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Partially identified heteroskedastic SVARs
\defcitealias{LMNS20}{L\"{u}tkepohl et al. (2020)} \defcitealias{CMT23WP}{Carriero et al. (2023)} \defcitealias{RWZ10RES}{Rubio-Ram{\'{\i}}rez et al. (2010)} \defcitealias{GMMO18}{Gafarov et al. (2018)} \defcitealias{GMS18}{Granziera et al. (2018)} \defcitealias{ARW18}{Arias et al. (2018)}
\doublespacing
In recent years, there has been growing interest in the use of external information to identify the parameters of econometric models. This external information, derived from features of the data or from proxies designed to capture specific structural shocks, is often combined with restrictions on the parameter space suggested by economic theory. A case in point is the literature that exploits heteroskedasticity to identify simultaneous equations systems Rigobon03 and structural vector autoregressive (SVARs) models LanneLutkepohl08JMCB. When the data exhibit volatility clusters that can be attributed to shifts in the variance of structural innovations -- while holding the parameters of the conditional mean constant -- there are important gains in terms of identification of the structural parameters (see, for example, KLbook, KLbook, Chapter 14, or the recent empirical work of BPSS21, BPSS21).
While heteroskedastic SVAR models have become a standard tool in macroeconometrics, it is important to recognize that two caveats apply when using them. First, identification through heteroskedasticity is a purely statistical identification strategy. In other words, shocks have a structural interpretation only if they give rise to impulse responses with a credible economic interpretation. Second, the size of the shifts in the variance of different shocks must be sufficiently heterogeneous for the identification to be valid.\footnote{\citetalias{LMNS20} developed a formal test for identification by heteroskedasticity. Lewis22 proposes an alternative test of weak identification in the context of heteroskedastic SVARs (HSVARs). This test, based on the IV literature, can be applied in very specific specifications.}
On the contrary, if some of the shifts in the variances are not distinct, heteroskedasticity cannot be used to identify structural shocks. In this case, the literature does not provide a specific solution for continuing to use HSVAR models, and other identification schemes - provided they are credible - must be considered to solve the identification problem. Interestingly, \citetalias{CMT23WP} propose a blended approach, where heteroskedasticity can be combined with sign restrictions, narrative restrictions, and external instruments.
This paper presents a new strategy that allows researchers to keep using HSVAR models even in the absence of information to identify the shocks of interest. Our approach starts where statistical tests would suggest stopping: namely, when some shifts in the variances of the structural shocks are suspected to be statistically indistinguishable from each other. Although in a rather different setup, based on simultaneous equation models for cross-sectional data, LMYJoE20 also investigate the situation where (conditional) heteroskedasticity involves only a subset of equations. They propose tests for the heteroskedasticity rank and a way for estimating and doing classical inference on the parameters of the equations associated with conditional heteroskedasticity.
Instead, our idea is to combine the presence of heteroskedasticity with some zero and/or sign restrictions on parameters or functions of them. This strategy allows us to deal with HSVAR models that are not point, but only set identified. The paper makes three main contributions. The first concerns the development of analytical results on identification. We show that, apart from normalisation constraints, a combination of heteroskedasticity and zero restrictions allows point identification in HSVAR models even in the absence of heterogeneous variance shifts. Importantly, the number of zero restrictions in this case is much smaller than that generally used for point identification in traditional SVAR models.
Our second contribution, closely related to the first, is to extend the topological analysis of the identified set offered in GK18 for SVAR models to the HSVAR literature, and to derive analytical results for point identification and for set identification with convex identified sets. The mapping between the reduced and structural form parameters facilitates the extension of the literature on set identification, largely used in standard SVAR models, to the HSVAR framework.
Finally, another contribution concerns estimation and inference on the identified set. In this respect, we adapt the robust Bayesian approach of GK18 to our setup. In fact, this approach is perfectly suited to the peculiarities of HSVARs, where the identifying assumptions are violated due to heterogeneous variance shifts in the structural shocks. We provide a useful algorithm to implement our strategy for estimation and inference of the identified set. This provides applied economists and econometricians with a new tool for their empirical analyses when clusters of volatility provide a useful but insufficient source of information for the identification of structural shocks.
An empirical example about the identification of structural shocks driving the real price of crude oil illustrates our methodology. In this example, we show how to set or point identify structural shocks when the standard HSVAR approach fails.
The rest of the paper is organized as follows. The next section briefly surveys the literature; Section (ref) introduces the econometric framework and provides some preliminary results on the identification of HSVARs. Section (ref) is dedicated to the theory of identification in HSVARs. Section (ref) focuses on the inferential analysis of identified sets through a Robust Bayesian approach. Section (ref) presents the empirical example and Section (ref) concludes. An appendix with the proofs and many other results completes the paper.
This paper is strongly related to the identification through heteroskedasticity literature both for simultaneous equations systems Rigobon03,KV2010,Lewbel12,LMYJoE20 and SVARs LanneLutkepohl08JMCB,Bacchiocchi17,KLbook,LN2017. However, to the best of our knowledge, the idea that heteroskedasticity can be helpful in identifying econometric models has been firstly proposed by SF01JoE in a context of factor models, nesting SVAR models as well. Our contribution builds on the results of the statistical tests for heterogeneous variance shifts by \citetalias{LMNS20} and Lewis22, in the sense that our approach can be implemented when any statistical tests suggest a failure of the identifying information from heteroskedasticity to point identify the structural shocks.\footnote{Different approaches exploiting heteroskedasticity for the identification of structural shocks are the recent contribution by Lewis21, that does not require volatility clusters, but simply needs for time-varying volatility of unknown form, as well as lutkepohl2021heteroscedastic, who analyse identification through heteroskedasticity in the context of proxy SVAR models. See also Sims22, Sims22, for a recent contribution on heteroskedastic SVARs with misspecified regimes.}
As previously mentioned, a paper that relates to our research is \citetalias{CMT23WP}, where the authors combine different identification strategies, each of which can contribute resolving the critical issues of the others. Specifically, they propose different algorithms that combine sign restrictions, heteroskedasticity and external instruments. In our approach we focus on weak identification due to lack of (or, better, not enough) heterogeneity in the variances, where zero and sign restrictions are tools for recovering the identification issue. In their approach, instead, heteroskedasticity is mainly intended as a tool to reduce the identified set obtained by sign restrictions, narrative restrictions, and external instruments approaches.
