<p>You are right.</p>
<p>Ordering a variable first reflects the identifying assumption that this variable will not respond contemporeneously to the other shocks in the system. You could obtain the exact same shock series via an OLS regression that includes as explanatory variables only the lags of all the variables in the VAR. Since you were happy to &quot;assume away&quot; all endogeneity concerns, your shock is neatly identified, and you don't care about the others. (You don't even need to impose the other zeros in the impact matrix: any rotation of the bottom-right (n-1)x(n-1) submatrix will leave the first shock unaffected.)</p>
<p>Ordering a variable last allows this variable to contemporaneously respond to all other shocks in the system, but it imposes the assumption that none of the variables in the system can contemporaneously respond to the shock in the last equation. Again, you could obtain the same shock series via an OLS regression including all lagged and contemporaneous values of all the variables in the VAR, since, once again, you were happy to assume away all endogeneity concerns. (Also, you don't even need to impose the other zeros in the impact matrix: any rotation of the top-left (n-1)x(n-1) submatrix will leave the last shock unaffected.)</p>
<p>Intuitively, the OLS-counterparts show that the ordering of the other variables doesn't matter when you're interested in exclusively the first or the last shock:</p>
<ul>
<li>either you don't includy <em>any</em> of the other variables contemporaneously in the OLS regression (first shock);</li>
<li>or you include <em>all of them</em> contemporaneously (last shock).</li>
</ul>
<p>For the shocks in between, the OLS regressions that would deliver the same shock series as the shock series obtained via cholesky include the variables ordered before the shock of interest, but not the variables order after. So here the ordering of the variables in the VAR does affect what the shock series will look like, because changing the ordering for these shocks would include or exclude some of the rearranged contemporaneous variables in the OLS regression.</p>

