# Using a deterministic trend in OLS

I have a time series data, see jpg. It goes straight up and then straight down, later tailing off. This seems to fit my textbook’s description of a deterministic trend being an almost exact function of time on the way up and on the way down. Using the D-F test I found that I could reject the existence of a unit root at the 5% level but not at the 1% level, and if I split the series into two parts, up and down, I cannot reject the unit root at even the 10% level. Is it valid to use this time series, as it is (ie not differenced), in an OLS regression on the basis that it is a deterministic trend and therefore ‘trend stationary’? I want to run it as an explanatory variable against another series which is difference stationary. When I did it, I found the answer to be reasonable (ie something similar to what I was expecting), the residuals did not show autocorrelation in a D-W test and I can reject a unit root in a D-F test at the 1% level.

• It strange things you write about unit root testing by DF. Your null is unit root, if you reject it at 1% you will definitely do the same at 5% level. I think you have to go from the nature of your process, may be it is well described by some differential (differences) equation. I would also consider some structural breaks. Aug 8 '11 at 10:23
• @Dmitrij Celov. I've edited the question to clear-up the D-F explanation. Sorry for the confusion.
– paul
Aug 8 '11 at 10:56
• Hi! You can't run a regression where dependent and indepedent varables (one independent variable) have different degree of integration. Seems to me that there is a regime shift about t = 18. According to the figure there is no overall deterministic trend, but change in indeterminstic trend can be modelled by using a dummy-variable. Try a model where both variables are differenced. Is it yearly data you are using? Aug 10 '11 at 8:12
• @Pantera. Hi and thanks for the comment. Do you mean that I can't run a regression of delta Y on X but must run a regression of delta Y on delta X? A D-F test on delta X (in this case delta of the time series in the graph) says that I can't reject a unit root even at the 10% level. Monthly data.
– paul
Aug 10 '11 at 22:59
• Hi Paul, The point is that you can't run a valid regression between variables which have different degrees of integration. If the variable has a deterministic trend, you have to take out or model the deterministic part. If the variables are non-stationary (integrated of first order), they could be cointegrated and shape a stationary relationship. If they are not cointegrated, than you have to take the first difference and you estimate a short run-model. Make sure you include at least season dummies when you check for unit roots - and not least, check for autocorrelation between the residuals. Aug 11 '11 at 8:15