While in our contribution the main assumption is that only the variances of the shocks are subject to breaks, other authors find evidence of structural shifts among the structural parameters of the model, too SimsZha06AER,InoueRossi11RESTATS,BG06RESTATS. This literature, that does not focus solely on SVAR models, allows for impulse responses to be different in the different regimes, while they are the same when only the variances do change.\footnote{However, only few papers constructively use the presence of regime shifts to solve the identification issue MM14ECTA,BacchiocchiFanelli15,BK20SVARWB.}
This paper contributes also to the literature on point and set identified SVARs. For point identification, we exploit the general criteria in \citetalias{RWZ10RES}, with subsequent modifications proposed by BKglob20, for global identification on SVARs. As for set identification, we use sign restrictions to set identify the structural impulse response functions of interest RWZ10RES,Uhlig05JME.
Strictly connected to this last point is the literature on how to do inference on set identified models. As stated in the introduction, our approach builds on GK18, but other approaches have been proposed in the literature to pursue this purpose.\footnote{\citetalias{GMMO18} and \citetalias{GMS18} provide results based on a frequentist setting, while, among others, BH15 and \citetalias{ARW18} adopt Bayesian inference.}
Consider the following Structural Vector Autoregressive (SVAR) model
where $y_{t}$ is a $n$-dimensional vector of observable variables, $\varepsilon_{t}$ is a vector of mutually orthogonal white noise processes, normally distributed with mean zero and time-varying covariance matrix. Specifically, let the covariance matrix of the structural shocks $\varepsilon_{t}$ be as follows
where $1 < T_B < T$ is the break date, $I_n$ is the $(n\times n)$ identity matrix, and $\Lambda$ is a $(n\times n)$ diagonal matrix made of strictly positive numbers.\footnote{We consider the initial conditions for the first regime, $y_{0},\ldots,y_{1-l}$, as given, while for the second regime they are fixed as the last $l$ observations of the former, in order to guarantee the contiguity of the regimes on the whole sample.} The $n\times 1$ vector $a$ contains the intercepts and the $n\times n$ matrices $A_{i}$, with $i=0,\ldots,l$, collect the structural parameters. The structural parameters can be indicated as $\theta=\left(A_0, A_+, \Lambda\right)\in \Theta \subset \ensuremath\mathbb{R}^{\left(n+m\right)n+n}$, with $m=nl+1$, and where the $n\times m$ matrix $A_+=\left(a, A_{1},\ldots, A_{l}\right)$. We denote the open dense set of all structural parameters by $\mathbb{P}^S\subset \ensuremath\mathbb{R}^{\left(n+m\right)n+n}$. The model in Eq.s ((ref))-((ref)) is a standard SVAR model with structural shocks characterized by different volatility regimes. As shown in Eq. ((ref)), structural innovations have unit variance before the break, and variance equal to the diagonal elements of $\Lambda$, denoted as $\lambda_i$ after the break. Hereafter, this model is referred to as heteroskedastic SVAR (HSVAR).
The reduced-form VAR model can be written as
where $b=A_{0}^{-1} a$, $B_{i}=A_{0}^{-1} A_{i}$. Furthermore, for both the regimes, the vector of error terms is defined as $u_t=A_{0}^{-1}\varepsilon_t$, with
The VAR model, thus, presents different covariance matrices of the error terms $\Omega_1$ and $\Omega_2$, and thus heteroskedasticity, as in, among others, Rigobon03, LanneLutkepohl08JMCB, and BacchiocchiFanelli15. The reduced-form parameters are denoted by $\phi=(B, \Omega_1, \Omega_2)\in \Phi \subset \ensuremath\mathbb{R}^{n+n^2l}\times \Omega_n \times \Omega_n$, where the $n\times m$ matrix $B=\left(b,B_{1},\ldots,B_{l}\right)$ and $\Omega_n$ is the space of positive-semidefinite matrices of dimension $n\times n$. The set of all reduced-form parameters is denoted by $\mathbb{P}^R\subset \ensuremath\mathbb{R}^{nm+n\left(n+1\right)}$. The reduced form will be denoted HVAR.
Conditional on the restrictions of the domain $\Phi$ such that all the roots of the characteristic polynomial lie outside the unit circle, there exists an equivalent VMA$(\infty)$ representation for the HVAR in Eq. ((ref)), assuming the form\footnote{The HVAR in Eq.s ((ref))-((ref)) is characterized by the same parameters for the conditional mean over the two regimes, therefore breaks are confined to second moments parameters. As a consequence, the absence of unit roots is a characteristic of the model in the whole sample, and not within each regime. Furthermore, as shown in Eq. ((ref)), the VMA representation is unique and not regime-specific. It follows that, if shocks have the same magnitude, impulse response functions in the two regimes are equivalent. If instead one considers a one standard deviation structural shock, impulse responses will be of different magnitude, but have exactly the same shape in different regimes.}
where $C_{j}$ is the j-th coefficient matrix of $\bigg(I_n-\sum_{i=1}^{l}B_{i}L^i\bigg)^{-1}$. Based on the VMA representation, the long-run impulse response, $IR^{\infty}$, and the one at any $h$, $IR^h$, are, respectively
whose $(i,j)$-element represents the response of the i-th variable of $y_{t+h}$ to a unit shock on the j-th element of $\varepsilon_{t}$, independently of the regime considered.
