I seem to misunderstand how the dwtest of the lmtest package in R works.
> r1 <- runif(1000, 0.0, 1.0)
> r2 <- runif(1000, 0.0, 1.0)
> dwtest(lm(r2 ~ r1 ))
Durbin-Watson test
data: lm(r2 ~ r1)
DW = 1.9806, p-value = 0.3789
alternative hypothesis: true autocorrelation is greater than 0
>
> r1 <- seq(0, 1000, by=1)
> r2 <- seq(0, 1000, by=1)
> dwtest(lm(r2 ~ r1 ))
Durbin-Watson test
data: lm(r2 ~ r1)
DW = 2.2123, p-value = 0.8352
When i understand everything right, I first test 2 sets of random numbers with each other (which do not correlate - correct)
Then i correlate the numbers from 1 to 1000 incrementing with them self (which does not correlate - uhm... what)
Can someone point me to the obvious error i do?
Looking on Wikipedia, it seems like the Durbin-Watson test is for autocorrelation of residuals, not for correlation. So, if I define r2 <- r1 + sin(r1), then I get a significant result from the DW test:
> r1 <- seq(0, 1000, by=1)
> r2 <- r1 + sin(r1)
> dwtest(lm(r2 ~ r1))
Durbin-Watson test
data: lm(r2 ~ r1)
DW = 0.91956, p-value < 2.2e-16
alternative hypothesis: true autocorrelation is greater than 0
Here's the reason. The value of r2[i], predicted from the linear model, is r1[i]. The "residual", which is the difference between the actual and predicted values, is r2[i]-r1[i]. If this is above zero, then r2[i+1]-r1[i+1] is probably also above zero, since they are neighboring values of the sine function. Therefore, there's "autocorrelation" in the residuals, meaning correlation between neighboring values.