Long-term relation between time series with deterministic trends

say I have two time series that move together but both seem to be characterised by a deterministic trend.

I have two questions:

1. How can I test whether the trend is deterministic or stochastic?
2. How would I determine the long-run relationship between the two series in the event they both have deterministic trend? I know that in case of two stochastic trend you might want to use VECM (in case they are co-integrated), but I am not sure if VECM also applies to deterministic trends?

• VECM would not be applicable if the variables do not have unit roots. – Richard Hardy Mar 13 '18 at 13:48
• Hi Richard, thanks for the clarification. What would be the alternative then. I so far could not detect that they have a unit root. – clog14 Mar 14 '18 at 9:29

How would I determine the long-run relationship between the two series in the event they both have deterministic trend?

Assuming the trends are both linear, here are some options:

1. Simple regression $y_t=\beta_0+\beta_1 x_t+\varepsilon_t$. Due to superconsistency the estimator for $\beta_1$ will be converging at a rate $t^{(3/2)}$ rather than $t^{(1/2)}$ and any autocorrelated errors or the like will be have a negligible effect on the estimator given a sufficiently large sample.
2. Regression with ARMA errors $y_t=\beta_0+\beta_1 x_t+u_t$ where $u_t$ is an ARMA process -- if one of the variables is exogenous. This is similar to 1. but could be more efficient in presence of autocorrelated errors, especially if the sample is not that large.
3. VAR model with exogenous time trends -- if both variables are endogenous.
• Hi Richard, I suppse this is all non-detrended variables right? This is because I "believe" that there is no issue of spurious relation here. For option 3, when you write that both are endogenous: do you mean that the causal effect can run both ways, or coud this also be a confounding factor that makes t hem endogenous? – clog14 Mar 14 '18 at 9:44
• @clog14, yes, I mean the original variables, not transformations thereof. Regarding "endogenous" I would not exclude the possibility of a confounding factor. – Richard Hardy Mar 14 '18 at 9:54
• @clog14 and regarding the first question, an idea would be to investigate the variance of the series. The variance grows faster in case of a linear trend than a stochastic trend. I guess there exist some tests based on that. Related threads: stats.stackexchange.com/questions/103193/…, stats.stackexchange.com/questions/286440/… – Richard Hardy Mar 14 '18 at 18:04

try them both and compare the results.

For more see Statistics for time series trend in R

• The language in your quotation is bizarre, because it attempts to define a model in terms of an estimator ("least squares coefficients") of a model! – whuber Mar 13 '18 at 14:42
• e.g. deterministic trend model Y(t)=B0 + B1*T where T is the counting numbers (1,2,...T) ; e.g. stochastic trend(adaptive) Yt)=Y(t-1) +B2 – IrishStat Mar 13 '18 at 15:04
• Hi, I did some checking and I cannot conclude that they have a stochastic trend. Suppose I believe really hard that the two deterministic trends are related, how would I quantify their relation? I assume that is the equivalent to a VECM, but w/o stochastic trends... – clog14 Mar 14 '18 at 9:28
• I would construct a transfer function between the two series. onlinecourses.science.psu.edu/stat510/node/75 and math.cts.nthu.edu.tw/… (ignoring corner method) . If you wish you can post your data in a csv file and I will look at it. – IrishStat Mar 14 '18 at 11:40