ECMT6002 Chap.3 Diagnostics and Robust Inference
Diagnostics and Robust Inference
This chapter diagnoses what happens when the disturbance variance is not constant or disturbances are correlated over time. OLS coefficients can remain centred on the target under suitable exogeneity conditions while conventional standard errors fail. Robust and autocorrelation consistent covariance estimators repair inference, not a biased conditional mean.
The decision sequence is therefore to identify the failure, decide whether coefficients or only inference are threatened, and choose a remedy appropriate to the sampling structure. Heteroskedasticity means conditional error variance changes across observations, often producing a funnel pattern in residual plots. Serial correlation means errors at different dates remain related after the model conditions on its regressors.
Both violate the spherical error covariance used by the conventional OLS variance formula. Under the maintained conditional mean, the fitted coefficients can still target the same slopes, but standard t and F calculations become unreliable. A sandwich covariance estimator changes estimated uncertainty without refitting the line.
Time-ordered data require a covariance estimator that also allows cross-date dependence, with the lag window chosen deliberately. Feasible GLS takes a different route by specifying and estimating a covariance structure, transforming the regression and seeking efficiency. Its gain is conditional on that structure being credible. Residual evidence should therefore be read together with the design and timing.
None of these covariance remedies repairs omitted variable bias, simultaneity or measurement error, because those problems concern correlation between a regressor and the structural disturbance rather than the disturbance variance alone.
What this chapter covers
- 01
Heteroskedasticity
- 02
Serial correlation
- 03
Residual diagnostics
- 04
Heteroskedasticity robust errors
- 05
Autocorrelation consistent errors
- 06
Efficiency and feasible GLS
Worked example · free
Comparing conventional and robust inference
- +1Use estimate minus null divided by standard error.
- +1The conventional statistic is 0.80 divided by 0.20, which equals 4.
- +1The robust statistic is 0.80 divided by 0.32, which equals 2.5.
Key terms
- Heteroskedasticity
- A conditional disturbance variance that changes across observations.
- Serial correlation
- Correlation between disturbances indexed by different time periods.
- Robust covariance
- A covariance estimator valid under a broader variance structure than the classical formula.
- Feasible GLS
- A transformed estimator that uses an estimated disturbance covariance structure.
Diagnostics and Robust Inference FAQ
Do robust errors fix endogeneity?
No. They change estimated sampling uncertainty under variance misspecification. If a regressor is correlated with the structural disturbance, the coefficient target itself is compromised and a different identification strategy is needed.
When is serial correlation especially plausible?
It is common in time ordered data because omitted persistent forces and dynamic adjustment can make nearby disturbances related. The lag pattern and sampling frequency should guide both diagnosis and covariance estimation.
Why not always use feasible GLS?
Its efficiency gain depends on specifying the disturbance covariance structure well. A poor structure can be fragile, whereas robust covariance estimation often offers safer inference without changing the conditional mean estimator.
What should follow a diagnostic rejection?
Name the violated covariance assumption, identify whether coefficient consistency is still credible, and select a remedy that matches the sampling structure. A robust covariance estimate may repair inference, while a misspecified conditional mean requires redesign. Report the diagnostic as evidence about a particular failure rather than declaring the entire regression invalid without qualification.
Exam move
Use a diagnostic matrix with rows for heteroskedasticity, serial correlation and endogeneity and columns for coefficient bias, conventional variance validity and suitable remedy. Work problems by naming the sampling structure first. Then state explicitly whether you are correcting the estimator, the covariance matrix or both.
Sketch residuals against fitted values for constant spread, widening spread and curved conditional mean, and explain why each picture suggests a different concern. Take one coefficient and recompute its t statistic under conventional and robust standard errors so the unchanged numerator and changed denominator are visible.
For time series, draw a lag diagram and identify which error pairs a covariance estimator must allow to move together. Compare robust OLS with feasible GLS in a decision table that records consistency conditions, efficiency claims and sensitivity to covariance misspecification.
When a test rejects homoskedasticity, avoid saying the whole regression is invalid; state precisely which conventional inference is no longer justified. Conversely, never let robust standard errors conceal a questionable regressor. Finish every diagnostic response with three clauses: what was detected, what conclusion is affected, and what remedy is supported by the data structure.
This language keeps variance problems separate from failures of identification.
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