Monetary Policy, Self-Fulfilling Expectations and the U.S. Business Cycle

May 05, 2020 at 04:44 pm

Finance and Economics Discussion Series

Divisions of Research & Statistics and Monetary Aairs

Federal Reserve Board, Washington, D.C.

Monetary Policy, Self-Fullling Expectations and the U.S.

Business Cycle

Giovanni Nicol

2020-035

Please cite this paper as:

Nicolo, Giovanni (2020). Monetary Policy, Self-Fullling Expectations and the U.S. Business Cycle," Finance and Economics Discussion Series 2020-035. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.035.

NOTE: Sta working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research sta or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Monetary Policy, Self-Fulﬁlling Expectations and

the U.S. Business Cycle

Giovanni Nicolò

April 28, 2020

Abstract

I estimate a medium-scaleNew-Keynesian model and relax the conventional assumption that the central bank adopted an active monetary policy by pursuing inﬂation and output stability over the entire post-war period. Even after accounting for a rich struc- ture, I ﬁnd that monetary policy was passive prior to the Volcker disinﬂation. However, sunspot shocks did not represent quantitatively relevant sources of volatility. By con- trast, such passive interest rate policy accommodated fundamental productivity and cost shocks that de-anchored inﬂation expectations, propagated via self-fulﬁlling inﬂation expectations and constituted the primary sources of the run-up in inﬂation from the 1960s through the late 1970s.

JEL: C11, C52, C54, E31, E32, E52.

Keywords: Monetary policy, business cycle, expectations, indeterminacy, Bayesian methods.

Federal Reserve Board, 20th Street and Constitution Avenue N.W., Washington, DC 20551. Email: giovanni.nicolo@frb.gov. The views expressed in this paper are those of the author and do not necessarily reﬂect the views of the Federal Reserve Board or the Federal Reserve System. I am grateful to Roger E.A. Farmer, Francesco Bianchi, Aaron Tornell and Vincenzo Quadrini for invaluable support and advice. I also thank Andrea Ajello, Steven Ambler, Andy Atkeson, Alessandro Barattieri, Martin Bodenstein, Matteo Crosignani, Pablo A. Cuba-Borda, Pablo Fajgelbaum, Leland Farmer, Francesco Ferrante, Etienne Gagnon, Ana Galvao, Francois Geerolf, Michela Giorcelli, Christopher J. Gust, Jinyong Hahn, Edward P. Herbst, Matteo Iacoviello, Cosmin Ilut, Benjamin K. Johannsen, Mariano Kulish, Robert J. Kurtzman, Eric Leeper, Edward Nelson, Lee E. Ohanian, Anna Orlik, Alain Paquet, Matthias O. Paustian, Giovanni Pellegrino, Andrea Prestipino, Andrea Raﬀo, John Rogers, Robert J. Tetlow, Pierre-Olivier Weill, Fabian Winkler, seminar participants at UCLA, the Monetary Aﬀairs Workshop of the Federal Reserve Board, UC Irvine, the University of Québec at Montréal, the University of Virginia, Banque de France, and conference participants at the Asian Meeting of the Econometric Society, the Applications of Behavioral Economics and Multiple Equilibrium Models to Macroeconomic Policy Conference, the European Winter Meeting of the Econometric Society, the Midwest Econometrics Group conference, the Conference on Advances in Applied Macro-Finance, the Theories and Methods in Macroeconomics Conference and the XXI Annual Inﬂation Targeting Conference for valuable comments.

Introduction

Most previous studies that identify the sources of U.S. business cycles using medium-scaleNew-Keynesian (NK) models rely on a conventional assumption about the conduct of monetary policy. The central bank is assumed to implement an active1monetary policy that systematically stabilizes inﬂation and output during the entire post-war period.2This assumption cannot be easily reconciled with the behavior of the data prior to the Volcker disinﬂation. From the late 1950s through the 1970s, the U.S. economy experienced high volatility, and inﬂation was unstable and rising. Moreover, assuming that the central bank operated under an active monetary policy substantially impacts the ﬁndings about the drivers of U.S. business cycles.

I estimate the standard medium-scale NK model by Smets and Wouters (2007) (henceforth SW), in which I relax the key assumption that the central bank pursued an active monetary policy both before and after 1979. If monetary policy is passive, the model is indeterminate and characterized by multiple equilibrium paths. Two features of the model become relevant to explain the persistence and volatility of the data. First, unexpected changes in expectations constitute non-fundamental 'sunspot' disturbances that induce an additional source of volatility. Second, the propagation of structural shocks depends on self-fulﬁlling expectations that alter the transmission and generate additional persistence.

To the best of my knowledge, this paper is the ﬁrst study that investigates the quantitative relevance of non-fundamental disturbances and the propagation of shocks based on self- fulﬁlling expectations for the U.S. macroeconomic instability prior to 1979. As previous studies generally adopted small-scale NK models (Clarida et al., 2000; Lubik and Schorfheide, 2004; Boivin and Giannoni, 2006; Bhattarai et al., 2016), the novelty of this paper is to address such question in the context of a medium-scale model and assess the consequences of allowing for a passive monetary policy on the identiﬁcation of the sources of U.S. business

1The terminology of active and passive policies originated with Leeper (1991).

Smets and Wouters (2003; 2007), Christiano et al. (2005), Del Negro and Eusepi (2011) and Arias et al. (2019) represent some of the key references.

cycles.

I ﬁnd ﬁve main results. First, the stance of U.S. monetary policy changed in the post-war period. Monetary policy was passive between 1955 and 1979, while it pursued active inﬂation targeting since 1984. Compared to previous studies that use medium-scale models, this result rejects the imposed assumption that monetary policy was active before 1979.

Second, the evidence of a passive monetary policy from 1955 to 1979 substantially aﬀects the identiﬁcation of the sources of U.S. business cycles. According to the estimated model, fundamental productivity and cost shocks were the main drivers of the run-up in the inﬂation rate from the early 1960s through the late 1970s. Under a passive monetary regime, positive technology shocks in the 1960s de-anchored inﬂation expectations and generated persistent inflationarypressures via self-fulﬁlling expectations.3Mark-up shocks only account for the sudden inﬂationary episodes related to the oil crises during the 1970s.4On the contrary, previous studies that impose an active monetary policy before 1979 exclude the role of self- fulﬁlling expectations for the transmission of structural shocks. The persistent rise in inﬂation from the early 1960s through the 1970s would be entirely and erroneously attributed to mark- up shocks.

Third, fundamental shocks were the sources of the high volatility of inﬂation and output before 1979. In a passive monetary policy regime, sunspot shocks potentially lead to additional macroeconomic instability. By contrast, my estimation of the SW model shows that non-fundamental disturbances were not signiﬁcant drivers of volatility between 1955 and 1979.

Fourth, I use the SW model to revisit the question on the sources of the reduction in U.S. macroeconomic volatility starting from the 1980s to 2007. I investigate whether the observed decrease in volatility is explained by a change in monetary policy to a more active stance since the early 1980s, as opposed to smaller structural shocks. Based on the SW model, I ﬁnd that

Fernald (2014b) and Gordon (2000) among others document that the U.S. economy experienced a period
of exceptional growth in productivity since World War II until the early 1970s.
4As the SW model does not explicitly include oil markets, unexpected ﬂuctuations in oil prices are captured by mark-up shocks.

the reduction in macroeconomic uncertainty was a combination of both the implementation of an active monetary policy anda lower volatility of the shocks, while the latter plays a relatively more important role.

Finally, I document that the propagation mechanism of the shocks depends on the conduct of monetary policy also when adopting a reduced-form approach. By estimating a medium- scale Bayesian vector autoregression (BVAR) model using the hierarchical approach, a laGiannone, Lenza, and Primiceri (2015), I show that the transmission of productivity and monetary policy shocks is substantially altered in the context of alternative monetary policy regimes. The reduced-form diﬀerences in the shock propagation are analogous to those resulting from the estimation of the structural SW model.

To solve the medium-scale model of SW with a passive monetary policy, I use the methodology developed in Bianchi and Nicolò (2019), which simpliﬁes technical complexities that hamper the implementation of existing solution methods to medium-scale models (Lubik and Schorfheide, 2003; Farmer et al., 2015).

The adoption of a medium-scale model provides two advantages. First, the richer structure constitutes a suitable framework to identify and quantify the sources of business cycles in the U.S. post-war period. Second, a passive monetary policy generates an additional source of persistence via self-fulﬁlling expectations and an additional source of volatility via non- fundamental disturbances. In line with the insights from Beyer and Farmer (2007a), the persistence and volatility of the U.S. macroeconomy could mistakenly favor the evidence of a passive monetary policy in the context of a small-scale model because of its parsimonious dynamicand stochasticstructure. Empirically, I show that the estimation of a small-scale model that does not account for the richer structure of the SW model overturns the evidence of an active monetary policy over the period between 1984 and 2007. The more parsimonious structure mistakenly interprets the persistence and volatility of the data as evidence of a passive monetary policy.

The rest of the paper is organized as follows. Section 2 highlights the contributions of the paper to the related literature. Section 3 motivates the adoption of medium-scale models

to properly assess the role of U.S. monetary policy to explain business cycles. Section 4 describes the main features of the SW model and the data used to conduct the estimation using Bayesian techniques. Section 5 explains the methodology developed in Bianchi and Nicolò (2019) and its implementation to solve and estimate the SW model, allowing for indeterminacy. Section 6 presents the ﬁndings. Section 7 concludes.

