# Durbin-Watson test

The **Durbin-Watson test** is a statistical test with which one tries to check whether a first-order autocorrelation is present, i.e. That is, whether the correlation between two consecutive residuals in a regression analysis is non-zero. The test was developed by the British statistician James Durbin and the Australian Geoffrey Watson.

## Action

### Hypotheses

In the case of first-order autocorrelation, the interference terms are modeled as follows . In the Durbin-Watson test, a null hypothesis is set up that states that there is no autocorrelation ( ) and its counter-hypothesis that states that there is autocorrelation ( ).

### Test statistics

The test statistic is:

Here each denotes the residuals of the regression in the -th period. If the difference between the residual quantities is very small or very large, then there is positive or negative autocorrelation. This means that the Durbin-Watson value tends towards zero or four.

### Test decision

Test statistic value | correlation | meaning |
---|---|---|

no autocorrelation | ||

perfect positive autocorrelation | ||

perfect negative autocorrelation |

The acceptance and rejection areas can be determined in a table. For there is positive autocorrelation, for negative autocorrelation, between and there is no autocorrelation. In the intervals and there are fuzzy areas in which no statements can be made.

## Durbin h statistics

In autoregressive models , this test statistic is skewed towards the value two, so that the autocorrelation is underestimated. However, the Durbin h statistic, which is standard normal distributed and undistorted for large samples, can easily be derived from the above statistic :

- ,

where the estimated variance of the regression coefficient of the time-lagged endogenous variable is and must be.

## Durbin-Watson test for panel data

For panel data , the above test statistic can be generalized as follows:

- , with = residuals of the Within regression

These test statistics are then compared with the acceptance and rejection ranges tabulated as a function of *T* (length of the balanced panel data set ), *K* (number of regressors) and *N* (number of observed individuals) [see, for example, Bhargava et al. (1982), p. 537]. A variant of this statistic for unbalanced panel data was developed by Baltagi and Wu (1999).

## literature

- Gujarati, Damodar N. (1995): Basic Econometrics, 3 ed., New York et al .: McGraw-Hill, 1995, pp. 605f.
- Eckey, Hans-Friedrich / Kosfeld, Reinhold / Dreger, Christian (2004): Ökonometrie, 3rd, revised. and exp. Ed., Wiesbaden: Gabler, 2004, page 114ff.
- Verbeek, Marno (2004): A Guide to Modern Econometrics, 2nd ed., Chichester: John Wiley & Sons, 2004, pp. 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, pp. 533-549.

## Individual evidence

- ↑ Compare to this paragraph: Eckey, Hans-Friedrich / Kosfeld, Reinhold / Dreger, Christian (2004): Ökonometrie, 3., revised. and exp. Ed., Wiesbaden: Gabler, 2004.
- ↑ d1 in Formula 16 in Baltagi / Wu (1999), Unequally spaced panel data regressions with AR (1) disturbances. Econometric Theory, 15 (6), pp. 814-823.