As is well known in the SVAR literature, without any restriction it is impossible to uniquely pin down the structural parameters based on the reduced form of the model. If, instead, we suppose the parameters of the conditional mean in the HSVAR in Eq. ((ref)) to remain stable across the two regimes, then Rigobon03, for a bivariate case, and LanneLutkepohl08JMCB, for the general case, proved there is some gain in terms of identification. In this section we introduce some general theoretical results for the HSVAR in Eq. ((ref)). All the results will be formally presented in Appendix (ref).
Consider an $n$-variable HSVAR model as in Eq.s ((ref))-((ref)). Following the parametrization and notations we have been using so far, we analyze identification of the $n \times n$ matrix $C$ that represents the inverse of structural coefficient matrix $A_0$, i.e. $C\equiv A_0^{-1}$, and $\Lambda$, $n \times n$ diagonal matrix with strictly positive elements. Given the reduced-form covariance matrix at regime 1 and 2, denoted by $\Omega_1$ and $\Omega_2$, respectively, $C$ and $\Lambda$ solve
The first important result on the identification of the HSVAR is about the set of solutions, in terms of $(C,\Lambda)$, of the system of equations in ((ref)). In Theorem (ref) in Appendix (ref) we show that the solution is not unique, but any permutations and change of signs of the column vectors in $C$, as far as the same permutations are applied to the diagonal elements of $\Lambda$, remain observationally equivalent. In order to solve this indeterminacy, one possibility is to fix a specific ordering for the equations, that corresponds to fixing a specific ordering for the variances of the structural shocks in the second regime, i.e. the diagonal elements in $\Lambda$. Of course, this becomes problematic when some of the variances are equivalent.
In this direction, the second important result, reported in Theorem (ref) in Appendix (ref), states how point identification is possible only once the solution for $\Lambda$ is characterized by all distinct elements on the main diagonal. In this case, fixing a specific ordering of the structural shocks, as well as the standard sign normalization, leads the solution $(C,\Lambda)$ to be unique, and thus point identified.
Finally, the third important result concerns the representation of the system in ((ref)) as an eigen-decomposition problem. To see this, let $\Omega_{1,tr}$ be a lower triangular Cholesky decomposition of $\Omega_1$. Following Proposition A.1 of Uhlig05JME, the set of non-singular matrices solving Eq. ((ref)) can be expressed as $C=\Omega_{1,tr} Q$, $Q \in \mathcal{O}(n)$, where $\mathcal{O}(n)$ is the set of $n \times n$ orthogonal matrices. Plugging this representation of $C$ into Eq. ((ref)), leads to
Symmetry of $\Omega_{1,tr}^{-1} \Omega_2 \Omega_{1,tr}^{-1\prime}$ and orthogonality of $Q$ implies that solving Eq. ((ref)) is precisely the eigen-decomposition problem. Identification of $(C,\Lambda)$ can be therefore cast as uniqueness of the eigen-decomposition of $\Omega_{1,tr}^{-1} \Omega_2 \Omega_{1,tr}^{-1\prime}$ into the diagonal matrix of eigenvalues and the corresponding eigenvectors collected in $Q$. According to the previous result, an HSVAR can be point identified, up to permutations and sign changes, if the eigenvalues of $\Omega_{1,tr}^{-1} \Omega_2 \Omega_{1,tr}^{-1\prime}$ are all distinct. A nice geometric interpretation for this result, for the simple bivariate case, is discussed in Appendix (ref).
In this section we extend the preliminary results reported in Section (ref) where, conditional on the reduced-form parameters, the identification issue was addressed as an eigen-decomposition problem. Specifically, let $\Lambda=(\lambda_1,\,\lambda_2,\,\ldots,\,\lambda_n)$ be the eigenvalues of the eigen-decomposition problem in Eq. ((ref)), where $C=A_0^{-1}$ collects impact responses, $Q$ is the orthogonal matrix containing the eigenvectors, and $\Omega_i$, $i=\{1,\,2\}$, are the reduced-form covariance matrices with $\Omega_{i,tr}$ the related Cholesky lower triangular matrices. Theorem (ref) shows that a necessary condition for point identification of the structural parameters is the absence of multiplicity in eigenvalues in $\Lambda$. In the case of multiple eigenvalues, in fact, the identification of $C$ fails. In particular, if two (or more) eigenvalues are equal, say $\lambda_j=\lambda_{j+1}$, then the corresponding columns of $Q$ -- i.e. the eigenvectors associated to $\lambda_j$ and $\lambda_{j+1}$ -- denoted by $q_j$ and $q_{j+1}$, are not unique. In fact they represent a basis for the two-dimensional vector space in $\ensuremath\mathbb{R}^{n}$, but any other couple of orthogonal unitary vector belonging to such a space could be an acceptable candidate to enter in the $Q$ matrix. The matrix $Q$, thus, will not be a singleton in $\mathcal{O}(n)$ anymore, but will be a set of admissible orthogonal matrices solving the eigen-decomposition problem in Eq. ((ref)).
More suitable notations and formalization are thus necessary. We start by formalizing the eigen-decomposition problem, with the possibility of multiple eigenvalues.
According to Definition (ref), let $Q_\lambda=Q(\lambda_1)\times\cdots\times Q(\lambda_k)$. It is possible to introduce the set of all admissible matrices $Q$ as follows
As in the case of multiplicities $\mathcal{Q}(\phi)$ is not a singleton in $\ensuremath\mathcal{O}\left(n\right)$, one could think of imposing restrictions, likewise it is traditionally done in SVARs. This will be the topic of the next section.
