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Identification and Inference in Proxy-SVARs with non-Gaussian Shocks
\setcounter{page}{1}
\title[\tiny Proxy-SVARs with non-Gaussian shocks]{Identification and Inference in Proxy-SVARs with non-Gaussian Shocks}
\author[]{Paritosh Shankarrao Junare}
\thanks{Correspondence: Department of Economics, University of Bologna, Italy. Email: [email removed]. I am immensely grateful to Luca Fanelli, Giuseppe Cavaliere and Giovanni Angelini for their guidance and support throughout this work. I am also thankful to Markku Lanne, Jani Luoto, Mika Meitz, Savi Virolainen and Filippo Ferroni for their insightful comments. Discussions with Raffaela Giacomini and Lutz Kilian at Nordic Econometric Meeting 2026; Silvia Sarpietro and Kirill Ponomarev at CEPR Job Market Bootcamp 2026, Tuscany; Giovanni Caggiano as a discussant at SIdE WEEE 2026, and other participants at these conferences are gratefully acknowledged.}
\dedicatory{\today}
\begin{abstract}
Two frequent approaches for identifying structural VARs are external instruments, which carry economic content but are often weak, and non-Gaussianity of the shocks which provides statistical identification but carries no economic meaning. We combine the two strategies in a single generalized method of moments framework that stacks proxy exclusion restrictions with higher-order moment conditions of the structural shocks. This hybrid approach point-identifies the target shocks while also identifying the non-target shocks up to sign and ordering. Under suitable rank conditions, the higher-order moments anchor the identification uniformly over the instrument strength. Consequently, under local-to-zero proxy relevance, estimators of dynamic causal effects remain consistent, and standard asymptotic inference remains valid. Moreover, the Anderson-Rubin confidence sets are substantially narrower than their instrument-only counterparts. The hybrid estimator is also more efficient than either source of identification used in isolation: at any fixed proxy relevance, even a weak instrument increases efficiency of the estimator through its covariance with the non-Gaussian moment block. Under local deviations from proxy exogeneity, we provide asymptotic bias bounds and show that stronger non-Gaussianity of the shocks compresses the bias. Finally, the over-identified structure yields two mutually orthogonal specification tests, for proxy exogeneity and validity of higher-order moment conditions. We derive their limiting distributions and provide a bootstrap procedure for finite-sample critical values. Monte Carlo simulations and two applications with identification of oil news-shock and a Euro-area MP shock demonstrate the potential of our framework.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:introduction}
The identification and estimation of dynamic causal effects is crucial to understand the transmission mechanisms of structural shocks in empirical macroeconomics. Identification through structural vector autoregressions (SVARs) generally requires restrictions on the structural impact matrix, denoted by $B$. It captures the instantaneous effects of the structural shocks on variables, and hence it is essential for the estimation of impulse response functions (IRFs). Two identification strategies have become increasingly prevalent. The first strategy employs economic theory-based external instruments or proxies: variables which are \enquote{external} to the VAR estimation, and are correlated with the target shocks (relevance condition) and orthogonal to other structural shocks (exogeneity condition), see \citet{mertens_2013, stock_identification_2018}. The second strategy exploits the non-Gaussianity of the structural shocks to obtain statistical identification: with at most one Gaussian shock, certain higher-order moment conditions recover $B$ up to sign and column permutations without additional restrictions, see \citet{comon_independent_1994, lanne_gmm_2021, keweloh_generalized_2021}.
On one hand, in proxy-SVARs, an external instrument is only as good as its correlation with the shock it is
meant to isolate. Under local-to-zero relevance, see \citet{staiger_instrumental_1997}, the estimates are inconsistent,
and standard asymptotic inference is unreliable as detailed in \citet{montiel_olea_inference_2021}. \citet{ramey_chapter_2016} notes that weak instruments are pervasive in the literature, and \citet{lewis_weak_2026} show that conventional
critical-value thresholds are ill-suited for assessing weak-instrument bias in
IRFs. Weak instruments are therefore empirically typical, and routinely
under-diagnosed; Appendix~\ref{appendix:evidencelit}
(Table~\ref{tab:weak_instruments}) lists prominent applications in which this
sensitivity is visible.
On the other hand, a large body of evidence in macro-financial literature finds that the structural shocks are pronouncedly non-Gaussian. The recovered structural shocks reject Gaussianity in favor of heavy-tailed distributions as shown in \citet{gourieroux_statistical_2017}, \citet{ludvigson_uncertainty_2021}; see
Appendix~\ref{appendix:evidencelit} (Table~\ref{tab:nongaussianity}). However, as noted earlier, this is a source of statistical identification where we still rely on labeling conventions to obtain meaningful economic interpretation of the shocks. Both of these identification strategies therefore face an obstacle that the other can overcome.
We propose a \enquote{hybrid} identification\footnote{We term the identification \emph{hybrid} because the two merged identification channels are of different kinds: one economic (external instruments), and one statistical (the non-Gaussianity of the shocks).} scheme where we combine the two channels within a single generalized method of moments (HGMM) framework that stacks proxy exclusion restrictions with higher-order moment conditions into a single
over-identified system. With $k$ instruments, under standard sign normalizations, we show global point identification of the target $k$ columns of $B$, and the non-target columns are identified up to signed permutations. The identification is {global} in the sense of \citet{komunjer_global_2012}: non-Gaussianity
eliminates the continuous rotational manifold of $B$, reducing identification
to a finite discrete set $\mathcal{B}$; and then the valid proxies select a
unique element from $\mathcal{B}$ delivering global point
identification of the target columns of $B$.
We provide three results that directly motivate the
HGMM estimator as a preferred alternative to proxy-only approach for applied
researchers.
\textit{First}, the HGMM estimator remains robust to weak proxies. Under
local-to-zero relevance, the proxy-only
estimator is inconsistent, whereas under certain rank conditions, the
HGMM estimator remains consistent with uniformly valid standard asymptotic inference as the non-Gaussian (NG) channel acts as an identification anchor. Since the Jacobian matrix of the NG moment block can remain full rank regardless of proxy strength, the joint information matrix remains bounded away from
zero even as the proxy relevance degenerates. This allows practitioners to rely on standard inference methods even in the presence of weak instruments where proxy-SVAR inference breaks down.
\textit{Second}, the joint estimator is asymptotically at least as efficient as the
proxy-only estimator: the difference in their asymptotic variances is positive semidefinite. The efficiency gain has a closed form. The joint information matrix decomposes additively into the
information obtained from the proxy exclusion conditions and the information the
non-Gaussian moments add {after} partialing out the identifying directions
already spanned by the valid proxies. The gain is strict and uniform whenever the non-Gaussian moments identify parameters along
directions the proxy block does not capture. One might expect the hybrid estimator to collapse onto the non-Gaussian GMM estimator if the proxy is very weak. It does not. The covariance between the two moment blocks continues to sharpen the weighting matrix, so at any fixed proxy relevance its variance is smaller than that of the non-Gaussian estimator alone. This generates an efficiency gain that a researcher who entirely discards the proxies would forego.
\textit{Third}, when the instrument is locally endogenous, the resulting first-order bias is inversely proportional to the minimum eigenvalue of the
NG information matrix. Stronger deviations from Gaussianity therefore compress the
upper bound on the bias directly, delivering a partial mis-specification
robustness that proxy-only approaches do not have.
On all three margins, therefore, augmenting a proxy-SVAR with non-Gaussian identification weakly dominates the standalone alternatives: precision improves under correct specification, consistency and standard inference survive a weak proxy, and the bias is bounded when the proxy is locally invalid.
Another contribution is of separate, valid specification
tests for each identification channel. The standard incremental
$J$-statistic is invalid in this setting: restricting the criterion
to the NG block alone admits $n!\cdot 2^n$ observationally equivalent
solutions, so the rotation parameters lack a unique probability limit
and the $\chi^2$ limit of the difference-in-$J$ statistic is not
restored without an anchor. For both identification channels, we construct
mutually orthogonal test statistics from the \emph{joint} GMM estimate. By orthogonalizing each block's sample moments
against the other, we obtain statistics $J_{\mathrm{prx}}$ and $J_{\mathrm{NG}}$
that are first-order insensitive to estimation error in the
complementary moment block \citep{newey_generalized_1985}. We also correct for the shared parameter estimation uncertainty in their asymptotic variances. This implies that, under the joint null of valid identification, the limiting distributions of the test statistics follow $\chi^2$ with degrees of freedom equal to
the number of moment conditions in the relevant block.
However, we note that asymptotic $\chi^2$ critical values can be unreliable as higher-order cumulant estimation in the
optimal weighting matrix may induce substantial finite-sample distortion.
Hence, we adopt the residual-based
moving block bootstrap of \citet{bruggemann_inference_2016} that jointly
resamples blocks of estimated residuals-proxy pairs,
preserving the cross-dependence structure that proxy identification
exploits. It also allows for conditional heteroskedasticity in the structural shocks. \citet{hall_bootstrap_1996} recentering ensures
that the bootstrap data-generating process satisfies the over-identifying
restrictions exactly at the estimated value, so that bootstrap
critical values adapt to the effective degrees of freedom.
Monte Carlo evidence from a three-variable VAR model validates the theoretical
results. The estimates are consistent and standard inference remains valid irrespective of the strength of the proxies. For identification robust inference, we also show the HGMM estimator provides narrower Anderson-Rubin (AR) confidence sets relative to the inference based only on proxy exclusion conditions. The joint criterion eliminates the failure of proxy identification without sacrificing
the validity of the identification-robust coverage. For the specification tests, under the null hypothesis of valid identification, all three test statistics
$J_{\mathrm{joint}}$, $J_{\mathrm{prx}}$, and $J_{\mathrm{NG}}$
exhibit close to nominal size. Under proxy endogeneity, the power ordering
$J_{\mathrm{prx}} \gg J_{\mathrm{NG}}$ holds
uniformly across sample sizes: the dominance of $J_{\mathrm{prx}}$
is consistent with proxy endogeneity as the primary source of
mis-specification, with $J_{\mathrm{NG}}$ inheriting relatively lower, albeit false, power
only through contamination via the shared estimate. Also, $J_{\mathrm{prx}}$ provides a valid formal test of
instrument orthogonality even when the instrument is weak.
We demonstrate the framework by revisiting identification of an oil-news shock and a Euro-area monetary policy shock. For the oil news shock of
\citet{kanzig_macroeconomic_2021}, estimated under the instrument correction of
\citet{kilian_how_2024}, the hybrid criterion delivers substantially narrower
Anderson--Rubin confidence sets than the proxy alone where the median reduction in set length is $71.6\%$, which are sharp enough to
characterize the shock as predominantly oil {demand} news shock. Notably for the real oil price, the confidence set for the on-impact estimate contracts by a factor of nearly ten once the higher-order moment conditions are added. For a Euro-area monetary policy shock identified from the
high-frequency surprises of \citet{altavilla_measuring_2019}, the gains are more modest with median reduction in set length of $26\%$. Together the illustrations show that the non-Gaussianity of the structural shocks sharpen the inference when the higher-order
moment conditions are informative enough to offset the cost of the additional
degrees of freedom.
The remainder of the paper is organized as follows.
Section~\ref{sec:method} presents the hybrid
GMM identification, asymptotic properties of the estimator and derives joint limiting distributions of the structural parameters; Section \ref{sec:weak_misp} proves consistency with standard asymptotic inference under weak proxies, and asymptotic bias bounds under local proxy endogeneity;
Section~\ref{sec:tests} develops the mutually orthogonal specification
tests and the required bootstrap procedure for finite-sample critical values;
Section~\ref{sec:results} reports Monte Carlo evidence;
Section~\ref{sec:empirical} presents the empirical illustrations and
Section~\ref{sec:conclusion} concludes. Appendix \ref{appendix:preliminaries} consolidates preliminary notations and conventions used in the paper. A brief review of related
literature follows next.
\subsection{Related Literature} \label{sec:related_lit}
Several recent papers combine multiple SVAR identification strategies. Within a
Bayesian framework, \citet{baumeister_2015} combine sign restrictions with prior
information; \citet{arias_inference_2021} develop an algorithm for proxy-SVARs with
traditional zero and sign restrictions; \citet{giacomini_narrative_2022} provide
robust inference for proxy-SVARs with narrative restrictions;
\citet{carriero_blended_2024} blend heteroskedasticity with sign/narrative
restrictions and instrumental variables; and \citet{angelini_invalid_2024} combine external instruments with changes in volatility regimes.
Most closely related studies are the over-identified GMM proxy-SVAR of
\citet{gregory_us_2024}, which uses second-moment uncorrelatedness of the
structural shocks and requires $n-1$ valid proxies, and its
generalization by \citet{bruns_avoiding_2025}, which relaxes the proxy count
by focusing the criterion on the proxy-targeted columns. Both, however,
draw their over-identifying content solely from \emph{second-moment}
uncorrelatedness, which vanishes when only a single shock is targeted
($\tfrac12 k(k-1)=0$ at $k=1$) and offers no identifying power for the
non-targeted columns. We instead draw identifying information from
higher-order moments under non-Gaussianity: even a single valid proxy point-identifies
its target column while the remaining columns are identified up to signed
permutation, and the non-Gaussian block supplies
over-identification and a test for proxy validity.
We further show robustness to weak proxy relevance where uncorrelatedness-based
approaches lose identifying content entirely.
Relative to the Bayesian non-Gaussian SVAR of \citet{keweloh_estimating_2025},
which combines statistical identification with potentially endogenous proxies,
ours is a frequentist approach with explicit weak instrument robustness,
efficiency results, formal specification tests, and an explicit bias bound under
local exogeneity violations. On testing, \citet{bruns_testing_2024} claim that the narrative arguments used to establish a proxy's validity typically imply a strong exogeneity condition in the form of conditional mean independence and not mere uncorrelatedness, whose measurable transformations generate over-identifying restrictions and hence a testable implication from a single proxy. Their moment conditions are, at the true parameter, the co-skewness and asymmetric co-kurtosis restrictions involving the target shock; our non-Gaussian block contains these as the subset indexed by the target column and adds the co-skewness conditions among the non-target shocks together with the symmetric co-kurtosis conditions.
When instruments are weak, \citet{montiel_olea_inference_2021} document severe
under-coverage of standard proxy-SVAR confidence sets and propose Anderson--Rubin
(AR) test inversion for robust inference, and \citet{angelini_identification_2024}
restore standard asymptotics for weakly instrumented target shocks by requiring
strong auxiliary instruments for the non-target shocks. The HGMM framework remains
robust under local-to-zero proxy relevance through a different mechanism: the
non-Gaussian Jacobian can retain full column rank as the proxy relevance
collapses, keeping the GMM information matrix bounded away
from zero in the limit. We consequently obtain substantially narrower AR confidence sets than the proxy-only criterion, comparable in length to the standard HGMM intervals. An applied researcher therefore needs neither additional instruments nor the heavy computation of AR sets to retain valid inference.
In statistical identification, \citet{comon_independent_1994} establishes that
non-singular linear mixing matrices are identified up to sign and column
permutation when the sources are mutually independent with at most one Gaussian.
\citet{keweloh_generalized_2021} provides a unified co-skewness and co-kurtosis
moment structure, and \citet{lanne_gmm_2021} construct a GMM estimator based on
fourth-order co-kurtosis conditions; \citet{lanne_identifying_2023} show that
$n(n-1)/2$ symmetric co-kurtosis conditions globally identify the impact matrix
when at least $n-1$ shocks have excess kurtosis of a common sign; the condition
we adopt for global identification.
In our moment set the co-skewness conditions deliver local identification and the
common-sign co-kurtosis conditions deliver global identification. We combine them
with proxy exclusion restrictions into a joint criterion and develop
Neyman-orthogonal specification tests. \citet{lewis_identification_2024} provides a
comprehensive review of statistical identification.
\section{Hybrid GMM methodology}\label{sec:method}
This section develops the hybrid GMM (HGMM) framework. We parameterize the impact matrix $B$ as a function of Givens rotation angles, $\phi$, as in \citet{matteson_dynamic_2011}, and estimate the
structural parameters by minimizing an efficiently weighted moment
criterion using residuals from a first-stage OLS estimation.
Combining the two channels delivers global point identification of the target
columns of $B$ (Proposition~\ref{prop:identification_unified}). The joint limiting distribution of the full parameter vector,
including autoregressive coefficients, is derived. We characterize the efficiency gains (Proposition \ref{prop:efficiency}) of the hybrid estimator relative to the
proxy-only estimator. The
asymptotic variance of $\hat{\phi}$ is corrected for the
generated-regressor distortion from first-stage estimation error
as noted in \citet{murphy_estimation_1985}, and the standard errors for all elements of $B$ and the structural
impulse-response functions follow with standard derivations from \citet{lutkepohl_2005}.
\subsection{SVAR setup} \label{subsec:svarparam}
We consider a structural VAR(p) model:
\begin{equation}
y_t = \nu + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p}
+ B\varepsilon_t, \qquad t = 1, \dots, T,
\end{equation}
where $y_t$ is an $n \times 1$ vector of endogenous variables, $\nu$ is a vector of intercepts and $A_j$ are $n \times n$ autoregressive
matrices. The reduced-form innovations, $u_t$ and structural shocks are linearly related through the
$n \times n$ non-singular impact matrix $B$:
\begin{equation}
u_t = B \varepsilon_t
\end{equation}
where $u_t$ is a sequence of serially uncorrelated reduced-form
disturbances with $\mathbb{E}[u_t] = 0$ and covariance matrix $\Sigma_u := \mathbb{E}(u_t u^{'}_t)$.
We further assume $y_t$ is stable with standard white noise, i.e.
\begin{assumption}
\label{as:stable_var}
$\det A(z) = \det (I_n - A_1z - \cdots - A_pz^p) \neq 0, \quad |z| \leq 1$
\end{assumption}
The structural shocks are assumed to be mean zero
and unit variance, so $\mathbb{E}[\varepsilon_t \varepsilon_t'] = I_n$ and
$\Sigma_u = B B'$.
To span the space of observationally equivalent impact matrices, let $H$
denote the lower-triangular Cholesky factor satisfying $HH' = \Sigma_u$.
We adapt the notation of \citet{matteson_dynamic_2011} and parameterize $B = HQ(\phi)$,
where $\phi \in \Phi \equiv (-\pi, \pi]^{p_\phi}$, $Q(\phi) \in SO(n)$ is an orthogonal rotation matrix governed by the
$p_\phi \times 1$ ($p_\phi = n(n-1)/2$) vector of rotation angles $\phi$. $SO(n)$ is a subgroup within the group of all $n \times n$ orthogonal matrices $O(n)$, with determinant equal to $+1$. There exists a unique mapping of $Q \in SO(n)$ into $\phi \in (-\pi, \pi]^{p_{\phi}}$, continuous if either all elements on the main diagonal of $Q$ are positive or all elements of $Q$ are non-zero. For a detailed exposition, see \citet{matteson_dynamic_2011,matteson_independent_2013}. Hence, the structural
shocks are recovered as
\begin{equation}
\varepsilon_t(\phi) = Q(\phi)' H^{-1} u_t = B^{-1} u_t.
