Invalid proxies and volatility changes

Angelini, Giovanni; Fanelli, Luca; Neri, Luca Working Paper Invalid proxies and volatility changes Quaderni - Working Paper DSE, No. 1193 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Angelini, Giovanni; Fanelli, Luca; Neri, Luca (2024) : Invalid proxies and volatility changes, Quaderni - Working Paper DSE, No. 1193, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/7606 This Version is available at: https://hdl.handle.net/10419/300135 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ ISSN 2282-6483 Invalid proxies and volatility changes Giovanni Angelini Luca Fanelli Luca Neri Quaderni - Working Paper DSE N°1193 INVALID PROXIES AND VOLATILITY CHANGES Giovanni Angelinia, Luca Fanellia, Luca Neria,b March 2024 Abstract When in proxy-SVARs the covariance matrix of VAR disturbances is subject to exogenous, permanent, nonrecurring breaks that generate target impulse response functions (IRFs) that change across volatility regimes, even strong, exogenous external instruments can result in inconsistent estimates of the dynamic causal effects of interest if the breaks are not properly accounted for. In such cases, it is essential to explicitly incorporate the shifts in unconditional volatility in order to pointidentify the target structural shocks and possibly restore consistency. We demonstrate that, under a necessary and sufficient rank condition that leverages moments implied by changes in volatility, the target IRFs can be point-identified and consistently estimated. Importantly, standard asymptotic inference remains valid in this context despite (i) the covariance between the proxies and the instrumented structural shocks being local-to-zero, as in Staiger and Stock (1997), and (ii) the potential failure of instrument exogeneity. We introduce a novel identification strategy that appropriately combines external instruments with “informative” changes in volatility, thus obviating the need to assume proxy relevance and exogeneity in estimation. We illustrate the effectiveness of the suggested method by revisiting a fiscal proxy-SVAR previously estimated in the literature, complementing the fiscal instruments with information derived from the massive reduction in volatility observed in the transition from the Great Inflation to the Great Moderation regimes. Keywords: External instruments, Fiscal multipliers, Identification, ProxySVARs, Structural breaks, Shifts in volatility, Weak instruments. JEL Classification: C32, C51, C52, E62 aDepartment of Economics, University of Bologna, Italy. bCarlo Giannini Fellow in Econometrics. Correspondence to: Luca Fanelli, Department of Economics, University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy; email: [email protected]. 1 Non-technical summary This research investigates the extent to which proxy-SVAR (SVAR-IVs) methods are effective. Such methods are used to understand how economic policies affect the economy. For instance, in the empirical application, we study what happens when governments implement polices that change their spending or tax revenues. Historically, however, the parameters of interest change with volatility regimes due to changes in market structures, preference parameters, or policy conduct. Then, standard proxy-SVAR methods fail to pinpoint the effect of such policies. In the presence of volatility regimes, we find that proxySVAR methods can yield reliable results if two conditions are met. First, the way policies affect the economy must stay the same, even when the overall economic environment changes. Second, the method must focus on relative effects of these policies, rather than their absolute value. This paper addresses this challenge and establishes that, by meeting a technical necessary and sufficient rank condition, informed by volatility regimes, we can identify and accurately estimate parameters of interest, even when faced with irrelevant or contaminated external instruments, and standard inference applies. The study demonstrates that properly combining volatility changes with external instruments can significantly improve inference quality. We employ this novel approach to revisit the analysis of a seminal fiscal proxy-SVAR estimated for the US economy, augmenting the set fiscal instruments with the change in the unconditional VAR error covariance matrix. This captures the decline in volatility observed in the transition from the Great Inflation to the Great Moderation period. Our findings reveal that: (i) the estimated fiscal multipliers are larger than one; (ii) the peak tax multiplier appears smaller than the estimated peak fiscal spending multiplier, albeit not dramatically; and (iii) the uncertainty around the dynamic tax multiplier substantially reduces when we consider the volatility change, as opposed to using the fiscal proxy-SVAR across the entire estimation sample disregarding the shift. 2 1 Introduction The last decades have witnessed significant advancements in the development of novel methods for the identification of macroeconomic shocks in Structural Vector Autoregressions (SVARs). Among these, “the identification by external instruments” and “the identification by heteroskedasticity” play an important role, see e.g. Stock and Watson (2017). In the identification by external instruments, the focus is on specific structural shocks which are identified through the use of variables external to the VAR, henceforth referred to as instruments or proxies interchangeably. External instruments address a “partial identification” problem, following the approach proposed by Stock (2008), Stock and Watson (2012,2018), and Mertens and Ravn (2013). The proxies must be relevant, i.e. correlated with the structural shock(s) of interest, and exogenous, i.e., uncorrelated with the non-instrumented shocks. The fulfillment of both relevance and exogeneity conditions ensures that, under regular conditions, the target impulse response functions (IRFs) can be point-identified, consistently estimated, and standard asymptotic inference applies. Montiel Olea, Stock, and Watson (2021) have extended asymptotic inference to cases where SVARs feature “weak” proxies as in Staiger and Stock (1997). Their contribution emphasizes that even external variables which are poorly correlated with the target structural shocks can offer valuable information for identification. In this paper, we show that external instruments provide valuable information for identification even in seemingly problematic situations, such as the occurrence of shifts in the unconditional volatility of the variables associated with changes in the dynamic causal effects of interest. As is known, economic relationships are affected by structural breaks, typically induced by changes in underlying structural behavior, market conditions or changes in policy conduct. These breaks typically lead to shifts in the dynamics and volatility of the variables of interest. The identification through heteroskedasticity approach, first introduced in SVARs by Lanne and L¨utkepohl (2008) and inspired by the seminal work of Rigobon (2003) (also see Sentana and Fiorentini, 2001), relies on the information present in the data through variations in the unconditional covariance matrix of the VAR. This method is typically based on the assumption that breaks in volatility do not alter the impact and propagation of structural shocks but only shift the variance of these shocks.1Bacchiocchi and Fanelli (2015), Bacchiocchi, Castelnuovo, and Fanelli (2018) and Angelini, Bacchiocchi, Caggiano, and Fanelli 1This approach is commonly acknowledged as a “statistical” identification method, given that the shocks can only be labeled ex-post, i.e., after the model is estimated, typically by examining the signs of estimated on-impact coefficients or the implied shape of IRFs. 3 (2019) have shown that when there is a rationale to believe that volatility shifts may be induced by breaks in structural parameters, the heteroskedasticity approach can be extended to scenarios where IRFs change across volatility regimes.2In such cases, point identification of the target IRFs can be achieved by incorporating a limited set of theory-driven or institutionally-knowledgebased constraints into the model, while still benefiting from the identification power provided by shifts in volatility. These constraints, referred to as “stability restrictions” by Magnusson and Mavroeidis (2014), involve specifying particular structural parameters to vary across volatility regimes while keeping other structural parameters constant; see also Bacchiocchi and Kitagawa (2022b). This paper contributes to the literature on SVARs by exploring how changes in unconditional volatility contribute to the identification of the target structural shocks with external instruments. These models will be denoted as proxySVARs (SVAR-IVs) throughout. Specifically, we focus on proxy-SVARs where the covariance matrix of VAR errors exhibits exogenous, permanent, nonrecurring breaks, leading to changes in the target IRFs. In such cases, it is essential to explicitly incorporate the shifts in unconditional volatility into the analysis because even strong, exogenous external instruments may produce inconsistent estimates of the target IRFs if the breaks are not properly taken into account. The combination of external instruments with the shifts in volatility ensures the consistency of the estimator of dynamic responses and the use of standard asymptotic inference. This result marks an important difference relative to the scenario in which the target IRFs remain constant across volatility regimes. In that scenario, indeed, we show that relative (normalized) IRFs can be estimated consistently using strong and exogenous instruments even if the breaks in volatility are not accounted for. We establish that under a necessary and sufficient rank condition derived from moments induced by changes in volatility, identification is achieved even in the presence of proxies that are neither relevant nor exogenous. Notably, standard asymptotic inference continues to hold despite: (i) the covariance between proxies and instrumented structural shocks are local-to-zero as in Staiger 2SVARs, whose IRFs change across macroeconomic regimes in correspondence to different levels of unconditional volatility, are denoted as “SVAR-WB” (with WB standing for “with breaks”) in Bacchiocchi, Castelnuovo, et al. (2018) and Bacchiocchi and Kitagawa (2022b). These models are intended to reflect shifts in key structural parameters related to the behavior of economic agents, market functioning, and/or policy conduct. Similar phenomena have been studied in the literature, for example, by Lubik and Schorfheide (2004) and Castelnuovo and Fanelli (2015) in the context of solutions generated by monetary DSGE models, and by Clarida, Gal´ı, and Gertler (2000) and Boivin and Giannoni (2006) to explain macroeconomic phenomena such as the Great Inflation and the Great Moderation. 4 and Stock (1997); and (ii) the potential breakdown of instrument exogeneity, wherein instruments, beyond their correlation with the target shocks, also exhibit correlation with some or all non-target shocks, a phenomenon we refer to as “contamination”. In this context, the investigator has the flexibility to deduce the role of external instruments in the identification process directly from the data. In the least favorable scenario, external instruments serve as a labeling device for the target structural shocks. Based on the above results, we develop a novel identification strategy that integrates external instruments with shifts in volatility regimes, eliminating the necessity to assume proxy relevance and exogeneity prior to estimation. Within this framework, the Classical Minimum Distance (CMD) estimator emerges as a natural choice.3We call external variables that are both credibly relevant and exogenous as “valid instruments” or “valid proxies”. Conversely, we use the terms “invalid instruments” or “invalid proxies” for external variables where the conditions of relevance and/or exogeneity is not satisfied. A more comprehensive characterization is offered in Definition 1, Section 2. Our analytic results, supported by extensive Monte Carlo simulations, demonstrate that using external instruments in a framework where IRFs change across volatility regimes is generally advantageous. In fact, when strong and exogenous instruments complement identification based on shifts in volatility, there are considerable gains in estimation precision. Notably, even with contaminated yet strong instruments consistency is not affected and there are significant gains in estimation precision relative to the case in which only changes in volatility are leveraged. Remarkably, even in the worst-case scenario where weak, contaminated instruments are included in the analysis, the estimator of the target IRFs remains consistent and asymptotically Gaussian. Intuitively, the relevance condition is not strictly necessary to meet in our framework as point identification of the target structural shocks can be achieved, under the derived necessary and sufficient rank condition, by the moment conditions implied by the shifts in unconditional volatility. This result parallels the findings of Antoine and Renault (2017) regarding the “relevance of weaker instruments”, i.e. the scenario that emerges in Generalized Method of Moments estimation when the moment conditions characterized by localto-zero instruments as in Staiger and Stock (1997) complement the moment conditions associated with strong instruments. Thus, the changes in the VAR covariance matrix assume a role akin to relevant instruments in the external instruments approach. 