AN IDENTIFICATION AND TESTING STRATEGY FOR PROXY-SVARs WITH WEAK PROXIES Giovanni Angelinia Giuseppe Cavaliereab Luca Fanellia

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AN IDENTIFICATION AND TESTING STRATEGY FOR

PROXY-SVARs WITH WEAK PROXIES

Giovanni Angelinia, Giuseppe Cavalierea,b, Luca Fanellia

First draft: September 2021.First revision: September 2022.

Second revision: July 2023.This version: October 2023

Abstract

When proxies (external instruments) used to identify target struc-

tural shocks are weak, inference in proxy-SVARs (SVAR-IVs) is nonstan-

dard and the construction of asymptotically valid conﬁdence sets for the

impulse responses of interest requires weak-instrument robust methods.

In the presence of multiple target shocks, test inversion techniques re-

quire extra restrictions on the proxy-SVAR parameters other those im-

plied by the proxies that may be diﬃcult to interpret and test. We show

that frequentist asymptotic inference in these situations can be conducted

through Minimum Distance estimation and standard asymptotic meth-

ods if the proxy-SVAR can be identiﬁed by using ‘strong’ instruments for

the non-target shocks; i.e. the shocks which are not of primary interest

in the analysis. The suggested identiﬁcation strategy hinges on a novel

pre-test for the null of instrument relevance based on bootstrap resam-

pling which is not subject to pre-testing issues. Speciﬁcally, the valid-

ity of post-test asymptotic inferences remains unaﬀected by the test out-

comes due to an asymptotic independence result between the bootstrap

and non-bootstrap statistics. The test is robust to conditionally het-

eroskedastic and/or zero-censored proxies, is computationally straight-

forward and applicable regardless of the number of shocks being instru-

mented. Some illustrative examples show the empirical usefulness of the

suggested identiﬁcation and testing strategy.

Keywords: Proxy-SVAR, Bootstrap inference, external instruments,

identiﬁcation, oil supply shock.

JEL Classification: C32, C51, C52, E44

aDepartment of Economics, University of Bologna, Italy. bDepartment of Economics, Uni-

versity of Exeter Business School, UK. Correspondence to: Giuseppe Cavaliere, Depart-

ment of Economics, University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy; email:

giuseppe.cavaliere@unibo.it.

arXiv:2210.04523v4 [econ.EM] 19 Oct 2023

1 Introduction

Proxy-SVARs, or SVAR-IVs, popularized by Stock (2008), Stock and Wat-

son (2012, 2018) and Mertens and Ravn (2013), have become standard tools

to track the dynamic causal eﬀects produced by macroeconomic shocks on

variables of interest. In proxy-SVARs, the model is complemented with ‘ex-

ternal’ variables – which we call ‘proxies’, ‘instruments’ or ‘external variables’

interchangeably; such variables carry information on the structural shocks of

interest, the target shocks, and allow to disregard the structural shocks not

of primary interest in the analysis, the non-target shocks. Recent contribu-

tions on frequentist inference in proxy-SVARs include Montiel Olea, Stock

and Watson (2021) and Jentsch and Lunsford (2022); in the Bayesian frame-

work, Arias, Rubio-Ramirez and Waggoner (2021) and Giacomini, Kitagawa

and Read (2022) discuss inference in the case of set-identiﬁcation.

Inference in proxy-SVARs depends on whether the proxies are strongly

or weakly correlated with the target shocks. If the connection between the

proxies and the target shocks is ‘local-to-zero’, as in Staiger and Stock (1997)

and Stock and Yogo (2005), asymptotic inference is non-standard. In such

case, weak-proxy robust methods can be obtained by extending the logic of

Anderson-Rubin tests (Anderson and Rubin, 1949), see Montiel Olea et al.

(2021). Grid Moving Block Bootstrap Anderson-Rubin conﬁdence sets (‘grid

MBB AR’) for normalized impulse response functions [IRFs] (Br¨uggemann,

Jentsch and Trenkler, 2016; Jentsch and Lunsford, 2019) can also be applied

in the special case where one proxy identiﬁes one structural shock; see Jentsch

and Lunsford (2022).

When proxy-SVARs feature multiple target shocks, further inferential diﬃ-

culties arise. First, (point-)identiﬁcation requires additional restrictions, other

than those provided by the instruments; see Mertens and Ravn (2013), An-

gelini and Fanelli (2019), Arias et al. (2021), Montiel Olea et al. (2021) and

Giacomini et al. (2022). Second, in the frequentist setup the implementation

of weak-instrument robust inference as in Montiel Olea et al. (2021) may im-

ply a large number of additional restrictions on the parameters of the proxy-

SVAR relative to those needed under strong proxies. These extra restrictions

are not always credible, and may be diﬃcult to test; see Montiel Olea et al.

