I am trying to understand how to interpret the dynamic and static effect from coefficients in regression models.
$GDP\_growth\_rate_{t,i} = \beta_1GCF_{t,i} +\beta_2GCF_{t-1,i}+\beta_3GCF_{t-2,i} +\beta X_{t,i} +u_{t,i}$
where GCF is Gross Capital Formation and the model is estimated using OLS.
My question is am I correct in interpreting $\beta_1$ as the impact multiplier /immediate effect of GCF on GDP and $\beta_1+\beta_2+\beta_3$ as the long-run multiplier/effect?
yes the way how your model is set up $\beta_1$ would be immediate effect/multiplier and $\beta_1+\beta_2+\beta_3$ the long-run one.
However, an important caveat is that this is due to the way how you set up your model and not a general result. For example, in an ARDL model with stationary variables of the following form:
$$y_t = \alpha + \beta_1 y_{t-1} + \gamma_1 x_t + \gamma_2 x_{t-1}+ e_t$$
the long run multiplier would actually become: $ \frac{\gamma_1 + \gamma_2}{1 - \beta_1}$
or in more general case
$$y_t = \alpha + \sum_{p=1} \beta_p y_{t-p} + \sum_{q=1} \gamma_q x_{t-q+1} +e_t$$
the long run multiplier would be given by: $\frac{\gamma_1+\gamma_2+...+ \gamma_q}{1-\beta_1-\beta_2-...-\beta_p}$.
In your case you do not include any lags of dependent variable so you have a special case where the denominator is 1 and hence it is enough to add in the coefficients but I thought it might be good to mention that as long as you include lagged dependent variable the calculation of the long run multiplier changes (see Verbeek (2008) guide to modern econometrics for more details).