3An alternative Quasi Maximum Likelihood (QML) approach, where the process governing VAR innovations and proxies is assumed conditionally normal within each volatility regime, is developed in the associated supplementary material. 5 The exogeneity condition can be relaxed due to the inherent “full identification” nature of the identification through changes in volatility, which delivers also information concerning the non-target shocks. The comprehensive point identification of both target and non-target shocks, a distinctive characteristic of the changes in volatility approach, gives rise to both advantages and limitations when external instruments are employed. The principal benefit lies in the fact that external instruments can be correlated with non-target structural shocks, other than the target shocks. Working with macroeconomic, aggregate data, there is a growing consensus that even when researchers carefully pick instruments that are plausibly exogenous, it is still unlikely that an instrument perfectly satisfies the orthogonality condition. Our approach empowers applied researchers to make reliable inferences in setups where instruments are nearly exogenous, not perfectly exogenous and, in general, correlated with the non-target shocks. In this regard, we note that, similar to Ludvigson, Ma, and Ng (2020,2021), our analysis does not primarily focus on relaxing exogeneity per se. Instead, our broader objective is to leverage the properties of external variables to facilitate identification Recently, Schlaak, Rieth, and Podstawski (2023) underscored the advantages that changes in volatility offer, under specific conditions, for testing instrument exogeneity in point-identified proxy-SVARs. In their framework, only the variances of structural shocks change across volatility regimes, while IRFs remain constant. A thorough comparison of our approach with that of Schlaak et al. (2023) will be presented in Section 3. Ludvigson et al. (2021), and Braun and Br¨uggemann (2023) have demonstrated the possibility, in principle, of handling proxies akin to the concept of “plausibly exogenous” instruments, as discussed by Conley, Hansen, and Rossi (2012).4Although we refer to point identification, our framework yields a similar result in the sense that the exogeneity condition need not to be imposed in estimation. Failure of the exogeneity condition as well as the strength of the proxies can be directly inferred from the data when the stability restrictions are correctly specified. Our framework also shares the same flexibility highlighted by Keweloh, Klein, and Pr¨user (2024) for models identified through the combination of proxy variables and non-Gaussian shocks. Remarkably, a significant advantage of our approach is that, given the stability restrictions, we do not rely on any assumptions regarding the distribution and/or cross-independence of the structural shocks. Our framework operates under the assumption that the structural 4In the microeconometric literature on Instrumental Variable (IV) regressions, there is a growing consensus that even when researchers carefully pick instruments that are plausibly exogenous, it is unlikely that an instrument perfectly satisfies the orthogonality condition; see, e.g. Berkowitz, Caner, and Fang (2012) and references therein. 6 shocks are cross-uncorrelated, see Ramey (2016, Section 2.1). Furthermore, in comparison to the aforementioned contributions, the suggested stability restrictions approach requires the specification of a set of (minimal) restrictions to achieve identification. Therefore, other than combing external instruments with changes in volatility, it addresses the challenges and limitations regarding the causal interpretation of purely statistical approaches to identification based on heteroskedasticity or non-Gaussian shocks, as emphasized by Montiel Olea, Plagborg-Møller, and Qian (2022) We present the key aspects of our approach by revisiting the fiscal proxySVAR estimated in Mertens and Ravn (2014) on US quarterly data covering the period 1950:Q1-2006:Q4. We use two fiscal instruments, one for the tax shock, which coincides with the narrative tax instrument of Mertens and Ravn (2014), and the other for the fiscal spending shock, taken from Angelini, Caggiano, Castelnuovo, and Fanelli (2023). Additionally, we consider a structural break in the VAR covariance matrix, capturing the substantial reduction in volatility observed during the transition from the Great Inflation to the Great Moderation period. The paper is organized as follows. Section 2covers our baseline proxySVAR specification (Section 2.1), the data generating process (DGP) and assumptions (Section 2.2), and our main results on identification, estimation and check of identifiability (Section 2.3). Section 3discusses connections and differences with contributions in the recent literature. Section 4applies the methodology to the identification of US fiscal multipliers. Section 5concludes. A supplement complements the paper in various dimensions, including formalization, comprehensive Monte Carlo experiments, proofs of propositions, additional empirical results, and the extension of the analysis to the case of multiple volatility regimes and QML estimation. 2 Proxy-SVARs with shifts in unconditional volatility In this section, we introduce the baseline proxy-SVAR specification within a DGP that incorporates a break (M= 1) in the error covariance matrix, resulting in M+ 1 = 2 volatility regimes in the data. We extend the analysis to more than two structural breaks in the supplementary material. Section 2.1 presents the proxy-SVAR as an augmented SVAR model and defines proxy properties. Section 2.1 outlines the DGP and the assumptions underpinning the analysis. Section 2.3 introduces the stability restrictions approach, covering identification conditions, CMD estimation, informal methods to check 7 2.3 The stability restrictions approach The main implication of Assumptions 1-2is that the subsets of observations (W1, ..., WTB) and (WTB+1, ..., WT) are characterized by two distinct VAR covariance matrices, Ση,2and Ση,1, respectively. The modeling of Ση,26= Ση,1 holds critical importance in this framework. A pertinent, related question is whether the target IRFs (3) can still be estimated consistently by using the instruments ztalone, despite the shift in volatility. We tackle this issue in Section 3and the supplementary material (see Section S.3) when we directly compare our approach with Schlaak et al. (2023). Our primary finding is that external instruments can be exclusively employed for inference, even when volatility breaks are disregarded, under two conditions: (i) the target IRFs remain constant across volatility regimes, indicating that the break solely affects the variance of the structural shocks while leaving their impact and propagation unchanged; (ii) “relative”, not absolute responses are estimated, i.e., responses obtained by imposing “unit effect” normalizations (see Proposition S.1). Conversely, when the target IRFs change across volatility regimes due to shifts in the impact of the target structural shocks, even strong and exogenous proxies fail to result in consistent estimation if the breaks in volatility are disregarded. Therefore, in general, shifts in volatility need to be incorporated in proxy-SVAR analysis. Our novel approach to model the change in volatility, based on target IRFs that change across volatility regimes is summarized in the next three sections. The supplementary material, Section S.4, provides a more “conventional” alternative where IRFs are assumed constant across volatility regimes. 2.3.1 Target IRFs and their identifiability We supplement Assumptions 1-2with a crucial condition under which the target IRFs deviate from those in equation (3). Assumption 3 (Regime-dependent IRFs) The dynamic causal effects produced by the target shocks can be summarized, for 1≤j≤k, by the IRFs: IRF•j(t, h) := (S0 n(Cy,1)hSn)H•1ej, t ≤TB, (S0 n(Cy,2)hSn)(H•1+ ∆H•1)ejt≥TB+ 1 (12) where ∆H•1denotes an n×kmatrix whose non-zero coefficients capture possible changes in the on-impact parameters in H•1in the shift from the first to the second volatility regime; ∆H•1:= H(2) •1−H•1,H(2) •1being the analogous of the matrix H•1in the second volatility regime. 14 Note that for ∆H•16= 0n×k, Assumption 3implies Ση,26= Ση,1, hence it does not conflict with Assumption 1(iv). Furthermore, the companion matrices Cy,i in (12) depend, for i= 1,2, on the autoregressive (slope) parameters in Γi, see (11), hence the slope parameters can remain possibly constant under Assumption 1.8Moreover, (12) is formulated to depict responses to one-standard deviation target shocks in both volatility regimes. We elaborate on this concept later in the paper. Throughout, for any matrix A, the notation ∆Awill refer to a matrix with the same dimensions as A, where non-zero elements represent potential parameter changes from the first to the second volatility regime. Formally, ∆Ais defined as ∆A:= A(2) −A, with A(2) corresponding to Ain the second volatility regime. We use the same notation for elements of a matrix, i.e. ∆ai,j := a(2) i,j −ai,j, for ai,j (i, j)-element of A, and a(2) i,j (i, j)-element of A(2). To point-identify and estimate the target IRFs (12) in system (11), we model the relationship between the vector collecting VAR innovations and proxies, ηt, and the vector that includes the structural shocks and the measurement errors associated with the proxies, ξtby: ηt=Gξt+ ∆G·I(t>TB)ξt,(13) where the matrix Ghas the structure discussed in Section 2.1, see (9), and the term ξthere is such that it is respected the condition E(ξtξ0 t) = In+r. In its general form, the structure of the matrix G+ ∆Greads: G+ ∆G=H0 RΦΩω | {z } G +∆H0 ∆RΦ∆Ωω | {z } ∆G := H•1H•20 Φ Υ Ωω | {z } G +∆H•1∆H•20 ∆Φ∆Υ∆Ωω | {z } ∆G (14) where it is seen that ∆Φ, ∆Ψand ∆Ωωcapture possible changes in the parameters governing proxy properties, while ∆H•2captures possible changes in the on-impact coefficients associated with the non-target shocks. Recall that the top-right blocks of zeros in Gand ∆Gpertain to the impact of measurement errors associated with the instruments on the variables, which must be zero in both volatility regimes by construction. Under (14), the dynamics of the 8It turns out that the target IRFs may change across the two volatility regimes under Assumption 1either because ∆H•16= 0, or possibly because both conditions ∆H•16= 0 and Cy,26=Cy,1hold. 15 proxies ztcan be described by the linear measurement error model: zt=RΦεt+ Ωωωt | {z } first volatility regime + [∆RΦεt+ ∆Ωωωt]I(t>TB) | {z } second volatility regime = [Φ + ∆ΦI(t>TB)] | {z } ˜ Φt ε1,t + [Υ + ∆ΥI(t>TB)] | {z } ˜ Υt ε2,t + [Ωω+ ∆ΩωI(t>TB)] | {z } ˜ Ωω,t ωt(15) where, under drifting DGPs characterized by sequences of models in which E(ztε0 1,t) = ˜ ΦT→˜ Φ = (Φ + ∆Φ), and for ˜ Υt= 0r×(n−k)or ˜ Υt6= 0r×(n−k), the model accounts for instrument properties as featured in Definition 1. Equation (15) remarks that the relevance and exogeneity of zt, as well as the variance of measurement errors, are allowed to change across volatility regimes. It also suggests that in line with Definition 1above and as in, e.g., Ludvigson et al. (2020,2021), the external instrument ztcan be correlated with shocks other than the target shocks, via nonzero elements in Υ and/or ∆Υ. As is well-known, the proxy-SVAR approaches developed by Mertens and Ravn (2013) and Stock and Watson (2018) achieve point identification by assuming that the external variables exhibit zero correlations with the non-target shocks. Conversely, the methodology proposed in the frequentist framework by Ludvigson et al. (2020,2021) for set-identified SVARs allows the external variables to display departures from exogeneity; see Braun and