Question 1: (i). State Gauss Markov theorem. [3 Marks] (ii). Consider…

1. (i). State Gauss Markov theorem. [3 Marks]

(ii). Consider a linear model as

$y_i = \beta_0 + \epsilon_i \quad ; \quad 1 \leq i \leq n$

where, $\epsilon_i \sim N(0, \sigma^2)$.

Let $\hat{\beta}_{01} = \dfrac{\displaystyle\sum_{i=1}^{n} y_i}{n}$ and $\hat{\beta}_{02} = \dfrac{\displaystyle\sum_{i=1}^{n}(1 + k_i)y_i}{n}$ be two linear estimators of $\beta_0$, where $k_i$ are constants and $i = 1, 2, \ldots, n$.

(a) Check whether $\hat{\beta}_{02}$ is an unbiased estimator of $\beta_0$. If it is not, find the condition under which it would be unbiased. [2 Marks]

(b) Prove that $\text{Var}(\hat{\beta}_{01}) \leq \text{Var}(\hat{\beta}_{02})$. [3 Marks]

1. Consider the linear model $(\underset{\sim}{Y}, X\underset{\sim}{\beta}, I_{3\times3})$ for the following data

(i). Find the least square estimate for $\underset{\sim}{\beta}$. [4 Marks]

(ii). Write the fitted linear regression model. [2 Marks]

(iii). Define $R^2$ and calculate the value of $R^2$ and interpret it. [5 Marks]