Check the assumptions before trusting the stars
The standard errors in a regression summary are only right if three things hold: the residuals are not correlated with each other, their spread does not change over time, and the regressors are not nearly copies of one another. statsmodels gives a test for each. The Durbin–Watson statistic looks at neighbouring residuals. The variance inflation factor, VIF, looks at how well each regressor can be rebuilt from the others. The Breusch–Pagan test asks whether the spread of the residuals depends on the regressors, and the Jarque–Bera test asks whether skewness and fat tails are large enough to matter. A bad result does not make the coefficients useless, it tells you which standard errors to distrust.
(1)
Residuals and two correlated regressorsTake the residuals from the five-month fit. The differences of neighbours are , their squares add to and , so and : the residuals tend to flip sign. A value near means no first-order correlation, near means positive correlation and near negative. If two regressors have correlation , each is explained by the other with and . With correlation the VIF is about , and with it is , which stretches the standard errors by . On real data, Apple's three-factor regression ( months from March 2005) has and a Breusch–Pagan p-value of , so there is little to worry about, though Jarque–Bera rejects normality (, kurtosis ). Now add the SPY fund's excess return next to the market factor: the two have correlation , both VIFs jump to , the standard error of the market loading goes from to , and its estimate flips to while barely moves (). Finally, regress log HD on log LOW over days: but . A DW far below warns that two trending prices are being matched by chance.
A warning is not a verdictEach diagnostic points at one assumption. Fix the matching problem: robust standard errors for correlated or uneven residuals (next topic), dropping or combining collinear regressors, or differencing trending series.
- Durbin–Watson near 2 is clean, near 0 or 4 means neighbouring residuals are linked.
- VIF above about 10 means the regressors tangle and individual t-statistics cannot be trusted.
- A high with a tiny DW is the classic sign of a regression between trending series.
Lab · what's graded
- A Durbin–Watson statistic20 pts
- The implied autocorrelation10 pts
- A variance inflation factor20 pts
- How far a standard error stretches20 pts
- Match the diagnostics10 pts
- Collinear regressors20 pts