Br¨uggemann (2023) for a Bayesian perspective. Ludvigson et al. (2020) note that the focus of their analysis is not on relaxing exogeneity per se, but on the broader objective of using the external instruments to aid in identification. Here, we demonstrate that breaks in unconditional volatility enable us to address a situation akin to Ludvigson et al. (2020), with the crucial distinction being that we achieve point identification. Importantly, this methodology offers the potential for the data to inform us about the extent of the correlations between the proxies zt and (part of) the non-target shocks. The moment conditions implied by model (13) are: Ση(t) = Ση,1=GG0t≤TB, Ση,2= (G+ ∆G)(G+ ∆G)0t≥TB+ 1 (16) and in light of the structure of the matrices Gand ∆Gin (14), system (16) implies that even when proxy exogeneity holds in both volatility regimes, Υ = 0r×(n−r)and ∆Υ= 0r×(n−r), the moment conditions supply information not 16 only on the parameters of direct interest H•1and ∆H•1, see (12), but also on elements in H•2and ∆H•2, that are not of direct interest in the analysis. We discuss the “full/partial” nature of the approach that incorporates volatility changes to external instruments in this and in the next section. It is important to highlight that the parameterization in (16) is more general than one might expect. Consider, e.g., an alternative parameterization given by: Ση(t) = (Ση,1=˘ GV(1) ˘ G0t≤TB, Ση,2= ( ˘ G+˘ ∆G)V(2)(˘ G+˘ ∆G)0t≥TB+ 1 (17) where the structural shocks and proxy measurement errors, ξt:= (ε0 t, ω0 t)0, are now intended to have variance E(ξtξ0 t) = V(1) in the first volatility regime, and E(ξtξ0 t) = V(2) in the second volatility regime. Here V(1) and V(2) are diagonal matrices with positive elements on the diagonal, respectively, and ˘ G and ˘ G+˘ ∆Gdiffer from Gand G+ ∆Gin (14) only in having “1” on their main diagonals. One interpretation of (17) is that the change in the covariance matrices Ση,2and Ση,1is now explained by changes in the variances of the structural shocks, captured by V(2) 6=V(1), as well as changes in on-impact coefficients, captured by the nonzero elements in the matrix ˘ ∆G. However, taken (17) and ˘ G,˘ ∆G,V(1) and V(2) as DGP, it is always possible to find matrices Gand ∆Gsuch that the following equalities hold: GG0=˘ GV(1) ˘ G0t≤TB, (G+ ∆G)(G+ ∆G)0= ( ˘ G+˘ ∆G)V(2)(˘ G+˘ ∆G)0t≥TB+ 1. For example, the equations above hold with G:= ˘ GV 1/2 (1) and ∆G= ( ˘ G+ ˘ ∆G)V1/2 (2) −˘ GV 1/2 (1) . In light of this equivalence, we prefer to rely on the moment conditions in (16) whose implied IRFs refer to one-standard deviation shocks in both volatility regimes. Defined the vectors ση,1=vech(Ση,1) and ση,2=vech(Ση,2), then the moment conditions can be expressed in the more compact form: ση,1=vech(GG0) ση,2=vech(G+ ∆G)(G+ ∆G)0(18) and, as in Magnusson and Mavroeidis (2014) identification can be attained through the following set of (linear) constraints on Gand ∆G: vec(G) = SGγ, (19) 17 vec(∆G) = S∆Gδ. (20) In (19)-(20), SGis an (n+r)2×aselection matrix of full column rank, a < (n+r)2,γ:= (γ0 H, γ0 R, γ0 Ωω)0is the a-dimensional vector collecting the free nonzero parameters entering the matrix G, with γH, γR,and γΩωcontaining the nonzero elements in H,RΦ, and Ωω, respectively; S∆Gis an (n+r)2×b selection matrix of full column rank b,b < (n+r)2, and δ:= (δ0 H, δ0 R, δ0 Ωω)0is the b-dimensional vector of free nonzero parameters in the matrix S∆G, with δH, δRand δΩωhaving analogous interpretation as γH,γRand γΩω, respectively. Thus, since the elements in the vector δcapture changes from the first to the second volatility regime, the restrictions on ∆Gshould be strictly interpreted as stability restrictions. The stability restrictions in (19)-(20) serve a dual purpose. On the one hand, the constraints imposed on matrix Gare instrumental in identifying k target structural shocks, alongside n−knon-target shocks possibly under a parsimonious set of restrictions (more on this below).9On the other hand, the constraints on ∆Gdetermine which of the non-zero and zero proxy-SVAR parameters contained in matrix Gundergo changes in the transition from the first to the second volatility regime. As it will be shown in the empirical illustration in Section 4, the specifications of the matrices G(γ), ∆G(δ), SGand S∆Gin (14)-(20) are tailored to the specific problem and the scopes of the analysis. Importantly, besides leveraging the change in volatility, the stability restrictions that the investigator specifies in (19)-(20) are grounded in the underlying theory or knowledge of the phenomenon under study. They do not rely on statistical information such as the distribution of the structural shocks or their cross-independence. It can be noticed that, jointly equation (18) through (20) jointly characterize a “full identification” problem in the sense that, as they stand, the moment conditions and identification restrictions affect not only the target structural shocks but also the non-target ones. The decision to take a (partial) stance on the non-target shocks arises when (n−k)>1 (see Section 4).10 In the 9It is worth noting that, for k > 1, even in a “conventional” proxy-SVAR with no structural breaks, achieving point-identification necessitates at least 1 2k(k−1) additional restrictions beyond the instruments in, e.g., Mertens and Ravn (2013) and Angelini and Fanelli (2019). 10In principle, our approach can be extended to the case in which identification and estimation are developed by partialing out the influence of non-target shocks from the analysis. Specifically, it is possible to maintain that changes in volatility can be exclusively attributed to parameters related to the impact of the target shocks on the variables. One advantage of this solution is that it relieves the investigator from taking a stance on the non-target shocks. A drawback is that one must assume proxy exogeneity in estimation, leading to the loss of one of the benefits of our suggested approach. 18 remainder of the paper, we delve into the (local) point identification and estimation of the proxy-SVAR under the general specification defined by equations (18)-(20), emphasizing the strengths of the proposed methodology. The parameters associated with the target IRFs in (12) are elements of γand δ, that we collect in the vector θ. More precisely, θ:= (γ0 H•1, δ0 H•1)0, with γH•1,δH•1, being subvectors of γHand δH, respectively. θis referred to as the vector of parameters of interest. The moment conditions (18) feature (n+r)(n+r+ 1) reduced-form coefficients, ση,1and ση,2, and a+bfree parameters that we collect in the vector ς:= (γ0, δ0)0. We can conveniently summarize these moment conditions by the distance function: m(ση, ς) := m1(ση,1, ς) m2(ση,2, ς)=ση,1−vech(GG0) ση,2−vech(G+ ∆G)(G+ ∆G)0,(21) where it is intended that the matrices Gand ∆Gare constrained as in (19)-(20). Equation (21) shows that the point identification problem of θis equivalent to the problem of uniquely recovering the vector ς, comprising some nuisance parameters, from the reduced-form covariance parameters in ση,1and ση,2, respectively. The next proposition establishes the necessary and sufficient conditions for this to happen. We denote with ς0the true value of ς. Proposition 1 (Identification under changing IRFs) Given the proxySVAR from Assumptions 1-3, consider the moment restrictions in (21) with Gand ∆Grestricted as in (19)-(20). Assume ς0is a regular point in the parametric space Pς. Then, irrespective of whether the proxies ztsatisfy one of the conditions in Definition 1: (i) a necessary and sufficient rank condition for the identification of ςin a neighborhood of ς0is that rank[J(ς0)] = a+b, where J(ς0)is the (n+r)(n+r+ 1) ×(a+b)Jacobian evaluated at ς0, given by: J(ς0) := ∂m(ση, ς) ∂ς0ς=ς0 , ∂m(ση, ς) ∂ς0= 2 I2⊗D+ n+r(G⊗In+r) 0(n+r)2×(n+r)2 (G+ ∆G)⊗In+r(G+ ∆G)⊗In+rSG0 0S∆G; (22) (ii) a necessary order condition is: (a+b)≤(n+r)(n+r+ 1).(23) 19 The main message from Proposition 1is that in the presence of a shift in unconditional volatility, the stability restrictions in (19)-(20) allow to pointidentify the parameters in a neighborhood of ς0, hence the proxy-SVAR parameters, θ, and the target IRFs. The result holds regardless of the properties of the external instruments outlined in Definition 1. The possible breakdown of proxy relevance and exogeneity does not affect the necessary and sufficient rank condition in Proposition 1.11 Intuitively, relevance is not strictly necessary because regardless of the local rank properties of Φ,the rank of the Jacobian matrix J(ς) in (22) remains unaffected by sequences of matrices ΦTconverging to Φ; this implies that even in cases where the proxies satisfy the conditions in Definition 1.(ii) and 1.(iv), identification of the proxy-SVAR can still be achieved through the shift in volatility. On the other hand, the exogeneity condition can be potentially relaxed due to the “full identification” nature of the approach through changes in volatility, which inherently delivers information concerning the non-target shocks, other than the target shocks. Provided the necessary and sufficient rank condition holds, the target structural shocks can be recovered even when the instruments are correlated with some non-target shocks. This flexibility, however, comes at the cost of the investigator needing to take a stance, at least partially, on how non-target shocks impact the variables, which requires a few constraints on H•2and ∆H•2, via (19)-(20). As shown in the empirical illustration in Section 4where we estimate a fiscal proxy-SVAR for the US economy with a shift in unconditional volatility, this stance can often be established by leveraging insights from other studies.12 2.3.2 Estimation To estimate the target IRFs under Proposition 1, we adopt the CMD approach. Assumptions 1-3suffice to guarantee the consistency and asymptotic normality of the CMD estimator, as indicated in Proposition 2below. Notably, no distribution assumption is required for ηt. 11It is noteworthy that the necessary and sufficient rank condition in Proposition 1remains valid, as expected, even when the restrictions imposed by the investigator Gand ∆Gimply “sub-sample identification”. With this term we mean that the identification of parameters γ and δcan be achieved through two SVAR analyses conducted separately on the two volatility regimes; see e.g. Blanchard and Gal´ı (2009). It is evident that the utility of the identification conditions outlined in Proposition 1extends well beyond the phenomenon of sub-sample identification. 12We establish an analogous proposition to Proposition 1for cases in which the target IRFs are assumed to remain constant across volatility regimes; see supplementary material, Section S.4, Proposition S.3. This proposition provides a theoretical foundation for the results that Schlaak et al. (2023) documented only through simulation studies for the case r=k= 1. 20 To introduce the estimation method, it may be useful to preliminary start from the asymptotic properties of the estimator of the reduced-form covariance matrices of the proxy-SVAR in (11), ˆση:= (ˆσ0 η,1,ˆσ0 η,2)0. Under Assumptions 1-2, it holds the asymptotic normality result: √T(ˆση−ση,0) = √Tˆση,1−ση,1,0 ˆση,2−ση,2,0d →N(0, Vση) , Vση:= Vση,10 0Vση,2 (24) where ση,0:= (σ0 η,0,1, σ0 η,0,2)0is the true value of ση, and the structure of the asymptotic covariance matrices Vση,i ,i= 1,2 is discussed in detail by e.g. Br¨uggemann et al. (2016); see also references therein. Henceforth, we assume the existence of a consistent estimator for Vση, denoted ˆ Vση.13 Under the conditions of Proposition 1, the estimator of the parameters ς, hence of θ, is obtained by solving the minimization problem: ˆςT:= arg min ς∈Pς mT(ˆση, ς)0ˆ V−1 σηmT(ˆση, ς) (25) where mT(ˆση, ς)0:= mT,1(ˆση,1, ς)0, mT,2(ˆση,2, ς)00is the distance function in (21) with σηreplaced with the estimator ˆση.14 Let ˆ θT:= (ˆγ0 H•1,T ,ˆ δ0 H•1,T )0be the subvector of the CMD estimator ˆςT:= (ˆγ0 T,ˆ δ0 T)0. The next proposition establishes asymptotic properties. Proposition 2 (Asymptotic properties CMD estimator) Let ˆςTbe the CMD estimator of the parameters ςobtained from (25), and ˆ θT:= (ˆγ0 H•1,T ,ˆ δ0 H•1,T )0 be the corresponding subvector of ˆςT. Let ς0be an interior of Pς(assumed compact), with θ0subset of ς0(θ0∈ Pθ⊆ Pς). Under the conditions of Proposition 1: ˆςT p →ς0,√T(ˆςT−ς0)d →N(0, Vς) ˆ θT p →θ0,√T(ˆ θT−θ0)d →N(0, Vθ) where Vς:= J(ς0)0V−1 σηJ(ς0)−1and Vθis the corresponding block of Vς. 13Under Assumptions 1-2and mild additional auxiliary conditions, the residual-based moving block bootstrap (MBB), as introduced by Br¨uggemann et al. (2016) (and outlined by Jentsch and Lunsford (2022) in the context of proxy-SVAR models), is consistent. This means that the regime-dependent covariance matrices Vση,i that form Vσηin (24) can be estimated by applying the MBB separately within each volatility regime. 