A regression analysis of the monthly returns of a sales company on the market return over ten years gives an intercept of $\hat{\beta}_0 = 0.65$, the slope $\hat{\beta} = 1.65$. Other quantities include: $s^2 = 20.45$, $\hat{\sigma}^2_X = 18.65$ and $\hat{\mu}_X = 0.61$. The analyst wishes to test whether the slope coefficient is different from 0. What is the 99% confidence interval for $\hat{\beta}$? | Financial Risk Manager Part 1 Quiz

A regression analysis of the monthly returns of a sales company on the market return over ten years gives an intercept of $\hat{\beta}_0 = 0.65$ , the slope $\hat{\beta} = 1.65$ . Other quantities include: $s^2 = 20.45$ , $\hat{\sigma}^2_X = 18.65$ and $\hat{\mu}_X = 0.61$ . The analyst wishes to test whether the slope coefficient is different from 0. What is the 99% confidence interval for $\hat{\beta}$ ?_

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Explanation:

Explanation

The confidence interval for the slope coefficient $\hat{\beta}$ is calculated using the formula:

[\hat{\beta} - C_t \times \text{SEE}_{\hat{\beta}}, \hat{\beta} + C_t \times \text{SEE}_{\hat{\beta}}]

Step 1: Determine the sample size and degrees of freedom

10 years × 12 months = 120 months (n = 120)
Degrees of freedom = n - 2 = 120 - 2 = 118

Step 2: Find the critical t-value

For a 99% confidence interval, α = 0.01, so α/2 = 0.005
With 118 degrees of freedom, the critical t-value (C_t) = 2.617

Step 3: Calculate the standard error of the slope coefficient

\text{SEE}_{\hat{\beta}} = \sqrt{\frac{s^2}{n\hat{\sigma}^2_X}} = \sqrt{\frac{20.45}{120 \times 18.65}} = \sqrt{\frac{20.45}{2238}} = \sqrt{0.00914} = 0.0956

Step 4: Construct the confidence interval

[1.65 - 2.617 \times 0.0956, 1.65 + 2.617 \times 0.0956] = [1.65 - 0.2502, 1.65 + 0.2502] = [1.3998, 1.9002]

Therefore, the 99% confidence interval for $\hat{\beta}$ is [1.3998, 1.9002], which corresponds to option D._

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