Question

Consider the simple linear regression model \[ Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad i = 1, 2, \dots, n \quad (n \geq 3), \] where \( \beta_0 \) and \( \beta_1 \) are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance \( \sigma^2>0 \). Suppose that \( \hat{\beta}_0 \) and \( \hat{\beta}_1 \) are the ordinary least squares estimators of \( \beta_0 \) and \( \beta_1 \), respectively. Define \( \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \), \( S_1 = \sum_{i=1}^n (x_i - \bar{x})^2 \), where \( y_i \) is the observed value of \( Y_i, i = 1, 2, \dots, n \). Then for a real constant \( c \), the variance of \( \hat{\beta}_0 + c \) is

• A. \( \sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_2} \right) \)
• B. \( \sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_1} \right) \)
• C. \( \frac{\sigma^2}{n} \)
• D. \( \sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_2} \right) + c^2 \)

Hint:

The variance of \( \hat{\beta}_0 \) depends on the sample size \( n \), the average of the \( x_i \)'s, and the variability of the \( x_i \)'s around their mean.

Answer 1

The Correct Option is B

We are given a simple linear regression model, and we need to compute the variance of \( \hat{\beta}_0 + c \).
Step 1: Understanding the properties of linear regression.
In linear regression, the variance of \( \hat{\beta}_0 \) is given by: \[ \text{Var}(\hat{\beta}_0) = \frac{\sigma^2}{n} + \frac{\bar{x}^2}{S_1}. \] Step 2: Applying the formula.
The variance of \( \hat{\beta}_0 + c \) will be: \[ \text{Var}(\hat{\beta}_0 + c) = \text{Var}(\hat{\beta}_0) = \sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_1} \right). \] Final Answer: \[ \boxed{\sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_1} \right)}. \]