The most-used tool in quant research and therefore heavily interviewed - usually on the assumptions rather than the mechanics.
The basics
For a simple regression of y on x, the OLS slope is
beta = Cov(x, y) / Var(x)
and the intercept passes the line through the sample means. It minimises the sum of squared residuals, and under the standard assumptions it estimates the conditional expectation E[y | x].
The assumptions, and what breaks
Interviewers work through these:
Linearity. If the true relationship is not linear, the coefficient is a best linear approximation and may be meaningless.
Independence of errors. Financial time series violate this constantly through autocorrelation, which leaves coefficients unbiased but makes standard errors far too small - so things look significant that are not.
Homoscedasticity. Constant error variance. Volatility clustering breaks it; use robust standard errors.
No perfect multicollinearity. Correlated predictors inflate coefficient variance and make individual coefficients unstable even when the model as a whole predicts well.
Exogeneity. Errors uncorrelated with predictors. If violated, coefficients are biased - this is the one that matters most and is hardest to check.
R-squared
The fraction of variance explained. Three things to say about it:
- It never decreases when you add a variable, so it cannot be used for model selection. Use adjusted R-squared, or better, out-of-sample performance.
- High R-squared does not imply a useful model - two trending series regress beautifully on each other and mean nothing (spurious regression).
- In finance, a genuinely predictive signal often has an R-squared of a fraction of a percent. Candidates who dismiss low R-squared as failure are revealing they have not worked with real return data.
Common interview questions
"What happens if you add an irrelevant variable?" R-squared rises slightly; coefficient variance increases; bias is unaffected.
"Regressing y on x versus x on y - same line?" No. The two slopes multiply to R-squared, so they coincide only in a perfect fit. This catches a lot of people.
"Correlation versus regression?" Correlation is symmetric and unit-free; regression is directional and in the units of the variables.
Practise in statistics.