What Impulse Response Do Instrumental Variables Identify?

What Impulse Response Do Instrumental Variables Identify?

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Introduction

In contemporary applied macroeconometrics, instrumental variables (IVs) are frequently employed as a crucial source of external variation for identifying impulse response functions to macroeconomic shocks, which are central to policy analysis. Two common approaches utilizing IVs are IV estimation within the local projection (LP-IV) framework and the estimation of extended structural vector autoregressions (SVARs), where external IVs are incorporated directly into the system, as seen in the proxy SVAR models. While the extended SVAR approach offers a more straightforward interpretation of estimates due to its specification of a complete system, the LP-IV approach, by contrast, is inherently partial. This partial nature can obscure the interpretation of its estimates, particularly when the standard identification assumptions are not fully met.

The econometric literature has devoted substantial effort to developing a general framework under which IV estimands can be properly interpreted, as evidenced by the local average treatment effect (LATE) developed by Imbens and Angrist (1994). A common identification assumption for the LP-IV estimand is the so-called IV exclusion restriction, as outlined in Plagborg-Møller and Wolf’s (2021) Eq.(16) or Stock and Watson’s (2018) Condition LP-IV, which requires that the IV is correlated with only a single structural shock in the endogenous observable policy variable. However, this assumption is often too restrictive. For instance, a government spending shock may consist of a combination of sectoral spending shocks with varying magnitudes, while a monetary policy surprise might include expectations of policy tightening over different time horizons or a mix of pure monetary shocks and the central bank’s assessment of economic conditions. In contrast, various proxy SVAR models have accommodated the composite nature of structural shocks. For example, Jarociński and Karadi (2020) and Giacomini, Kitagawa, and Read (2022) assume that $k$-dimensional proxies are correlated with an equivalent number of structural shocks within the SVAR model. Miranda-Agrippino and Ricco (2023) extend the external IV-based identification of shocks as explored by Stock and Watson (2018) under the notion of partial invertibility.

Nevertheless, much of the LP-IV literature including key contributions by Stock and Watson (2018) and Plagborg-Møller and Wolf (2021, 2022) continues to operate under the assumption that the shocks identified through IVs are homogeneous. This significantly constrains the applicability of the LP-IV approach. Recognizing this limitation, Miranda-Agrippino and Ricco (2021) have highlighted the potential bias in LP-IV estimates due to the composite nature and proposed modifications to the IVs themselves, utilizing more information to better align the instruments with the exclusion restrictions required for valid identification. Miranda-Agrippino and Ricco (2023) further emphasized that the IVs developed in their 2021 paper could identify the shock of interest when some shocks are invertible provided that the shock of interest can be recovered from reduced-form shocks.

We start by deriving an explicit representation of the LP-IV estimand by employing a potentially non-invertible vector moving average (SVMA) model. This model allows for the IVs to be correlated with multiple component shocks within the policy variable, reflecting the composite nature of macroeconomic shocks. We show that the LP-IV estimand is expressed as a weighted sum of impulse responses to the individual components of the composite structural shock. These weights are determined by the correlation between the IV and each component shock. Importantly, since this correlation is not restricted to be non-negative, the LP-IV estimand does not necessarily identify an informative quantity like a proper weighted average of the impulse responses.

To fix ideas, suppose we aim to estimate the response of GDP at horizon $h$ to a unit change in government spending. Since observed changes in government spending are unlikely to be exogenous, an IV is required to identify the government spending multiplier. Ramey and Zubairy (2018) use narrative military news as an IV to provide exogenous variation in government spending. The cumulative government spending multiplier is then estimated using LP-IV. The LP-IV estimand (the population counterpart of the estimator) is given by

where $y_{t}$ is the GDP, $x_{t}$ is the government spending variable, and $z_{t}$ is the military news shock\footnote{All the variables are real and divided by the trend real GDP.}. Since government spending consists of defense and non-defense spending, we find that

where $ \theta_{h,\text{defense}}$ and $\theta_{h,\text{non-defense}}$ are the (cumulative) defense and non-defense spending multipliers, respectively, and $w$ is the weight whose sign depends on the correlation of the military news shock and the sectoral spending shock. $\beta_{h}$ is causal only if $w$ is between zero and one. It may be interpreted as the aggregate spending multiplier when $w$ is the proportion of sectoral spending\footnote{Ramey and Zubairy (2018) recognize that the response of GDP to defense and non-defense spending shocks may differ. While they touch upon this issue in relation to the local average treatment effect and the average treatment effect, they do not provide a formal analysis.}. However, in our replication of Ramey and Zubairy (2018) in Section (ref), we find that for $h=18$ (eighteen quarters) the LP-IV estimator is decomposed as

Due to the negative weight, the estimated government spending multipler is much smaller than either the defense or non-defense spending multiplier. This would lead to an inaccurate empirical conclusion for researchers and policy makers even when the instrument and the estimation procedure are legitimate and valid.

The issue arising from the negative weights has received much attention recently in the microeconometrics literature. For instance, de Chaisemartin and D’Haultfoeuille (2020) show that the regression coefficient in the two-way fixed effects model can be expressed as a weighted sum of the average treatment effects where the weight can be negative due to different timing of treatment receipts across units. Mogstad, Torgovitsky, and Walters (2021) show that the two-stage least squares estimand using multiple instruments may lose a causal interpretation due to negative weights assigned to some subpopulation under general treatment effect heterogeneity.

Nonetheless, estimates derived from instruments with negative weights should not be dismissed. Although they cannot be interpreted structurally on their own, they retain informational value and---when analyzed jointly with estimates from other instruments---can help bound or partially identify the component-wise responses of interest.

While combining multiple IVs can aid identification, conventional approaches like two-stage least squares (2SLS) or generalized method of moments (GMM) may compromise structural interpretation. For example, Ramey and Zubairy (2018) report 2SLS estimates using both the narrative military news and spending shock instruments. However, these estimators yield weighted averages of impulse responses, and the weights may be negative due to correlation between the IVs. Consequently, the resulting LP-2SLS or LP-GMM estimands may lack a valid structural interpretation. In addition, the over-identifying restrictions test is not a valid test of joint instrument validity in this setting.

We propose two strategies to achieve identification of structural parameters using multiple IVs. Our goal here is to identify the impulse responses to the component shocks within the structural shock of interest.

The first strategy is to use disaggregated data to identify the weights. For example, quarterly data on defense and non-defense spending in the US after the WWII are available and we can use these variables to identify the weights for the defense and non-defense spending multipliers. When the number of available IVs is at least as many as the number of components in the structural shock, the impulse responses to each component shock can be identified regardless of the signs of the weights.

In Section (ref) we estimate the defense and non-defense spending multipliers using the sectoral spending data and two IVs: the military news shock of Ramey and Zubairy (2018) and the current defense spending shock of Blanchard and Perotti (2002).

Our second strategy uses sign restrictions, which are commonly used in the SVAR literature (e.g., Uhlig, 2005) and have been discussed by Plagborg-Møller and Wolf (2021) and Alpanda, Granziera, and Zubairy (2021) in the LP framework. Specifically, our approach builds on sign restrictions on the correlations between shocks and IVs rather than directly on the structural impulse responses. These restrictions are coherent with the IV-based identification and easy to justify. This types of sign restrictions are used in work focusing on Bayesian SVARs, such as Jarociński and Karadi (2020), Arias, Rubio-Ram\'{i}rez, and Waggoner (2021), and Giacomini, Kitagawa, and Read (2022).

Moreover, our strategy requires no additional computation since the LP-IV estimates for each IV are directly used as bound estimates, unlike Bayesian approaches that often necessitate extensive computation or simulation. Surprisingly, an IV that is correlated with the component shocks with opposing signs can be more useful than one correlated with the component shocks in the same direction. When two IVs with opposing signs are combined, the resulting identified set is the most informative, as shown in Proposition (ref). Lastly, our approach easily accommodates sign restrictions with many IVs, as a tighter identified set can be obtained by intersecting the bounds.

In Section (ref), we revisit the analysis by Jarociński and Karadi (2020), who examined the heterogeneous impulse responses of the information shock and the pure monetary policy shock within the context of the central bank’s monetary policy announcements. By utilizing high-frequency financial market surprises as IVs, they explored these responses, but our approach imposes sign restrictions within the LP-IV model to obtain the identified set for the impulse response to the pure monetary policy shock.

In addition to proposing new identification strategies, we assess whether the LP-IV based identified sets can be further sharpened under weak stationarity. Focusing on a finite-order SVMA model with two IVs and two component shocks, and imposing sign restrictions as in Jarociński and Karadi (2020), we show that the LP-IV identified set for the component-wise impulse responses takes the form of a half-rectangle. We demonstrate that the identified set is inevitably unbounded under certain sufficient conditions. Separately, we provide alternative sufficient conditions under which the identified set for the other component-wise impulse response is sharp. This unboundedness result is empirically relevant and has not been fully recognized in the existing literature.

As argued by Wolf (2022), the standard Bayesian procedure employed in such analyses often fails to adequately address the fact that sign restrictions lead to partial or set identification of the impulse response rather than point identification. Giacomini, Kitagawa, and Read (2022) propose a method to robustify the Bayesian approach to better handle this set identification issue. However, a significant challenge remains: the identified set is not always bounded, which violates a key regularity condition required for the validity of their proposed confidence region. The possibility of unbounded identified sets is implicitly suggested by Baumeister and Hamilton (2015) and is analyzed explicitly by Read (2024) who also demonstrates the empirical relevance of unboundedness in set-identified SVARs when impulse responses are normalised to a unit effect.

We emphasize that our findings have broader implications beyond LP-IV. Plagborg-M\o ller and Wolf (2021) have shown that LP-IV and SVAR with the IV ordered first in the triangular system estimate the same impulse responses asymptotically. Since their result is nonparametric, our findings extend to impulse responses estimated by SVAR with IVs. Although Plagborg-M\o ller and Wolf (2021) did not study the relationship between LP and VAR with multiple IVs, their approach of adding IVs to the top of the VAR system would be related to an LP-GMM type estimator.