\end{equation}
\subsection{Proxy identification}\label{subsec:proxyidentification}
Consider $m_t$ as a $k \times 1$ vector of external instruments. Identification through instruments requires that each proxy $m_{l,t}$ satisfies a relevance
condition, i.e. nonzero correlation with its target shock $\varepsilon_{j,t}$ and
an exogeneity condition to all remaining shocks \citep{stock_identification_2018}:
\begin{equation}
\mathbb{E}[m_{l,t}\, \varepsilon_{j,t}] = \psi_l \neq 0, \qquad
\mathbb{E}[m_{l,t}\, \varepsilon_{i,t}] = 0 \quad \text{for } i \neq j.
\label{eq:proxy_conditions}
\end{equation}
Hence, we formally assume:
\begin{assumption} \label{as:validproxy}
\textit{There exists a $k\times1$
instrument vector $m_t$, with $k \geq 1$, satisfying
\[
\mathbb{E}[m_t \varepsilon_t'] = \bigl(\,\Psi \; ,\;
0_{k\times(n-k)}\,\bigr),
\]
with targeted shocks indexed $1,\ldots,k$ and
$\Psi = \operatorname{diag}(\psi_1,\ldots,\psi_k)$ is non-singular}
\end{assumption}
These conditions yield $q_{\mathrm{prx}} = k(n-1)$ strict exclusion restrictions. The corresponding sample moment block is
\begin{equation}
g_{P,t}(\phi) =
\begin{pmatrix}
\cdots\\
m_{l,t}\cdot\varepsilon_{i,t}(\phi)\\
\cdots
\end{pmatrix},
\qquad l \in \{1,\dots,k\},\quad i \neq \mathrm{target}(l).
\end{equation}
\subsection{Identification through non-Gaussianity}\label{subsec:nongaussianconditions}
The statistical identification of SVAR models via non-Gaussianity exploits the information contained in the higher-order moments of the structural shocks.
This approach yields identification of $B$ up to sign and column permutation, as established by \citet{comon_independent_1994}. For a detailed review, we refer the reader to \citet{hyvarinen_independent_2013} and \citet{lewis_identification_2024}. Usually, the key maintained assumption is that the structural shocks, $\varepsilon_t$ are mutually independent, but recent literature has shown that it is not a necessary condition for identification, see \citet{mesters_non-independent_2024,lanne_gmm_2021, guay_identification_2021}. The identifying restrictions require only that the structural shocks be mutually uncorrelated with vanishing higher-order co-moments, together with a non-Gaussianity condition on their marginals; full statistical independence is not needed\footnote{Mutual independence would additionally force all cross-cumulants of every order to vanish, whereas only the third- and fourth-order co-moments enter the moment conditions, so these are imposed directly. This is strictly weaker than independence and does not require recovering independent components from the reduced-form innovations \citep{kilian_structural_2017}; see Remark~\ref{rem:scope_nongaus}.}. Formally, we assume:
\begin{assumption}\label{as:indep}
\textit{The structural shocks $\varepsilon_{i,t}$, $i=1,\dots,n$, are strictly stationary with mean zero, unit variance, mutual uncorrelatedness $\mathbb{E}[\varepsilon_{i,t}\varepsilon_{j,t}]=0$ for $i\neq j$, finite fourth moments, and diagonal third- and fourth-order cumulant tensors,}
\begin{equation}
\operatorname{cum}(\varepsilon_{i,t},\varepsilon_{j,t},\varepsilon_{k,t})=0 \ \ \text{unless } i=j=k,
\end{equation}
\begin{equation}
\operatorname{cum}(\varepsilon_{i,t},\varepsilon_{j,t},\varepsilon_{k,t},\varepsilon_{l,t})=0 \ \ \text{unless } i=j=k=l,
\end{equation}
\textit{with diagonal elements $\kappa_{3,k}=\mathbb{E}[\varepsilon_{k,t}^3]$ and $\kappa_{4,k}=\mathbb{E}[\varepsilon_{k,t}^4]-3$.}
\end{assumption}
\begin{assumption}\label{as:gau}
\textit{The structural shocks satisfy: \\
\textnormal{(a)} at most one has zero
skewness, i.e.\ $\kappa_{3,m}=\mathbb{E}[\varepsilon_{m,t}^3]\neq 0$ for at least
$n-1$ shocks; and \\
\textnormal{(b)} at least $n-1$ have nonzero excess kurtosis of a
common sign, i.e.\ $\kappa_{4,m}=\mathbb{E}[\varepsilon_{m,t}^4]-3$ is of equal sign
across the non-mesokurtic shocks.}
\end{assumption}
Both parts of Assumption \ref{as:gau} can be read as deviations from the Stein
identity: for a standard Gaussian $X$, $\mathbb{E}[Xh(X)]-\mathbb{E}[h'(X)]=0$
for every continuously differentiable $h$, the quantities $\kappa_{3,k}$ and $\kappa_{4,k}$ in Assumption
\ref{as:indep} are exactly the second and third-order deviations. Part (i) of the Assumption \ref{as:gau} is standard, Part (ii) is slightly stronger but mild in practice. Apart from the listed studies in Table \ref{tab:nongaussianity} and as noted by \citet{lanne_identifying_2023}, macro-financial data are mostly leptokurtic with few exceptions. Excess kurtosis is
positive for asset returns, exchange rates and high-frequency policy surprises,
as well as for output and commodity prices at monthly and quarterly
frequency, over essentially any sample containing a
disruption to the global economy, such as Global Financial Crisis 2007-08, COVID-19. Also, with conditionally heteroskedastic shocks, positive excess kurtosis follows mechanically\footnote{For $\varepsilon_{m,t}=\sigma_{m,t}
z_{m,t}$ with $z_{m,t}$ independent of the scale, $\kappa_{4,m}\ge
\kappa_{4,z_m}$ with strict inequality whenever $\sigma_{m,t}$ is
non-degenerate, and $\kappa_{4,m}>0$ when $z_{m,t}$ is Gaussian. The variations in shocks' volatility therefore shift excess kurtosis upward for every shock and cannot by
itself generate a mixture of signs.}. Negative excess kurtosis instead requires
short-tailed or bounded shocks, and part (b) fails only if at least two of these
occur in the same system, since $n-1$ shocks need only share a sign.
We operationalize these assumptions by mapping the relevant set of higher-order moment conditions into vector $g_{NG,t}(\phi)$. For an $n$-variable VAR, we adopt a baseline parameterization with
$q_{\mathrm{NG}} = n(n-1) + n(n-1)/2$ moment conditions:
\begin{itemize}
\item \textit{Co-skewness (Group 1):} The $n(n-1)$ third-order
cross-moment conditions
$\mathbb{E}[\varepsilon_{i,t}^{2}\,\varepsilon_{j,t}] = 0$
for all ordered pairs $(i,j)$ with $i \neq j$.
\item \textit{Symmetric co-kurtosis (Group 2):} The
$n(n-1)/2$ fourth-order conditions
$\mathbb{E}[\varepsilon_{i,t}^{2}\,\varepsilon_{j,t}^{2}] = 1$
for all $i < j$.
\end{itemize}
Two properties of these moment conditions are worth making explicit. First, they are {unconditional} moment restrictions that hold under Assumption~\ref{as:indep} without independence: uncorrelatedness and the diagonal third- and fourth-order cumulant tensors give $\mathbb{E}[\varepsilon_{it}^2\varepsilon_{jt}]=0$ and $\mathbb{E}[\varepsilon_{it}^2\varepsilon_{jt}^2]=1$ directly. Together with the mixing condition of Section~\ref{ass:regularity}, we allow for idiosyncratic conditional heteroskedasticity in the shocks (ARCH/GARCH, stochastic volatility): each $\varepsilon_{i,t}$ is then a martingale difference sequence, serially uncorrelated but its conditional variance is serially dependent\footnote{These moment conditions exclude only {common} (cross-shock) volatility, which violates $\mathbb{E}[\varepsilon_{it}^2\varepsilon_{jt}^2]=1$, and can be verified by testing the estimated shocks $\hat\varepsilon_t$ for cross-dependence in their squares. We refer the reader to Remark \ref{rem:scope_nongaus} for a discussion on the scope of these higher-order moment conditions.}. Second, the variance of the quartic conditions requires finite eighth-order cross-moments, $\mathbb{E}[\varepsilon_{it}^4\varepsilon_{jt}^4]<\infty$; under cross-sectional independence these factorize as $\mathbb{E}[\varepsilon_{it}^4]\,\mathbb{E}[\varepsilon_{jt}^4]$ and only $4+2\delta$ marginal moments are needed. The general requirement is satisfied by the non-Gaussian parametric families considered (normal-inverse Gaussian, $\chi^2$, and Student-$t$ with degrees of freedom exceeding $8+4\delta$).
\begin{remark}[Scope of the non-Gaussian moment block] \label{rem:scope_nongaus}
Assumption~\ref{as:indep} already relaxes full statistical independence: it requires only mutual uncorrelatedness and diagonal third- and fourth-order cumulant tensors. There are other sets of moment conditions in the literature, resting on different maintained assumptions: \citet{lanne_gmm_2021} use co-kurtosis conditions under mutual orthogonality, and \citet{keweloh_generalized_2021} uses all third- and fourth-order moments under independence.
We do not rank these strategies by identification strength. The central claim is that \emph{any} non-Gaussian identified SVAR, once expressed as sufficiently identifying moment conditions under appropriate assumptions, admits augmentation to proxy exclusion conditions within a unified GMM criterion. To illustrate, consider the assumptions of \citet{liu_non-gaussian_2025}: (i) all structural shocks have zero co-skewness and zero co-kurtosis, except that symmetric co-kurtoses may be nonzero and no two shocks share identical fourth-moment relationships, i.e.\ $\sum_{i=1}^{n}\mathbb{E}(\varepsilon_{it}^2 \varepsilon_{jt}^2) \neq \sum_{i=1}^{n}\mathbb{E}(\varepsilon_{it}^2 \varepsilon_{kt}^2)$ for all $j \neq k$; and (ii) at most one structural shock has both zero skewness and zero excess kurtosis. Under these assumptions, our symmetric co-kurtosis conditions can be replaced by asymmetric co-kurtosis conditions. The resulting moment conditions identify $B$ up to sign and column permutation, see \citet{liu_non-gaussian_2025, mesters_non-independent_2024}. These assumptions are strictly weaker than full independence and also allow for common volatility processes where the structural shocks exhibit correlated volatilities.
The analysis of the next section therefore takes $g_{NG,t}(\phi)$ as given under Assumptions \ref{as:indep} and \ref{as:gau}, and characterizes its incremental identifying information relative to the proxy exclusion conditions.
\end{remark}
\subsection{Global identification of $B$}\label{subsec:indeterminacy}
By combining the proxy exclusion and sufficient higher-order moment conditions within the GMM objective, we achieve global point identification of the target columns of $B$, as
established in Proposition \ref{prop:identification_unified}.
\begin{lemma}[Non-Gaussian identification]\label{lem:ng_set}
Let $B_0=HQ_0$ with $Q_0\in O(n)$ and $\varepsilon_t(\phi)=Q(\phi)'H^{-1}u_t$.
Under Assumptions~\ref{as:indep} and \ref{as:gau}, the Jacobian $G_{NG}(\phi_0)$ has
full column rank $p_\phi$, and the population non-Gaussian conditions are satisfied
only on the signed-permutation group
$\mathcal{B}=\{B_0P\Lambda:P\in\mathcal{P}_n,\ \Lambda\in\mathcal{D}_n^{\pm}\}$
($0<|\mathcal{B}|\le n!\,2^n$); hence the non-Gaussian identified set is $\mathcal{B}$.
\end{lemma}
\begin{proposition}[Point Identification with external instruments and non-Gaussian shocks]
\label{prop:identification_unified}
Let $u_t = B_0\varepsilon_t$ where $B_0 \in \mathbb{R}^{n\times n}$
is non-singular, the VAR satisfies Assumption \ref{as:stable_var},
and $\varepsilon_t$, $m_t$ satisfy
Assumptions \ref{as:validproxy}--\ref{as:gau}. By Lemma~\ref{lem:ng_set}, the
non-Gaussian moment conditions identify $B_0$ up to the signed-permutation set
\[
\mathcal{B} = \{B_0 P\Lambda : P \in \mathcal{P}_n,\;
\Lambda \in \mathcal{D}_n^{\pm}\},
\qquad 0 < |\mathcal{B}| \leq n!\cdot 2^n.
\]
Further assume:
\begin{enumerate}[label=(\alph*)]
\item\label{ass:proxy_u} \textbf{(Valid proxies)} The
instrument vector $m_t$ is $k \times 1$ with $1 \leq k \leq n$,
and satisfies Assumption \ref{as:validproxy}.
\item\label{ass:norm_u} \textbf{(Sign normalization)} The
diagonal elements of the impact matrix $B$, i.e. $B_{ii} > 0$ for all
$i = 1,\ldots,k$.
\end{enumerate}
Partition $B_0 = [B_{0,1}\;\; B_{0,2}]$ where $B_{0,1}$ collects the $k$ target columns and
$B_{0,2}$ collects the remaining
$n-k$ columns. Then:
\begin{enumerate}[label=(\roman*)]
\item\label{res:target} \textbf{(Target identification)}
$B_{0,1}$ is globally point-identified for all
$1 \leq k \leq n$.
\item\label{res:residual} \textbf{(Non-target identification)}
For $k = n$ or $k = n-1$, $B_{0,2}$ is also globally
point-identified, so $B_0$ is globally point-identified. For
$1 \leq k < n-1$, $B_{0,2}$ is identified only up to the
sign and permutation
\[
\mathcal{E}_{n-k} = \{B_{0,2}P_{22}\Lambda_2 :
P_{22}\in\mathcal{P}_{n-k},\;
\Lambda_2\in\mathcal{D}_{n-k}^{\pm}\},
\qquad |\mathcal{E}_{n-k}| \leq (n-k)!\cdot 2^{n-k}.
\]
\end{enumerate}
\end{proposition}
\noindent \textit{Proof}: See Appendix \ref{appendix:proof_identification_unified}.
Proposition \ref{prop:identification_unified} subsumes both full
and partial identification\footnote{The notion of identification employed is that of
{global} identification in the sense of \citet{komunjer_global_2012}: $\phi_0$ is globally identified if the population moment
conditions admit $\phi_0$ as their unique solution over all of $\Phi$. This
is strictly stronger than the local identification condition of
\citet{rothenberg_identification_1971}, which only rules out a continuum through $\phi_0$} as special cases of a unified
framework\footnote{\citet{angelini_exogenous_2019} establish
necessary and sufficient conditions for point identification in
SVAR models with $r$ instruments identifying $g \leq r$ structural
shocks. We do not pursue the general case.}.
When $k < n-1$,
the target sub-matrix $B_{0,1}$ is point-identified while the
remaining $B_{0,2}$ remains identified only up to a finite signed
permutation equivalence class. In both cases, identification of
the target columns is achieved through the joint restrictions
imposed by proxy exclusion and higher-order moment conditions. As we will see in Section \ref{subsec:weakproxy}, this result
remains valid under local-to-zero proxy relevance, that is,
when the proxy instrument is arbitrarily weak provided that the
Jacobian of the non-Gaussian moment block retains full column rank at the true parameter.
This proposition also resolves an interpretive limitation of pure
statistical identification. Through non-Gaussianity, we recover the
columns of $B_0$ up to signed permutation indeterminacy, with
no intrinsic mapping between statistically identified components
and economically meaningful structural shocks. The proxy
instrument conditions eliminate this indeterminacy for
the target columns. The external instrument, whose economic
interpretation is grounded in theoretical
prior knowledge, selects the unique permutation and sign
normalization consistent with both the higher-order moment
conditions and the proxy relevance and orthogonality
conditions. Economic identification of the target shocks is
therefore achieved as a direct consequence of the moment
conditions themselves, rather than through post-estimation
labeling conventions applied to anonymous statistical
components.
\subsection{Two-Step GMM estimation}\label{subsec:twostepgmm}
The unified, over-identified system combines $q = q_{NG} +
q_{\mathrm{P}}$ moment equations into the stacked average:
\begin{equation}\label{eq:moment_conditions}
\bar{g}_T(\phi) = \frac{1}{T} \sum_{t=1}^T
\begin{pmatrix}
g_{NG,t}(\phi) \\[2pt] g_{P,t}(\phi)
\end{pmatrix}.
\end{equation}
We can estimate it using standard \citet{hansen_1982} two-step GMM. The
first-step estimator minimizes the unweighted criterion:
$\hat{\phi}_1 = \arg\min_\phi\, \bar{g}_T(\phi)'\bar{g}_T(\phi)$.
The first-step residuals are used to construct the optimal weighting matrix:
\begin{equation}
\hat{S} = \frac{1}{T} \sum_{t=1}^T
\bigl(g_t(\hat{\phi}_1) - \bar{g}_T(\hat{\phi}_1)\bigr)
\bigl(g_t(\hat{\phi}_1) - \bar{g}_T(\hat{\phi}_1)\bigr)'.
\end{equation}
The second-step estimator minimizes the efficiently weighted objective:
\begin{equation}\label{eq:estimator}
\hat{\phi} = \arg\min_\phi\; \bar{g}_T(\phi)'\,\hat{S}^{-1}\,\bar{g}_T(\phi).
\end{equation}
Under the mixing condition of Section~\ref{ass:regularity}, the moment vector is serially dependent (the co-kurtosis block is not a martingale difference), so the optimal weighting matrix is the \citet{newey_simple_1987} HAC\footnote{$\hat{S}_{HAC} = \hat{\Gamma}_0 + \sum_{i=1}^{T-1}\omega_{i,T}(\hat{\Gamma}_i+ \hat{\Gamma}'_i)$, where $\hat{\Gamma}_i$ is a consistent estimator of ${\Gamma}_i$, the $i$th autocovariance matrix of the moment conditions, ${g}_t(\phi_0)$.} estimator with data-dependent bandwidth of \citet{andrews_heteroskedasticity_1991}. Under serial independence of the shocks, the autocovariances vanish and $\hat S$ reduces to the covariance matrix above.
