Identification Testing

Before relying on statistical identification, the practitioner must verify that the identifying conditions hold in the data. This page covers the diagnostic tests for non-Gaussian and heteroskedasticity-based SVAR identification: multivariate normality tests, shock gaussianity and independence tests, likelihood ratio tests, and bootstrap identification strength assessments.

using MacroEconometricModels, Random
Random.seed!(42)

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Quick Start

Recipe 1: Normality test suite

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)

suite = normality_test_suite(model)
report(suite)

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Recipe 2: Multivariate Jarque-Bera

jb = jarque_bera_test(model)
report(jb)

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Recipe 3: Shock gaussianity test

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

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Recipe 4: Shock independence test

ica = identify_fastica(model)
indep = test_shock_independence(ica; max_lag=10)
report(indep)

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Recipe 5: Gaussian vs non-Gaussian LR test

lr = test_gaussian_vs_nongaussian(model; distribution=:student_t)
report(lr)

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Multivariate Normality Tests

Testing for multivariate normality of VAR residuals is the essential first step. If residuals are Gaussian, non-Gaussian identification is not possible –- the Darmois-Skitovich theorem requires at most one Gaussian component for uniqueness. Rejecting normality supports the use of ICA or ML methods.

Multivariate Jarque-Bera

The multivariate Jarque-Bera test (Lutkepohl 2005, Section 4.5) combines skewness and kurtosis measures:

\[JB = T \cdot \frac{b_{1,k}}{6} + T \cdot \frac{(b_{2,k} - k(k+2))^2}{24k}\]

where:

• $b_{1,k} = T^{-2} \sum_{i,j} (u_i' \Sigma^{-1} u_j)^3$ is multivariate skewness
• $b_{2,k} = T^{-1} \sum_i (u_i' \Sigma^{-1} u_i)^2$ is multivariate kurtosis
• Under $H_0$: $JB \sim \chi^2(2k)$

The :component method applies univariate JB tests to each standardized residual. The component-wise p-values in the result's component_pvalues field pinpoint which variables drive non-Gaussianity.

# Joint and component-wise tests
jb = jarque_bera_test(model)
report(jb)

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jb_comp = jarque_bera_test(model; method=:component)
report(jb_comp)

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Mardia's Tests

Mardia (1970) proposed separate tests for multivariate skewness and kurtosis:

\[b_{1,k} = \frac{1}{T^2} \sum_{i,j} (u_i' \Sigma^{-1} u_j)^3, \quad b_{2,k} = \frac{1}{T} \sum_i (u_i' \Sigma^{-1} u_i)^2\]

Under $H_0$: $T \cdot b_{1,k}/6 \sim \chi^2(k(k+1)(k+2)/6)$ and $(b_{2,k} - k(k+2)) / \sqrt{8k(k+2)/T} \sim N(0,1)$. Three modes are available: mardia_test(model; type=:skewness), :kurtosis, or :both. Rejecting kurtosis but not skewness suggests heavy-tailed shocks (Student-t appropriate). Rejecting skewness suggests asymmetric shocks (skew-normal may be preferred).

Doornik-Hansen and Henze-Zirkler

The Doornik-Hansen (2008) omnibus test transforms each component's skewness and kurtosis via the Bowman-Shenton transformation: $DH = \sum_{j=1}^k (z_{1j}^2 + z_{2j}^2) \sim \chi^2(2k)$. The Henze-Zirkler (1990) test is based on the empirical characteristic function and is consistent against all alternatives. Call via doornik_hansen_test(model) and henze_zirkler_test(model).

Normality Test Suite

The normality_test_suite runs all 7 tests simultaneously: multivariate JB, component-wise JB, Mardia skewness, Mardia kurtosis, Mardia combined, Doornik-Hansen, and Henze-Zirkler. Consistent rejection across multiple tests provides strong evidence against Gaussianity. Individual results are accessible via suite.results[i].

Return value (NormalityTestResult):

Shock Gaussianity Test

After recovering structural shocks via ICA or ML, this test verifies that shocks are non-Gaussian. A univariate Jarque-Bera test is applied to each shock:

\[JB_j = T \left( \frac{\hat{s}_j^2}{6} + \frac{\hat{\kappa}_j^2}{24} \right) \sim \chi^2(2)\]

where $\hat{s}_j$ is sample skewness and $\hat{\kappa}_j$ is excess kurtosis. The Darmois-Skitovich theorem requires at most one Gaussian shock for identification. The test counts how many shocks fail to reject at the 5% level.

ica = identify_fastica(model)
result = test_shock_gaussianity(ica)
report(result)
println("Gaussian shocks: ", result.details[:n_gaussian], " (need <= 1)")

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The test also accepts NonGaussianMLResult: test_shock_gaussianity(ml_result).

