Autocorrelation and the Durbin-Watson statistic

Autocorrelation (or serial correlation) means the errors of different observations are correlated: Cov(u_t, u_s) ≠ 0 for t ≠ s. The classical assumption says it is zero. The usual model, the first-order autoregressive or AR(1) scheme, says each error is part of the last plus a fresh shock:

Where does it come from? Inertia in economic series (output, prices, consumption adjust slowly); an omitted variable that is itself persistent, or the wrong functional form, whose effect sits in the error (lesson 7); data that have been smoothed or interpolated; and lagged adjustment. It matters mostly for time series, where observations have a natural order. In a random cross-section there is no order to follow.

Positive autocorrelation shows up as long runs of residuals above and then below zero; negative as an alternating zigzag; none as a patternless scatter. The lab below draws all three.

As with heteroscedasticity (lesson 9), unbiasedness needs only E(u | X) = 0, so with exogenous regressors OLS remains unbiased and consistent. What fails:

• OLS is no longer efficient. It ignores the information in the last residual about the next. Prais-Winsten estimation, which uses it, has a smaller variance.
• The usual standard errors are wrong, because the formula assumes uncorrelated errors. With positive autocorrelation and a persistent regressor they are typically too small, so t statistics are too large and R² is overstated: spurious significance.

The lab simulates the model with a persistent regressor. Raise ρ and see the usual standard error fall short of the real spread and the interval’s coverage drop below 95%. Then set the regressor’s persistence φ to zero: with a regressor that has no memory, the usual standard error holds up even with strongly autocorrelated errors. Autocorrelation does its damage to the standard error when both the error and the regressor are persistent, which is the usual state of macro time series.

The Durbin-Watson (DW) statistic compares each residual with the one before it:

DW has no single critical value. Its distribution depends on the regressors, so tables give two bounds, dL and dU, for each sample size and number of regressors. The decision runs along the line from 0 to 4:

Two caveats the exam asks about. The test needs a constant in the regression and non-random regressors; with a lagged dependent variable among the regressors DW is biased towards 2, so it can miss autocorrelation that is there (use Durbin’s h or the Breusch-Godfrey test). And it examines only first-order autocorrelation.

The Breusch-Godfrey (BG) test fixes DW’s weaknesses. Regress the residuals on the original regressors and p lagged residuals, and take LM = nR² of that regression, which is chi-square with p degrees of freedom under H₀ of no autocorrelation up to order p. It allows higher orders, lagged dependent variables and other regressors, and gives a p-value instead of a zone.

The Ljung-Box statistic tests whether a whole set of autocorrelation coefficients (of residuals or of a series) is jointly zero; it is used mostly in time-series modelling (lesson 16). A runs test counts how often the residuals change sign: too few runs means positive autocorrelation. Only DW, BG and the Ljung-Box appear regularly in exams.

The consumption function of macroeconomics: regress the log of India’s private consumption on the log of GDP, 1960 to 2025 (World Bank, constant rupees, read 2026-10-24). The slope is 0.88, with a standard error of 0.0041 and R² = 0.9986. Everything looks wonderful. But the Durbin-Watson statistic is 0.165, ρ̂ = 0.937, and the Breusch-Godfrey test gives LM = 54.1 (p < 0.0001). The residuals move in long, slow swings, which is autocorrelation as strong as it gets.

That standard error of 0.0041 cannot be believed. Prais-Winsten estimation (which models the autocorrelation) gives a slope of 0.875 with a standard error of 0.0183 and ρ̂ = 0.961, more than four times as large, and Cochrane-Orcutt gives 0.895 with a standard error of 0.0172. Differencing the logs (annual growth rates, 65 observations) gives a slope of 0.772 with a standard error of 0.0526, R² = 0.774 and a Durbin-Watson statistic of 1.78 (Breusch-Godfrey p = 0.47): the autocorrelation is gone, and the elasticity of consumption growth to income growth is about 0.77.

Why is the autocorrelation so strong? Both series trend upwards through the whole period, and regressing a trending series on another trending series can produce a high R² and a low DW even when nothing links them. Economic theory says consumption and income are related, but the same pattern is the signature of a spurious regression (lesson 15), where DW well below R² is the warning (the Granger-Newbold rule of thumb), and whether these two series really share a long-run relationship is a question for cointegration tests (lesson 16), which in these data do not find one.

• Look for a specification error first. A missing variable, the wrong functional form or missing dynamics can look exactly like autocorrelation (lesson 7). Adding the omitted persistent variable, or a lag, may remove it.
• Generalised differencing (feasible GLS). If u_t = ρu_t−1 + ε_t, subtract ρ times the lagged equation:
  $$y_t-\rho y_{t-1}=\alpha (1-\rho )+\beta (x_t-\rho x_{t-1})+{\epsilon }_t$$
  The new error is white noise, so OLS on the transformed data is BLUE. ρ is unknown, so it is estimated from the residuals: Cochrane-Orcutt iterates between estimating ρ and the transformed regression and drops the first observation; Prais-Winsten keeps it by scaling it by √(1 − ρ²).
• Newey-West (HAC) standard errors keep the OLS estimate and correct the standard errors for both heteroscedasticity and autocorrelation, as robust standard errors do for heteroscedasticity alone.
• First differences, when ρ is close to 1: the regression of Δy on Δx. Used in the growth-rate regression above.

The lagged dependent variable. A model such as y_t = α + βx_t + γy_t−1 + u_t with autocorrelated u has y_t−1 correlated with u_t−1, hence with the error: a regressor correlated with the error. Then OLS is biased and inconsistent, not just inefficient, and DW is biased towards 2. That is a problem of endogeneity (lesson 11).

This lesson has a short question set, because few past papers ask about autocorrelation; the exercises above cover the calculations. The matching questions on Durbin-Watson, Ljung-Box and Breusch-Godfrey are in the test atlas.