Difference between Distributed Lags and VAR Models

Question

What would be the difference of estimating a variable inside and outside a VAR model? Namely, if I know the relevant explanatory variables to model a certain variable in a time series framework, what's wrong with estimating a distributed lag instead of a VAR?

Edit: For me it's clear the limitations in terms of forecasting and impulse-response analysis. I'm not sure about the impacts on coefficients and residuals.

EDIT: I believe that an answer for my question is that it would be a specification error if we estimated an ADL model for a set of variables (in which only one would enter endogenously) that we know to affect each other in a contemporaneous way. In that case, some type of VAR is the correct approach since we'd need to take into account the multivariate covariance matrix of the residuals for appropriate inference.

Answer 1

I am having a difficulty understanding the actual question, but let me provide some clarification that could be helpful. Take two time series, $x_t$ and $y_t$.

Distributed lag DL($q$) model:

$$ y = \beta_0 + \alpha_1 x_{t-1} + \dotsc + \alpha_q x_{t-q} + \varepsilon_t. $$

Autoregressive distributed lag ARDL($p,q$) model:

$$ y = \beta_0 + \alpha_1 x_{t-1} + \dotsc + \alpha_q x_{t-q} + \beta_1 y_{t-1} + \dotsc + \beta_p y_{t-p} + \varepsilon_t. $$

One equation of vector autoregressive VAR($r$) model:

$$ y = \beta_0 + \alpha_1 x_{t-1} + \dotsc + \alpha_r x_{t-r} + \beta_1 y_{t-1} + \dotsc + \beta_r y_{t-r} + \varepsilon_t. $$

Take $r=\max(p,q)$, set $\alpha_j=0$ for $j>q$ and set $\beta_j=0$ for $j>p$; you get that one equation of this restricted VAR($r$) is the same as ARDL($p,q$). Furthermore, set $\beta_j=0$ for $j=1,\dotsc,p$ to get an DL($q$) model.

From this perspective, the models are not that different. If you do not care about forecasting (which is straightforward with VAR but less so with DL or ARDL because the latter two do not give forecasts for $x_t$), the DL, ARDL and one equation of a VAR allow you to do essentially the same thing.

(The coefficients of a DL or an ARDL model may be restricted, e.g. by Koyck or Almon lag structures, but in principle this could also be done in a VAR model.)