14As is known, the optimization problem in (25) can be influenced by multiple local minima, aligning with the fact that the necessary and sufficient rank condition in Proposition 1holds for local, not global identification; we refer to Bacchiocchi and Kitagawa (2022a) for SVARs in which identification is local and not global. When two or more local minima arise from (25), one reasonable selection criterion is to inspect ex-post the signs of estimated coefficients and compare them with the signs expected from the theory. 21 Proposition 2establishes that the target IRFs can be consistently estimated, and standard asymptotic inference can be applied regardless of the proxy properties formalized in Definition 1. The simulation studies, summarized in the supplementary material, emphasize the benefits arising from the results in Propositions 1-2in terms of gains in efficiency in the estimation of target IRFs relative to not including external instruments in the analysis. Proposition 2ensures that when (n+r)(n+r+ 1) >(a+b), the number of overidentifying restrictions resulting from the estimation problem (25), (n+ r)(n+r+ 1) −(a+b), can be empirically evaluated using the overidentifying restrictions test. Notably, as evidenced by our simulation studies, the test for overidentifying restrictions appropriately rejects when the stability restrictions include incorrect constraints; for instance, when proxy exogeneity is imposed in estimation but the DGP of the instrument belongs to model (15) with Υ or (Υ + ∆Υ) different from zero. Jointly, Propositions 1-2provide the foundation for our approach to the identification and estimation of proxy-SVARs with permanent, nonrecurring breaks in the error covariance matrix. Essentially, the suggested approach (i) does not necessitate pre-testing proxy strength and exogeneity; (ii) does not need to rely on weak-instrument robust methods; (iii) does not require imposing proxy exogeneity in estimation. 2.3.3 Checks of identifiability The identification and estimation approach discussed in the previous section holds if and only if the Jacobian matrix J(ς) in (22) is full column rank. When the rank condition rank[J(ς)] = a+bholds locally, the shift in unconditional volatility captured under Assumption 1(iv) by the distance Ση,2−Ση,1, conveys sufficient information to compensate for potential instrument weakness without compromising the inference on the target structural shocks and the validity of standard asymptotics. Even contaminated instruments do not affect estimates’ consistency when stability restrictions are correctly specified. Conversely, when the shift Ση,2−Ση,1is “weak”, the shift in unconditional volatility does not provide sufficient information for identification. We study such phenomenon in Section S.6 of the supplement, focusing in detail on how changes in volatility impact the necessary and sufficient rank condition for identification derived in Proposition 1. In particular, we approximate scenarios characterized by weak changes in volatility with the phenomenon of “shrinking shifts”, which arises when Assumption 1(iv) does not hold because the distance Ση,2−Ση,1tends to vanish as T→ ∞. We postpone to future research the elaboration of a comprehensive robust approach to proxy-SVARs that fully 22 integrates potentially weak external instruments in a context in which the identification information stemming from heteroskedasticity is scant. In practical situations practitioners can, in principle, test the rank of the Jacobian matrix ex-post, meaning after estimating the model and substituting the elements in Gand ∆Gin (22) with their estimates, obtaining J(ˆςT); see, e.g., Kleibergen and Paap (2006) and Al-Sadoon (2017) and references therein. Alternatively, one can apply the pre-test for the null hypothesis that IRFs do not change, against the alternative that they change across a finite number of volatility regimes, as recently introduced by L¨utkepohl and Schlaak (2022) for the one instrument-one shock case, generalized in Bruns and L¨utkepohl (2024) to the multiple instruments setup. However, while our approach accommodates invalid proxies, Bruns and L¨utkepohl (2024) test requires strong, exogenous instruments. Hence, it can be interpreted as an implicit assessment of the validity of the rank condition of the Jacobian J(ς) only when the investigator has no doubts about instrument strength and exogeneity. In any case, if a pre-test of the identification conditions is undertaken, post-test inference must be adjusted accordingly. In the empirical illustration discussed in Section 4, following the indications arising from our Monte Carlo sperimentations, we assess the quality of identification by inspecting the smallest singular value implied by the estimated Jacobian matrix J(ˆςT) relative to its associated uncertainty as captured by a bootstrap confidence interval, emphasizing that no rigorous inference is being drawn in this process.15 3 Connections and differences with the literature In this section, we review contributions in the existing literature where external instruments are explicitly combined with changes in volatility, highlighting the main differences with our approach. While the current literature sees a growing number of articles driven by the idea that “blending” different methods enhances the (pointor set-) identification of the structural shocks of interest, 15In our Monte Carlo experiments, we evaluate the identifiability of the analyzed proxySVARs with changes in volatility by examining the distribution of the smallest eigenvalue of the matrix J(ˆςT)0J(ˆςT), which corresponds to the smallest singular value of the estimated Jacobian matrix J(ˆςT).Results suggest that in situations in which the necessary and sufficient conditions for identification in Proposition 1holds, the variability surrounding the estimated smallest singular value of the Jacobian matrix J(ˆςT) is considerably smaller compared to cases in which the distance between covariance matrices across volatility regimes tends to shrink. 23 +          ∆h(1) 1,1 ∆h(1) 1,2 0 0 0 ∆h(1) 2,1 ∆h(1) 2,2 0 0 0 0 ∆h(1) 3,2 ∆h(2) 3,1 0 0 ∆ϕ1,1∆ϕ1,20 ∆σω,10 0 ∆ϕ2,20 0 ∆σω,2          | {z } ∆G:=  ∆H•1 ∆Φ ∆H•1 ∆Υ 0 ∆Ωω   I(t>TB)       εtax t εg t εgdp t ωtax t ωg t        | {z } ξt (28) and is based on the following set of hypotheses. The first two columns of the matrix G, which pertain to the instantaneous impact of the fiscal shocks on the variables in the first volatility regime, reproduce exactly the structure of the matrix (H0 •1,Φ0)0considered in the specification (26)-(27) for which the proxy-SVAR was estimated on the whole sample period. Given the full identification nature of the changes in volatility approach, the third column of the matrix Grefers to the instantaneous impact of the output (non-target) shock on the variables. In this case, we borrow the restriction h(2) 2,1= 0 from Mertens and Ravn (2014). This restriction, considered an uncontroversial tenet in the US fiscal empirical literature, posits that fiscal spending does not respond instantaneously to the output shock; see also Blanchard and Perotti (2002). Furthermore, we relax a-priori the exogeneity of the two fiscal proxies ztwith respect to the output shock, leaving the contamination parameters Υ1,1and Υ2,1unrestricted in (28), letting the data to inform us about possible violation of the exogeneity condition. Finally, we compensate for the zero restriction in the position (2,1) of the matrix of relevance parameters Φ by allowing the measurement error affecting the tax proxy to potentially influence the variance of fiscal spending instrument through the parameter σω,2,1. In the second volatility regime, i.e. for the sample starting from t≥TB+1, we specify a single stability restriction pertaining to the impact of the tax shock. In particular, we posit that the instantaneous impact of the tax shock on output remains constant across the two regimes, implying ∆h(1) 3,1 = 0 in the first column of ∆H•1in (28). To motivate this constraint, note that the inspection of the bottom-right graph Figure 1, which plots the “great ratio” TAXt− GDPt, suggests that the relative dynamics of tax revenues and GDP remains substantially stable over the sample period 1950:Q1–2006:Q4, as it should be expected, e.g., under a sustainable debt policy. It can be therefore argued that the change in volatility occurring at TB= 1984:Q1, affects both time series simultaneously, which is unsurprising given the strict cyclical relationship connecting tax revenues and output. Also, considering the nature of the proxy 30 used to instrument the latent tax shock which is highly zero-censored and does not display marked changes across the two volatility regimes, the stability restriction ∆h(1) 3,1 = 0 appears a reasonable one in our context. All other nonzero on-impact coefficients in H•1, as well as the nonzero relevance parameters in Φ are permitted to change across the two volatility regimes. As for the on-impact coefficients associated with the non-target output shock, we specify the matrix ∆H•2similar to H•2, i.e., keeping the restriction borrowed from Mertens and Ravn (2014) also valid in the second volatility regime (∆h(2) 2,1 = 0) and letting all other coefficients vary. Lastly, we keep the contamination parameters Υ1,1 and Υ2,1unchanged relative to the Great Inflation. Finally, we allow for a shift in the variance of the fiscal instrument’s measurement error. The proxy-SVAR specified in (28) involves (a+b)=27 parameters, collected in the vector ς, which are spread across the matrices Gand ∆Gin (28), and is based on (n+r)(n+r+1)=30 moment conditions. The model is therefore overidentified if the necessary and sufficient rank condition in Proposition 1holds. The CMD estimates ˆςTresulting from problem (25) are summarized in the upper panel of Table 2along with 68% MBB confidence intervals. The overidentifying restrictions test reported in the bottom panel of Table 2strongly supports the estimated model with a p-value of 0.90, highly supportive of the chosen specification. An informal check of the quality of the identification of the estimated proxy-SVAR with the change in volatility is also summarized in the bottom part of Table 2, which displays the estimated smallest singular value of the Jacobian matrix, J(ˆςT), with associated 68% MBB confidence interval. As observed above, we refrain from interpreting the fact that the bootstrap confidence interval for the smallest singular value does not contain zero as conclusive statistical evidence that the rank identification condition in Proposition 1is met for the estimated model. At the same time, however, we do not observe clear-cut signs suggesting a potential lack of identification due to an insufficiently informative shift in volatility. Therefore, we can reasonably maintain that the reduction in volatility of the data observed in the shift from the Great Inflation to the Great Moderation regime suffices to point-identify the model, and standard asymptotic inference can be used in this framework. The CMD estimates in Table 2reveal important information about the properties and quality of the instruments used to estimate fiscal proxy-SVAR. Two main considerations arise. First, apparently, the relevance of the fiscal spending instrument zg tcaptured by the parameters ϕ2,2(Great Inflation) and ϕ2,2+ ∆ϕ2,2(Great Moderation), tends to decline because of the negative and significantly estimated parameter ∆ϕ2,2. However, in line with the Great Moderation phenomenon, this decline is compensated by a decrease in the 31 variance of the associated proxy measurement error (due to the significantly and negative estimated parameter ∆σω,2).Overall, the estimated correlation between the proxy zg tand the identified fiscal spending shock εg tremains high across the two volatility regimes; see the columns (ii) and (iii) of Table 1. Interestingly, the contamination parameter Υ2,1is estimated at a very low magnitude and is not statistically significant, supporting the hypothesis that the instrument used for the fiscal spending shock is exogenous and relevant on both volatility regimes. Second, the tax instrument ztax tis poorly correlated with the tax shock εtax tin the Great Inflation period, where the relevance parameter ϕ1,1is not statistically significant. However, relevance increases markedly in the Great Moderation regime, where the relevance parameter change, ∆ϕ1,1, is significant and the magnitude and statistical significance of ϕ1,1+ ∆ϕ1,1become substantial. To illustrate, examining columns (ii) and (iii) of Table 1, we observe that the implied correlation between the tax proxy ztax tand the estimated tax shock εtax tswings from 15% to 46% across the two volatility regimes. This marked change in the relevance condition is not surprising given the zero-censored nature of the narrative tax instrument, which, by construction, contains many zeros that inherently tend to weaken strength. A simple count shows that the number of zeros characterizing the tax instrument in the Great Inflation period, where volatility is higher, is considerably higher than the number of zeros in the Great Moderation, where ztax tseems to more accurately approximate the latent tax shock. Moreover, the 68% MBB confidence interval for the contamination parameter, Υ1,1, suggests that the tax proxy is negatively linked, albeit not dramatically, with the output shock. A similar finding is also documented in Keweloh et al. (2024), leveraging the non-normality of structural shocks in a Bayesian approach. The implied “contamination correlations” in Table 1, specifically in columns (ii) and (iii), vary from -11% in the Great Inflation to -9.8% in the Great Moderation. As emphasized, in our framework inference is reliable also when instruments are nearly exogenous and not necessarily perfectly exogenous. Specifically, the potential breakdown of the exogeneity condition does not compromise the consistency of the parameter estimator under correctly specified stability restrictions. Consequently, the dynamic fiscal multipliers derived from the estimates in Table 2, while fully capitalizing on and reflecting instrument properties, can be deemed robust to instruments being weak and/or contaminated. 