Interpretation ARDL all variables in log

I was hoping someone could help me confirm or correct me on my interpretation of my estimated coefficients of ARDL model.

All my variables are converted into natural log form, ln, both Y and Xs.

My interpretation of say a coefficient on varibale Y first lag, L1, of .726 will be: a 1% change in the first lag of Y is associated with a .73% increase in Y on average.

or a L2 of an X variable with coefficient -.432 will be interpreted as follows: a 1% change in the second lag of X is associated with a .43% decrease in Y on average.

If you have lags of $$Y$$ on the RHS of the ARDL, then the interpretation you describe is not correct because the ARDL is then dynamic. What you've given is the regression interepretation which doesn't hold for the dynamic ARDL.

Let me see if I can find a good reference for you because it will explain why it's not correct a lot more clearly than I could. The political science community has a couple of researchers who have some insightful papers on the ARDL. I just have to find what I'm thinking of.

Here is a paper ( link at bottom ) but it's not the one I was thinking of so let me explain it with ARDL(1,0) so really just an AR(1)

Suppose we have: $$Y_t = \alpha Y_{t-1} + \epsilon_t$$.

Note that any movement in $$Y_{t-1}$$ above has an infinite because the initial effect is $$\alpha$$, then, during the next period, it's $$\alpha^2$$, then $$\alpha^3$$ and so on and so forth.

Why ? The easiest way to see it is to re-write the AR(1) ( assume observations start at $$t= 0$$ ) as :

$$Y_t = \sum_{i=1}^{\infty} \alpha^{i} \epsilon_{t-i}$$.

So, using the re-written model, a shock, $$\epsilon_0 = 1$$ at time $$t=0$$ ( and assume no more new shocks after that ) causes effects that last as one goes further out in time but the effect gets smaller and smaller as one goes further out.

Notice that one can use this same argument, if one has an ARDL that includes past $$X's$$. The effect of any lagged $$X$$ will last out into the distant future because the lagged $$Y$$ carries the effect into the future.

Still, the paper below discusses this is a little bit ( it's not the one I was looking for ) so might be worth looking at.

https://www.researchgate.net/publication/44885834_Dynamic_Models_for_Dynamic_Theories_The_Ins_and_Outs_of_Lagged_Dependent_Variables

• Your answer is very helpfull! would it be correct to interpret it as stated, but these are the immediate effects for the variables that are not lagged e.g., Xt on Yt ?
– Gus
May 15, 2023 at 9:44
• And then also, the coefficient of Xt-1 is the % effect on Yt.
– Gus
May 15, 2023 at 9:46
• Hi Gus: Could you write out your the full model because if you have multiple variables that are lagged then you need to view the others as fixed as see what happens when there is a change to just one of them. But the concept is still the same as I explained above. May 16, 2023 at 16:01
• Hi Gus: Here is the paper I was looking for earlier but couldn't find. www-personal.umich.edu/~franzese/…. I think it does a better job of explaining what I was trying to explain earlier. See if it helps. May 16, 2023 at 16:09
• Highly appreciate it! I have read the paper before but after you explanation I understand it better! Considering the models in the paper I have a general model with none of the restrictions: Yt = α0 + α1Yt−1 + β0Xt + β1Xt−1 + εt. I understand that I can interpret the immidiate effects from the impact multipliers β0 and β1. But they will have effects, beyond the immediate effect, that is distributed over the other points in time. Thus, in the long run which is the dynamic multiplier k1? And this long run effect can be calculated using ardl (no need for cointegration and estimating ECM)? :)
– Gus
May 21, 2023 at 17:51