<p>The terms &quot;ordered last&quot; or &quot;ordered first&quot; do not necessarily relate to the place of the variable in the <span class="math-container">$y_t$</span> 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.</p>
<p>Let the structural model be: <span class="math-container">$y_t = c + Ay_{t-1} + Bu_t$</span></p>
<p><span class="math-container">$y_t= \begin{bmatrix}
n_t\\
m_t\\
q_t\\
\end{bmatrix}$</span>
and
<span class="math-container">$B = \begin{bmatrix}
b_{11} &amp; b_{12} &amp; b_{13}\\
b_{21} &amp; b_{22} &amp; b_{23}\\
b_{31} &amp; b_{32} &amp; b_{33}\\
\end{bmatrix}= \begin{bmatrix}
b_{11} &amp; 0 &amp; 0\\
b_{21} &amp; b_{22} &amp; 0\\
b_{31} &amp; b_{32} &amp; b_{33}\\
\end{bmatrix}$</span>.</p>
<p>That is <span class="math-container">$B$</span> <em><strong>coincides</strong></em> with the lower triangular of a Choleski decomposition.</p>
<p>&quot;Ordered first&quot; means variables like <span class="math-container">$n_t$</span>, that have only zeros in their row vectors where <span class="math-container">$i \neq j$</span>. &quot;Ordered last&quot; are variables like <span class="math-container">$q_t$</span>, that have only zeros in their column vectors where <span class="math-container">$i \neq j$</span> .</p>
<p>So &quot;ordering a variable first&quot; means assuming that it is not affected by the other variables contemporaneously and &quot;ordering a variable last&quot; means assuming that it does not affect the other variables contemporaneously.</p>
<p>It is not necessary that the model where we order some variable last actually has a lower triangular <span class="math-container">$B$</span>. Perhaps we want to use partial identification (just like Cristiano, Eichenbaum, Evans (1999)) and we are not interested in the entire <span class="math-container">$B$</span>.</p>
<p>In the specification you are showing, we wouldn't call any variable &quot;ordered last&quot; or &quot;ordered first&quot;.</p>
<p>I once again encourage you to consult
<a href="https://econweb.ucsd.edu/%7Evramey/research/Shocks_HOM_Ramey.pdf" rel="nofollow noreferrer">Prof. Ramey's excellent exposition on identification</a>
.</p>
<p>Consider especially Sections 2.3.1. and 2.3.2.</p>
<p>Presenting Blanchard Perotti (2002)  in section 2.3.2 on page 10 she writes:</p>
<blockquote>
<p>Let ܻ<span class="math-container">$Y_1$</span> be net taxes, ܻ<span class="math-container">$Y_2$</span> be government spending, and ܻ<span class="math-container">$Y_3$</span> be GDP. They identify the shock  to  government  spending  using  a  Cholesky  decomposition  in  which  government  spending  is ordered first (i.e. ܾ<span class="math-container">$b_{21} = b_{23} = 0$</span>). They identify exogenous shocks to net taxes by setting ܾ<span class="math-container">$b_{13}= 2.08$</span>,  an  outside  estimate  of  the  cyclical  sensitivity  of  net  taxes.</p>
</blockquote>
<p>The variable that is &quot;ordered first&quot; - government spending is actually the second component of <span class="math-container">$Y_t$</span>:</p>
<p><span class="math-container">$B= \begin{bmatrix}
b_{11} &amp; b_{12} &amp; 2.08\\
0 &amp; b_{22} &amp; 0\\
b_{31} &amp; b_{32} &amp; b_{33}\\
\end{bmatrix}$</span>.</p>