(2021, Section A.7) and Section S.9 of our supplement.1Fourth, the theory for

1From the perspective of Bayesian inference, one can in principle make the usual argument

that weak identiﬁcation issues do not matter. For instance, Caldara and Herbst (2019) discuss

how it is still possible to obtain numerical approximations of the exact ﬁnite-sample posterior

distributions of the parameters of proxy-SVARs when instruments are weak. Giacomini et

al. (2022) show that for set-identiﬁed proxy-SVARs with weak instruments, the Bernstein-

von Mises property fails for the estimation of the upper and lower bonds of the identiﬁed set.

the grid bootstrap Anderson-Rubin conﬁdence sets does not extend to cases

where multiple instruments identify multiple target shocks.

This paper is motivated by these inferential diﬃculties. In particular, we

design an identiﬁcation and (frequentist) estimation strategy intended to cir-

cumvent, when possible, the use of weak-instrument robust methods. The idea

we pursue is to identify the proxy-SVAR through an ‘indirect’ approach, where

a vector of proxies (say, wt), correlated with (all or some of) the non-target

shocks of the system and uncorrelated with the target shocks (say, zt), is used

to infer the IRFs of interest indirectly. We call this strategy ‘indirect identi-

ﬁcation strategy’ or ‘indirect-MD’ approach, as opposed to the conventional

‘direct’ approach based on instrumenting the target shock(s) directly with the

(potentially weak) proxies zt. As highlighted by our empirical illustrations, the

indirect approach can prove more useful to a practitioner than one might think.

The proxies wtcontribute to deﬁning a set of moment conditions upon

which we develop a novel Minimum Distance [MD] estimation approach (Newey

and McFadden, 1994). We derive novel necessary order conditions and neces-

sary and suﬃcient rank condition for the (local) identiﬁability of the proxy-

SVAR. If the proxies wtare strong for the non-target shocks and the model is

identiﬁed, asymptotically valid conﬁdence intervals for the IRFs of interest ob-

tain in the usual way; i.e., either by the delta-method or by bootstrap methods.

Interestingly, the idea of using instruments for the non-target shocks to iden-

tify and infer the eﬀects of structural shocks of interest was initially pursued

via Bayesian methods in Caldara and Kamps (2017), where two ﬁscal (target)

shocks are recovered by instrumenting the non-ﬁscal (non-target) shocks of the

system. We defer to Section 5 a detailed comparison of our method with Cal-

dara and Kamps (2017).

Key to the indirect identiﬁcation strategy is the availability of strong prox-

ies for the non-target shocks. In particular, it is essential that the investigator

can screen ‘strong’ from ‘weak’ instruments, and that such screening does not

aﬀect post-test inference. To do so, we further contribute by designing a novel

pre-test for strong against weak proxies based on bootstrap resampling.

Inspired by the idea originally developed in Angelini, Cavaliere and Fanelli

(2022) for state-space models, we show that the bootstrap can be used to infer

the strength of instruments, other than building valid conﬁdence intervals for

IRFs. In particular, we exploit the fact that under mild requirements, the

MBB estimator of the proxy-SVAR parameters is asymptotically Gaussian

when the instruments are strong while, under weak proxies `a la Staiger and

Stock (1997), the distribution of MBB estimator is random in the limit (in

the sense of Cavaliere and Georgiev, 2020) and, in particular, is non-Gaussian.

This allows to show that a test for the null of strong proxies can be designed

as a normality test based on an appropriate number of bootstrap repetitions;

such test is consistent against proxies which are weak in Staiger and Stock’s

(1997). An idea that echoes this approach in the Bayesian setting can be found

in Giacomini et al. (2022), who suggest using non-normality of the posterior

distribution of a suitable function of proxy-SVAR parameters to diagnose the

presence of weak proxies. This idea is not pursued further in their paper.

Our suggested test has several important features. First, it controls size un-

der general conditions on VAR disturbances and proxies, including the case of

conditional heteroskedasticity and/or zero-censored proxies. Second, with re-

spect to extant tests such as Montiel Olea and Pﬂueger’s (2013) eﬀective ﬁrst-

stage F-test for IV models with conditional heteroskedasticity,2our test can be

applied in the presence of multiple structural shocks; as far as we are aware, no

test of strength for proxy-SVARs with multiple target shocks has been formal-

ized in the literature. Third, it is computationally straightforward, as it boils

down to running multivariate/univariate normality tests on the MBB replica-

tions of bootstrap estimators of the proxy-SVAR parameters. Fourth, it can

be computed in the same way regardless of the number of shocks being instru-

mented. Fifth, and most importantly, the test does not aﬀect second-stage

inference, meaning that regardless of the outcome of the test, post-test infer-

ences are not aﬀected. This property marks an important diﬀerence relative

to the literature on weak instrument asymptotics, where the negative conse-

quences of pretesting the strength of proxies are well known and documented

(see, inter alia, Zivot, Startz and Nelson, 1998; Hausman, Stock and Yogo,

2005; Andrews, Stock and Sun, 2019; Montiel Olea et al., 2021).