For subsequent derivations, we conformably partition the optimal inverse-weighting matrix as:
\begin{equation}
S = \begin{pmatrix} S_{NN} & S_{NP} \\ S_{PN} & S_{PP} \end{pmatrix},
\qquad S_{NP} = S_{PN}',
\end{equation}
Also, we define the population Jacobian matrices as
\begin{equation} \label{eq:jacobian}
G(\phi) = \frac{\partial g(\phi)}{\partial\phi'}
\in \mathbb{R}^{q\times p_\phi}, \qquad
G \equiv G(\phi_0) = \bigl(G_{NG}', G_P' \bigr)',
\end{equation}
where $G_P = \partial g_P(\phi_0)/\partial\phi'$ and
$G_{NG} = \partial g_{NG}(\phi_0)/\partial\phi'$, and a consistent sample Jacobian is $\hat{G}_T(\phi)$.
\subsection{Joint inference with VAR parameters}\label{sec:inference}
With a standard set of regularity conditions, we show
consistency and asymptotic normality of the HGMM estimator of the rotation parameters $\phi$, and its efficiency gains relative to proxy-only criterion. Also, we also provide joint asymptotic distribution of the estimates of autoregressive and structural rotation parameters. Throughout, let $\psi = (\theta', \phi')'$ denote the full
parameter vector, with $\theta = \mathrm{vec}(\nu, A_1, \ldots, A_p)$
collecting the reduced-form VAR coefficients and $\phi$ the structural
rotation parameters.
\subsubsection{Regularity conditions}\label{ass:regularity}
\begin{enumerate}
\item The parameter space for the rotation angles $\phi \in (-\pi, \pi]^{p_\phi}$ is compact.
\item The process $\{(\varepsilon_t,m_t)\}$ is $\alpha$-mixing with mixing coefficients satisfying $\sum_{h}\alpha(h)^{\delta/(2+\delta)}<\infty$, and $\mathbb{E}\|g_t(y_t,\phi)\|^{2+\delta}<\infty$ for some $\delta>0$.
\item The long-run variance $S=\sum_{h}\mathbb{E}\big[g_t(y_t,\phi_0)g_{t-h}(y_t,\phi_0)'\big]$ is positive definite.
\end{enumerate}
Condition~1 guarantees that the extremum estimator is
well-defined and that a uniform LLN applies to the criterion, delivering
consistency once $\phi_0$ is the unique minimizer of the criterion function, as provided by the
identification result (Assumptions~\ref{as:indep} - \ref{as:gau}, Lemma~\ref{lem:ng_set}). Condition~2 governs the {temporal} dependence of the data, where $\alpha$-mixing together with the moment bound conditions
is needed for a central limit theorem for
dependent arrays to apply to $T^{-1/2}\sum_t g_t(\phi_0)$. Importantly, this does not place any restriction on cross-sectional dependence beyond the co-moment structure of
Assumption~\ref{as:indep}: it accommodates serially dependent shocks such as (idiosyncratic)
ARCH/GARCH or stochastic volatility, for which the higher-order moment conditions are serially
correlated, rather than martingale difference sequence. Condition~3 concerns the resulting optimal
weighting matrix. Since the moment conditions are serially correlated, the normalized sum has the {long-run}
variance $S=\sum_h\mathbb{E}[g_t(\phi_0)g_{t-h}(\phi_0)']$; positive definiteness of $S$ ensures the efficient weighting $S^{-1}$ and the asymptotic variance
$(G'S^{-1}G)^{-1}$ are well-defined.
Under the regularity conditions in \ref{ass:regularity} and
Assumptions \ref{as:validproxy} - \ref{as:gau}, the estimator \eqref{eq:estimator} is
consistent and asymptotically normal, i.e.
\begin{equation} \label{eq:can}
\sqrt{T}\,(\hat\phi - \phi_0)
\;\xrightarrow{d}\;
\mathcal{N}\!\bigl(0,\; \mathcal{V}_{joint}\bigr)
\end{equation}
where, $\mathcal{V}_{joint} = \bigl(G'S^{-1}G\bigr)^{-1}$ is the asymptotic variance\footnote{This result assumes the reduced-form innovations ($u_t$) are known, however in practice the innovations are estimated and hence, we include the generated-regressor correction while deriving the asymptotic distribution of $\psi$ in Section \ref{sec:joint_asymp}.} of the estimator.
The proof is a direct application of \citet{hall_generalized_2005}.
\subsubsection{Efficiency gains}\label{subsec:efficiencygains}
The hybrid GMM estimator combines two distinct identification strategies, yielding efficiency gains relative to proxy-based identification alone\footnote{Analogous efficiency gains are obtained relative to non-Gaussian identification alone, provided the proxy instruments satisfy $\operatorname{rank}(\widetilde{G}_{P}) = p_\phi$. This condition is generically restrictive, as sufficient valid proxies to fully identify a moderate-dimensional VAR system are rarely available in practice.}.
\begin{proposition}[Efficiency of joint GMM]
\label{prop:efficiency}
Under Assumptions \ref{as:validproxy}-\ref{as:gau}, with $G_{NG}$ of full column rank (Lemma~\ref{lem:ng_set}) and
$\operatorname{rank}(G_P) = p_\phi$:
\begin{equation}
\mathcal{V}_P \;-\; \mathcal{V}_{joint} \;\succeq\; 0.
\label{eq:eff_ineq}
\end{equation}
where,
\begin{equation}
\mathcal{V}_{joint} \;\equiv\; \mathcal{I}_{joint}^{-1} \;=\; \bigl(G'S^{-1}G\bigr)^{-1} \qquad \mathcal{V}_P \equiv \bigl(G_P'S_{PP}^{-1}G_P\bigr)^{-1}
\label{eq:Vjoint}
\end{equation}
and $\mathcal{V}_P$ is the analogous variance of the proxy-only GMM estimator.
\end{proposition}
\begin{corollary}[Strict efficiency gain]\label{cor:strict_eff}
$ \mathcal{V}_P - \mathcal{V}_{joint} \succ 0$ iff $\operatorname{rank}(\tilde{G}_{NG})
= p_\phi$
\end{corollary}
where,
\begin{equation}\label{eq:GNG_tilde}
\tilde{G}_{NG}\equiv G_{NG} - S_{NP}S_{PP}^{-1}G_P
\end{equation}
is the partialed NG Jacobian. The corollary states strict efficiency gains only if the non-Gaussian moment conditions identify $\phi$ in directions not already identified by the proxy moments.
\noindent The proof rests on an exact additive decomposition of the joint information matrix:
\begin{equation}
\mathcal{I}_{joint}
\;=\; \underbrace{G_P'S_{PP}^{-1}G_P}_{\mathcal{I}_P}
\;+\; \underbrace{\tilde{G}_{NG}'\Sigma_{NG}^{-1}\tilde{G}_{NG}}_{\Delta \mathcal{I}_{NG}},
\label{eq:info_decomp}
\end{equation}
where $\Sigma_{NG} \equiv S_{NN} - S_{NP}S_{PP}^{-1}S_{PN} \succ 0$ is
the Schur complement of $S_{PP}$ in $S$, adopted from \citet{newey_generalized_1985}. Since $\Sigma_{NG}^{-1} \succ 0$,
the increment $\Delta \mathcal{I}_{NG} \succeq 0$, and matrix monotonicity of
inversion delivers \eqref{eq:eff_ineq}. For the detailed proof, see Appendix \ref{appendix:efficiency}.
The decomposition \eqref{eq:info_decomp} is the GMM analog of the
Frisch-Waugh-Lovell theorem: $\Delta \mathcal{I}_{NG}$ is the information the NG
moments contribute for $\phi$ {after} projecting out the identifying
directions already spanned by the proxy moments in the $S$-inner product.
The efficiency gain is thus governed by the extent to which the two channels
identify $\phi$ along distinct directions.
\subsubsection{Joint asymptotic distribution}\label{sec:joint_asymp}
Before the two-step estimation of Section \ref{subsec:twostepgmm}, we obtain the autoregressive estimates $\theta$ by OLS which yield the reduced-form innovations $\hat{u}_t$ and the Cholesky
factor of the covariance matrix $\hat{\Sigma}_u$, $\hat{H} = \mathrm{chol}(\hat{\Sigma}_u)$. The second stage minimizes
the HGMM criterion over $\phi$ using the generated residuals
$\hat{\varepsilon}_t(\phi) = Q(\phi)'\hat{H}^{-1}\hat{u}_t$. Since,
the second-stage moment conditions depend on first-stage autoregressive estimates, the asymptotic
variance of $\hat{\phi}$ requires a generated-regressor
correction of \citet{murphy_estimation_1985}.
\paragraph{First-Stage Asymptotics}
Under the regularity conditions in \ref{ass:regularity} and Assumptions \ref{as:stable_var} and \ref{as:indep}, standard OLS
theory from \citet{lutkepohl_2005} gives
\begin{equation}
\sqrt{T}\,(\hat{\theta} - \theta_0)
\xrightarrow{d} \mathcal{N}(0,\, \mathcal{V}_\theta),
\qquad
\mathcal{V}_\theta = \mathbb{E}[\zeta_t\zeta_t'],
\end{equation}
where $\zeta_t \equiv (\mathbb{E}[x_t x_t'])^{-1}x_t \otimes u_t$, $\otimes$ denotes the Kronecker product, $x_t = (y_{t-1}',\ldots,y_{t-p}',1)' \in \mathbb{R}^{np+1}$ is $\mathcal{F}_{t-1}$-measurable with $\mathcal{F}_{t-1} :=
\sigma(y_{t-1}, y_{t-2}, \ldots)$ is the natural filtration of $\{y_t\}$. In the case of homoscedastic residuals, the variance simplifies to $\mathcal{V}_\theta = \bigl(\mathbb{E}[x_t x_t']\bigr)^{-1} \otimes \Sigma_u$; under conditional heteroskedasticity $\mathcal V_\theta$ retains the general
long-run form above.
\paragraph{Second-Stage Asymptotics}
The joint estimator $\hat{\psi}$ solves a stacked moment system identifying $\phi$ via the proxy exclusion and higher-order moment conditions. Following the two-step estimator theory from
\citet{murphy_estimation_1985} and \citet{newey_mcfadden_1994} for GMM framework, under
regularity conditions \ref{ass:regularity} and Assumptions \ref{as:stable_var} - \ref{as:gau}:
\begin{equation}
\sqrt{T}
\begin{pmatrix} \hat{\theta} - \theta_0 \\ \hat{\phi} - \phi_0 \end{pmatrix}
\xrightarrow{d}
\mathcal{N}\!\left(0,\;
\mathcal{V}_\psi \equiv
\begin{bmatrix} \mathcal{V}_\theta & \mathcal{V}_{\theta\phi} \\ \mathcal{V}_{\theta\phi}' & \mathcal{V}_\phi \end{bmatrix}
\right).
\label{eq:joint_asymp}
\end{equation}
Let $\sqrt{T}(\hat\theta-\theta_0) = T^{-1/2}\sum_t \zeta_t + o_p(1)$ be the OLS asymptotic linear representation with
$\mathcal{V}_\theta = \mathbb{E}[\zeta_t\zeta_t']$. Since the second-stage
moments depend on $\theta$ through the generated residuals, the first-stage
estimation error propagates into $\hat\phi$, and the relevant asymptotic
object is the long-run variance of the {combined} score:
$g_t + \Gamma_\theta \zeta_t$ where $\Gamma_\theta \equiv \mathbb{E}[\partial g_t(\psi_0)/\partial\theta']$:
\begin{equation}
\Omega_\psi \;\equiv\; \operatorname{Var}\!\big(g_t + \Gamma_\theta \zeta_t\big)
\;=\; S \;+\; \Gamma_\theta \mathcal{V}_\theta \Gamma_\theta'
\;+\; \Gamma_\theta \mathcal{C}' \;+\; \mathcal{C}\Gamma_\theta',
\qquad
\mathcal{C} \equiv \mathbb{E}[g_t(\psi_0)\,\zeta_t'],
\label{eq:Omega_corrected}
\end{equation}
where $S=\mathbb{E}[g_tg_t']$ and $\mathcal{C}$ is the cross-covariance
between the structural moments and the first-stage score. The
generated-regressor corrected variance block is
\begin{equation}
\mathcal{V}_\phi
= (G_0'S^{-1}G_0)^{-1}\,G_0'S^{-1}\,\Omega_\psi\,
S^{-1}G_0\,(G_0'S^{-1}G_0)^{-1}.
\label{eq:vphi_corrected}
\end{equation}
The off-diagonal block does {not} vanish in general:
\begin{equation}
\mathcal{V}_{\theta\phi}
= -\,\big(\mathcal{V}_\theta\Gamma_\theta' + \mathcal{C}'\big)\,
S^{-1}G_0\,(G_0'S^{-1}G_0)^{-1}.
\label{eq:vthetaphi}
\end{equation}
The cross-covariance $\mathcal{C}$ is generically nonzero under non-Gaussian
identification. For instance, for the co-skewness coordinate
$g=\varepsilon_{i}^2\varepsilon_{j}$ ($i\neq j$) and $u_t=B_0\varepsilon_t$,
$\mathbb{E}[u_{k,t}\,\varepsilon_{i,t}^2\varepsilon_{j,t}]
= B_{0,kj}\,\mathbb{E}[\varepsilon_{j,t}^2\varepsilon_{i,t}^2]=B_{0,kj}\neq 0$,
so the higher-order structural moments are correlated with the reduced-form
innovations that drive the first-stage error. This implies that both $\mathcal{C}$
and $\mathcal{V}_{\theta\phi}$ must be retained.
A consistent estimator of $\mathcal{V}_\phi$ is
\begin{equation}
\hat{\mathcal{V}}_\phi
= (\hat{G}'\hat{S}^{-1}\hat{G})^{-1}\,\hat{G}'\hat{S}^{-1}\,
\hat{\Omega}_\psi\,
\hat{S}^{-1}\hat{G}\,(\hat{G}'\hat{S}^{-1}\hat{G})^{-1},
\end{equation}
\begin{equation}
\hat{\Omega}_\psi=\hat S+\hat\Gamma_\theta\hat{\mathcal V}_\theta\hat\Gamma_\theta'
+\hat\Gamma_\theta\hat{\mathcal C}'+\hat{\mathcal C}\hat\Gamma_\theta',
\label{eq:vphi_est}
\end{equation}
where $\hat{G}=T^{-1}\sum_t \partial g_t(\hat\psi)/\partial\phi'$,
$\hat\Gamma_\theta=T^{-1}\sum_t \partial g_t(\hat\psi)/\partial\theta'$,
$\hat\zeta_t=(T^{-1}\sum_s x_sx_s')^{-1}x_t\otimes\hat u_t$, and
$\hat{\mathcal C}=T^{-1}\sum_t g_t(\hat\psi)\,\hat\zeta_t'$.
\subsubsection{Inference on the impact matrix}
\label{sec:b0_inference}
Since $B = H(\theta)Q(\phi)$ is a smooth differentiable function of $\psi$ under
the regularity conditions and stable VAR, we can apply the delta method to \eqref{eq:joint_asymp}. Consider the composite Jacobian
\begin{equation}
\mathcal{J}_{B} =
\frac{\partial\,\mathrm{vec}(B)}{\partial\psi'} =
\left(\frac{\partial\,\mathrm{vec}(B)}{\partial\theta'}
\;,\;
\frac{\partial\,\mathrm{vec}(B)}{\partial\phi'}\right)
\in \mathbb{R}^{n^2 \times (n^2 p + n + p_\phi)}.
\label{eq:jac_B0}
\end{equation}
Both blocks are analytically tractable. For the rotation component:
\begin{equation}
\frac{\partial\,\mathrm{vec}(B)}{\partial\phi'}
= (I_n \otimes H)\,\frac{\partial\,\mathrm{vec}(Q(\phi))}{\partial\phi'},
\label{eq:jac_B0_phi}
\end{equation}
where the Givens Jacobian $\partial\,\mathrm{vec}(Q(\phi))/\partial\phi'$
has explicit closed form as a sum of antisymmetric generator matrices, as shown in
\citet{matteson_dynamic_2011}. For the Cholesky component:
\begin{equation}
\frac{\partial\,\mathrm{vec}(B)}{\partial\theta'}
= (Q(\phi)'\otimes I_n)\,
\frac{\partial\,\mathrm{vec}(H)}{\partial\,\mathrm{vech}(\Sigma_u)'}\cdot
\frac{\partial\,\mathrm{vech}(\Sigma_u)}{\partial\theta'},
\label{eq:jac_B0_theta}
\end{equation}
where the Cholesky derivative uses the implicit differentiation formula of
\citet{magnus_neudecker_1989} via the duplication matrix $D_n$, and
$\partial\,\mathrm{vech}(\Sigma_u)/\partial\theta'$ follows from standard
theory.
The delta method applied to \eqref{eq:joint_asymp} with Jacobian $\mathcal{J}_{B}$ in
\eqref{eq:jac_B0} yields
\begin{equation}
\sqrt{T}\,\mathrm{vec}(\hat{B}_0 - B)
\xrightarrow{d}
\mathcal{N}\bigl(0,\; \mathcal{J}_{B} \mathcal{V}_\psi \mathcal{J}_{B}'\bigr).
\label{eq:b0_asymp}
\end{equation}
We can estimate the standard errors for $[\hat{B}_0]_{ij}$ as the square roots of the corresponding diagonal entries of $\mathcal{J}_{B}\hat{\mathcal{V}}_\psi \mathcal{J}_{B}'$. Once we have established the inference for the impact matrix $B$, the derivation of the asymptotic distribution of the structural impulse response functions (SIRFs) follows directly from \citet{lutkepohl_2005}, Chapter 3. We defer it to the Appendix \ref{subsec:structural_irfs}.
\section{Behavior under weak and endogenous proxies}\label{sec:weak_misp}
In this section, we provide three sets of results. First, under local-to-zero weak proxies, the HGMM estimator remains consistent and asymptotically normal, with the non-Gaussian block acting as the identifying anchor. It also attains a smaller asymptotic variance than the standalone non-Gaussian GMM estimator, the gain arising from the cross-block long-run covariance between the proxy exclusion and higher-order moment conditions (Proposition~\ref{prop:localtozero}). Second, under a uniform non-Gaussian
identification condition, standard inference is uniformly valid over proxy strength
(Proposition~\ref{prop:uniform}). Third, when the proxies are locally endogenous through
correlation with non-target structural shocks, the estimator incurs a finite
asymptotic bias, for which we derive an explicit bound in terms of the degree of
endogeneity and the strength of the non-Gaussian identification
(Proposition~\ref{prop:bias_bound_local}).
\subsection{Robust identification with irrelevant proxies} \label{subsec:weakproxy}
In a GMM framework, proxy instruments contribute identifying
information through the relevance conditions $\mathbb{E}[m_t \varepsilon_{1,t}] = \psi_1$,
where $\psi_1$ denotes the first-stage population regression coefficient.
