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I have a OLS model that I try to prove it has cointegration between two regressors and the dependent variable. The model fits well, with a very high $R^2$. The residuals don't seem to be autocorrelated. enter image description here

enter image description here

FYI, I only have 135 data points for the dependent variable. The model residuals pass KPSS, ADF and PP test to be stationary, but Durbin–Watson test fails:

> tseries::adf.test(bettermodel$res)
  Augmented Dickey-Fuller Test
    data:  bettermodel$res
            Dickey-Fuller = -4.2461, Lag order = 5, p-value = 0.01
    alternative hypothesis: stationary

    Warning message:
    In tseries::adf.test(bettermodel$res) :
      p-value smaller than printed p-value

    > tseries::kpss.test(bettermodel$res)
    	KPSS Test for Level Stationarity
    data:  bettermodel$res
    KPSS Level = 0.0843, Truncation lag parameter = 2, p-value = 0.1

    Warning message:
    In tseries::kpss.test(bettermodel$res) :
      p-value greater than printed p-value

    > tseries::pp.test(bettermodel$res)
    	Phillips-Perron Unit Root Test
    data:  bettermodel$res
    Dickey-Fuller Z(alpha) = -53.7486, Truncation lag parameter = 4, p-value= 0.01
    alternative hypothesis: stationary

    Warning message:
In tseries::pp.test(bettermodel$res) : p-value smaller than printed p-value

    > lmtest::dwtest(bettermodel)

        Durbin-Watson test

    data:  bettermodel
    DW = 0.8288, p-value = 0.000000000003394
    alternative hypothesis: true autocorrelation is greater than 0

I feel confused about this and I am not sure whether my model selection satisfies the condition of cointegration. Any thoughts?

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  • $\begingroup$ Are you sure about putting the emphasis on test results for stationarity (ADF, PP, KPSS) vs. autocorrelation (DW)? They need not be related. Your actual question could be whether your model selection satisfies the condition of cointegration -- rather than what to do when ADF+PP+KPSS says on thing while DW says another thing. (So I suggest you to edit the post and especially its title accordingly.) $\endgroup$
    – Richard Hardy
    Commented Apr 13, 2016 at 7:33
  • $\begingroup$ @RichardHardy Thank you Richard. I have changed the title. $\endgroup$
    – chl111
    Commented Apr 13, 2016 at 16:16
  • $\begingroup$ The ACF plot sure looks like the residuals are autocorrelated to me. $\endgroup$
    – gung - Reinstate Monica
    Commented Apr 13, 2016 at 17:30
  • $\begingroup$ @gung The model passes the Engle–Granger cointegration test, from package "egcm", but it doesn' pass Durbin-Watson test. I am not sure how "strong" the autocorrelation can lead to the conclusion of spurious regression? $\endgroup$
    – chl111
    Commented Apr 13, 2016 at 17:33
  • $\begingroup$ Even though you changed the title, you still maintained the same idea: contrasting stationarity/nonstationarity versus autocorrelation. But it seems you already accepted an answer, so my comments are not that relevant anymore. $\endgroup$
    – Richard Hardy
    Commented Apr 13, 2016 at 17:58

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I agree with Richard that it is not clear from your question how your strategy is to answer the question of whether there is cointegration.

Sticking to your title, however, there is nothing weird about the pattern of results for ADF, KPSS and DW that you observe.

A rejection of ADF (or PP) and a non-rejection of KPSS both "agree" in that one gets a rejection in favor of stationarity and a non-rejection of the null of stationarity. Now, DW tests the null of lack of (first-order) serial correlation. That null is to be expected to be rejected for stationary processes like AR(1) processes $Y_t=\phi Y_{t-1}+ \epsilon_t$ with $|\phi|<1$ and $\phi\neq0$.

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