Raman's Four-Factor Return Model: A High R-squared Built on Unreliable Standard Errors
Priya Raman models three-month forward returns on a developed-market equity index using four regressors: dividend yield, the change in the high-yield credit spread, the change in the manufacturing PMI, and the term spread. She has 96 quarterly observations from 2000 to 2023. OLS gives R-squared of 0.61 and adjusted R-squared of 0.57. Coefficient t-statistics are 3.1 (dividend yield), -2.4 (credit spread), 2.9 (PMI) and 0.6 (term spread). Variance inflation factors are 8.5 for dividend yield and 7.9 for the term spread; the other two are below 1.5. A Breusch-Pagan test returns a p-value of 0.02, and the Durbin-Watson statistic is 0.90. Raman concludes the model is robust because adjusted R-squared exceeds 0.5 and three of four coefficients are significant, and she proposes to size equity exposure using the fitted values. A colleague notes the model was estimated on overlapping three-month forward returns sampled quarterly.
1. Which regression assumptions are violated, and what is the separate consequence of each violation for the coefficients, the standard errors and the R-squared?
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Multicollinearity (VIFs of 8.5 and 7.9). Dividend yield and the term spread are highly linearly related. The consequence is not biased coefficients; OLS remains unbiased and consistent. The consequence is inflated standard errors, which understate t-statistics and make the model's coefficients unstable across samples. R-squared is inflated in the sense that the model appears to explain more than it does, although the joint explanatory power is genuine.
Heteroskedasticity (Breusch-Pagan p = 0.02). Reject the null of constant variance. Coefficients remain unbiased and consistent, but the conventional OLS standard errors are wrong, so every t-statistic, confidence interval and p-value in the table is untrustworthy. The direction of bias is unknown without a robust covariance estimator.
Serial correlation (Durbin-Watson 0.90). With roughly 96 observations and four regressors, a value near 2 indicates no autocorrelation; 0.90 indicates strong positive first-order autocorrelation. This makes OLS standard errors too small, so t-statistics are overstated and R-squared is inflated. It does not bias the coefficients.
Overlapping returns. Three-month forward returns sampled quarterly do not overlap, so the observation window is not the source of the serial correlation; it is more likely persistent omitted variables or a slow-moving dependent variable. The analyst should verify the sampling frequency before attributing the Durbin-Watson result.
Net effect. Coefficients may be usable, standard errors are not, and the model's apparent fit overstates its forecasting power.
2. The term spread has a t-statistic of 0.6. Should Raman drop it from the model? Explain the inferential trap and state what she should do instead.
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No, not on this evidence. Dropping a variable because its t-statistic is 0.6 is the classic multicollinearity trap. When two regressors share most of their variation, the model struggles to attribute explanatory power to either, so both t-statistics shrink even if both belong in the model. The correct interpretation is 'the data cannot separate these two effects', not 'the term spread has no effect'.
Also note the specification issue. The term spread and dividend yield are both proxies for expected returns and monetary conditions. If theory suggests the term spread belongs, dropping it will bias the retained dividend-yield coefficient (omitted variable bias) and make the model look better than it is.
What she should do instead.
- Re-specify: replace the two collinear levels with an economically motivated combination, or orthogonalise one against the other.
- Test the joint restriction: an F-test on the two collinear coefficients together, rather than two separate t-tests.
- Check stability: estimate the model on two subsamples and compare coefficients.
- Report robust (heteroskedasticity-consistent, and if needed HAC) standard errors before making any keep/drop decision.
- Only after these steps should a variable be removed, and then on economic as well as statistical grounds.
3. Given the diagnostics, what is a defensible conclusion for the investment committee, and what would Raman have to do before the fitted values could be used to size positions?
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What is defensible now. Only a qualitative statement: dividend yield, credit spreads and PMI changes are jointly associated with forward index returns in this sample. The precise coefficient values, their individual significance, and the 0.61 R-squared should not be quoted as evidence of predictive power.
Before using fitted values for sizing, she needs:
- Correct inference: re-estimate with White or Newey-West (HAC) standard errors to handle the heteroskedasticity and serial correlation. Recheck which coefficients remain significant.
- Address collinearity: joint tests and a re-specified model, as above.
- Out-of-sample validation: estimate on 2000-2015, test on 2016-2023. A model with in-sample R-squared of 0.61 that collapses out of sample is not a sizing tool.
- Stability and regime checks: coefficients in crisis versus expansion periods; the credit-spread coefficient in particular is likely to behave differently in stress.
- Practical sizing link: convert fitted returns to expected excess returns, feed them into a risk model with a tracking-error budget, and cap the position implied by a single model.
Committee message. The R-squared is not evidence of robustness; it is inflated by collinearity and serial correlation, and the standard errors are invalid. Three of four t-statistics being significant is consistent with a model that has real signal but unreliable individual inference. Do not size on fitted values until the model has been re-estimated with robust errors and validated out of sample.