Linear Regression Assumptions and How They Fail on Market Data

Each classical regression assumption fails on market data in a distinct way, and each failure points at a distinct, named fix.

Ordinary least squares regression rests on assumptions about linearity, independence, and constant variance that market data routinely breaks, and each specific violation has a specific consequence for how much a fitted coefficient or its reported significance can be trusted. Knowing which assumption failed points directly at which fix applies, rather than treating every unstable regression as the same generic problem.

Ordinary least squares assumes linearity in the parameters, meaning the model is a linear combination of the coefficients even if the underlying variables are transformed. It assumes strict exogeneity, meaning the error term is uncorrelated with the regressors. It assumes independence of the errors across observations, meaning no autocorrelation. It assumes homoskedasticity, meaning the error variance is constant rather than changing with the level of the regressors or with time. It assumes no perfect multicollinearity among the regressors, meaning no predictor is an exact linear function of the others. It assumes normally distributed errors, though this last assumption matters mainly for…

Each assumption supports a different part of the result: violating exogeneity biases the coefficients themselves, while violating homoskedasticity or independence leaves the coefficients unbiased but corrupts the standard errors used to judge whether they are significant. Knowing which category a given violation falls into determines whether the fix is a different model specification or simply a different formula for the standard errors.

Market data has variance that changes across regimes, calm stretches followed by turbulent ones, so a regression of one return series on another is close to guaranteed to have heteroskedastic residuals. The coefficients stay unbiased, but the standard formula for their standard errors, which assumes constant variance, understates the true uncertainty during volatile subperiods and overstates it during calm ones, distorting every t-statistic computed from it. The fix, heteroskedasticity-consistent standard errors, is covered in full in heteroskedasticity and robust standard errors.

Detecting it before trusting a regression's significance takes one extra diagnostic step, typically a Breusch-Pagan or White test run on the residuals. Skipping that step and reporting the classical standard errors as though the variance were constant is one of the most common inference errors in return-based regressions, and it is an easy one to catch once a researcher knows to look for it.