Lesson M09.L03: Correcting for Serial Correlation

Field	Detail
Module	M09 — Serial Correlation and Time Series Inference
Level	Graduate (ECMT5001 S1 2027)
Duration	35 min
Learning Objective	Apply HAC standard errors and feasible GLS to obtain valid inference when serial correlation is present; understand when each approach is appropriate and what each changes
Provenance	Wooldridge, Introductory Econometrics, Ch 12.3

Explanation

Once serial correlation is detected, two broad remedies exist:

Keep OLS, fix the standard errors → HAC (Newey-West)
Change the estimator → GLS / feasible GLS (Cochrane-Orcutt, Prais-Winsten)

1. HAC (Newey-West) Standard Errors

HAC stands for Heteroskedasticity and Autocorrelation Consistent. It corrects OLS standard errors for both heteroskedasticity and serial correlation simultaneously, analogous to White (robust) standard errors in cross-section settings.

Key property: HAC SEs adjust the variance-covariance matrix of \(\hat{\boldsymbol{\beta}}^{\text{OLS}}\) without changing the point estimates.

The HAC estimator replaces the usual \(\hat{\sigma}^2(\mathbf{X}'\mathbf{X})^{-1}\) with a sandwich estimator:

\[\widehat{\text{Var}}_{\text{HAC}}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}'\mathbf{X})^{-1} \hat{\mathbf{S}} (\mathbf{X}'\mathbf{X})^{-1}\]

where \(\hat{\mathbf{S}}\) is the Newey-West long-run variance estimator that uses a kernel to weight autocovariances of the score:

\[\hat{S} = \hat{\Gamma}_0 + \sum_{j=1}^{L} w_j \left(\hat{\Gamma}_j + \hat{\Gamma}_j'\right)\]

with \(\hat{\Gamma}_j = \frac{1}{n}\sum_{t=j+1}^{n} \hat{u}_t \hat{u}_{t-j} \mathbf{x}_t \mathbf{x}_{t-j}'\) and Bartlett kernel weights \(w_j = 1 - j/(L+1)\).

Bandwidth choice: The lag truncation \(L\) is typically chosen as \(L = \lfloor 4(n/100)^{2/9} \rfloor\) (Newey-West rule of thumb) or \(L = \lfloor 0.75 \cdot n^{1/3} \rfloor\).

Practical recommendation: Use HAC SEs as the default for any time-series regression. They are valid under very general conditions — you don't need to know the exact form of the serial correlation.

2. GLS with AR(1) Errors: Quasi-Differencing

When the error process is known (or assumed) to be AR(1), Generalized Least Squares (GLS) eliminates the serial correlation by transforming the model.

Population model:

\[y_t = \beta_0 + \beta_1 x_t + u_t, \qquad u_t = \rho u_{t-1} + e_t, \quad e_t \sim \text{i.i.d.}(0, \sigma_e^2)\]

Quasi-differencing transformation: Subtract \(\rho\) times the lagged equation from the current equation:

\[y_t - \rho y_{t-1} = \beta_0(1-\rho) + \beta_1(x_t - \rho x_{t-1}) + e_t\]

Define the quasi-differenced variables:

\[\tilde{y}_t = y_t - \rho y_{t-1}, \qquad \tilde{x}_t = x_t - \rho x_{t-1}, \qquad \tilde{\beta}_0 = \beta_0(1-\rho)\]

Then the transformed model is:

\[\tilde{y}_t = \tilde{\beta}_0 + \beta_1 \tilde{x}_t + e_t\]

where \(e_t\) is i.i.d. — so OLS on the quasi-differenced model is BLUE (Gauss-Markov applies).

Note on the first observation: Under Prais-Winsten, the first observation is transformed as \(\tilde{y}_1 = \sqrt{1-\rho^2} \cdot y_1\) (preserves it). Under Cochrane-Orcutt, the first observation is dropped.

3. Feasible GLS (FGLS)

In practice \(\rho\) is unknown and must be estimated. The Feasible GLS (or FGLS) procedure:

Step 1: Estimate the original model by OLS, obtain residuals \(\hat{u}_t\).

Step 2: Estimate \(\rho\) by regressing \(\hat{u}_t\) on \(\hat{u}_{t-1}\) (without intercept):

\[\hat{\rho} = \frac{\sum_{t=2}^{n} \hat{u}_t \hat{u}_{t-1}}{\sum_{t=1}^{n-1} \hat{u}_t^2}\]

Step 3: Form quasi-differenced variables using \(\hat{\rho}\):

\[\tilde{y}_t = y_t - \hat{\rho}\, y_{t-1}, \qquad \tilde{x}_t = x_t - \hat{\rho}\, x_{t-1}\]

Step 4: Regress \(\tilde{y}_t\) on \(\tilde{x}_t\) by OLS — these are the Cochrane-Orcutt estimates.

Iterative Cochrane-Orcutt repeats Steps 2–4 until \(\hat{\rho}\) converges.

