Lesson M09.L02: Testing for Serial Correlation

Field	Detail
Module	M09 — Serial Correlation and Time Series Inference
Level	Graduate (ECMT5001 S1 2027)
Duration	30–35 min
Learning Objective	Apply the Durbin-Watson and Breusch-Godfrey tests for serial correlation; understand the limitations of each and when to prefer BG over DW
Provenance	Wooldridge, Introductory Econometrics, Ch 12.2

Explanation

1. The Durbin-Watson Test

The Durbin-Watson (DW) statistic is the classical test for first-order serial correlation in OLS residuals:

\[\text{DW} = \frac{\sum_{t=2}^{n}(\hat{u}_t - \hat{u}_{t-1})^2}{\sum_{t=1}^{n}\hat{u}_t^2}\]

Relationship to the sample autocorrelation $\hat{\rho}$:

Expanding the numerator and simplifying:

\[\text{DW} \approx 2(1 - \hat{\rho})\]

where $\hat{\rho} = \frac{\sum_{t=2}^n \hat{u}_t \hat{u}_{t-1}}{\sum_{t=1}^n \hat{u}_t^2}$ is the first-order sample autocorrelation of the residuals.

Interpretation:

DW value	$\hat{\rho}$ implied	Conclusion
$\approx 2$	$\approx 0$	No serial correlation
$< 2$ (e.g., 0.8)	$> 0$	Positive serial correlation
$> 2$ (e.g., 3.2)	$< 0$	Negative serial correlation
$\approx 0$	$\approx 1$	Strong positive SC
$\approx 4$	$\approx -1$	Strong negative SC

Critical values: The DW test compares the computed statistic against lower ($d_L$) and upper ($d_U$) bounds from the DW table (which depend on $n$, $k$, and significance level):

\[\begin{cases} \text{Reject } H_0 \text{ (pos. SC)} & \text{if } \text{DW} < d_L \\ \text{Inconclusive} & \text{if } d_L \leq \text{DW} \leq d_U \\ \text{Fail to reject } H_0 & \text{if } \text{DW} > d_U \end{cases}\]

Limitations of the DW Test

Strict exogeneity required: DW is only valid when no lagged dependent variables appear as regressors. If $y_{t-1} \in \mathbf{x}_t$, the DW statistic is biased toward 2 (under-rejects).
Inconclusive region: There is a range $[d_L, d_U]$ where the test gives no answer. This is the most frustrating feature for practitioners.
Only tests AR(1): DW is not designed to detect higher-order serial correlation, e.g., AR(2) or seasonal patterns.
Doesn't test for negative SC by default: Most tables are set up for positive SC; negative SC test requires comparing $(4 - \text{DW})$ to the same critical values.

2. The Breusch-Godfrey Test

The Breusch-Godfrey (BG) LM test addresses all three DW limitations. It: - Allows lagged dependent variables in the regression - Tests for serial correlation up to any pre-specified order $p$ - Has a clean $\chi^2$ distribution under $H_0$ (no inconclusive region)

Hypotheses:

\[H_0: \rho_1 = \rho_2 = \cdots = \rho_p = 0 \quad \text{(no serial correlation up to order } p\text{)}$$ $$H_1: \text{at least one } \rho_j \neq 0\]

Steps for the BG Test:

Step 1: Estimate the original model by OLS and save the residuals $\hat{u}_t$:

\[y_t = \beta_0 + \beta_1 x_{1t} + \cdots + \beta_k x_{kt} + u_t \quad \Rightarrow \quad \hat{u}_t = y_t - \hat{y}_t\]

Step 2: Run the auxiliary regression: regress $\hat{u}_t$ on the original regressors and $p$ lags of the residuals:

\[\hat{u}_t = \delta_0 + \delta_1 x_{1t} + \cdots + \delta_k x_{kt} + \gamma_1 \hat{u}_{t-1} + \gamma_2 \hat{u}_{t-2} + \cdots + \gamma_p \hat{u}_{t-p} + \text{error}\]

The regressors $\mathbf{x}_t$ are included to partial out the effect of the design matrix.

Step 3: Compute the LM statistic:

\[\text{LM} = n \cdot R^2_{\text{aux}} \xrightarrow{d} \chi^2_p \quad \text{under } H_0\]

where $R^2_{\text{aux}}$ is from the auxiliary regression and $n$ is the usable sample size.

Step 4: Reject $H_0$ at level $\alpha$ if $\text{LM} > \chi^2_{p,\alpha}$.

Key advantage: The BG test only requires contemporaneous exogeneity (TS.3′) — it is valid even with $y_{t-1}$ in the model.

