# What are the differences between long and short term impacts in ARDL model?

Here I am using ARDL as suggested by Pesaran and Shin, 1999 to deal with variables that are integrated of a different order. They derive, assuming you reject the null of no cointegration, long and short term effects (the later through an error correction model). But I don't know how to interpret these, are they similar to the interpretation of slopes in say OLS? And what is the practical difference between short term and long term effects (short and long term relationships)? I am not really sure what these terms mean in practice.

• What is the difference between no constant and restricted, and unrestricted ECM in ARDL.What doe sir represent? Also,why some short run variables are missing in the ECM results? What can be the issue and how it can be resolved? One more question, how principal component analysis ,from where I can learn it for making the index.I would appreciate if someone can provide prompt response. thankyou Nov 9, 2023 at 10:55
• What's your confusion? If you understand the autoregressive model, and if you understand the distributed lag, then you put the concepts together. "I got a pen, I got an apple. Unh! Apple pen!" Dec 12, 2023 at 22:06

Hi: Suppose you had a simple ARDL(1,1). ( they might refer to this as a (0,1). I forget tthe convention )

$$y_{t} = \rho y_{t-1} + \beta x_{t} + \epsilon_t$$.

This is probably the simplest ARDL that there is and it's also called a Koyck distributed lag model. But the concept carries over to the more general case.

The short term effect of $$x_{t}$$ in this model is $$\beta$$ because a 1 unit impulse in $$x_{t}$$ increases the response by $$\beta$$ immediately.

To calculate the long term effect, one can re-write the model as follows:

$$y_{t} = \frac{x_{t}}{(1 - \rho L)} + \frac{\epsilon_{t}}{(1-\rho L)} =$$

$$= \beta \sum_{i=0}^{\infty} \rho^{i} x_{t-i} + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i}$$

The infinite sum of the first term on the RHS can be written as $$\frac{1}{(1-\rho)}$$. Therefore, the long term effect of a one unit impulse in $$x_{t}$$ is $$\frac{\beta}{(1-\rho)}$$.

The pdf at this link contains more of the details on this.

https://www.reed.edu/economics/parker/312/tschapters/S13_Ch_3.pdf

• thank you. I actually read that link but did not understand. Part of the problem is that I really don't understand what "long term" means substantively and how this varies from short term. In the type of models I know there is no long or short term effect just an effect. The ARDL I am working from uses an error correction model - would that change the answer you gave? Aug 7, 2020 at 19:31
• Hi:The idea is that the effect of the impulse response of the variable ( in this case $x_t$) doesn't come all at once but over time. At the first time, $t=1$, the effect is $\beta$. The next time at time at $t=2$, it's $\rho \beta$, the next time at $t=3$ it's $\rho^2 \beta$. This goes on ad-infinitum and, when you add them all up, you get $\frac{\beta}{(1-\rho)}$. They call it a distributed lag because the effect is spread out over time. There's a ton of literature on the koyck distributed lag. If you google for "koyck distributed lag" , there are some good explanations. Aug 7, 2020 at 23:11
• Actually, for me, writing the ARDL out as an error correction model makes the "distributed effect" less explicit. When I have a minute, I will try to find what I think is a good explanation of the koyck distributed lag. Ironically, it's some political science papers that I have found to be the best at getting the idea across. The one I linked to already is decent (it's not the poli sci ones I'm referring to ) but I've read better ones. Aug 7, 2020 at 23:17
• Hi: See if this is helpful. www-personal.umich.edu/~franzese/… Aug 7, 2020 at 23:18