When $\psi_1 \neq 0$, the proxy moment conditions augment the rank of the
expected Jacobian, extending identification beyond the non-Gaussian identified
set $\mathcal{B}$ to point identification of the full rotation matrix $Q(\phi_0)$.
The strength of this contribution is monotone in $\|\psi_1\|$ i.e., as the relevance weakens, the proxy moment conditions become asymptotically
collinear with the score, the Jacobian loses rank in finite samples, and the
GMM estimator inherits the well-known issues of weakly identified systems, as shown in
\citet{newey_generalized_2009, montiel_olea_inference_2021}. However, under hybrid GMM identification, this vulnerability is absent. We show that the estimates remain consistent and weakly efficient to standalone NG-GMM estimator, even if proxy relevance is infinitesimally non-zero, formally of Pitman order $T^{-1/2}$.
\begin{proposition}[Behavior under local-to-zero proxy relevance]\label{prop:localtozero}
Suppose the conditions of Proposition \ref{prop:identification_unified}
hold, but proxy relevance follows the local-to-zero (Pitman) sequence
\[
\mathbb{E}[m_{t,T}\varepsilon_t']
= \frac{1}{\sqrt{T}}\bigl(\,C\;,\;0_{k\times(n-k)}\,\bigr),
\qquad C = \operatorname{diag}(c_1,\ldots,c_k),\ \ c_i > 0,
\]
and let $\Sigma_{NG}$ be the Schur complement defined in
\eqref{eq:info_decomp}. Define
$\mathcal{I}^{*}\equiv G_{NG}'\,\Sigma_{NG}^{-1}\,G_{NG}$. Then,
\begin{enumerate}[label=(\roman*)]
\item \label{localtozero_infolimit} The population
proxy Jacobian satisfies $G_{P,T}=O(T^{-1/2})$, and the joint
information matrix converges in operator norm,
\begin{equation}
\mathcal{I}_{\mathrm{joint},T}\ \longrightarrow\ \mathcal{I}^{*}\succ 0
\end{equation}
\item \label{localtozero_can} The hybrid
GMM estimator satisfies
\begin{equation}
\hat{\phi}\ \xrightarrow{p}\ \phi_0,
\end{equation}
\begin{equation}
\sqrt{T}\,(\hat{\phi}-\phi_0)\ \xrightarrow{d}\
\mathcal{N}\!\bigl(0,\ \mathcal{V}_{NG}^{*}\bigr),
\end{equation}
\begin{equation}
\mathcal{V}_{NG}^{*}=(\mathcal{I}^{*})^{-1}
=\bigl(G_{NG}'\,\Sigma_{NG}^{-1}\,G_{NG}\bigr)^{-1}.
\end{equation}
\item \label{localtozero_effoverng}
$\mathcal{V}_{NG}^{*}\preceq \mathcal{V}_{NG}
\equiv (G_{NG}'\,S_{NN}^{-1}\,G_{NG})^{-1}$, the asymptotic variance of
the optimal standalone NG-GMM estimator, with equality if the
cross-block long-run covariance $S_{NP}=0$.
\end{enumerate}
\end{proposition}
\noindent\textit{Proof.} See Appendix \ref{appendix:proof_localtozero}.
Apart from the consistency and asymptotic normality of the estimator under weak proxies in Part \ref{localtozero_can}, Part \ref{localtozero_effoverng} is another crucial result. A natural conjecture is that, as proxy relevance is weak, the hybrid
estimator should revert to the standalone NG-GMM, recovering the variance
$\mathcal{V}_{NG}$. Proposition \ref{prop:localtozero} shows this conjecture
to be conservative. The variance $\mathcal{V}_{NG}^*$ satisfies
$\mathcal{V}_{NG}^* \preceq \mathcal{V}_{NG}$ since the joint estimator continues to exploit the cross-block
long-run covariance $S_{NP}$ between the non-Gaussian and proxy moment
conditions, conditioning the higher-order estimating equations on the proxy moment
space. Hence, at any fixed proxy relevance, it provides efficiency through its covariance structure with the higher-order
moments, even when its direct identifying contribution is negligible. We further prove the uniform validity of the standard inference for impact matrix over the proxy strength.
\subsection{Uniform validity of standard inference}\label{subsec:uniformvalid}
Let $\gamma=(\phi_0,\theta_0,\Psi,F)\in\mathcal G$ denote the data-generating
process, where $F$ is the joint distribution of the structural shocks and the
proxy. All population objects below ($G_{NG}$, $\Sigma_{NG}$, $S$) are
functionals of $\gamma$.
\begin{assumption}[Uniform non-Gaussian identification]\label{as:uniform_ng}
Under Assumptions \ref{as:stable_var}--\ref{as:gau}, there exist constants $\underline g>0$, $0<\underline s\le\bar s<\infty$ and $\underline\psi>0$ such that
inference is taken uniformly over the class of data-generating processes
\[
\mathcal{P}=\Big\{\gamma:\
\sigma_{\min}\!\big(G_{NG}(\phi_0)\big)\ge \underline g,\
\operatorname{eig}\!\big(S(\gamma)\big)\subset[\underline s,\bar s],\
\min_{l\le k}|\psi_l|\ \ge\ \underline\psi \Big\}.
\]
\end{assumption}
where, $\sigma_{\min}(\cdot)$ is the smallest singular value and
$\operatorname{eig}(\cdot)$ is the set of eigenvalues. The proxy is relevant\footnote{In the Appendix, Corollary \ref{cor:irrelevance} establishes that strict proxy irrelevance ($\Psi = 0$)
collapses identification to the non-Gaussian identified set $\mathcal{B}$. So indeed, the proxy is still required to select the target column, but only out of a discrete (not a continuum) number of choices. This allows the proxy relevance to be arbitrarily close to zero and NG moment block anchors the identification with uniformly valid standard inference.} but $\underline\psi>0$ can be arbitrarily small. Thus, the non-Gaussian block is required to be {uniformly} strongly
identifying, while the proxy block is allowed to be arbitrarily weak. Lemma~\ref{lem:ng_set} already guarantees
$\sigma_{\min}(G_{NG}(\phi_0))>0$ under Assumptions~\ref{as:indep}-\ref{as:gau};
Assumption~\ref{as:uniform_ng} strengthens this to a uniform positive lower bound
over $\mathcal{P}$.
\begin{lemma}[Uniform information bound]\label{lem:uniform_info}
Under Assumption \ref{as:uniform_ng},
\[
\inf_{\gamma\in\mathcal P}\ \lambda_{\min}\!\big(G'S^{-1}G\big)\ \ge\
\underline\kappa\ \equiv\ \underline g^{\,2}/\bar s\ >\ 0 .
\]
\end{lemma}
\begin{proposition}[Uniform validity of Wald inference over proxy strength]\label{prop:uniform}
Suppose Assumption \ref{as:uniform_ng} and the regularity conditions of
Section \ref{ass:regularity} hold. Let $R$ be a fixed $d\times p_\phi$ matrix of
full row rank, and let $\hat{\mathcal V}_\phi$ be the generated-regressor
corrected variance estimator of \eqref{eq:vphi_est}. Then the Wald statistic
\[
W_T \;=\; T\,(R\hat\phi-r)'\,\big(R\,\hat{\mathcal V}_\phi\,R'\big)^{-1}\,(R\hat\phi-r)
\]
is asymptotically $\chi^2_d$ \emph{uniformly over proxy strength}:
\[
\lim_{T\to\infty}\ \sup_{\gamma\in\mathcal P}\
\sup_{x\in\mathbb R}\
\big|\,\mathbb P_\gamma(W_T\le x)-F_{\chi^2_d}(x)\,\big|\;=\;0 .
\]
Consequently, the Wald confidence set $\mathcal C_{1-\alpha}=\{r:W_T(r)\le \chi^2_{d,1-\alpha}\}$
has correct uniform asymptotic size,
$\displaystyle \lim_{T\to\infty}\inf_{\gamma\in\mathcal P}\mathbb P_\gamma\big(R\phi_0\in\mathcal C_{1-\alpha}\big)=1-\alpha.$
\end{proposition}
\noindent\textit{Proof.} See Appendix \ref{appendix:proof_uniform}
Proposition~\ref{prop:uniform} is the inferential counterpart of
Proposition~\ref{prop:localtozero} where the non-Gaussian block restores {standard} inference of the point estimates,
uniformly in the strength of the instrument. Because the non-Gaussian
Jacobian is uniformly strongly identifying over $\mathcal P$, the joint information
matrix $G'S^{-1}G$ stays bounded away from singularity even under local-to-zero proxy relevance. The estimator, therefore, remains asymptotically
normal and the generated-regressor corrected variance estimator
$\hat{\mathcal V}_\phi$ is uniformly consistent. Hence, as long as the
structural shocks are non-Gaussian\footnote{See Appendix \ref{appendix:evidencelit} for its empirical standing.}, the practitioner may report conventional
delta-method standard errors for the structural parameters without recourse to grid-based identification-robust methods.
\begin{remark}[The case of doubly-weak identification]
Proposition \ref{prop:uniform} is uniform over proxy strength while the
non-Gaussian block is uniformly strongly identifying
($\sigma_{\min}(G_{NG})\ge\underline g$). If non-Gaussianity is itself weak or
absent ($\sigma_{\min}(G_{NG})\to0$, i.e. near-Gaussian shocks), the DGP leaves
$\mathcal P$, both identification channels degenerate, and the limit is
non-standard; Wald inference is then not uniformly valid. For that regime we can rely
on the identification-robust Anderson--Rubin confidence sets which retain correct coverage uniformly
over the strength of {both} channels \citep{stock_gmm_2000,
montiel_olea_inference_2021}.
\end{remark}
\subsection{Bounded asymptotic bias under local proxy endogeneity}\label{subsec:proxycontam}
In applied macroeconomic and financial research, external instruments may be
endogenous by narrative mis-classification, or the
inadvertent capture of confounding information, as discussed in
\citet{bauer_reassessment_2022, miranda-agrippino_transmission_2021}. When a
proxy co-varies with non-target shocks, the orthogonality condition fails and
the proxy moment conditions are contaminated, i.e. the population expectation
$\mathbb{E}[g_P(\phi_0)] = \mu \neq 0$, inducing a non-vanishing asymptotic
bias in the structural parameter estimates.
The hybrid GMM framework permits a precise characterization of this bias and
its determinants. Because identification operates through two distinct
channels, the asymptotic bias is not governed solely by the degree of proxy
endogeneity but sufficiently strong non-Gaussian identification attenuates the
bias. The following proposition makes this
relationship explicit.
\begin{proposition}[Asymptotic bias bound under local proxy endogeneity]\label{prop:bias_bound_local}
Suppose Assumptions~\ref{as:indep}--\ref{as:gau} hold, so $G_{NG}(\phi_0)$ has
full column rank at $\phi_0$, and let the estimator use the efficient weight
$W=S^{-1}$. The proxy moment block is locally contaminated,
\[
\mathbb{E}[g_P(\phi_0)]=\mu_T=c/\sqrt{T},\quad \|c\|<\infty,
\qquad \mathbb{E}[g_{NG}(\phi_0)]=0 .
\]
With the partialed proxy Jacobian $\tilde G_P=G_P-S_{PN}S_{NN}^{-1}G_{NG}$ and the
Schur complement $\Sigma_P=S_{PP}-S_{PN}S_{NN}^{-1}S_{NP}$, let $\phi_{*,T}$
minimize $\mathcal Q_T(\phi)=\mathbb{E}[g(\phi)]'S^{-1}\mathbb{E}[g(\phi)]$ and
$\mathbb{B}_T=\sqrt{T}(\phi_{*,T}-\phi_0)$. Then
\begin{equation}
\mathbb{B}_T=-(G'S^{-1}G)^{-1}\,\tilde G_P'\,\Sigma_P^{-1}\,c+O(T^{-1/2}),
\label{eq:local_bias}
\end{equation}
\begin{equation}
\sqrt{T}(\hat\phi-\phi_0)\xrightarrow{d}
\mathcal{N}\!\Big(-(G'S^{-1}G)^{-1}\tilde G_P'\Sigma_P^{-1}c,\ (G'S^{-1}G)^{-1}\Big),
\label{eq:local_limit}
\end{equation}
so $\hat\phi$ is consistent ($\phi_{*,T}\to\phi_0$). The Euclidean norm of the
scaled bias is bounded by
\begin{equation}
\|\mathbb{B}_T\|_2\ \le\
\frac{\big\|\tilde G_P'\,\Sigma_P^{-1}\,c\big\|_2}
{\lambda_{\min}\!\big(G_{NG}'S_{NN}^{-1}G_{NG}\big)}
\ +\ O(T^{-1/2}).
\label{eq:local_bias_bound}
\end{equation}
\end{proposition}
\noindent\textit{Proof.} See Appendix \ref{appendix:bias_bound_local}.
The two sides of \eqref{eq:local_bias_bound} separate the problem from the remedy. The numerator $\big\|\tilde G_P'\,\Sigma_P^{-1}\,c\big\|_2$ is the raw contamination, transmitted only through the proxy moment conditions. The denominator $\lambda_{\min}\!\big(G_{NG}'S_{NN}^{-1}G_{NG}\big)$ is the curvature the non-Gaussian block alone contributes to the population criterion. The bias ceiling is therefore strictly decreasing in the strength of non-Gaussian identification: as $\lambda_{\min}\!\big(G_{NG}'S_{NN}^{-1}G_{NG}\big)$ grows, the higher-order moment conditions anchor the estimator more firmly at $\phi_0$ and the distortion from the mis-specified proxy shrinks. A proxy-only estimator has no such denominator. There $\mathcal{I}_{NG} = 0$, and the analogous bound is unbounded whenever $c \neq 0$.
\section{Specification tests}\label{sec:tests}
The asymptotic theory of Section \ref{sec:inference} assumes valid
specification of both the non-Gaussian moment block and the proxy exogeneity
conditions. In practice, neither can be taken as given. Proxy exogeneity is
not directly testable in isolation. The exclusion restriction
$\mathbb{E}[m_{l,t}\varepsilon_{i,t}] = 0$ for $i \neq \mathrm{target}(l)$
involves unobserved structural shocks and cannot be evaluated without auxiliary
identifying assumptions.
The hybrid GMM framework of Section \ref{sec:method} resolves the attribution
problem that may affect each identification channel individually. As the
joint moment vector $\bar{g}_T(\phi)$ is over-identified beyond the $p_\phi =
n(n-1)/2$ rotation angles, the framework admits formal tests of the proxy
exogeneity conditions conditional on the higher-order moments of the structural shocks, and
vice versa. This section motivates and constructs the mutually orthogonal test statistics, establishes their
asymptotic distributions, and describes the bootstrap procedure used to
obtain valid finite-sample critical values. Specifically, we construct three test statistics: $J_{\mathrm{joint}}$, $J_{\mathrm{prx}}$, and $J_{NG}$.
\subsection{The joint null hypothesis}\label{sec:joint_null}
Let $\bar{g}_T(\phi)$ denote the sample moment vector of \eqref{eq:moment_conditions} with $q = q_{NG} + q_P$ components. The
joint null hypothesis assumes that the proxy instruments satisfy the exogeneity
conditions \eqref{eq:proxy_conditions} and that the structural
innovations satisfy the non-Gaussian moment conditions (\ref{as:indep} and \ref{as:gau}). Under this null, evaluated at the efficient
two-step estimator $\hat{\phi}$ of Section \ref{sec:method}, the
over-identifying restrictions test statistic of \citet{hansen_1982} is
\begin{equation}
J_{\mathrm{joint}}
= T\cdot\bar{g}_T(\hat{\phi})'\,\hat{S}^{-1}\,\bar{g}_T(\hat{\phi})
\xrightarrow{d} \chi^2(q - p_\phi),
\label{eq:j_joint}
\end{equation}
where $\hat{S}$ is the consistent long-run covariance estimator from
\eqref{eq:vphi_est}. The degrees of freedom equal $q - p_\phi =
(q_{NG} + q_P) - p_\phi$. Rejection of \eqref{eq:j_joint} signals mis-specification
of at least one moment block, but it does not identify its origin. We develop conditional orthogonal tests to address this lack of attribution.
\subsection{Neyman-Orthogonal tests}\label{sec:subset_tests}
Identification through non-Gaussianity recovers the rotation matrix $Q(\phi)$ only up
to right-multiplication by any element of the signed permutation group
$\mathcal{SP}(n)$, so the unrestricted non-Gaussian estimator, which excludes
the proxy moment conditions, admits $2^n \cdot n!$ observationally equivalent
solutions and does not possess a unique probability limit. The standard incremental $J$-statistic requires the unrestricted estimator
to be asymptotically normal around a unique $Q(\phi_0)$. Without a column anchor provided by proxy exogeneity conditions to select a single element of the $\mathcal{SP}(n)$-orbit, this
condition fails and the $\chi^2$ limit distribution of the incremental $J$-statistic is not
restored. Therefore, the standard test is invalid in this setting.
Instead, we propose to construct the test statistics from the same joint GMM estimator $\hat{\phi}$, which is point-identified under the joint null and has a stable representation. The key challenge is that the raw sample moments $\bar{g}_{NG,T}$ and $\bar{g}_{P,T}$ are correlated through the shared data, identification space and the common parameter estimate $\hat{\phi}$, so their naive quadratic forms do not have standard chi-squared limits under the null.
We address these difficulties by constructing mutually orthogonalized score statistics following \citet{newey_generalized_1985}.
The notations for the partitioned moment vectors, variance, and Jacobian matrices remain the same as Section \ref{subsec:twostepgmm}.
To construct a test for the proxy conditions that
is asymptotically insensitive (under the null) to the higher-order moment conditions, we partial out $\bar{g}_{P,T}$ with respect to the linear span of $\bar{g}_{NG,T}$. This yields the orthogonalized proxy moment vector
\begin{equation}
\tilde{g}_P = \bar{g}_{P,T} - S_{PN}S_{NN}^{-1}\bar{g}_{NG,T}.
\label{eq:gphat}
\end{equation}
Similar to the decomposition in Proposition \ref{prop:efficiency}, equation \eqref{eq:gphat} is the GMM analog of the Frisch-Waugh-Lovell
theorem. The coefficient $S_{PN}S_{NN}^{-1}$ is the best linear predictor of
$g_P$ given $g_{NG}$ under the long-run variance $S$, so subtracting it
removes the component of $\bar{g}_{P,T}$ linearly predictable from $\bar{g}_{NG,T}$.