Our paper is related to program evaluations under treatment effect heterogeneity. In their influential paper, Imbens and Angrist (1994) demonstrated that a valid IV can only identify the LATE in the potential outcomes framework. Since it is natural to consider a macroeconomic shock as the treatment and the impulse response as the treatment effect, our result can be seen as an extension of LATE to impulse response analysis. However, there are several important differences between our framework and the potential outcomes framework. First, unlike the potential outcomes framework, where the set of treatments is a singleton or finite, we define the composition of the macroeconomic shock as the treatment, which is naturally a continuum. For example, a one-unit exogenous change in government spending may consist entirely of defense spending or some combination of defense and non-defense spending. Second, unlike the potential outcomes framework, where everyone gets homogeneous treatment but their individual treatment effects are heterogeneous, the composition of a macro shock provides heterogeneous treatment, but the response of the macro variables is homogeneous conditional on the composition. We discuss the similarities and differences in the identified structural parameters and identifying conditions between our framework and the potential outcomes framework in detail in Online Appendices.

Lastly, we provide a brief literature review. There is a significant body of work that uses IVs in macroeconometrics. Some notable examples include Stock and Watson (2012), Mertens and Ravn (2013), Gertler and Karadi (2015), and Jaroci\'{n}ski and Karadi (2020), who employ IVs for identification in SVAR models. The LP method was introduced by Jord\`{a} (2005), and LP-IV have been used in several studies, such as Jord\`{a}, Schularick, and Taylor (2015, 2020), Stock and Watson (2018), and Ramey and Zubairy (2018). For a recent overview of LP-IV, we refer to Jord\`{a} and Taylor (2024).

Our paper is organized as follows. Section (ref) introduces the underlying structural model and shows what the LP-IV identifies. We develop identification strategies for the component-wise impulse responses in Section (ref), leveraging disaggregated data or employing sign restrictions. The empirical applications concerning the analysis of the fiscal multiplier and monetary policy are given in Sections (ref) and (ref). Section (ref) concludes. The proofs of Propositions, supplementary inference procedures, and their theoretical justifications are collected in the Appendix.

Model and Identification

To motivate our approach, suppose that a researcher considers a bivariate SVAR model for the government spending multiplier:

where $x_{t}$ is government spending, $y_{t}$ is GDP, $(\varepsilon_{xt},\varepsilon_{yt})$ are the structural shocks, and $A(L)$ is the lag polynomial. The structural shocks are uncorrelated with each other. The researcher is interested in identifying the impulse response of GDP to the government spending shock, which comprises defense and non-defense spending shocks. We may write $\varepsilon_{xt}=\varepsilon_{xt}^{D}+\varepsilon_{xt}^{ND}$. It is plausible that defense and non-defense spending shocks generate heterogeneous responses of GDP but this cannot be accommodated by replacing $\varepsilon_{xt}$ with $\varepsilon_{xt}^{D}+\varepsilon_{xt}^{ND}$ in the SVAR model.

To analyze the heterogeneous effects of the components in the structural shock on the endogenous variables, we adopt a general non-invertible structural vector moving average (SVMA) model. SVMA is useful due to its flexible nature such that it allows for a larger number of structural shocks than the observable endogenous variables.

Structural Vector Moving Average

Let $\boldsymbol{Y}_{t}$ be an $n\times1$ vector of observed endogenous variables and let $\boldsymbol{\varepsilon}_{t}$ be an $m\times1$ vector of unobserved structural shocks. The endogenous variables $\boldsymbol{Y}_{t}$ is written as a linear combination of current and past $\boldsymbol{\varepsilon}_{t}$'s:

where $L$ is the lag operator, $\boldsymbol{\Theta}(L) = \boldsymbol{\Theta}_{0} + \boldsymbol{\Theta}_{1}L + \boldsymbol{\Theta}_{2}L^{2}+\cdots$, and $\boldsymbol{\Theta}_{h}$ for $h=0,1,2,...$ is an $n\times m$ matrix of impulse response coefficients. We assume that $\boldsymbol{\varepsilon}_{t}$'s are i.i.d., $E[\boldsymbol{\varepsilon}_{t}]=0$, and $E[\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t}']\equiv \boldsymbol{\Lambda}>0$ where $\boldsymbol{\Lambda}$ is a diagonal matrix (i.e., the shocks are mutually uncorrelated). A word on notation: we reserve the index `$ s $' to indicate an element in the system (ref), e.g., $ \varepsilon_{s,t} $ then denotes the $ s $-th element of $ \boldsymbol{\varepsilon}_t $.

Let $y_{t}$ be the last element of $\boldsymbol{Y}_{t}$ without loss of generality. We define the impulse response of $y_{t}$ at horizon $h$ to the shock $\varepsilon_{s,t}$ as

Here we are slightly abusing notation by letting $\theta_{h,ys}$ be the $(n,s)$-th element of $\boldsymbol{\Theta}_{h}$. This is to emphasize that $y_{t}$ is the main variable of interest.

Suppose that a researcher is interested in the impulse response of $y_{t}$ at horizon $h$ to a macroeconomic policy shock $\xi_{t}$, which may consist of multiple component shocks $\varepsilon_{s,t}$ in $\boldsymbol{\varepsilon}_{t}$ for $s\in \mathcal{S}_{\xi}\subseteq \{1,2,...,m\}$. Without loss of generality, let the elements of $\mathcal{S}_{\xi}$ be the first $S$ elements of $\boldsymbol{\varepsilon}_{t}$, so that we may write \[\xi_{t}=\sum_{s=1}^{S}\varepsilon_{s,t}.\] Conventionally in the LP literature, the shock $\xi_{t}$ has often been treated as a single unit as if the components' coefficients are almost similar row-wise.

Quite a few papers document the composite nature of a macroeconomic policy shock, which consists of either more disaggregated policy shocks or shocks with distinct, often opposite natures and impacts. For fiscal policy shocks, Auerbach and Gorodnichenko (2012) show that disaggregated spending behave differently relative to an aggregate fiscal policy shock. Cox, M\"uller, Pasten, Schoenle, Weber (2020) and Bouakez, Rachedi and Santoro (2020) note that the composition of aggregate government spending heavily affects the aggregate spending multiplier and discuss sector-specific government spending multipliers. Meanwhile, monetary policy shocks, unlike government spending, which can be easily categorized by sectors, are categorized by their impact on the economy. For instance, Jaroci\'{n}ski and Karadi (2020) decompose a monetary policy announcement into a pure monetary policy shock and an information shock. Kaminska, Mumtaz, and \u{S}ustek (2021), on the other hand, decompose it into three components: shocks to the short term policy rate (action shocks), shocks due to communication about future economic conditions or policy intentions (path shocks), and shocks to risk premia due to the effect of communication on uncertainty (premia shocks).

The shock $\xi_{t}$ is typically unobserved and its scale is indeterminate. A common solution to this problem is to measure the magnitude of $\xi_{t}$ by means of an observable endogenous variable $x_{t}$ and then to use an IV $z_{t}$ correlated with $x_{t}$ to get external variation. As a result, the response of $y_{t+h}$ to $z_{t}$ relative to the response of $x_{t}$ to $z_{t}$ can be interpreted as the average response of $y_{t+h}$ to $\xi_{t}$ of a magnitude that corresponds to a unit change in $x_{t}$. Stock and Watson (2018) provide an example that the causal effect of GDP growth ($y_{t}$) to a monetary policy shock ($\xi_{t}$) can be identified by the ratio of the impulse responses of GDP growth and the federal fund rate ($x_{t}$) to a monetary policy announcement ($z_{t}$). Ramey and Zubairy (2018) calculate the government spending multiplier as a ratio of the impulse responses of GDP ($y_{t}$) and total government spending ($x_{t}$) to the military news shock ($z_{t}$).

Let $x_{t}$ be the first element of $\boldsymbol{Y}_{t}$ without loss of generality. Similar to $y_{t}$, we define the impulse response of $x_{t}$ at horizon $h$ to the shock $\varepsilon_{s,t}$ as $\theta_{h,xs}$, slightly abusing notation. We assume $\theta_{0,xs}>0$ for all $s = 1, 2, \dots, S$, which we refer to as the sign normalization, throughout the paper. It is sometimes convenient to assume $\theta_{0,xs} = 1$ for all $s = 1, 2, \dots, S$, a convention referred to as the unit effect normalization in the literature (Stock and Watson, 2018). In the government spending shock example, the unit effect normalization implies that a one-unit change in a sectoral spending shock results in a one-unit change in total spending.

Local Projections with an Instrumental Variable

Let $z_{t}$ be an IV that satisfies the following assumptions.

Define $\alpha_{s} = E[z_{t}\varepsilon_{s,t}]$. Assumption (ref)(i) implies that $\alpha_{s}\neq0$ for some $s=1,2,...,S$. Assumption (ref) is an extension of Condition LP-IV of Stock and Watson (2018) allowing that more than one structural shocks are correlated with the IV and the correlations are heterogeneous across the shocks.

The following proposition establishes that the LP-IV estimand is an affine combination of the normalized component-wise impulse responses. The proof is in Appendix (ref).

An important implication of Proposition (ref) is that the effect of a macroeconomic shock ($\xi_{t}$) identified by LP-IV is instrument-specific when there are at least two components ($S\geq2$) in the shock with heterogeneous component-wise impulse responses.