Related Literature

A vast literature rationalizes the behavior of the data in the post-war period with changes in interest-rate policies (Clarida et al., 2000; Lubik and Schorfheide, 2004; Boivin and Giannoni, 2006; and Bhattarai et al., 2016). These papers show that U.S. monetary policy failed to implement active inﬂation targeting before 1979. However, they use univariate or small-scale linear rational expectations (LRE) models for which the identiﬁcation and quantiﬁcation of the relevant drivers of business cycles is limited. On the other hand, another strand of the literature argues that changes in business cycle ﬂuctuations stem from the diﬀerent properties of the exogenous shocks that hit the U.S. economy (Primiceri, 2005; Sims and Zha, 2006; Canova and Gambetti, 2009). In particular, Primiceri (2005) ﬁnds evidence of time variation in U.S. monetary policy, but little causal link between changes in the systematic component of monetary policy and the high inﬂation and unemployment rates over the post- war period. Nevertheless, the conclusion of the paper also warns that "In order to explore the true behavioral sources of these sequences of particularly bad shocks, a larger and fully structural model is needed."5

The contribution of this paper is to provide an interpretation of U.S. business cycle ﬂuc- tuations in the United States based on the conduct of monetary policy and the role of self-fulﬁlling expectations. The novelty is to quantify the implications of passive monetary policy for U.S. business cycle ﬂuctuations, especially in the period prior to 1979. Under such regime, structural shocks are accommodated by the central bank and induce self-fulﬁlling

See Primiceri (2005), pag. 844.

expectations that alter the propagation mechanism and identify diﬀerent determinants of business cycles. The upward trend in the inﬂation rate observed in the 1960s is attributed to persistent technology shocks that generated strong economic activity and self-fulﬁlling inﬂationary expectations. Together with an economic expansion, a productivity shock generates inﬂationary expectations that passive monetary policy fails to keep anchored. These inﬂationary pressures more than compensate the drop in marginal cost, and this mechanism thus results into a self-fulﬁlling inﬂationary eﬀect. Moreover, I show that sunspot shocks play no quantitative role in explaining the volatility observed before 1979.6In section 6.5, I also provide additional empirical evidence for the diﬀerence in propagation depending on the conduct of monetary policy by estimating a BVAR with the hierarchical approach, a laGiannone, Lenza, and Primiceri (2015).

The adoption of a medium-scale LRE model raises two technical complexities. First, the partition of the parameter space into a determinate and indeterminate region is unknown for richer models. Second, existing methods require to construct the indeterminate solution following the seminal contribution of Lubik and Schorfheide (2003) or to rewrite the model based on the existing degree of indeterminacy (Farmer et al., 2015). Given the technical complexities, previous studies that adopt medium-scale models mainly abstract from the possibility of observing policies that lead to indeterminate outcomes and restrict a priorithe parameter space to the unique, determinate region (Smets and Wouters, 2003; Christiano et al., 2005; Smets and Wouters, 2007; Justiniano et al., 2010; Justiniano et al., 2011; Del Negro and Eusepi, 2011; Arias et al., 2019).7

Alternative explanations for the run-up of U.S. inﬂation since the early 1960s relate to the possibility that policymakers overestimated potential output (Orphanides, 2002) or underestimated both natural rate of unemployment and the persistence of inﬂation in the Phillips curve (Primiceri, 2005). In this paper, I focus on understanding the mechanisms through which the de-anchoring of inﬂation expectations due to structural
shocks could have played a relevant role to explain the macroeconomic instability in the period prior to 1979. 7A notable exception is the work of Justiniano and Primiceri (2008) who consider changes in the role of
monetary policy in a medium-scale model as a possible explanation for the Great Moderation. To solve the

model under indeterminacy, they adopt the method of Lubik and Schorfheide (2004) and set the priors for the

additional parameters Maround the "continuity solution" that guarantees that the impact of fundamental shocks on endogenous variables is continuous at the boundary of the indeterminacy region. However, such identiﬁcation assumption could be restrictive for two reasons. First, as shown in section 6.2, the data strongly favor diﬀerences in the transmission of fundamental disturbances between the alternative monetary policy regimes. Second, section 6.1 shows that the correlation of the sunspot shocks with the mark-up shock is crucial in capturing the inﬂation dynamics in the pre-Volcker period. However, in Justiniano and Primiceri

In this paper, I implement the methodology we developed in Bianchi and Nicolò (2019) to relax this assumption and estimate the medium-scale model by SW over the entire parameter space. I ﬁnd that the assumption imposed in SW is rejected for the period before 1979, during which the central bank adopted a passive monetary policy. Importantly, I verify that the assumption of a unique, determinate equilibrium has quantitative implications for the identiﬁcation of the main drivers of U.S. business cycles. Moreover, I show both theoretically and empirically that the study of the conduct of monetary policy requires the adoption of a rich dynamic and stochastic structure (Beyer and Farmer, 2007a).8

The paper also contributes to the literature that studies the sources of the reduction in U.S. macroeconomic volatility from the early 1980s to 2007. I investigate the validity of two prominent theories that have been advocated to explain this empirical phenomenon. First, several studies show that the behavior of the data changed because of a decrease in the variance of the shocks driving the economy in the period subsequent to the Volcker disinﬂation (Primiceri, 2005; Sims and Zha, 2006; Justiniano and Primiceri, 2008; Justiniano et al., 2010; Fernandez-Villaverde et al., 2010; Justiniano et al., 2011). This strand of the literature ﬁnds that the reduction in volatility is not related to monetary policy and it can therefore be considered as "good luck". Second, the work of Clarida et al. (2000) and Lubik and Schorfheide (2004) among others indicates that monetary policy acted more systematically since the 1980s, therefore suggesting a view related to "good policy". In this paper, I ﬁnd that the data supports both theories. Both a change in the conduct of monetary policy to a more active stance and a signiﬁcant drop in the volatility of structural shocks account for the decrease in U.S. macroeconomic uncertainty.

Finally, the paper contributes to the literature that studies the empirical implications of dy-

(2008), the mean of the posterior distribution for the corresponding parameter, M, lies on the tail of the prior distribution, potentially limiting the scope of properly ﬁtting the inﬂation dynamics. Compared to Justiniano and Primiceri (2008), this paper abstracts from stochastic volatility, but by adopting the method of Bianchi and Nicolò (2019), it guarantees a ﬂexible approach to understand the relationship between the conduct of monetary policy, the transmission of the shocks and the identiﬁcation of the sources of business cycle ﬂuctuations.

8A closely related literature also discusses the concerns due to model misspeciﬁcation for the empirical performance of Dynamic Stochastic General Equilibrium models and provides policy analysis approaches to deal with it (Del Negro et al., 2007, Del Negro and Schorfheide, 2009).

namic indeterminacy.9The seminal contributions of Farmer and Guo (1994) and Farmer and Guo (1995) focus on the relevance of sunspot shocks to explain business cycle ﬂuctuations. More recently, Lubik and Schorfheide (2004) empirically evaluate the possibility that monetary policy could lead to indeterminate outcomes, and Bhattarai et al. (2016) enrich this analysis by accounting for a nontrivial interaction between monetary and ﬁscal policy. This paper considers the richer dynamic and stochastic structure of the SW model to empirically study the implications of dynamic indeterminacy for U.S. business cycles in the post-war period.

While the current paper represents a ﬁrst step in the quantitative assessment of the implications of a passive monetary policy for macroeconomic outcomes and the identiﬁcation of the sources of business cycles, relevant questions are still open and deserve future research. In line with the work of Kiley (2007), Ascari and Ropele (2009), Coibon and Gorodnichenko (2011), Hirose et al. (2017), and Haque et al. (2019) among others, studies on the conduct of monetary policy would need to account for the positive trend inﬂation observed in the U.S. macroeconomic history. While the ﬁndings of these papers are based on parsimonious models, Arias et al. (2019) transparently describe the diﬃculties of including such analysis in the context of a medium-scale model, and possible extensions of the current work could help overcome such obstacles. Moreover, further research could address whether the adoption of a rich stochastic and dynamic structural model is also consistent with the presence of temporarily unstable paths to explain the Great Inﬂation, as suggested by Ascari et al. (2019) in the context of a stylized three-equation NK model. Finally, Del Negro and Eusepi

(2011) point to the role of inﬂation expectations in the data as relevant information to discipline the understanding of the relationship between alternative interest-rate policy and the formation of expectations for macroeconomic outcomes. The current work could potentially be generalized to also address such questions.

Further extensions of this work would also explore the possibility of accounting not only for dynamic indeterminacy, but also for static indeterminacy (i.e., multiplicity of steady

In this paper, I will refer to indeterminacy only as the dynamic properties of the model in the neighbor- hood of the unique steady state of a model.

states). Based on the cointegrating properties of the data (Beyer and Farmer, 2007b), Farmer and Platonov (2019) develop a micro-founded model that accounts for the possibility of observing multiple steady-state unemployment rates. The three-equations version of the model corresponds to the standard three-equation NK model in which the NK Phillips curve is replaced by a "belief function" that describes how agents form expectations about future nominal income growth. In Farmer and Nicolò (2018; 2019), we show that the reduced- form representation corresponds to a cointegrated vector error correction model (VECM) and the model outperforms the conventional three-equation NK model in ﬁtting the data before and after 1979. An interesting avenue of research extends the proposed alternative framework to a medium-scale model that also displays multiplicity of steady states and therefore maps into a VECM in reduced-form. The purpose would therefore be to study whether the cointegrating properties of the proposed model would better explain the data in the post-war period relative to a baseline NK model that displays self-stabilizing properties around the unique steady state.