32 4.2.1 Fiscal spending multipliers The dynamic fiscal multipliers resulting from the proxy-SVAR estimated in Table 2are plotted in Figure 2. Solid red lines refer to the Great Inflation period and are surrounded by red shaded 68% MBB confidence intervals; solid blue lines pertain to the Great Moderation period and are surrounded by blue shaded 68% MBB confidence intervals. While differences in terms of size and uncertainty for dynamic fiscal spending multipliers (lower panel) are mild across the two volatility regimes, they appear more pronounced for dynamic tax multipliers (top panel). Focusing first on the dynamic spending multipliers in the lower panel of Figure 2, we observe that, despite some statistically significant differences emerging between the blue and red lines at some initial horizons, magnitudes and associated uncertainty captured by 68% MBB confidence intervals appear not dramatically different across the two volatility regimes. The estimated peak fiscal spending multiplier, Mpeak g, summarized in the columns (ii) and (iii) of Table 1, is 2.38 in the Great Inflation and remains 2.38 in the Great Moderation period. These point estimates are surrounded by comparable 68% MBB confidence intervals, namely (1.2, 2.5) and (1.3, 2.9), respectively. The only notable difference that emerges between the two volatility regimes is that the peak effect is achieved 4 quarters after the shock in the Great Moderation and 2 quarters after the shock in the Great Inflation. These findings on the US fiscal spending multiplier diverge from those of Lewis (2021) on the one hand and share contact points with Fritsche et al. (2021) on the other hand. Lewis (2021), who considers our same estimation sample, exploits the nonparametric heteroskedasticity in fiscal data while keeping the target IRFs constant across volatility regimes. He detects a fiscal spending multiplier peaking at 0.75 after two quarters, very imprecisely estimated. Instead, among their many specifications, Fritsche et al. (2021) also rely on Markov Switching dynamics across high and low volatility states, allowing IRFs to change across these two states. Considering an estimation sample that partially covers the period after the Global Financial Crisis, Fritsche et al. (2021) confirm changes in the impact of government spending shocks between high and low volatility regimes, with the high volatility state essentially matching our Great Inflation period, and the low volatility state essentially covering our Great Moderation sample. They establish that the fiscal spending multiplier is significantly higher in the low volatility state (where it peaks around 2.5-3) compared to the high volatility state (where it peaks around 1.72-2). Our results align with those in Fritsche et al. (2021) and further strengthen their findings, as we complement the identification arising from the change in volatility with the information stemming 33 from a relevant, exogenous, fiscal spending instrument. 4.2.2 Tax multipliers Focusing on the dynamic tax multipliers plotted in the upper panel of Figure 2, noticeable differences across the two volatility regimes become apparent. Relative to the case in which the proxy-SVAR is estimated on the entire sample ignoring the break in volatility (black solid line), we observe in both volatility regimes a significant reduction in the magnitude of estimated dynamic multipliers, accompanied by a substantial decline in associated uncertainty. Specifically, our estimates of the tax multiplier in column (ii) of Table 1,Mpeak tax , peak at 1.99 (6 quarters after the shock) during the Great Inflation and decline to a peak of 1.66 (2 quarters after the shock) during the Great Moderation. In both cases, 68% MBB confidence intervals deliver considerably more precise estimates relative to the case in which the change in volatility is ignored. The estimated output elasticity of tax revenues, ψtax y, is 2.28 in the Great Inflation and 4.92, and imprecisely estimated, in the Great Moderation.22 These results suggest that the peak tax multipliers obtained with the proxy-SVAR approach on the whole estimation sample, approximately 3 in Mertens and Ravn (2014) and 2.62 in our framework, are likely to reflect a bias induced by the narrative tax instrument being weak, in addition to being contaminated by the output shock. Once we account for the shift in volatility, estimate consistency is restored, accompanied by a remarkable increase in precision. Overall, our analysis reveals four crucial findings: (i) the relevance of the narrative tax instrument of Mertens and Ravn (2014) shifts from “weak-like” to “strong-like” during the transition from the Great Inflation to the Great Moderation period; (ii) the exogeneity condition fails in the sense that we detect some non-negligible correlation between the tax instrument and the output shock; (iii) once instrument properties in (i) and (ii) are accounted for, the estimated peak tax multiplier is smaller than the estimated peak fiscal spending multiplier, though not dramatically so; (iv) the uncertainty surrounding the estimated dynamic tax multipliers reduces substantially compared to applying the fiscal proxy-SVAR on the entire sample without accounting for the change in volatility. Finding (ii) is also documented in Keweloh et al. (2024), who leverage the non-Gaussianity of structural shocks in a Bayesian context. 22As is known, there exists a direct link between the magnitude of the parameter ψtax yand the on-impact tax multiplier, as discussed in Caldara and Kamps (2017). In our case, the on-impact tax multiplier associated is not significant. 34 4.2.3 Forcing regime-invariant IRFs: The Proxy-SVAR-H approach For comparative purposes, we conclude our empirical analysis by forcing the target IRFs to be constant across the two volatility regimes. Hence, we implement Schlaak et al.’s (2023) approach, denoted Proxy-SVAR-H, whose analytic and empirical results have been summarized in the supplementary material, see Sections S.4.1 and S.4.2. The implied peak fiscal multipliers are summarized in column (iv) of Table 1. Figure 3summarizes all dynamic fiscal multipliers estimated in this paper, without reporting confidence intervals to improve readability. Colors are the same as in Figure 2. Dynamic multipliers implied by the Proxy-SVAR-H approach are plotted in green. 5 Concluding remarks We have developed an identification and estimation strategy for proxy-SVARs in cases where changes in economic behavior, market conditions, policy conduct, and institutional mechanisms induce permanent and nonrecurring shifts in the unconditional VAR error covariance matrix, leading to breaks in the target IRFs across volatility regimes. In such settings, even when instruments are relevant and exogenous, they may fail to produce consistent, asymptotically Gaussian estimates of both absolute and relative target IRFs if changes in volatility are not appropriately considered. We have introduced a novel methodology to address inference in proxy-SVARs in these cases. Our results emphasize that if the moment conditions arising from changes in volatility are sufficiently informative and allow for the point identification of target IRFs through stability restrictions, and if these stability restrictions are correctly specified by the econometrician, even invalid external instruments can contribute to identifying the structural shocks of interest. In general, external instruments improve estimation efficiency even when the exogeneity condition fails, and weak instruments can still convey information on the target shocks without the need to rely on weak-instruments robust methods. The comprehensive review on fiscal multipliers of Ramey (2019) highlights the significant lack of consensus on the tax multiplier, primarily attributed to the inherent difficulty in identifying exogenous tax shocks, a task more challenging than identifying fiscal spending shocks. Our estimator of the U.S. tax multiplier is robust to the tax proxy being weak in one regime and strong in the other, as well as to contamination from the output shock. Our analysis shows that the identification of the effects of fiscal policy can be substantially 35 improved by properly combining external instruments with changes in volatility. Acknowledgements We thank Ralph Br¨uggemann, Efrem Castelnuovo, Rustam Ibragimov, Madina Karamysheva, Sascha Keweloh, Damian Kozbur, Helmut L¨utkepohl, Mirela Mirescu, Anton Skrobotov, Tommaso Tornese, Rainer Winkelmann, Micheal Wolf, as well as seminar participants at the First VTSS Virtual Workshop for Junior Researchers in Time Series (April 2023), CEBA Talks (November 2023), UZH Seminars (November 2023), UEA Seminars, University of East Anglia (December 2023), and participants to the IAAE 2022 Annual Meeting (Oslo, June 2023), the SEM 2023 Conference (Milan, July 2023), the CFE 2023 Conference (Berlin, December 2023) and the SIdE-IWEEE 2024 Workshop (Bolzano, January 2024). We gratefully acknowledge financial support from MIUR (PRIN 2022, Grant 20229PFAX5) and the University of Bologna (RFO grants). References Angelini, G., Bacchiocchi, E., et al. (2019). “Uncertainty across volatility regimes”. In: Journal of Applied Econometrics 34.3, pp. 437–455. 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(2011). “Understanding the Aggregate Effects of Anticipated and Unanticipated Tax Policy Shocks”. In: Review of Economic Dynamics 14.1, pp. 27–54. 39 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 Tax multipliers Proxy-SVAR, 1950:Q1-2006:Q4 1st regime, 1950:Q1-1984:Q1 2nd regime, 1984:Q2-2006:Q4 Proxy-SVAR-H, break at 1984:Q1 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 Spending multipliers Proxy-SVAR, 1950:Q1-2006:Q4 1st regime, 1950:Q1-1984:Q1 2nd regime, 1984:Q2-2006:Q4 Proxy-SVAR-H, break at 1984:Q1 Figure 3: Estimated dynamic fiscal multipliers without confidence intervals at a 20-quarters horizon. Tax multipliers are in the upper panel; fiscal spending multipliers in the lower panel. Black solid lines refer to multipliers estimated on the whole sample 1950:Q1–2006:Q4 without accounting for a break in volatility. Red solid lines refer to multipliers estimated on the first volatility regime 1950:Q1–1984:Q1 (Great Inflation). Blue solid lines refer to multipliers estimated on the second volatility regime, 1984:Q2–2006:Q4 (Great Moderation). Green lines refer to multipliers obtained from the Proxy-SVAR-H approach (see supplement, Section S.4), i.e. accounting for a break in volatility while maintaining that IRFs remain constant across volatility regimes. 