The resulting score satisfies the Neyman orthogonality condition:
\begin{equation}
\frac{\partial}{\partial\theta_{NG}}
\,\mathbb{E}\!\left[\tilde{g}_P(\phi_0,\theta_{NG})\right] = 0,
\label{eq:neyman}
\end{equation}
where $\theta_{NG}$ parameterizes local perturbations of the non-Gaussian
block. Condition \eqref{eq:neyman} ensures that first-order mis-specification
or estimation error in the non-Gaussian block does not contaminate the
limiting distribution of the test statistic for the proxy conditions.
An identical construction delivers the orthogonalized non-Gaussian moment
vector:
\begin{equation}
\tilde{g}_{NG} = \bar{g}_{NG,T} - S_{NP}S_{PP}^{-1}\bar{g}_{P,T}.
\label{eq:gnghat}
\end{equation}
Both $\tilde{g}_P$ and $\tilde{g}_{NG}$ are evaluated at the jointly
estimated $\hat{\phi}$, so parameter estimation uncertainty must
be accounted for in their asymptotic variances.
The asymptotic variance of $T^{1/2}\tilde{g}_P$ is governed by the conditionally
projected Jacobian $\tilde{G}_P = G_P - S_{PN}S_{NN}^{-1}G_{NG}$ and the Schur
complement $\Sigma_P = S_{PP} - S_{PN}S_{NN}^{-1}S_{NP}$ introduced in
Proposition~\ref{prop:bias_bound_local} (the proxy-block counterparts of
$\tilde{G}_{NG}$ in \eqref{eq:GNG_tilde} and $\Sigma_{NG}$ in \eqref{eq:info_decomp}).
Here, $\tilde{G}_P$ measures the sensitivity of $\tilde{g}_P$ to perturbations in
$\phi$ after netting out the component collinear with the non-Gaussianity block,
while $\Sigma_P$ captures the marginal long-run variance of the orthogonalized
proxy moments after projecting out that block. This variance consists of two
parts: the Schur complement $\Sigma_P$, less a correction for estimation error in
$\hat{\phi}$ via the delta method.
The full restricted variance matrix is then
\begin{equation}
\mathcal Q_P = \underbrace{\Sigma_P}_{\text{Schur complement}}
\;-\;
\underbrace{\tilde{G}_P\bigl(G'\hat{S}^{-1}G\bigr)^{-1}\tilde{G}_P'}_{
\text{parameter estimation correction}}.
\label{eq:QPP}
\end{equation}
The second term subtracts from $\Sigma_P$ the variance asymptotically
attributable to the sampling fluctuation of $\hat{\phi}$ through
$\tilde{G}_P$. $\mathcal Q_P$ equals the Schur
complement of the $(q_{NG} \times q_{NG})$ block of the full efficient
weight matrix $\hat{S}^{-1}$ after projecting out the parameter estimation
variance, delivering the minimum-variance matrix for the test statistic.
The symmetric construction yields $\mathcal Q_{NG}$.
Formally, the two specification tests are
\begin{equation}
J_{\mathrm{prx}} = T\,\tilde{g}_P'\,\mathcal Q_P^{+}\,\tilde{g}_P,
\qquad
J_{NG} = T\,\tilde{g}_{NG}'\,\mathcal Q_{NG}^{+}\,\tilde{g}_{NG},
\label{eq:subset_stats}
\end{equation}
where $(\cdot)^+$ denotes the Moore-Penrose pseudo-inverse.
\subsection{Limiting distributions}\label{subsec:limitingdist}
To test the validity of the proxy exogeneity conditions, we require the maintained assumption that the non-Gaussian moment block identifies $\phi_0$ i.e., formally:
\begin{equation}
H_0^{prx}: \mathbb{E}(\tilde{g}_P) = 0 | rank({G}_{NG}) = p_\phi
\end{equation}
under the assumptions \ref{as:indep} and \ref{as:gau}. When evaluated at the joint efficient estimator $\hat{\phi}$, the orthogonalized sample moments $T^{1/2}\tilde{g}_P$ converge in distribution to $\mathcal{N}(0, \mathcal Q_P)$. The variance matrix $\mathcal Q_P$, which explicitly accounts for the parameter estimation error through the projected Jacobian $\tilde{G}_P$, has generic rank $q_P$. Therefore, with analogous reasoning for the non-Gaussianity block, both test statistics satisfy:
\begin{equation}\label{eq:subset_chi2}
J_{\mathrm{prx}} \xrightarrow{d} \chi^2(q_P), \qquad J_{\mathrm{NG}} \xrightarrow{d} \chi^2(q_{\mathrm{NG}}).
\end{equation}
The degrees of freedom equal the number of restrictions in each specific block: $q_P$ and $q_{NG}$ respectively. The penalty for estimating the $p_\phi$ parameters is already integrated into the construction of $\mathcal Q_P$ and $Q_{\mathrm{NG}}$ via the second term in equation \eqref{eq:QPP}. This allows the generalized inverse to operate on the full $q_P$-dimensional (or $q_{NG}$-dimensional) covariance structure. While the statistics are asymptotically insensitive under the joint null, mis-specification under the alternative, (for instance, endogenous instruments or dependent structural shocks) in one block will transmit contamination to both statistics through the shared parameter estimate $\hat{\phi}$. This property is utilized as a diagnostic tool in
Section \ref{sec:results}. We formalize the limiting distributions in the following proposition:
\begin{proposition}[Limiting distributions of the test statistics]
\label{prop:limitdist}
Under Assumptions \ref{as:validproxy}-\ref{as:gau} where $G_{NG}$ has full column rank, with $H_0\colon
\mathbb{E}[g(\phi_0)] = 0$, and consistent estimators
$\hat{\mathcal Q}_P \xrightarrow{p} \mathcal Q_P$ and
$\hat{\mathcal Q}_{NG} \xrightarrow{p} \mathcal Q_{NG}$:
\begin{enumerate}[\upshape(i)]
\item $J_{\mathrm{prx}} \equiv T\,\tilde{g}_{P,T}'\hat{\mathcal Q}_P^{-1}\tilde{g}_{P,T}
\xrightarrow{d} \chi^2(q_P)$.
\item If $\mathrm{rank}(G_P) = p_\phi$, then
$J_{NG} \equiv T\,\tilde{g}_{NG,T}'\hat{\mathcal Q}_{NG}^{-1}
\tilde{g}_{NG,T} \xrightarrow{d} \chi^2(q_{NG})$.
\item In general, let $\hat\Pi$ be the orthogonal projector onto
the column space of $\hat{\mathcal Q}_{NG}$, consistently estimated from
the spectral decomposition of $\hat{\mathcal Q}_{NG}$ by retaining
eigenvectors with eigenvalues above a threshold $\tau_T \to
0$ with $T^{1/2}\tau_T \to \infty$. Then,
\begin{equation}
J_{NG} \equiv T\,\tilde{g}_{NG,T}' \ \hat{\Pi} \ \hat{\mathcal Q}_{NG}^+ \
\hat{\Pi} \ \tilde{g}_{NG,T}
\xrightarrow{d} \chi^2(r_{NG}),
\qquad r_{NG} \equiv q_{NG} - (p_\phi - \mathrm{rank}(G_P)).
\label{eq:JNG}
\end{equation}
\end{enumerate}
\end{proposition}
\noindent \textit{Proof}: See Appendix \ref{app:limitingdistributions}.
\begin{remark}[Compression of degrees of freedom] \label{rem:df_compress}
Both the test statistics, $J_{\mathrm{prx}}$ and $J_{NG}$, have standard chi-squared limits with degrees of freedom equal to the number of restrictions in the tested block, only if the other block delivers full identification of $\phi_0$. If the proxy block fails to identify $\phi_0$ on its own due to insufficient instruments, the non-Gaussian block must drive the identification, and the effective degrees of freedom of $J_{NG}$ compress below $q_{NG}$. The same logic applies symmetrically to $J_{\mathrm{prx}}$ when the non-Gaussian block fails to identify $\phi_0$ on its own. This is because the orthogonalization in \eqref{eq:gphat} and \eqref{eq:gnghat} projects out the component of the tested moment vector that is linearly predictable from the other block. If the other block has full column rank $p_\phi$, then all local perturbations of $\phi$ are detectable through it, so the projection removes all directions along which $\phi$ can fluctuate without violating the tested block's restrictions, leaving $q_P$ (or $q_{NG}$) effective restrictions. However, if the other block has deficient rank, then some local perturbations of $\phi$ are undetectable through it, so the projection preserves those directions in the tested moment vector, reducing its effective rank and degrees of freedom by the dimension of the unidentified subspace.
This implies that effective, generalized asymptotic distribution of the test statistics are:
\begin{equation}
J_{\mathrm{prx}} \xrightarrow{d} \chi^2(q_P - (p_\phi - \mathrm{rank}(G_{NG}))),
\qquad
J_{\mathrm{NG}} \xrightarrow{d} \chi^2(q_{NG} - (p_\phi - \mathrm{rank}(G_P))).
\label{eq:df_subset}
\end{equation}
This characteristic can be salient in SVAR applications with partial identification through external instruments, where the number of rotation angles $p_\phi$ grows quadratically with the number of variables, while the available valid instruments are typically small. As a result, in empirical applications, the test statistic $J_{NG}$ becomes conservative, and is not always a reliable standalone test of the non-Gaussian
specification. In this regime $J_{\mathrm{prx}}$ is the operative diagnostic: the
non-Gaussian block retains full column rank (Lemma~\ref{lem:ng_set}) and anchors
estimation regardless of proxy strength, so the test for proxy exogeneity stays valid
even when the instruments are used for partial identification.
\end{remark}
\subsection{Residual-based moving block bootstrap}\label{sec:bootstrap}
The chi-squared approximations in \eqref{eq:j_joint} and
\eqref{eq:subset_chi2} rely on the Gaussian limit of the standardized sample moments. However, the weighting matrix involves up to fourth-order cumulants. Also, as discussed in Remark \ref{rem:df_compress}, the asymptotic degrees of freedom for the test statistics also depend on the rank of the Jacobian matrices of each moment block. This invalidates the use of critical values from asymptotic $\chi^2$ distributions in empirical settings with finite samples.
We adopt the residual-based moving block bootstrap procedure of \citet{bruggemann_inference_2016}, which also allows for conditional heteroskedasticity. To preserve the empirical dependence structure between the residuals and proxies $(\hat{u}_t, m_t)$, we resample
overlapping blocks of the joint residual-proxy pairs \citep{jentsch_asymptotically_2022}.
Let $\hat{\phi}$ denote the original-data two-step estimator and
$\bar{g}_T(\hat{\phi})$ the associated sample moment average. To efficiently obtain the bootstrap analog of the sample moments,
we use the $k$-step iterative estimation procedure of \citet{andrews_2002}.
Moment recentering through \citet{hall_bootstrap_1996} subtracts $\bar{g}_T(\hat{\phi})$ from
each bootstrap moment average, so that the bootstrap data-generating
process satisfies the over-identifying restrictions exactly at
$\hat{\phi}$. With recentering, the bootstrap correctly mimics the non-standard limit
distribution due to the
degrees-of-freedom compression.
The detailed procedure can be found in Appendix \ref{appendix:mbb}.
By repeating the bootstrap procedure over $N_B$ replications, we obtain the empirical distributions of $(J_{\mathrm{joint}}^*, J_{\mathrm{prx}}^*, J_{NG}^*)$.
Hence, we can compute bootstrap $p$-values as the proportion of bootstrap replications
exceeding the corresponding sample statistic.
\section{Monte Carlo evidence}
\label{sec:results}
We assess the finite-sample properties of the hybrid GMM
estimator and the specification tests via a Monte Carlo simulation study.
\subsection{Data-generating process}
The baseline DGP is a stable VAR(2) with $n = 3$ variables. The true
structural impact matrix is $B_0 = H_0 Q(\phi_0)$, with $H_0 = I_3$ and
rotation angles $\phi_0 = (0.80,\, {-0.40},\, 1.20)'$.
Under the null hypothesis of valid specification, the structural shocks\footnote{The choice of the underlying distributions is not consequential; the simulation results still hold with other skewed, heavy-tailed distributions including generalized Student-$t$ and other hyperbolic distributions.} are mutually independent\footnote{Independence is imposed only in the simulation design; it is a special case of Assumption~\ref{as:indep}, which requires only uncorrelatedness and diagonal third- and fourth-order cumulant tensors.} and
non-Gaussian, satisfying the moment conditions for identification via
co-skewness and co-kurtosis, see Section \ref{subsec:nongaussianconditions}. Each shock is normalized to zero mean and
unit variance:
\begin{itemize}
\item {Shock 1} is drawn from a Normal-Inverse Gaussian
distribution with parameters $(\alpha, \beta, \delta) =
(1.6,\, 0.8,\, 1.6)$, yielding theoretical skewness
$\kappa_3 \approx 1.01$ and excess kurtosis $\kappa_4 \approx 2.70$.
\item {Shock 2} is drawn from a standardized $\chi^2(7)$
distribution, with $\kappa_3 = \sqrt{8/7} \approx 1.07$ and
$\kappa_4 = 12/7 \approx 1.71$.
\item {Shock 3} is drawn from a standardized $\chi^2(3)$
distribution, with $\kappa_3 = \sqrt{8/3} \approx 1.63$ and
$\kappa_4 = 4.00$.
\end{itemize}
The instruments are constructed as $m_{it} = \psi_{i}\,\varepsilon_{it}
+ \sigma_v v_{it}$, where $v_{it} \overset{\text{i.i.d.}} {\sim}
\mathcal{N}(0,1)$. The relevance matrix is $\Psi = \operatorname{diag}(0.70,\, 0.60,\, 0.80)$ and the noise standard deviation is
$\sigma_v = 0.50$ for all the instruments.
Therefore, the dimension of the rotational angle parameters is $p_{\phi} = 3$, with moment conditions $q_{NG} = 9$, $q_P = 6$, giving us the total of $q = 15$.
After estimating the rotation angles $\hat{\phi}$, see Section \ref{subsec:twostepgmm}, we construct the test statistics $J_{\mathrm{joint}}$, $J_{\mathrm{prx}}$, and $J_{\mathrm{NG}}$ as described in Section \ref{sec:tests}. To obtain bootstrap critical values, we implement a residual-based moving block bootstrap procedure, detailed in Appendix \ref{appendix:mbb}.
We report the Monte Carlo results over sample sizes $T \in \{200,\, 300,\, 500,\,
1000\}$, with $M = 500$ replications and $N_{B} = 999$ bootstrap replications.
\begin{remark}[Computational Implementation and Asymptotic Invariance]
\label{rem:comp_invariance}
Proposition \ref{prop:identification_unified} establishes global point identification
over the unconstrained rotation space $SO(n)$. In finite samples, however,
optimization over higher-order moments is sensitive to initial conditions near
Givens gimbal lock boundaries ($\phi = \pi/2$). We initialize
the numerical solver at the unweighted non-Gaussian identified preliminary estimate
$\hat{\phi}_{\mathrm{prelim}} = \arg \ \min_\phi \bar{g}_{NG}(\phi)'\bar{g}_{NG}(\phi)$
and restrict the search to the principal geometric branch
$\phi \in \Phi_0 \equiv (-\pi/2,\, \pi/2)^{p_\phi}\subset\Phi$, which eliminates
topologically isomorphic aliases without any loss of generality. The criterion is
then minimized by a constrained solver on $\Phi_0$, started both at
$\hat{\phi}_{\mathrm{prelim}}$ and at draws from the interior of the branch, so
that the reported estimate is the best of several local solutions rather than the
first one reached.
\end{remark}
\begin{remark}[Spectral Regularization]
The matrices $\mathcal Q_P$ and $\mathcal Q_{NG}$ for the test statistics \eqref{eq:subset_stats} are constructed from higher-order moment conditions whose finite-sample covariance is usually ill-conditioned.
Consequently, inversion without regularization may amplify the smallest eigenvalues,
inflating the test statistics and distorting size. We regularize via spectral
truncation: eigenvalues of $\mathcal Q$ below the threshold $\tau = 10^{-4}$ are set
to zero prior to inversion, retaining only the subspace along which the matrix
is well-conditioned \citep{antoine_efficient_2007}. This truncation controls
the effective rank of $\mathcal Q^+$ without distorting the limiting distribution along
the retained eigenvalue directions.
\end{remark}
\subsection{Baseline results}
\label{sec:baseline}
\begin{table}[htbp]
\centering
\caption{Estimates of $\phi_0$ under $H_0$: Non-Gaussian Shocks with Valid Proxies}
\label{tab:phi_estimates_h0}
\begin{tabular}{lccccc}
\toprule
& True Value & $T = 200$ & $T = 300$ & $T = 500$ & $T = 1000$ \\
\midrule
\multicolumn{6}{l}{\textit{Estimates and their standard errors}} \\[3pt]
$\hat{\phi}_1$ & $0.80$ & 0.8046 & 0.8005 & 0.8007 & 0.8006 \\
& & (0.0300) & (0.0250) & (0.0201) & (0.0145) \\
$\hat{\phi}_2$ & $-0.40$ & $-$0.3958 & $-$0.3955 & $-$0.3996 & $-$0.3985 \\
& & (0.0315) & (0.0266) & (0.0212) & (0.0153) \\
$\hat{\phi}_3$ & $1.20$ & 1.1991 & 1.2017 & 1.2025 & 1.2002 \\
& & (0.0310) & (0.0259) & (0.0207) & (0.0150) \\[6pt]
\bottomrule
\end{tabular}
\begin{minipage}{0.92\textwidth}
\vspace{4pt}
\scriptsize
\textit{Notes:} Average estimates and their standard errors are based on $M = 500$ Monte Carlo replications with $B = 999$ bootstrap replications. The DGP is a stationary VAR(2) with $n = 3$ variables and true rotation parameters $\phi_0 = (0.80, -0.40, 1.20)'$. Structural shocks are drawn from NIG, $\chi^2(7)$, and $\chi^2(3)$ distributions. Proxies are valid with relevance $\Psi = (0.70, 0.60, 0.80)'$ and noise $\sigma_v = 0.50$.