When there is only one component in $\xi_{t}$ (i.e., $S=1$), (ref) simplifies to Equation (8) of Stock and Watson (2018). In this case, $\xi_{t} = \varepsilon_{1,t}$ and the LP-IV estimand $\beta_{h} =\theta_{h,y1}$ with the unit effect normalization, $\theta_{0,x1}=1$. Thus, Proposition (ref) generalizes the previous identification result to cases when the instrument is correlated with multiple shocks in the SVMA model (ref).\footnote{Our model accommodates the general case where the researcher focuses on a composite macroeconomic shock, such as a monetary policy shock or a fiscal policy shock, and an IV is used to identify the shock. Technically, our model can equivalently handle situations where an IV is correlated with arbitrary multiple shocks in the SVMA model, which would typically render it an invalid IV in conventional settings.}

Proposition (ref) does not provide a complete identification result because the right-hand side of (ref) is not a convex combination---that is, a weighted average with non-negative weights---but rather an affine combination. To see this, let $S=2$. Then (ref) becomes

It is clear from (ref) that without restrictions on the signs of $\alpha_{s}$, we cannot interpret $\beta_{h}$ as a structural parameter, as it could take any real value. For example, suppose $\theta_{h,y1}=1$, $\theta_{h,y2}=2$, and $\theta_{0,x1} = \theta_{0,x2} = 1$, so the normalized impulse responses are both positive. However, if $\alpha_{1} = 0.2$ and $\alpha_{2} = -0.1$, then $\beta_{h} = 0$. Such a situation arises when the IV is correlated with the two structural shocks, $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$, with opposite signs.

Under the following assumption, we may interpret $\beta_{h}$ as a meaningful average of the underlying component-wise impulse responses.

In addition to Assumption (ref), this assumption imposes an additional requirement for the LP-IV estimand to admit a structural (causal) interpretation. Given the sign normalization for $\theta_{0,xs}$'s, this is equivalent to imposing the same sign conditions on the $\alpha_s$'s. Economic theory can often justify this assumption for a given IV and its relation to the component shocks within the structural shock. For example, Jarociński and Karadi (2020) use high-frequency surprises in fed funds futures ($z_{t}^{ff}$) and in stock prices ($z_{t}^{sp}$) as the IVs for the monetary policy shock ($\xi_{t}$), which is decomposed into the pure monetary policy shock ($\varepsilon_{1,t}$) and the central bank information shock ($\varepsilon_{2,t}$). Under the sign normalization, they impose a restriction that $\alpha_{1}>0$ and $\alpha_{2}>0$ for $z_{t}^{ff}$, but $\alpha_{1}<0$ and $\alpha_{2}>0$ for $z_{t}^{sp}$. Since $z_{t}^{ff}$ satisfies Assumption (ref), the impulse response identified by $z_{t}^{ff}$ is causal, while the impulse response identified by $z_{t}^{sp}$ is not.

Assumption (ref) constrains the instrument to load with the same sign on every structural shock $\varepsilon_{s,t}$ that drives the endogenous variable $x_t$. Conceptually, this mirrors the monotonicity restriction that identifies LATE in the potential outcomes framework (Imbens and Angrist 1994). Whereas monotonicity rules out “defiers” at the individual level, Assumption Same-Sign forces all the components of the composite macro policy shock to move in the same direction. A detailed comparison of the two assumptions is provided in the Online Appendices.

Since $E[\boldsymbol{Y}_{t}]=0$, $\boldsymbol{Y}_{t}$ is typically specified in differences or changes, but researchers may be interested in the level of $\boldsymbol{Y}_{t}$. The difference in the levels between $t$ and $t+h$ can be written as the cumulative changes over the $h$ periods: $\sum_{j=0}^{h}\boldsymbol{Y}_{t+j}$. Write $\widetilde{y}_{t+h} = \sum_{j=0}^{h}y_{t+j}$ and $\widetilde{x}_{t+h} = \sum_{j=0}^{h}x_{t+j}$. Define the cumulative impulse responses as $\widetilde{\theta}_{h,ys}=\sum_{j=0}^{h}\theta_{j,ys}$ and $\widetilde{\theta}_{h,xs}=\sum_{j=0}^{h}\theta_{j,xs}$, respectively.

Corollary (ref) is relevant for identification and estimation of the cumulative impulse responses, such as the cumulative government spending multiplier using external instruments as in Ramey and Zubairy (2018). Their LP-IV estimand takes the form of the LHS of (ref) after controlling for the lagged variables, where $y_{t}$ is the GDP, $x_{t}$ is government spending, $z_{t}$ is the military news shock, all relative to the trend GDP.

For the LP-IV estimand to have a structural interpretation, a version of Assumption (ref) is required to ensure non-negative weights. Note that the weight depends on $h$ in Corollary (ref). For illustration, suppose that the government spending shock consists of defense ($s=1$) and non-defense ($s=2$) spending shocks: $\xi_{t} = \varepsilon_{1,t}$ + $\varepsilon_{2,t}$. We can check if either $\alpha_{s}\widetilde{\theta}_{h,xs}\geq0$ or $\alpha_{s}\widetilde{\theta}_{h,xs}\leq0$ for $s=1,2$ and $h\geq0$. First, it is reasonable to assume that positive sectoral spending shocks have positive impacts on the cumulative government spending: $\widetilde{\theta}_{h,xs}>0$ for $s=1,2$ and $h\geq0$. Next, it is reasonable to assume that the military news shock is positively correlated with the defense spending shock ($\alpha_{1}>0$). Thus, the weight for the non-defense spending multiplier would be non-negative if the military news shock is non-negatively correlated with the non-defense spending shock ($\alpha_{2}\geq0$). If this is justified, then the estimand is a weighted average of the cumulative sectoral spending multiplier. On the other hand, if $\alpha_{2}<0$, the estimand may not provide meaningful information about the government spending multiplier because the estimand can be close to zero or even negative while the defense and non-defense spending multipliers are strictly greater than one. We find empirical evidence supporting $\alpha_{2}<0$ in Section (ref).

Identification with Multiple Instruments

It is NOT okay to use 2SLS

In the previous section, we established an identification result when a single IV is used to identify the impulse response of a structural shock via LP. In practice, it is common to have multiple IVs available to identify the same structural shock. Therefore, it is important to explore how the identification result from the previous section extends to cases with multiple IVs.

When more than one IV is available, conventional wisdom suggests combining the IVs to achieve efficiency. Stock and Watson (2018) propose a GMM-type estimator to combine multiple IVs in the LP model. Ramey and Zubairy (2018) estimate the government spending multiplier using LP-2SLS\footnote{Ramey and Zubairy (2018) report the LP estimate using “combined instruments” alongside other LP estimates using a single instrument. While Ramey and Zubairy do not explicitly mention that their estimates using “combined instruments” are based on 2SLS, this was verified by inspecting their Stata code and replicating their results using our own Matlab code.} using two IVs together.

However, Proposition 1 implies that using a GMM-type estimator, including 2SLS, to combine IVs should be avoided if the correlation structure between the IV and the components of a structural shock is heterogeneous across instruments. Since an LP-IV estimand is a linear combination of the structural impulse responses with possibly negative weights, combining such LP-IV estimands does not yield identification of a meaningful parameter.

This point is made clear using the result of Andrews (2019), who shows that the 2SLS estimand is a linear combination of the IV estimands using one instrument at a time with weights summing to one but possibly being negative. Consider two instruments, $z_{t}^{A}$ and $z_{t}^{B}$, and two components in the structural shock, $\xi_{t} = \varepsilon_{1,t}+\varepsilon_{2,t}$. By Proposition (ref), the LP-IV estimand is expressed as

for $j=A,B$, where $\beta^{A}_{h}$ and $\beta^{B}_{h}$ are the LP-IV estimands using $z_{t}^{A}$ and $z_{t}^{B}$ one at a time, respectively. Let $\beta^{2SLS}_{h}$ be the 2SLS estimand using both instruments together. According to Andrews (2019), $\beta^{2SLS}_{h} = p\beta^{A}_{h} + (1-p)\beta^{B}_{h}$ where

and $Z_{t} = (z_{t}^{A},z_{t}^{B})'$. Substituting this result into (ref), we get:

Here, the weights sum to one but can be negative. Since $p$ depends on the correlation between the IVs, $\beta^{2SLS}_{h}$ may not be a proper weighted average even when $z_{t}^{A}$ and $z_{t}^{B}$ satisfy Assumption (ref), in which case each LP-IV estimand $\beta_{h}^{A}$ and $\beta_{h}^{B}$ is causal. The example below illustrates such a case. Instead of combining the IVs using 2SLS, we present a more constructive way to combine the IVs in Section (ref) in the context of set identification under sign restrictions.

\noindentExample: a non-causal 2SLS as a linear combination of causal LP-IV estimands\newline Suppose that for $-1<\rho<1$,

Assumption (ref) is satisfied for both IVs. We calculate that $w_{1}^{A} = 1/2$, $w_{1}^{B}=1/13$, and \[\beta^{2SLS}_{h} = \frac{15(3-4\rho)\theta_{0,x1}}{233-208\rho}\cdot\frac{\theta_{h,y1}}{\theta_{0,x1}} + \frac{4(47-37\rho)\theta_{0,x2}}{233-208\rho}\cdot\frac{\theta_{h,y2}}{\theta_{0,x2}}.\] Then the weight for $\theta_{h,y1}/\theta_{0,x1}$ is negative if $\rho>3/4$.

Another implication of Proposition (ref) concerning the use of 2SLS with multiple IVs is that the over-identifying restrictions test (also known as Hansen's J test or Sargan test) will reject the null hypothesis of jointly valid IVs. The test tests whether the IV estimates based on a single IV converge in probability to a unique value, but this is not the case if each IV identifies a different weighted average (Lee, 2018). To be precise, the null hypothesis is given by the moment condition $H_{0}: 0 = E[Z_{t}(y_{t}-x_{t}\beta^{*})]$ where $\beta^{*}$ is the presumed true value estimated by 2SLS. Thus, the null hypothesis of the test can be written as

However, the LP-IV estimands $\beta_{h}^{A}$ and $\beta_{h}^{B}$ uniquely satisfy $0=E[z_{t}^{A}(y_{t+h}-x_{t}\beta_{h}^{A})]$ and $0=E[z_{t}^{B}(y_{t+h}-x_{t}\beta_{h}^{B})]$ by Proposition (ref)\footnote{Since $E[y_{t+h}]=0$ in the SVMA model (ref), $\operatorname*{cov}(y_{t+h},z_{t})=E[y_{t+h}z_{t}]$. The same applies to $x_{t}$.}. Therefore, it follows from (ref) that the null hypothesis of the test does not hold, provided that $p\neq0$ and $\beta_{h}^{A}\neq\beta_{h}^{B}$. Indeed, Ramey and Zubairy (2018) report significant p-values of the test statistic (ranging from 0.03 to 0.05) in their footnote 22. The rejection may be due to either the invalidity of a subvector of the IVs under the conventional assumption (e.g. Stock and Watson, 2018), or the composite nature of the structural shock given in this paper. To the best of our knowledge, there is no test to distinguish the two in the macroeconometrics literature.

