Durbin-Watson statistic

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The Durbin-Watson statistic is a test statistic used to detect the presence of autocorrelation in the residuals from a regression analysis. It is named after James Durbin and Geoffrey Watson.

If e_t is the residual associated with the observation at time t, then the test statistic is $d = {\sum_{t=2}^T (e_t - e_{t-1})^2 \over {\sum_{t=1}^T e_t^2}}$ . Its value always lies between 0 and 4.

A value of 2 indicates there appears to be no autocorrelation. Small values of d indicate successive error terms are, on average, close in value to one another, or positively correlated. Large values of d indicate successive error terms are, on average, much different in value to one another, or negatively correlated.

To test for postitive autocorrelation at significance α, the test statistic d is compared to lower and upper critical values (d_L,α and d_U,α):

If d < d_L,α, there is statistical evidence that the error terms are positively autocorrelated.
If d > d_U,α, there is statistical evidence that the error terms are not positively autocorrelated.
If d_L,α < d < d_U,α, the test is inconclusive.

To test for negative autocorrelation at significance α, the test statistic (4 - d) is compared to lower and upper critical values (d_L,α and d_U,α):

If (4 − d) < d_L,α, there is statistical evidence that the error terms are negatively autocorrelated.
If (4 - d) > d_U,α, there is statistical evidence that the error terms are not negatively autocorrelated.
If d_L,α < (4 − d) < d_U,α, the test is inconclusive.

The critical values, d_L,α and d_U,α, vary by level of significance (α), the number of observations, and the number of predictors in the regression equation. Their derivation is complex—statisticians typically obtain them from the appendices of statistical texts (e.g. Bounds for Critical Values for the Durbin-Watson Statistic: Λ = number of regressors, N = number of observations).

[edit] Durbin h-statistic

This statistic is biased for autoregressive moving average models, so that autocorrelation is underestimated. But for big samples one can easily compute the unbiased normally distributed h-statistic:

$h=(1-\frac {1} {2} d) \sqrt{\frac {T} {1-T \cdot \hat Var(\hat\beta_1\,)}}$ , with the estimated variance $\hat Var(\hat\beta_1)$ of the regression coefficient of the lagged dependend variable for $T \cdot \hat Var(\hat\beta_1)<1 \,$ .

[edit] Durbin-Watson test for panel data

For panel data this statistic can be generalized as follows:

If e_i,t is the residual associated with the observation in panel i at time t, then the test statistic is $d_{pd}=\frac{\sum_{i=1}^N \sum_{t=2}^T (e_{i,t} - e_{i,t-1})^2} {\sum_{i=1}^N \sum_{t=1}^T e_{i,t}^2}$

This statistic can be compared with tabulated rejection values [compare for example Bhargava et al. (1982), page 537]. These values are calculated dependent on T (length of the balanced panel—time periods the individuals were surveyed), K (number of regressors) and N (number of individuals in the panel).

[edit] References

Durbin, J., and Watson, G. S., "Testing for Serial Correlation in Least Squares Regression, I." Biometrika 37 (1950): 409-428.
Durbin, J., and Watson, G. S., "Testing for Serial Correlation in Least Squares Regression, II." Biometrika 38 (1951): 159-179.
Gujarati, Damodar N. (1995): Basic Econometrics, 3. ed., New York et al.: McGraw-Hill, 1995, page 605f.
Verbeek, Marno (2004): A Guide to Modern Econometrics, 2. ed., Chichester: John Wiley & Sons, 2004, Seite 102f.
Bhargava, A./Franzini, L./Narendranathan, W. (1982): Serial Correlation and the Fixed Effects Models, in: Review of Economic Studies, Vol. 49 Iss. 158, 1982, page 533-549.

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Categories: Econometrics | Statistical tests | Time series analysis