SVAR Identification through Heteroskedasticity

Python · R · Part 10 of the VAR arc — identification without restrictions

Overview

Part 10 of the VAR arc: letting the volatility do the identifying. Every structural VAR so far bought identification with an assumption — a recursive timing zero (Part 2, Part 8a), a sign restriction, an external instrument, a long-run neutrality, an elasticity bound or prior (Part 8b). Each is an economic belief imposed from outside. Rigobon (2003) showed a route that needs none of them: if the structural shock variances change over time while the contemporaneous structure stays put, the structure is identified by the change in volatility alone — a purely statistical identification. This closes the loop with Part 4: the stochastic volatility modelled there as a nuisance to be captured turns out to be exactly the variation that identifies the shocks. Built from scratch on the same monetary VAR, and used to test — rather than assume — the recursive ordering.

\Sigma_1 = B_0\,D_1\,B_0', \quad \Sigma_2 = B_0\,D_2\,B_0' \;\Longrightarrow\; B_0 = \Sigma_1^{1/2}W, \ \ W = \text{eigvecs}\big(\Sigma_1^{-1/2}\Sigma_2\,\Sigma_1^{-1/2}\big)

Two volatility regimes pin the structure — no restriction needed

A single reduced-form covariance $\Sigma = B_0 D B_0'$ gives $m(m{+}1)/2$ equations for the $m^2$ elements of the impact matrix $B_0$ — short by exactly $m(m{-}1)/2$ , the number of restrictions every other scheme has to add. Rigobon's observation: if the sample splits into two volatility regimes with the same $B_0$ but different structural variances, $\Sigma_1=B_0 D_1 B_0'$ and $\Sigma_2=B_0 D_2 B_0'$ , two covariances double the equations and $B_0$ is identified with no restriction at all — provided the variance ratios on the diagonal of $D_2 D_1^{-1}$ are distinct (each shock's variance must change by a different proportion). The solution is a simultaneous diagonalisation: whiten by $\Sigma_1$ and take the eigenvectors of $\Sigma_1^{-1/2}\Sigma_2\Sigma_1^{-1/2}$ , so $B_0=\Sigma_1^{1/2}W$ diagonalises both regimes at once (a generalised eigenproblem). Two regimes are the minimal case; the continuous version uses the full Part 4 stochastic-volatility path as the identifying heteroskedasticity (Lewis 2021). Here we take the cleanest cut — the Great Moderation — on the Part 4 monetary system (GDP growth, inflation, funds rate), with a diffuse inverse-Wishart posterior over the two regime covariances for bands.

Results

The Great Moderation identifies the model. The three structural-shock variances fell to 0.11, 0.26 and 0.44 of their pre-1984 levels — three different proportions, so Rigobon's distinctness condition holds and $B_0$ is identified with no economic restriction. (Had all variances fallen by the same factor, the volatility change would have been uninformative.) The data reject the recursive ordering. The heteroskedasticity-identified impact matrix is not lower-triangular: the contemporaneous effect of the inflation shock on GDP growth is 1.19, with a 68% band [0.38, 1.97] that excludes zero — yet a Cholesky scheme with output ordered first forces that element to be exactly 0. Where the recursive and heteroskedasticity-identified monetary responses diverge, it is the recursive assumption, not the data, doing the work. This is a fourth kind of identification — alongside recursive zeros, sign restrictions, and instruments — that imposes nothing and so can test what the others assume: the honest counterpart to the oil-market debate of Part 8b, letting the second moments pin the structure rather than arguing over which prior to hold. Volatility is signal, not just nuisance: the same heteroskedasticity Part 4 modelled for calibrated uncertainty is enough to identify the shocks, uniting the identification thread (Part 2) and the volatility thread (Part 4) in one estimator. Cross-checked in R (the same generalised eigenproblem).

Notebooks

Python Identification through heteroskedasticity from scratch — Rigobon simultaneous diagonalisation across the Great-Moderation regimes, and a test of the recursive ordering (Python) Open → R R cross-check — base-R generalised eigenproblem recovering the impact matrix (R) Open →

Downloads

hetid.py The from-scratch heteroskedasticity-identification module — regime covariances and the simultaneous-diagonalisation generalised eigenproblem (new self-contained module) bvar_sv_data.csv The Part 4 monetary system — US quarterly GDP growth, GDP-deflator inflation, and the Fed funds rate, 1959–2019 (FRED-QD)

References

Rigobon, R. (2003). Identification through heteroskedasticity. Review of Economics and Statistics 85(4), 777–792. — the restriction-free identification reproduced here
Sentana, E. & Fiorentini, G. (2001). Identification, estimation and testing of conditionally heteroskedastic factor models. Journal of Econometrics 102(2), 143–164. — identification from time-varying variances
Lanne, M., Lütkepohl, H. & Maciejowska, K. (2010). Structural vector autoregressions with Markov switching. Journal of Economic Dynamics and Control 34(2), 121–131. — regime-based heteroskedasticity identification of SVARs
Lütkepohl, H. & Netšunajev, A. (2017). Structural vector autoregressions with heteroskedasticity: a review of different volatility models. Econometrics and Statistics 1, 2–18. — survey of the identifying volatility models
Lewis, D. J. (2021). Identifying shocks via time-varying volatility. Review of Economic Studies 88(6), 3086–3124. — the continuous stochastic-volatility version of the identification
Rigobon, R. & Sack, B. (2004). The impact of monetary policy on asset prices. Journal of Monetary Economics 51(8), 1553–1575. — an applied use of the heteroskedasticity identification