CC BY-SA 4.0
2021-04-04T03:03:13.820Z
Hi Grada: At a glance, it looks like a really useful document, especially since the identification issue has always confused me.  thank you.
mark leeds
CC BY-SA 4.0
2021-04-04T03:11:16.937Z
@markleeds It is safe to say that I cannot tell you almost anything about frequentist SVAR/symmetric LP, that is not there.
Grada Gukovic
CC BY-SA 4.0
2021-04-04T09:02:29.787Z
So simply put, this is not Cholesky identification but rather structural identification.
Emmanuel Ameyaw
CC BY-SA 4.0
2021-04-04T09:51:58.710Z
@EmmanuelAmeyaw, "Choleski identification" is structural. When you use Choleski decomposition you assume that 
is lower triangular when ordered accordingly. I.e. The first component of  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.
Grada Gukovic
CC BY-SA 4.0
2021-04-04T13:19:26.967Z
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.
mark leeds
CC BY-SA 4.0
2021-04-05T11:19:39.297Z
@Grada, thanks for the comment. So in a nutshell, the model I posted is using a structural Cholesky triangular identification...but when one writes down a model with pen and paper, the order does not matter, right? Like the only thing that matters is which variable affects some other variable contemporaneously. So for example, in STATA, I could do 'svar MI, RF, Y, CPI' and then put the appropriate restrictions in the A  matrix? Anyway, I was reading a paper that said the authors put RF just before MI as a penultimate variable. So I thought order matters....
Emmanuel Ameyaw
CC BY-SA 4.0
2021-04-05T11:34:08.867Z
...but it seems it does not. Like they could put RF anywhere and modify the A_0 matrix accordingly, yeah?
Emmanuel Ameyaw
CC BY-SA 4.0
2021-04-05T11:51:41.000Z
@Mark, Yeah I was asking a question, I just forgot to add ?...:). Anyway, it seems recursive (or Cholesky) identification is structural but not all structural identifications are Cholesky (or recursive), right?
Emmanuel Ameyaw
CC BY-SA 4.0
2021-04-05T13:45:30.330Z
Emmanuel: I don't know if it's an if and only if and plan on reading the paper that Grada linked to.  I have a feeling that answers a lot of those types of questions. And sorry for being no help there.
mark leeds
CC BY-SA 4.0
2021-04-06T14:40:28.410Z
Emmanuel: There may be multiple meanings of "structural",  I think structural means that there is some economic interpretation to the model parameters,  as opposed to say a reduced form where the parameters have purely statistical interpretations. Note that you might have a structural representation from which a reduced form cannot be obtained. So, although I'm not positive, I don't think it's an if and only if. For example,  setting some error terms to zero ( due to some economic argument ) can result in a structural representation that isn't Choleski. Maybe Grada can say something more.
mark leeds
CC BY-SA 4.0
2021-04-04T01:54:51.803Z
2021-04-04T05:36:46.193Z
2021-04-04T05:36:46.193Z
Grada Gukovic
<p>In my understanding of Cholesky identification, <span class="math-container">$RF$</span> in the model below should be ordered last since it is contemporaneously affected by all variables in the model. So why does <strong>Leeper EM, Sims CA, Zha T, Hall RE, Bernanke BS (1996)</strong> put <span class="math-container">$RF$</span> before <span class="math-container">$MI$</span>? 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?</p>
<p><a href="https://i.sstatic.net/RBu5r.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/RBu5r.png" alt="enter image description here" /></a></p>

<macroeconomics><svar>
what is the Intuition behind Cholesky identification in VAR/SVAR
Emmanuel Ameyaw
# what is the Intuition behind Cholesky identification in VAR/SVAR

(Asked by: username_0 on 2021-04-03)

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?

[![enter image description here](https://i.sstatic.net/RBu5r.png)](https://i.sstatic.net/RBu5r.png)

**Tags:** macroeconomics, svar

---

## 1 answers

### Answer by username_1 on 2021-04-04. Score: 4

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](https://econweb.ucsd.edu/%7Evramey/research/Shocks_HOM_Ramey.pdf)
.

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 $Y\_t$:

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

> **Comments:**
> **username_2**: Hi Grada: At a glance, it looks like a really useful document, especially since the identification issue has always confused me.  thank you.  
> **username_1**: @markleeds It is safe to say that I cannot tell you almost anything about frequentist SVAR/symmetric LP, that is not there.  
> **username_0**: So simply put, this is not Cholesky identification but rather structural identification.  
> **username_1**: @EmmanuelAmeyaw, "Choleski identification" is structural. When you use Choleski decomposition you assume that  is lower triangular when ordered accordingly. I.e. The first component of  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.  
> **username_2**: 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.  
> **username_0**: @Grada, thanks for the comment. So in a nutshell, the model I posted is using a structural Cholesky triangular identification...but when one writes down a model with pen and paper, the order does not matter, right? Like the only thing that matters is which variable affects some other variable contemporaneously. So for example, in STATA, I could do 'svar MI, RF, Y, CPI' and then put the appropriate restrictions in the A  matrix? Anyway, I was reading a paper that said the authors put RF just before MI as a penultimate variable. So I thought order matters....  
> **username_0**: ...but it seems it does not. Like they could put RF anywhere and modify the A_0 matrix accordingly, yeah?  
> **username_0**: @Mark, Yeah I was asking a question, I just forgot to add ?...:). Anyway, it seems recursive (or Cholesky) identification is structural but not all structural identifications are Cholesky (or recursive), right?  
> **username_2**: Emmanuel: I don't know if it's an if and only if and plan on reading the paper that Grada linked to.  I have a feeling that answers a lot of those types of questions. And sorry for being no help there.  
> **username_2**: Emmanuel: There may be multiple meanings of "structural",  I think structural means that there is some economic interpretation to the model parameters,  as opposed to say a reduced form where the parameters have purely statistical interpretations. Note that you might have a structural representation from which a reduced form cannot be obtained. So, although I'm not positive, I don't think it's an if and only if. For example,  setting some error terms to zero ( due to some economic argument ) can result in a structural representation that isn't Choleski. Maybe Grada can say something more.  

---