One of the characteristics of HSVARs is that the identification is obtained from a statistical point of view, without imposing restrictions on the parameters. However, we have seen in Section (ref) that normalization restrictions are important and play a relevant role. Moreover, we will see that in some cases, imposing equality or sign restrictions can be interesting to improve the results obtained through HSVARs, especially when some of the assumptions in Theorems (ref) to (ref) are no longer valid, as the presence of multiplicities. In this section we discuss normalization restrictions first, then we move to the equality restrictions before concluding with sign restrictions.
Normalization restrictions
The normalization issue has been largely debated in econometrics. Specifically for SVAR models, we refer to WZ03 and, more recently, to HWZ07. They show that a poor normalization rule can invalidate statistical inference on the parameters. In our setup, the first normalization restriction consists in imposing the covariance matrix of the structural shocks to be the identity matrix in the first regime, i.e. $E(\varepsilon_t \varepsilon_t^\prime)=I_n$, and, as a consequence, the $\Lambda$ matrix in the second regime, i.e. $E(\varepsilon_t \varepsilon_t^\prime)=\Lambda$, as already introduced in Eq. ((ref)). However, as discussed in Section (ref), indeterminacy of the solutions also arises in terms of the sign of the columns of $C$ and the particular ranking of the variaces in $\Lambda$. The normalization rule used in this paper is summarized here below.
For the sake of simplicity, concerning the ordering of the elements in the diagonal matrix $\Lambda$, we assume
It is important to stress that all the results developed in the paper can be rephrased in terms of different normalization rules coherent with Definition (ref).
Equality restrictions
The classical approach to address the identification issue in SVARs is to impose equality restrictions on the structural parameters or on particular linear and non-linear functions of them. Although not common in the literature of heteroskedastic SVARs, we do not preclude this possibility and allow for possible equality and sign restrictions. We first consider the former, while the latter will be presented in the next section.
GK18 and \citetalias{ARW18}, in the context of SVARs, stress that imposing constraints on the structural parameters, or on suitable functions of them, such as on the impulse responses, corresponds to restrict the columns of the orthogonal matrix $Q\in \mathcal{Q}(\phi)$. Here below, we show that it also happens in the context of HSVARs. In fact, as shown in Eq. ((ref)), the structural-form parameters can be defined as the product of the orthogonal matrix $Q$ and quantities coming from the reduced form. As these latter elements are unrestricted, imposing restrictions is equivalent to constrain the columns of $Q$. The set of equality restrictions we consider, in compact notation, are as follows:
where $F_{i}(\phi)$, of dimension $f_{i}\times n$, depends on the reduced-form parameters $\phi=(B, \Omega_1, \Omega_2)$ only, while $q_{i}$ is the i-th column of $Q$. The total number of restrictions characterizing the HSVAR is given by $f=f_1+\cdots+f_n$.
Focusing our attention on the i-th eigenspace as in the eigen-decomposition in Definition (ref), let the set of restrictions on the vectors $\big(q_1^i,\,\ldots,\,,q_{m_i}^i\big)\in Q(\lambda_i)$ be contained in the $f^i\times n$ matrix $\textbf{F}^i(\phi,Q)$, with $f^i$ denoting the total number of restrictions on the vectors $\big(q_1^i,\,\ldots,\,,q_{m_i}^i\big)$. Moreover, let the $f_j^i$ restrictions on the j-th vector $q_j^i$ be defined as $F_j^i(\phi) q_j^i=0$. This allows us to introduce the following definition:
BKglob20 introduced first this definition of non redundant restrictions to complement the result in Theorem 7 in RWZ10RES on the identification of SVARs. In the same way, we will use it for developing conditions for point identification in our HSVARs.
Sign restrictions
Uhlig05JME, among others, proposes sign restrictions to impulse responses in order to obtain identified sets rather than point identification. GK18 and \citetalias{ARW18} combine sign and zero restrictions to tighten the impulse response identified sets.
As for the equality restrictions, sign restrictions can be seen as constraints on the columns of the $Q$ matrix. Suppose to impose a set of $s_{h,i}$ restrictions on the impulse responses to the i-th shock at the h-th horizon. We can write the sign restrictions as $S_{h,i}(\phi)q_i\,\geq\,\textbf{0}$, where, given the definition of the impulse response provided in Eq. ((ref)), $S_{h,i}\equiv D_{h,i}\,C_h(B) \Omega_{1,tr}$ is a $s_{h,i}\times n$ matrix with $D_{h,i}$, of dimension $s_{h,1}\times n$, a selection matrix made of $1$ and $-1$ elements indicating the restricted impulse responses. A compact notation for all the sign restrictions can be defined by
Admissible structural parameters and identified set
Based on the parametrization of the model and the set of all possible restrictions considered above, it is now possible to formally define when a point in the parametric space can be indicated as admissible.
At the same time, it is interesting to focus on the set of all the admissible matrices $Q$. We thus provide the following definition.
Finally, given that the attention could not be limited to the structural parameters but on transformations of them, like impulse response functions, it is also important to define the so called identified set.
As discussed in the previous sections, point identification in HSVAR can be achieved only if the eigen-decomposition problem in Eq. ((ref)) is characterized by $n$ distinct eigenvalues. This feature does correspond to non proportional shifts in the variances of the structural shocks among the two regimes.
If this is not the case, or, practically speaking, we do not have credible evidence on structural breaks on the second moments of some of the variables in our HSVAR, point identification can still be reached by combining heteroskedasticity with zero restrictions. The next theorem formalizes this intuition.
