Hey Muhammad,

What you have is more appropriately studied as an ARIMAX model. That is, an ARIMA model which includes time-varying or time-invariant covariates.

Here is an example:https://forecasters.org/wp-content/uploads/gravity_forms/7-2a51b93047891f1ec3608bdbd77ca58d/2013/07/Kongcharoen_Chaleampong_ISF2013.pdf

Robert

Hi Robert: I'm not absolutely certain because I'm totally clear on the question but I don't think Muhammad needs an arimax. In he following, I am assuming that   by the way that Muhammad has stated the relations,  he meant that

X_(t +1) = X_t* (1 + growthx_(t+1) )

and

Y_(t+1) = Y_t * (1 + growthy_(t+1))

If the above is correct ( I would think it should be growthx_t and growthy_t but if that's the case, what I say below still holds with the appropriate correction ), then this implies ( approximately) that

Delta(X_t) = growthx_(t+1)

Delta(Y_t) = growthy_(t+1)

Since growth is not observable ( I assume ), then the best model might be some random walk + noise model for each, assuming they ( the X s and Y's  are unrelated ). So, say

Delta(X_t) = mux_t+1 + epsilon_t+1

mux_t+1 = mux_t + gamma_t

and the same for Y_t.

A model like above is called an random walk + noise model and the mux_t is the unobserved component. The model can be fit using a kalman filtering approach. The approach will return estimates of the mux_t for each t. If you're not familar with it, check out Andrew Harvey's books. I hope this helps. Obviously, if I have the equations wrong in a more serious way than described, then none of it is true.

Mohammed: I do have a mistake in my answer in that I should have written

Delta(log(Y_t+1) and Delta(log(X_t+1). wherever I wrote  Delta(Y_t) and Delta(X_t).

I'm just using the approximation that log(Y_t+1) - log(Y_t) = growth rate  from time t time t + 1. So I left out the log and had the subscripting wrong also. My apologies for errors

Muhammed: One last thing that I neglected to mention. The random walk plus noise model" that I described earlier can be shown to be equivalent to an ARIMA(0,1,1) ( again see one of Harvey's books for details on this )  so, if you prefer, you can fit an ARIMA(0,1,1)  on Delta(log(X_t+1)) and this is equivalent. But,  for me, doing this causes one to lose the understanding-intuition  of what's going on so I would leave it the way I had it and estimate the random walk plus noise model  as I have shown it. All the best.

                                                                                    Mark

Thank you so much Mark Leeds for such a detail discussion and elaborated Answers

Hi Muhammad: your welcome but andrew harvey has written 3 books. 2 of them forecasting structural time series models and the 1980 one ( I forget the title. maybe "time series models" ),explain it all in much more clearly and in much greater detail than  I could. all the best and good luck with project.