept and $\boldsymbol{\theta}=\left(\boldsymbol{\beta}^{\prime}, \boldsymbol{\gamma}^{\prime}, \sigma^{2}\right)^{\prime}$ is th vector of unknown parameters.1. Write the first-order conditions for the maximum likelihood estimation of $\boldsymbol{\theta}$.2. Identify the set of restrictions that makes $V\left(\varepsilon_{i} \mid \boldsymbol{x}_{i}, \boldsymbol{z}_{i}\right)$ constant across observations.3. Write down the formulation of the LM test statistic for the null hypothesis corresponding to the above set of restrictions.

pset 5.1 Consider the linear regression model$$y_{i}=x_{i} \beta+u_{i}$$where the only independent variable $x_{i}$ is determined as$$x_{i}=y_{i} \gamma+v_{i}$$and $V\left(u_{i}\right)=\sigma_{u}^{2}$ and $V\left(v_{i}\right)=\sigma_{v}^{2}, \forall i$. Prove that the OLS estimator of $\beta$ based on the first equation does not converge in probability to $\beta$. (hint: write $x_{i}$ as a function of only $u_{i}$ and $v_{i}$.)

pset 6.1 Prove that the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model with individualspecific intercepts, that is$$y_{i t}=\boldsymbol{x}_{i t}^{\prime} \boldsymbol{\beta}+\alpha_{i}+u_{i t},$$where $\alpha_{i}$ is a parameter to be estimated, for $i=1, \ldots, n$ is equal to $\hat{\boldsymbol{\beta}}_{W G}$.

pset 5.2 Prove that $\hat{\boldsymbol{\beta}}_{G I V E}$ can be derived as the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model$$\boldsymbol{y}=X \boldsymbol{\beta}+\hat{U} \boldsymbol{\gamma}+\boldsymbol{\eta}$$where $\hat{U}$ is the matrix of first-stage residuals.

pset 4.1 Consider the $\operatorname{rv} X \sim \operatorname{Gamma}(\alpha, p)$, where $\alpha>0$ is the scale parameter and $p>0$ is the shape parameter, and with$$\begin{aligned}\mathrm{E}(X) & =\frac{p}{\alpha} \\\mathrm{E}\left(X^{2}\right) & =\frac{p(p+1)}{\alpha^{2}}\end{aligned}$$and assume you have the random sample $X_{1}, \ldots, X_{n}$.1. Find the method of moment estimators of $\alpha$ and $p$.2. By noticing that$$\begin{aligned}\mathrm{E}\left(\frac{1}{X}\right) & =\frac{\alpha}{p-1} \\\mathrm{E}(\log (X)) & =\Psi(p)-\log (\alpha)\end{aligned}$$write the criterion function for the two-step GMM estimator of $\boldsymbol{\theta}=(\alpha, p)^{\prime}$.

请你为我分析题目并且一步一步和我一起解答这道题可以吗？“Consider the full rank linear modelyi = β0 + β1x1i + β2x2i + εi, (i = 1, . . . , n.)Assume that the errors are independent, normally distributed with mean 0 and variance σ2.Derive an expression for a joint 100(1 − α)% confidence region for parameter β1 and β2.(Hint: You can use the result that the marginal distribution over a subset of multivariatenormal random variables is multivariate normal, and the mean and covariance matrix are ob-tained by dropping the irrelevant variables (the variables that one wants to marginalize out)from the mean vector and the covariance matrix.)”