In summary, it is generally not advisable to combine multiple IVs using conventional estimators such as 2SLS. Instead, there are more constructive approaches to leverage the information provided by multiple IVs. One possibility is to explore additional sources of information that can assist in identifying the impulse responses to the components of the structural shock. In the remainder of this section, we propose two such strategies: (i) using disaggregated data and restrictions on the inter-component impulse responses, and (ii) imposing sign restrictions on the correlation between the IV and the component-wise shocks. For simplicity, our identification strategies focus on the case where $S=2$ (two components), which covers the main applications discussed in Sections (ref) and (ref).\footnote{The strategies developed in this section can be extended to the cases with $S\geq3$ in principle, but with considerably more complex notations.}

Identification by Disaggregated Data

The first strategy involves using additional observed endogenous variables by augmenting the SVMA model. These variables correspond to each component-wise shock, $\varepsilon_{s,t}$. An example of such variables is a disaggregated series of $x_{t}$. For example, government spending ($x_{t}$) is disaggregated into defense ($x_{1,t}$) and non-defense ($x_{2,t}$) spending, which correspond to defense and non-defense spending shocks, respectively.

The augmented SVMA model (given in Appendix (ref)) specifies that the disaggregated variables $x_{1,t}$ and $x_{2,t}$ are part of the vector moving average system so that for $s=1,2$,

where $\psi_{0,sr} = E[x_{s,t}|\varepsilon_{r,t}=1]-E[x_{s,t}|\varepsilon_{r,t}=0]$ and $\{\cdots\}$ is a linear combination of the elements in the curly brackets. We assume the unit effect normalization, $\psi_{0,ss}=1$ for $s=1,2$, so that a unit change in a shock results in a one-unit increase in its corresponding disaggregated variable. Since $x_{t} = x_{1,t}+x_{2,t}$, the baseline model (ref) is a reduced SVMA model. The LP-IV estimand under the augmented SVMA model is

A key identifying restriction is on the contemporaneous inter-component impulse response, $\psi_{0,rs}$ for $r\neq s$ in (ref). This is the impact response of $x_{r,t}$ to a unit increase in $\varepsilon_{s,t}$ and they have precise economic interpretations. For example, let $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ be the defense and the non-defense spending shock, respectively. If $\psi_{0,21}=-0.2$, then a unit increase in the defense spending shock increases defense spending by the same amount ($\psi_{0,11}=1$) but decreases non-defense spending by 0.2. When $0=\psi_{0,12}=\psi_{0,21}$, the defense shock does not affect non-defense spending on impact, and vice versa. With a restriction on the inter-sectoral impulse responses, the impulse responses to the defense and nondefense shocks, $\theta_{h,y1}$ and $\theta_{h,y2}$, are identified.

To see how this restriction achieves identification, observe that under Assumption (ref), $\operatorname*{cov}(x_{s,t},z_{t}) = \alpha_{s}+\psi_{0,sr}\alpha_{r}$ for $s\neq r$. By solving for $\alpha_{s}$, we obtain:

for $s=1,2$ and $r\neq s$, provided that $\psi_{0,21}\psi_{0,12}\neq1$. By imposing restrictions on $\psi_{0,sr}$, the coefficients $\alpha_s$ are identified. For the second IV, we impose the same restriction on $\psi_{0,sr}$ and identify the corresponding instrument-specific coefficients $\alpha_s$. Combining (ref) across the two IVs then identifies $\theta_{h,y1}$ and $\theta_{h,y2}$.

It is worth mentioning that assuming no contemporaneous inter-component impulse responses, $\psi_{0,12}=\psi_{0,21}=0$, imposes a structural restriction similar to the recursive causal ordering in the triangular SVAR model to identify the impulse responses. In the analysis of the causal effects of personal and corporate income tax changes using SVAR with external IVs, Mertens and Ravn (2013) use a similar condition to isolate the effect of a change in only one of the tax rates.

The following proposition formally establishes identification of $\theta_{h,y1}$ and $\theta_{h,y2}$ using disaggregated series $x_{1,t}$ and $x_{2,t}$ and a restriction on $\psi_{0,sr}$, $s\neq r$.

The condition that $W$ is invertible corresponds to the standard rank condition for identification in models with two endogenous variables and two IVs. In our case, $W$ is not invertible if both IVs are correlated with only one component shock.

The identification strategy proposed in this subsection is illustrated in Section (ref) to obtain the defense and non-defense spending multipliers from the LP-IV estimates of the aggregate multipliers.

Identification Bounds by Sign Restrictions

The second strategy is to use the LP-IV estimand as bounds for the component-wise impulse responses by imposing sign restrictions on the correlation between the IV and the component shocks. This strategy is particularly powerful when some IVs violate Assumption (ref).

For instance, consider a case with two IVs and two component shocks, where the first one is positively correlated with both component shocks, while the second one has correlations of opposing signs. According to Proposition (ref), the first LP-IV estimand is a proper weighted average of the two impulse responses to each shock, whereas the second one lacks meaningful causal interpretation. However, this set of sign restrictions provides a new interpretation for the second LP-IV estimand. In this case, one of the two impulse responses can be shown to lie between the two LP-IV estimands. Therefore, while an LP-IV estimand in isolation fails to be informative when the IV exhibits opposing correlations with the components of the structural shock of interest, it can serve, in conjunction with another LP-IV estimand, as an enlightening bound for component-wise impulse responses.

Recall Proposition (ref) with $S=2$:

Rearranging terms, we obtain:

Under the sign normalization, $\theta_{0,xs}>0$, sign restrictions on $\alpha_{s}$ provide an inequality relationship between the LP-IV estimand and the component-wise impulse responses. An example of sign restrictions on the correlation between the IV and the component shock is Jaroci\'{n}ski and Karadi (2020). We apply our identification strategy given in this section to Jaroci\'{n}ski and Karadi (2020) in Section (ref).

Assuming that $\alpha_{s}\neq0$ for $s=1,2$, there are four different cases about the signs of $\alpha_{1}$ and $\alpha_{2}$ but we need not consider all the four cases. Since $\operatorname*{cov}(x_{t},z_{t})$ is estimable, we assume without loss of generality that $\operatorname*{cov}(x_{t},z_{t})=\alpha_{1}\theta_{0,x1}+\alpha_{2}\theta_{0,x2}>0$. Since $\theta_{0,xs}>0$, we rule out $\alpha_{1}<0$ and $\alpha_{2}<0$. We do not consider $\alpha_{s}=0$ because if $\alpha_{1}=0$ then $\beta_{h} = \theta_{h,y2}/\theta_{0,x2}$ and if $\alpha_{2}=0$, then $\beta_{h} = \theta_{h,y1}/\theta_{0,x1}$, which corresponds to the conventional LP-IV identification. In addition, we also rule out $\theta_{h,y1}/\theta_{0,x1}=\theta_{h,y2}/\theta_{0,x2}$, which trivially implies $\beta_{h}=\theta_{h,y1}/\theta_{0,x1}=\theta_{h,y2}/\theta_{0,x2}$.

Corollary (ref) formally gives the identified set of $(\theta_{h,y1}/\theta_{0,x1},\theta_{h,y2}/\theta_{0,x2})$ by sign restrictions on $\alpha_{1}$ and $\alpha_{2}$. The identified sets are illustrated in Figure (ref). In each panel, the sets that correspond to $\Theta_{++}$, $\Theta_{+-}$, and $\Theta_{-+}$ are shown as shaded areas.

Our sign restrictions strategy can handle multiple IVs straightforwardly. Applying Corollary (ref) to two or more IVs, we construct an identified set by intersection. The shape of the identified set differs depending on the set of sign restrictions imposed.

We illustrate the intersections in Figure (ref), where the identified set by two IVs (denoted by the superscript $A$ or $B$ and positively correlated with $x_t$) are shown as dark shaded areas. Let $\beta_{h}^{A}$ and $\beta_{h}^{B}$ be the IV estimand using each of the instrument $A$ and $B$ one at a time, respectively. Panels (a)-(c) show the identified set when (a) $\alpha_{1}^{j}>0$ and $\alpha_{2}^{j}>0$, (b) $\alpha_{1}^{j}>0$ and $\alpha_{2}^{j}<0$, (c) $\alpha_{1}^{j}<0$ and $\alpha_{2}^{j}>0$ for $j=A,B$, respectively. These cases arise when the same sign restrictions are imposed for the instruments. In these cases, $min(\beta_{h}^{A},\beta_{h}^{B})$ and $max(\beta_{h}^{A},\beta_{h}^{B})$ serve as the upper or lower bounds depending on the inequality relationship between $\theta_{h,y1}/\theta_{0,x1}$ and $\theta_{h,y2}/\theta_{0,x2}$. In other words, if we can impose a restriction that $\theta_{h,y1}/\theta_{0,x1}>\theta_{h,y2}/\theta_{0,x2}$ or vice versa, then we can obtain the one-sided identified set for $\theta_{h,y1}/\theta_{0,x1}$ and $\theta_{h,y2}/\theta_{0,x2}$. For example, if $\theta_{h,y1}/\theta_{0,x1}>\theta_{h,y2}/\theta_{0,x2}$, then $max(\beta_{h}^{A},\beta_{h}^{B})<\theta_{h,y1}/\theta_{0,x1}$ and $\theta_{h,y2}/\theta_{0,x2}<min(\beta_{h}^{A},\beta_{h}^{B})$ in Panel (a).

