# what is the Intuition behind Cholesky identification in VAR/SVAR

In my understanding of Cholesky identification, $$RF$$ in the model below should be ordered last since it is contemporaneously affected by all variables in the model. So why does Leeper EM, Sims CA, Zha T, Hall RE, Bernanke BS (1996) put $$RF$$ before $$MI$$? Or maybe order does not matter as long as we can put zero restrictions however we want to ensure exact identification or overidentification? Does order only matter in Cholesky identification without zero restrictions?

The terms "ordered last" or "ordered first" do not necessarily relate to the place of the variable in the $$y_t$$ vector. Rather they describe the column (row) vector of the contemporaneous impact matrix pertaining to the variable. Once the restrictions have been set and the model is identified, where you exactly place the variables is an uninteresting computational issue that concerns only the implementation of the model in your software package. Here is what I mean in equations.

Let the structural model be: $$y_t = c + Ay_{t-1} + Bu_t$$

$$y_t= \begin{bmatrix} n_t\\ m_t\\ q_t\\ \end{bmatrix}$$ and $$B = \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23}\\ b_{31} & b_{32} & b_{33}\\ \end{bmatrix}= \begin{bmatrix} b_{11} & 0 & 0\\ b_{21} & b_{22} & 0\\ b_{31} & b_{32} & b_{33}\\ \end{bmatrix}$$.

That is $$B$$ coincides with the lower triangular of a Choleski decomposition.

"Ordered first" means variables like $$n_t$$, that have only zeros in their row vectors where $$i \neq j$$. "Ordered last" are variables like $$q_t$$, that have only zeros in their column vectors where $$i \neq j$$ .

So "ordering a variable first" means assuming that it is not affected by the other variables contemporaneously and "ordering a variable last" means assuming that it does not affect the other variables contemporaneously.

It is not necessary that the model where we order some variable last actually has a lower triangular $$B$$. Perhaps we want to use partial identification (just like Cristiano, Eichenbaum, Evans (1999)) and we are not interested in the entire $$B$$.

In the specification you are showing, we wouldn't call any variable "ordered last" or "ordered first".

I once again encourage you to consult Prof. Ramey's excellent exposition on identification .

Consider especially Sections 2.3.1. and 2.3.2.

Presenting Blanchard Perotti (2002) in section 2.3.2 on page 10 she writes:

Let ܻ$$Y_1$$ be net taxes, ܻ$$Y_2$$ be government spending, and ܻ$$Y_3$$ be GDP. They identify the shock to government spending using a Cholesky decomposition in which government spending is ordered first (i.e. ܾ$$b_{21} = b_{23} = 0$$). They identify exogenous shocks to net taxes by setting ܾ$$b_{13}= 2.08$$, an outside estimate of the cyclical sensitivity of net taxes.

The variable that is "ordered first" - government spending is actually the second component of $$Y_t$$:

$$B= \begin{bmatrix} b_{11} & b_{12} & 2.08\\ 0 & b_{22} & 0\\ b_{31} & b_{32} & b_{33}\\ \end{bmatrix}$$.

• Hi Grada: At a glance, it looks like a really useful document, especially since the identification issue has always confused me. thank you. Apr 4, 2021 at 3:03
• @markleeds It is safe to say that I cannot tell you almost anything about frequentist SVAR/symmetric LP, that is not there. Apr 4, 2021 at 3:11
• So simply put, this is not Cholesky identification but rather structural identification. Apr 4, 2021 at 9:02
• @EmmanuelAmeyaw, "Choleski identification" is structural. When you use Choleski decomposition you assume that 𝐵 is lower triangular when ordered accordingly. I.e. The first component of 𝑦 is not affected by any other variable contemporaneously, the second is affected only by the first and itself and so on. If you cannot justify why this assumption should hold you should not be imposing it, as the results you are going to obtain by imposing a wrong assumption are rubbish. Apr 4, 2021 at 9:51
• Emmanuel: I'm not sure if you're asking a question in your comment above but I would think of Cholesky decomposition as an approach used in structural identification. Bayesian assumptions are another approach. I'm not sure if there are others. Apr 4, 2021 at 13:19