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Lesson M08.L03: Serial Correlation and Spurious Regression

Field Detail
Module 08 — Time Series Basics
Level Graduate (ECMT5001 S1 2027)
Duration 30–35 min
Learning Objective Define serial correlation and its consequences for OLS; derive why spurious regression occurs with non-stationary series; apply diagnostic checks for both problems.
Provenance Wooldridge, Introductory Econometrics, Ch 10.5, 11.1

Explanation

1. Serial Correlation (Autocorrelation)

Serial correlation (autocorrelation) exists when the error terms at different time periods are correlated:

\[\text{Corr}(u_t, u_s) \neq 0 \quad \text{for some } t \neq s\]

This violates Gauss-Markov assumption MLR.5 (no serial correlation in errors), which assumes:

\[\text{Cov}(u_t, u_s) = 0 \quad \forall\; t \neq s\]

Serial correlation is the norm, not the exception, in economic time series — residuals that reflect omitted cyclical forces, adjustment costs, habit formation, or policy persistence will be correlated over time.

2. The AR(1) Error Process

The most common model for serial correlation is the AR(1) (first-order autoregressive) process:

\[\boxed{u_t = \rho u_{t-1} + e_t}\]

where: - \(\rho\) is the autocorrelation coefficient, \(|\rho| < 1\) for covariance stationarity. - \(e_t \sim \text{i.i.d.}(0, \sigma_e^2)\) is white noise (serially uncorrelated). - When \(\rho > 0\): positive serial correlation — positive errors tend to follow positive errors (smooth, persistent residuals). - When \(\rho < 0\): negative serial correlation — errors alternate in sign (oscillating residuals, less common in macro).

Under this AR(1) process, the error variance and autocovariances are:

\[\text{Var}(u_t) = \frac{\sigma_e^2}{1 - \rho^2}, \quad \text{Cov}(u_t, u_{t-h}) = \rho^h \cdot \frac{\sigma_e^2}{1 - \rho^2}\]

3. Consequences of Serial Correlation for OLS

If the regressors satisfy strict exogeneity, OLS remains:

  • Unbiased: \(E(\hat{\beta}) = \beta\) — serial correlation in errors does not bias the coefficient estimates.
  • Consistent: \(\text{plim}(\hat{\beta}) = \beta\).

However, OLS is:

  • Inefficient: GLS (or FGLS) with the correct error structure is more efficient.
  • Invalid standard errors: The OLS variance formula \(\hat{V}(\hat{\beta}) = \hat{\sigma}^2(X'X)^{-1}\) is incorrect when errors are serially correlated. This means t-statistics, F-statistics, and confidence intervals are all invalid.

The direction of standard error bias with positive serial correlation (\(\rho > 0\)):

OLS standard errors are typically understated (too small), leading to over-rejection of \(H_0\) and spuriously significant t-statistics.

Remedies: 1. HAC standard errors (Newey-West): correct the variance-covariance matrix to account for serial correlation (and heteroskedasticity) without modelling the specific error structure. 2. FGLS / Cochrane-Orcutt: estimate \(\rho\) from residuals, transform data to remove AR(1) correlation, run OLS on the transformed model. 3. Include lagged regressors / lagged \(y\) to absorb the serial correlation structurally.

4. Testing for Serial Correlation

Visual inspection: Plot OLS residuals against time. Positive serial correlation: runs of positive residuals followed by runs of negative residuals. Negative: alternating patterns.

Breusch-Godfrey (BG) LM test: Regress OLS residuals \(\hat{u}_t\) on \(\hat{u}_{t-1}, \ldots, \hat{u}_{t-p}\) and all original regressors. Test \(H_0\): \(\rho_1 = \cdots = \rho_p = 0\) using LM statistic \(\sim \chi^2_p\).

Durbin-Watson statistic (for AR(1) only, no lagged \(y\) in the model):

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

Interpretation: \(DW \approx 2\) (no AC), \(DW < 2\) (positive AC), \(DW > 2\) (negative AC).

5. Spurious Regression

Spurious regression is one of the most dangerous traps in time series econometrics. It occurs when two unrelated time series, each exhibiting a trend or unit root (non-stationarity), are regressed on each other:

The result: Very high \(R^2\), highly significant t-statistics, and large coefficients — even though the two series have no genuine economic relationship.

Classic example:

Regress Australian GDP (in levels, \(A\)) on US population (in millions, \(P\)):

\[A_t = \beta_0 + \beta_1 P_t + u_t\]

Both series trend upward over time. OLS may find \(R^2 > 0.95\) and a highly significant \(\hat{\beta}_1\) — but this is entirely spurious. Australian GDP is not caused by US population size; both simply trend upward over the 20th century.

