Question 1

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] 2. Consider the linear model $(\underset{\sim}{Y}, X\underset{\sim}{\beta}, I_{3\times3})$ for the following data | $\underset{\sim}{x}$ | $\underset{\sim}{y}$ | |---|---| | 5 | 7 | | 4 | 11 | | 6 | 9 | (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] R code with scatter plot of Y against X and a fitted red regression line (Figure 1), followed by questions 5. Consider a linear model as $$y_i = \beta_0 + \beta_1 x_i + \epsilon_i \quad ; \quad 1 \le i \le n$$ where, $\epsilon_i \sim N(0, \sigma^2)$. For the given linear model, we want to perform hypothesis testing at 5% significance level, i.e. $\alpha = 0.05$, to check if there is a relationship between the variables $x$ and $y$. The following output has been obtained for a particular model : Call: lm(formula = y ~ x) Residuals: Min 1Q Median 3Q Max -14.039 -6.295 -2.028 5.343 31.007 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 101.88323 1.88248 54.12 <2e-16 *** x -2.03429 0.03236 -62.86 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 9.342 on 98 degrees of freedom Multiple R-squared: 0.9758, Adjusted R-squared: 0.9756 F-statistic: 3951 on 1 and 98 DF, p-value: < 2.2e-16 (a) Define null and alternative hypothesis. [1 Mark] (b) Write the fitted linear regression model. [2 Marks] (c) Based on the output obtained, justify if we can reject the null hypothesis. Also, what can you conclude about the relationship between $x$ and $y$? [2 Marks]