# ARDL Model in First Differences For Non-Cointegrated Time Series?

I would like to regress $$Y_t$$ on $$X_t$$. I have concluded that the series are each $$I(1)$$ and not cointegrated. I am curious as to if I can still use an ARDL model to capture any possible long-term effects. Concretely, is there any reason that the model $$\Delta Y_t = \alpha_0 + \alpha_1 \Delta Y_{t-1} + \beta_0 \Delta X_t + \beta_1 \Delta X_{t-1}$$ is incorrectly specified? Because if not, the long-run multiplier $$(\beta_0 + \beta_1)/(1- \alpha_1)$$ is what I would be looking for.

The reason for my hesitation is the following passage from this paper by Andrew Phillips on ARDL models (emphasis mine)

before estimating any model, had we found that both series were I(1) but not cointegrating, we would have known that all four specifications [investigated in the paper, including ARDL] were inappropriate; instead, both series would need to be first-differenced, excluding the possibility of any long-run effect.

But why should the possibility of a long-run effect be excluded? I cannot see what is wrong attempting to estimate the model as written above. It may of course turn out that the estimate for $$\alpha_1$$ is statistically insignificant, which would point to the absence of a long-run effect. But to my mind this is more instructive than simply first differencing and fitting $$\Delta Y_t = \alpha_0 + \beta_0 \Delta X_t\ ,$$ which seems to be what Phillips is suggesting.

On the other hand, your proposed model uses first-differenced variables, just as the paper suggests to do. (It does not seem to me the paper suggests excluding lags, as you seem to be interpreting it.) Also, it is fairly standard to fit VAR on first-differences if the variables are I(1) but not cointegrated. Your suggested ARDL model is almost the same as a single equation of that (aside from including a contemporaneous $$\Delta X_t$$ term); I do not see why it would not work.