\end{minipage}
\end{table}
\begin{table}[htbp]
\centering
\caption{Size of Specification Tests under $H_0$: Non-Gaussian Shocks with Valid Proxies}
\label{tab:size_h0}
\begin{tabular}{lcccc}
\toprule
& $T = 200$ & $T = 300$ & $T = 500$ & $T = 1000$ \\
\midrule
\multicolumn{5}{l}{\textit{Panel A: Empirical Rejection Rates (nominal size $= 0.05$)}} \\[3pt]
Joint specification test & 0.034 & 0.048 & 0.044 & 0.056 \\
Proxy validity test & 0.044 & 0.028 & 0.036 & 0.036 \\
Non-Gaussianity test & 0.024 & 0.044 & 0.028 & 0.044 \\[6pt]
\multicolumn{5}{l}{\textit{Panel B: Mean $J$-statistics}} \\[3pt]
$J_{\text{joint}}$ (bootstrap) & 24.471 & 21.352 & 17.236 & 12.714 \\
$J_{\text{joint}}$ (estimate) & 20.361 & 18.001 & 14.077 & 12.218 \\[3pt]
$J_{\text{prx}}$ (bootstrap) & 18.379 & 13.680 & 10.315 & 6.555 \\
$J_{\text{prx}}$ (estimate) & 14.101 & 12.455 & 8.421 & 6.069 \\[3pt]
$J_{\text{NG}}$ (bootstrap) & 32.052 & 24.031 & 17.549 & 11.941 \\
$J_{\text{NG}}$ (estimate) & 20.170 & 17.535 & 13.823 & 10.989 \\
\bottomrule
\end{tabular}
\begin{minipage}{0.92\textwidth}
\vspace{4pt}
\footnotesize
\textit{Notes:} Panel A reports the fraction of $M = 500$ Monte Carlo replications in which the null hypothesis is rejected at the 5\% nominal level. Panel B reports average $J$-statistics across replications. ``Bootstrap'' denotes the average bootstrap $J$-statistic ($N_B = 999$); ``estimate'' denotes the average of the $J$-statistic estimates. The joint test evaluates all overidentifying restrictions; the NG test evaluates non-Gaussianity moment conditions conditional on proxy validity; the proxy validity test evaluates proxy validity conditional on non-Gaussianity.
\end{minipage}
\end{table}
\begin{table}[htbp]
\centering
\caption{Estimates of $\phi_0$ under $H_1$: Invalid Proxy}
\label{tab:phi_estimates_h1}
\begin{tabular}{lccccc}
\toprule
& True Value & $T = 200$ & $T = 300$ & $T = 500$ & $T = 1000$ \\
\midrule
\multicolumn{6}{l}{\textit{Estimates and their standard errors}} \\[3pt]
$\hat{\phi}_1$ & $0.80$ & 0.8277 & 0.8372 & 0.8228 & 0.8034 \\
& & (0.0384) & (0.0296) & (0.0219) & (0.0152) \\
$\hat{\phi}_2$ & $-0.40$ & $-$0.4014 & $-$0.3671 & $-$0.3539 & $-$0.3427 \\
& & (0.0475) & (0.0402) & (0.0328) & (0.0234) \\
$\hat{\phi}_3$ & $1.20$ & 0.6723 & 0.8381 & 0.9966 & 1.1371 \\
& & (0.0494) & (0.0401) & (0.0309) & (0.0223) \\[6pt]
\bottomrule
\end{tabular}
\begin{minipage}{0.92\textwidth}
\vspace{4pt}
\footnotesize
\textit{Notes:} Average estimates and their standard errors are based on $M = 500$ Monte Carlo replications with $B = 999$ bootstrap replications. The DGP is identical to the $H_0$ specification except that proxy $m_2$ is mis-specified: $m_{2t} = 0.4 \, \varepsilon_{1t} + \sigma_v \nu_{2t}$, violating exogeneity by responding to shock 1 instead of shock 2. The substantial bias in $\hat{\phi}_3$ reflects the contamination of the proxy moment conditions.
\end{minipage}
\end{table}
\begin{table}[htbp]
\centering
\caption{Power of Specification Tests under $H_1$: Invalid Proxy}
\label{tab:power_h1}
\begin{tabular}{lcccc}
\toprule
& $T = 200$ & $T = 300$ & $T = 500$ & $T = 1000$ \\
\midrule
\multicolumn{5}{l}{\textit{Panel A: Empirical Rejection Rates (nominal size $= 0.05$)}} \\[3pt]
Joint specification test & 0.708 & 0.894 & 0.972 & 1.000 \\
Proxy validity test & 0.606 & 0.810 & 0.926 & 0.994 \\
Non-Gaussianity test & 0.072 & 0.288 & 0.642 & 0.942 \\[6pt]
\multicolumn{5}{l}{\textit{Panel B: Mean $J$-statistics}} \\[3pt]
$J_{\text{joint}}$ (bootstrap) & 28.821 & 24.334 & 19.889 & 14.223 \\
$J_{\text{joint}}$ (estimate) & 90.325 & 116.339 & 165.058 & 292.212 \\[3pt]
$J_{\text{prx}}$ (bootstrap) & 30.403 & 22.313 & 17.051 & 9.107 \\
$J_{\text{prx}}$ (estimate) & 94.347 & 117.791 & 218.212 & 293.421 \\[3pt]
$J_{\text{NG}}$ (bootstrap) & 57.127 & 37.248 & 23.962 & 14.197 \\
$J_{\text{NG}}$ (estimate) & 46.739 & 61.246 & 88.026 & 124.158 \\
\bottomrule
\end{tabular}
\begin{minipage}{0.92\textwidth}
\vspace{4pt}
\scriptsize
\textit{Notes:} Panel A reports the fraction of $M = 500$ Monte Carlo replications in which the null hypothesis is rejected at the 5\% nominal level. Panel B reports average $J$-statistics across replications. ``Bootstrap'' denotes the average bootstrap $J$-statistic ($N_B = 999$); ``estimate'' denotes the average of the $J$-statistic estimates. Under $H_1$, the proxy validity and joint test statistics diverge sharply from their bootstrap counterparts, reflecting growing power as $T$ increases. The NG bootstrap critical values are heavily inflated at small $T$ due to the contamination of the rotation estimates, suppressing NG rejection rates despite the underlying shocks remaining non-Gaussian.
\end{minipage}
\end{table}
\subsubsection{Estimates under $H_0$ and $H_1$}
Table \ref{tab:phi_estimates_h0} reports the average estimates and standard errors for $\phi_0$, over the simulations, under the
baseline DGP. The estimator is effectively
unbiased at all sample sizes and
standard errors decline monotonically.
Table \ref{tab:phi_estimates_h1} reports estimates under proxy
invalidity, where instrument $m_2$ violates the exogeneity condition:
$m_{2t} = 0.4\,\varepsilon_{1t} + \sigma_v v_{2t}$, i.e. it loads on shock 1 rather
than its designated target, shock 2. Table~\ref{tab:phi_estimates_h1} shows
severe bias in $\hat{\phi}_2$ and $\hat{\phi}_3$. Both are consistent with the standard result that GMM estimators converge to a
pseudo-true value rather than $\phi_0$ under moment mis-specification, as shown in
\citet{white_1982, hall_inoue_2003}.
\subsubsection{Size under $H_0$}
Table \ref{tab:size_h0}, Panel A reports empirical rejection rates at
the 5\% nominal level. All three tests are slightly conservative at modest sample sizes. The large mean bootstrap $J$-statistics in small samples reflect the inefficiency of the estimates when the underlying shocks are heavy-tailed; they converge toward their sample counterparts as $T$ grows. The mean $J$ statistics, both estimated and bootstrap, track the means of their limiting distributions closely: $J_{\mathrm{prx}}$ against $\chi^2(q_P)$ and $J_{\mathrm{NG}}$ against $\chi^2(q_{NG})$, as stated in Proposition~\ref{prop:limitdist}.
\subsubsection{Power under $H_1$}
In Table \ref{tab:power_h1}, all three tests accumulate power rapidly as $T$ increases. At $T=500$,
rejection rates are $0.972$, $0.926$, and $0.642$ for the joint, proxy
validity, and non-Gaussianity tests respectively.
A remark regarding the rejection rates of the non-Gaussianity test is in order.
The test for non-Gaussianity rejects despite the
true shocks are non-Gaussian. The cause is parameter contamination, and it is expected. The joint GMM objective pools proxy and non-Gaussianity
moment conditions, so the invalid proxy moments displace $\hat{\phi}$ from
$\phi_0$. The recovered structural shocks $\hat{\varepsilon}_t =
\hat{B}_0^{-1}(\hat{\phi})\,u_t$ do not match the true shocks,
and their higher-order cross-moments violate the moment conditions even though the underlying DGP is non-Gaussian. This implies that
$g_{NG}(y_t, \hat{\phi})$ has nonzero population expectation under $H_1$,
generating power in the non-Gaussianity test through an indirect spillover
channel. This mechanism is consistent with the partitioned $J$-statistic
structure: the Schur-complement decomposition yields orthogonal test statistics
only under the joint null. Under a mis-specification of the proxy moment conditions, only parameter contamination propagates through the shared space to all components \citep{newey_generalized_1985}.
The power ordering $J_{\mathrm{prx}} \gg J_{\mathrm{NG}}$ is uniform across all sample sizes and supports the theoretical interpretation. The ordering is even more pronounced for the point-estimates of $J$-statistics, which diverge sharply from their bootstrap counterparts as $T$ increases. The proxy validity statistic reaches $218.1$ at $T=500$ against a bootstrap mean of $17.1$, whereas the non-Gaussianity statistic reaches $88.0$ against a bootstrap mean of $23.9$. When all three
tests reject and the $J_{\mathrm{prx}}$ dominates, the pattern is consistent with
proxy invalidity as the primary source of moment mis-specification.
\subsection{The case of weak instruments}
\label{sec:weakproxy}
One of the central concerns in a proxy-SVAR is the sensitivity of inference
to the strength of the external instruments. Weak instruments can distort the sampling distribution of GMM-based estimators and invalidate standard asymptotic
approximations, see \citet{staiger_instrumental_1997, stock_gmm_2000}. While the relevance of instrument is
routinely verified in applied work through first-stage $F$-statistics, limitations of such pre-tests in terms of low power and distorted subsequent inference are documented in \citet{roth_pretest_2022}. We show that even under weak proxies, the hybrid framework continues to recover the parameters consistently (Proposition~\ref{prop:localtozero}), because the higher-order moment conditions anchor the identification. The specification tests, however, exhibit asymmetric size distortions caused by anchor failure in the conditional testing procedure. $J_{\mathrm{NG}}$ becomes severely conservative; $J_{\mathrm{prx}}$ retains reliable calibration and remains a valid test of proxy orthogonality.
We adopt the local-to-zero framework for proxy
relevance i.e., a Pitman drift $c/\sqrt{T}$ with
$c = 2.24$ across the sample sizes. At these magnitudes, the proxy
Jacobian $G_{P,T} = O(T^{-1/2})$ contributes negligible identifying
information. Results\footnote{Similar results with strong proxies and weak non-Gaussian shocks can be found in Appendix \ref{appendix:weak_ng_sim}. We also provide results for doubly weak-identification, i.e. weak proxies and weak non-Gaussianity. In this case, the estimates remain unbiased with increasing sample size, albeit with larger standard errors compared to strong identification case.} are reported in
Tables \ref{tab:phi_estimates_weak} and \ref{tab:size_weak}.
\begin{table}[htbp]
\centering
\caption{Parameter Estimates under $H_0$: Weak Proxies (Pitman Drift $c = 2.24$)}
\label{tab:phi_estimates_weak}
\begin{tabular}{lccccc}
\toprule
& True Value & $T = 200$ & $T = 300$ & $T = 500$ & $T = 1000$ \\
\midrule
\multicolumn{6}{l}{\textit{Estimates and their standard errors}} \\[3pt]
$\hat{\phi}_1$ & $0.80$ & 0.7851 & 0.7754 & 0.8019 & 0.8007 \\
& & (0.0597) & (0.0526) & (0.0439) & (0.0332) \\
$\hat{\phi}_2$ & $-0.40$ & $-$0.3621 & $-$0.3956 & $-$0.3872 & $-$0.3959 \\
& & (0.0639) & (0.0580) & (0.0507) & (0.0387) \\
$\hat{\phi}_3$ & $1.20$ & 1.1789 & 1.1870 & 1.2013 & 1.2024 \\
& & (0.0563) & (0.0492) & (0.0405) & (0.0307) \\[6pt]
\bottomrule
\end{tabular}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize
\textit{Notes:} Results based on $M = 500$ Monte Carlo replications with $N_B = 999$ bootstrap replications. Proxy relevance follows a Pitman drift: $\Psi_i = c / \sqrt{T}$ with $c = 2.24$ for all instruments, yielding $\Psi \approx (0.158, 0.129, 0.100, 0.071)$ for $T = (200, 300, 500, 1000)$. All other DGP parameters are identical to the strong-proxy $H_0$ specification.
\end{minipage}
\end{table}
\subsubsection{Consistent estimation}
Table \ref{tab:phi_estimates_weak} shows that the joint GMM estimator recovers the parameters consistently despite severe proxy weakness. Though we do note some finite-sample bias at $T=200$, it
reflects the elevated variance of higher-order sample moments under
heavy tails. The estimates converge toward $\phi_0$ as $T$ increases, albeit with higher standard errors than the baseline estimates of Table \ref{tab:phi_estimates_h0} with strongly relevant proxies.
This is the mechanism of Proposition~\ref{prop:localtozero}. As the proxy block Jacobian $G_{P,T} \xrightarrow{T \to \infty} 0$, the identification burden shifts entirely to the
non-Gaussian block. The co-skewness and symmetric co-kurtosis conditions
provide full-rank local curvature in a neighborhood of $\phi_0$,
ensuring the variance matrix, $\mathcal{V}_{NG}^* =\bigl(G_{NG}'\,\Sigma_{NG}^{-1}\,G_{NG}\bigr)^{-1}$ remains asymptotically non-singular. Simultaneously, the
shrinking, but a nonzero proxy relevance imposes a strictly positive
penalty on each element of the finite discrete set
$\mathcal{B} \setminus \{\phi_0\}$, resolving the global permutation
ambiguity that the higher-order moments alone cannot eliminate.
\subsubsection{Robust Anderson-Rubin confidence intervals}
\label{subsec:sim_bootstrap_ci}
In the presence of weak instruments, standard Wald-type confidence intervals around the point estimates are not asymptotically valid. Anderson-Rubin (AR) confidence
sets, see \citet{montiel_olea_inference_2021,
jentsch_asymptotically_2022}, constructed by inverting the AR statistic over the
parameter space, provide a valid alternative. They maintain
correct asymptotic coverage uniformly over proxy strength,
including the local-to-zero regime.
To assess the inferential contribution of the non-Gaussian
moment conditions under weak proxy identification, we compare the AR confidence sets for the rotation parameters $\phi$ under two criteria, \textit{Proxy-only} and \textit{Proxy+NG}. The former involves only the proxy orthogonality moment conditions in the GMM criterion for the AR statistic whereas the latter includes all the moment conditions (non-Gaussian and proxy). Consequently, the asymptotic critical values are obtained from $\chi^2(q_p)$ and $\chi^2(q_p + q_{NG})$ distributions. This allows us to compare the effect of additional identifying strength of the higher-order moments conditions, in terms of the length of weak identification robust confidence sets, under local-to-zero proxy relevance.
For each simulated dataset, the AR statistic is evaluated over a fine $p_{\phi}$-dimension grid of candidate rotation angle vectors $\phi \in \Phi_0$. At each grid point, the AR statistic takes the form
\begin{equation}
AR_T(\phi_{(i)}) = T \cdot \bar{g}(\phi_{(i)})' \hat{S}(\phi_{(i)})^{-1} \bar{g}(\phi_{(i)}),
\end{equation}
where $\bar{g}(\phi_{(i)})$ is the sample moment vector at $i^{th}$ grid point and $\hat{S}(\phi_{(i)}) = T^{-1}\sum_{t=1}^T (g_t(\phi_{(i)}) - \bar{g}(\phi_{(i)}))(g_t(\phi_{(i)}) - \bar{g}(\phi_{(i)}))'$ is the consistent variance estimator\footnote{In the empirical application, $\widehat{S}(\phi_{(i)})$ is replaced by the residual based moving block bootstrap estimator $\widehat{S}_{MBB}(\phi_{(i)})$
to account for serial dependence.}. Under the null that $\phi_{(i)}$ is the true rotation, $AR_T(\phi_{(i)})$ is asymptotically distributed as $\chi^2(q)$, where $q$ is the number of moment conditions.
The AR confidence set is the collection of parameter values not
rejected by the test at $1 - \alpha$ level :
\begin{equation}
\mathcal{C}_{1-\alpha} = \bigl\{\phi_{(i)} \in \Phi_0 :
AR_T(\phi_{(i)}) \leq \chi^2_{q,\,1-\alpha}\bigr\}.
\label{eq:AR_set}
\end{equation}
In practice, $\mathcal{C}$ is approximated by
evaluating $AR_T(\phi_{(i)})$ over a fine grid of $1,000,000$ ($100^{p_{\phi}}$)
candidate vectors covering $\Phi_0$, and retaining all grid
points at which the criterion does not exceed the critical
value. Given that such grid-based inferential methods are computationally prohibitive, we include the simulation results for a sample size $T = 1000$, for $100$ Monte Carlo replications.
\begin{table}[htbp]
\centering
\caption{Anderson-Rubin 90\% Confidence Sets with HGMM standard CIs}\label{tab:compare_sim_ARCI}
\begin{tabular}{lcccc}
\toprule
Parameter & Mean Estimate & Proxy-only & Proxy+NG & HGMM standard CI \\
\midrule
$0.80$ & $0.8007$ & $[0.2221, 1.3328]$ & $[0.7140, 0.9044]$ & $[0.7447, 0.8567]$\\[1pt]
\quad & & $1.1107$ & $0.1904$ & 0.1120 \\[4pt]
$-0.40$ & $-0.3959$ & $[-0.8409, 0.0793]$ & $[-0.5077, -0.2697]$ & $[-0.4603, -0.3305]$\\[1pt]
\quad & & $0.9202$ & $0.2380$ & 0.1298 \\[4pt]
$1.20$ & $1.2024$ & $[-1.5708, 1.5708]$ & $[1.1265, 1.2852]$ & $[1.1509, 1.2539]$\\[1pt]
\quad & & $3.1416$ & $0.1587$ & 0.1030 \\
\bottomrule
\end{tabular}
\vspace{8pt}
\begin{minipage}{0.95\textwidth}
\scriptsize
\textit{Notes.} The 90\% AR confidence sets are median sets, across the monte carlo simulations. (with their lengths below). The main comparison is between the \textit{Proxy-only} and \textit{Proxy+NG} criterions. The former involves only the proxy orthogonality moment conditions in the AR statistic whereas the latter includes all the moment conditions (non-Gaussian and proxy). The HGMM standard CI is computed as: $[\hat{\phi} - 1.645 \times \hat{\sigma}_{\phi}, \hat{\phi} + 1.645 \times \hat{\sigma}_{\phi}]$, where $\hat{\phi}$ and $\hat{\sigma}_{\phi}$ are the average of the estimates (across the monte carlo replications) and their standard errors, respectively. Results based on $M = 100$ Monte Carlo replications with $N_B = 999$ bootstrap replications. Proxy relevance follows a Pitman drift: $\Psi_i = c / \sqrt{T}$ and $c = 2.24$, with sample size, $T = 1000$. All other DGP parameters remain unchanged from the $H_0$ specification.