Comparison: DW vs BG

Feature	Durbin-Watson	Breusch-Godfrey
Requires strict exogeneity	Yes	No (contemporaneous sufficient)
Valid with $y_{t-1}$ as regressor	No	Yes
Tests higher-order SC	No (AR(1) only)	Yes (up to order $p$)
Inconclusive region	Yes	No
Null distribution	Non-standard (DW tables)	$\chi^2_p$ (simple)
Sample requirement	Moderate	Larger samples preferred

Bottom line: In most applied work, prefer the BG test. DW is still reported in many software outputs but should not be your primary diagnostic when lagged $y$ is present.

Worked Example

Data: Annual data on retail sales (in $bn) regressed on GDP and a trend, $n = 80$ observations.

\[\text{sales}_t = \beta_0 + \beta_1 \text{GDP}_t + \beta_2 t + u_t\]

After OLS estimation, the researcher computes:

$\text{DW} = 1.12$
First-order residual autocorrelation: $\hat{\rho}_1 = (2 - 1.12)/2 = 0.44$

Step 1: DW check

From DW tables with $n=80$, $k=2$, $\alpha = 5\%$: $d_L = 1.59$, $d_U = 1.73$.

Since $\text{DW} = 1.12 < d_L = 1.59$ → reject $H_0$, evidence of positive AR(1) serial correlation.

Step 2: BG test (order $p = 2$)

Run auxiliary regression:

\[\hat{u}_t = \delta_0 + \delta_1 \text{GDP}_t + \delta_2 t + \gamma_1 \hat{u}_{t-1} + \gamma_2 \hat{u}_{t-2} + \text{error}\]

Results: $R^2_{\text{aux}} = 0.21$, effective $n = 78$ (lost 2 observations for lags).

\[\text{LM} = 78 \times 0.21 = 16.38\]

Critical value: $\chi^2_{2, 0.05} = 5.99$.

Since $16.38 > 5.99$ → reject $H_0$ at the 5% level. There is significant serial correlation up to order 2.

Step 3: Interpret and compare

Both tests reject $H_0$. However, the BG test additionally tells us that the serial correlation extends beyond AR(1) (the AR(2) lag is also significant). DW would have missed this nuance.

Software note (Stata): estat bgodfrey, lags(2) after regress; in Python: statsmodels.stats.diagnostic.acorr_breusch_godfrey(result, nlags=2).

Common Misconception

"DW ≈ 2 guarantees there is no serial correlation."

This is false on two counts:

DW ≈ 2 means no evidence of AR(1) serial correlation — it says nothing about higher-order patterns. An AR(2) process with $\rho_1 = 0$ and $\rho_2 = 0.5$ can produce DW ≈ 2 while BG(p=2) clearly rejects.
Even for AR(1), the DW has an inconclusive region. DW in $[d_L, d_U]$ means the test is uninformative, not that $H_0$ is true.

Always use BG as the primary test, and choose $p$ based on the data frequency (e.g., $p = 4$ for quarterly data, $p = 12$ for monthly) or economic reasoning.

Practice Prompts

An analyst regresses log(employment) on log(output) and one lag of log(employment), obtaining DW = 1.85. They conclude "no serial correlation." Explain why this conclusion is unreliable. What test should they use instead?
You run a BG test with $p = 4$ on monthly data and compute $\text{LM} = 9.3$. At the 5% level, do you reject $H_0$? (Recall $\chi^2_{4,0.05} = 9.49$.) Interpret the result carefully — what can and cannot you conclude?
A time-series regression produces $\hat{\rho}_1 = -0.35$ from OLS residuals. Compute the approximate DW statistic. What direction of serial correlation does this suggest, and is it more or less common in macroeconomic data than positive serial correlation?

Further Resources

Wooldridge, Introductory Econometrics (7th ed.), Ch 12.2
Breusch, T.S. (1978), "Testing for Autocorrelation in Dynamic Linear Models," Australian Economic Papers
Godfrey, L.G. (1978), "Testing Against General Autoregressive and Moving Average Error Models," Econometrica
Python: statsmodels.stats.diagnostic.acorr_breusch_godfrey; Stata: estat bgodfrey

DW value	\(\hat{\rho}\) implied	Conclusion
\(\approx 2\)	\(\approx 0\)	No serial correlation
\(< 2\) (e.g., 0.8)	\(> 0\)	Positive serial correlation
\(> 2\) (e.g., 3.2)	\(< 0\)	Negative serial correlation
\(\approx 0\)	\(\approx 1\)	Strong positive SC
\(\approx 4\)	\(\approx -1\)	Strong negative SC

Feature	Durbin-Watson	Breusch-Godfrey
Requires strict exogeneity	Yes	No (contemporaneous sufficient)
Valid with \(y_{t-1}\) as regressor	No	Yes
Tests higher-order SC	No (AR(1) only)	Yes (up to order \(p\))
Inconclusive region	Yes	No
Null distribution	Non-standard (DW tables)	\(\chi^2_p\) (simple)
Sample requirement	Moderate	Larger samples preferred