The two time series are non stationary at level but stationary on first difference - I(1). I have already performed Engle Granger and the Johansen Test and found them to be cointegrated in both the cases.
Thanks for the reply. I have already established cointegration through both the Johansen and the Engle Granger procedures, I want to further test the parametric restrictions in the cointegrating regression, I was wondering whether running an OLS on differenced ( Stationary Variables) would be appropriate?
Salim, of course your solution is fine. If your variables are non-stationary and they have unit roots (ADF test, etc.) one solution is to use a First Difference transformation (either % change from one period to the next or a First Difference in natural log of the levels). This will work just fine. But, you will observe that when you transform your variables into First Difference-like transformation much of their relationships break down. People will argue that what remains is the true relationship between the variables. At times, this does not leave you much to work with. And, it can be interesting to keep your variable in their original forms and develop a Cointegration model. For that, you just have to test the residuals of your regression and confirm that they are indeed stationary. I think the Cointegration model route can be very interesting and superior when your "level" variable are constrained. Let's say you are modeling or estimating a Y variable like Unemployment rate, or Interest rate, or charge-off rate, or maybe Debt/GDP ratio. All those variables can be over certain time period non-stationary. Yet, they can be very interesting in their original form. Such variables would be worth attempting to do a Cointegration model. Other variables are true nominal level variables and I feel are inappropriate for Cointegration models. I am thinking here of GDP level in $ term or inflation as a CPI level, etc. Such variables obviously trend upward forever and are way too non-stationary even for a Cointegration model. Even if residuals are non-stationary, I would advance such Cointegration models would be misspecified. That's because a simple trend variable in all those cases would get you a regression model with an R Square close to 1. It also would get you an autocorrelation of residuals that is way too high (Durbin Watson much below 1). That's the type of models Clive Granger was really concerned about when he considered them as "spurious regression" absent any real economic meaning.
Hello Gaetan, Thanks for your reply. Yes . I agree , What I have done , is that I have established cointegration between the two I(1) variables through various tests. What I want to do now is conduct parametric tests on the cointegration regression, Should I then in such a case use OLS on differenced variables which are stationary or prefer dynamic OLS which would be run on non stationary ( at level ) but cointegrated variables?
Your parametric test on the cointegration regression will be simply to test the residuals of that regression using the ADF test or the sllightly more conservative Phillips-Ouliaris test. If the residuals pass that test, you are in good shape you have a cointegration model. You actually do not need to difference those variables. I am not very familiar with Dynamic OLS. I am concerned those models are vulnerable to overfitting and do not work well in forecasting because you rerun them constantly only on a small sample of the time series.
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