46 SUPPLEMENT TO INVALID PROXIES AND VOLATILITY CHANGES By Giovanni Angelini, Luca Fanelli, Luca Neri March 2024 Abstract This supplement complements the paper along many dimensions as described in the Introduction below. S.1 Introduction In this Supplement, we extend and complete the paper along several dimensions. Section S.2 introduces special matrices utilized in the paper and supplement. Section S.3 discusses the conditions under which the proxy-SVAR approach, which does not explicitly incorporate breaks in the VAR covariance matrices when these breaks are present in the DGP, yields consistent estimates of the target IRFs and when it does not. Section S.4 focuses on the case in which the shifts in unconditional volatility are assumed to solely affect the variance of the structural shocks, not their impact and transmission mechanisms. This implies constant IRFs across volatility regimes. Subsection S.4.1 formalizes the underlying theory, and Subsection S.4.2 applies it to infer the US fiscal multipliers, complementing the results in the paper. Section S.5 extends the identification and estimation approach outlined in the paper along two important dimensions. Firstly, it considers the case where the number of breaks in volatility is M≥2, resulting in M+ 1 volatility regimes. Secondly, it explores QML estimation as an alternative to the CMD estimation considered in the paper. Section S.6 investigates how the phenomenon of “shrinking shifts” impacts the necessary and sufficient identification rank condition derived in Proposition 1. Section S.7 summarizes the results of comprehensive Monte Carlo experiments that investigate the finite sample performance of the stability restrictions approach developed in the paper. In particular, we examine the performance of the stability restrictions approach in terms of relative efficiency of estimated target IRFs with respect to other estimation approaches, under different properties of external instruments. Furthermore, we examine the performance of the overidentifying restrictions test when the econometrician incorrectly imposes proxy exogeneity in estimation. We also evaluate methods to indirectly 1 assess the identifiability of the proxy-SVAR when the changes in volatility are “weak”. Unless differently specified, hereafter all references – except those starting with ‘S.’ – refer to sections, assumptions, equations and results in the main paper. S.2 Special matrices In the paper and in what follows, we often make use of the following matrices (Magnus and Neudecker, 1999): Dnis the n-dimensional duplication matrix (Dnvech(A) = vec(A), Abeing an n×nmatrix) and D+ n:= (D0 nDn)−1Dn is the Moore-Penrose generalized inverse of Dn.Kns is the ns-dimensional commutation matrix (Knsvec(A) = vec(A0), A being n×s). We simply use Knin place of Knn when n=s. Moreover, Nn:= 1 2(In2+Kn), and note that in the proof of propositions we often exploit the result: D+ nNn=D+ n. Finally, we denote with vecd(A) the vector containing the diagonal elements of the square matrix A. Then, given the p×pdiagonal matrix A, the p2×p derivative FA:= ∂vec(A) ∂vecd(A)0contains by construction ‘0’ and ‘1’. Specifically, the matrix FAis such that rank[FA] = pif the diagonal elements of Aare distinct. Conversely, rank[FA] = p−cwhen there are crepeated elements on the diagonal of A. Hence, the rank of FAdepends on whether there are repeated elements on the diagonal of Aor not. S.3 Proxy-SVARs estimation disregarding volatility breaks The covariance matrix Σu,z =E(utz0 t) encountered in (6) plays a crucial role in proxy-SVAR estimation. In the absence of structural breaks, Σu,z can be estimated by its sample analog: ˆ Σu,z := 1 T T X t=1 ˆutz0 t where ˆut,t= 1, ..., T, are the VAR residuals. As shown by Jentsch and Lunsford (2022) and Angelini, Cavaliere, and Fanelli (2024), under fairly general conditions on the process ηt:= (u0 t, z0 t)0, which encompasses the α-mixing hypothesis as specified in point (i) of Assumption 1, and regardless of proxy properties, the estimator ˆ ζT:= vec(ˆ Σu,z) is a √T-consistent, asymptotically Gaussian estimator of ζ:= vec(Σu,z). Hence, subject to standard regularity 2 conditions, proxy-SVAR estimation relies on the following results: ˆ ζT p →ζ0,√T(ˆ ζT−ζ0)d →N(0, Vζ) (S.1) where ζ0is the true value of ζ:= vec(Σu,z) and Vζis a positive definite covariance matrix. From equation (6), it follows that in the presence of “well behaved” proxies as captured by Definition 1.(i), Σu2,z(Σu1,z)−1=Hrel 2,1:= H2,1(H1,1)−1, which implies that the relative on-impact effects of the target shocks on the variables can be estimated by: ˆ Hrel 2,1:= ˆ Σu2,z(ˆ Σu1,z)−1(S.2) where ˆ Σu2,z and ˆ Σu1,z are the corresponding blocks of ˆ Σu,z. It turns out that under relevant and exogenous instruments, ˆ Σu2,z p →H2,1Φ0and ˆ Σu1,z p → H1,1Φ0,rank[Φ] = k, implying that ˆ Hrel 2,1in (S.2) is a consistent estimator of the true Hrel 2,1:= H2,1(H1,1)−1. Consistency, however, is no longer guaranteed when the instruments fail to be relevant and/or exogenous as in Definitions 1(ii)-1(iv). In the special case where r=k= 1 (one instrument is used for one target structural shock), robust asymptotically correct inference on the coefficients in Hrel 2,1:= H2,1(H1,1)−1can be grounded on weak-instrument robust techniques as outlined in the test inversion methodologies discussed by Montiel Olea, Stock, and Watson (2021). In these cases, the instrument can possibly be weak according to Definition 1.(ii) yet still informative. However, when k > 1, it is not evident how test inversion methods should be handled in the absence of further restrictions; see, e.g., Montiel Olea et al. (2021) and Angelini et al. (2024) for discussions. In this section, we investigate whether and under which conditions the estimators considered in (S.1) and (S.2) are consistent despite Σu,26= Σu,1, where Σu,1and Σu,2are the unconditional covariance matrices of VAR disturbances in the two volatility regimes. Assumption 1implies: Σu(t):=Σu,1·I(t≤TB)+Σu,2·I(t≥TB+ 1) , Σu,26= Σu,1.(S.3) To simplify the analysis, we consider the following auxiliary assumptions. Assumption 4 The DGP for the external instruments ztbelongs to model (7), meaning that the parameters in (RΦ,Ωω) are constant across volatility regimes. Assumption 5 The VAR slope parameters in the companion matrix Cy= Cy(Π) remain constant across volatility regimes 3 With Assumptions 4-5, we define a “favorable” scenario in which the structural break solely impacts the covariance matrix of VAR disturbances. This impact does not affect VAR dynamics, the relevance parameters, the exogeneity condition, and instrument measurement error. There are two ways by which we can incorporate condition (S.3) in SVAR analysis. A common solution is to exploit the simultaneous factorization (see, e.g., Magnus and Neudecker, 1999, Theorem 23): Σu,1=HH0=H•1H0 •1+H•2H0 •2t≤TB, Σu,2=HPH0=H•1P•1H•1+H•2P•2H0 •2t≥TB+ 1 (S.4) where Pis a diagonal matrix with distinct positive elements on the main diagonal, and P•1and P•2are diagonal matrices such that: P=P•1 P•2,HP1/2=H•1P1/2 •1, H•2P1/2 •2. This standard modeling of a change in volatility (Lanne and L¨utkepohl, 2008) implicitly assumes that the underlying structural specification is defined as follows: ut=HεtI(t≤TB) + HP1/2εtI(t>TB) ={H•1ε1,t +H•2ε2,t}I(t≤TB) +nH•1P1/2 •1ε1,t +H•2P1/2 •2ε2,toI(t > TB) so that, recalling that Σε:= E(εtε0 t) = In, the diagonal elements in Pcan be interpreted as the variances of the structural shocks in the second volatility regime relative to the first volatility regime (where variances are normalized to 1). It turns out that the on-impact responses to one-standard deviation shocks are captured by the matrix Hin the first volatility regime and the matrix HP1/2in the second volatility, simply indicating a proportionate rescaling of IRFs. In this scenario, the absolute target IRFs are given by the expression: IRF•j(t, h) := ((Sn(Cy)hS0 n)H•1ejt≤TB, (Sn(Cy)hS0 n)H•1P1/2 •1ejt≥TB+ 1 , 1 ≤j≤k(S.5) so that, for e.g. k= 1, the relative IRFs are: IRF•1(t, h) IRF1,1(t, 0) = (Sn(Cy)hS0 n)1 Hrel 2,1,t= 1, ..., T. (S.6) 4 Equations (S.5) and (S.6) show that while the absolute target IRFs (to one standard deviation shocks) vary between the two volatility regimes due to the re-scaling of the target structural shocks, the relative target IRFs remain unaltered. The next proposition establishes the conditions under which the target IRFs can be estimated by external instruments ignoring the break in the covariance matrix. Proposition S.1 (Constant IRFs) Assume that the DGP belongs to the proxy-SVAR (8)under Assumptions 1-2and Assumptions 4-5. Consider a drifting DGP characterized by sequences of models in which E(ztε0 1,t) = ΦT, where the external instruments ztsatisfy the condition in Definition 1.(i). Further, assume that the DGP belongs to (S.4). Then (i) the estimator ˆ Σu,z := 1 TPT t=1 ˆutz0 tis such that: ˆ Σu,z p →τ(0) BH•1Φ0+ (1 −τ(0) B)H•1P1/2 •1Φ0 =hτ(0) BH•1+ (1 −τ(0) B)H•1P1/2 •1iΦ0, τ(0) Bbeing the true fraction of observation in the first volatility regime; (ii) for k= 1 and rank[P] = n,ˆ Σu2,z ˆ Σ−1 u1,z p →Hrel 2,1. Proposition S.1 suggests that while the absolute target IRFs (S.5) cannot be estimated consistently, the relative target IRFs in (S.6) in principle can, despite the break in volatility (but in Σualone). This result, however, is not guaranteed to hold when the external instruments do not satisfy the condition in Definition 1.(i), or Assumptions 4-5do not hold. It turns out that, in general, correct asymptotic inference on the absolute target IRFs must take the break in volatility into explicit account; see Schlaak, Rieth, and Podstawski (2023). The alternative method one can exploit to incorporate condition (S.3) in SVAR analysis, is to consider the parameterization: Σu,1=HH0=H•1H0 •1+H•2H0 •2, t ≤TB, Σu,2= (H+ ∆H)(H+ ∆H)0 = (H•1+ ∆H•1)(H•1+ ∆H•1)0+ (H•2+ ∆H•2)(H•2+ ∆H•2)0, t ≥TB+ 1 (S.7) where, as in the paper, ∆H= (∆H•1,∆H•2) denotes an n×nmatrix whose non-zero coefficients capture possible changes in the on-impact parameters in Hin the shift from the first to the second volatility regime. The modeling 5 of the change in unconditional volatility in (S.7) implies that the underlying structural specification is defined as follows (Bacchiocchi and Fanelli, 2015): ut=HεtI(t≤TB) + {H+ ∆H}εtI(t>TB) ={H•1ε1,t +H•2ε2,t}I(t≤TB) +{[H•1+ ∆H•1]ε1,t + [H•2+ ∆H•2]ε2,t}I(t>TB) so that the IRFs change in the shift from the first to the second volatility regime, because the break modifies the magnitude of the responses of the variables to the structural shocks.1In this scenario, the (absolute) target IRFs are given in equation (12), here reported for convenience: IRF•j(t, h) := (Sn(Cy)hS0 n)H•1ejt≤TB (Sn(Cy)hS0 n)(H•1+ ∆H•1)ejt≥TB+ 1 , 1 ≤j≤k. (S.8) Notice that the scenario depicted by the equations in (S.7)-(S.8) coincides with the framework analyzed in the paper but includes Assumptions 4-5. For k= 1, the relative target IRFs now are: IRF•1(t, h) IRF1,1(t, 0) =       (Sn(Cy)hS0 n)1 H2,1(H1,1)−1t≤TB, (Sn(Cy)hS0 n)1 (H2,1+ ∆H2,1)(H1,1+ ∆H1,1)−1t≥TB+ 1 (S.9) hence it is evident that for ∆H•1= (∆0 H1,1,∆0 H2,1)06= 0, both absolute and relative target IRFs change after the break. We have implicitly proved the next proposition. Proposition S.2 (Regime-varying IRFs) Assume that the DGP belongs to the proxy-SVAR (8)under Assumptions 1-2and Assumptions 4-5. Consider a drifting DGP characterized by sequences of models in which E(ztε0 1,t) = ΦT, where the external instruments ztsatisfy the condition in Definition 1.(i). Further, assume that the DGP belongs to (S.7). Then, both absolute and relative target IRFs can not be estimated consistently if the break in volatility is not taken into account. 1Note that for ∆Hchosen as ∆H:= H(P1/2−In), Pbeing a diagonal matrix with positive (distinct) elements on the diagonal, the moment conditions (S.7) collapse to (S.4), showing that (S.7) nests (S.4). 