Questions: Vector Autoregression (VAR) Models and Impulse Responses

Question 1

An economist builds a VAR with GDP growth and inflation and reports: 'A shock to GDP has no contemporaneous effect on inflation, but a shock to inflation immediately affects GDP.' What best explains this asymmetric result?

• A. This follows necessarily from the OLS estimation of the VAR equations
• B. This reflects a specific Cholesky ordering where GDP was listed first, imposing the assumption that GDP shocks are contemporaneously exogenous to inflation
• C. This is the standard empirical finding confirmed by all identification strategies
• D. This means GDP growth Granger-causes inflation but not vice versa

Answer: (B)

In a Cholesky-identified VAR, the variable listed first is assumed to respond only to its own shock contemporaneously — shocks to variables listed later cannot affect it in the same period. By placing GDP first, the researcher implicitly assumes GDP does not respond to inflation shocks within a period, while inflation can respond to GDP shocks. Swapping the order would reverse this asymmetry and change all impulse responses. The result is not a neutral mathematical output — it is a consequence of an identification assumption.

Question 2

In a VAR(1) model, the stability condition requires:

• A. All individual autoregressive coefficients to be less than 1 in absolute value
• B. All eigenvalues of the companion matrix to have modulus less than 1
• C. The residuals of all equations to be uncorrelated with each other
• D. The number of lags p to equal the number of variables in the system

Answer: (B)

Stability in a multivariate system is not captured by individual coefficient bounds — it requires that all eigenvalues of the companion (block) matrix lie strictly inside the unit circle. This generalizes the univariate AR stability condition |ρ| < 1 to the matrix setting. If any eigenvalue has modulus ≥ 1, the system is explosive — shocks grow without bound rather than damping out. Option A is a sufficient condition only in special cases, not generally.

Question 3

Swapping the order of variables in a Cholesky-identified VAR changes the impulse response functions because the ordering imposes assumptions about which variables can react to which shocks contemporaneously.

• T. True
• F. False

Answer: True

The Cholesky decomposition assigns all contemporaneous correlation to the variable listed first — it cannot respond to variables below it in the same period, while variables below it can respond to it immediately. Reordering changes which variable 'causes' contemporaneous movements in others and thus changes the IRFs. This non-neutrality is why sophisticated identification strategies use economic theory (long-run restrictions, sign restrictions) rather than relying on an arbitrary ordering.

Question 4

Forecast error variance decomposition (FEVD) shows that at most forecast horizons, a variable's forecast uncertainty is dominated by its own past shocks — cross-variable contributions remain small and stable over time.

• T. True
• F. False

Answer: False

Cross-variable contributions to FEVD typically grow with the forecast horizon. At short horizons (e.g., one quarter ahead), most of a variable's forecast uncertainty comes from its own shocks because there has been little time for other variables' shocks to propagate through the system. As the horizon extends, the dynamic transmission channels activate and other variables' shocks account for increasing shares of forecast error variance. The claim that cross-variable contributions remain small and stable conflates short-run with long-run dynamics.

Question 5

Explain why impulse response functions from a reduced-form VAR cannot be interpreted causally without an identification strategy, and what the Cholesky ordering assumes.

Model answer: Reduced-form VAR residuals are typically correlated across equations — a simultaneous shock to GDP and inflation makes it impossible to say which caused which contemporaneous movement. To compute IRFs, you need to decompose this joint shock into orthogonal components and assign causal direction. The Cholesky ordering assumes a triangular structure: the variable listed first is hit by a 'pure' own shock that cannot be caused by variables below it in the same period, while variables lower in the ordering can respond contemporaneously to those above. This is an identifying assumption about causal priority, not a statistical result.

Without identification, an 'inflation shock' in a VAR is actually a mixture of true inflation shocks and all other contemporaneously correlated shocks. The Cholesky ordering is one way to resolve this ambiguity by imposing a recursive causal structure. Its validity depends entirely on whether the causal ordering chosen matches reality — which is an economic judgment, not a statistical one. This is why economists say 'the VAR is identified up to an orthogonalization choice.'