Common mistake 1 with Structural VAR

The big one: reporting recursive impulse responses without saying what the ordering was, or checking whether it matters. Two researchers with the same data and different orderings both get correct arithmetic and different conclusions.

Common mistake 2 with Structural VAR

Ordering by what looks natural rather than by an argument. "Most exogenous first" needs a reason - a policy variable that reacts to news within the quarter does not belong first.

Common mistake 3 with Structural VAR

Treating sign-restricted bands as confidence intervals. They are identification uncertainty, which no amount of data removes.

Common mistake 4 with Structural VAR

Using Blanchard–Quah on more than two variables. The long-run restriction as usually stated only pins down a 2×2 system; this module says so and falls back rather than pretending.

Common mistake 5 with Structural VAR

Running a structural VAR on non-stationary levels. The stationarity check is on by default for a reason.

EconStatsCraft › Structural VAR

Structural VAR

A VAR gives correlated residuals, which are not shocks anyone can name. This adds the assumption that turns them into named shocks - a recursive ordering, a long-run restriction, or a set of signs - and reports what changes if the assumption is wrong.

What it needs

Endogenous variables - Order matters under the recursive scheme: most exogenous first.

The intuition

A VAR describes how the variables move together but cannot say which shock caused what, because the residuals of the different equations are correlated - they move at the same instant and the data cannot untangle simultaneity on its own. A structural VAR adds an identifying assumption from outside the data. Recursive ordering is the commonest: assert that some variables cannot react within the period, and the arithmetic does the rest.

When to use it

When the question is causal rather than predictive: what happens to output after a rate rise, how much of inflation variance is a supply shock, does a fiscal expansion crowd out investment. If you only want forecasts, a plain VAR is enough and needs no identifying assumption.

Where the estimator comes from

The reduced form y_t=A_1y_{t-1}+\dots+u_t has E[u_tu_t']=\Sigma, and the structural shocks satisfy u_t=B\varepsilon_t with E[\varepsilon_t\varepsilon_t']=I, so \Sigma=BB'. That is n(n+1)/2 equations in n^2 unknowns, so n(n-1)/2 restrictions are needed. A RECURSIVE scheme takes B lower triangular, which the Cholesky factor delivers uniquely - and lower triangular means variable 1 does not respond within the period to any shock but its own, variable 2 responds only to shocks 1 and 2, and so on. Long-run identification (Blanchard and Quah, 1989) instead restricts the cumulative response \left(I-\sum A_i\right)^{-1}B, and sign restrictions merely require responses to have specified signs, which identifies a SET of models rather than one.

The model

A_0 y_t = \sum_{i=1}^{p} A_i y_{t-i} + \varepsilon_t,\qquad \varepsilon_t \sim \text{WN}(0, I),\qquad u_t = A_0^{-1}\varepsilon_t,\qquad \Sigma_u = A_0^{-1}A_0^{-\prime}

What it assumes

Everything a reduced-form VAR assumes: stationary series, correct lag length, stable coefficients over the sample.
The identifying restriction is TRUE. This is not testable from the data - it is the assumption that makes the shocks separable, and different assumptions give different answers from identical estimates.
Under the recursive scheme: the variable ordered first responds to nothing else within the period. Order is the assumption.
Under Blanchard–Quah: one shock has no permanent effect on one variable. That is a restriction on the cumulative response, not on impact.
Under sign restrictions: the signs you impose are the ones economic theory actually implies - and you accept that the answer is a set, not a number.

Statistical properties

The ordering is an economic assumption, not a technical setting. Placing a variable first asserts it is unaffected within the period by everything else, and that claim must be argued.
How much it matters depends on the residual correlation. With near-zero correlation the ordering barely changes anything; with high correlation it changes a great deal, so report the correlation alongside the impulse responses.
The zeros imposed are exact and unfalsifiable. Whichever variable comes first WILL show a zero impact response to every other shock, because you put it there.
Impulse response confidence bands from the bootstrap are typically wide and are often omitted from published charts. A response that looks decisive with the point estimate alone frequently is not.
Sign restrictions avoid the arbitrariness of an ordering at the cost of set identification: the answer is a range of admissible responses, and the median of that range is not an estimate of anything.
Forecast error variance decompositions inherit the same identification. They are as arbitrary as the ordering that produced them.

How to read the output

Read the impact matrix first. Column j is what shock j does to everything in the same period. In the recursive scheme the zeros above the diagonal are your assumption, not a finding - do not report them as evidence.
The ordering-sensitivity table is the one that decides whether you have a result. If the impacts change materially when the order is reversed, your impulse responses are an artefact of the ordering.
Impulse responses answer "what happens over time after one shock". Cumulative effects answer "does it settle back". A cumulative response that stays away from zero is a permanent level shift.
The variance decomposition answers "how much of the movement in this variable is that shock" - at a given horizon. Shares change with the horizon, so quote it.
Under sign restrictions the band is not a confidence interval. It is the range of answers consistent with your restrictions, and it does not shrink with more data.

A worked reading

Three series, one lag, recursive identification. The impact matrix is triangular by construction: a shock to growth moves inflation by 0.0349 and the interest rate by 0.0504 on impact, while shocks to inflation and to the interest rate move growth by exactly 0.0000. Those zeros were IMPOSED, not estimated, and they are the whole content of the identifying assumption. The ordering-sensitivity table makes the cost visible: reverse the order and all three of those cross-effects become 0.0000 instead - the entire impact response of inflation to a growth shock disappears, because the assumption about which variable can react within the period has been reversed. No test can adjudicate between the two orderings; only an argument about timing can. The variance decomposition shows where it matters least and most: growth is 97.7% its own shock at horizon 1 and still 86.9% at horizon 12, so that row is robust, while the interest rate goes from 64.4% own-shock at horizon 1 to 23.2% at horizon 12 and is the row an ordering change would rewrite.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Results changing sharply with the ordering	Highly correlated residuals	Report the correlation; justify the ordering or use sign restrictions
Impact response of exactly zero	Imposed by the recursive scheme	Not a finding
Impulse responses exploding	A non-stationary VAR	Check the companion matrix eigenvalues
Very wide bootstrap bands	Normal, and usually hidden	Show them. Most published responses are not significant
A price puzzle - prices rising after a tightening	Classic sign of an omitted variable	Add a commodity price index, as Sims suggested
Long-run restriction giving implausible responses	Sensitive to the estimated long-run matrix	This scheme is fragile with persistent data

Common mistakes

The big one: reporting recursive impulse responses without saying what the ordering was, or checking whether it matters. Two researchers with the same data and different orderings both get correct arithmetic and different conclusions.
Ordering by what looks natural rather than by an argument. "Most exogenous first" needs a reason - a policy variable that reacts to news within the quarter does not belong first.
Treating sign-restricted bands as confidence intervals. They are identification uncertainty, which no amount of data removes.
Using Blanchard–Quah on more than two variables. The long-run restriction as usually stated only pins down a 2×2 system; this module says so and falls back rather than pretending.
Running a structural VAR on non-stationary levels. The stationarity check is on by default for a reason.

Where to read this properly

Sims (1980), "Macroeconomics and Reality", Econometrica 48(1).
Blanchard and Quah (1989), American Economic Review 79(4), for long-run restrictions.
Uhlig (2005), "What are the Effects of Monetary Policy on Output?", Journal of Monetary Economics 52(2), on sign restrictions.
Kilian and Lutkepohl, Structural Vector Autoregressive Analysis (2017).

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Structural VAR

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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