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. ${\displaystyle d<d_{u}}$ ${\displaystyle d>4-d_{u}}$ ${\displaystyle d_{o}}$ ${\displaystyle 4-d_{o}}$ ${\displaystyle \lbrack d_{u};d_{o}\rbrack }$ ${\displaystyle \lbrack 4-d_{o};4-d_{u}\rbrack }$

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.

Web links

• Link to a table for high n and k values