\end{minipage}
\end{table}
The results demonstrate a breakdown of proxy-only
identification under weak instruments, and its recovery
through non-Gaussianity. For $\phi_3$, the \textit{Proxy-only}
AR set spans the entire feasible
rotation parameter space, indicating that the proxy criterion
fails to reject any parameter value and is entirely
uninformative. As shown in Table \ref{tab:compare_sim_ARCI}, all three \textit{Proxy+NG}
sets are centered near the true parameter values ($0.80$,
$-0.40$, $1.20$) and are comparable in length to the HGMM
Wald intervals, indicating that the NG conditions restore the informativeness regarding
the true parameter. The
\textit{Proxy+NG} AR criterion remains informative and
well-centered because the higher-order moments
conditions identify the rotation parameters independently
of proxy strength, as established in
Proposition \ref{prop:localtozero}. The joint hybrid
criterion eliminates the identification failure without sacrificing the validity of the identification-robust coverage.
\subsubsection{Asymmetric size distortions in specification tests}
While point identification is preserved, Table \ref{tab:size_weak}
reveals a pronounced asymmetry in the
size of the specification tests. The asymmetry
originates in the conditional structure of each test, i.e., which
block of the joint moment system anchors the parameter estimation.
Panel A of Table \ref{tab:size_weak} shows that $J_{\mathrm{NG}}$ almost never rejects at all sample sizes, against a nominal significance level of $5\%$. Panel B identifies the
mechanism where the mean sample statistic $\hat{J}_{\mathrm{NG}}$ is stable between
$12.5$ and $9.9$, while their bootstrap analogs are massively inflated
at small samples. The bootstrap mean of $J_{\mathrm{NG}}$ falls from $147.0$ at
$T=200$ to $49.6$ at $T=1000$.
\begin{table}[htbp]
\centering
\caption{Size of Specification Tests under $H_0$: Weak Proxies (Pitman Drift $c = 2.24$)}
\label{tab:size_weak}
\begin{tabular}{lcccc}
\toprule
& $T = 200$ & $T = 300$ & $T = 500$ & $T = 1000$ \\
\midrule
\multicolumn{5}{l}{\textit{Panel A: Empirical Rejection Rates (nominal size $= 0.05$)}} \\[3pt]
Joint specification test & 0.000 & 0.008 & 0.006 & 0.040 \\
Proxy validity test & 0.018 & 0.032 & 0.030 & 0.040 \\
Non-Gaussianity test & 0.000 & 0.000 & 0.000 & 0.006 \\[6pt]
\multicolumn{5}{l}{\textit{Panel B: Mean $J$-statistics}} \\[3pt]
$J_{\text{joint}}$ (bootstrap) & 25.313 & 21.309 & 17.749 & 14.020 \\
$J_{\text{joint}}$ (estimate) & 16.524 & 16.017 & 13.967 & 12.958 \\[3pt]
$J_{\text{prx}}$ (bootstrap) & 11.527 & 9.218 & 7.757 & 6.814 \\
$J_{\text{prx}}$ (estimate) & 9.384 & 8.522 & 7.344 & 6.794 \\[3pt]
$J_{\text{NG}}$ (bootstrap) & 147.073 & 113.351 & 79.632 & 49.601 \\
$J_{\text{NG}}$ (estimate) & 12.571 & 11.910 & 10.714 & 9.939 \\
\bottomrule
\end{tabular}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\scriptsize
\textit{Notes:} Panel A reports the fraction of $M = 500$ Monte Carlo replications in which the null hypothesis is rejected at the 5\% nominal level. Panel B reports average $J$-statistics across replications. ``Bootstrap'' denotes the average bootstrap $J$-statistic ($N_B = 999$); ``estimate'' denotes the average of the $J$-statistic estimates. Under weak proxy asymptotics ($\Delta_i = c/\sqrt{T}$, $c = 2.24$), the proxy moments become asymptotically uninformative. The NG bootstrap critical values are massively inflated due to the near-singular contribution of the proxy block to the long-run covariance matrix.
\end{minipage}
\end{table}
This failure to reject is a direct consequence of {anchor failure} in the
conditional testing procedure. The $J_{\mathrm{NG}}$ statistic evaluates the higher-order
moment conditions with a maintained hypothesis that the identifying strength is provided by the proxy block\footnote{This is a specific case where there are sufficient instruments to identify the entire system. As noted in Remark \ref{rem:df_compress}, if (instruments) $k < n-1$ (structural shocks), the proxy block cannot provide the entire identification and the distribution of the $J_{\mathrm{NG}}$ statistic faces compression in its (asymptotic) degrees of freedom.}.
Under strong proxy relevance, $G_P$ is full rank and absorbs the full
parameter estimation penalty, leaving the higher-order moments free to
test for over-identification. Under weak proxy relevance, $G_{P,T} \approx 0$, the joint GMM criterion obtains identifying curvature from the non-Gaussian
block alone. The sample statistic $\hat{J}_{NG}$ is
evaluated over a subspace overfitted to the higher-order moments,
compressing the realized statistic toward $q_{NG}-(p_{\phi}-rank(G_{P}))$ (Proposition \ref{prop:limitdist}).
Simultaneously, since the
proxy block contributes near-zero information to the joint long-run
covariance matrix, the restricted variance matrix $\mathcal Q_{NG}^{*}$
inherits extreme instability from these higher-order moments, generating
bootstrap values that are orders of magnitude larger than the
compressed sample statistic.
In contrast, Panel A of Table \ref{tab:size_weak} shows that $J_{\mathrm{prx}}$
maintains substantially better size, improving with $T$. The test
is moderately conservative at small $T$ but improves steadily. This is because, analogous to the $J_{\mathrm{NG}}$ test, the $J_{\mathrm{prx}}$ test relies on
the non-Gaussian block as its identifying anchor. The true DGP is strongly
non-Gaussian, so $G_{NG}$ maintains full column rank independently of
the proxy validity, identifying $\hat{\phi}$ without any contribution from $G_{P}$.
Hence, $J_{\mathrm{prx}}$ evaluates proxy orthogonality
against a correctly specified parameter space. The bootstrap distribution
of the test statistic accurately reflects the true sampling variation of
the proxy moments, providing valid inference.
\section{Empirical Applications}
\label{sec:empirical}
We demonstrate the potential of our HGMM framework with two empirical applications. The first identifies an \emph{oil news shock} from the OPEC-window
oil futures surprise of \citet{kanzig_macroeconomic_2021} after adopting the corrections suggested by \citet{kilian_how_2024}
(Section~\ref{subsec:emp_oil}); the second identifies a Euro-area
\emph{monetary policy shock} from the high-frequency interest-rate surprise of
\citet{altavilla_measuring_2019} (Section~\ref{subsec:emp_mp}). In each case, we
report identification-robust confidence sets for the impulse responses under two
criteria, \textit{Proxy-only} (the exclusion conditions alone) and \textit{HGMM} (the
exclusion conditions combined with the higher-order moment conditions), and we subject the
proxy exclusion restriction to a formal specification test. Each illustration is accompanied by a robustness
exercise, reported in the appendix: for the oil shock, we re-run the analysis on
the \citet{kanzig_macroeconomic_2021} baseline (original instrument and specification) in Appendix~\ref{appendix:kanzig_rob},
and for the monetary policy shock on the \citet{gertler_monetary_2015} specification in Appendix~\ref{appendix:gk}.
\subsection{Identification-robust inference}
\label{subsec:emp_arsets}
The headline comparison in both illustrations is between the
identification-robust confidence sets of the two criteria. For each rotation in
the target-shock subspace, we construct the Anderson--Rubin statistic and retain the
rotation if the statistic does not exceed its per-candidate bootstrap quantile. The confidence set for a given impulse response is the projection of this
acceptance region onto the corresponding coordinate, obtained by optimizing the impulse
response function subject to the acceptance constraint. Because the target-shock
responses depend only on the first column of the impact matrix, the inversion
reduces to the $n-1$ rotation angles that build that column, and the remaining
columns are not individually relevant. This is analogous to the
procedure validated in Section~\ref{subsec:sim_bootstrap_ci}; the full
construction is given in Appendix~\ref{sec:emp_arsets_cons}. The dimension of the search is
therefore $n-1$ rather than $p_\phi=n(n-1)/2$, and it grows linearly rather than
quadratically in the size of the system. For the six-variable oil system the
inversion runs over $5$ angles with $16$ moment conditions rather than over the
$15$ angles of the point estimate, and for the four-variable monetary system over
$3$ angles with $10$ conditions. Throughout,
bootstrap quantiles use a residual-based moving-block bootstrap (MBB) that jointly
resamples the recentered residuals and the proxy \citep{bruggemann_inference_2016, jentsch_asymptotically_2022}
and re-estimates the VAR within each draw, so that first-stage uncertainty
propagates into the structural objects. Confidence sets are reported at the
nominal $90\%$ level.
\subsection{Oil news shock}
\label{subsec:emp_oil}
We revisit the identification of oil news shocks in
\citet{kanzig_macroeconomic_2021}, who use changes in oil futures prices around
pre-scheduled OPEC announcements as an external instrument for the structural
shock to oil price expectations. The setting is well suited to our framework
because the relevance and construction of the monthly instrument have been
questioned, most directly by \citet{kilian_how_2024} but also by
\citet{mori_estimating_2024, cavaliere_bootstrap_2025}, so it is an ideal case
in which the identifying information from the higher-order moment conditions can be
consequential.
\subsubsection{Specification}\label{subsubsec:oil_spec}
Our main specification adopts the corrections of \citet{kilian_how_2024}, so
that the instrument is constructed consistently with the timing of the
reduced-form data. The instrument aggregates daily futures-price surprises
around OPEC announcements into a monthly series. However, a naive within-month sum is inconsistent with a VAR built on monthly
{average} prices, because a permanent surprise of size $\delta$ on trading
day $d_k$ of a month with $T_t$ trading days shifts the current month's average
by $\frac{T_t-d_k+1}{T_t}\,\delta$ and the next month's average by
$\frac{d_k-1}{T_t}\,\delta$. The corrected proxy weights each surprise by its
within-month timing and carries the residual weight into the following month. We
adopt this temporally weighted instrument, restrict the sample to 1989:M4
onwards where the trading-day data are reliable, and estimate a log-levels VAR
with the \citet{kilian_role_2014} oil-market lag order $p=24$.
The system has the six baseline variables: real oil price, world oil
production, world oil inventories, world industrial production, U.S. industrial
production, and U.S. CPI, so that $n=6$ and $k=1$. The reduced-form VAR is
\begin{equation}
y_t = c + \sum_{j=1}^{p} A_j\, y_{t-j} + u_t,
\qquad u_t \sim (0,\Sigma_u),
\label{eq:VAR_emp}
\end{equation}
estimated on 1989:M4-2017:M12; the structural innovations satisfy
$u_t = B\varepsilon_t$, where $\varepsilon_t$ meets Assumption~\ref{as:indep}.
Following Section~\ref{subsec:svarparam}, $B = H\,Q(\phi)$ with $H$ the Cholesky
factor of $\Sigma_u$ and $Q(\phi)\in SO(n)$ governed by the
$p_\phi = n(n-1)/2 = 15$ Givens angles, so $\varepsilon_t(\phi)=Q(\phi)'H^{-1}u_t$.
Indexing the oil news shock as $j=1$ and letting $m_t$ denote the instrument (robust $F = 8.9$)
the stacked moment vector combines the higher-order moment conditions of
Section~\ref{subsec:nongaussianconditions} with the proxy exclusion conditions,
giving $q_P = k(n-1)=5$ proxy conditions and $q_{NG}=75$ non-Gaussian conditions:
the co-skewness and symmetric co-kurtosis blocks of
Section~\ref{subsec:nongaussianconditions}, together
with the asymmetric co-kurtosis\footnote{\textit{Asymmetric co-kurtosis:} the
$n(n-1)$ fourth-order conditions
$\mathbb{E}[\varepsilon_{i,t}^{3}\,\varepsilon_{j,t}]=0$ for all ordered pairs
$(i,j)$ with $i\neq j$. They are not necessary for identification and are not part
of the moment block of Section~\ref{subsec:nongaussianconditions}; appending them
raises the count to $2n(n-1)+n(n-1)/2=75$. Their sample variance involves the cross moment
$\mathbb{E}[\varepsilon_{i,t}^{6}\varepsilon_{j,t}^{2}]$, which reduces to the
sixth-order marginal requirement $\mathbb{E}|\varepsilon_{i,t}|^{6+3\delta}<\infty$
under the factorization discussed in Section~\ref{subsec:nongaussianconditions};
some evidence in Appendix~\ref{appendix:evidence_nongauss} support the requirement
in this specification. Figure~\ref{fig:kl_arsets_nogrp2} shows the
identification-robust sets when the asymmetric co-kurtosis conditions are excluded.} conditions appended as a further source of information, so a total of
$q=80$ moment conditions and $p_\phi=15$ parameters. We estimate $\hat\phi$ by continuous-updating
GMM and normalize the impact matrix to a
positive diagonal; the impulse responses at horizon $i$ are
\begin{equation}
\hat{\Theta}_i = \bigl(R\,\hat{\mathcal A}^{\,i}\,R'\bigr)\hat{B},
\qquad i = 0,1,\dots,h,
\label{eq:IRF_emp}
\end{equation}
with $R=[I_n,\mathbf 0_{n\times np}]$ and $\hat{\mathcal A}$ the companion matrix
of \eqref{eq:VAR_emp}. The identification-robust sets compare \textit{Proxy-only}
(the five proxy conditions that identify the target shock) with \textit{HGMM}
(addition of the target shock's higher-order moment conditions, $16$ conditions in
total), inverting the AR statistic over the five-dimensional target-shock
subspace at $10^5$ candidate rotations; the bootstrap uses block length
$\ell=36$ and $9999$ replications. Responses are normalized to a $10\%$ increase in the
real oil price on impact. Appendix~\ref{appendix:evidence_nongauss} shows some evidence of non-Gaussianity in the reduced-form innovations, and reports the
third and fourth moments of the estimated structural shocks where the joint Gaussianity of the
vector is rejected by the pooled Jarque-Bera test ($JB=35.14$ with $12 \ d.f.$ and $p=0.0004$).
\subsubsection{Results}\label{subsubsec:oil_results}
Figure~\ref{fig:kl_mbb_vs_ar} plots the 90\% identification-robust confidence
sets along with their standard MBB confidence bands. An adverse oil \emph{supply
news} shock and a positive oil \emph{demand news} shock both raise the real price
of oil, so the two are distinguished only by the accompanying responses of oil
production, global real activity, and inventories \citep{kilian_role_2014}. Under
supply news, the anticipation of future scarcity raises storage demand on
impact, so inventories build immediately, production declines sluggishly but
persistently, and global industrial production falls. Under flow demand news, the oil price rises {together with}
global activity, production responds positively to the price incentive, and
inventories need not build.
Our estimates match the demand-news pattern, and the inventory response is
decisive. World oil inventories are drawn {down}: the response is
broadly negative over the first $12$ months, and turns significantly positive only after two years. This is the opposite of the immediate accumulation that defines a
supply news shock. World industrial production rises on impact by $0.23\%$ and is
significantly positive; U.S. industrial production is significantly positive over first ten
months and U.S. consumer prices increase for more than four years. World oil
production shows no persistent decline: it falls on impact, then rises over the following six months, consistent with
producers responding to the higher price rather than with an anticipated supply
shortfall.
\begin{figure}[H]
\centering
\includegraphics[width=0.98\textwidth]{images/results_main_text/KL_HGMM/figures/fig4_mbb_vs_ar.png}
\caption{90\% identification-robust confidence sets with \textit{Proxy-only} and \textit{HGMM} criterions, with standard MBB percentile bands.}
\label{fig:kl_mbb_vs_ar}
\begin{minipage}{0.92\textwidth}
\vspace{2pt}
\scriptsize
\textit{Notes:} Each panel superimposes three $90\%$ interval estimates for the impulse response of one variable to the oil news shock. The shaded band is the residual-based moving-block-bootstrap (MBB) percentile confidence interval for the HGMM {point} estimator. The two dashed lines are the identification-robust AR sets: \textcolor{blue}{\textit{Proxy-only}} (blue, dashed) inverts the AR statistic using only the five proxy orthogonality conditions that identify the target shock; \textcolor{red}{\textit{HGMM}} (red, dashed) additionally imposes the co-skewness, asymmetric co-kurtosis, and co-kurtosis-aggregate conditions of the target shock ($16$ conditions in total). The solid black line is the HGMM point estimate. Critical values from a residual-based moving-block bootstrap ($9999$ replications, block length $\ell = 36$). Responses are normalized to a $10\%$ increase in the real oil price on impact. The $x$-axis is the horizon in months; the $y$-axis is the response in percent.
\end{minipage}
\end{figure}
Table~\ref{tab:arlen_kilian} reports the lengths of the confidence sets under
both criteria and Table~\ref{tab:arlen_kilian_ratio} the corresponding length
ratios at horizons $0$ to $4$. The identifying contribution of the higher-order
moment conditions is large: the median reduction in set length is $71.6\%$, and
the \textit{HGMM} sets are narrower than the \textit{Proxy-only} sets at every
variable-horizon pair, with a median length ratio of $3.5$ and
a minimum of $1.6$. The gain is largest for the real oil price itself, where the
impact set contracts from a length of $7.41$ under the instrument alone to $0.75$
once the higher-order moment conditions are added, a ratio of nearly ten; world
industrial production contracts from $0.87$ to $0.18$ on impact and U.S. CPI from
$0.34$ to $0.06$.