Reasons for the Adoption ofMedium-Scale Models

3.1 Identiﬁcation Problem

Previous studies that allow for indeterminacy in U.S. monetary policy mainly adopt small- scale NK models and rationalize the empirical properties of the data before 1979 with a passive monetary policy (Clarida et al., 2000, Lubik and Schorfheide, 2004). If monetary policy is 'passive', two features of the model become relevant to explain the persistence in inﬂation dynamics and the high volatility of U.S. macroeconomic data over this period. First, the propagation of structural shocks depends on self-fulﬁlling expectations that alter the transmission mechanism and generate an additional source of persistence. Second, unexpected changes in expectations constitute non-fundamental 'sunspot' disturbances that generate an additional source of volatility.

I argue that the ﬁndings in earlier studies are potentially susceptible to the choice of parsi-

monious models (Beyer and Farmer, 2007a). Small-scale models impose restrictions on the structure of the underlying economy. By excluding richer models, the restrictions favor the result of a passive monetary policy as missing propagation mechanisms are misinterpreted as evidence of indeterminate dynamics. The identiﬁcation problem relates to the possibility that a model with a richer dynamic and stochastic structure could explain the macroeconomic volatility and inﬂation persistence before 1979, even when monetary policy is active. Adopting the medium-scale NK model of SW allows me to verify whether previous ﬁndings carry over to a richer structure.10

In the spirit of Beyer and Farmer (2007a), the following analytic example provides an intuition of the identiﬁcation problem that an econometrician faces when testing for indeterminacy. Suppose that a researcher studies the dynamics of the inﬂation rate using two alternative univariate LRE models. One model explains current inﬂation only as a function of expected inﬂation, t= aEt(t+1). As the endogenous variable is expectational, the model is well speciﬁed when the associated one-step ahead forecast error is also deﬁned ttEt 1(t). Considering the case of jaj > 1, the model is indeterminate, and the solution corresponds to any process for inﬂation and its expectation that takes the following form

t
<
Et (t+1)
:

t 1+ t;

(1)

2t 1+ t;

where a 1< 1. According to this model, the dynamics are explained by lagged inﬂation rate, and the non-fundamental shock, t.

The alternative univariate LRE model considered by the econometrician is richer in both its dynamic and stochastic structure. The model describes current inﬂation as a function not

10Lubik and Schorfheide (2004) acknowledge that their results are sensitive to model misspeciﬁcation as missing propagation mechanisms would favor the result of model indeterminacy. Their robustness check consists in comparing the ﬁt of a small-scale NK model for the pre-Volcker period with a richer model to account for missing propagation mechanisms. However, the comparison is between two structurally diﬀerent models, and the robustness check could therefore be sensitive to the choice of which propagation mechanism are included in the richer model. In this paper, I am instead considering the SW model for both the deter- minate and the indeterminate regions, while aiming at reducing the identiﬁcation problem that is inherent to the question by considering a medium-scale model.

only of expected inﬂation but also of lagged inﬂation and a fundamental shock, "t, that is

t= aEt(t+1)+bt 1+"t. Given the deﬁnition of the forecast error, ttEt 1(t), the

dynamics of the model depend on the two roots of the model denoted by and .11When one root is unstable, the model has a unique, determinatesolution. By assuming without loss of generality that jj > 1and jj < 1, the determinate solution is

8	t=	t 1 +	(+)	"t;
	(2)
>	Et(t+1) =	2t 1	+	(+)	"t:

<
>
:

The identiﬁcation problem arises because of the observational equivalence of the two alternative models. Without further information about the true variance of the shocks tand "t, the indeterminatemodel in (1) and the determinatemodel in (2) are characterized by the same likelihood function.

However, the choice of a parsimonious structure aﬀects the inference of the econometrician by erroneously favoring the indeterminate model in (1). Suppose that the true data generating process for the inﬂation rate is the richer, determinate model, t= aEt(t+1) + bt 1+ "t. Also, suppose the researcher chooses the parsimonious dynamic structure, t= aEt(t+1), where lagged inﬂation and the exogenous shock are omitted. The inference would therefore mistakenly lead the econometrician to conclude that the data is consistent with the dynamics of the indeterminate model in (1) because of the observational equivalence.

I provide empirical relevance of this concern in section 6.1. By estimating a small-scale model that does not account for the richer dynamic and stochastic structure of the SW model, the evidence of an active monetary policy over the period between 1984:1 and 2007:3 is overturned.

11It can be shown that the roots of the model are related to the structural parameters of the model as follows: a = 1=( + ) and b = =( + ).

3.2 Digging into the Mechanisms

A second advantage of adopting a medium-scale model such as SW is to provide a suitable framework to quantitatively assess the implications that a passive monetary policy has on the macroeconomy. In this section, I use a simple classical monetary model to show that, if monetary policy is passive, the dynamic and stochastic properties of the model diﬀer in two dimensions. First, the propagation of fundamental shocks through the economy diﬀers due to the formation of self-fulﬁlling expectations in response to the shocks. Second, the model is subject to non-fundamental sunspot disturbances. While small-scale models are not suﬃciently detailed, medium-scale models provide a suitable framework to quantitatively assess the relative importance in the data.

To provide an intuition, I consider a classical monetary model described by the Fisher equa-

tion, Rt= rt+ Et(t+1), and a simple Taylor rule, Rt=t, where Rtand tdenote

the deviations of the nominal interest rate and the inﬂation rate from their target level.12I assume that the real interest rate rtis given and follows a mean-zero Gaussian i.i.d. dis- tribution.13To properly specify the model, I also deﬁne the one-step ahead forecast error associated with the expectational variable, t, as ttEt 1(t). Combining the Fisher equation and the Taylor rule, I obtain the univariate model Et(t+1) =trt.

In this simple model, the monetary authority is active if it responds to changes in the inﬂation rate by more than one for one. By recalling the Taylor rule, this condition can be equivalently expressed as jj > 1. The solution in this region of the parameter space is determinate, and described by the following system

8	t= 1rt;	(3)
>

Et(t+1) = 0:
:

For expositional purposes, I abstract from the possibility that the nominal interest rate responds to changes in output.
In the classical monetary model, the real interest rate results from the equilibrium in labor and goods market and it depends on the technology shocks. I am considering an exogenous process for the technology shocks and therefore I take the process for the real interest rate as given.

The strong response of the monetary authority ensures that inﬂation is pinned down as a function of the exogenous real interest rt, and the expectations that agents hold about the future inﬂation rate are constant at its steady state, Et(t+1) = 0. Conversely, a passive monetary policy, jj 1, signiﬁcantly aﬀects the dynamic and stochastic properties of the model. The indeterminate solution corresponds to the following system of equations14

8	t	=		t 1 +t rt 1;	(4)
>	Et(t+1) =	2	t 1+ t(rt+ rt 1) :
<
>
:

The comparison of the solutions in (3) and (4) shows that a change in monetary policy substantially aﬀects the properties of the model and the interpretation of business cycle ﬂuc- tuations in at least two dimensions. First, the impact and transmission of the samestructural shock, rt, on the dynamics of the model diﬀers between the two speciﬁcations. While under determinacy, the inﬂation rate also follows an i.i.d. process, under indeterminacy, the shock de-anchors agents' expectations from the central bank's long-run target and transmits via self-fulﬁlling inﬂation expectations.15This is clearly not the case for the determinate solution where expectations are constant at the long-run inﬂation rate and play no role for the dynamics of the model.

Second, if the monetary authority is passive, the economy is subject to an additional, non- fundamental disturbance related to unexpected changes in agents' expectations, t. The sunspot shock therefore provides an additional source of uncertainty which could potentially help the model in matching the high volatility of the data in the period prior to the appointment of Paul Volcker as the Chairman of the Federal Reserve System.

14The solution is obtained by combining the deﬁnition of the forecast error, t, with the univariate model

such thatt= Et 1(t)+t= t 1+t	rt 1. Therefore, expectations about future inﬂation are therefore
described byEt(t+1) = trt= 2t	1 +t (rt +rt 1).

15The structural shock to the real interest rate does not aﬀect the inﬂation rate whenever it is assumed to be uncorrelated with thenon-fundamentalshock,t.

The Model and Data

Dynamic stochastic general equilibrium (DSGE) models are useful tools to conduct quantitative policy analysis. To this purpose, a branch of the literature focused on developing richer models that could provide a better match with the data. Based on the standard three- equation NK model, the work by Smets and Wouters (2003) and Christiano et al. (2005) expands the framework to account for relevant frictions and shocks. The model presented in Smets and Wouters (2007) now constitutes the heart of the structural DSGE models that are adopted by most central banks in advanced economies. While the reader is referred to the original paper for the details about the derivation of the model, this section describes its relevant features as well as the measurement equations and the data used to estimate the model using Bayesian techniques.

