Suppose you are estimating the CAPM beta for All-Trick, a publicly traded brokerage for day-traders. You use 32 months of data and regress the stock performance of All-Trick against that of the overall market. You find that the slope coefficient estimated from the regression (b₁) equals 1.98, with a standard error of 0.33.

Find the 95% confidence interval for the beta of All-Trick.

Correct Answer: [1.3061, 2.6539]

The example provides us all the data we need except for the critical value for t. Since the confidence interval will exclude data in the extremes of both tails, we look up for t-values based on significance level of 0.05/2 = 0.025. The degrees of freedom will be 32 - 2 = 30, since the regression will estimate two parameters. The critical t-value for 30 degrees of freedom and a 0.025 level of significance is 2.042.

Therefore, using the formula above, our 95% confidence interval will be: 1.98 +or- (2.042 x 0.33) = 1.98 +or- 0.6739 = [1.3061, 2.6539].

We are 95% confident that the true beta lies within this range.