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. 1

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION In large samples It can be shown that in large samples d tends to 2 – 2r, where r is the parameter in the AR(1) relationship ut = rut–1 + et. 2

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION In large samples No autocorrelation If there is no autocorrelation, r is 0 and d should be distributed randomly around 2. 3

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION In large samples No autocorrelation Severe positive autocorrelation If there is severe positive autocorrelation, r will be near 1 and d will be near 0. 4

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION In large samples No autocorrelation Severe positive autocorrelation Severe negative autocorrelation Likewise, if there is severe positive autocorrelation, r will be near –1 and d will be near 4. 5

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation 2 4 Thus d behaves as illustrated graphically above. 6

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dcrit 2 dcrit 4 To perform the Durbin–Watson test, we define critical values of d. The null hypothesis is H0: r = 0 (no autocorrelation). If d lies between these values, we do not reject the null hypothesis. 7

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dcrit 2 dcrit 4 The critical values, at any significance level, depend on the number of observations in the sample and the number of explanatory variables. 8

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dcrit 2 dcrit 4 Unfortunately, they also depend on the actual data for the explanatory variables in the sample, and thus vary from sample to sample. 9

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dcrit dU 2 dcrit 4 However Durbin and Watson determined upper and lower bounds, dU and dL, for the critical values, and these are presented in standard tables. 10

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dcrit dU 2 dcrit 4 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. 11

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dcrit dU 2 dcrit 4 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.) 12

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dcrit dU 2 dcrit 4 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. 13

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.43 1.62 (n = 45, k = 3, 5% level) Here are dL and dU for 45 observations and two explanatory variables, at the 5% significance level. 14

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.43 1.62 2.38 2.57 (n = 45, k = 3, 5% level) 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. 15

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.43 1.62 2.38 2.57 (n = 45, k = 3, 5% level) So if d < 1.43, we reject the null hypothesis and conclude that there is positive autocorrelation. 16

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.43 1.62 2.38 2.57 (n = 45, k = 3, 5% level) If 1.43 < d < 1.62, the test is indeterminate and we do not come to any conclusion. 17

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.43 1.62 2.38 2.57 (n = 45, k = 3, 5% level) If 1.62 < d < 2.38, we do not reject the null hypothesis of no autocorrelation. 18

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.43 1.62 2.38 2.57 (n = 45, k = 3, 5% level) If 2.38 < d < 2.57, we do not come to any conclusion. 19

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.43 1.62 2.38 2.57 (n = 45, k = 3, 5% level) If d > 2.57, we conclude that there is significant negative autocorrelation. 20

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION No autocorrelation Severe positive autocorrelation Severe negative autocorrelation positive autocorrelation no autocorrelation negative autocorrelation dL dU 2 4 1.24 1.42 2.58 2.76 (n = 45, k = 3, 1% level) Here are the bounds for the critical values for the 1% test, again with 45 observations and two explanatory variables. 21

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION 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. 22

DURBIN–WATSON TEST FOR AR(1) AUTOCORRELATION ============================================================ 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 6.359334 Adjusted R-squared 0.998515 S.D. dependent var 0.437527 S.E. of regression 0.016859 Akaike info criter-5.263574 Sum squared resid 0.011937 Schwarz criterion -5.143130 Log likelihood 121.4304 F-statistic 14797.05 Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000 dL dU 1.24 1.42 (n = 45, k = 3, 1% level) 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. 23

Copyright Christopher Dougherty 2000–2006 Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use. 15.03.06