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Econometrics and Financial Time Series / Regression Diagnostics and Generalized Least Squares

Worked Examples — Diagnosing a Factor or CAPM Regression

A reported CAPM alpha is only useful if the regression that produced it has survived a credible diagnostic process. A point estimate can be numerically stable while its classical tt-statistic is invalid, its factor attribution is unidentified in practice, or its conditional mean has been misspecified. This capstone turns the preceding diagnostic lessons into one desk-ready sequence: fit OLS, inspect residuals, test the covariance and mean specification, select HC or HAC inference when OLS remains the estimand, and rebuild the model when the diagnostics say that the regression itself is no longer answering the economic question.
The examples use invented but realistic monthly excess returns. They connect the classical linear model, heteroscedasticity, serial correlation, specification failure, and multicollinearity into a single decision rather than five disconnected software menus.
Learning objectives. By the end of this note you should be able to:
  • execute a diagnostic checklist for a CAPM or APT-style return regression;
  • separate a wrong covariance matrix from a wrong conditional mean;
  • choose classical, HC, or HAC standard errors from residual evidence;
  • use RESET, a break test, and VIF to decide whether factor interpretation remains defensible;
  • route a failed regression to the appropriate Stage 2–7 econometric tool.

In this lesson

  1. The regression under review
  2. A diagnostic algorithm before reporting alpha
  3. Example 1: a CAPM fit that needs HC, not a new beta
  4. Example 2: HAC protects a static answer, but cannot explain persistence
  5. Example 3: RESET and a known break say the mean must change
  6. Example 4: VIF changes the attribution claim, not the fitted return
  7. When diagnostics require a model rebuild

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