A factor regression can report a statistically precise alpha while its residuals remember yesterday's surprise. That memory means the sample contains fewer independent pieces of information than the textbook OLS calculation assumes. Under the conditional-mean assumption, the coefficient can remain consistent, but its usual standard error and -statistic are not licensed. This note turns the serial-correlation warning in the classical linear model into a workflow: describe AR(1) disturbances, diagnose residual dependence with Durbin–Watson or Breusch–Godfrey, then choose between Newey–West HAC inference and a deliberately specified GLS model.
The choice is substantive. HAC protects inference without pretending to know the full disturbance process; GLS buys efficiency only when that process is credibly specified. Often, however, correlated residuals are evidence that the conditional mean has omitted dynamics. In that case the right Stage 2 response is an ARMA or dynamic-regression model, not a more elaborate correction to a static equation.
Learning objectives. By the end of this note you should be able to:
- state the AR(1) disturbance process and derive its stationary autocorrelation pattern;
- distinguish OLS consistency from valid classical standard errors under serial correlation;
- compute and interpret Durbin–Watson, including when it is not a suitable diagnostic;
- carry out and interpret a Breusch–Godfrey LM test of residual autocorrelation through a chosen lag order;
- construct the Newey–West HAC covariance estimator and distinguish it from GLS;
- explain when an AR(1) GLS transformation is defensible and when a dynamic mean model is the better repair.
In this lesson
- Residual memory changes the covariance, not automatically the mean
- Durbin–Watson: a fast first-order screen
- Breusch–Godfrey: test the residual process you care about
- Newey–West HAC: preserve OLS, repair large-sample inference
- GLS versus modelling the conditional mean
Before you start
- The Classical Linear Model and Where Finance Breaks Gauss–Markov
- Heteroscedasticity: White's Test, Robust Standard Errors, and Weighted Least Squares
The full lesson, with worked examples and exercise solutions, is for members.
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