Statistical Identification

Statistical identification recovers the structural impact matrix $B_0$ from higher-moment information –- time-varying variances (heteroskedasticity) or non-Gaussian shock distributions –- without imposing recursive orderings, sign restrictions, or zero restrictions. The classification follows Lewis (2025), the definitive survey of higher-moment identification in macroeconometrics.

• 13 identification methods: 5 ICA + 4 ML (non-Gaussianity) and 4 heteroskedasticity estimators
• 5 diagnostic tests: normality suite, shock Gaussianity, LR test, independence, identification strength
• Full pipeline integration: all methods produce a rotation $Q$ consumed by irf(), fevd(), historical_decomposition()
• Sub-pages: Non-Gaussian Methods | Heteroskedasticity | Testing

using MacroEconometricModels, Random
Random.seed!(42)

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Quick Start

Recipe 1: FastICA identification

fred = load_example(:fred_md)
Y = to_matrix(apply_tcode(fred[:, ["INDPRO", "CPIAUCSL", "FEDFUNDS"]]))
Y = Y[all.(isfinite, eachrow(Y)), :]
model = estimate_var(Y, 2)
ica = identify_fastica(model)
report(ica)

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Recipe 2: Student-t ML identification

ml = identify_student_t(model)
report(ml)

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Recipe 3: Markov-switching heteroskedasticity

ms = identify_markov_switching(model; n_regimes=2)
report(ms)

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Recipe 4: Normality test suite

suite = normality_test_suite(model)
report(suite)

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Recipe 5: Shock Gaussianity test

ica = identify_fastica(model)
result = test_shock_gaussianity(ica)
report(result)

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Recipe 6: IRF via statistical identification

irfs = irf(model, 20; method=:fastica)
report(irfs)

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plot_result(irfs)

The SVAR Setting

The structural VAR decomposes reduced-form residuals into orthogonal structural shocks:

\[u_t = B_0 \varepsilon_t, \quad \Sigma = B_0 B_0'\]

where

• $u_t$ is the $n \times 1$ vector of reduced-form residuals
• $\varepsilon_t$ is the $n \times 1$ vector of structural shocks (unit variance, mutually independent)
• $B_0$ is the $n \times n$ structural impact matrix

The covariance $\Sigma = B_0 B_0'$ provides $n(n+1)/2$ equations for $n^2$ unknowns, leaving $n(n-1)/2$ free parameters. Statistical identification resolves this gap without economic restrictions:

• Heteroskedasticity: regime-dependent covariances $\Sigma_k = B_0 \Lambda_k B_0'$ supply additional equations. See Heteroskedasticity.
• Non-Gaussianity: independence beyond uncorrelatedness (coskewness, cokurtosis) identifies $B_0$ from a single sample. See Non-Gaussian Methods.

Method Comparison

All 13 methods return a rotation matrix $Q$ and structural impact matrix $B_0 = L Q$ where $L = \text{chol}(\Sigma)$.

IRF Pipeline Integration

All 13 methods integrate with irf(), fevd(), and historical_decomposition() via compute_Q(). Pass the method name as a symbol:

irfs_ica = irf(model, 20; method=:fastica)
irfs_ml  = irf(model, 20; method=:student_t)
irfs_ms  = irf(model, 20; method=:markov_switching)
decomp   = fevd(model, 20; method=:jade)
nothing # hide

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Supported symbols: :fastica, :jade, :sobi, :dcov, :hsic, :student_t, :mixture_normal, :pml, :skew_normal, :markov_switching, :garch.

The Labeling Problem

Statistical identification recovers $B_0$ only up to column permutation and sign. The data alone cannot determine which column corresponds to which economic shock –- economic information is still required to label shocks. The package normalizes $B_0$ to have a positive diagonal (sign convention). The test_identification_strength bootstrap measures column-assignment stability across replications via Procrustes distance. See Lewis (2025, Section 6.4) for a thorough discussion.

Sub-Page Guide

• Non-Gaussian Methods –- ICA (FastICA, JADE, SOBI, distance covariance, HSIC), ML (Student-t, mixture normal, PML, skew-normal), Darmois-Skitovich theorem, contrast functions, unified dispatcher
• Heteroskedasticity –- Eigendecomposition identification, Markov-switching, GARCH, smooth transition, external volatility instruments, result field tables
• Testing –- Normality suite (7 tests), shock Gaussianity, LR test, independence, identification strength, overidentification, weak identification diagnostics

Common Pitfalls

1. Non-Gaussianity is a prerequisite, not an assumption. ICA and ML methods require at most one Gaussian shock. Run normality_test_suite(model) first. If residuals are Gaussian, the problem is unidentified.

2. Column permutation differs across bootstrap replications. The Procrustes distance in test_identification_strength accounts for this, but naive bootstrap CIs on individual $B_0$ entries do not. Verify identification stability before interpreting shock-by-shock results.

3. Heteroskedasticity requires distinct eigenvalues. If two shocks have identical variance ratios across regimes, the eigendecomposition of $\Sigma_1^{-1}\Sigma_2$ does not uniquely identify the corresponding columns. Check that Lambda values are well-separated.

4. Weak identification is common in practice. When variance changes are small or deviations from Gaussianity are mild, Wald tests have poor size properties (Lewis 2022). Run test_identification_strength as a diagnostic.

5. Smooth transition needs an external variable. Unlike Markov-switching and GARCH, identify_smooth_transition requires a transition variable s of the same length as the residuals (e.g., a lagged endogenous variable).