The model contains both real and nominal frictions. On the real side, households are assumed to form habit in consumption. By renting capital services to ﬁrms, households also face an adjustment cost and optimally choose the capital utilization rate with an increasing cost. Firms incur a ﬁxed cost in production and are subject to nominal price rigidities, à la Calvo, while indexing the optimized price to past inﬂation. The model also displays nominal wage frictions that allow for indexation to past wage inﬂation.

The economy follows a deterministic, balanced growth path, along which seven shocks drive the dynamics of the model. Three shocks aﬀect the demand-side of the economy. A risk premium shock aﬀects the household's intertemporal Euler equation by impacting the spread between the risk-free rate and the return on the risky asset. The investment-speciﬁc shock has an eﬀect on the investment Euler equation that the household considers when choosing the amount of capital to accumulate. The third demand-side shock is an exogenous spending shock that impacts the aggregate resource constraint. Similarly, the supply-side of the economy is subject to three shocks: a productivity shock as well as a price and wage mark-up shock. Finally, the monetary authority follows a Taylor rule as described in the following

equation,

Rt= Rt1+ (1 ) frt+ ry(ytytp)g + ry(ytytp) yt1ytp

1+"tR:

The monetary authority chooses the nominal interest rate, Rt, by allowing for some degree of interest rate inertia as measured by the parameter . Changes in the inﬂation rate, t, and the output gap, deﬁned as the deviations of actual output from its fully ﬂexible price and wage counterpart, also generate a response by the monetary authority. The Taylor rule also accounts for changes in the output gap, while any unexpected deviation in the policy instrument is deﬁned as a monetary policy shock, "Rt.16

To estimate the model, I use Bayesian techniques and the measurement equations that relate the macroeconomic data to the endogenous variables of the model are deﬁned below

2	dlGDPt	3		2		3		2 yt	yt	1	3
6	dlCONSt	7		6		7		6 ct	ct	1	7
6	dlINV	t	7		6		7		6 i	t	i	1	7
6		7		6		7		6	t	7
6			7		6		7		6				7
6	dlW AGt	7	=	6		7	+	6		wt	1	7	;
6	7		6	7		6wt	7
6			7		6		7		6				7
6	lHourst	7		6	7		6		lt		7
6	7		6 l	7		6			7
6			7		6		7		6				7
6			7		6		7		6				7
6	dlPt		7		6		7		6		t		7
6			7		6		7		6				7
6			7		6		7		6				7
6F EDF UNDSt	7		6R7		6		Rt		7
6			7		6		7		6				7
4			5		4		5		4				5

where dl denotes the percentage change measured as log diﬀerence and l denotes the log. The observables are the seven quarterly U.S. macroeconomic time series used in SW, and they match the number of shocks that aﬀect the economy. The series considered are: the growth rate in real GDP, consumption, investment and wages, log hours worked, inﬂation rate measured by the GDP deﬂator, and the federal funds rate.17

The deterministic balanced growth path is deﬁned in terms of four parameters: , the quar-

terly trend growth rate common to real GDP, consumption, investment and wages; l, the

16The model also assumes that the monetary policy shock follows an autoregressive process deﬁned by

"tR= R"tR1+ uR, where uR iid	0; 2	. The same assumption also holds for the other structural shocks
of the model.	t	tN	R

17Price controls set in the early 1970s may have impacted the dynamics of the inﬂation rate. However, the estimation of the model successfully captures the oil price shocks that characterize this period in the U.S.

steady-state hours worked (normalized to zero); , the quarterly steady-state inﬂation rate;

R, the steady-state nominal interest rate. Hence, the measurement equations in (4) relate the macroeconomic time series with the corresponding endogenous variables of the model fyt; ct; it; wt; lt; t; Rtg, while accounting for a balanced growth path.

While the full sample of SW ends in the fourth quarter of 2004, I updated the time series and estimated the model over three subsamples. The ﬁrst period starts in 1955:4, which corresponds to one year after the end of the Korean War, and it ends in 1969:4, the date in which the chairmanship of William Martin ends.18The second sample considers the chairmanships of both Arthur Burns and William Miller, and it spans from 1970:1 until 1979:2. As I show empirically in section 6.1, it is relevant to distinguish the ﬁrst two subsamples because, in line with the evidence documented by Fernald (2014b) among others, the second period is characterized by slower productivity growth, resulting into a distinct balanced growth path. Finally, the third period begins in 1984:1, date in which Kim and Nelson (1999) identify a structural break in the U.S. business cycle. The last observation coincides with the onset of the Great Recession in 2007:3.

Methodology

The adoption of medium-scale DSGE models to study the conduct of monetary policy raises technical complexities. First, while the partition of the parameter space into a determinate and indeterminate region can be derived analytically for small-scale models, it is generally unknown for larger models. Second, the model could be characterized by regions of the parameters space associated with multiple degrees of indeterminacy, and the researcher has to test for the potential degrees of indeterminacy of the model.19Third, standard software

18As argued in the work of Bernanke and Blinder (1992) and Bernanke and Mihov (1998), the federal funds rate has been the main policy tool in the United States during the post-war period considered in this paper, even if the Federal Reserve varied its operational procedures.

19A grid point method could be used to numerically identify the region of the parameter space associated with the indeterminate solution and the degrees of indeterminacy. However, this method does not provide a mapping between the dynamic properties of the model and its structural parameters.

packages do not allow for indeterminacy.20In practice, most papers simply rule out the possibility of indeterminacy by estimating the model exclusively in the determinate region of the parameter space. Among others, SW also adopt this approach and assume a priori a unique, determinate solution of the model.

Bianchi and Nicolò (2019) develop a new method to solve and estimate LRE models allowing for indeterminacy of the model solution. While the paper builds on Lubik and Schorfheide (2003, 2004) and Farmer et al. (2015), the novelty is to provide an approach that, using the information in the data, endogenously partitions the parameter space into the determinate and indeterminate region, and deals with the possibility of multiple degrees of indeterminacy. Hence, this methodology substantially simpliﬁes the approach to test for indeterminacy in U.S. monetary policy.

5.1 Building the Intuition

I consider a simple analytical example to present the technical complexities that a researcher faces when dealing with indeterminacy and to provide an intuition for how the methodology developed in Bianchi and Nicolò (2019) simpliﬁes the construction of the solution under indeterminacy. Recalling the classical monetary model in section 3.2, the corresponding univariate representation is Et(t+1) = trt.

As previously described, the solution to this model depends on the conduct of monetary pol- icy. If the monetary authority is active, jj > 1, the determinate solution is t= (1=)rt. Alternatively, if the monetary authority is passive, jj 1, the indeterminate solution is any process that takes the following form t= t 1rt 1+ t. The problem that a researcher faces when dealing with the indeterminate solution of a LRE model is the following. The equilibrium dynamics are uniquely determined if the Blanchard-Kahn condition is satisﬁed (Blanchard and Kahn, 1980). The condition requires the number of expectational variables of the model to equal the number of its unstable roots. The endogenous variable, t, of the

20Examples of standard solution algorithm are the code Gensys developed by Sims (2001), the toolkit by Uhlig (1999) and the algorithm of Anderson and Moore (1985) among others.

univariate model is expectational and the dynamic properties of the model depend on the value assumed by . When jj > 1, the model has a unique solution as it has a suﬃcient number of unstable roots to match the number of expectational variables. However, when jj 1, the model is indeterminate because it is missing one explosive root.

Bianchi and Nicolò (2019) propose to augment the original model by appending an independent process which could be either stable or unstable. The key insight consists of choosing this auxiliary process in a way to deliver the correct solution. When the original model is determinate, the auxiliary process must be stationary so that also the augmented representation satisﬁes the Blanchard-Kahn condition. When the model is indeterminate, the additional process should, however, be explosive so that the Blanchard-Kahn condition is satisﬁed for the augmented system, even if it is not for the original model.

The methodology of Bianchi and Nicolò (2019) proposes to solve the following augmented system of equations

Et(t+1)

rt;

(5)

!t 1 +t

where

[0; 2]

tis a newly deﬁned mean-zero

tis an auxiliary autoregressive process,

sunspot shock with standard deviation

and t

still denotes the forecast errors, t

t Et

1(t)as in the original model.21

When the original LRE model is determinate, jj > 1, the Blanchard-Kahn condition for the augmented representation in (5) is satisﬁed when j1=j 1. Indeed, the original model has the same number of unstable roots as the number of expectational variables. The methodology thus suggests to append a stable autoregressive process, and standard solution methods deliver the determinate solution t= (1=)rt. As the coeﬃcient j1=jis smaller than 1, the solution for the augmented representation also includes the autoregressive process !t. Importantly, its dynamics do not impact the endogenous variable t.

Considering the case of indeterminacy (i.e. jj 1), the original model has one ex-

21The choice of parametrizing the auxiliary process with 1= instead of induces a positive correlation between and that facilitates the implementation of the method when estimating a model.

pectational variable, but no unstable root, thus violating the Blanchard-Kahn condition. If the autoregressive process is explosive (i.e. j1=j > 1), the augmented representation satisﬁes the Blanchard-Kahn condition and delivers the same indeterminate solution t=t 1rt 1+t. Moreover, to guarantee boundedness, the solution imposes conditions such that !tis always equal to zero, and the solution for tdoes not depend on the appended autoregressive process.

