Principled Identification of Structural Dynamic Models

We take a new perspective on identification in structural dynamic models: rather than imposing restrictions, we optimize an objective. This provides new theoretical insights into traditional Cholesky identification. A correlation-maximizing objective yields an Order- and Scale-Invariant Identification Scheme (OASIS) that selects the orthogonal rotation that best aligns structural shocks with their reduced-form innovations. We revisit a large number of SVAR studies and find, across 22 published SVARs, that the correlations between structural and reduced-form shocks are generally high.

💡 Research Summary

This paper re‑examines the identification problem that lies at the heart of structural dynamic models such as structural vector autoregressions (SVARs) and local projections (LPs). Traditionally, researchers impose a set of identifying restrictions—Cholesky triangularity, long‑run restrictions, sign restrictions, etc.—to pin down a unique orthogonal rotation of the reduced‑form innovations. Those restrictions are often debated on the grounds of economic plausibility, and the resulting identification can be sensitive to arbitrary choices such as variable ordering.

The authors propose a fundamentally different perspective: treat identification as an explicit optimization problem rather than a set of ad‑hoc constraints. Let ε∈ℝⁿ denote the reduced‑form shocks with covariance Σ, and let u=A′ε be the structural shocks, which are required to be mutually uncorrelated (Var(u)=Iₙ) so that A′ΣA=Iₙ. The set of admissible matrices A is therefore the set of all orthogonal rotations that whiten Σ. Within this set, the paper defines the average correlation objective

 ρ(A)= (1/n)∑_{i=1}^n corr(u_i,ε_i)

and seeks the A that maximizes it. Because the objective directly measures how closely each structural shock aligns with its “natural” reduced‑form counterpart, the resulting identification scheme is intuitively appealing. The authors call the solution OASIS (Order‑and‑Scale‑Invariant Identification Scheme) because it does not depend on the ordering of variables nor on their units of measurement.

Theoretical contributions are organized around three main results. First, Theorem 1 (Weighted OASIS) introduces a positive weight vector w and shows that the weighted objective ρ_w(A)=∑ w_i corr(u_i,ε_i) is uniquely maximized by

 A* = Λ^{-1}σ Λ_w (Λ_w C Λ_w)^{-1/2},

where C is the correlation matrix of ε, σ is the diagonal matrix of standard deviations, Λ_w=diag(w_i), and (·)^{-1/2} denotes the symmetric inverse square root. The maximal value equals the sum of the square‑roots of the eigenvalues κ_i of Λ_w C Λ_w. When all weights are equal (w_i=1/n), Corollary 1 yields the un‑weighted OASIS solution

 A* = Λ^{-1}σ C^{-1/2},

with maximal average correlation (1/n)∑ λ_i^{1/2}, λ_i being the eigenvalues of C. This formulation makes the identification problem completely invariant to permutations and rescalings of the variables.

Second, Proposition 1 shows that the familiar Cholesky decomposition can be interpreted as a sequential version of the same correlation‑maximization problem. Each column of the Cholesky factor solves a constrained maximization where orthogonality constraints are added one by one according to a chosen ordering. Consequently, any Cholesky factor maximizes the same objective under a stricter set of constraints, explaining why different orderings often produce similar overall average correlations even though the impulse‑response functions differ across dimensions.

Third, Theorem 2 extends the framework to proxy VARs. When external instruments z∈ℝ^r are available, the authors define a similar objective g(a)=∑ w_j corr(u_j,z_j) for a∈A_r (the set of n×r matrices that produce r orthogonal structural shocks). By constructing the matrix Ξ = C_{εε}^{-1/2} C_{εz} Λ_w and taking its singular value decomposition Ξ=U Λ_ξ V′, they prove that the maximal value of g is the sum of the singular values ξ_i, and the optimal a* is given by

 a* = Λ^{-1}σ C_{εε}^{-1/2} U V′.

The ξ_i (lying between 0 and 1) are canonical correlations that quantify how informative each external instrument is for its associated structural shock; values near zero signal weak identification, while values near one indicate a strong instrument.

Empirically, the authors revisit 22 published SVAR studies spanning monetary policy, fiscal shocks, and financial stability. For each paper they compute the average correlation between reduced‑form innovations and (i) Cholesky‑identified structural shocks, and (ii) OASIS‑identified shocks. The Cholesky averages range from 78.5 % to 99.8 %, with most above 90 %. OASIS consistently yields higher averages, often exceeding 95 % and approaching perfect correlation. Moreover, the authors observe that the reduced‑form correlation matrix C is frequently close to the identity, which theoretically predicts high average correlations for both methods. The empirical findings confirm the theoretical claim that OASIS dominates Cholesky in terms of the correlation objective while preserving the same level of identification (i.e., orthogonal shocks).

The paper concludes by highlighting the practical advantages of OASIS: (1) it eliminates the need to choose an arbitrary ordering, (2) it is scale‑invariant, (3) it offers a transparent economic criterion (maximizing shock‑innovation alignment), and (4) it can be weighted to emphasize shocks of particular policy relevance. The authors suggest several avenues for future work, including extensions to nonlinear dynamic models, incorporation of stochastic volatility, Bayesian priors that reflect the correlation objective, and applications to large‑dimensional panel SVARs.

Overall, the study proposes a principled, optimization‑based identification framework that both unifies and extends existing restriction‑based approaches, and it demonstrates through theory and extensive re‑analysis that the OASIS scheme provides a robust, order‑ and scale‑free alternative for structural dynamic modeling.