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I have a time series for which I have built a linear regression, say $Y(t)=\beta X(t)$.

A regression implies that $Y$ is actually a function of $X$ (that is, $Y(X)$), but not the other way around ($X(Y)$), right? (since $X$ is assumed to be exogenous to the model, and $Y$ is the endogenous variable.)

Is the Durbin-Watson a good test to see if $X$ truly is exogenous? I am not sure I understood that test... All the explanations of the test about correlated errors are rather abstract; if someone knows of a time series graphical examples showing when the Durbin-Watson statistic says 0 or 2, it would be great to make intuitive sense of it.

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1.

A regression implies that $Y$ is actually a function of $X$ (that is, $Y(X)$), but not the other way around ($X(Y)$), right? (since $X$ is assumed to be exogenous to the model, and $Y$ is the endogenous variable.)

Yes, a regression treats $Y$ as a function of $X$ and not the other way around. There is an additive random noise component there, too, which you forgot to include on the right hand side of the equation in the first line of your post.

2.

I do not think Durbin-Watson test is used for assessing exogeneity. You probably mixed it up with Durbin-Wu-Hausman test (or just the Hausman test).

Regarding testing exogeneity in time series, the Hausman test is among the better-known ones; here is a thread explaining how it works. There is another short thread here explaining why it is difficult to test for exogeneity; essentially, you have to examine all possible sources of endogeneity and reject all of them to establish exogeneity, but this is not possible in practice. I may add that this is similar to testing independence: you can never empirically prove that two variables are independent, you can just reject a particular form of dependence between them.

Besides the Hausman test, you may also look at Granger's block exogeneity test mentioned here.

Since the Durbin-Watson test was mentioned, let me add something, even though it is unrelated to testing exogeneity. The Durbin-Watson test might be too specific as it tests for autocorrelation at lag order 1 and not higher order lags; more general tests such as Breusch-Godfrey or in some instances Ljung-Box could be used instead; here is a good overview and comparison of the two tests. But this is all about testing for autocorrelation, not exogeneity.

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  • $\begingroup$ What do you imply by "in some instances" for LB test? what are the restrictions? is BG better in this sense? $\endgroup$
    – mugen
    Sep 19, 2015 at 17:14
  • $\begingroup$ An excellent answer to these questions is given by Alecos Papadopoulos in the link I provided (in the second line from the bottom). I highly recommend reading it! $\endgroup$
    – Richard Hardy
    Sep 19, 2015 at 17:17
  • $\begingroup$ Just found the question, I can only say I'm super impressed. Fascinating, thank you! $\endgroup$
    – mugen
    Sep 19, 2015 at 17:22
  • $\begingroup$ Thank you for the thorough answer. I took an econometrics class awhile ago and was probably confusing the test, and the online information seemed weird. Anyhow, is there a function you recommend for testing the hypothesis in R? $\endgroup$
    – Ricardo Magalhães Cruz
    Sep 20, 2015 at 10:40
  • $\begingroup$ I have not used the test(s) myself, so I know as much about the relevant R functions as you do. $\endgroup$
    – Richard Hardy
    Sep 20, 2015 at 11:27

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