Ordering in VAR models

In Enders' 'Applied Econometric Time Series', I repeatedly stumbled upon the notion of the "ordering of a VAR model" and I am not sure I understand the concept right. As far as I understand it, the concept relates to the problem of recovering the true $\epsilon$ of the underlying structural equations.

Looking e.g. at the following VAR in standard form:

$y_t = a_{10} + a_{11}y_{t-1} + a_{12}z_{t-1} + e_{1t}$

$z_t = a_{20} + a_{21}y_{t-1} + a_{22}z_{t-1} + e_{2t}$

The problem is that the two errors are actually composites of the underlying shocks $\epsilon_{yt}$ and $\epsilon_{zt}$ of the structural equations, which we do not know. Here, Enders mentions that "(...) if the correlation coefficient between the $e_{1t}$ and $e_{2t}$ is low, the ordering is not likely to be important", and vice versa.

My problem is that I do not grasp the concept of ordering and how it would affect the identification problem of the disturbances in the structural model. Does ordering refer to the order of the independent variables in the VAR?

• It seems that with "ordering", Enders refers to the ordering of the Choleski decomposition, meaning that whether $b_{12}$ or $b_{21}$ are set equal to zero to restrict the VAR model and consequently identify the underlying model in structural for. Maybe someone can confirm/clarify this for me? Commented Mar 17, 2013 at 15:33
• Here is a tutorial that can help you! youtu.be/SbE8ns0oOTs Regards! Commented Jun 12, 2021 at 2:55

Ordering means placing the variables (all) in the decreasing order of exogenity . For example, if y1,y2, and y3 are three variables in the system and if we have from economic theory (or previous empirical findings) that y2 is relatively more exogeneous than y1 and y3 and y1 is relatively more exogeneous than y3 but less exogenous than y2, then we have the order as y2 y1 y3. To put in simple words, y2 (say weather) is more likely to influence y1 (GDP) and y3 (traveling) but the reverse is not true and y1 (GDP) is more likely to influence y3 (traveling) but the reverse is not true.Note that we are talking here of relative exogenity not the absolute one.
• If I understand you correctly, ordering the variables by exogeneity, i.e. causal priority will change the order of the coefficients in $B^{-1}$ matrix and, hence, the Choleski decomposition. We thereby attach economic meaning to our choice of restriction. The order of the dependent variables is then of course only an issue when $n>2$ and the correlation coefficient between the variables is high enough. Commented Mar 18, 2013 at 15:29
• Well, my thought was that when $n=2$, $B = \begin{pmatrix}1 & b_{12}\\ b_{21} & 1 \end{pmatrix}$ and I can let $b_{12}=0$ or $b_{21}=0$ without changing the ordering, but by simply changing the Choleski decomposition. Commented Mar 19, 2013 at 15:13