Panels (d)-(i) show the identified sets when the different sign restrictions, varying across the instruments, are imposed. In these cases multiple IV's provide more informative bounds for the component-wise impulse responses without the inequality relationship between $\theta_{h,y1}/\theta_{0,x1}$ and $\theta_{h,y2}/\theta_{0,x2}$. Furthermore, the intersection of two identified sets with opposing sign restrictions gives the identified set bounded both above and below. This can happen when $\Theta_{+-}^{A}$ and $\Theta_{-+}^{B}$ intersect. If $\beta_{h}^{A}>\beta_{h}^{B}$, then the identified set for $(\theta_{h,y1}/\theta_{0,x1},\theta_{h,y2}/\theta_{0,x2})$ is given by

This set is illustrated in Panel (f).

The following Proposition gives conditions under which the identified set for $\theta_{h,y1}/\theta_{0,x1}$ and $\theta_{h,y2}/\theta_{0,x2}$ are one-sided or two-sided. It turns out that the projected (rather than joint) identified sets are characterized by four types of sets, $UL$, $U$, $L$, and $T$, where

for $s=1,2$.

Proposition (ref) shows that the identified set is most (least) informative when the sign restrictions are opposite (the same) for the IVs. For practical reference, we give a full characterization of the identified set for each combination of signs allowing for $\text{cov}(z_{t}^{j},x_{t})<0$ for $j=A, B$ and $\beta_{h}^{A}<\beta_{h}^{B}$ in Appendix (ref).

In empirical applications, the bounds for the identified set is given by the LP-IV estimates. Since the bounds are estimated, an appropriate confidence set should be calculated. We provide the detailed procedure of obtaining confidence sets for the identified set in the spirit of Imbens and Manski (2004) detailed in Appendix (ref).\footnote{While there are some similar aspects between our approach to conducting inference specified in Appendix D.2 and the one by Imbens and Manski (2004), there are notable differences. Imbens and Manski (2004) assume the existence of the upper and lower bounds of the parameter of interest $\theta$, there is no guarantee that there exist the upper and lower bounds for the parameter except for certain combinations of the weights $w_1$ and $w_2$.}

We give a brief literature review to highlight the contribution of our set identification approach relative to the existing studies. There exists a substantial body of research dedicated to constructing identified sets for impulse responses under sign restrictions within fully specified SVAR models, encompassing the IV equations within the system, e.g. Giacomini, Kitagawa, and Read (2022) and references therein. Moon and Schorfheide (2012) and Granziera, Moon, and Schorfheide (2018) have analyzed the asymptotic difference between the Bayesian and frequentist approaches to the set estimation and shown that the Bayesian highest posterior density sets exclude parts of the estimated identified sets while the frequentist confidence sets extend beyond the boundaries of the estimated identified set.

On the other hand, prior to this paper, while sign restrictions have been utilized within the LP framework, for instance, Plagborg-M\o ller and Wolf (2021) and Alpanda, Granziera, and Zubairy (2021), the LP-IV has not been implemented with sign restrictions. Nor has its inference been investigated let alone the bounds for impulse responses identified by the LP-IV. In contrast to previous SVAR-based approaches, which are computationally more intensive, our approach imposes sign restrictions without requiring constrained optimization, making it particularly attractive as the number of available IVs increases. Although Plagborg-Møller and Wolf (2021) establish the equivalence between LP-IV and SVAR estimates under a single IV, this equivalence does not naturally extend to the multiple-IV case---unless one estimates separate VAR systems to compute impulse responses for each IV, which essentially replicates the strategy we propose.

Sharpness of Sign-Restricted LP-IV Identification

In Section (ref), we show that combining multiple sign-restricted IVs yields informative bounds for the component-wise impulse responses of interest. In particular, when different sign restrictions are imposed on two IVs, the joint identified set for the impulse responses to $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ takes the form of either a rectangle unbounded in only one direction, or a right- or left-angled triangle, depending on the combination of sign restrictions and the ordering of the two LP-IV estimands.

This identified set utilizes only partial information from the observable moments, specifically $(E[x_{t}z_{t}^{A}], E[x_{t}z_{t}^{B}], E[y_{t+h}z_{t}^{A}], E[y_{t+h}z_{t}^{B}])$. Incorporating additional available moments could potentially lead to a smaller identified set. In this section, we show that, under conditions described later, the marginal sign-restricted LP-IV identified set for each impulse response is sharp, in the sense that every parameter value in the set is consistent with the observed autocovariances. Therefore, this identified set already incorporates all information implied by the second moments of the observables. We focus on the case where one IV is positively correlated with both $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$, and the other IV is correlated positively with $\varepsilon_{1,t}$ and negatively with $\varepsilon_{2,t}$. This scenario is highly relevant in practice, as this set of sign restrictions appears to be natural, e.g. when IVs capture on high-frequency surprises for monetary policy shocks, as in our empirical study in Section (ref).

While we state sharp identification results for a bivariate VMA$(1)$ model, this is solely to simplify the exposition; our results hold for a more general VMA model, as discussed in the Supplemental Appendix.

Plagborg-M\o ller and Wolf (2022), in a similar vein, establish sharp identification of variance decompositions--that is, the contribution of a specific shock to the fluctuation of an endogenous variable of interest at a given horizon--based on the autocovariances of the observables. In contrast, ours is the first work to establish the sharpness of (relative) impulse responses, arguably the primary causal parameters in empirical macroeconomics. While they allow for multiple IVs (Supplemental Appendix B3., p.9), they do not impose sign restrictions, which are the key source of identification for impulse responses in our framework.

Model Setup

Consider the SVMA$(1)$ model for the pair $(\mathbf{Y}_{t}^{\prime},\mathbf{Z}_{t}^{\prime})^{\prime}$ of $2 \times 1$ endogenous and instrumental variable vectors $\mathbf{Y}_{t}=(x_{t},y_{t})'$ and $\mathbf{Z}_{t}=(z_{t}^{A},z_{t}^{B})'$ with respect to the $3 \times 1$ structural shocks $\boldsymbol{\varepsilon}_{t}$ and the $2 \times 1$ projection error vector $\mathbf{v}_{t}$:

where $\boldsymbol{\varepsilon}_{t}\sim WN(\mathbf{0}_{3\times1},\text{diag}(\boldsymbol{\lambda}))$ and $\mathbf{v}_{t}\sim WN(\mathbf{0}_{2\times1},\boldsymbol{I})$. Assume $\text{diag}(\boldsymbol{\lambda})>0$ where $\boldsymbol{\lambda}=(\lambda_1,\lambda_2,\lambda_3)^{\prime}$. The matrix $\mathbf{D}$ is the projection coefficients of $\mathbf{Z}_{t}$ on $(\varepsilon_{1,t},\varepsilon_{2,t})$ and is given by

Note that this specification imposes the exclusion restriction that $\alpha_{3}^{j}=0$ for $j=A,B$, and $\alpha_{s}^{j}=d_{s}^{j}\lambda_{s}$ for $j=A,B$ and $s=1,2$. The parameters of interest are $\theta_{1,y1}/\theta_{0,x1}$ and $\theta_{1,y2}/\theta_{0,x2}$, the relative impulse responses of $y_{t}$ with respect to $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ at horizon $1$. We may consider the impulse responses at horizon $0$ instead, and the results in Section (ref) remain valid.

As in Section (ref), we assume without loss of generality that $E[z_{t}^{j}x_{t}]>0$ or

Each IV satisfies the following sign restrictions:

As shown in Proposition (ref) and Figure (ref)(d) and (g), the identified set for $(\theta_{1,y1}/\theta_{0,x1}, \allowbreak \theta_{1,y2}/\theta_{0,x2}) $ based on LP-IV estimands is given by

where $\beta^{j}:=\beta_{1}^{j}=E[z_{jt}y_{t+1}]/E[z_{jt}x_{t}],\:j=A,B$.

Sharpness under Weak Stationarity

The identified set for $(\theta_{1,y1}/\theta_{0,x1},\theta_{1,y2}/\theta_{0,x2})$ in ((ref)) is based on the LP-IV estimands, which depend only on the covariance between $\mathbf{Z}_{t}$ and $(x_{t},y_{t+1})$, namely $(E[x_{t}z_{t}^{j}],E[y_{t+1}z_{t}^{j}])$ for $j=A,B$. Yet it does not utilize other features of the underlying weakly stationary process $\{(\mathbf{Y}_{t}^{\prime},\mathbf{Z}_{t}^{\prime})^{\prime}\}_{t}$ that are consistently estimable. In this section, we define observational equivalence with respect to the autocovariance function of $\{(\mathbf{Y}_{t}^{\prime},\mathbf{Z}_{t}^{\prime})^{\prime}\}_{t}$, which we denote by $\text{AC}_{P}$, and investigate whether the LP-IV identified set is sharp, i.e., it incorporates all the information contained in $\text{AC}_{P}$. Notice that the $\text{AC}_{P}$ implied by the model depends only on the finite-dimensional component $\boldsymbol{\tau}$ of the model, defined on the parameter space

where $\mathbb{R}_{++}$ denotes the positive real line. An element $\boldsymbol{\tau}$ of $\boldsymbol{\Xi}$ is said to be observationally equivalent to another model parameter value $\grave{\boldsymbol{\tau}} \in \boldsymbol{\Xi}$ if they imply the same $\text{AC}_{P}$.

Let $\Pi=(\Pi_1,\Pi_2)^\prime$ be a function from $\boldsymbol{\Xi}$ to $\mathbb{R}^{2}$, defined as

Then, given an observed $\text{AC}_{P}$, the identified set for $(\theta_{1,y1}/\theta_{0,x1},\theta_{1,y2}/\theta_{0,x2})$ is given by

The corresponding (marginal) identified set for $\theta_{1,yj}/\theta_{0,xj},\:j=1,2$, is given by

A candidate identified set for $\theta_{1,y1}/\theta_{0,x1}$ (or $\theta_{1,y2}/\theta_{0,x2}$) is said to be sharp in $\boldsymbol{\Xi}$ if it coincides with $ID_{1}(\text{AC}_{P}\:;\:\boldsymbol{\Xi})$ (or $ID_{2}(\text{AC}_{P}\:;\:\boldsymbol{\Xi})$).

