Financial Risk Manager Part 1

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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 $\bar{x} = 0.61$ . The analyst wishes to test whether the slope coefficient is different from 0. What is 99% confidence interval for $\hat{\beta}$ ?

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NNikitesh

Last updated: February 2, 2026 at 10:22

[1.6034, 1.8906]

[1.3034, 1.8966]

[1.3997, 1.9002]

[1.5034, 1.6976]

Explanation:

The confidence interval is given by:

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

The critical value ( $C_t$ ) at 1% (0.5% on each tail) level is 2.618. (From the standard normal table)

Now, $\text{SEE}_{\hat{\beta}}$ is given by,

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

So,

= [1.3997, 1.9002]$$

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