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.)

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.)

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.

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.)