It may be of interest to investigate sharpness under additional normalizations that further restrict the parameter space. To this end, we consider the following subsets of $\boldsymbol{\Xi}$:

and

where $\boldsymbol{\Xi}^{(\text{\normalfont EN})}$ and $\boldsymbol{\Xi}^{(\text{\normalfont VN})}$ correspond to the parameter space further restricted by the unit-effect normalization for $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ with respect to $x_{t}$, and the unit-variance normalization for $\boldsymbol{\varepsilon}_{t}$, respectively. We refer to the restrictions of the function $\Pi$ to $\boldsymbol{\Xi}^{(\text{\normalfont EN})}$ and $\boldsymbol{\Xi}^{(\text{\normalfont VN})}$ also as $\Pi$ for brevity. Then, the joint and marginal identified sets under the unit-effect and unit-variance normalizations are given by ((ref))-((ref)) with $\boldsymbol{\Xi}$ replaced by $\boldsymbol{\Xi}^{(\text{\normalfont EN})}$ and $\boldsymbol{\Xi}^{(\text{\normalfont VN})}$, respectively.

The following proposition characterizes the sharpness and unboundedness of the marginal identified sets for the relative impulse responses under the sign restriction (ref). We separate the case $\beta^{A}=\beta^{B}$, in which $(\theta_{1,y1}/\theta_{0,x1},\theta_{1,y2}/\theta_{0,x2})$ is point-identified. In that case, the identified set is a singleton and thus sharp by definition.

It should be noted that Proposition (ref) holds regardless of whether the unit-effect or unit-variance normalization is imposed. While the unit-variance normalization is commonly used in practice, the unit-effect normalization has primarily been applied to point-identified models in the literature. Which normalization provides a more natural interpretation of the estimands depends on the context and the nature of the data; the unit-effect normalization can be more appropriate even in set-identified SVMA models. For instance, when $x_{t}$ corresponds to total government spending, and $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ are defense and non-defense spending shocks, respectively, the unit-effect normalization $\theta_{0,x1}=\theta_{0,x2}=1$ implies that both shocks are scaled in the same monetary units.

To understand what restrictions ((ref)) and ((ref)) impose on the underlying data generating process through the autocovariance, notice that $d_{1}^{A}/\theta_{0,x1}$ and $d_{2}^{A}/\theta_{0,x2}$ can be interpreted as the relative impulse responses of $z_{t}^{A}$ to $\varepsilon_{1,t}$ and $\varepsilon_{2,t},$ respectively. Then, ((ref)) states that there should exist a data generating process that is compatible with $\text{AC}_P$ and the relative contribution of $\varepsilon_{2,t}$ to the variation of $z_{t}^{A}$ must be as large as that of $\varepsilon_{1,t}$. In this case, the identified set for the impulse response to the second component is unbounded. Likewise, if the relative contribution of first component is as large as the second, that is, (ref) holds, the LP-IV based identified set for the impulse response to the first component is sharp.

Next, note that ((ref)) is a component in the decomposition of the covariance between $z_{t}^{A}$ and $z_{t}^{B}$:

where $\boldsymbol{\Gamma}=[\gamma_{1}^{A}\quad\gamma_{2}^{A}\:;\:\gamma_{1}^{B}\quad\gamma_{2}^{B}]$. If we can assume $\boldsymbol{\Gamma}=0$, then ((ref)) is empirically testable. Indeed, we note that Jarociński and Karadi (2020) put the same structures on IVs. In their empirical model, the sample correlation coefficient between the IVs is negative ($-.4261$,\: s.e.$=.1391$), which indicates that the premise of Proposition (ref).3 holds, in which case and the marginal identified set for $\theta_{1,y2}/\theta_{0,x2}$\footnote{This corresponds to the impulse response to the central bank information shock in Jarociński and Karadi (2020).} is unbounded. This entails that any bounded confidence set has zero coverage probability (Dufour, 1997). Since their procedure always produces a bounded credible set, its frequentist size must also be zero. On the other hand, our LP-IV inference framework accommodates unbounded identified sets, providing valid one-sided confidence intervals as discussed in Appendix (ref).

As shown in the proof of the proposition (see (ref)), ((ref)) implies that ((ref)) holds with strict inequality. ((ref)) further requires that the relative contribution of $\varepsilon_{2,t}$ to the variation of $z_{t}^{B}$ is reasonably large, to the degree of which depends on how tight the inequality in ((ref)) is.

On the other hand, ((ref)) implies that $d_{1}^{A}d_{1}^{B}\lambda_{1}+d_{2}^{A}d_{2}^{B}\lambda_{2}>0$ so that if $\boldsymbol{\tau}\in\overline{\boldsymbol{\Xi}}$ satisfies ((ref)), then the same $\boldsymbol{\tau}$ cannot satisfy ((ref)). Since $\boldsymbol{\Xi}$ must contain at least one element (the “true” parameter value) and thus is nonempty, for any value of $\text{AC}_{P}$, there always exists some parameter value that generates $\text{AC}_{P}$ and either ((ref)) or ((ref)) holds. Thus, for any value of $\text{\normalfont AC}_P$, either the LP-IV marginal identified set for $\theta_{1,y1}/\theta_{0,x1}$ is sharp, or the identified set for $\theta_{1,y2}/\theta_{0,x2}$ is unbounded. Both LP-IV marginal identified sets are sharp if there exist two distinct elements $\boldsymbol{\tau},\widetilde{\boldsymbol{\tau}}\in\overline{\boldsymbol{\Xi}}$ such that both generate $\text{AC}_{P}$ and $\boldsymbol{\tau}$ satisfies ((ref)) and $\widetilde{\boldsymbol{\tau}}$ satisfies ((ref)).

This raises the question of whether such conditions hold for any data generating process and the corresponding $\text{AC}_{P}$. Furthermore, if the answer is negative, can sharp identification of marginal identified sets still be achieved using only LP-IV estimands? The answer to both questions is negative.

In Appendix (ref), we present an example of a value of $\text{AC}_{P}$ such that either ((ref)) or ((ref)) holds for every $\boldsymbol{\tau}\in\overline{\boldsymbol{\Xi}}$ that generates $\text{AC}_{P}$. For both Proposition (ref).2 and (ref).3 to hold for a given $\text{AC}_{P}$, additional restrictions must be imposed on the true data generating process. In this example, we consider $\boldsymbol{\tau}\in\boldsymbol{\Xi}$ such that $\boldsymbol{\Gamma}=\boldsymbol{0}_{2\times2}$, and show that for $\overline{\boldsymbol{\tau}}$ to be observationally equivalent, it is necessary that $\widetilde{\boldsymbol{\Gamma}}=\boldsymbol{0}_{2\times2}$. This example illustrates that when the IVs are subject to only a low level of noise, their moments provide additional identification power for one of the relative impulse responses of interest, in which case the corresponding marginal LP-IV identified set can be improved. It may be of interest to further explore what conditions on the data generating process ensure sharpness of LP-IV identification for both relative responses; we leave this for future work.

Government Spending Multiplier in the U.S.

The government spending multiplier is the ratio of the change in GDP to the change in government spending. Understanding the magnitude of the multiplier is crucial for making fiscal policies, but there is still a debate in the literature about whether it is larger than one. For instance, studies by Blanchard and Perotti (2002), Barro and Redlick (2011), Ramey (2011), Auerbach and Gorodnichenko (2012), Nakamura and Steinsson (2014), and Ramey and Zubairy (2018) have explored this question extensively.

Most studies focus on estimating the aggregate or defense spending multipliers by examining variations in defense spending. This is because non-defense spending varies less than defense spending, and more importantly, non-defense spending is likely to be endogenous concerning GDP (Barro and Redlick, 2011).

We estimate the non-defense spending multiplier in the United States in the post WWII period. Building on the identification results from the previous sections, we break down the aggregate spending multipliers, which were estimated by external IVs (LP-IV estimates), into sectoral spending multipliers.\footnote{It may be tempting to estimate the defense and non-defense spending multipliers directly using sectoral spending variables. This would be only possible with an IV which is only correlated with a particular sectoral spending shock. Otherwise, the resulting LP-IV estimand is still a weighted average of defense and non-defense spending multipliers.} To achieve this, we use disaggregated sectoral spending data and two IVs, as outlined in Section (ref). We focus on the post-WWII period because quarterly sectoral spending data are only available after WWII.

The LP-IV model is

where $y_{t}$ is GDP, $x_{t}$ is government spending, $\boldsymbol{R}_{t}$ is a set of control variables, and $\boldsymbol{\phi}_{h}(L)$ is a coefficient vector of polynomial in the lag operator of order 4. Since $\sum_{j=0}^{h}y_{t+j}$ is the sum of GDP over $h$ periods and $\sum_{j=0}^{h}x_{t+j}$ is the sum of government spending over $h$ periods, the parameter $\beta_{h}$ is the cumulative government spending multiplier. The instrument is $z_{t}$. The control variables include lagged values of $y_{t}$, $x_{t}$, and $z_{t}$.

We use two IVs in our analysis: the military news shock from Ramey and Zubairy (2018), referred to as `RZ news shock', and the current defense spending shock from Blanchard and Perotti (2002), labeled as `BP defense shock'. The BP defense shock is (the residual of) current defense spending.\footnote{We conducted the weak IV test of Montiel Olea and Pflueger (2013) to accommodate possible serial correlation in the errors. We did not find any statistical evidence of weak IVs for both IVs.} When using the 'BP defense shock' as an IV, our control variables consist of the lagged values of GDP, government (aggregate) spending, and defense spending. This approach aligns with the construction of the Blanchard-Perotti (2002) shock in Ramey and Zubairy (2018) using current government spending.\footnote{Following Ramey and Zubairy (2018), the identification of the BP defense shock is equivalent to the SVAR defense spending shock of Blanchard and Perotti (2002). The SVAR system in Section IX. B. of Blanchard and Perotti (2002) includes four variables: taxes, defense spending, non-defense spending, and the GDP. Our main specification does not include taxes, but we obtained a similar result when we included taxes in the control variables.}

Figure (ref) presents the cumulative government spending multipliers for each horizon from two quarters to five years out. The bands are the pointwise 90% confidence bands using Newey-West (1987) standard errors. The LP-IV estimates are based on two different IV's: the RZ news shock and the BP defense shock. These estimates correspond to the LHS of (ref) in Corollary (ref). We also report LP-2SLS estimates using both IVs together.

