# 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 ( ). ${\ displaystyle \ epsilon _ {t} = \ rho _ {1} \ epsilon _ {t-1} + \ nu _ {t}}$${\ displaystyle \ rho _ {1} = 0}$${\ displaystyle \ rho _ {1} \ neq 0}$

### Test statistics

The test statistic is:

${\ displaystyle d: = {\ frac {\ sum _ {t = 2} ^ {T} (\ epsilon _ {t} - \ epsilon _ {t-1}) ^ {2}} {\ sum _ {t = 1} ^ {T} \ epsilon _ {t} ^ {2}}} \ approx 2 (1 - {\ hat {\ rho}} _ {1})}$

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. ${\ displaystyle \ epsilon _ {t}}$${\ displaystyle t}$${\ displaystyle d}$

### Test decision

Test statistic value correlation meaning
${\ displaystyle d = 2 \,}$ ${\ displaystyle {\ hat {\ rho}} _ {1} = 0 \,}$ no autocorrelation
${\ displaystyle d = 0 \,}$ ${\ displaystyle {\ hat {\ rho}} _ {1} = 1 \,}$ perfect positive autocorrelation
${\ displaystyle d = 4 \,}$ ${\ displaystyle {\ hat {\ rho}} _ {1} = - 1 \,}$ 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. ${\ displaystyle d ${\ displaystyle d> 4-d_ {u}}$${\ displaystyle d_ {o}}$${\ displaystyle 4-d_ {o}}$${\ displaystyle \ lbrack d_ {u}; d_ {o} \ rbrack}$${\ displaystyle \ lbrack 4-d_ {o}; 4-d_ {u} \ rbrack}$

## 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 :

${\ displaystyle h = \ left (1 - {\ frac {1} {2}} d \ right) {\ sqrt {\ frac {T} {1-T \ cdot {\ widehat {\ mathrm {Var}}} \ left ({\ hat {\ beta}} \ right)}}}}$,

where the estimated variance of the regression coefficient of the time-lagged endogenous variable is and must be. ${\ displaystyle {\ widehat {\ mathrm {Var}}} \ left ({\ hat {\ beta}} \ right)}$${\ displaystyle T \ cdot {\ widehat {\ mathrm {Var}}} \ left ({\ hat {\ beta}} \ right) <1}$

## Durbin-Watson test for panel data

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

${\ displaystyle d_ {pd} = {\ frac {\ sum _ {i = 1} ^ {N} \ sum _ {t = 2} ^ {T} (\ epsilon _ {it} - \ epsilon _ {i, t-1}) ^ {2}} {\ sum _ {i = 1} ^ {N} \ sum _ {t = 1} ^ {T} \ epsilon _ {it} ^ {2}}}}$, with = residuals of the Within regression${\ displaystyle \ epsilon _ {it}}$

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

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