Read a regression output: Kalindor Motors' monthly returns on the market over 60 months give beta 1.35 (standard error 0.15), alpha 0.2% a month (standard error 0.4%) and R-squared 0.58. Does beta differ from 1, what is its 95% interval, and what do you make of alpha?

Beta is significantly above 1: t = (1.35 - 1) / 0.15 = 2.33, beyond the 5% critical value of 2.00, and the 95% interval is 1.05 to 1.65. The printed t of 9.0 only says beta is not zero. Alpha of 0.2% a month has t = 0.5 and an interval of -0.6% to 1.0% a month: indistinguishable from zero. R-squared of 0.58 leaves 42% of the variance stock-specific.

If a friend claims his car does better than 15 km a litre, you check his mileage against 15, not against zero; nobody doubts the car moves. Software tests every coefficient against zero by default, so the printed t of 9.0 answers a question nobody asked: whether Kalindor moves with the market at all. The useful question is whether it moves more than the market, a test against 1. That t-statistic is (1.35 - 1) / 0.15 = 2.33. With 58 degrees of freedom, the 5% two-sided critical value is 2.00, so beta is above 1 at the 5% level, with a p-value of about 0.023.

Alpha is 0.2% a month, which sounds like 2.4% a year. Its t-statistic is 0.2 / 0.4 = 0.5 and its 95% interval runs from -0.6% to 1.0% a month, roughly -7% to +12% a year, so the data cannot tell Kalindor's alpha from zero. That is typical: five years of monthly data almost never pin down a single stock's alpha. R-squared of 0.58 means the market explains 58% of the variance of Kalindor's returns and 42% is company-specific, the part a market hedge cannot remove. For a hedger that matters as much as the beta: a perfect beta hedge would still leave most of the stock's month-to-month swings.

A quick consistency check catches typos and misread columns. The alpha's standard error is roughly the residual volatility over the square root of 60, which implies a residual standard deviation of about 3.1% a month. The beta's standard error is roughly residual volatility over the square root of 60 times market volatility, which implies market volatility of about 2.67% a month. Those two imply an R-squared of 0.57, matching the printed 0.58, so the output is internally consistent. The market volatility it implies, about 9% a year, describes a fairly calm five years.

Close with how you would use the beta. For hedging, 1.35 is the best single estimate, but the interval from 1.05 to 1.65 is wide, and betas estimated this way tend to drift towards 1 in later periods. A common adjustment weights the estimate two-thirds and 1 one-third, giving 1.23. The limitation: the test assumes the residuals are independent with constant variance; with autocorrelated or volatile-clustered residuals, use robust standard errors before leaning on the 2.33.