The paper is organized as follows. In Section 2 we motivate our approach

with a simple illustrative example. In Section 3 we introduce the proxy-SVAR

and rationalize the suggested identiﬁcation strategy. The assumptions are

summarized in Section 4, while we present our indirect-MD approach in Sec-

tion 5. Section 6 deals with the novel approach to testing for strong prox-

ies. To illustrate the practical implementation and relevance of our approach,

we present in Section 7 two illustrative examples that reconsider models al-

ready estimated in the literature. Section 8 concludes. An accompanying sup-

plement complements the paper along several dimensions, including auxiliary

lemmas and their proofs, the proofs the propositions in the paper and an ad-

ditional empirical illustration based on a ﬁscal proxy-SVAR.

2See Montiel Olea et al. (2021) for an overview on ﬁrst-stage regressions in proxy-SVARs

or, alternatively, Lunsford (2016) for tests based on regressing the proxy on the reduced-form

residuals.

2 Motivating example: a market

(demand/supply) model

In this section we outline the main ideas in the paper by considering a ‘toy’

proxy-SVAR, where we omit the dynamics without loss of generality. We con-

sider a model that comprises a demand and supply function for a good with

associated structural shocks, given by the equations

qt

pt

| {z }

=β1,1β1,2

β2,1β2,2

| {z }

εd,t

εs,t 

| {z }

εt

≡β1,1εd,t +β1,2εs,t

β2,1εd,t +β2,2εs,t (1)

where qtand ptare quantity and price at time t, respectively. The nonsingular

matrix Bcaptures the instantaneous impact, on Yt:= (qt, pt)′, of the structural

shocks εd,t, εs,t, which are assumed to have unit variance and to be uncorre-

lated. We temporary (and conventionally) label εd,t as the ‘demand shock’ and

εs,t as the ‘supply shock’, and assume that the objective of the analysis is the

identiﬁcation and estimation of the instantaneous impact of the demand shock

on Ytthrough the ‘external variables’ approach. Hence, εd,t is the target shock,

εs,t is the non-target shock, and the parameters of interest are the on-impact

responses ∂Yt

∂εd,t =B•1:= (β1,1, β2,1)′; here B•1denotes the ﬁrst column of B.

Since the two equations in (1) are essentially identical for arbitrary param-

eter values, nothing distinguishes a demand shock from a supply shock in the

absence of further information/restrictions. The typical ‘direct approach’ to

this partial identiﬁcation problem is to consider an instrument ztcorrelated

with the demand shock, E(ztεd,t) = ϕ̸= 0 (relevance condition), and uncorre-

lated with the supply shock, E(ztεs,t) = 0 (exogeneity condition). Now, con-

sider the case where the investigator strongly suspects that ztis a weak proxy

(meaning that ϕcan be ‘small’), but they also know that there exists an exter-

nal variable wt, correlated with the non-target supply shock and uncorrelated

with the demand shock; formally, E(wtεs,t) = λ̸= 0 and E(wtεd,t) = 0. Then,

the proxy wtcan be used to recover the parameters of interest (i.e., B•1) ‘in-

directly’; i.e., by instrumenting the non-target supply shock εs,t, rather than

the target demand shock εd,t. To show how, let A:= B−1and consider the

alternative representation of (1):

α1,1α1,2

α2,1α2,2

| {z }

qt

pt

| {z }

=A1•Yt

A2•Yt=εd,t

εs,t 

| {z }

εt

where A1•:= (α1,1, α1,2) and A2•denote the ﬁrst row and the second row of

A, respectively. Since wtis correlated with ptbut uncorrelated with εd,t, it is

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ANIDENTIFICATIONANDTESTINGSTRATEGYFORPROXY-SVARsWITHWEAKPROXIESGiovanniAngelinia,GiuseppeCavalierea,b,LucaFanelliaFirstdraft:September2021.Firstrevision:September2022.Secondrevision:July2023.Thisversion:October2023AbstractWhenproxies(externalinstruments)usedtoidentifytargetstruc-turalshocksareweak,i...

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AN IDENTIFICATION AND TESTING STRATEGY FOR PROXY-SVARs WITH WEAK PROXIES Giovanni Angelinia Giuseppe Cavaliereab Luca Fanellia

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