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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. nbl full

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.

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  • $\begingroup$ 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. $\endgroup$
    – Dmitrij Celov
    Commented Aug 8, 2011 at 10:23
  • $\begingroup$ @Dmitrij Celov. I've edited the question to clear-up the D-F explanation. Sorry for the confusion. $\endgroup$
    – paul
    Commented Aug 8, 2011 at 10:56
  • $\begingroup$ 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? $\endgroup$
    – Pantera
    Commented Aug 10, 2011 at 8:12
  • $\begingroup$ @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. $\endgroup$
    – paul
    Commented Aug 10, 2011 at 22:59
  • $\begingroup$ 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. $\endgroup$
    – Pantera
    Commented Aug 11, 2011 at 8:15

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Your series would appear to have a number of trends and possibly some asymptotic ARIMA structure. You can incorporate this series "as is" in a regression model. Detecting and identifying any needed lag structure using this variable would probably need suitable differencing.

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    $\begingroup$ Thanks for this answer and also for the very interesting and informative discussion by telephone earlier. The Autobox forecasting environment is remarkably sophisticated. $\endgroup$
    – paul
    Commented Aug 9, 2011 at 7:15
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If this data come from the social sciences, I will try to figure out what made the trend change its direction at 19. Fitting a linear trend to the entire data seems to produce a misleading interpretation of the time dependent behavior. You may try interrupted time series analysis. Alternatively, you may divide the data into two parts, and estimate linear upward and downward trends, although the small number of cases is a problem then.

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    $\begingroup$ You might have been fooled by superfluous material in the question. It's only asking whether this time series could be used as an explanatory variable in a regression: it doesn't ask to model this time series. $\endgroup$
    – whuber ♦
    Commented Mar 9, 2018 at 16:45

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