The results obtained in Section (ref) allow to point identify all the columns of $Q\in\ensuremath\mathcal{O}\left(n\right)$ associated with eigenvalues without multiplicity. For all the other columns, they can be point identified according to the particular pattern of zero restrictions suggested by Theorem (ref). Let $\lambda_i$ be an eigenvalue with algebraic multiplicity $g(\lambda_i)=m_i$, in this section we consider restrictions that make the $(q_1^i,\ldots,q_{m_i}^i)$ columns of $Q$ only set identified, being
with strict inequality for at least one $j=\{1,\ldots,m_i\}$.
As in many empirical applications, suppose we are interested in one single shock, i.e. one column of $Q$. Moreover, according to Remark (ref), it is crucial to understand whether the point identified impulse responses obtained through the eigenvalues without multiplicity can be compatible with the shock of interest. If this is not the case, it is likely to suppose this latter to be associated with the eigenspace generated by $\lambda_i$, with multiplicity $m_i$. We first introduce a specific ordering of the shocks according to the identifying restrictions in Eq. ((ref)), and then provide the conditions for the identified set to be convex.
The next theorems, based on Proposition 3 in GK18, provides sufficient conditions for the impulse response identified set $IS(\phi\,|F,S)$ to be convex. Precisely, we first consider the case of zero restrictions only, and then extend to the case of zero and sign restrictions.
The previous theorem just consider zero restrictions on the vectors of $Q(\lambda_i)$. The following one, instead, also allows for sign restrictions, although these last can be imposed on the vector $q_{j^\ast}^{i}$ associated with the shock of interest.
Taken jointly, the results of Theorems (ref) and (ref) generalize Proposition 3 in GK18 to the case of a structural break on the variances of the shocks, with potential eigenvalue multiplicity.
On the other side, the two theorems provide important insights on the possibility to apply the standard identification-through-heteroskedasticity approach to the case in which some of the switches in the variances are the same. As we will see in the next section, these new results represent the foundations for developing an estimator for the bounds of the identified set and produce the related inference.
In this section we present a completely brand new approach to conduct inference on set-identified HSVARs, where the set identification comes from the fact that not all the shifts in the variances of the shocks are statistically different. Details on how to estimate the reduced-form parameters and on how to check for proportional variance shifts are reported in Appendix (ref) and Appendix (ref), respectively. In this section, instead, we deal with all situations in which some of such variances are not significantly different each other, and we introduce our Robust Bayes approach to conduct inference on the identified set of interest.
Now suppose some of the eigenvalues obtained by the eigen-decomposition in Eq. ((ref)) present potential multiplicity. This evidence, for example, could be statistically checked by the \citetalias{LMNS20} or Lewis22 tests. Once this evidence is “statistically confirmed”, a natural way of proceeding is to impose such eigenvalues to be effectively equal. This choice, however, imposes implicitly restrictions on the covariance matrices of the reduced form. While ML estimator subject to constraints on the parameters is generally implementable, it is rather problematic in the specific case of imposing equality restrictions among the eigenvalues. In such particular case, in fact, firstly, imposing the restrictions makes the model no longer identified, and thus creating convergence problems of the algorithm maximizing the likelihood function and, secondly, it is technically difficult to impose restrictions on some parameters that are observationally equivalent to permutations, as highlighted in Theorem (ref) in the Appendix.
Our strategy, instead, is based on the following lemma, that, according to evidence of potential eigenvalue multiplicity, suggests imposing the multiplicities as the result of a minimization problem about the unrestricted and restricted covariance matrices of the HVAR.
The previous lemma provides a theoretical ground for fixing the common eigenvalues, when they are not statistically distinct, such that the unrestricted and restricted reduced-form covariance matrices are as close as possible, according to a specific metric. The assumption considered in the previous lemma is that the matrix $Q$, containing the eigen-vectors, is common in the two matrices $\Omega$ and $\tilde{\Omega}$. This assumption is completely reasonable for our problem in that it states that for the eigenvalues without multiplicity the eigenvectors are common in $\Omega$ and $\tilde{\Omega}$. Those associated to the eigenvalue with multiplicity, say $\lambda_i$, being not identified, must simply lay on the sub-space $Q(\lambda_i)$, orthogonal to $Q^{\perp}(\lambda_i)$. In this respect, the eigenvectors $(q_1^i,\ldots,q_m^i)$ obtained through the eigen-decomposition of $\Omega$ share this feature and can be used also as eigenvectors of $\tilde{\Omega}$. This explains the common $Q$ matrix used in Lemma (ref) both for $\Omega$ and $\tilde{\Omega}$.
Let $\tilde{\pi}_\phi$ be a probability measure on the space $\Phi$ of reduced-form parameters. In order to obtain a prior distribution for $\phi$ we need to restrict the support of $\tilde{\pi}_\phi$ such that its elements satisfy the sign, normalization and equality restrictions, as well as the fact that they show eigenvalue multiplicities as in Eq. ((ref)). In this respect, we define the prior distribution for the reduced-form parameters as follows
that, by construction, assigns probability one to the distribution of data that admits eigenvalue multiplicity and is consistent with the identifying restrictions. As the structural parameters are a function of $\big(\phi,Q\big)\in \Phi\times \ensuremath\mathcal{O}\left(n\right)$, we define a joint prior for the two sets of parameters $\big(\phi,Q\big)$ as $\pi_{\phi,Q}=\pi_{Q|\phi}\:\pi_\phi$, where $\pi_{Q|\phi}$ is supported on $\ensuremath\mathcal{Q}(\phi\,|F,S)\in \ensuremath\mathcal{O}\left(n\right)$.
In the case of no multiplicity, sign and permutation normalizations allow to pin down just one admissible $Q$, and $\pi_{Q|\phi}$ becomes a degenerate distribution centered on such $Q$. In the case of multiplicity and zero restrictions satisfying the pattern in Eq. ((ref)), instead, the HSVAR will be only set identified and the prior $\pi_{Q|\phi}$ has to be specified in order to obtain a posterior distribution for the structural parameters and impulse responses, as desired in standard Bayesian approach. Other than being a challenging task for applied economists to specify $\pi_{Q|\phi}$, it has been shown that the choice of such a prior, being never updated by the data, can have non-negligible impact on the posterior inference even asymptotically (BH15, BH15).