Summarizing, the choice of the additional parameter should be made as follows. For values of jjoutside the unit circle, the Blanchard-Kahn condition for the augmented representation is satisﬁed for values of j1=jsmaller than 1. Conversely, under indeterminacy (i.e. jj 1) the condition is satisﬁed when j1=jis greater than 1. Also, note that under both determinacy and indeterminacy, the exact value of 1=is irrelevant for the law of motion of t. Under determinacy, the auxiliary process !tis stationary, but its evolution does not aﬀect the law of motion of the model variables. Under indeterminacy, !tis always equal to zero. Hence, the introduction of the auxiliary processes does not aﬀect the properties of the solution in either case. These processes only serve the purpose of providing the necessary explosive roots under indeterminacy and creating the mapping between the sunspot shocks and the expectational errors.

5.2 Implementation to Smets and Wouters (2007)

To solve and estimate indeterminate medium-scale models such as SW, a researcher faces the following technical complexities. It is not possible to derive analytically the partition of the parameter space, and the researcher does not know the exact properties of the indeterminacy region. Also, the adoption of a medium-scale model implies that a researcher does not know the degree of indeterminacy which characterizes the model.

To overcome these complexities, I implement the method proposed by Bianchi and Nicolò (2019). First, for any model with pexpectational variables, the maximum degree of indeterminacy also corresponds to p. Deﬁning fi;tgpi=1to be the forecast error associated with each expectational variable, I augment the original LRE model by appending up to pex-

ogenous processes !i;t=	1	!i;t 1+ i;ti;tfor i = 1; ::; p. Second, for a given draw of

	i

the structural parameters of the model, I wish to make draws of ismaller or greater than 1 with equal probabilities, as the partition of the parameter space is unknown. Therefore, I assume a uniform distribution over the interval [0:9; 1:1] as a prior distribution.22Third, while the newly deﬁned shocks, fi;tgpi=1; are independent, they are potentially related to the structural shocks of the model. Hence, I assume a uniform distribution over the interval [0; 1] for the newly deﬁned shocks,fi;tgpi=1, and over the interval [ 1; 1] for their correlations with the seven structural shocks described in section 4.23To estimate the model, I use the Kalman ﬁlter to generate the likelihood function and a Markov Chain Monte Carlo to explore the posterior.24

The data favors a speciﬁcation with one degree of indeterminacy and, as proven theoretically in Bianchi and Nicolò (2019), it is without loss of generality that I include the forecast error

associated with the inﬂation rate, ;ttEt 1(t), as the non-fundamental shock in

the augmented representation. Therefore, I estimate the SW model augmented with the exogenous process !t= 1!t 1+ t;t. For the parameter , I assume a uniform prior distribution over the interval [0:9; 1:1] and I specify a uniform prior over the interval [0; 1] for the standard deviation of the sunspot shock, .25The newly deﬁned sunspot shock, t, could potentially be correlated with the seven structural shocks of the model and I set a uniform prior distribution over the interval [ 1; 1] for its correlations with the seven structural shocks.

Importantly, I verify that the standard deviation of the newly deﬁned shock, t, and its

22Any symmetric interval around 1 guarantees an equal probability of drawing greater or smaller than

I refer the reader to Bianchi and Nicolò (2019) for a detailed technical discussion that shows the existence of a unique mapping between the parametrization used in this paper and the one adopted in Lubik and Schorfheide (2004) to characterize the full set of indeterminate equilibria. Therefore, by setting uniform priors over the well-deﬁned interval [ 1; 1] for such correlations, the methodology of Bianchi and Nicolò (2019) accounts for the parametrization of Lubik and Schorfheide (2004) as a special case.
Alternative algorithms, such as the sequential Monte Carlo of Herbst and Schorfheide (2015), could also be implemented. However, the richness of the model, the complexity of the boundary of the determinacy region as a function of the deep model parameters and the evidence of two local maxima that are interior to the determinate and indeterminate region respectively could pose technical challenges also in the implementation of such algorithms.
As shown in table 3, the posterior distribution for the sunspot shock is not at the boundary but rather interior to the interval over which the uniform prior distribution.

correlations with the structural shocks are locally identiﬁed using the method proposed by Iskrev (2009).26Each parametrization of the correlations has strong predictions for the transmission of each shock, and the richness of the data used to estimate the SW model crucially identiﬁes them. I ﬁnd that only the correlation between the newly deﬁned shock and the price mark-up shock statistically diﬀers from zero, implying that the price mark-up shock has a contemporaneous eﬀect on inﬂation, while the remaining shocks have a lagged impact. Hence, I report below the results of the estimation when I set the correlations with the remaining shocks to zero, as this speciﬁcation is favored by the data.

Main Findings

6.1 U.S. Monetary Policy in the Post-War Period

I ﬁnd that the conduct of monetary policy changed during the post-war period from a passive stance before 1979 to an active inﬂation targeting since the early 1980s. This result has two implications. First, the assumption imposed in SW about an active monetary policy both before and after 1979 is rejected. Second, the ﬁndings in previous studies that adopted small-scale models (Clarida et al., 2000; Lubik and Schorfheide, 2004) carry over to the SW model.

By considering the model and the data described in section 4, I apply the methodology presented in section 5 to estimate the SW model over three subsamples. The ﬁrst period starts in 1955:4, which corresponds to one year after the end of the Korean War, and it ends in 1969:4 as the chairmanship of William Martin terminates in early 1970. The second sample considers the chairmanships of both Arthur Burns and William Miller, and it spans from 1970:1 until 1979:2. Finally, the beginning of the third period corresponds to 1984:1, in which Kim and Nelson (1999) initially identify a structural break in the U.S. business cycle,

26In a related paper, Qu and Tkachenko (2017) document that indeterminacy can help identify struc- tural parameters and that identiﬁcation properties can diﬀer substantially between small- and medium-scale models.

while the end is marked by the Great Recession in 2007:3.27

Appendix A reports the prior distributions for the structural parameters of the model and the exogenous processes that drive the dynamics of the economy. Relative to SW, the only diﬀerence relates to the prior distribution of the Taylor rule coeﬃcient associated with the response of the monetary authority to changes in the inﬂation rate. While SW specify a normal distribution truncated at 1, centered at 1.50 and with standard deviation 0.25, I set a ﬂatter normal prior distribution centered at 1 and with standard deviation 0.35.

Table 1 reports the results of the estimation for each subsample. Relative to SW, the novelty is to relax the a prioriassumption of equilibrium uniqueness. The method described in section 5 allows to estimate the model over the entire parameter space. For each period, the Metropolis-Hastings algorithm ﬁnds two local maxima, one associated with the determinate solution and the other with the indeterminate representation. It is therefore possible to compute the corresponding marginal data density using the modiﬁed harmonic mean estimator proposed by Geweke (1999) and the posterior model probabilities associated with each local maxima. Focusing on the ﬁrst two samples that cover the period from 1955:4 to 1979:2, the data strongly favors the representation associated with indeterminacy, therefore rejecting the assumption of equilibrium uniqueness imposed in SW. On the contrary, the period subsequent to the Volcker disinﬂation is associated with a determinate, unique representation.

The evidence of a change in the monetary policy stance since 1984:1 is presented in table 2, where the posterior distributions of the structural parameters in the three sub-periods are compared.28Considering the Taylor rule coeﬃcient associated with the response of the monetary authority to changes in the inﬂation rate, r, it is clear that the monetary authority was passive prior to 1979, thus consistent with a weak response of the monetary authority

27The ﬁndings in this section for the period prior to 1979:2 are quantitatively unchanged when considering a sample spanning from 1955:4 until 1979:2. However, studying the two samples separately is relevant to understand the connection between diﬀerent steady state properties between the two periods and the exceptional growth in productivity until the early 1970s documented in Fernald (2014b).

28I consider the posterior estimates to be unchanged when the posterior mean of a parameter estimated in either of the two sample periods is within the 90% probability interval associated with the posterior distribution obtained in the alternative sample.

Table 1: Model comparison for each sample period

		Determinacy	Indeterminacy
Martin (55Q4 - 69Q4)	Log data density	-338.1	-319.2
	Posterior Model Prob (%)	0%	100%

Burns-Miller (70Q1 - 79Q2)	Log data density	-277.9	-272.5
	Posterior Model Prob (%)	0%	100%
Post-Volcker (84Q1 - 07Q3)	Log data density	-399.7	-407.2
	Posterior Model Prob (%)	100%	0%

The table reports the (log) data densities and the posterior model probabilities obtained for each sample period.

to changes in the inﬂation rate. Table 2 also suggests that for the period subsequent to the Volcker disinﬂation, the monetary authority changed its stance and acted more aggressively to stabilize inﬂation, therefore ensuring equilibrium uniqueness.

