Time Seriesadvanced

Serial Correlation in the Errors

Serial correlation (also called autocorrelation) means the regression errors are correlated across time, the canonical case being a first-order autoregressive process $u_t = \rho u_{t-1} + \varepsilon_t$ with $|\rho| < 1$ . Crucially, like heteroskedasticity, serial correlation does not bias OLS: under strict exogeneity the coefficients stay unbiased and consistent. What it breaks is the usual variance formula, so the standard errors are wrong, and the $t$ and $F$ statistics built on them are invalid. The practical message is that you can keep the OLS point estimates but must repair the inference.

Why it matters

If a shock in one quarter tends to carry into the next, the errors move in runs rather than independently. OLS still finds the right line on average, but it overstates how much independent information the sample contains, so the reported standard errors are typically too small and the $t$ statistics look more impressive than they should. The estimates are honest; the precision is not.

Formulas

AR(1) errors

u_t = \rho u_{t-1} + \varepsilon_t, \quad |\rho| < 1

rho is the autocorrelation of the errors; rho = 0 means no serial correlation.

Consequence for inference

\widehat{\mathrm{Var}}(\hat{\beta}_{\text{OLS}}) \ \text{is biased}

Default OLS standard errors are invalid under serial correlation, even though the slope is unbiased.

Worked examples

Scenario

After fitting a static time-series model you suspect the errors are positively autocorrelated.

Solution

Estimate `regress y x, robust`, predict the residuals with `predict uhat, resid`, then regress them on their own lag: `regress uhat L.uhat`. A large, significant coefficient on `L.uhat` estimates rho and signals AR(1) serial correlation. The OLS slope on `x` is still unbiased, but its default standard error is not.

NoteRegressing residuals on their lag is the intuition behind the formal tests covered next.

Common mistakes

✗Serial correlation biases the OLS coefficients. Under strict exogeneity the coefficients remain unbiased and consistent; only the standard errors are affected.
✗Positive serial correlation makes standard errors too large. It typically makes the default standard errors too small, so $t$ statistics are overstated.
✗Serial correlation and heteroskedasticity are unrelated problems. They are parallel: both leave OLS unbiased but invalidate the usual variance estimator, and both have robust-standard-error fixes.
✗If $\rho$ is small the problem can be ignored. Even modest autocorrelation distorts inference, and with a lagged dependent variable serial correlation can also threaten consistency.

Revision bullets

•Serial correlation = errors correlated over time
•Canonical case is AR(1), $u_t = \rho u_{t-1} + \varepsilon_t$
•OLS stays unbiased under strict exogeneity
•The usual standard errors are wrong, so $t$ and $F$ tests fail
•Parallel to heteroskedasticity: fix the inference, keep the estimates

Quick check

When the errors are serially correlated but strict exogeneity holds, OLS coefficient estimates are:

In the AR(1) error process $u_t = \rho u_{t-1} + \varepsilon_t$ , the parameter $\rho$ measures:

Connected topics

Heteroskedastic Robust SE TS.1-TS.6 Test Serial Fix Serial Unit Roots

Sources

Wooldridge (2019), §12.1
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Defines serial correlation, the AR(1) model, and shows that OLS stays unbiased while its standard errors become invalid.
Greene (2018), Ch. 20
Greene, William H. Econometric Analysis. 8th ed. Pearson, 2018.
Detailed treatment of autocorrelated disturbances and their effect on OLS efficiency and inference.

How to cite this page

Dr. Phil's Quant Lab. (2026). Serial Correlation in the Errors. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-serial-correlation

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