Consistent with Ramey and Zubairy (2018), the estimated multipliers using either of the RZ and BP shocks are below one after the first year. When the BP defense shock is used, the initial multiplier is smaller than the RZ shock but shows more prolonged effects after three years. The LP-2SLS estimates are quite similar to those estimated using the BP defense shock. This is because 2SLS implicitly puts more weight on the stronger IV (in this case, the BP defense shock). However, it is evident that the 2SLS estimates are not necessarily a weighted average of the two LP-IV estimates. Therefore, even when the validity of each LP-IV estimand is justified, combining multiple IVs via 2SLS may lead to a loss of causal interpretation, as demonstrated in Section (ref).

We combine information from the two IVs to estimate the cumulative non-defense and defense spending multipliers using Proposition (ref). To estimate the sectoral spending multipliers, we need to calibrate the cumulative inter-sectoral effect\footnote{This is the cumulative version of the inter-component impact responses in Section (ref). The cumulative version of Proposition (ref) is given in Appendix X.}, which is the cumulative effect of a sectoral shock $\varepsilon_{s,t}$ on $\sum_{j=0}^{h}x_{r,t+j}$, where $s\neq r$. For simplicity, we assume that the cumulative inter-sectoral effects are symmetric, i.e., the effect of a defense spending shock on non-defense spending is equal to that of a non-defense spending shock on defense spending. We set the cumulative inter-sectoral effects to have non-positive values of 0, -0.2, -0.4, or -0.6. For example, if the cumulative effect is -0.2 then a unit increase in the defense spending shock decreases non-defense spending by 0.2 over $h$ quarters, and vice versa. In the budget-neutral case, the cumulative inter-sectoral effect is -1: a one-unit increase in the defense spending shock raises defense spending by one unit, but decreases non-defense spending by an equivalent amount, resulting in no change in total government spending. In this case, the sectoral spending multipliers are not identified.

In Figure (ref), we present the cumulative sectoral spending multipliers derived from the two LP-IV estimates of the government spending multipliers shown in Figure (ref). The shaded areas represent pointwise 90% confidence intervals, calculated using Newey-West (1987) standard errors. Details of the estimation procedure are provided in Appendix (ref). The defense spending multiplier generally remains below one and decreases as the cumulative inter-sectoral effect becomes larger in absolute terms. In contrast, the cumulative non-defense spending multiplier follows a different pattern: although all estimates start below one, they increase steadily after approximately six quarters. This suggests a prolonged effect of non-defense spending on GDP. Notably, the non-defense spending multiplier exceeds one after roughly 18 quarters when the cumulative inter-sectoral effect is zero. This is surprising, given that the LP-IV estimates using the RZ news shock and the BP defense shock--both of which are used to obtain the sectoral spending multipliers--are below one. This is due to the negative weight assigned to the non-defense spending multiplier in the decomposition of the government spending multiplier.

To understand the role of weights in the government spending multiplier, we examine the decomposition of the cumulative government spending multiplier for eighteen-quarters ($h=18$) when there is no cumulative inter-sectoral effects. Let $s=1$ denote defense spending and $s=2$ denote non-defense spending. The government spending multiplier estimated by the RZ news shock, is then decomposed as

and the multiplier estimated by the BP defense shock is decomposed as

where $\widetilde{\theta}_{h,ys}$ and $\widetilde{\psi}_{h,ss}$ ($s = 1, 2$) are the cumulative impulse response of $y_{t}$ and $x_{s,t}$, respectively, over $h$ periods in response to the component shock $\varepsilon_{s,t}$. The weights are consistently estimated using sectoral spending data by

where $j$ denotes the RZ news shock or the BP defense shock and $v_{t}^{\perp(j)}$ denotes the residual of $v_{t}$ after regressing it on the set of control variables including the lagged values of the instrument $j$. The estimated weights show similar magnitudes and signs across different $h$ for both IVs.

The decomposition (ref) reveals that the government spending multiplier estimated using the RZ news shock is smaller than both sectoral spending multipliers due to the negative weight assigned to the non-defense spending multiplier. This indicates a violation of Assumption (ref) for the RZ news shock as an IV. The positive military news shock leads to a positive impact on defense spending, but at the same time, it has a negative impact on non-defense spending. Consequently, the positive non-defense spending multiplier has a negative effect on GDP, thereby underestimating the sectoral spending multipliers. This raises concerns about the structural interpretation of the government spending multiplier estimated using the RZ news shock.

In contrast, the decomposition (ref) for the government spending multiplier estimated using the BP defense shock, both weights have positive signs, but the weight for the non-defense spending multiplier is very small (0.03). As a result, the government spending multiplier estimated by the BP defense shock closely resembles the defense spending multiplier.

Figure (ref) shows the impulse response of defense spending (dash-dotted line) and non-defense spending (solid line) to the RZ news shock (left panel) and to the BP defense shock (right panel). Note that the dependent variables are sectoral spending ($x_{s,t+h}$), rather than the cumulative sectoral spending ($\sum_{j=0}^{h}x_{s,t+j}$). The RZ news shock has a relatively small but statistically significant negative effect on non-defense spending. This may be due to the government budget constraint. In contrast, the response of non-defense spending to the BP defense shock is statistically insignificant for all $h$.

Monetary Policy and Central Bank Information Shock

Jarociński and Karadi (2020; hereinafter JK2020) argue that central bank monetary policy shocks contain valuable information about economic conditions and monetary policy. They disentangle the central bank information shock from the composite monetary policy shock using Bayesian structural VAR with sign restrictions.

We apply the sign restrictions approach from Section (ref) to JK2020's dataset, using their restrictions and IVs but implementing them with LP-IV to obtain set-identified impulse responses to both pure monetary policy and central bank information shocks.

Let $\varepsilon_{1,t}$ be the pure monetary policy shock, and $\varepsilon_{2,t}$ be the central bank information shock. We assume that the central bank announcements ($\xi_{t}$) contain $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$, along with other structural shocks and measurement errors, that belong to the right-hand side of the SVMA model (ref). The IVs are the high-frequency surprises (the change between 10 minutes before and 20 minutes after the announcements) in the fed funds futures ($z_{t}^{ff}$) and in the stock price ($z_{t}^{sp}$). These IVs react to the central bank announcements in the very short time period, so they are assumed to be uncorrelated with any other structural shocks except for $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$. The sign restrictions we impose are the same as those in Table 1 of JK2020, given by:

Since the scale of $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ is indeterminate, the monthly average of the one-year constant-maturity Treasury yield ($x_{t}$) is used to fix the scale of the shocks. Let $z_{t}$ be either $z_{t}^{ff}$ or $z_{t}^{sp}$. The LP-IV estimand is decomposed as:

Consider $z_{t}=z_{t}^{ff}$. Since Assumption (ref) is satisfied by the sign restrictions (ref), the LP-IV estimand $\beta_{h}^{ff}$ is interpreted as a structural impulse response to a monetary policy shock. In contrast, $\beta_{h}^{sp}$ using $z_{t}^{sp}$ as the IV does not have a structural interpretation due to the opposite signs as in (ref). Following the sign of the sample covariances, we assume $\operatorname*{cov}(z_{t}^{ff},x_{t})>0$ and $\operatorname*{cov}(z_{t}^{sp},x_{t})<0$.

The econometric model for LP-IV is given by:

where $y_{t}$ is a macro variable of interest: the monthly average of the one-year Treasury yield, the monthly average of the S&P 500 index in log levels, the real GDP and the GDP deflator in log levels, or the excess bond premium (EBP), $\mu_{h}$ is a constant, $x_{t}$ is the monthly average of the one-year Treasury yield, and $\boldsymbol{R}_{t-1}$ is a set of control variables. These control variables include lagged values of all of the macro variables included in the impulse response analysis, as well as $x_{t}$ and $z_{t}$. $\boldsymbol{\phi}_{h}(L)$ is a coefficient vector of polynomial in the lag operator of order 12 following JK2020.

Figure (ref) displays the LP-IV and LP-2SLS estimates, along with the identified set of impulse responses to the pure monetary policy shock (left column) and the central bank information shock (right column), obtained by imposing sign restrictions (ref)-(ref). The blue shaded areas represent the identified sets with 68% pointwise confidence bands, calculated using Proposition (ref) in Appendix (ref). In each panel, the solid line, dot-dashed line, and x marker denote the LP-IV estimate $\widehat{\beta}_{h}^{ff}$, $\widehat{\beta}_{h}^{sp}$, and the LP-2SLS estimate using both IVs together, respectively. Impulse responses are calculated for a 25 basis-point (BP) change in the one-year government bond yield.\footnote{JK2020's analysis of the same data shows an impact response of the one-year government bond yield of about five basis points increase to one standard deviation change in the monetary policy shock, and about ten basis points increase to one standard deviation change in the central bank information shock, with shocks normalized to unit variance. The different shock scaling affects response magnitudes.}

Let us first examine the point estimates. The LP-IV estimate $\widehat{\beta}_{h}^{ff}$ (solid line) is calculated using high-frequency surprises in the fed funds futures ($z_{t}^{ff}$) as the IV. On impact, the one-year government bond yield increases by 25BP by construction, and this effect is quite persistent. Stock prices initially decline but begin to recover after approximately one year. The negative impact on real GDP and the price level exhibits even greater persistence. In contrast to these long-lasting effects on GDP and price level, the effect on the excess bond premium is notably short-lived.