Table 2: Posterior distribution of structural parameters

		1955:4-1969:4	1970:1-1979:2	1984:1-2007:3
Coeﬃcient	Description	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]
	Adjustment cost	4.75	[2.90,6.63]	3.41	[2.01,4.69]	6.92	[5.04,8.79]
c	IES	1.11	[0.86,1.37]	0.91	[0.67,1.12]	1.58	[1.36,1.85]
h	Habit Persistence	0.62	[0.49,0.74]	0.62	[0.48,0.76]	0.64	[0.56,0.73]
l	Labor supply elasticity	1.98	[0.89,3.07]	1.27	[0.25,2.14]	2.30	[1.29,3.24]
w	Wage stickiness	0.74	[0.63,0.85]	0.70	[0.59,0.81]	0.68	[0.53,0.83]
p	Price Stickiness	0.60	[0.50,0.67]	0.58	[0.50,0.65]	0.75	[0.67,0.83]
w	Wage Indexation	0.33	[0.13,0.53]	0.56	[0.35,0.77]	0.44	[0.20,0.68]
p	Price Indexation	0.30	[0.12,0.47]	0.47	[0.24,0.70]	0.28	[0.10,0.44]
	Capacity utiliz. elasticity	0.52	[0.31,0.74]	0.47	[0.24,0.71]	0.71	[0.57,0.86]
	Share of ﬁxed costs	1.59	[1.46,1.71]	1.34	[1.18,1.51]	1.60	[1.46,1.75]
	Share of capital	0.24	[0.20,0.29]	0.18	[0.13,0.23]	0.22	[0.19,0.26]
	S.S. inﬂation rate (quart.)	0.60	[0.44,0.76]	0.62	[0.46,0.78]	0.69	[0.55,0.80]
100( 11)	Discount factor	0.17	[0.06,0.26]	0.20	[0.08,0.31]	0.13	[0.05,0.21]
	S.S. hours worked	1.33	[0.17,2.54]	-2.42	[-3.63,-1.17]	1.52	[0.36,2.80]
l
	Trend growth rate (quart.)	0.57	[0.38,0.76]	0.36	[0.31,0.41]	0.45	[0.42,0.48]

r	Taylor rule inﬂation	0.64	[0.35,0.99]	0.75	[0.54,0.99]	1.83	[1.44,2.24]
ry	Taylor rule output gap	0.13	[0.05,0.20]	0.16	[0.09,0.23]	0.09	[0.04,0.15]
ry	Taylor rule (output gap)	0.11	[0.07,0.15]	0.18	[0.12,0.24]	0.15	[0.10,0.19]
	Taylor rule smoothing	0.87	[0.81,0.94]	0.73	[0.61,0.86]	0.84	[0.80,0.88]

The table compares the posterior estimates of structural parameters under indeterminacy for the pre-Volcker period and under determinacy for the post-Volcker period.

Importantly, these results provide evidence that, even when accounting for the richer propagation mechanisms, the equilibrium was indeterminate before 1979, and the ﬁndings of Clarida

et al. (2000) and Lubik and Schorfheide (2004) among others carry over to a medium-scale model. Appendix B reports predictive checks, further supporting these ﬁndings, and shows that the imposition of an active monetary policy, as in the original paper by SW, is not innocuous and delivers erroneous estimates of the structural parameters.

Table 2 also provides evidence in support of Fernald (2014b) who documents that the U.S. economy experienced a period of exceptional growth in productivity in the post-war period until the early 1970s. Both the trend growth rate of the economy, , and the (steady state)

hours worked, l, drop signiﬁcantly in the period between 1970 until 1979 relative to the previous period. The posterior distributions also show that the post-Volcker period is characterized by a mildly higher degree of price stickiness, p, and a more persistent process of the price-markup shock measured by pin table 3. This ﬁnding is supported by Galí and Gertler (1999), who provide evidence of an increased average price duration over this period due to the lower and more stable inﬂation rate. Also, the post-Volcker period is associated with a larger adjustment cost faced by the representative agent that chooses a higher degree of capital utilization rate.

The comparison in table 3 of the properties of the exogenous processes between the period before and after 1979 provides an additional ﬁnding. In line with a large literature, the volatility of the shocks driving macroeconomic ﬂuctuations are signiﬁcantly smaller starting from the mid-1980s (Stock and Watson, 2003; Primiceri, 2005; and Sims and Zha, 2006). This result and the evidence of the change in the conduct of monetary policy are clearly linked to the discussion on the possible explanations for the sources of the reduction in U.S. macroeconomic volatility from the early 1980s to 2007. In section 6.4, I show that, according to the SW model, both the change in the monetary policy stance andthe lower size of the shocks explain this empirical observation for U.S. macro data. In line with the results of the local identiﬁcation test by Iskrev (2009) described in section 5.2, the posterior distribution for the standard deviation of the sunspot shock, , and its correlation with the mark-up shock, p, are both tightly estimated, even when assuming a uniform distribution as a prior.

Table 3: Posterior distribution of exogenous processes

		1955:4-1969:4	1970:1-1979:2	1984:1-2007:3
Coeﬃcient	Description	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]

a	Technology shock	0.52	[0.44,0.61]	0.56	[0.45,0.67]	0.36	[0.31,0.40]
b	Risk premium shock	0.19	[0.11,0.27]	0.17	[0.11,0.23]	0.18	[0.14,0.22]
g	Government sp. shock	0.50	[0.42,0.58]	0.55	[0.44,0.65]	0.41	[0.36,0.46]
I	Investment-speciﬁc shock	0.61	[0.42,0.79]	0.38	[0.24,0.51]	0.35	[0.28,0.43]
r	Monetary policy shock	0.11	[0.09,0.12]	0.22	[0.18,0.27]	0.12	[0.10,0.14]
p	Price mark-up shock	0.24	[0.20,0.28]	0.31	[0.23,0.39]	0.09	[0.07,0.11]
w	Wage mark-up shock	0.24	[0.18,0.28]	0.31	[0.23,0.38]	0.31	[0.24,0.37]
	Sunspot shock	0.14	[0.07,0.21]	0.19	[0.06,0.33]	-	-
a	Persistence technology	0.95	[0.92,0.99]	0.74	[0.60,0.87]	0.92	[0.87,0.96]
b	Persistence risk premium	0.58	[0.34,0.83]	0.77	[0.62,0.92]	0.22	[0.05,0.35]
g	Persistence government sp.	0.86	[0.77,0.94]	0.85	[0.77,0.94]	0.96	[0.94,0.98]
I	Persistence investment-speciﬁc	0.50	[0.30,0.70]	0.66	[0.47,0.84]	0.66	[0.52,0.78]
r	Persistence monetary policy	0.50	[0.31,0.69]	0.31	[0.11,0.51]	0.35	[0.19,0.51]
p	Persistence price mark-up	0.24	[0.04,0.41]	0.37	[0.11,0.65]	0.83	[0.72,0.95]
w	Persistence wage mark-up	0.62	[0.34,0.90]	0.43	[0.14,0.68]	0.81	[0.66,0.95]
p	MA price mark-up	0.64	[0.43,0.85]	0.68	[0.45,0.95]	0.66	[0.48,0.84]
w	MA wage mark-up	0.50	[0.25,0.74]	0.55	[0.25,0.88]	0.62	[0.38,0.82]
ga	Cov(a; g)	0.57	[0.38,0.76]	0.62	[0.40,0.84]	0.40	[0.22,0.57]
p	Corr( ; p)	0.92	[0.82,0.99]	0.66	[0.29,0.99]	-	-

The table compares the posterior estimates of the parameters associated with the exogenous processes under indeterminacy for the pre-Volcker period and under determinacy for the post-Volcker period.

Finally, I show that by estimating a small-scale model that does not account for the richer structure of the SW model, the evidence of an active monetary policy over the period between 1984:1 and 2007:3 is overturned. This result is consistent with the identiﬁcation problem described in section 3.1 for which missing propagation mechanisms could be misinterpreted as evidence of indeterminacy.

To this purpose, I estimate the small-scale model in Del Negro and Schorfheide (2004), who adapted the model by Lubik and Schorfheide (2004) to account for a balanced growth path.29Table 4 reports the (log) data densities and the corresponding posterior model probabilities for the determinate and indeterminate representations using the two alternative models. The ﬁrst row shows that by estimating the small-scale model of Del Negro and Schorfheide (2004) using the post-Volcker data from SW, the evidence of indeterminacy is pervasive. Nevertheless, the conclusion is reversed once richer and more relevant propagation

29Appendix C reports the details about the model, the data, the prior and the posterior distributions of the model parameters of the model by Del Negro and Schorfheide (2004).

Table 4: Model comparison: Del Negro and Schorfheide (2004) and SW

Sample: 1984:1-2007:3		Determinacy	Indeterminacy
Del Negro and Schorfheide (2004) model	Log data density	-58.09	-26.56
	Posterior Model Prob (%)	0%	100%

SW model	Log data density	-399.670	-407.16
	Posterior Model Prob (%)	100%	0%

The table reports the (log) data densities and the posterior model probabilities for the Del Negro and Schorfheide (2004) model and the SW model over the period 1984:1-2007:3.

mechanisms and structural shocks are included. According to the SW model, monetary policy was active and consistent with a determinate equilibrium over the period between 1984:1 and 2007:3.

6.2 Monetary Policy, Expectations, and the Propagation Mechanism

In this section, I document the eﬀects of a change in monetary policy on the dynamics of the economy and the transmission of structural shocks. When monetary policy is passive, the propagation of structural disturbances is altered and more persistent due to the formation of self-fulﬁlling expectations in response to shocks. In particular, I study the propagation of three shocks that, as highlighted in section 6.3, explain most of U.S. business cycle ﬂuctua- tions in the period prior to 1979: productivity, risk-premium and monetary policy shocks.30Importantly, the results presented in this section are further supported by the empirical evidence in section 6.5 that also documents the diﬀerences in the transmission of shocks depending on the conduct of monetary policy by estimating a BVAR model, a laGiannone, Lenza, and Primiceri (2015).