Formal explanation (Granger and Newbold, 1974):

When \(y_t\) and \(x_t\) are independent random walks (unit root processes):

\[y_t = y_{t-1} + e_t, \quad x_t = x_{t-1} + v_t \quad (e_t \perp v_t)\]

The true slope is \(\beta_1 = 0\), but the OLS \(t\)-statistic for \(\hat{\beta}_1\) does not follow a \(t\)-distribution. It diverges as \(T \to \infty\):

\[t \to \pm\infty \text{ in distribution as } T \to \infty\]

In finite samples, you will almost always reject \(H_0: \beta_1 = 0\) even when the truth is \(\beta_1 = 0\). The standard critical values (e.g. ±1.96) are entirely wrong.

The residuals are highly serially correlated (DW statistic near 0) — this is in fact the telltale sign of a spurious regression.

Rule: Always check for spurious regression when working with non-stationary time series: 1. Plot the series: do they trend? 2. Compute DW on the regression residuals: if DW ≪ 2, suspect spurious regression. 3. Test each series for a unit root (ADF test — see M08.L04). 4. If both series have unit roots and are not cointegrated, do not run OLS in levels. First-difference the series instead.

Worked Example

Identifying a Spurious Regression

Data: Australian nominal GDP (\(A_t\), billions AUD) and US population (\(P_t\), millions), annual 1960–2023.

Step 1. Estimate OLS in levels:

\[\hat{A}_t = -2{,}850 + 72.4\, P_t, \quad R^2 = 0.98, \quad t_{\hat{\beta}} = 54.3, \quad DW = 0.12\]

Step 2. Red flags:

Indicator Value Interpretation
\(R^2\) 0.98 Unusually high — both series trend
\(t\)-statistic 54.3 Enormous — but invalid
DW statistic 0.12 Extremely low → severe positive serial correlation → spurious regression

Step 3. Why R² is misleading here.

The high \(R^2\) reflects a shared trend, not economic causation. If we fit a common time trend \(t\) to both series and remove it, the residuals would have near-zero correlation. The regression is capturing "both things went up over time", nothing more.

Step 4. Solution — use growth rates (first differences).

\[\Delta A_t = \beta_0 + \beta_1 \Delta P_t + u_t\]

After differencing, \(R^2\) drops dramatically and the t-statistic becomes insignificant (\(p > 0.5\)), correctly indicating no relationship between Australian GDP growth and US population growth.

Conclusion: The original regression was entirely spurious. The DW statistic near zero was the critical warning sign.

Common Misconception

"A high R² in a time series regression indicates a strong causal relationship."

This is false, especially when the series are non-stationary (trending or have a unit root).

In a spurious regression, \(R^2\) can approach 1 even when the two series are completely independent random walks. The high \(R^2\) reflects nothing more than common trending behaviour — both series are moving in the same direction over time, driven by their own dynamics, not by each other.

The correct interpretation of \(R^2\) in time series requires: 1. Checking that both series are stationary (or cointegrated if non-stationary). 2. Confirming that regression residuals show no severe serial correlation (DW not near 0). 3. Economic plausibility — does theory support the relationship?

High \(R^2\) is a necessary but far-from-sufficient condition for a valid time series relationship.

Practice Prompts

  1. Consequences of AR(1) errors. A regression of quarterly consumer spending on disposable income yields an OLS \(\hat{\rho} = 0.72\) from the residuals. (a) Is OLS biased? Justify your answer (assume strict exogeneity holds). (b) What is the consequence for the reported t-statistics? (c) Describe two approaches to obtain valid inference despite the serial correlation.

  2. Spurious regression detection. You regress monthly Australian house prices on monthly global oil prices over 1990–2023. Results: \(R^2 = 0.91\), \(DW = 0.18\), \(t\)-statistic on oil price = 38.7. (a) Is this likely a spurious regression? List the evidence. (b) What would you test next, and what would you do if the test confirms your suspicion? (c) If both series are I(1), under what condition could you still run OLS in levels without it being spurious?

  3. DW statistic interpretation. Three regressions on different time series datasets yield DW statistics of 0.24, 1.95, and 3.61. (a) For each DW value, state whether it suggests positive autocorrelation, no autocorrelation, or negative autocorrelation. (b) For the regression with DW = 0.24, what is the approximate estimated \(\hat{\rho}\) (using the approximation \(DW \approx 2(1-\hat{\rho})\))? (c) What follow-up steps would you take for the regression with DW = 0.24?

Further Resources

  • Wooldridge, Introductory Econometrics (7e), §10.5 (Serial Correlation), §11.1 (OLS under Serial Correlation).
  • Granger, C.W.J. & Newbold, P. (1974). "Spurious regressions in econometrics." Journal of Econometrics, 2(2), 111–120. — Foundational paper on spurious regression.
  • Newey, W.K. & West, K.D. (1987). "A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix." Econometrica, 55(3), 703–708. — HAC standard errors.
  • Hamilton, J.D. (1994). Time Series Analysis, Ch 8 — Serial correlation and GLS.
  • See also: M08.L04 (Stationarity and Unit Root Tests — solution to the spurious regression problem).