6 It is important to note that if in correspondence of the break in volatility the IRFs change, even under the simplifying Assumptions 4-5, the estimators ˆ Σu,z and ˆ Σu2,z ˆ Σ−1 u1,z do not maintain consistency, as explained in detail in the paper. S.4 Constant IRFs: proxy-SVAR-H approach In this section, we delve into an alternative modeling approach for the condition Ση,16= Ση,2relative to the stability approach developed in Section 2.3. Specifically, we concentrate on the scenario where it is assumed that the shift in unconditional volatility solely affects the variance of the structural shocks, leaving their impact and transmission mechanisms unchanged; that is, IRFs remain constant across volatility regimes. For simplicity, throughout, we adopt terminology from Carriero, Marcellino, and Tornese (2023) and refer to this approach as the Proxy-SVAR-H approach, with “H” standing for heteroskedasticity. Section S.4.1 deals with the methodology, and Section S.4.2 applies it to the estimation of US fiscal multipliers, complementing the results in Section 4.2 of the paper. S.4.1 Methodology We keep Assumption 1valid, but focus on the scenario in which the target IRFs are as in (S.5), i.e. regime-invariant. Let Gun be the (n+r)×(n+r) unrestricted nonsingular counterpart of the matrix Gin (9), i.e. Gun does not incorporate any zero entry. We exploit the simultaneous factorization (see, e.g., Magnus and Neudecker, 1999, Theorem 23): Ση,1=GunG0 un (S.10) Ση,2=GunΛG0 un = (GunΛ1/2) | {z } G∗ un (GunΛ1/2)0 | {z } G∗0 un (S.11) where Λ is an (n+r)×(n+r) diagonal matrix with positive elements. A typical interpretation of (S.10)-(S.11) is that the simultaneous factorization captures situations in which, given constant on-impact coefficients, the variances of the elements in ξt:= (ε0 t, ω0 t)0are equal to the identity matrix in the first volatility regime and change in relative terms to Λ in the second volatility regime. Hence, the diagonal elements of Λ can be interpreted as the variances of the elements in ξtrelative to the first volatility regime, where variances were normalized to 7 one; see, e.g., Lanne and L¨utkepohl (2008) and Sims (2021).2 A well known result from the conventional literature on identificationthrough-heteroskedasticity is that the moment conditions (S.10)-(S.11) suffice to point-identify the parameters in Gun, up to column permutation and sign normalization. This implies that the identification of structural shocks can occur, at most, ex-post, meaning once the investigator observes the estimated parameters in the columns of Gun and the resulting IRFs. Subsequently, based on the significance and sign of these estimates, the investigator can assign labels to the corresponding structural shocks. However, as argued in Schlaak et al. (2023), the proxies ztmay help to solve, at least partially, the “up-tocolumn-permutation” issue implicit in (S.10)-(S.11). Here we demonstrate analytically what Schlaak et al. (2023) show through simulation studies, namely that also in this framework, possibly invalid proxies do not affect the identification achieved by sufficiently strong changes in volatility. Consider the notation Gun =Gun(γ, ¯γ), where γis the vector of a parameters that enter the matrix Gin (9) and ¯γis the vector of (n+r)2−a parameters that are set to zero in the matrix Gbut not in Gun. In other words, G=Gun(γ, 0) for ¯γ:= 0. Let ρ= (γ0,¯γ0, λ0)0,λ=vecd(Λ), the (n+r)(n+r+ 1) ×1 vector containing all parameters contained in Gun and Λ.The identifiability of ρensures that also the parameters of interest and the target IRFs in (S.5) are identifiable. ρ0denotes the true value of ρ. Proposition S.3 Given the SVAR in (11) and Assumptions 1-2, consider the simultaneous factorization (S.10)-(S.11). Then a necessary and sufficient condition for the identification of ρin a neighborhood of ρ0in the parameter space Pρ(ρ0∈ Pρ, regular point) is that det[J(ρ0)] 6= 0, where J(ρ0)is the (n+r)(n+r+ 1) ×(n+r)(n+r+ 1) Jacobian matrix evaluated at ρ0, given by: J(ρ) = ∂mc(ση, ρ) ∂ρ0ρ=ρ0 , mc(ση, ρ) = ση,1−vech(GunG0 un) ση,2−vech(GunΛG0 un), ∂mc(ση, ρ) ∂ρ0:= 2(I2⊗D+ (n+r))(Gun ⊗I(n+r)) 0 (GunΛ⊗I(n+r)) (Gun ⊗Gun)In+r0 01 2FF. (S.12) 2Note that with Gun unrestricted, the diagonal elements of Λ coincide with the eigenvalues of the symmetric matrix Ση,2Σ−1 η,1. Obviously, if the investigator wishes to consider responses to one-standard deviation shocks in both volatility regimes, the on-impact responses to onestandard deviation shock in the second volatility regimes are captured by the matrix GunΛ1/2, which differs from Gun for simple scaling factors. 8 Some remarks are in order. First, the stated necessary and sufficient condition for identification in Proposition S.3 demonstrates analytically that identification can be achieved regardless of the proxy properties specified in Definition 1. This result emphasizes that an informative change in volatility tends to enhance identification irrespective of proxy properties. This holds regardless of whether IRFs change or remain constant across volatility regimes. Moreover, Proposition S.3 indirectly rationalizes the findings documented in Schlaak et al. (2023) through simulation methods, i.e. that external instruments positively contribute to identification when the data exhibit changes in unconditional volatility. Second, equation (S.12) shows that the rank of the Jacobian J(ρ) in (S.12) may collapse when the matrix FΛis not full column rank, situation that may occur when some diagonal elements of Λ are not distinct, which in turn implies that the differences in the covariance matrices Ση,2and Ση,1are not “sufficiently strong” to ensure the point identification of all structural shocks in the system.3In their application to the monetary policy framework, Schlaak et al. (2023) estimate the diagonal elements of Λ and verify that confidence intervals constructed for the diagonal elements of Λ, using one standard deviation around point estimates, do not overlap. The checks of identifiability discussed in Section 2.3 can be also applied in this context. S.4.2 Empirical results: fiscal multipliers revisited In this section we apply the methodology discussed in the previous section to the estimation of US fiscal multipliers. The general framework is the same discussed in Section 4. The implied peak fiscal multipliers, summarized in column (iv) of Table 1, are the ones with the lowest magnitude across all estimated models. In particular, the point estimate of the peak fiscal spending multiplier, Mpeak g, 1.5, and associated confidence intervals are very close to their counterparts obtained by the fiscal instruments alone, ignoring the break in volatility. Since, in our framework, fiscal multipliers reflect, by construction, the effect of relative IRFs, and considering that the consequences of the change in volatility appear evident on the tax shock and are less marked on the fiscal spending shock, this result is somewhat expected in light of our discussion in Section 3. 3Bacchiocchi, Bastianin, Kitagawa, and Mirto (2024) consider a partial identification approach in a Bayesian framework in situations like these, but without considering external instruments. 9 ine situations in which relevance is met, meaning that the correlation with the target shock is strong on the estimation sample, and scenarios in which the external instrument is local-to-zero as in Staiger and Stock (1997), i.e. ϕ:= cT−1/2, with |c|<∞. In general, the design covers all possible properties for the proxies as per Definition 1. The DGP values for ϕ, Υ, σωand ∆σω are specified below. By combining the VAR with the external instrument for ε1,t, the covariance matrices satisfy, under Assumption 1, the moment conditions: Ση,1=GG0 Ση,2= (G+ ∆G)(G+ ∆G)0 with DGP values for Gand ∆Ggiven by: G=H•1H•202×1 ϕΥσω=  1.00 0.40 0 0.70 0.90 0 ϕΥ 1   ∆G=∆H•1∆H•202×1 ∆ϕ∆Υ∆σω= −0.50 0 0 0 0 0 ∆ϕ∆Υ−0.04  . The true vector of structural parameters, ς0:= (γ0 0, δ0 0)0, comprises γ0:= (1,0.7,0.40,0.90, ϕ0,Υ0,1)0and δ0:= (−0.5,∆ϕ,0,∆Υ,0,−0.040)0. In this design, the target IRFs in (12) change across the two volatility regimes solely because of changes in the on-impact parameters H•1:= (1,0.7)0, as captured by the elements in ∆H•1:= −0.5.Overall, the total number of structural parameters to estimate when Υ 6= 0,∆Υ6= 0 (exogeneity fails) is 11, while there are (n+r)(n+r+ 1) =12 moment conditions. Therefore, the proxy-SVAR incorporates d=1 testable overidentifying restriction when Υ6= 0,∆Υ6= 0 (exogeneity fails), and d=3 testable overidentifying restrictions when Υ = 0,∆Υ= 0 (exogeneity holds) and is imposed in estimation. The necessary and sufficient rank condition implied by Proposition 1is satisfied for the specified values of (ϕ0,∆ϕ,0) and (Υ0,∆Υ,0) we consider below. In all experiments, we generate N=10,000 samples of lengths T={250,500,1,000}, respectively, under the hypothesis that the structural shocks εt:= (ε1,t, ε2,t)0 and the proxy’s measurement error ωtare drawn from iidN(0,1) processes.4 When dealing with strong proxies, the DGP values of ϕand ∆ϕare such that corr(ε1,t, zt)=0.58 for the full sample. Instead, when dealing with local-to4We can relax both Gaussianity and the iid hypothesis provided the process ηt:= (u0 t, z0 t)0 respects the α-mixing conditions stated in Assumption 1. 16 zero proxies, the correlations vary with the sample size, namely corr(ε1,t, zt) = {0.045,0.0318,0.0225}, depending on whether the sample length Tis equal to 250,500 or 1,000 respectively. Relative performance We start by examining whether there are gains or losses from complementing the identification of the target shock via the change in volatility with the external instrument. We first present the models which are compared and then define the measure of relative efficiency used throughout. In Table S.1 we call “Model.q”, q= 1,2,3,4,5 the five model involved in the comparison. With Model.1 we denote results obtained by our stability restrictions, namely by estimating the proxy-SVAR parameters ςby the CMD approach discussed in Section 2.3, assuming that the econometrician correctly specifies the VAR lag length and knows the break date TB. Model.1 is used as a benchmark in the comparison; hence, relative efficiency measures in Table S.1 are set to 1 for this model. Model.2 is the same as Model.1 but with the contamination parameters Υ and ∆Υset to zero, meaning imposing proxy exogeneity. Model.3 denotes result obtained by the change in volatility approach alone, i.e. without leveraging the instrument for identification. Model.4 denotes results obtained by the proxy-SVAR-H approach, see Section S.4, i.e. assuming that the target IRFs remain constant across the two volatility regimes. Model.5 denotes results obtained by the external instrument, ignoring the break in volatility, i.e. a “conventional” proxy-SVAR approach. Numbers in Table S.1 correspond to measures of relative efficiency based on Mean Squared Error (MSE) obtained in samples of length T= 500 as follows. For q≥2, we have: rel-MSEModel.q Model.1:= τB×rel-MSEModel.q Model.1(t)I(t≤TB) + (1 −τB)×rel-MSEModel.q Model.1(t)I(t>TB) (S.22) where τB:= bTB/Tcis the fraction of the sample covering the first volatility regime and: rel-MSEModel.q Model.1(t) := 1 25 25−1 X h=0          1 NPN j=1 \ IRFModel.q i,1,j (t, h)−IRF0 i,1(t, h)2 1 NPN j=1 \ IRFModel.1 i,1,j (t, h)−IRF0 i,1(t, h)2         . (S.23) In (S.22)-(S.23), N=10,000 is the number of Monte Carlo simulations, i= {1,2}denotes the response variable considered in Yt= (Y1,t, Y2,t)0,IRF0 i,1(t, h) 17 is the true value of the absolute response of Yi,t+hto the target shock ε1,t (see equation (12)), and \ IRFModel.q i,1,j (t, h) the corresponding estimate obtained from Model.qon the sample of observation generated at the iteration j. Note that the measures in (S.22)-(S.23) are opportunely adapted to considering the whole sample of length Tfor Model.4 and Model.5, where the target IRFs are kept constant across the two volatility regimes. For q≥2, measures obtained from (S.22)-(S.23) greater than 1 indicate that Model.qperforms worse in terms of MSE than the benchmark Model.1. Conversely, values less than 1 indicate that there are relative gains in efficiency. It is worth remarking, however, that in this DGP, the estimators of the target IRFs in the numerator of (S.23) are not consistent for Model 4 and Model.5, as these models maintain that the IRFs do not change across volatility regimes. In these cases, therefore, comparisons drawn from Table S.1 must be interpreted with caution. Panel (a) of Table S.1 focuses on the case where a strong external instrument is used, while panel (b) refers to a local-to-zero instrument. For both cases, the instrument can be exogenous to the non-target shock (corr(zt, ε2,t) = 0) or can be contaminated to various extents (corr(zt, ε2,t) = {0.05, 0.15, 0.25}). We notice from Panel (a) of Table S.1 that the incorporation of a strong and exogenous instrument to the identification based on a shift in volatility leads to considerable gains in performance. As expected, only the model which correctly imposes proxy exogeneity in estimation (other than the stability restrictions) performs better than the benchmark. In general, even when the exogeneity condition fails, a strong instrument tends to increase the accuracy with which the target IRFs are estimated, on average. Interestingly and, as expected, Panel (b) of Table S.1 shows that the gains relative to only leveraging the shift in volatility vanish in the presence of local-to-zero instruments. However, even in the scenario characterized by invalid proxies as in Definition 1.(iv), no alternative approach to the stability restrictions approach proves to be better. Overidentifying restrictions test Next we focus on the rejection frequency of the overidentifying restrictions test resulting from the CMD estimation approach discussed in the paper. On each of the N=10,000 generated samples of lengths T={250,500,1000}, we estimate the parameters ςof the proxy-SVAR by the CMD approach discussed in Section 2.3, assuming that the econometrician correctly specifies the VAR lag length and knows the break date TB. This approach corresponds to Model.1 in the comparisons discussed above. Then, we investigate the rejection frequency of the implied overidentifying restriction test under two main cases. 