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I am estimating an error correction model of the following form, using panel data where $i$ are countries and $t$ are years:

$\Delta y_{it} = \alpha + \phi_1 y_{it-1} + \phi_2 y_{it-2} + \gamma x_{it-1} + \mu_i + \epsilon_{it}$

(Note that I do not estimate the immediate impact of $x_{it}$ on $\Delta y_{it}$, only the effect of the lagged value.)

Alternatively, one can express the model in the ADL form:

$y_{it} = \alpha + (\phi_1+1) y_{it-1} + \phi_2 y_{it-2} + \gamma x_{it-1} + \mu_i + \epsilon_{it}$

Under either specification, $\gamma$ is the short run effect of $x$ on $\Delta y_{it}$.

My first question: is the long run effect as follows?

$\frac{-\gamma}{\phi_1 + \phi_2}$

And my second question: how would I calculate the yearly impact of (a) a temporary 1-unit increase in $x$ and (b) a permanent 1-unit increase in $x$? I am ultimately interested in plotting these effects over a period of time (e.g. 25 years), so I need the effects in year 1, 2, 3, and so on.

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I eventually found a method of calculating, and plotting, long run effects of dynamic panel models. It is described here

Williams, Laron K., and Guy D. Whitten. 2012. “But Wait, There’s More! Maximizing Substantive Inferences from TSCS Models.” Journal of Politics 74(3): 685–693.

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