6.2.1 Productivity Shock

The impact of a productivity shock has implications that diﬀer depending on the conduct of U.S. monetary policy. The four panels on the right of ﬁgure 1 show the transmission of a

30As discussed in section 5.2, the data favors a speciﬁcation in which the non-fundamental shock, t, is correlated only with the markup shock. This implies that the inﬂation rate responds on impact only to a markup shock while with a lag to all the remaining fundamental shocks, including those analyzed in this section.

1		Output			0.2		Inflation

0.8					0.15
0.6					0.1
0.4					0.05
0.2					0
0	5	10	15	20	-0.05	5	10	15	20
0	0

Interest Rate

Marginal Cost

0.2					0
0.15

0.1					-0.2
0.05					-0.4
0
				-0.6		Mean
-0.05
					90% Prob. Interval
0	5	10	15	20	0	5	10	15	20

1		Output			0.2		Inflation

0.8					0.15
0.6					0.1
0.4					0.05
0.2					0
0	5	10	15	20	-0.05	5	10	15	20
0	0

Interest Rate

Marginal Cost

0.2					0
0.15

0.1					-0.2
0.05					-0.4
0
				-0.6
-0.05
0	5	10	15	20	0	5	10	15	20

Figure 1: Productivity shock

Mean impulse responses to a productivity shock are denoted by solid lines, while dashed lines represent the associated 90% probability intervals.

(one standard deviation) productivity shock in the post-Volcker period on output, inﬂation rate, nominal interest rate, and marginal cost incurred by ﬁrms.31The shock generates economic activity and deﬂationary pressures because of a drop in marginal cost. Under the active inﬂation targeting, the monetary authority responds by lowering the policy rate by more than one-for-one. Conversely, the four panels on the left are associated with the passive monetary policy of the Burns and Miller chairmanship.32The shock still results into a drop in marginal cost and an economic expansion. However, the productivity shock also generates inﬂationary expectations that are accommodated by the passive monetary authority. Firms observe a contemporaneous drop in marginal cost, but also expect future marginal costs to be persistently higher than the steady state level. This mechanism thus results into a self- fulﬁlling rise of the inﬂation rate. The corresponding increase in the nominal interest rate is gradual and not aggressive enough to anchor inﬂation expectations and stabilize the inﬂation rate, therefore allowing for persistent eﬀects on the economy.

6.2.2 Risk-Premium Shock

The risk-premium shock represents a wedge between the policy rate set by the central bank and the return that households receive to hold their assets. As ﬁgure 2 suggests, the shock

31The size of the shock depends on the standard deviation estimated in each of the two samples. As found in table 3, the size of the shock in the two samples before 1979 is larger than the standard deviation estimated for the post-Volcker period.

32I obtain similar results for the period that covers the Martin chairmanship.

Burns-Miller

1		Output					Inflation		0.2
				0
0.8

				-0.05				0.1
0.6
				-0.1
0.4								0
				-0.15
0.2
				-0.2				-0.1
0
				-0.25				-0.2
-0.2					-0.3
0	5	10	15	20	0	5	10	15	20

Interest Rate

		Mean
		90% Prob. Interval
0	5	10	15	20

Post-Volcker

1		Output					Inflation		0.2
				0
0.8

				-0.05				0.1
0.6
				-0.1
0.4								0
				-0.15
0.2
				-0.2				-0.1
0
				-0.25				-0.2
-0.2					-0.3
0	5	10	15	20	0	5	10	15	20

Interest Rate

Figure 2: Risk-premium shock

Mean impulse responses to a risk-premium shock are denoted by solid lines, while dashed lines represent the associated 90% probability intervals.

has similar eﬀects on the real economy regardless of the conduct of monetary policy. A (one standard deviation) negative shock increases consumption as the required rate of return on assets is lower. Also, the decrease in the cost of capital further stimulates economic activity due to larger investments by ﬁrms. However, the inﬂation response to the risk-premium shock depends on the conduct of monetary policy. When monetary policy is active, ﬁrms face a higher marginal cost that maps into inﬂationary pressures. When monetary policy is passive, agents observe a rise in the real interest rate and form self-fulﬁlling deﬂationary expectations that more than oﬀset the increase in marginal cost. In this case, the risk-premium shock therefore generates deﬂationary eﬀects.

6.2.3 Monetary Policy Shock

The bottom three panels of ﬁgure 3 describe the predictions of a contractionary monetary policy shock under the active regime of the post-Volcker period. Output and inﬂation drop and revert to the steady state of the economy. When monetary policy is indeterminate, the responses to a contractionary monetary policy shock are reported in the top three panels. Economic activity is depressed. However, in line with the empirical ﬁndings of Lubik and Schorfheide (2004), the unexpected tightening of monetary policy is associated with a persistent inﬂationary eﬀect due to self-fulﬁlling inﬂationary expectations. Therefore, ﬁgure 3

Burns-Miller

		Output
0				0.1

-0.1				0.05
-0.2

-0.3				0
-0.4

-0.5				-0.05
-0.6
5	10	15	20
0

		Inflation
				0.2
				0.15
				0.1
				0.05
				0
0	5	10	15	20

Interest Rate

		Mean
		90% Prob. Interval
0	5	10	15	20

Post-Volcker

		Output					Inflation
0					0.1				0.2
								0.15
-0.1					0.05
-0.2								0.1

-0.3
				0				0.05
-0.4

-0.5					-0.05				0
-0.6	5	10	15	20	0	5	10	15	20
0

Interest Rate

Figure 3: Monetary policy shock

Mean impulse responses to a monetary policy shock are denoted by solid lines, while dashed lines represent the associated 90% probability intervals.

highlights the diﬀerences in the impact and the transmission of structural shocks such as an unexpected monetary policy tightening.

6.3 The History of U.S. Business Cycles

The interpretation of U.S. business cycle ﬂuctuations relies on the conduct of monetary pol- icy. The passive interest rate policy in the period from 1955 to 1979 meant that monetary policy accommodated the transmission of fundamental shocks and failed to insulate the in- ﬂation rate from persistent deviations relative to the target. Persistent technology shocks explain the initial upward trend in the inﬂation rate observed since the early 1960s. This result is supported by the evidence of Fernald (2014b), according to whom the U.S. economy experienced a period of exceptional growth in productivity since World War II until the oil crisis in 1973. Mark-up shocks account for the sudden inﬂationary episodes related to the oil price shocks during the 1970s but do not explain the persistent rise in the inﬂation rate. Sunspot shocks play a minor role to explain the high macroeconomic volatility observed before 1979. These ﬁndings indicate that the strong evidence of a passive monetary policy before 1979 lies in the distinctive transmission mechanism of structural shocks because of self-fulﬁlling expectations rather than in the quantitative relevance of non-fundamental dis- turbances. Regarding the period after the Volcker disinﬂation, the results are in line with

previous ﬁndings in the literature and reported in appendix E.

6.3.1 Martin and the Post-Korean War Period

I ﬁrst focus on the post-Korean War period that starts in 1955 and during which William Martin has been the Chairman of the Fed until the beginning of 1970.

Figure 4 plots the historical decomposition of the deviations of the inﬂation rate from its long run for two alternative speciﬁcations.33The left panel plots the decomposition under indeterminacy that is favored by the data.34While sunspot shocks could have potentially contributed for the model to better match the high volatility in the inﬂation rate, the historical decomposition indicates that they played no quantitative role. The rise in inﬂation since the early 1960s is associated with a sequence of productivity shocks. In line with the analysis in section 6.2, the left panel indicates that the cumulated impact of positive productivity shocks explains the upward trend in the inﬂation rate.35As discussed below, this result is supported by the empirical evidence presented in Fernald (2014b) among others.36

The right panel reports the decomposition that results by imposing equilibrium uniqueness as conducted in SW. A comparison with the left panel suggests that the assumption substantially aﬀects the interpretation of the data.37The upward trend in the inﬂation rate during the 1960s is erroneously attributed to mark-up shocks. However, the results in section 6.1

33The historical decomposition of output growth for the two sample periods prior to 1979 is provided in appendix D and shows minor diﬀerences between the determinate and the indeterminate representation.

34To conduct the historical decompositions, I use the posterior means estimated for the pre-Volcker period for each of the two local maxima found during the estimation and that are reported in table 2 and 3. Also, to simplify the analysis, I group the exogenous spending shock, the investment-speciﬁc shock and the risk- premium shock as "demand" shocks. Similarly, price and wage mark-up shocks are grouped as "mark-up" shocks.

Monetary policy shocks had a minor impact that resulted into mildly deﬂationary pressures during the early 1960s. It is useful to recall that the historical decomposition cumulates the eﬀect of a given shock on the inﬂation rate until a given date. Given the persistence of the monetary policy shocks under indeterminacy, as described in section 6.2, a monetary policy shock can be identiﬁed as the change in the contribution to explain the dynamics.
While the SW model does not account for a ﬁscal block explicitly, it is relevant to note that government spending shocks possibly associated with the Vietnam War are not quantitatively relevant to explain inﬂation dynamics over this time period.
Minor diﬀerences in the historical decomposition at a given year across the two panels are explained by diﬀerences in the contribution of the initial conditions for each representation.