The consequence of augmenting non-Gaussian identification is that the demand-shock characterization above exists only under the
hybrid criterion. Table~\ref{tab:arsign_kilian} in
Appendix~\ref{appendix:extra_kilian} records the horizons at which each criterion
determines the sign of a response. The instrument alone never resolves the sign
of the world oil production or U.S. industrial production response; it also never detects the inventory drawdown, since its confidence set for
inventories never lies strictly below zero at any horizon and turns positive only
after two years. A proxy-only analysis of this specification would therefore be unable
to distinguish supply news from demand news at all. Furthermore, the proxy exclusion restriction is not rejected
($\hat J_{\text{prx}}=2.87$, with asymptotic and bootstrap $p$-values of $0.71$
and $0.97$, respectively).
\begin{table}[htbp]
\centering
\caption{Length comparison of 90\% Anderson--Rubin confidence sets: \textit{Proxy-only} vs \textit{HGMM} (Kilian-corrected specification)}
\label{tab:arlen_kilian}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcccccccccc}
\toprule
& \multicolumn{2}{c}{$h=0$} & \multicolumn{2}{c}{$h=1$} & \multicolumn{2}{c}{$h=2$} & \multicolumn{2}{c}{$h=3$} & \multicolumn{2}{c}{$h=4$} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}\cmidrule(lr){10-11}
Variable & Proxy & HGMM & Proxy & HGMM & Proxy & HGMM & Proxy & HGMM & Proxy & HGMM \\
\midrule
Real oil price & 7.41 & \textbf{0.75} & 8.65 & \textbf{0.63} & 8.55 & \textbf{0.88} & 7.33 & \textbf{0.77} & 5.11 & \textbf{0.91} \\
World oil production & 1.21 & \textbf{0.40} & 0.97 & \textbf{0.43} & 0.70 & \textbf{0.28} & 0.71 & \textbf{0.37} & 0.71 & \textbf{0.29} \\
World oil inventories & 0.95 & \textbf{0.30} & 0.69 & \textbf{0.21} & 0.67 & \textbf{0.24} & 0.59 & \textbf{0.18} & 0.56 & \textbf{0.24} \\
World IP & 0.87 & \textbf{0.18} & 0.73 & \textbf{0.17} & 0.95 & \textbf{0.20} & 0.93 & \textbf{0.21} & 0.92 & \textbf{0.24} \\
U.S. IP & 0.41 & \textbf{0.13} & 0.42 & \textbf{0.12} & 0.53 & \textbf{0.16} & 0.55 & \textbf{0.17} & 0.71 & \textbf{0.22} \\
U.S. CPI & 0.34 & \textbf{0.06} & 0.44 & \textbf{0.07} & 0.34 & \textbf{0.06} & 0.30 & \textbf{0.07} & 0.27 & \textbf{0.05} \\
\bottomrule
\end{tabular}}
\begin{minipage}{0.96\textwidth}
\vspace{4pt}
\footnotesize
\textit{Notes:} Each entry is the length (upper minus lower bound) of the 90\% identification-robust Anderson--Rubin confidence set for the impulse response of the row variable to the target shock at horizon $h$ (months). \textit{Proxy} inverts the AR statistic using only the five proxy exclusion conditions that identify the target shock; \textit{HGMM} (Proxy+NG) additionally imposes the target shock's higher-order moment conditions, for $16$ conditions in total. The smaller length in each (variable, horizon) pair is in \textbf{bold}. Responses are normalized to a $10\%$ increase in the real oil price on impact, so entries are in percent. Sets are obtained by inverting the AR statistic over the target-shock rotation subspace with per-candidate moving-block-bootstrap critical values ($9999$ replications, block length $\ell=36$); see the construction in Appendix~\ref{sec:emp_arsets_cons}.
\end{minipage}
\end{table}
\begin{table}[htbp]
\centering
\caption{Ratio of 90\% Anderson--Rubin confidence-set lengths: \textit{Proxy-only} / \textit{HGMM} (Kilian-corrected specification)}
\label{tab:arlen_kilian_ratio}
\resizebox{0.80\textwidth}{!}{
\begin{tabular}{lccccc|cc}
\toprule
& \multicolumn{5}{c|}{Ratio at horizon $h$} & \multicolumn{2}{c}{Over $h=0,\dots,50$} \\
\cmidrule(lr){2-6}\cmidrule(lr){7-8}
Variable & $h=0$ & $h=1$ & $h=2$ & $h=3$ & $h=4$ & Median & Max \\
\midrule
Real oil price & 9.90 & 13.82 & 9.76 & 9.48 & 5.61 & 9.76 & 13.82 \\
World oil production & 3.01 & 2.25 & 2.47 & 1.94 & 2.46 & 2.46 & 3.01 \\
World oil inventories & 3.21 & 3.35 & 2.81 & 3.26 & 2.34 & 3.21 & 3.35 \\
World IP & 4.90 & 4.42 & 4.84 & 4.38 & 3.90 & 4.42 & 4.90 \\
U.S. IP & 3.26 & 3.42 & 3.36 & 3.29 & 3.28 & 3.29 & 3.42 \\
U.S. CPI & 5.42 & 5.97 & 5.64 & 4.55 & 5.56 & 5.56 & 5.97 \\
\bottomrule
\end{tabular}}
\begin{minipage}{0.96\textwidth}
\vspace{4pt}
\footnotesize
\textit{Notes:} Each entry is the ratio $\text{len(Proxy)}/\text{len(HGMM)}$ of the length of the 90\% identification-robust Anderson--Rubin confidence set under \textit{Proxy-only} to the corresponding length under \textit{HGMM}, for the impulse response of the row variable to the target shock. Values above one indicate that the \textit{Proxy-only} set is longer. The last two columns summarize the ratio over all $51$ horizons $h=0,\dots,50$. Construction as in Table~\ref{tab:arlen_kilian}.
\end{minipage}
\end{table}
Other results with the specification without the asymmetric co-kurtosis moment block, the MBB
percentile bands for the proxy-SVAR estimator,
and sensitivity to the bootstrap block length are reported in
Appendix~\ref{appendix:extra_kilian}. A robustness exercise with application to the \citet{kanzig_macroeconomic_2021} baseline instrument and
specification is reported in Appendix~\ref{appendix:kanzig_rob}.
\subsection{Euro-area monetary policy shock}\label{subsec:emp_mp}
Our second illustration identifies a Euro-area monetary policy shock using the
high-frequency interest-rate surprises of \citet{altavilla_measuring_2019} as external instruments. They
measure surprises as changes in overnight-index-swap (OIS) rates in narrow
windows around ECB policy communications and rotate the resulting surprises into
four orthogonal factors: \emph{target}, \emph{timing}, \emph{forward
guidance}, and \emph{quantitative easing}. For the detailed construction of the
surprises and the factor rotation, we refer the reader to
\citet{altavilla_measuring_2019}.
\subsubsection{Specification}\label{subsubsec:mp_spec}
We follow the specification of four variables ordered with
the policy indicator first: the two-year OIS rate (the target), the log Euro
Stoxx 50 equity index, the log EUR/USD exchange rate, and the two-year
inflation-linked swap rate, so that $n=4$ and $k=1$. The VAR has the form
\eqref{eq:VAR_emp} with $p=15$ lags, estimated at daily frequency over
2005:M3-2007:M12. We use the \emph{forward guidance} factor from the press-conference window as
the external instrument $m_t$ (robust $F = 9.9$), since every variable in this VAR is a
forward-looking asset price, and all four respond to news about the expected path
of policy. A surprise
that is orthogonal to the contemporaneous policy decision isolates this
path component. With $n=4$, the rotation has $p_\phi=n(n-1)/2=6$ Givens angles. We estimate
$\hat\phi$ by continuous-updating GMM, stacking $q_P=k(n-1)=3$ proxy exclusion
conditions with the $q_{NG}=n(n-1)+n(n-1)/2=18$ higher-order moment conditions\footnote{Unlike the oil-news specification of
Section~\ref{subsubsec:oil_spec}, we do not append the asymmetric co-kurtosis
conditions here. Matching a Student-$t$ to the estimated excess kurtosis of the
third structural shock gives $\hat\nu=4.7$, which does not clear the $6+3\delta$ moment bound required
by the asymmetric conditions under the factorization of
Section~\ref{subsec:nongaussianconditions}, whose sample variance would therefore not be
consistently estimable. Identification holds on the conditions above in either
case; see Appendix~\ref{appendix:evidence_nongauss}.} of
Section~\ref{subsec:nongaussianconditions}. Responses are reported for a
one-standard-deviation monetary policy shock. Appendix~\ref{appendix:evidence_nongauss} reports the corresponding moments of the structural shocks, where all four excess kurtoses are positive, and the pooled Jarque-Bera test clearly rejects their joint Gaussianity.
\subsubsection{Results}\label{subsubsec:mp_results}
Figure~\ref{fig:al_mbb_vs_ar} plots the 90\% identification-robust confidence sets along with their standard MBB confidence bands. A contractionary forward guidance surprise raises the two-year OIS rate by
$2.7$ basis points on impact for a one-standard-deviation shock and decays slowly: the response is still
$0.7$ basis points after $200$ business days. Equity prices fall on impact
($-0.09$ log points), and then rise after two weeks. The euro appreciates from about five months onward, the only horizons at which the exchange rate response is signed. Though the medium-run pattern of a
persistent increase in expected rates and a
stronger euro is the conventional transmission of a tightening surprise, the delayed {increase} in equity prices is the
information-effect, which the literature associates with hawkish communication that also
conveys a more favorable assessment of the future economic outlook
\citep{nakamura_high-frequency_2018, jarocinski_deconstructing_2020,
miranda-agrippino_transmission_2021}.
Table~\ref{tab:arlen_altavilla} reports the lengths of confidence sets at both criteria: \textit{HGMM} and \textit{Proxy-only}, and
Table~\ref{tab:arlen_altavilla_ratio} the corresponding length ratios at horizons
of $0$, $50$, $100$, $150$, and $200$ business days.
The identifying contribution of the higher-order moment conditions is modest relative to oil-news shock in Section \ref{subsec:emp_oil}, but still substantial with a median reduction
in length of $26.4\%$; by
variable, the median reduction is $20\%$ for the two-year OIS rate, $30\%$ for
the equity index, $29\%$ for the exchange rate, and $39\%$ for the
inflation-linked swap, where the sets are more than halved at some horizons. The \textit{HGMM} confidence sets are narrower than the \textit{Proxy-only} sets
at every variable-horizon pair. The two sets are not nested as the per-candidate bootstrap critical value rises with the number of conditions imposed, and Appendix~\ref{appendix:gk} reports a specification in which the added conditions are not enough to offset the additional degrees of freedom. Also, Table \ref{tab:arsign_altavilla} in Appendix \ref{appendix:extra_altavilla} shows these narrower confidence sets allow us to interpret the signs of the point estimates in longer horizons. Furthermore, the proxy exclusion restriction is not rejected ($\hat{J}_{prx} = 3.18$, with asymptotic and bootstrap $p$-values of $0.36$ and $0.75$, respectively). Other results with the
MBB percentile bands (including for the proxy-estimator), AR sets with alternate bootstrap block length are reported in Appendix \ref{appendix:extra_altavilla}. A robustness
exercise on the \citet{gertler_monetary_2015} specification is included in Appendix \ref{appendix:gk}.
\begin{table}[htbp]
\centering
\caption{Length comparison of 90\% Anderson--Rubin confidence sets: \textit{Proxy-only} vs \textit{HGMM} (Altavilla monetary-policy specification)}
\label{tab:arlen_altavilla}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcccccccccc}
\toprule
& \multicolumn{2}{c}{$h=0$} & \multicolumn{2}{c}{$h=50$} & \multicolumn{2}{c}{$h=100$} & \multicolumn{2}{c}{$h=150$} & \multicolumn{2}{c}{$h=200$} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}\cmidrule(lr){10-11}
Variable & Proxy & HGMM & Proxy & HGMM & Proxy & HGMM & Proxy & HGMM & Proxy & HGMM \\
\midrule
2Y OIS & 0.0180 & \textbf{0.0162} & 0.0308 & \textbf{0.0264} & 0.0302 & \textbf{0.0245} & 0.0238 & \textbf{0.0183} & 0.0193 & \textbf{0.0141} \\
Euro Stoxx 50 & 1.021 & \textbf{0.799} & 0.484 & \textbf{0.278} & 0.363 & \textbf{0.230} & 0.337 & \textbf{0.259} & 0.328 & \textbf{0.258} \\
EUR/USD & 0.485 & \textbf{0.349} & 0.231 & \textbf{0.189} & 0.188 & \textbf{0.111} & 0.192 & \textbf{0.122} & 0.211 & \textbf{0.159} \\
2Y infl.-linked swap & 0.0191 & \textbf{0.0086} & 0.0158 & \textbf{0.0120} & 0.0064 & \textbf{0.0049} & 0.0056 & \textbf{0.0025} & 0.0036 & \textbf{0.0016} \\
\bottomrule
\end{tabular}
}
\begin{minipage}{0.96\textwidth}
\vspace{4pt}
\footnotesize
\textit{Notes:} Each entry is the length (upper minus lower bound) of the 90\% identification-robust Anderson--Rubin confidence set for the impulse response of the row variable to the target shock at horizon $h$ (business days). \textit{Proxy} inverts the AR statistic using only the $q_P=3$ proxy exclusion conditions that identify the target shock; \textit{HGMM} additionally imposes the higher-order moment conditions, for $7$ conditions in total. The smaller length in each (variable, horizon) pair is in \textbf{bold}. The interest-rate and swap responses are in percentage points; the equity and exchange-rate responses are in log points. Sets are obtained by inverting the AR statistic over the target-shock rotation subspace with per-candidate moving-block-bootstrap critical values ($9999$ replications, block length $\ell=50$); see the construction in Appendix~\ref{sec:emp_arsets_cons}.
\end{minipage}
\end{table}
\begin{table}[htbp]
\centering
\caption{Ratio of 90\% Anderson--Rubin confidence-set lengths: \textit{Proxy-only} / \textit{HGMM} (Altavilla monetary-policy specification)}
\label{tab:arlen_altavilla_ratio}
\resizebox{0.80\textwidth}{!}{
\begin{tabular}{lccccc|cc}
\toprule
& \multicolumn{5}{c|}{Ratio at horizon $h$} & \multicolumn{2}{c}{Over $h=0,\dots,200$} \\
\cmidrule(lr){2-6}\cmidrule(lr){7-8}
Variable & $h=0$ & $h=50$ & $h=100$ & $h=150$ & $h=200$ & Median & Max \\
\midrule
2Y OIS & 1.11 & 1.17 & 1.23 & 1.30 & 1.37 & 1.25 & 1.38 \\
Euro Stoxx 50 & 1.28 & 1.74 & 1.58 & 1.30 & 1.27 & 1.43 & 1.86 \\
EUR/USD & 1.39 & 1.23 & 1.69 & 1.56 & 1.32 & 1.41 & 1.87 \\
2Y infl.-linked swap & 2.22 & 1.32 & 1.31 & 2.24 & 2.25 & 1.64 & 2.33 \\
\bottomrule
\end{tabular}
}
\begin{minipage}{0.96\textwidth}
\vspace{4pt}
\footnotesize
\textit{Notes:} Each entry is the ratio $\text{len(Proxy)}/\text{len(HGMM)}$ of the length of the 90\% identification-robust Anderson--Rubin confidence set under \textit{Proxy-only} to the corresponding length under \textit{HGMM}, for the impulse response of the row variable to the target shock. Values above one indicate that the \textit{Proxy-only} set is longer. The last two columns summarize the ratio over all $201$ horizons $h=0,\dots,200$. Construction as in Table~\ref{tab:arlen_altavilla}.
\end{minipage}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{images/results_main_text/AL_HGMM/figures/baseline/fig4_mbb_vs_ar.png}
\caption{90\% identification-robust confidence sets with \textit{Proxy-only} and \textit{HGMM} criteria, with standard MBB percentile bands.}
\label{fig:al_mbb_vs_ar}
\begin{minipage}{0.92\textwidth}
\vspace{2pt}
\scriptsize
\textit{Notes:} Each panel superimposes three $90\%$ interval estimates for the impulse response of one variable to the monetary policy shock. The shaded band is the residual-based moving-block-bootstrap (MBB) percentile confidence interval for the HGMM {point} estimator. The two dashed lines are the identification-robust AR sets: \textcolor{blue}{\textit{Proxy-only}} (blue, dashed) inverts the AR statistic using only the three proxy-exclusion conditions that identify the target shock; \textcolor{red}{\textit{Proxy+NG}} (red, dashed) is the HGMM criterion which additionally imposes the three co-skewness conditions of the target shock and the fourth-order aggregate, for $7$ conditions in total. Critical values are from a residual-based moving-block bootstrap ($9999$ bootstrap replications, block length $\ell = 50$), computed per candidate rotation. The solid black line is the HGMM point estimate. Responses are to a one standard deviation contractionary monetary policy shock; the $x$-axis is the horizon in business days.
\end{minipage}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
This paper combines identification through external instruments with identification through non-Gaussianity, stacking higher-order moment conditions and proxy exclusion restrictions into a single over-identified GMM criterion. With non-Gaussian structural shocks, the higher-order moment conditions eliminate the continuous rotational indeterminacy in $\mathcal{O}(n)$ and reduce the identified set to the finite discrete group $\mathcal{B}$; a sufficient number of valid proxies then selects a unique element from $\mathcal{B}$. The proxy-targeted columns of $B_0$ are globally point-identified and the remaining columns are identified up to signed permutation. Neither channel achieves this alone: the estimator is robust to weak instruments, and it does not require an arbitrary labeling convention to give the estimated shocks an economic interpretation.
Under local-to-zero proxy relevance the estimator remains consistent, and the identification-robust Anderson--Rubin confidence sets are substantially narrower than those based on the proxy orthogonality conditions alone. The hybrid estimator is also more efficient than the standalone non-Gaussian GMM estimator, which is less obvious: even when the proxies lose their identifying power, their long-run covariance with the non-Gaussian moment conditions continues to sharpen the effective weighting of the non-Gaussian block. In the local-to-zero limit this gain
vanishes and the hybrid estimator attains the non-Gaussian bound exactly, so
carrying a weak instrument in the criterion never costs efficiency.
We propose two mutually orthogonal specification tests, $J_{\mathrm{prx}}$ and $J_{\mathrm{NG}}$, which assess the maintained assumptions of each identification channel separately. Under the joint null both follow $\chi^2$ limits. The effective degrees of freedom compress below the nominal count by $p_\phi - \mathrm{rank}(G_{\text{complement}})$ whenever the complementary Jacobian block is rank-deficient. A residual-based moving block bootstrap with Hall--Horowitz recentering supplies critical values that adapt both to this compression and to the fourth-order cumulants in the weighting matrix.
Some extensions can be pursued. The first is to explore the estimator when either of the identification source is weak: both channels do not enter the criterion at the same informative level, so this problem does not reduce to a familiar weak-identification analysis with two sources, and requires further research. The second is partial identification through non-Gaussianity, where only some shocks satisfy the higher-order moment conditions; this would involve estimation of the target column directly. The reduction used for the
identification-robust sets shows that the target-shock responses are a function of
the unit vector $q_1$ alone. This implies that a criterion built from the moment conditions
involving the target shock could be minimized over the sphere $\mathcal{S}^{n-1}$
without ever forming the full rotation. This would make the dimension of the
estimation problem grow linearly rather than quadratically in $n$, at the cost of
giving up the joint identification of the remaining columns. It is the natural
route to applying the framework to applications larger than those considered here.
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