The other LP-IV estimates, $\widehat{\beta}_{h}^{sp}$ (dot-dash), calculated using the high-frequency surprises in the S&P 500 index ($z_{t}^{sp}$), show markedly different responses. The one-year government bond yield moves in the opposite direction, with a negative effect persisting for over two years. For other macro variables, the direction of the responses is similar to $\widehat{\beta}_{h}^{ff}$, but the magnitude is generally much larger.

Our theoretical result provides credible guidance on how to interpret these two different impulse responses. $\widehat{\beta}_{h}^{ff}$ is structurally interpretable if the sign restrictions (ref) hold. In contrast, $\widehat{\beta}_{h}^{sp}$ lacks such an interpretation because the sign restrictions (ref) do not satisfy Assumption (ref).

While $\widehat{\beta}_{h}^{ff}$ provides a structural impulse response estimate, it alone cannot inform about the relative magnitude of effects from pure monetary policy shocks versus central bank information shocks. The LP-2SLS method, using both $z_{t}^{ff}$ and $z_{t}^{sp}$ instruments (denoted by x markers), is not a solution. As shown in Section (ref), it neither identifies structural component-wise impulse responses nor allows for general structural interpretation. In Figure (ref), the LP-2SLS estimates closely mirror $\widehat{\beta}_{h}^{ff}$ due to $z_{t}^{ff}$'s greater strength compared to $z_{t}^{sp}$.\footnote{The effective F statistics for both IVs (Montiel Olea and Pflueger, 2013) are approximately 16 for $z_{t}^{ff}$ and 3 for $z_{t}^{sp}$, varying little over $h$. The 5% and 10% critical values for testing whether the IV estimator bias exceeds 10% of the OLS bias are 23.1 and 19.7, respectively. Neither IV passes the weak IV test, with $z_{t}^{sp}$ being particularly weak.}

Applying the sign restrictions approach from Section (ref), we use LP-IV estimates as identification bounds for impulse responses to component shocks within monetary policy announcements. Specifically, $z_{t}^{ff}$ gives $\Theta_{++}$ and $z_{t}^{sp}$ gives $\Theta_{+-}$\footnote{We use $-z_{t}^{sp}$ to apply Corollary (ref) since $\operatorname*{cov}(z_{t}^{sp},x_{t})<0$.} in Corollary (ref), with their intersection corresponding to cases (ii) and (iii) in Proposition (ref). Thus, for each $h$, the identified set of responses to the monetary policy shock is $\{\theta_{h,y1}:\min(\beta_{h}^{ff},\beta_{h}^{sp})<\theta_{h,y1}<\max(\beta_{h}^{ff},\beta_{h}^{sp})\}$, while for the central bank information shock it is $\{\theta_{h,y2}:\theta_{h,y2}>\beta_{h}^{ff}\}$ if $\beta_{h}^{ff} >\beta_{h}^{sp}$ and $\{\theta_{h,y2}:\theta_{h,y2}<\beta_{h}^{ff}\}$ if $\beta_{h}^{ff} <\beta_{h}^{sp}$.

The corresponding 68% pointwise confidence region for the identified set is shown in Figure (ref) as blue shaded areas. The results in the left column are highly informative. In response to a monetary policy shock, the interest rate response is less persistent, and the negative effects on the stock market and price level are stronger compared to the composite monetary policy announcement shock. This finding aligns with the Bayesian posterior of the responses presented in the first column of Figure 2 in JK2020.

The confidence region for responses to the central bank information shock (right column in Figure (ref)) is less informative because the sign restrictions only provide a one-sided identified set for $\theta_{h,y2}$. It is important to note that the lower and upper percentiles of the Bayesian posterior for responses to the information shock (second column of Figure 2 in JK2020) might be misinterpreted as a two-sided identified set. Since we use the same sign restrictions as JK2020, it is likely that their Bayesian posterior severely undercovers the one-sided identified set. The robust credible region proposed by Giacomini and Kitagawa (2021), which adopts a multiple-prior Bayesian approach, also assumes that identified sets are bounded to achieve correct frequentist coverage.

Imposing a Cross-IV Restriction

To obtain more informative identified sets for the responses to the information shock, we impose an additional cross-IV restriction. This is implemented by constructing a new IV that satisfies the desired sign restrictions using the existing ones. Alternatively, if an additional IV with the appropriate sign restrictions is available, the intersection of the identified sets can be directly computed.

Let $z_{t}^{*}$ denote this new IV and $\beta_{h}^{*}$ the LP-IV estimand. If $z_{t}^{*}$ satisfies

then these sign restrictions correspond to $\Theta_{-+}$. The intersection $\Theta_{++}^{ff}\cap\Theta_{+-}^{sp}\cap\Theta_{-+}^{*}$ yields a two-sided identified set for $\theta_{h,y2}$:

Note that the identified set for $\theta_{h,y1}$ remains unchanged.

We construct $z_{t}^{*}$ satisfying the sign restrictions in (ref) and the ordering in (ref) as a linear combination of existing IVs: $z_{t}^{*} = z_{t}^{ff} + \delta^{*}z_{t}^{sp}$ where $\delta^{*}\in\mathbb{R}$ satisfies

Since $\operatorname*{cov}(z_{t}^{ff},\varepsilon_{1,t})/(-\operatorname*{cov}(z_{t}^{sp},\varepsilon_{1,t}))<\operatorname*{cov}(z_{t}^{ff},x_t)/(-\operatorname*{cov}(z_{t}^{sp},x_{t}))$ under the sign restrictions, such a $\delta^*$ exists\footnote{The data support $\operatorname*{cov}(z_{t}^{ff},x_t)/(-\operatorname*{cov}(z_{t}^{sp},x_{t}))>0$.}. The third inequality in (ref) is not a restriction on unobservable moments. The second inequality in (ref) is equivalently written as

Setting a value of $\delta^{*}$ explicitly in (ref) amounts to imposing a restriction on the relative magnitude of the covariance between the instrument and the structural shock across the IVs. This type of restriction is also considered in Arias, Rubio-Ram\'{i}rez, and Waggoner (2021) and Giacomini, Kitagawa, Read (2022). In addition, note that

by construction. Since $\delta^{*}(-\operatorname*{cov}(z_{t}^{sp},x_{t})/\operatorname*{cov}(z_{t}^{*},x_{t}))>0$, $\beta_{h}^{*}>\beta_{h}^{ff}$ if $\beta_{h}^{ff}>\beta_{h}^{sp}$, and $\beta_{h}^{*}<\beta_{h}^{ff}$ if $\beta_{h}^{ff}<\beta_{h}^{sp}$, which aligns with the ordering in (ref).

If we set $\delta^{*}$ close to its upper bound, the constructed instrument $z_{t}^{*}$ becomes weaker in the sense that $\operatorname*{cov}(z_{t}^{*},x_{t})\approx0$. As shown in (ref), this implies that $\beta_{h}^{*}\rightarrow \infty$ if $\beta_{h}^{ff}>\beta_{h}^{sp}$, and $\beta_{h}^{*}\rightarrow -\infty$ if $\beta_{h}^{ff}<\beta_{h}^{sp}$. In this case, the identified set in (ref) becomes less informative, essentially reverting to the original one-sided set without using $z_{t}^{*}$. On the other hand, a smaller value of $\delta^{*}$ imposes a stronger restriction, and the identified set becomes tighter, since $|\beta_{h}^{*}-\beta_{h}^{ff}|=C\delta^{*}$ for some constant $C>0$.

Figure (ref) illustrates the confidence regions for the responses to the central bank information shock using three IVs: $z_{t}^{ff}$, $z_{t}^{sp}$, and $z_{t}^{*}$ with varying values of $\delta^{*}$. In the JK2020 dataset, we observe $\widehat{\operatorname*{cov}}(z_{t}^{\perp (ff)},x_{t}^{\perp (ff)})\approx 0.12$ and $\widehat{\operatorname*{cov}}(z_{t}^{\perp (sp)},x_{t}^{\perp (sp)})\approx -0.04$, implying an upper bound for $\delta^{*}$ of approximately three. Therefore, we experiment with $\delta^{*}=1,1.5,2$. Since the estimated variances of $z_{t}^{\perp(ff)}$ and $z_{t}^{\perp(sp)}$ are similar, (ref) approximately holds in terms of the correlation. Then $\delta^{*}=1$ restricts that the pure monetary policy shock is negatively correlated with the high-frequency surprise in the stock market at least as much as it is positively correlated with the high-frequency surprise in the fed fund futures. When $\delta^{*}=2$, the negative correlation can be as small as half of the positive correlation. Comparing these results with the right column of Figure (ref), we observe a significant tightening of the identified sets through the use of the additional IV $z_{t}^{*}$.

This section proposed a transparent approach to tightening the identified set and confidence intervals for impulse responses by imposing restrictions on the relative magnitudes of the covariances between the shocks and the instruments. Specifically, this method transforms a one-sided identified set into a two-sided one. In light of this, the standard Bayesian approach in JK2020--which produces bounded credible sets for all impulse responses--may have unknowingly imposed restrictions on both the relative magnitudes and the signs of these covariances.

In summary, our sign restrictions approach based on LP-IV offers a simple and honest alternative to existing Bayesian VAR methods. Moreover, when the LP-IV model (ref) is extended to include higher-order or non-linear control terms, our approach remains applicable.

Conclusion

This paper presents a formal analysis of the identification of the LP-IV approach in a more empirically relevant setting, along with a systematic method for using disaggregated data and incorporating sign restrictions within the LP-IV framework.

On the one hand, our findings caution that the structural interpretation of the impulse response obtained by LP-IV depends on the correlation between the instrument and the potential components of the structural shock, and problematic cases can arise in practical applications. On the other hand, we demonstrate that the problematic LP-IV estimand can be transformed into valuable information when multiple IVs are available, as it facilitates the generation of informative identified points or intervals for structural parameters.

Given the simplicity, flexibility, and widespread use of the LP-IV approach in empirical studies, our discovery that it can effectively handle multiple IVs, datasets with sectoral data, and sign restrictions will enable researchers to apply it in a diverse range of cases and interpret outcomes reasonably. This is likely to foster greater adoption of the LP-IV approach in various research contexts.