<p>In economic models (not econometrics) contemporaneous effects are a common rule. From that point of view, there's no issue to see that relation between <span class="math-container">$y_{1,t}$</span> and <span class="math-container">$y_{2,t}$</span>. Also, no issues on the lagged CS from too.</p>
<p>From an econometric point of view, we're dealing with the dilemma between structural form vs reduced from. Which a very common discussion in Econometric Textbooks. The one you call <em>econometrics form</em> looks like the reduced form (no endogenous regressors) and the <em>CS form</em> looks like the structural.</p>
<p>The advantage is to change from one to the other, for estimation and for interpretation purposes, and also to estimate Impulse Response Functions. Thus the econometric form allows you to create IRF, which are quite frequent in economic papers.</p>

CC BY-SA 4.0
2022-05-17T17:21:51.363Z
Many thanks for the response! "The one you call econometrics form looks like the reduced form" -> I don't agree with this. To me both (1) and (2) seem "structural" as the difference is just moving the endogenous regressor term C * y_t to different side of the equality sign. We'd obtain the reduced form version if we pre-multiplied (1) by inverse of B_0 (see Killian & Lütkepohl 2017, p. 109); then indeed we'd have no endogenous regressor. Due to this, your answer does not seem to answer the question I am after.
vvv
CC BY-SA 4.0
2022-05-11T15:39:13.023Z
2022-05-11T15:39:13.023Z
Lima_Institute_of_Econometrics
<p>Consider a structural vector autoregressive (SVAR) model. One way to define a SVAR model is</p>
<p><span class="math-container">$$
\begin{aligned}
B_0 y_t = B_1 y_{t-1} + \cdots + B_p y_{t-p} + \omega_t \quad (1) \ ,
\end{aligned}
$$</span></p>
<p>where <span class="math-container">$\omega_t$</span> are the structural disturbances (or &quot;shocks&quot; in economic parlance). We make the usual assumptions regarding <span class="math-container">$\omega_t$</span> (see e.g. Killian &amp; Lütkepohl 2017, p. 109). I call (1) the <em>&quot;econometrics form&quot;</em>, as it is most typically encountered in econometric texts. An alternative way to write a SVAR model is</p>
<p><span class="math-container">$$
\begin{aligned}
y_t = C y_t + B_1 y_{t-1} + \cdots + B_p y_{t-p} + \omega_t \quad &amp; (2) \ ,
\end{aligned}
$$</span></p>
<p>where <span class="math-container">$B_0 = I - C$</span>. I call this the <em>&quot;CS form&quot;</em>, as I've encountered it often in the computer science literature (see e.g. Hyvärinen et. al. 2010).</p>
<p>Consider an example (from Moneta et. al. 2013) with two variables and one lag. Assume we have already solved the <em>identification problem</em>, that is, we have found the &quot;true&quot; matrix <span class="math-container">$B_0$</span> (or equivalently <span class="math-container">$C$</span>):</p>
<p><span class="math-container">$$
\begin{aligned}
    B_0 = \begin{bmatrix}
        1 &amp; -0.5 \\
        0 &amp; 1
    \end{bmatrix} , \
\end{aligned}
    B_0^{-1} = \begin{bmatrix}
        1 &amp; 0.5 \\
        0 &amp; 1
    \end{bmatrix}, \
    C = \begin{bmatrix}
        0 &amp; 0.5 \\
        0 &amp; 0
    \end{bmatrix} \ .
$$</span></p>
<p>Assume further that <span class="math-container">$B_p$</span> with <span class="math-container">$p\geq 2$</span> are zero matrices and</p>
<p><span class="math-container">$$
\begin{aligned}
    B_1 = \begin{bmatrix}
        -0.1 &amp; 0 \\
        0.2 &amp; 0.3
    \end{bmatrix}
\end{aligned} \ .
$$</span></p>
<p>Now pre-multiplying (1) with <span class="math-container">$B_0^{-1}$</span> and performing matrix multiplications leaves us with (<em>&quot;econometrics form&quot;</em>)</p>
<p><span class="math-container">$$
\begin{aligned}
y_{1,t} &amp;= 0.15 y_{2, t-1} + \omega_{1, t} + 0.5 \omega_{2, t} \\
y_{2,t} &amp;= 0.2 y_{1, t-1} +  0.3 y_{2, t-1} + \omega_{2, t} \ ,
\end{aligned}
$$</span></p>
<p>whereas in the <em>&quot;CS form&quot;</em> we have (no pre-multiplication, simply plugging in the identified matrices <span class="math-container">$C$</span> and <span class="math-container">$B_1$</span>)</p>
<p><span class="math-container">$$
\begin{aligned}
y_{1,t} &amp;= 0.5 y_{2, t} - 0.1 y_{1, t-1} + \omega_{1, t} \\
y_{2,t} &amp;= 0.2 y_{1, t-1} +  0.3 y_{2, t-1} + \omega_{2, t} \ .
\end{aligned}
$$</span></p>
<p>We can make following claims about <strong>contemporaneous</strong> causal effects:</p>
<ul>
<li>In <em>&quot;econometrics form&quot;</em>
<ul>
<li><span class="math-container">$\omega_{1,t}$</span> affects <span class="math-container">$y_{1,t}$</span></li>
<li><span class="math-container">$\omega_{2,t}$</span> affects both <span class="math-container">$y_{1,t}$</span> and <span class="math-container">$y_{2,t}$</span></li>
</ul>
</li>
<li>In <em>&quot;CS form&quot;</em>
<ul>
<li><span class="math-container">$\omega_{1,t}$</span> affects <span class="math-container">$y_{1,t}$</span></li>
<li><span class="math-container">$\omega_{2,t}$</span> affects <span class="math-container">$y_{2,t}$</span></li>
<li><span class="math-container">$y_{2,t}$</span> affects <span class="math-container">$y_{1,t}$</span>.</li>
</ul>
</li>
</ul>
<p>Although mathematically equivalent, from a more philosophical point of view it seems we are making different claims. In the <em>&quot;econometrics form&quot;</em>, we are adopting the view typically held by economists that external, abstract &quot;shocks&quot; influence the variables (according to Frisch-Slutsky paradigm). Notably, structural disturbances are in general not to be interpreted in terms of the units of measurement applied to the model variables (Killian &amp; Lütkepohl 2017, p. 112). In  the <em>&quot;CS form&quot;</em>, it seems to me that rather than putting abstract shock labels on <span class="math-container">$\omega_{i,t}$</span>, we consider it simply as an innovation to the corresponding model variable <span class="math-container">$y_{i}$</span>. On the other hand, now there exists contemporaneous causal effects between the variables themselves!</p>
<p>Is there some relevant difference between the two forms in terms of economic interpretation? Are the two sets of causal claims equally valid?</p>
<p>References:</p>
<ul>
<li>Hyvärinen, Zhang, Schimizu, Hoyer (2010): Estimation of a structural vector autoregression model using non-Gaussianity</li>
<li>Moneta, Entner, Hoyer, Coad (2013): Causal inference by independent component analysis: theory and applications</li>
<li>Killian &amp; Lütkepohl (2017): Structural vector autoregressive analysis</li>
</ul>

