1 DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION The standard test statistic for autocorrelation of the AR(1) type is the Durbin–Watson d statistic, computed

1

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION

The standard test statistic for autocorrelation of the AR(1) type is the Durbin–Watson d statistic, computed from the residuals as shown above. Most regression applications calculate it automatically and present it as one of the standard regression diagnostics.

T

tt

T

ttt

e

eed

1

2

2

21 )(

In large samples

2

It can be shown that in large samples d tends to 2 – 2, where is the parameter in the AR(1) relationship ut = ut–1 + t.

T

tt

T

ttt

e

eed

1

2

2

21 )(

22 d


In large samples

No autocorrelation

3

If there is no autocorrelation, is 0 and d should be distributed randomly around 2.

T

tt

T

ttt

e

eed

1

2

2

21 )(

22 d

2d


In large samples

No autocorrelation

Severe positive autocorrelation

4

If there is severe positive autocorrelation, will be near 1 and d will be near 0.

T

tt

T

ttt

e

eed

1

2

2

21 )(

22 d

2d0d


In large samples

No autocorrelation


Severe negative autocorrelation

5

Likewise, if there is severe positive autocorrelation, will be near –1 and d will be near 4.

T

tt

T

ttt

e

eed

1

2

2

21 )(

22 d

2d0d4d


No autocorrelation



6

Thus d behaves as illustrated graphically above.

2d0d4d

2 40

positiveautocorrelation

negativeautocorrelation

noautocorrelation


No autocorrelation



7

To perform the Durbin–Watson test, we define critical values of d. The null hypothesis is H0: = 0 (no autocorrelation). If d lies between these values, we do not reject the null hypothesis.

2d0d4d

2 40



noautocorrelation

dcrit dcrit


No autocorrelation



8

The critical values, at any significance level, depend on the number of observations in the sample and the number of explanatory variables.

2d0d4d

2 40



noautocorrelation

dcrit dcrit


No autocorrelation



9

Unfortunately, they also depend on the actual data for the explanatory variables in the sample, and thus vary from sample to sample.

2d0d4d

2 40 dcrit



noautocorrelation

dcrit


No autocorrelation



10

However Durbin and Watson determined upper and lower bounds, dU and dL, for the critical values, and these are presented in standard tables.

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit


No autocorrelation



11

If d is less than dL, it must also be less than the critical value of d for positive autocorrelation, and so we would reject the null hypothesis and conclude that there is positive autocorrelation.

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit


No autocorrelation



12

If d is above than dU, it must also be above the critical value of d, and so we would not reject the null hypothesis. (Of course, if it were above 2, we should consider testing for negative autocorrelation instead.)

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit


No autocorrelation



13

If d lies between dL and dU, we cannot tell whether it is above or below the critical value and so the test is indeterminate.

2d0d4d

2 40 dL dUdcrit



noautocorrelation

dcrit


No autocorrelation



14

Here are dL and dU for 45 observations and two explanatory variables, at the 5% significance level.

2d0d4d

2 40 dL dU



noautocorrelation

1.43 1.62(n = 45, k = 3, 5% level)


No autocorrelation



15

There are similar bounds for the critical value in the case of negative autocorrelation. They are not given in the standard tables because negative autocorrelation is uncommon, but it is easy to calculate them because are they are located symmetrically to the right of 2.

2d0d4d

2 40 dL dU



noautocorrelation

2.38 2.571.43 1.62


(n = 45, k = 3, 5% level)

No autocorrelation



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So if d < 1.43, we reject the null hypothesis and conclude that there is positive autocorrelation.

2d0d4d

2 40 dL dU



noautocorrelation

1.43 1.62 2.38 2.57


(n = 45, k = 3, 5% level)

No autocorrelation



17

If 1.43 < d < 1.62, the test is indeterminate and we do not come to any conclusion.

2d0d4d

2 40 dL dU



noautocorrelation

1.43 1.62 2.38 2.57


(n = 45, k = 3, 5% level)

No autocorrelation



18

If 1.62 < d < 2.38, we do not reject the null hypothesis of no autocorrelation.

2d0d4d

2 40 dL dU



noautocorrelation

1.43 1.62 2.38 2.57


(n = 45, k = 3, 5% level)

No autocorrelation



19

If 2.38 < d < 2.57, we do not come to any conclusion.

2d0d4d

2 40 dL dU



noautocorrelation

1.43 1.62 2.38 2.57


(n = 45, k = 3, 5% level)

No autocorrelation



20

If d > 2.57, we conclude that there is significant negative autocorrelation.

2d0d4d

2 40 dL dU



noautocorrelation

1.43 1.62 2.38 2.57


(n = 45, k = 3, 5% level)

No autocorrelation



21

Here are the bounds for the critical values for the 1% test, again with 45 observations and two explanatory variables.

2d0d4d

2 40 dL dU



noautocorrelation

1.24 1.42 2.58 2.76


(n = 45, k = 3, 1% level)

22

Here is a plot of the residuals from a logarithmic regression of expenditure on housing services on income and the relative price of housing services. The residuals exhibit strong positive autocorrelation.

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-0.03

-0.02

-0.01

0

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0.02

0.03

0.04

1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003


============================================================Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob.============================================================ C 0.005625 0.167903 0.033501 0.9734 LGDPI 1.031918 0.006649 155.1976 0.0000 LGPRHOUS -0.483421 0.041780 -11.57056 0.0000============================================================R-squared 0.998583 Mean dependent var 6.359334Adjusted R-squared 0.998515 S.D. dependent var 0.437527S.E. of regression 0.016859 Akaike info criter-5.263574Sum squared resid 0.011937 Schwarz criterion -5.143130Log likelihood 121.4304 F-statistic 14797.05Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000============================================================

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The d statistic is very low, below dL for the 1% significance test (1.24), so we would reject the null hypothesis of no autocorrelation.


dL dU

1.24 1.42(n = 45, k = 3, 1% level)

Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use.

15.03.06

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1 DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION The standard test statistic for autocorrelation of the AR(1) type is the Durbin–Watson d statistic, computed