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Let's say in linear regression, I got a fit and I can plot residuals to see whether there is any systematic trend in such a plot. How to quantitatively determine whether the residues are really random? Is Durbin-Watson test used for this purpose? How to interpret such test if so?

Please provide an example, preferably in R.

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This also depends on the type of data you have. Do you have a time series? Then looking at a ACF/PACF plot of the residuals may be one way of determining if these are white noise:

require(tseries)
tsdiag(rnorm(100))

you may find this PDF helpful.

If you have cross-section data an inspection of the residual plots yields very valuable information in my opinion:

data(cars)
plot(lm(speed~dist, data=cars))

There exist tests (see comments) but I very often find graphs more useful.

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