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2024-09-28T02:43:26.747Z
Hi: For your conclusions about what effects what, you can add that, in "CS form", $w_{2,t}$ effects $y_{1t}$ because $w_{2,t}$ effects $y_{2t}$ which effects $y_{1t}$. So, my point is that, once that one is added,  your conclusions are almost consistent in the respective forms. Also, do you have a typo because the second econometrics form equation contains two $y_{2,t-1}$'s ?
mark leeds
CC BY-SA 4.0
2024-10-09T11:57:52.407Z
Thanks for catching the typos, I corrected the equations!

Yes, one could add the conclusion that $y_{2, t}$ *mediates* the effect of $\omega_{2,t}$ to $y_{1,t}$. But in my view having a direct (causal) effect ($\omega_{2,t} \rightarrow y_{1,t}$) is something different than having a mediated effect ($\omega_{2,t} \rightarrow y_{2,t} \rightarrow y_{1,t}$). This leads in my head to the "philosophical" difference, although being numerically equivalent. In "CS" from $\omega_{2,t}$ can be interpreted explicitly in terms of the unit of measurement of $y_{2, t}$, but in "econometrics form" cannot.
vvv
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2024-10-10T04:01:20.243Z
okay. I'm not sure if your question was answered ? I hope so.
mark leeds
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2024-10-11T16:10:16.863Z
Not really, I’m afraid. I am still unsure how can it be that in the first form the set of causal claims is different from the latter form.
vvv
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2024-10-13T01:27:59.663Z
Hi vvv: Just so I understand: Are you saying that it's troubling because in the econometrics form, $w_{2t}$ effects $y_{1t}$ directly whereas, in the CS form, it's an indirect effect because $w_{2t}$ effects $y_{1t}$ indirectly through $y_{2t}$ ?
mark leeds
CC BY-SA 4.0
2024-10-13T03:38:05.753Z
vvv: Based on my reading the first couple of pages of the Moneta paper, the system of equations where you wrote "whereas in the CS form, we have" is actually the reduced form VAR. The bottom of page 3 atleast implies that. I haven't figured out what in the paper corresponds to the econometrics form of the model. I'll take a look at that tomorrow.
mark leeds
CC BY-SA 4.0
2024-10-18T08:09:59.823Z
Yes, that's what I'm saying.

