A regression analysis of 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 $\hat{\mu}_X = 0.61$. What is the standard error estimate of $\hat{\beta}_0$? | Financial Risk Manager Part 1 Quiz

A regression analysis of 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 $\hat{\mu}_X = 0.61$ . What is the standard error estimate of $\hat{\beta}_0$ ?

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Explanation:

Explanation

The standard error of the intercept ( $\hat{\beta}_0$ ) in a linear regression can be calculated using the formula:

\text{SEE}_{\beta_0} = \sqrt{\frac{s^2(\hat{\mu}_X^2 + \hat{\sigma}_X^2)}{n\hat{\sigma}_X^2}}

Where:

$s^2 = 20.45$ (residual variance)
$\hat{\mu}_X = 0.61$ (mean of the independent variable)
$\hat{\sigma}_X^2 = 18.65$ (variance of the independent variable)
$n = 120$ (number of monthly observations over 10 years: 12 months × 10 years)

Substituting the values:

\text{SEE}_{\beta_0} = \sqrt{\frac{20.45(0.61^2 + 18.65)}{120 \times 18.65}}

First, calculate the numerator:

$0.61^2 = 0.3721$
$0.3721 + 18.65 = 19.0221$
$20.45 \times 19.0221 = 388.902045$

Then calculate the denominator:

$120 \times 18.65 = 2238$

Now compute the fraction:

$388.902045 / 2238 = 0.1738$

Finally, take the square root:

$\sqrt{0.1738} = 0.4169$

Therefore, the standard error estimate of $\hat{\beta}_0$ is 0.4169._

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