Keating (1992) Structural Approaches to Vector Autoregressions

Summary

A pedagogical survey published in the Federal Reserve Bank of St. Louis Review (Sep/Oct 1992). Derives both contemporaneous and long-run structural vector autoregression (VAR) identification from the underlying simultaneous-equations model, makes the identification counting problem explicit, and motivates the structural program via the Cooley-LeRoy (1985) critique of the Cholesky decomposition. Concludes with an empirical comparison of the two identification strategies using a 4-variable US macro model (price level, output, interest rate, money) over 1959Q1–1991Q3.

Key Claims

• The VAR is not a stand-alone object but the reduced form of a linear simultaneous-equations system; its residuals are mixtures of structural shocks.
• The Cholesky ("atheoretical") decomposition is not atheoretical: it is algebraically equivalent to a recursive contemporaneous structural model, one of $n!$ possible orderings.
• With $\Sigma_\varepsilon$ diagonal, $D = I$, and the diagonal of $A$ normalized to unity, exactly $n(n-1)/2$ further restrictions on $A$ are required for exact identification.
• The long-run key equation $[I - \beta(1)]^{-1}\Sigma_e\,[I - \beta(1)]^{-\prime} = \Theta(1)\,\Sigma_\varepsilon\,\Theta(1)'$ relates observable reduced-form covariance to structural long-run multipliers, enabling long-run identification without contemporaneous restrictions.
• In the empirical comparison, the long-run model yields results broadly consistent with theory; the contemporaneous model produces several anomalies (e.g., money supply shocks with no price effect).
• Contemporaneous zero restrictions may be invalid under rational expectations (Keating 1990), because any observable variable can signal future events.

Concepts Introduced or Extended

• Structural Identification — explicit counting argument; Cooley-LeRoy critique of Cholesky
• Variance Decomposition — full formula derived from the moving-average (MA) representation
• Impulse Response Function — moving-average representation (MAR) derived for both levels and differenced VAR

Entities Mentioned

• John Keating
• Christopher Sims

Quotes

"The Choleski ordering can be interpreted as a recursive contemporaneous structural model. Unfortunately, most economic theories do not imply recursive contemporaneous systems."

"Long-run structural VAR models may provide superior results because they typically do not impose contemporaneous exclusion restrictions."

My Take

The paper's main lasting contribution is pedagogical clarity: the counting argument, the simultaneous-equations derivation of the VAR, and the Cooley-LeRoy critique are all made more transparent here than in Bernanke (1986) or Blanchard-Watson (1986). The empirical finding that long-run restrictions outperform contemporaneous ones is plausible but not surprising — it is partly an artifact of the specific model chosen (output driven solely by supply shocks long-run), and the contemporaneous model's poor performance may reflect misspecification rather than a general inferiority of short-run restrictions.