Lastrapes (2005) Estimating and Identifying Vector Autoregressions under Diagonality and Block Exogeneity Restrictions

Summary

Lastrapes (2005) shows how to estimate and identify a large-scale vector autoregression (VAR) — where $n_1$ (micro/industry variables) can be very large — by imposing two over-identifying restrictions: (1) the variables in $z_1$ are mutually independent conditional on common factors $z_2$ (diagonality of $A_{11}^h$ ), and (2) $z_2$ is block exogenous with respect to $z_1$ ( $A_{21}^h = 0$ ). Under these restrictions the system separates into two independent sub-systems estimable by equation-by-equation OLS, which is fully efficient because the residual covariance $H = \Omega_{11} - \Omega_{12}\Omega_{22}^{-1}\Omega_{12}'$ is diagonal. Structural identification of the full system requires restrictions only on the aggregate sub-system $z_2$ .

Key Claims

Diagonality restriction: $A_{11}^h$ diagonal for $h=0,1,\ldots,p$ — mutual independence of micro variables conditional on common factors. All cross-sectional correlation in $z_1$ is mediated by $z_2$ .
Block exogeneity restriction: $A_{21}^h = 0$ for all $h$ — aggregate variables $z_2$ are not driven by micro variables $z_1$ . Plausible whenever $z_2$ contains aggregate-level and $z_1$ contains industry- or region-level variables.
Block separation: Under both restrictions the VAR separates into (a) an unrestricted $n_2$ -dimensional VAR for $z_2$ , and (b) $n_1$ univariate equations each depending only on own lags plus $z_2$ and its lags (with contemporaneous $G_0 = \Omega_{12}\Omega_{22}^{-1}$ ).
OLS efficiency: The residual covariance from the $z_1$ sub-system is $H = (A_{11}^0)^{-1}(A_{11}^{0})'{}^{-1}$ , which is diagonal because $A_{11}^0$ is diagonal. Therefore equation-by-equation OLS is fully efficient — no generalized least squares (GLS) correction needed.
Identification falls entirely on $z_2$ : Given $D_{22}^0$ identified from $\Omega_{22}$ (e.g., Cholesky), the cross block $D_{12}^0 = \Omega_{12}(D_{22}^{0'})^{-1}$ follows automatically, and the diagonal $D_{11}^0$ is just-identified as positive square roots of diagonal elements of $\Omega_{11} - D_{12}^0 D_{12}^{0'}$ . No restrictions on micro-aggregate cross-responses are needed.
Long-run identification: Blanchard-Quah-style restrictions on $\tilde{D}_{22}$ from the $z_2$ sub-system are also sufficient. Once $\tilde{D}_{22}$ is identified from $\tilde{D}_{22}\tilde{D}_{22}' = \tilde{C}_{22}\Omega_{22}\tilde{C}_{22}'$ , the short-run matrix follows from $D_{22}^0 = \tilde{C}_{22}^{-1}\tilde{D}_{22}$ , and $D_{12}^0$ , $D_{11}^0$ are recovered as before.
Critique of equation-by-equation VARs (Carlino-Defina approach): Estimating a separate small VAR for each element of $z_1$ (each with its own aggregate variables but without imposing block exogeneity jointly) is misleading — the identification of aggregate shocks varies arbitrarily across the separate systems.
Scalability: The restricted system is feasible for large $n_1$ (e.g., 80 industries) as long as $n_2$ is small, because estimation of the $z_2$ sub-system requires only $T \gg n_2^2 p$ observations.

Concepts Introduced or Extended

Entities Mentioned

William D. Lastrapes

Quotes

"I show that least square methods are efficient and that identification of the structural dynamic (impulse) responses of the model requires restrictions only on the subset of common variables."

"Estimating separate VARs for each element in the cross-section … can be misleading since the estimation and identification of aggregate shocks under this strategy will vary across the separate VARs."

My Take

A compact, self-contained technical note that solves a real dimensionality problem cleanly. The two algebraic results — diagonal $H$ and the three-step identification chain — are tight and the derivations are short. The practical payoff is large: 80-industry systems become estimable from standard post-war samples. The paper is entirely classical (no Bayesian methods), so it sits orthogonally to most of the wiki's content but adds an important frequentist benchmark for large-scale VAR identification. The critique of the Carlino-Defina approach (state-by-state separate VARs) is well-taken. No empirical application is included — purely methodological.