18 In one scenario, the econometrician leaves the contamination parameters unrestricted in estimation, with the idea that the significance of Υ and ∆Υcan be inferred from the data provided the conditions in Proposition 1hold. The other scenario coincides with the case in which the econometrician imposes the exogeneity restriction in estimation (Υ = 0,∆Υ= 0). This implies that the estimated proxy-SVAR is misspecified when contamination (Υ 6= 0,∆Υ6= 0) holds in the DGP, i.e. for corr(zt, ε2,t) = {0.05, 0.15, 0.25}. Under the null that the stability restrictions hold, the overidentifying restrictions test statistic (see equation (25)) is asymptotically distributed in the stated DGP as χ2 drandom variable, with degree of freedom das defined above. Tests are conducted at the 5% nominal significance level and rejection frequencies are summarized in Table S.2. The right panel of Table S.2 pertains to the case where the econometrician does not impose proxy exogeneity in estimation. The left panel, instead, assumes that exogeneity is imposed. It is observed that, regardless of the correlation between the instrument and the non-target shock, when the external instrument is left free in estimation, the CMD estimation approach provides rejection frequencies compatible with finite sample size control regardless of strength. On the contrary, rejection frequencies tend to increase with the extent of contamination and the increase in sample size regardless of strength, revealing that the test tends to have finite sample power against the failure of instrument exogeneity. Overall, the results in Table S.2, combined with those in Table S.1, confirm that when identification of the proxy-SVAR is ensured by shifts in volatility, as implied by the necessary and sufficient rank conditions in Proposition 1, it is generally advantageous for the econometrician not to impose exogeneity in estimation. Violations of the exogeneity condition can be detected from the data regardless of whether the instrument is relevant or local-to-zero. Notably, even when the instrument is both relevant and contaminated, it proves to be useful for the inference on the target IRFs. However, the overidentifying restrictions test rejects the model’s validity when exogeneity is incorrectly imposed in estimation, and again, this holds irrespective of instrument strength. We turn on the performance of the overidentifying restrictions test at the end of this section, where we explore how the stability restrictions approach performs when the shifts in volatility provide limited information, resulting in near-rank failure for the Jacobian J(ς), rendering the results in Propositions 1-2invalid. Checks of identifiability and shrinking shifts Results in Table S.1 and Table S.2 are obtained under scenarios in which the proxy-SVAR with a 19 break in unconditional volatility is identified. Identifiability depends on the full column rank condition of the Jacobian matrix J(ς), as derived in Proposition 1; see equation (22). The validity of the necessary and sufficient rank ensures the use of standard asymptotic inference, as stated in Proposition 2. Here, we investigate to what extent the smallest singular values of J(ˆςT), given the CMD estimates ˆςTand associated measures of uncertainty are informative about the identifiability of the proxy-SVAR. First, we consider the case in which the change in VAR covariance matrices is sufficient to identify the model, consistent with the DGP considered so far. Table S.3 summarizes the average, across Monte Carlo simulations, of the estimated smallest singular values of the Jacobian matrix along with associated interquartile ranges (IQRs). IQRs are used in this context as broad approximations of confidence intervals. As previously, we explore scenarios with both relevant and local-to-zero instruments, and both exogenous and contaminated instruments. Results in Table S.3 indicate that in situations where the change in volatility is sufficient for identification, the smallest singular values of the estimated Jacobian matrix are far from zero and the associated IQRs tend not to include the zero. Another important finding from Table S.3 is that proxy properties do not affect the identifiability of the model, confirming the analytic results discussed in the paper and the figures in tables S.1 and S.2. The results outlined in Table S.3 also indirectly support the identifiability of the fiscal proxy-SVAR estimated in Section 4of the paper. In that section, in the bottom part of Table 2, we reported the estimated smallest singular value of the Jacobian matrix along with associated 68% MBB confidence interval, which reassuringly seem to rule out the case of a zero singular value. Secondly, we reexamine the performance of the stability restrictions approach under a different scenario. Specifically, we address cases where the distance between covariance matrices in the two volatility regimes, (Ση,2−Ση,1), shrinks according to equation (S.21) in the paper, reported here for convenience: Ση,2−Ση,1=%TΨT. In this equation, as the scalar %Tconverges to zero, %T→0, ΨT→Ψ = (G˜ ∆G+˜ ∆GG)6= 0 determining a near-rank failure setup for the Jacobian J(ς); see Section 2.3. Table S.4 summarizes the estimated smallest singular values of the Jacobian matrix and associated IQRs when VAR covariance matrices shrink at the rate %T∼o(T−1/2).We now observe a departure from the patterns seen in Table S.3. Unlike the scenarios presented there, where the ratio between the estimated average smallest singular values and the average length of IQRs is 20 consistently greater than 2, we now notice a distinct trend. Specifically, the magnitude of the estimated smallest singular values tends to be systematically smaller than 2 times the IQR, indicating a lack of identification resulting from the change in volatility. However, in line with expectations, strong proxies appear to sustain the identifiability of the proxy-SVAR when the exogeneity condition is imposed in estimation, although not when it is not. This phenomenon is explained by the fact that when deviations from exogeneity are allowed, the instruments also provide information about the non-target shocks. Consequently, if the change in volatility poorly identifies the non-target shocks, the failure in identification extends to the entire system. In contrast, when exogeneity is imposed in estimation, whether or not it holds in the DGP, any information originating from the non-target shocks is not transmitted to the target shocks. In this case, the Jacobian matrix maintains full column rank, ensuring identification. S.8 Proofs of propositions Proof of Proposition 1:(i) The result follows by deriving the moment conditions in (18)-(21) with respect to the parameter ς:= (γ0, δ0)0and then applying matrix derivative rules; see Bacchiocchi and Fanelli (2015, Proposition 1); (ii) the necessary order condition follows trivially from the dimensions of the Jacobian matrix: J(ς) := ∂m(ση, ς) ∂ς0 | {z } (n+r)(n+r+1)×(a+b) = ∂m1(ση,ς) ∂ς0 ∂m2(ση,ς) ∂ς0! in (22), where both ∂m1(ση, ς)/(∂ς0) and ∂m2(ση, ς)/(∂ς0) are of dimension (1 2(n+r)(n+r+ 1) ×(a+b)). Proof of Proposition 2:Let ˆ QT(ς) := mT(ˆση, ς)0ˆ V−1 σηmT(ˆση, ς) be the objective function upon which CMD estimation is computed in (25). We observe that: (a) under the conditions of Proposition 1,Q0(ς) := m(σ+ 0, ς)0V−1 σηm(σ+ 0, ς) is uniquely maximized at ς0in the neighborhood Nς0; (b) Pςis compact and Nς0⊆ Pς; (c) Q0(ς) is continuous; (d) ˆ QT(ς) converges uniformly in probability to Q0(ς).To prove that (d) holds, recall that under Assumptions 1-2 ˆση p →ση,0, hence mT(ˆση, ς)p →m(ση,0, ς) by the Slutsky Theorem. Also recall that it exists an estimator of the asymptotic covariance matrix Vσηsuch that ˆ Vση p →Vση, see (24). Then, with k·k denoting the Euclidean norm, by the 21 triangle and Cauchy-Schwartz inequalities: ˆ QT(ς)−Q0(ς)≤[mT(ˆση, ς)−m(ση,0, ς)]0ˆ V−1 ση[mT(ˆση, ς)−m(ση,0, ς)] +m(ση,0, ς)0[ˆ V−1 ση+ˆ V−10 ση][mT(ˆση, ς)−m(ση,0, ς)] +m(ση,0, ς)0[ˆ V−1 ση−V−1 ση]m(ση,0, α) ≤ kmT(ˆση, ς)−m(ση,0, ς)k2  ˆ V−1 ση   + 2 km(ση,0, ς)kkmT(ˆση, ς)−m(ση,0, ς)k  ˆ V−1 ση   +km(ση,0, ς)k2  ˆ V−1 ση−V−1 ση   so that supς∈Pςˆ QT(ς)−Q0(ς) p →0. Given (a), (b), (c), and (d), the consistency result follows from Newey and McFadden (1994, Theorem 2.1). To prove asymptotic normality, we start from the first-order conditions implied by the problem (25) in the paper: J(ˆςT)0ˆ V−1 σηmT(ˆση,ˆςT) = 0 (S.24) where J(ˆςT) denotes the Jacobian matrix J(ς) := ∂m(ση,ς) ∂ς0:= J(ση, ς) evaluated at the estimated parameters ˆσηand ˆςT, respectively. By expanding mT(ˆση,ˆςT) around ς0and solving, yields the expression (valid in Nς0): √T(ˆςT−ς0) = −nJ(ˆση,ˆςT)0ˆ V−1 σηJ(ˆση,¯ς)o−1J(ˆση,ˆςT)0ˆ V−1 ση√TmT(ˆση, ς0) (S.25) where ¯ςis a mean value. From (24) and the delta-method: √TmT(ˆση, ς0)d →N(0,J(ς0)0VσηJ(ς0)0) (S.26) where J(ˆση, ς0)p → J(ση,0, ς0) := J(ς0). From the consistency result in (i), as T→ ∞,J(ˆση,ˆςT)p → J(ση,0, ς0) := J(ς0) and J(ˆση,¯ς)p → J(ση,0, ς0) := J(ς0), respectively. Moreover, the matrix J(ς0)0V−1 σηJ(ς0) is nonsingular in Nς0because of Proposition 1. It turns out that −nJ(ˆση,ˆςT)0ˆ V−1 σηJ(ˆση,¯ς)o−1J(ˆση,ˆςT)ˆ V−1 ση p → −nJ(ς0)0V−1 σηJ(ς0)o−1J(ς0)0V−1 ση, so that the conclusion follows from (S.26) and the Slutsky theorem. 22 Proof of Proposition S.1:(i) Consider the sums: ˆ Σu,z := 1 T T X t=1 ˆutz0 t=1 T   TB X t=1 ˆutz0 t+ T X t=TB+1 ˆutz0 t   =1 T   TB TB TB X t=1 ˆutz0 t+T−TB T−TB T X t=TB+1 ˆutz0 t   =TB T(1 TB TB X t=1 ˆutz0 t)+T−TB T   1 T−TB T X t=TB+1 ˆutz0 t   . hence, for T→ ∞, TB T(1 TB TB X t=1 ˆutz0 t)p →τ(0) BEutz0 tI(t≤TB) =τ(0) BE{H•1ε1,t +H•2ε2,t}z0 t=τ(0) BH•1Eε1,tz0 t=τ(0) BH•1Φ0; T−TB T   1 T−TB T X t=TB+1 ˆutz0 t   p →(1 −τ(0) B)Eutz0 tI(t>TB) = (1−τ(0) B)EhnH•1P1/2 (1) ε1,t +H•2P1/2 (2) ε2,toz0 ti=H•1P1/2 (1) Eε1,tz0 t=H•1P1/2 (1) Φ0. (ii) It is seen that with Φ 6= 0 (rank[Φ] = k= 1) and rank[P(1)] = 1: Σu,z (Σu1,z)−1=Σu1,z Σu2,z (Σu1,z)−1 =         H1,1Φ0 H2,1Φ0(H1,1Φ0)−1t≤TB H1,1P1/2 (1) Φ0 H2,1P1/2 (1) Φ0!H1,1P1/2 (1) Φ0−1t≥TB+ 1 =           1 (H2,1Φ0) (H1,1Φ0)−1 1 (H2,1P1/2 (1) Φ0)H1,1P1/2 (1) Φ0−1!,t= 1, ..., T 23 =1 Hrel 2,1,t= 1, ..., T. As ˆ Σu2,z p →H2,1P1/2 (1) Φ0and ˆ Σu1,z p →H1,1P1/2 (1) Φ0, it follows that ˆ Σu2,z ˆ Σu1,z−1 is consistent for Hrel 2,1. Proof of Proposition S.3:(i) The nonlinear functional relationship between ση:= (vech(Ση,1)0, vech(Ση,2)0)0and ρis given by ση=σ(ρ), where σ(ρ) = (vech[GunGun0]0, vech[GunΛGun0])0. A necessary and sufficient condition for ρ to be uniquely recovered from σηis that rank[J(ρ)] = dim(ρ)=(n+r)(n+ r+ 1) locally, where J(ρ) = ∂σ(ρ) ∂ρ0is (n+r)(n+r+ 1) ×(n+r)(n+r+ 1). Using matrix algebra derivatives and the properties of duplication matrices: J(ρ) = ∂vech(GunG0 un) ∂ρ0 ∂vech(GunΛG0 un) ∂ρ0!= ∂vech(GunG0 un) ∂ρ0 ∂vech(GunG0 un) ∂λ0 ∂vech(GunΛG0 un) ∂ρ0 ∂vech(GunΛG0 un) ∂λ0! = D+ (n+r) ∂vec(GunG0 un) ∂ρ00 D+ (n+r) ∂vec(GunΛG0 un) ∂ρ0D+ (n+r) ∂vec(GunΛG0 un) ∂λ0! D+ (n+r) ∂vec(GunG0 un) ∂vec(Gun)0×∂vec(Gun) ∂ρ00 D+ (n+r) ∂vec(GunΛG0 un) ∂vec(Gun)0×∂vec(Gun) ∂ρ0D+ (n+r) ∂vec(GunΛG0 un) ∂vec(Λ)0×∂vec(Λ) ∂λ0!. Now, given N(n+r):= 1 2(I(n+r)2+K(n+r)) with K(n+r)commutation matrix, and since we have: ∂vec(GunG0 un) ∂vec(Gun)0= 2N(n+r)(Gun ⊗I(n+r)); ∂vec(GunΛG0 un) ∂vec(Gun)0= 2N(n+r)(GunΛ⊗I(n+r)); ∂vec(GunΛG0 un) ∂vec(Λ)0= (Gun ⊗Gun), and 2D+ (n+r)N(n+r)= 2D+ (n+r), the Jacobian reads: J(ρ) = 2(I2⊗D+ (n+r))(Gun ⊗I(n+r)) 0 (GunΛ⊗I(n+r)) (Gun ⊗Gun)1 2FΛ = 2(I2⊗D+ (n+r))(Gun ⊗I(n+r)) 0 (GunΛ⊗I(n+r)) (Gun ⊗Gun)SG0 01 2FΛ. Proof of Proposition S.4:See Bacchiocchi and Fanelli (2015), Supplemen24 tary Material. S.9 Sensitivity of the stability restrictions estimator Sensitivity analysis consists into measuring the effects of local misspecification of the moments of estimator onto first-order asymptotics of the ˆ θ. Let us consider misspecification of the moments in the sense of wrong assignment of the parameter τ. Namely, for any 6= 0, let us consider situations with 2 regimes, where the break date used in the estimation of θis TB, =b(τ0+)Tcas opposed to the true TB=bτ0Tc, such that as T→ ∞, TB, →TB.Let ˆσηbe a consistent estimator of ση. In the context of the stability restrictions estimator, local misspecifications may realize under the form of estimation error and/or incorrect assignment of a break in the DGP. Without loss of generality, the misspecified moment function (21) can be represented by m(ˆση,=) References Angelini, G., Cavaliere, G., and Fanelli, L. (2024). “An identification and testing strategy for proxy-SVARs with weak proxies”. 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