CC BY-SA 4.0
2021-04-06T15:10:15.943Z
That was a a beautiful answer but just to make sure I follow: The ordering of the variables ( in a cholesky decomp ) does matter when you are interested in all the shocks, correct ? It's only when you interested in the first or last, that the ordering does not matter. Thanks.
mark leeds
CC BY-SA 4.0
2021-04-07T09:19:02.173Z
Yes, if you're interested in all the shocks, then you should be comfortable with all the identifying restrictions that the choleky factorization imposes (i.e., all the upper-triangular zeros). And those restrictions (the zeros for each row) depend on the order in which you include the variables.
Wecon
CC BY-SA 4.0
2021-04-12T13:18:59.080Z
@Wecon, may I ask though. So are there just three ways to identification? (1). you are interested in the shock of the first ordered variable only. (2) you are interested in the shock of the last ordered variable only. (3) you are interested in all the shocks of each variable. Let's say you have a 5 variable model, [a, b, c, d, e]. 'a' and 'b' are different types of commodity variables so they are both completely exogenous. I am interested in shocks to 'a' and 'b' and how 'e' responds to these shocks. Is that possible?
Emmanuel Ameyaw
CC BY-SA 4.0
2021-04-12T13:23:30.607Z
[continuation]...so 'a' and 'b' are arranged arbitrarily but they must be arranged before 'c' and 'd'. 'c' and 'd' are arranged arbitrarily but it must be arranged before 'e'. Thus the identifying assumption is that [a, b] should be ordered before [c, d] and [c, d] must be ordered before [e]. But order of a and b does not matter, and order of c and d does not matter. This is possible right?
Emmanuel Ameyaw
CC BY-SA 4.0
2021-04-13T14:29:51.463Z
If your commodity prices are exogenous, then you can just estimate the causal effects with a local projection. Impossible to judge from the information you gave, but you may want to reconsider your assumption that commodity prices are exogenous: they're an equilibrium outcome affected by supply and demand; and especially demand may not be exogenous w.r.t. the other variables in the model (being financial variables or macro variables).  The ordering of [c,d,e] doesn't matter. But it will matter which of the two commodity prices you order first: [a,b,c,d,e] or [b,a,c,d,e] will be different.
Wecon
CC BY-SA 4.0
2021-04-13T14:31:49.060Z
Moreover, don't forget that you have two different identifying assumptions for the two different commodity prices. If you don't have an argumentation for this, you may want to look into other identification schemes than the recursive one implemented via Cholesky.
Wecon
CC BY-SA 4.0
2021-04-06T12:14:13.710Z
2021-04-06T12:14:13.710Z
Wecon
<p>For example, in a three-variable SVAR model in Favero, C. A. (2001), the author uses Cholesky decomposition to identify only money shocks by ordering it last. Thus, both <span class="math-container">$p_t$</span> and <span class="math-container">$y_t$</span> affect <span class="math-container">$m_t$</span> contemporaneously, and this is the only identification condition in the model.</p>
<p><a href="https://i.sstatic.net/1kXxB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1kXxB.png" alt="enter image description here" /></a></p>
<p>He mentions that the identification of shocks to <span class="math-container">$p_t$</span> and <span class="math-container">$y_t$</span> do not matter in the model. We can see here that <span class="math-container">$p_t$</span> affects <span class="math-container">$y_t$</span> contemporaneously but not the reverse.</p>
<p>In another example in the same book, <span class="math-container">$y_t$</span> contemporaneously affects <span class="math-container">$p_t$</span> which is in contrast to the example above. Again, he seeks to identify shocks only to <span class="math-container">$FF_t$</span>, and hence does not discuss the ordering of the non-policy block of the model. I guess it is arbitrary.</p>
<p><a href="https://i.sstatic.net/BvoSL.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BvoSL.png" alt="enter image description here" /></a></p>
<p>I see same thing in Primiceri, G. E. (2005). Interest rates are ordered last in the VAR. The interaction between inflation and unemployment is arbitrarily modeled in a lower triangular form, with inflation first, unemployment second. So I guess unemployment could also be first.</p>
<p>Another example is by Drechsel, T., &amp; Tenreyro, S. (2018). There are 4 variables in the VAR. The only identification condition is that commodity price is ordered first in the lower triangular Cholesky identification scheme. The author does not mention how the other 3 domestic variables are arranged in the model. So I guess, arbitrarily.</p>
<p><strong>So my question is, in a lower triangular Cholesky identification scheme, one does not necessarily need to have an economic theory for all the orderings of the variables in the model, right? I mean, some of the orderings can be arbitrary if one do not care about the shocks associated with these aribitrarily ordered variables in the model?</strong></p>

<macroeconomics><identification><svar>
Can a subset of variables in a lower triangular Cholesky identification scheme be ordered arbitrarily if we do not care about their shocks?
Emmanuel Ameyaw
# Can a subset of variables in a lower triangular Cholesky identification scheme be ordered arbitrarily if we do not care about their shocks?

(Asked by: username_0 on 2021-04-05)

For example, in a three-variable SVAR model in Favero, C. A. (2001), the author uses Cholesky decomposition to identify only money shocks by ordering it last. Thus, both $p\_t$ and $y\_t$ affect $m\_t$ contemporaneously, and this is the only identification condition in the model.

[![enter image description here](https://i.sstatic.net/1kXxB.png)](https://i.sstatic.net/1kXxB.png)

He mentions that the identification of shocks to $p\_t$ and $y\_t$ do not matter in the model. We can see here that $p\_t$ affects $y\_t$ contemporaneously but not the reverse.

In another example in the same book, $y\_t$ contemporaneously affects $p\_t$ which is in contrast to the example above. Again, he seeks to identify shocks only to $FF\_t$, and hence does not discuss the ordering of the non-policy block of the model. I guess it is arbitrary.

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

I see same thing in Primiceri, G. E. (2005). Interest rates are ordered last in the VAR. The interaction between inflation and unemployment is arbitrarily modeled in a lower triangular form, with inflation first, unemployment second. So I guess unemployment could also be first.

Another example is by Drechsel, T., & Tenreyro, S. (2018). There are 4 variables in the VAR. The only identification condition is that commodity price is ordered first in the lower triangular Cholesky identification scheme. The author does not mention how the other 3 domestic variables are arranged in the model. So I guess, arbitrarily.

