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Suppose there are two non-stationary time series of integrated order 1. The two time series are not cointegrated.

According to conventional econometric theory,

"In general, regression models for non-stationary variables give spurious results. [The] only exception is if the model eliminates the stochastic trends to produce stationary results.

For non-stationary variables we should always think in terms of cointegration. Only look at regression output if the results are stationary or cointegrate." [1]

Accordingly, in the absense of cointegration between the relevant two I(1) time series, a necessary condition for a correct regression model would be to use the first differences of each variable as input. [2]

Some voices, however, suggest to use an autoregressive distributed lag (ARDL) model in this case: Provided that enough lags for each of the two variables of interest are included, it is possible to use the raw (undifferenced) time series as input without running the risk of spurious regression results. [3, 4] In the example above, one would have to build an ARDL(1, 1) model, i.e. include 1 lag of each variable.

My question is: Can an ARDL(1, 1) model indeed be used as a "treatment" of spurious regression with two I(1) time series variables, which are not cointegrated?


Sources: [1] http://web.econ.ku.dk/metrics/econometrics2_05_ii/slides/10_cointegration_2pp.pdf [2] https://www.reed.edu/economics/parker/312/tschapters/S13_Ch_4.pdf [3] Cointegration - same thing as stationary residuals? [4] https://mpra.ub.uni-muenchen.de/83973/1/MPRA_paper_83973.pdf

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There is a form of ARDL which works if the variables are integrated of order 1 or 0 (they do not have to be integrated the same). There are specific requirements, however, (essentially they have to be co integrated). Error correction models are used to generate short and long term impact. I did a lot of threads on this in the last few weeks.:) This is one example of the analysis (not the threads I did). http://www.scienpress.com/Upload/JSEM%2FVol%205_4_3.pdf I have never read of an ARDL which did this if they were not co integrated. If they are not co integrated I am not sure how there can be a long term relationship.

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