In order to fix this unpleasant issue, we use the robust Bayes inference proposed by GK18. This approach consists in fixing a single prior $\pi_\phi$ for the reduced-form parameters, but a set of priors for $\pi_{Q|\phi}$. This strategy allows to obtain a class of posteriors for $(\phi,Q)$ and, as a consequence, for the impulse response of interest $r=r(\phi,Q)$.
According to GK18, the results of this procedure can be summarized by reporting the posterior mean bounds interval, that can be seen as an estimator for the identified set, and an associated robustified credible region measuring the uncertainty related to the former. In particular, if we define $\ell(\phi)=\mathrm{inf}\big\{r(\phi,Q):\,Q\in\ensuremath\mathcal{Q}(\phi\,|F,S)\big\}$ and $u(\phi)=\mathrm{sup}\big\{r(\phi,Q):\,Q\in\ensuremath\mathcal{Q}(\phi\,|F,S)\big\}$, the posterior mean bounds interval can be written as $\Big[\int_\Phi\,\ell(\phi)\mathrm{d} \phi_\phi\:;\:\int_\Phi\,u(\phi)\mathrm{d} \phi_\phi\Big]$. The robustified credible region, instead, consists in an interval $C_\alpha$ for which the posterior probability is greater than or equal to $\alpha$ uniformly, i.e. $\pi_{r|Y}(C_\alpha)\geq \alpha$.
In this subsection we present an algorithm to be used in the case of two volatility regimes in the data with known break date. This last assumption is rather standard in the literature, being the break dates associated to well documented changes in the policy conduct or to financial crises.\footnote{See, among many others, LanneLutkepohl08JMCB, BG06RESTATS, ABCF19, Rigobon03, Bacchiocchi17, CMT23WP. Moreover, Rigobon03 also shows consistency of the estimated parameters in the case of break date miss-specification.}
Some remarks about the algorithm are in order. The first one is about the prior for the two covariance matrices $\Omega_1$ and $\Omega_2$. As the aim of the analysis is to highlight the possible eigenvalue multiplicity, it would be preferable to use diffuse priors, like diagonal matrices with equal values on the main diagonal, that from one side are non-informative and on the other side consider all the eigenvalues to be equal and let the likelihood function to play the relevant role in this respect.
Second, the way the draws from the posterior distribution are obtained depends on the theoretical results of Appendix (ref). Using independent priors for $\Omega_1$, $\Omega_2$ and $\phi_{B}$ allows to develop a Gibbs sampler that is rather simple and permits to explore the joint posterior distribution in a very convenient way. Step 3 of our algorithm is based on this approach for generating the draws $\phi$ from the distribution $\tilde{\pi}_{\phi|Y}$. Any alternative way, however, can be performed without altering the other steps of the algorithm.
Third, checking for the emptiness of the identified set in the case of sign restrictions is performed in Step 4 by using linear projections starting from normal draws, as in GK18 and many other contributions in the Bayesian literature. As an alternative, we could use the QR decomposition as proposed by \citetalias{ARW18}.
Fourth, for each of the draws consistent with the zero and sign restrictions, we consider it as if there were eigenvalue multiplicities, regardless of whether it is effectively so from a statistical point of view, for the draw $\phi$. In this respect, the eigenvectors associated to the potential multiple eigenvalues act as a basis for the space of the not identified columns of $Q$. From one side, this way of proceeding is extremely conservative as it completely ignores the amount of information contained in all those draws where all the eigenvalues are substantially distinguished. From the other side, however, it avoids the consequences of a pre-testing step to be applied to each draw to statistically check for eigenvalue multiplicity. Our inference, thus, is robust to eigenvalue multiplicity in the sense that we apply the robust Bayesian approach to the set identified columns of $Q$ associated to the suspected multiple eigenvalues.
Fifth, the constrained nonlinear optimization problem in Step 5 is less demanding than the one in Algorithm 1 by GK18, as the argument is not the entire matrix $Q$ but just a subset of its columns. Even if the HVAR model is relatively large, the number of eigenvectors generating the subspace $Q(\lambda_i)$ is in general relatively small and we do not expect concerns about the convergence properties of the numerical optimization step. On the contrary, we could replace Step 5 by a new algorithm in the spirit of Algorithm 2 in GK18, where the constrained nonlinear optimization problem is substituted by iterating many times Step 4.1-Step 4.3 and approximate the interval $\big[\ell(\phi_m),\: u(\phi_m)\big]$ with the minimum and maximum values obtained in such iterations. If the number of iterations goes to infinity, such alternative bounds still provide a consistent estimator of the identified set.
Sixth, the algorithm works even in the case the zero restrictions allow to point identify the matrix $Q$. In this case, the set $\ensuremath\mathcal{Q}(\phi\,|F,S)$ is always non-empty, and the constrained nonlinear optimization problem simply returns $\ell(\phi)=u(\phi)$. The inference, then, becomes standard.
From a theoretical point of view, GK18 discuss the importance of convexity, continuity and differentiability of the identified set $IS(\phi\,|F,S)$ for the posterior means to have a valid frequentist interpretation. Obviously, the same has to be verified in our setup. If this is the case, they show that the set of posterior means is a consistent estimator of the true identified set and the robust credible region is an asymptotically valid confidence set for the true identified set.
Concerning convexity, we have already proved it in the previous Theorems (ref) and (ref). About continuity and differentiability, since we use the same set of zero and sign restrictions as in GK18, extending their Proposition 4 and Proposition 5 to our setup is straightforward. Definitely, appropriate choice of zero and sign restrictions associated with mild regularity conditions on the coefficient matrices of such restrictions guarantee our results on HSVARs to have a valid frequentist interpretation, as for traditional SVARs.