In Moneta et. al., equations 1 and 2 are both structural form. Going from 1 to 2, they even write 'This model can equivalently be expressed in the standard SVAR form'. Equation 4 is indeed reduced form.

I edited my original post. Essentially, the difference in the two forms comes from how one gets rid of $y_{j, t}$ in the equation for $y_{i,t}$. In the standard econometrics way one pre-multiplies by $B_0^{-1}$, in the 'CS form' one throws the extra contemp. term to the other side of the eq., yielding the difference in direct vs. mediated effects.
vvv
CC BY-SA 4.0
2024-10-18T08:21:01.387Z
Above it should say (hate the 5min edit limit on comments): Essentially, the difference in the two forms comes from how one gets rid of 'extra' $y_{i,t}$ in the equation for $y_{i,t}$. In the CS form, the equation for $y_{i,t}$ may contain terms $y_{j,t}$, $i\neq j$, on the RHS. This leaves room for (causal) effects between contemp. y's. In the econometrics form, all the contemp. (causal) effects are embedded in the effects $\omega \rightarrow y_t$.
vvv
CC BY-SA 4.0
2024-10-19T09:56:55.863Z
I've never seen it stated that way but it makes sense. My understanding is that the econometrics form says nothing about what causes what so your statement is consistent with that. Canova wrote a long paper on vector autogressions in 1995 but it was part of the Handbook of Econometrics. I was unable to get it for free.
mark leeds
CC BY-SA 4.0
2024-10-19T10:04:22.847Z
just saying one more thing in case anyone reads this down the road and is also not familar with the terms econometric form and cs form.  The econometrics form basically assumes that everything is correlated which is pretty much saying that nothing can be identified as cause and effect. The cs formulation, I think is the recursive var that tries to order the variables which allows one to separate the effects but that ends up being arbitrary also.
mark leeds
CC BY-SA 4.0
2024-11-17T08:52:30.017Z
I view the "econom." form as a way to introduce new, unmeasured variables into the system (namely, the shocks $\omega$) which are the underlying true causal variables. y's in Economics are often merely accounting measures (like GDP) that respond to more abstract driving forces (according to Frisch-Slutsky paradigm). In "CS" form (and also, e.g., in earth sciences) the y's are "real" system variables causing each other. When identification is done via contemp. causal ordering, the "econom." and "CS" forms coincide mathematically (but perhaps differ in interpretation, cf. my original question).
vvv
CC BY-SA 4.0
2024-11-17T18:35:49.287Z
vvv. I like your view and I think we are saying roughly the same thing. If you can get Canova's 1995 paper, he does a nice job of explaining it and I think what he's saying mostly coincides with your viewpoint.
mark leeds
<causality><identification><svar>
SVAR causal interpretation: shock effects vs effects between variables
vvv
# SVAR causal interpretation: shock effects vs effects between variables

(Asked by: username_0 on 2021-11-21)

Consider a structural vector autoregressive (SVAR) model. One way to define a SVAR model is

$$
\begin{aligned}
B\_0 y\_t = B\_1 y\_{t-1} + \cdots + B\_p y\_{t-p} + \omega\_t \quad (1) \ ,
\end{aligned}
$$

where $\omega\_t$ are the structural disturbances (or "shocks" in economic parlance). We make the usual assumptions regarding $\omega\_t$ (see e.g. Killian & Lütkepohl 2017, p. 109). I call (1) the *"econometrics form"*, as it is most typically encountered in econometric texts. An alternative way to write a SVAR model is

$$
\begin{aligned}
y\_t = C y\_t + B\_1 y\_{t-1} + \cdots + B\_p y\_{t-p} + \omega\_t \quad & (2) \ ,
\end{aligned}
$$

where $B\_0 = I - C$. I call this the *"CS form"*, as I've encountered it often in the computer science literature (see e.g. Hyvärinen et. al. 2010).

