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 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 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.

We use ------to derive the OLS betasa.Derivative and sum of residualsb.Sum of squared residualsc.Derivatived.Derivative and sum of squared residuals

Consider a sample of N individuals. Assume that y is the N × 1 binary dependentvariable and d1, d2, . . . , dM are the N × 1 dummy regressors for exhaustive andexclusive categories, meaning that each person in the population falls into one andonly one category (if for individual i dki = 1, all the other M − 1 regressors ofindividual i are equal to 0). Consider the linear regression, without the intercept,y on X, with X = [d1 d2 . . . dM ].Prove that:(a) The matrix X′X is diagonal.(b) The OLS estimates of the coefficients are all in the [0, 1] interval (0 ≤ ˆβj ≤ 1,for j = 1 . . . M ).(c) The fitted values from the OLS regression are in the [0, 1] interval for all indi-viduals (0 ≤ ˆyi ≤ 1, for i = 1 . . . N