We apply our methodology to the SVAR model for the global crude oil market of kilian200AER that includes three variables: the percent change in global crude oil production ($\Delta prod_t$), an index of global real economic activity ($rea_t$) and the logarithm of the real price of crude oil ($rpo_t$). Data are monthly and the sample period runs from January 1973 through December 2007. \footnote{We rely on exactly the same dataset as kilian200AER. See Appendix (ref) for details and additional results.} While kilian200AER identifies three structural shocks that drive the real price of crude oil using a simple recursive scheme, \citetalias{LMNS20}, LN2014JaE and LukAdvEcnm show that the same structural innovations can also be recovered by exploiting the existence of distinct volatility regimes.\footnote{Identification through heteroskedasticity has found other applications in studies of the crude oil market. See e.g. bruns2023have and kanzig2021macroeconomic.} We set the lag order of the VAR equal to 12 and follow \citetalias{LMNS20} that distinguish between two volatility regimes with -- an exogenously determined -- change point in October 1987.\footnote{Note that \citetalias{LMNS20}, LN2014JaE and LukAdvEcnm rely on a VAR of order 3, while kilian200AER stresses that a VAR of order 24 is necessary to capture long price cycles of crude oil and hence for accurately estimating the impulse responses in global oil market models. The selected lag order is thus a compromise between these two approaches.}
Table (ref) in Appendix (ref) reports the estimated $\lambda$s and the results of the test by \citetalias{LMNS20} for eigenvalue multiplicity. From the test it clearly emerges that $H_0: \lambda_2 = \lambda_3$ cannot be rejected by the data, implying that standard identification through heteroskedasticity fails. In fact, only one structural shock can be statistically identified relying on changes in volatility. If the identified shock cannot be given an interpretation that is consistent with economic theory, or if one wishes to identify other structural shocks, additional sign or exclusion restrictions are needed. In this case, Theorem (ref) and its implementation in Algorithm (ref) become extremely useful.
We write the relationship between structural innovations, $\varepsilon_t$ and reduced-form errors, $u_t = C \varepsilon_t$ as follows:
where $C$ corresponds to the impact response matrix ($A_0^{-1}$). We consider four SVARs based on different identifying restrictions:
Model $\mathcal{M}_0$ is the recursively identified SVAR model of kilian200AER used as a benchmark against which we compare results from HSVAR models. Three exclusion restrictions -- $c_{12}=c_{13}=c_{23}=0$ -- allow to point identify an oil supply shock and two demand shocks (i.e. aggregate and oil-specific demand shocks). Oil supply shocks represent innovations to the current physical availability of crude oil. Aggregate demand shocks capture unexpected changes of the demand for all industrial commodities driven by fluctuations in the global business cycle, while oil-specific demand shocks represent shifts in the precautionary demand for crude oil triggered by concerns about the future availability of supplies.
The HSVAR model $\mathcal{M}_1$ exploits changes in volatility for identifying structural shocks, without imposing additional exclusion or sign restrictions. We assume the eigenvalues to be all distinct, although results in Table (ref)(b) in the Appendix highlight the existence of eigenvalue multiplicity.
For this reason, in model $\mathcal{M}_2$ we impose the constraint $\lambda_2=\lambda_3$ in Step 3 of Algorithm (ref). With two distinct eigenvalues, we can point identify only one structural shock that, as will be shown in Section (ref), yields impulse responses consistent with those associated with an oil-specific demand shock. The remaining structural shocks are set identified combining static and dynamic sign restrictions. See Table (ref) in Appendix (ref). We postulate that a negative oil supply shock increases the real price of crude oil and depresses global real economic activity on impact. A positive aggregate demand shock is expected to raise oil price and production on impact. Notice that we also place sign normalizations on the main diagonal of $C\equiv A^{-1}_0$. Furthermore, we constrain the sign of the response of real crude oil price to oil supply disruptions to be positive for twelve months, starting from the impact response. These additional restrictions rule out models with the real price of crude oil decreasing below its starting level after a negative oil supply shock.
While specification $\mathcal{M}_2$ allows to point identify a single structural shock, because of eigenvalue multiplicity the HSVAR model remains set identified. In this case, Theorem (ref) shows that point identification of the HSVAR model can be achieved with a single exclusion restriction. In model $\mathcal{M}_3$, we add one exclusion restriction on the impact response matrix: $c_{21}=0$. This restriction implies that the first shock does not affect real economic activity within the same month. Compared to the recursively identified model, $\mathcal{M}_0$, when a volatility shift is exploited the identification scheme is less demanding. In fact in model $\mathcal{M}_3$ point identification is achieved -- at least in statistical sense -- combining the shift in the volatility of one of the structural shocks with a single zero restriction.
All specifications are estimated with Bayesian techniques drawing from the posterior of reduced-form parameters until we obtain 1000 realizations of the non-empty identified set. We estimate the whole set of structural impulse response functions over an horizon of 24 months. Notice that we report the implied response of world crude oil production obtained by cumulating that for $\Delta prod_t$. We focus on shocks that are expected to raise the real price of crude oil. Therefore, in the case of supply shocks we plot the responses to a negative shock representing a disruption of crude oil supply.