Consider an example (from Moneta et. al. 2013) with two variables and one lag. Assume we have already solved the *identification problem*, that is, we have found the "true" matrix $B\_0$ (or equivalently $C$):

$$
\begin{aligned}
B\_0 = \begin{bmatrix}
1 & -0.5 \\
0 & 1
\end{bmatrix} , \
\end{aligned}
B\_0^{-1} = \begin{bmatrix}
1 & 0.5 \\
0 & 1
\end{bmatrix}, \
C = \begin{bmatrix}
0 & 0.5 \\
0 & 0
\end{bmatrix} \ .
$$

Assume further that $B\_p$ with $p\geq 2$ are zero matrices and

$$
\begin{aligned}
B\_1 = \begin{bmatrix}
-0.1 & 0 \\
0.2 & 0.3
\end{bmatrix}
\end{aligned} \ .
$$

Now pre-multiplying (1) with $B\_0^{-1}$ and performing matrix multiplications leaves us with (*"econometrics form"*)

$$
\begin{aligned}
y\_{1,t} &= 0.15 y\_{2, t-1} + \omega\_{1, t} + 0.5 \omega\_{2, t} \\
y\_{2,t} &= 0.2 y\_{1, t-1} + 0.3 y\_{2, t-1} + \omega\_{2, t} \ ,
\end{aligned}
$$

whereas in the *"CS form"* we have (no pre-multiplication, simply plugging in the identified matrices $C$ and $B\_1$)

$$
\begin{aligned}
y\_{1,t} &= 0.5 y\_{2, t} - 0.1 y\_{1, t-1} + \omega\_{1, t} \\
y\_{2,t} &= 0.2 y\_{1, t-1} + 0.3 y\_{2, t-1} + \omega\_{2, t} \ .
\end{aligned}
$$

We can make following claims about **contemporaneous** causal effects:

* In *"econometrics form"*
  + $\omega\_{1,t}$ affects $y\_{1,t}$
  + $\omega\_{2,t}$ affects both $y\_{1,t}$ and $y\_{2,t}$
* In *"CS form"*
  + $\omega\_{1,t}$ affects $y\_{1,t}$
  + $\omega\_{2,t}$ affects $y\_{2,t}$
  + $y\_{2,t}$ affects $y\_{1,t}$.

Although mathematically equivalent, from a more philosophical point of view it seems we are making different claims. In the *"econometrics form"*, we are adopting the view typically held by economists that external, abstract "shocks" influence the variables (according to Frisch-Slutsky paradigm). Notably, structural disturbances are in general not to be interpreted in terms of the units of measurement applied to the model variables (Killian & Lütkepohl 2017, p. 112). In the *"CS form"*, it seems to me that rather than putting abstract shock labels on $\omega\_{i,t}$, we consider it simply as an innovation to the corresponding model variable $y\_{i}$. On the other hand, now there exists contemporaneous causal effects between the variables themselves!

Is there some relevant difference between the two forms in terms of economic interpretation? Are the two sets of causal claims equally valid?

References:

* Hyvärinen, Zhang, Schimizu, Hoyer (2010): Estimation of a structural vector autoregression model using non-Gaussianity
* Moneta, Entner, Hoyer, Coad (2013): Causal inference by independent component analysis: theory and applications
* Killian & Lütkepohl (2017): Structural vector autoregressive analysis