Gaussian vs Non-Gaussian LR Test

The likelihood ratio test compares Gaussian and non-Gaussian shock specifications:

\[LR = 2(\ell_1 - \ell_0) \sim \chi^2(p)\]

where:

• $\ell_0$ is the Gaussian log-likelihood (Cholesky)
• $\ell_1$ is the non-Gaussian log-likelihood (ML with distribution-specific parameters)
• $p = n \times p_{\text{dist}}$ is the number of extra distribution parameters

lr = test_gaussian_vs_nongaussian(model; distribution=:student_t)
report(lr)

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Rejecting $H_0$ supports non-Gaussian identification. Available distributions: :student_t, :mixture_normal, :pml, :skew_normal.

Shock Independence Test

Independence of recovered shocks is necessary for valid identification. This test combines two measures via Fisher's method:

1. Cross-correlation portmanteau: $Q = T \sum_{i < j} \sum_{\ell=0}^{L} r_{ij,\ell}^2 \sim \chi^2\bigl(\binom{n}{2}(L+1)\bigr)$
2. Distance covariance (Szekely et al. 2007): permutation-based p-value (199 replicates)

Fisher's method: $\chi^2_F = -2 \sum_k \ln p_k \sim \chi^2(2K)$. Failing to reject ($p \geq 0.05$) indicates independent shocks.

ica = identify_fastica(model)
result = test_shock_independence(ica; max_lag=10)
report(result)

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Identification Strength

The bootstrap identification strength test assesses robustness of the estimated $B_0$. The procedure resamples residuals $B$ times, re-estimates $B_0$ via the specified ICA method, and computes the Procrustes distance between each bootstrap $B_0$ and the original. Small distances indicate strong identification.

result = test_identification_strength(model; method=:fastica, n_bootstrap=499)
report(result)
println("Normalized distance: ", round(result.details[:normalized_distance], digits=4))

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Overidentification Test

When additional restrictions beyond non-Gaussianity are imposed on $B_0$, this bootstrap test checks consistency. It compares the discrepancy between $B_0 B_0'$ and $\Sigma$ to a bootstrap distribution under the null.

ica = identify_fastica(model)
result = test_overidentification(model, ica; n_bootstrap=499)
report(result)

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Return value (IdentifiabilityTestResult –- shared by all tests in this section):

Complete Example

# Step 1: Normality diagnostics
suite = normality_test_suite(model)
report(suite)

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# Step 2: LR test — Gaussian vs Student-t
lr = test_gaussian_vs_nongaussian(model; distribution=:student_t)
report(lr)

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# Step 3: ICA identification + shock diagnostics
ica = identify_fastica(model)
gauss = test_shock_gaussianity(ica)
report(gauss)
println("Gaussian shocks: ", gauss.details[:n_gaussian])

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indep = test_shock_independence(ica; max_lag=5)
report(indep)
println("Shocks independent: ", indep.identified)

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# Step 4: Identification strength
strength = test_identification_strength(model; method=:fastica, n_bootstrap=199)
report(strength)

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# Step 5: Compute structural IRFs if identification holds
if gauss.identified
    irfs = irf(model, 20; method=:fastica)
    report(irfs)
end

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Common Pitfalls

1. Confusing normality rejection with identification: Rejecting multivariate normality is necessary but not sufficient. The shock gaussianity test must also confirm at most one Gaussian shock. Normality tests operate on reduced-form residuals; shock tests operate on structural shocks.

2. Independence test direction: The shock independence test uses fail-to-reject logic. identified == true means independence is not rejected. This is the opposite convention from normality tests.

3. Bootstrap sample size: Use n_bootstrap=199 for preliminary diagnostics and n_bootstrap=999 for publication results. The identification strength test is computationally expensive.

4. Multiple testing: Running 7 normality tests inflates the family-wise error rate. Look for consistent rejection across multiple tests before concluding non-Gaussianity.

References

• Jarque, Carlos M., and Anil K. Bera. 1980. "Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals." Economics Letters 6 (3): 255–259. DOI

• Mardia, Kanti V. 1970. "Measures of Multivariate Skewness and Kurtosis with Applications." Biometrika 57 (3): 519–530. DOI

• Doornik, Jurgen A., and Henrik Hansen. 2008. "An Omnibus Test for Univariate and Multivariate Normality." Oxford Bulletin of Economics and Statistics 70: 927–939. DOI

• Henze, Norbert, and Bernhard Zirkler. 1990. "A Class of Invariant Consistent Tests for Multivariate Normality." Communications in Statistics - Theory and Methods 19 (10): 3595–3617. DOI

• Lewis, Daniel J. 2022. "Robust Inference in Models Identified via Heteroskedasticity." Review of Economics and Statistics 104 (3): 510–524. DOI

• Lutkepohl, Helmut. 2005. New Introduction to Multiple Time Series Analysis. Berlin: Springer. ISBN 978-3-540-40172-8.