Linear Statistical Models Quiz 2: 4 August 2024 (May 2024 term)

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]

1. 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 :

1Call:
2lm(formula = y ~ x)
3
4Residuals:
5    Min      1Q  Median      3Q     Max
6-14.039  -6.295  -2.028   5.343  31.007
7
8Coefficients:
9             Estimate Std. Error t value Pr(>|t|)
10(Intercept) 101.88323    1.88248   54.12   <2e-16 ***
11x            -2.03429    0.03236  -62.86   <2e-16 ***
12---
13Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
14
15Residual standard error: 9.342 on 98 degrees of freedom
16Multiple R-squared:  0.9758,    Adjusted R-squared:  0.9756
17F-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]