**Tags:** causality, identification, svar
> **Comments:**
> **username_1**: Hi: For your conclusions about what effects what, you can add that, in "CS form", $w_{2,t}$ effects $y_{1t}$ because $w_{2,t}$ effects $y_{2t}$ which effects $y_{1t}$. So, my point is that, once that one is added,  your conclusions are almost consistent in the respective forms. Also, do you have a typo because the second econometrics form equation contains two $y_{2,t-1}$'s ?  
> **username_0**: Thanks for catching the typos, I corrected the equations!  Yes, one could add the conclusion that $y_{2, t}$ *mediates* the effect of $\omega_{2,t}$ to $y_{1,t}$. But in my view having a direct (causal) effect ($\omega_{2,t} \rightarrow y_{1,t}$) is something different than having a mediated effect ($\omega_{2,t} \rightarrow y_{2,t} \rightarrow y_{1,t}$). This leads in my head to the "philosophical" difference, although being numerically equivalent. In "CS" from $\omega_{2,t}$ can be interpreted explicitly in terms of the unit of measurement of $y_{2, t}$, but in "econometrics form" cannot.  
> **username_1**: okay. I'm not sure if your question was answered ? I hope so.  
> **username_0**: Not really, I’m afraid. I am still unsure how can it be that in the first form the set of causal claims is different from the latter form.  
> **username_1**: Hi vvv: Just so I understand: Are you saying that it's troubling because in the econometrics form, $w_{2t}$ effects $y_{1t}$ directly whereas, in the CS form, it's an indirect effect because $w_{2t}$ effects $y_{1t}$ indirectly through $y_{2t}$ ?  
> **username_1**: vvv: Based on my reading the first couple of pages of the Moneta paper, the system of equations where you wrote "whereas in the CS form, we have" is actually the reduced form VAR. The bottom of page 3 atleast implies that. I haven't figured out what in the paper corresponds to the econometrics form of the model. I'll take a look at that tomorrow.  
> **username_0**: Yes, that's what I'm saying.  In Moneta et. al., equations 1 and 2 are both structural form. Going from 1 to 2, they even write 'This model can equivalently be expressed in the standard SVAR form'. Equation 4 is indeed reduced form.  I edited my original post. Essentially, the difference in the two forms comes from how one gets rid of $y_{j, t}$ in the equation for $y_{i,t}$. In the standard econometrics way one pre-multiplies by $B_0^{-1}$, in the 'CS form' one throws the extra contemp. term to the other side of the eq., yielding the difference in direct vs. mediated effects.  
> **username_0**: Above it should say (hate the 5min edit limit on comments): Essentially, the difference in the two forms comes from how one gets rid of 'extra' $y_{i,t}$ in the equation for $y_{i,t}$. In the CS form, the equation for $y_{i,t}$ may contain terms $y_{j,t}$, $i\neq j$, on the RHS. This leaves room for (causal) effects between contemp. y's. In the econometrics form, all the contemp. (causal) effects are embedded in the effects $\omega \rightarrow y_t$.  
> **username_1**: I've never seen it stated that way but it makes sense. My understanding is that the econometrics form says nothing about what causes what so your statement is consistent with that. Canova wrote a long paper on vector autogressions in 1995 but it was part of the Handbook of Econometrics. I was unable to get it for free.  
> **username_1**: just saying one more thing in case anyone reads this down the road and is also not familar with the terms econometric form and cs form.  The econometrics form basically assumes that everything is correlated which is pretty much saying that nothing can be identified as cause and effect. The cs formulation, I think is the recursive var that tries to order the variables which allows one to separate the effects but that ends up being arbitrary also.  
> **username_0**: I view the "econom." form as a way to introduce new, unmeasured variables into the system (namely, the shocks $\omega$) which are the underlying true causal variables. y's in Economics are often merely accounting measures (like GDP) that respond to more abstract driving forces (according to Frisch-Slutsky paradigm). In "CS" form (and also, e.g., in earth sciences) the y's are "real" system variables causing each other. When identification is done via contemp. causal ordering, the "econom." and "CS" forms coincide mathematically (but perhaps differ in interpretation, cf. my original question).  
> **username_1**: vvv. I like your view and I think we are saying roughly the same thing. If you can get Canova's 1995 paper, he does a nice job of explaining it and I think what he's saying mostly coincides with your viewpoint.  

---

## 1 answers

### Answer by username_2 on 2022-05-11. Score: 0

In economic models (not econometrics) contemporaneous effects are a common rule. From that point of view, there's no issue to see that relation between $y\_{1,t}$ and $y\_{2,t}$. Also, no issues on the lagged CS from too.

From an econometric point of view, we're dealing with the dilemma between structural form vs reduced from. Which a very common discussion in Econometric Textbooks. The one you call *econometrics form* looks like the reduced form (no endogenous regressors) and the *CS form* looks like the structural.

The advantage is to change from one to the other, for estimation and for interpretation purposes, and also to estimate Impulse Response Functions. Thus the econometric form allows you to create IRF, which are quite frequent in economic papers.

> **Comments:**
> **username_0**: Many thanks for the response! "The one you call econometrics form looks like the reduced form" -> I don't agree with this. To me both (1) and (2) seem "structural" as the difference is just moving the endogenous regressor term C * y_t to different side of the equality sign. We'd obtain the reduced form version if we pre-multiplied (1) by inverse of B_0 (see Killian & Lütkepohl 2017, p. 109); then indeed we'd have no endogenous regressor. Due to this, your answer does not seem to answer the question I am after.  

---