Impulse responses for $\mathcal{M}_0$ and $\mathcal{M}_1$, as not informative for the implementation of our methodology, are reported in Appendix (ref) to save space (Figures (ref) and (ref), respectively). Not surprisingly, the impulse responses for $\mathcal{M}_0$ are totally in line with the original ones in kilian200AER.\footnote{As for the estimation, we rely on a noninformative improper Jeffreys' prior that allows to draw reduced-form parameters from a normal-inverse-Wishart posterior.} Model $\mathcal{M}_1$, instead, builds on the standard identification through heteroskedasticity approach assuming implicitly that all eigenvalues are distinct. However, as the test by \citetalias{LMNS20} shows, changes in volatility alone here might not convey enough information to point identify all the structural shocks. We overcome this issue by imposing restrictions in $\mathcal{M}_2$ and $\mathcal{M}_3$. Impulse responses from HSVAR models $\mathcal{M}_2$-$\mathcal{M}_3$ are displayed in Figures (ref)-(ref).\footnote{Since HSVAR models normalize structural residuals to have identity covariance matrix in the first regime, the scaling of these figures is not the same as that of Figure (ref).}
Static and dynamic sign restrictions in model $\mathcal{M}_2$ allow to set identify supply and demand shocks that are expected to drive the real price of crude oil. Since the only distinct eigenvalue is associated with the oil-specific demand shock, sign restrictions are imposed on the remaining columns of $Q$. The impulse responses appearing in the first two columns of Figure (ref) specifically refer to those columns of $Q$ and, as such, we also present the set of posterior means (blue vertical bars) and the bounds of the robust credible region with credibility 68% obtained with Algorithm (ref).\footnote{Our implementation is based on the constrained nonlinear optimization problem highlighted in Step 5 of Algorithm (ref). We follow GK18 that mitigate possible convergence problems using five different starting values for the optimization problem in Step 5.}
Imposing sign restrictions we recover oil supply and aggregated demand shock whose effects on the endogenous variables of the VAR are consistent with expectations from economic theory and previous analyses kilian2012agnostic. The shape of impulse responses in the first two columns of Figure (ref) are in line with those implied by the recursively identified model, $\mathcal{M}_0$. Also notice that the combination of volatility changes and sign restrictions delivers reasonable impulse responses even in the presence of multiple eigenvalues and using less sign restrictions than what is usually done in the literature.\footnote{In Appendix E we show results based on the implementation of Algorithm (ref) while performing the \citetalias{LMNS20} test for heteroskedasticity on each draw of $\phi$ in the context of set identified impulse response functions. Figure (ref) shows how performing the test for identification thorugh heteroskedasticity does not affect the results.} In fact, we leave one of the columns of $Q$ completely unrestricted and exploit changes in volatility to point identify the corresponding shock. Contrary to the standard HSVAR, $\mathcal{M}_1$, oil supply shocks implied by $\mathcal{M}_2$ are associated with impulse responses that reasonably summarize the expected effects of such shocks on real economic activity and the real price of crude oil. Here we observe a positive response of the price of crude oil with a peak after 6 months.
The width of the HPD and of the robust credible regions for both oil supply and aggregate demand shocks are similar. We can thus draw essentially the same conclusions using any of them.\footnote{Results based on an alternative implementation of Algorithm (ref) are almost identical. In such implementation, we follow GK18 and substitute Step 5 with 10000 iterations of Step 4.1-Step 4.3. The interval $\big[\ell(\phi_m),\: u(\phi_m)\big]$ is then approximated by the minimum and maximum values over such iterations. This also confirms the convergence of the numerical algorithm in Step 5. See Appendix (ref).} GK18 propose a measure of the informativeness of the choice of an unrevisable prior for $Q$ that compares the width of such regions. The fact that in our case such measure is generally small (at any horizon), indicates that the fraction of the credible region tightened by choosing a particular unrevisable prior is very modest.\footnote{The informativeness of the prior with credibility $\alpha$ is defined as $\left\{1 - \left[\text{width highest posterior density}(\alpha)/\text{width robust credible region}(\alpha)\right]\right\}$. Such fraction is in the range 0.05-0.52 for the impact response to an oil supply shock (i.e. the largest fraction is that associated with the response of real economic activity) and in the range 0.03-0.37 for the impact response to an aggregate demand shock (i.e. the largest fraction is that associated with the response of world crude oil production). At horizon 12 such intervals become 0.03-0.21 and 0.06-0.22.}
Model $\mathcal{M}_3$ illustrates that our methodology allows to point identify HSVAR models in the presence of multiple eigenvalues and that this can be achieved with less zero restrictions than in the case of recursive identification. In $\mathcal{M}_3$ we impose that the first shock does not affect real economic activity within the same month. Interestingly, this restriction is consistent with the evidence in Figures (ref) and (ref) where we see that the impact response of real economic activity to an oil supply shock is close to zero. \\ Results for $\mathcal{M}_3$ are reported in Figure (ref). Here, the zero restriction has the effect of tightening the width of the HPD regions, when compared to the standard HSVAR model $\mathcal{M}_1$. Focusing on the response of real price of oil to an oil disruption, we see that in this case the response is slightly more significant, as the HPD region does not always contain the zero. One difference with the benchmark, $\mathcal{M}_0$, concerns the response of world crude oil production to an aggregate demand shock that here is positive, with highest posterior density region that does not contain the zero up to horizon 20.
This paper deals with SVAR models with structural breaks on the second moments of the structural shocks, offering some new contributions. We first study the identification theory and propose a set of results for easily checking whether the model is globally identified. Second, we study the consequences on the impulse response functions of HSVARs that do not satisfy such identifying conditions. We deal with a SVAR model with heteroskedasticity, where non distinct changes in the variance shifts raise an identification issue. We solve the identification problem by imposing equality and sign restrictions and provide a methodology that helps giving a structural economic interpretation to the set or point identified shocks, by requiring fewer restrictions to be imposed. A way to do inference on the model both in case of point and set-identification is also proposed, as well as an empirical application of our approach to the global crude oil market model. Some issues remain to be addressed by future research, such as extending the model to more than two volatility regimes and analysing the consequences of having an unknown break date.
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