References

Survey

• Lewis, D. J. (2025). Identification Based on Higher Moments in Macroeconometrics. Annual Review of Economics, 17, 665–693. DOI: 10.1146/annurev-economics-070124-051419

Heteroskedasticity

• Rigobon, R. (2003). Identification through Heteroskedasticity. Review of Economics and Statistics, 85(4), 777–792. DOI: 10.1162/003465303772815727
• Sentana, E. & Fiorentini, G. (2001). Identification, Estimation and Testing of Conditionally Heteroskedastic Factor Models. Journal of Econometrics, 102(2), 143–164. DOI: 10.1016/S0304-4076(01)00051-3
• Lanne, M. & Lutkepohl, H. (2008). Identifying Monetary Policy Shocks via Changes in Volatility. Journal of Money, Credit and Banking, 40(6), 1131–1149. DOI: 10.1111/j.1538-4616.2008.00151.x
• Normandin, M. & Phaneuf, L. (2004). Monetary Policy Shocks: Testing Identification Conditions under Time-Varying Conditional Volatility. Journal of Monetary Economics, 51(6), 1217–1243. DOI: 10.1016/j.jmoneco.2003.11.002
• Lutkepohl, H. & Netsunajev, A. (2017). Structural VARs with Smooth Transition in Variances. Journal of Economic Dynamics and Control, 84, 43–57. DOI: 10.1016/j.jedc.2017.09.001
• Lewis, D. J. (2021). Identifying Shocks via Time-Varying Volatility. Review of Economic Studies, 88(6), 3086–3124. DOI: 10.1093/restud/rdab009

Non-Gaussianity –- ICA

• Hyvarinen, A. (1999). Fast and Robust Fixed-Point Algorithms for Independent Component Analysis. IEEE Trans. Neural Networks, 10(3), 626–634. DOI: 10.1109/72.761722
• Cardoso, J.-F. & Souloumiac, A. (1993). Blind Beamforming for Non-Gaussian Signals. IEE Proceedings-F, 140(6), 362–370. DOI: 10.1049/ip-f-2.1993.0054
• Belouchrani, A. et al. (1997). A Blind Source Separation Technique Using Second-Order Statistics. IEEE Trans. Signal Processing, 45(2), 434–444. DOI: 10.1109/78.554307
• Comon, P. (1994). Independent Component Analysis, A New Concept? Signal Processing, 36(3), 287–314. DOI: 10.1016/0165-1684(94)90029-9
• Matteson, D. S. & Tsay, R. S. (2017). Independent Component Analysis via Distance Covariance. JASA, 112(518), 623–637. DOI: 10.1080/01621459.2016.1150851
• Gretton, A. et al. (2005). Measuring Statistical Dependence with Hilbert-Schmidt Norms. In Algorithmic Learning Theory, 63–77. Springer. DOI: 10.1007/11564089_7
• Szekely, G. J. et al. (2007). Measuring and Testing Dependence by Correlation of Distances. Annals of Statistics, 35(6), 2769–2794. DOI: 10.1214/009053607000000505

Non-Gaussianity –- ML

• Lanne, M., Meitz, M. & Saikkonen, P. (2017). Identification and Estimation of Non-Gaussian SVARs. Journal of Econometrics, 196(2), 288–304. DOI: 10.1016/j.jeconom.2016.06.002
• Gourieroux, C., Monfort, A. & Renne, J.-P. (2017). Statistical Inference for ICA: Application to Structural VAR Models. Journal of Econometrics, 196(1), 111–126. DOI: 10.1016/j.jeconom.2016.09.007
• Lanne, M. & Lutkepohl, H. (2010). SVARs with Nonnormal Residuals. Journal of Business & Economic Statistics, 28(1), 159–168. DOI: 10.1198/jbes.2009.06003
• Herwartz, H. (2018). Hodges-Lehmann Detection of Structural Shocks. Oxford Bulletin of Economics and Statistics, 80(4), 736–754. DOI: 10.1111/obes.12234
• Azzalini, A. (1985). A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics, 12(2), 171–178. https://www.jstor.org/stable/4615982
• Keweloh, S. A. (2021). A GMM Estimator for SVARs Based on Higher Moments. Journal of Business & Economic Statistics, 39(3), 772–882. DOI: 10.1080/07350015.2020.1730858
• Lanne, M. & Luoto, J. (2021). GMM Estimation of Non-Gaussian SVAR. Journal of Business & Economic Statistics, 39(1), 69–81. DOI: 10.1080/07350015.2019.1629940

Diagnostics

• Lewis, D. J. (2022). Robust Inference in Models Identified via Heteroskedasticity. Review of Economics and Statistics, 104(3), 510–524. DOI: 10.1162/resta00977
• Jarque, C. M. & Bera, A. K. (1980). Efficient Tests for Normality, Homoscedasticity and Serial Independence. Economics Letters, 6(3), 255–259. DOI: 10.1016/0165-1765(80)90024-5
• Mardia, K. V. (1970). Measures of Multivariate Skewness and Kurtosis with Applications. Biometrika, 57(3), 519–530. DOI: 10.1093/biomet/57.3.519
• Doornik, J. A. & Hansen, H. (2008). An Omnibus Test for Univariate and Multivariate Normality. Oxford Bulletin of Economics and Statistics, 70, 927–939. DOI: 10.1111/j.1468-0084.2008.00537.x
• Henze, N. & Zirkler, B. (1990). A Class of Invariant Consistent Tests for Multivariate Normality. Communications in Statistics, 19(10), 3595–3617. DOI: 10.1080/03610929008830400
• Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer. ISBN 978-3-540-40172-8.