# What is ordering of the variables in a SVAR model

I am currently working on my dissertation where I try to analyze how macroeconomic shocks affect firms financing decisions. The research paper that I am currently reading says:
"Overall, this suggests that the results using the recursive scheme can be highly sensitive to the ordering of the variables—that is, the impulse responses can be subject to large biases if the ordering is incorrect" What does this ordering of variables mean? I am confused. Thank you in advance!

• Please give reference to the paper you got the quote from. Jul 25, 2022 at 14:03

Let's take a hypothetical example of the SVAR with lag 1 (simplified specification for understanding):

$$Inc_t = \alpha_1 + Inc_{t-1} + Con_{t-1} + Inv_{t-1} + e_1$$

$$Inv_t = \alpha_2 + Inc_t + Inc_{t-1} + Con_{t-1} + Inv_{t-1} + e_2$$

$$Con_t = \alpha_3 + Inc_t + Inv_t + Inc_{t-1} + Con_{t-1} + Inv_{t-1} + e_3$$

This SVAR represents an ordering of Income, Investment and Consumption.

We can observe that because Current period Income is only affected by the lagged terms in the first equation. But, income is contemporaneously affecting both investment and consumption. This implies that the current period investment and consumption are affected by current period Income along with lags.

Similarly, investment is contemporaneously affecting only consumption, that is, current consumption depends on current investment along with the lags.

This is also referred to as Cholesky Decomposition. The different choices of ordering will completely change the model. You could order it as Investment, income and consumption and so on, which will change the specification of the contemporaneous terms (variables which affect the others in the same period).

Hence, it is up to the researcher to decide the correct ordering based on some economic theory or other considerations. The decision has to be made on which variables may have a delayed effect on others and which can affect the others instantly in the same period.

The relative endogeneity/exogeneity is important here. Relatively less endogeneity would mean that the variable can affect the others contemporaneously (in the same time period). In this hypothetical example, income is the most exogenous and consumption is the most endogenous.

• Thank you, I understand it better now :) Jul 26, 2022 at 15:17

The ordering means placing the variables in the (decreasing) order of exogeneity. For example, if $$y_1,y_2$$ and $$y_3$$ are variables in the system and economic theory says $$y_3$$ is relatively more exogeneous than $$y_1$$ and $$y_2$$ and $$y_1$$ is more exogeneous than $$y_2$$ but less exogenous than $$y_3$$, then they should be ordered $$y_3,y_1,y_2$$. It is also important to understand that while all variables are endogenous some are more/less endogenous, that is what I mean when I say relative exogeneity.

Exactly as the source says, if you mess up the order your results will be biased. To figure out the correct order you should survey theoretical literature on the issue you want to examine, or see past empirical papers that also used SVAR to examine same/similar topic and see their ordering (and justification for it).