# SVAR causal interpretation: shock effects vs effects between variables

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

\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_{2, t-1} + 0.30 y_{2, t-1} + \omega_{2, t} \ , \end{aligned}

whereas in the "CS form" we have

\begin{aligned} y_{1,t} &= 0.50 y_{2, t} - 0.10 y_{1, t-1} + \omega_{1, t} \\ y_{2,t} &= 0.2 y_{2, t-1} + 0.30 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

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.