4. First Differencing as a Special Case

If \(\rho \approx 1\) (the process is near a unit root), quasi-differencing with \(\hat{\rho} \approx 1\) becomes first differencing:

\[\Delta y_t = y_t - y_{t-1} = \beta_1 \Delta x_t + \Delta u_t\]

First differencing removes any I(1) non-stationarity and eliminates AR(1) serial correlation when \(\rho = 1\). It is common practice for trending macro series (GDP, price levels, employment). The cost: the constant \(\beta_0\) is absorbed, and you lose one observation.

Comparing HAC vs GLS

Feature	HAC SEs (OLS + corrected SEs)	FGLS (Cochrane-Orcutt)
Point estimates change?	No — only SEs change	Yes — both \(\hat{\beta}\) and SEs change
Requires knowing error structure?	No	Yes (assumes AR(1))
Efficiency	Less efficient if AR(1) is correct	More efficient under correct specification
Robustness	More robust to mis-specification	Sensitive to wrong \(\rho\) or wrong order
Practical recommendation	Default choice	Use only with strong AR(1) evidence

Worked Example

Setup: Annual data on interest rates and inflation, \(n = 50\) years.

\[\text{rate}_t = \beta_0 + \beta_1 \text{inflation}_t + u_t\]

OLS estimates (with standard SEs):

	Coeff	Std Err	\(t\)
Intercept	1.42	0.38	3.74
inflation	0.81	0.12	6.75

Diagnostics: BG test (p=1) gives LM = 8.3 > \(\chi^2_{1,0.05} = 3.84\) → serial correlation confirmed. \(\hat{\rho} = 0.52\).

Step 1: Apply Cochrane-Orcutt

Quasi-differenced variables (\(t = 2, \ldots, 50\)):

\[\widetilde{\text{rate}}_t = \text{rate}_t - 0.52\,\text{rate}_{t-1}, \quad \widetilde{\text{infl}}_t = \text{infl}_t - 0.52\,\text{infl}_{t-1}\]

Step 2: OLS on transformed model — FGLS estimates:

	Coeff	Std Err	\(t\)
\(\tilde{\beta}_0\) (intercept)	0.68	0.24	2.83
inflation	0.74	0.16	4.63

Step 3: OLS with HAC SEs (Newey-West, \(L = 3\)):

	Coeff	Std Err (HAC)	\(t\)
Intercept	1.42	0.52	2.73
inflation	0.81	0.19	4.26

Comparison:

Method	\(\hat{\beta}_1\) (inflation)	SE	\(t\)-stat
OLS (naive)	0.81	0.12	6.75
OLS + HAC	0.81	0.19	4.26
FGLS (CO)	0.74	0.16	4.63

Conclusions: - HAC keeps the OLS coefficient (0.81) but widens the SE from 0.12 to 0.19 — the \(t\)-stat falls from 6.75 to 4.26. Still significant but less dramatically so. - FGLS changes the coefficient estimate (0.81 → 0.74) and widens the SE. Different interpretation: the estimated effect of inflation on interest rates is smaller under FGLS. - Naive OLS \(t = 6.75\) is inflated — the true evidence is weaker.

Common Misconception

"Correcting for serial correlation changes the coefficient estimates."

This is only partially true — it depends on the method:

HAC SEs: The OLS point estimates are unchanged. Only the standard errors (and hence \(t\)-stats and confidence intervals) change. This is the most common and recommended correction.
GLS/FGLS (Cochrane-Orcutt): Both the coefficient estimates and the standard errors change, because the estimator itself changes — it minimises a different (weighted) objective function.

The key distinction: HAC is a correction to inference, not estimation. GLS is a correction to both.

Practice Prompts

A researcher estimates a wage-inflation model and finds \(\hat{\rho} = 0.65\) from OLS residuals. They apply Cochrane-Orcutt. After transformation, the coefficient on inflation changes from 0.52 to 0.39 and the standard error increases from 0.08 to 0.13. Compute the \(t\)-statistic for the OLS and FGLS estimates. Which should the researcher report, and why?
For a monthly financial time series with \(n = 240\), compute the Newey-West bandwidth \(L\) using the rule \(L = \lfloor 4(n/100)^{2/9} \rfloor\). How does this bandwidth affect the bias-variance tradeoff in the HAC estimator?
A researcher has annual GDP data and suspects the error process may be AR(1) with \(\rho\) close to 1. They apply Cochrane-Orcutt with \(\hat{\rho} = 0.97\). Explain why first differencing might be more appropriate than Cochrane-Orcutt in this case, and what additional diagnostic (unit root test) they should run first.

Further Resources

Wooldridge, Introductory Econometrics (7th ed.), Ch 12.3
Newey, W.K. & West, K.D. (1987), "A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix," Econometrica, 55(3), 703–708
Cochrane, D. & Orcutt, G.H. (1949), "Application of Least Squares Regression to Relationships Containing Auto-Correlated Error Terms," JASA
Python: sm.OLS(y, X).fit(cov_type='HAC', cov_kwds={'maxlags': L}); Stata: newey y x, lag(L)