**So my question is, in a lower triangular Cholesky identification scheme, one does not necessarily need to have an economic theory for all the orderings of the variables in the model, right? I mean, some of the orderings can be arbitrary if one do not care about the shocks associated with these aribitrarily ordered variables in the model?**

**Tags:** macroeconomics, identification, svar


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## 1 answers

### Answer by username_1 on 2021-04-06. Score: 3 | Accepted answer

You are right.

Ordering a variable first reflects the identifying assumption that this variable will not respond contemporeneously to the other shocks in the system. You could obtain the exact same shock series via an OLS regression that includes as explanatory variables only the lags of all the variables in the VAR. Since you were happy to "assume away" all endogeneity concerns, your shock is neatly identified, and you don't care about the others. (You don't even need to impose the other zeros in the impact matrix: any rotation of the bottom-right (n-1)x(n-1) submatrix will leave the first shock unaffected.)

Ordering a variable last allows this variable to contemporaneously respond to all other shocks in the system, but it imposes the assumption that none of the variables in the system can contemporaneously respond to the shock in the last equation. Again, you could obtain the same shock series via an OLS regression including all lagged and contemporaneous values of all the variables in the VAR, since, once again, you were happy to assume away all endogeneity concerns. (Also, you don't even need to impose the other zeros in the impact matrix: any rotation of the top-left (n-1)x(n-1) submatrix will leave the last shock unaffected.)

Intuitively, the OLS-counterparts show that the ordering of the other variables doesn't matter when you're interested in exclusively the first or the last shock:

* either you don't includy *any* of the other variables contemporaneously in the OLS regression (first shock);
* or you include *all of them* contemporaneously (last shock).

For the shocks in between, the OLS regressions that would deliver the same shock series as the shock series obtained via cholesky include the variables ordered before the shock of interest, but not the variables order after. So here the ordering of the variables in the VAR does affect what the shock series will look like, because changing the ordering for these shocks would include or exclude some of the rearranged contemporaneous variables in the OLS regression.

> **Comments:**
> **username_2**: That was a a beautiful answer but just to make sure I follow: The ordering of the variables ( in a cholesky decomp ) does matter when you are interested in all the shocks, correct ? It's only when you interested in the first or last, that the ordering does not matter. Thanks.  
> **username_1**: Yes, if you're interested in all the shocks, then you should be comfortable with all the identifying restrictions that the choleky factorization imposes (i.e., all the upper-triangular zeros). And those restrictions (the zeros for each row) depend on the order in which you include the variables.  
> **username_0**: @username_1, may I ask though. So are there just three ways to identification? (1). you are interested in the shock of the first ordered variable only. (2) you are interested in the shock of the last ordered variable only. (3) you are interested in all the shocks of each variable. Let's say you have a 5 variable model, [a, b, c, d, e]. 'a' and 'b' are different types of commodity variables so they are both completely exogenous. I am interested in shocks to 'a' and 'b' and how 'e' responds to these shocks. Is that possible?  
> **username_0**: [continuation]...so 'a' and 'b' are arranged arbitrarily but they must be arranged before 'c' and 'd'. 'c' and 'd' are arranged arbitrarily but it must be arranged before 'e'. Thus the identifying assumption is that [a, b] should be ordered before [c, d] and [c, d] must be ordered before [e]. But order of a and b does not matter, and order of c and d does not matter. This is possible right?  
> **username_1**: If your commodity prices are exogenous, then you can just estimate the causal effects with a local projection. Impossible to judge from the information you gave, but you may want to reconsider your assumption that commodity prices are exogenous: they're an equilibrium outcome affected by supply and demand; and especially demand may not be exogenous w.r.t. the other variables in the model (being financial variables or macro variables).  The ordering of [c,d,e] doesn't matter. But it will matter which of the two commodity prices you order first: [a,b,c,d,e] or [b,a,c,d,e] will be different.  
> **username_1**: Moreover, don't forget that you have two different identifying assumptions for the two different commodity prices. If you don't have an argumentation for this, you may want to look into other identification schemes than the recursive one implemented via Cholesky.  

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