Hi,
in a practice exam question, it says “Durbin Watson statistic is inappropriate in an autoregressive regression” & “DW statistic is appropriate for trend models but not autoregressive models”.
But in the textbook, “there are two methods that are commonly used to detect the presence of serial correlation(Autocorrelation” and it says DW is one of them.
I believe the former argument is wrong, could anybody correct me if I’m wrong please? thanks.
An autoregressive regression explicitly accounts for the autocorrelated errors. A trend model accounts for trending data, but not directly for the correlated errors, which are perhaps also seasonally linked (or even just trending over time) still leaving some correlation in the disturbance/error term (for example).
You may have also heard that it is inappropriate to use a DW test on non-time series data. This is true for the simple reason that you can calculate a “significant” DW statistic on data that are not temporally ordered (this has been done by randomizing data that are not time series), but this doesnt mean there is legitimate autocorrelation (remember, autocorrelation is correlated errors separated by a time period). In essence, you should only be performing the DW test when you have time series data and likely, autocorrelated errors. If you perform a DW test on non-time series data, the results of the test are meaningless (based on how the d-statistic is calculated). In a similar way, if you have explicitly accounted for the correlated errors (autoregressive model), you do not need to test for autocorrelated with DWs test.
Lastly, if you want to see why it is useless to use a DW test after explicitly accounting for the autocorrelated errors or in cross sectional data, I suggest you look at this:
Original regression (OLS) on time-series data (not autoregressive): E(Yt)= B0 + B1X1 + Rt where Rt is the residual for period t.
Residual correlation model: Rt = ρ1Ut-1 + Ut where Ut-1 is the residual from one period prior and Ut is the residual from the current period (in this model). Think, Ut is the part of Rt that is not correlated with the prior residual, Ut-1.
With this:
AR (1) model: E(Yt)= B0 + B1X1 + ρ1Ut-1 + Ut Where rho1 is the coefficient relating Rt to the prior residual (we have just substitued the prior equation for Rt) and Ut is the current period residual which is now UNCORRELATED with the terms in the model.
DW approximation is 2(1-ρ) where ρ is the correlation between Ut and Ut-1. If ρ is now zero (or much closer) our DW test pretty much 2, indicating no autocorrelation. From wikipedia, is the actual calculation, which allows you to see why data that are not time series are not appropriate to test, since the calculation involves a residual and the residual one period prior.
I hope this helps!