Chapter 12: Serial Correlation & Heteroskedasticity

Dealing with Complex Error Structures in Time Series

What is Serial Correlation?

The assumption of "no serial correlation" (TS.5) means that the errors in our regression are uncorrelated across time.

Corr(u_t, u_s) = 0 for all t ≠ s

Serial Correlation (or autocorrelation) is the violation of this assumption. It means the errors are correlated over time. The most common form is an AR(1) model for the errors:

u_t = ρu_t-1 + e_t

This is very common in static or FDL models where the dynamics are not fully captured.

Consequences of Serial Correlation

This should sound familiar! The consequences are very similar to those of heteroskedasticity.

What's Still OK?

OLS estimators are still unbiased and consistent (assuming TS.1'-TS.3').

What's Broken?

The OLS standard errors are biased and invalid.
Our t-stats, F-stats, and CIs are unreliable.
OLS is no longer BLUE.

The Bottom Line: Just like with heteroskedasticity, your coefficient estimates are fine, but your statistical inference is invalid.

Solution 1: Serial Correlation-Robust Standard Errors

The easiest and most modern solution is to use standard errors that are robust to both heteroskedasticity and serial correlation. These are often called HAC (Heteroskedasticity and Autocorrelation Consistent) or Newey-West standard errors.

This approach is very general and does not require us to know the specific form of the serial correlation.
It allows us to use OLS and still get valid inference in large samples, even if strict exogeneity is violated (as long as contemporaneous exogeneity holds).
The only tricky part is choosing a "truncation lag," but most software packages have a reasonable default.

Testing for Serial Correlation

It's still useful to test for serial correlation. The simplest test is for AR(1) serial correlation.

H₀: ρ = 0 in u_t = ρu_t-1 + e_t

A Simple t-Test for AR(1) Correlation:

Run your original OLS regression and get the residuals, û_t.
Run an auxiliary regression of the residuals on their lag: û_t on û_t-1.
The t-statistic on the coefficient of û_t-1 is a valid test for serial correlation.

Important: If your model includes a lagged dependent variable (or any regressor that isn't strictly exogenous), you must include all of the original x's in the auxiliary regression in step 2 for the test to be valid!

Check Your Understanding

You estimate a static model of the Phillips Curve and find strong evidence of AR(1) serial correlation. Your friend says, "This means your estimate of the unemployment coefficient is biased and inconsistent." Is your friend correct? Why or why not?

Answer:

Your friend is incorrect. Serial correlation alone does not cause bias or inconsistency in the OLS estimators. The key assumption for consistency is contemporaneous exogeneity (TS.3'), which is not violated by serial correlation in the errors.

The problem is with inference. You cannot trust the usual t-statistic on the unemployment coefficient. You should re-calculate it using a serial correlation-robust standard error.

Solution 2: Feasible Generalized Least Squares (FGLS)

If we assume the errors follow a specific model (like AR(1)) and the regressors are strictly exogenous, we can use FGLS to get more efficient estimates than OLS.

The Cochrane-Orcutt / Prais-Winsten Procedure:

Run OLS and get the residuals û_t.
Estimate ρ by regressing û_t on û_t-1. Call the estimate ρ̂.
Create quasi-differenced data: ỹ_t = y_t - ρ̂y_t-1 and x̃_tj = x_tj - ρ̂x_t-1,j.
Run OLS on the transformed variables. The resulting coefficients are the FGLS estimates.

This procedure purges the serial correlation, leading to more efficient estimates and valid standard errors (if the AR(1) model is correct).

Chapter 12 Summary

Serial correlation is a common feature of time series regressions, but we have a clear set of tools to diagnose and handle it.

Serial correlation violates the Gauss-Markov assumption TS.5.
It does not cause bias or inconsistency, but it invalidates our usual standard errors and tests.
The modern and most robust solution is to use OLS with HAC (Newey-West) standard errors, which corrects inference for both serial correlation and heteroskedasticity.
We can test for AR(1) serial correlation by regressing OLS residuals on their lag.
If regressors are strictly exogenous and the AR(1) model holds, FGLS (e.g., Prais-Winsten) is more efficient than OLS.