!1#

!2#

!1#

!2#

Sunspot#

Monetary#

policy#

Mark!up#

Demand#

Produc:vity#

reject the

relies

on the left panel.

Historical"Decomposi0on"2"Inﬂa0on Historical"Decomposi0on"2"Inﬂa0on"

2.5#

1.5#

0.5# 0#

!0.5#

!1#

3# 2.5# 2#

1.5# 1# 0.5#

0# !0.5#

!1#

Sunspot#

Monetary#

policy#

Mark!up#

Demand#

Produc:vity#

Figure 4: Historical decomposition of the inﬂation rate (1955-1969)

Historical decomposition of the inﬂation rate under indeterminacy (left) and determinacy (right) at quarterly rates.

The evidence that the U.S. economy experienced exceptional growth in productivity prior to the early 1970s is documented by Fernald (2014b) among others38. This literature points to a wave of technological innovations as the source of a rise in productivity growth and therefore economic activity. In particular, when considering the quarterly time series for total factor productivity produced by Fernald (2014a), the resulting (standardized) series is plotted in ﬁgure 5 together with the smoothed productivity shocks estimated using the SW model. The comparison indicates that the estimation of the SW model successfully identiﬁes the sequence of positive productivity shocks that the U.S. economy experienced starting from the early 1960s.39Importantly, in a passive monetary policy regime, productivity shocks generate persistent inﬂationary expectations that are consistent with the observed upward trend in inﬂation.

6.3.2 The Burns and Miller Chairmanships

The second period begins with the chairmanship of Arthur Burns in 1970 and ends in 1979 with the conclusion of the chairmanship of William Miller. Figure 6 presents the historical

Other work that supports this view is provided by Fernald (2014a), Gordon (2000), Davig and Wright (2000), and Field (2003).
The correlation between the two sequences of productivity shocks is 0.74, suggesting that the model does remarkably well in extracting productivity shocks that are in line with Fernald (2014a).

3#
2#
1#
0#
!1#	SW#model#
!2#	Fernald#(2014)#

!3#
!4#

Figure 5: Total factor productivity

Sample 1955:4-1969:4. Quarterly (standardized) series of total factor productivity (solid) from Fernald (2014a) and of the smoothed productivity shocks from SW model (dashed) in percentages at quarterly rates.

decomposition of the inﬂation rate over this sample period according to alternative monetary policy regimes. The left panel presents the decomposition associated with indeterminacy, as supported by the data, while the right panel is obtained by imposing the assumption that monetary policy was active. As explained in section 6.2, the conduct of a passive monetary policy is such that positive risk-premium shocks have a contractionary eﬀect on the economy but also lead agents to form inﬂationary expectations that are self-fulﬁlling and persistent. Hence, a combination of demand shocks and positive productivity shocks sustained the high inﬂation observed in the late 1970s, while the spike in 1979 is attributed to mark-up shocks.40Even for this sample period, sunspot shocks have no quantitative relevance for U.S. business cycles. Conversely, the right panel shows that the assumption of an active monetary policy mistakenly attributes the ﬂuctuations in the inﬂation rate exclusively to mark-up shocks.

6.4 What Changed in the Early 1980s?

The work of Kim and Nelson (1999) and McConnell and Perez-Quiros (2000) ﬁrst documented the lower volatility of U.S. real GDP since the early 1980s. Extensive research has been conducted to provide explanations for the reduction in U.S. macroeconomic volatility. Using the SW model, I investigate the validity of two prominent theories that have been advocated

40While ﬁgure 6 generally refers to demand shocks, the break down for each demand shock shows that the contribution of the risk-premium shock is the most relevant as opposed to the government spending or investment-speciﬁc shocks.

!1#				Demand#
	!1#
			Produc:vity#
!2#		!2#

Historical"Decomposi0on"2"Inﬂa0on Historical"Decomposi0on"2"Inﬂa0on"

2.5#

1.5#

0.5# 0#

!0.5#

!1#

			Sunspot#
2.5#

	2#		Monetary#

1.5#		policy#

		Mark!up#
	1#

0.5#		Demand#


0#


!0.5#		Produc:vity#


!1#

Figure 6: Historical decomposition of the inﬂation rate (1970-1979)

Historical decomposition of the deviation of the inﬂation rate from its steady state under indeterminacy (left) and determinacy (right) at quarterly rates.

to explain this empirical fact. The work by Sims and Zha (2006) suggests that the behavior of the data changed because of a decrease in the variance of the structural shocks since the Volcker disinﬂation. Primiceri (2005) ﬁnds evidence that policy also changed, but the role played by structural disturbances is more relevant. According to this strand of the literature, the reduction in volatility of U.S. macroeconomic data is not related to monetary policy and it can therefore be considered as "good luck".

An alternative theory has been supported by the work of Clarida et al. (2000) and Lubik and Schorfheide (2004), among others who ﬁnd evidence of an active inﬂation targeting since the Volcker disinﬂation. The reduction in volatility can be attributed to the "good policy" of the monetary authority.

To provide an intuition for the approach that I adopt using the SW model, I consider the standard three-equation NK model of Lubik and Schorfheide (2004). The model is described by a dynamic IS curve

xt= Et(xt+1) (RtEt(t+1)) + gt;

a NK Phillips Curve

t= Et(t+1) + (xtzt) ;

and a monetary policy reaction function

Rt= RRt 1+ (1 R) [ 1t+ 2(xtzt)] + "R;t;

where xtrepresents the deviation of output from its trend and the demand shock, gt, and supply shock, zt, are autoregressive processes of the form gt=ggt 1+"g;tand zt=zzt 1+

"z;t.

To test alternative theories using the SW model, I estimate diﬀerent model speciﬁcations by imposing restrictions on sets of parameters and volatilities. In model 1, "Policy and Shocks", I constrain the private sector parameters, f; ; g, to be the same across the period from 1955 to 1979 and the period from 1984 to 2007. I also allow the policy parameter, fR;1;2g, the variances of the shocks, fg;zg, and the autoregressive coeﬃcients, fg;zg, to vary across the two periods. This speciﬁcation considers a combination of "good luck" and "good policy" to explain the data. Relative to model 1, I then consider model 2, "Shocks only", by further restricting the policy parameter, fR;1;2g, thus considering the "good luck" view. Conversely, consistent with the "good policy" theory, model 3, "Policy only", allows for the policy parameter to vary across sub-periods, while constraining all the other structural parameters and variances to be constant. Following the intuition provided with the baseline three-equation NK model, I apply the same approach to estimate alternative model speciﬁ- cations of the SW model and test for the validity of the "good luck" and/or "good policy" theory in the data.

Table 5: Model comparison across model speciﬁcations

	Log data density	Posterior model prob
Policy and shocks	-986.85	100%	100%
Shocks only	-994.51	0%	-
Policy only	-1021.08	-	0%

The table reports the marginal data densities for the three alternative speciﬁcations and the pairwise posterior model probabilities relative to model 1, Policy and Shocks.

Table 5 reports the logarithm of the marginal data densities computed using Geweke's (1999)

modiﬁed harmonic mean estimator for each of the three model speciﬁcations. Unlike a standard likelihood ratio statistic, the marginal data density penalizes a model that is over- parameterized. Nevertheless, the posterior model probabilities derived as pairwise comparisons relative to model 1, Policy and Shocks, indicate that, based on the SW model, a combination of both good luck and good policy is the explanation for the observed reduction in the volatility since the mid-1980s. Also, in line with the ﬁndings of Primiceri (2005), model 2, Shocks only, has a more relevant role rather than the theory based exclusively on the change of monetary policy to an active stance, model 3 Policy only.

Table 6: Posterior distribution of common parameters

Coeﬃcient	Description	Mean	[ 5 , 95 ]
	Adjustment cost	6.59	[4.75,8.48]
c	IES	1.44	[1.25,1.63]
h	Habit Persistence	0.62	[0.51,0.73]
l	Labor supply elasticity	2.14	[1.37,2.91]
w	Wage stickiness	0.84	[0.79,0.89]
p	Price Stickiness	0.77	[0.70,0.83]
w	Wage Indexation	0.40	[0.24,0.56]
p	Price Indexation	0.18	[0.06,0.30]
	Capacity utiliz. elasticity	0.68	[0.55,0.81]
	Share of ﬁxed costs	1.63	[1.50,1.75]
	Share of capital	0.22	[0.19,0.25]
	S.S. inﬂation rate (quart.)	0.60	[0.48,0.71]
100( 11)	Discount factor	0.10	[0.04,0.16]
	S.S. hours worked	0.67	[-0.29,1.55]
l
	Trend growth rate (quart.)	0.42	[0.39,0.45]

The table reports the posterior estimates of the structural parameters constrained to be the same in the pre- and post-Volcker period.

Focusing on the estimation of model 1, Policy and Shocks, table 6 reports the posterior estimates of the constrained parameters across the two subsamples. As expected, the posterior distribution of the model parameters are in line with those estimated for the two samples separately and reported in section 6.1. The comparison of the posterior estimates of the structural parameters in table 7 indicates that the conduct of monetary policy changed toward a more aggressive stance in the post-Volcker period. Moreover, consistent with the ﬁndings of Sims and Zha (2006), among others, the magnitude of the volatility of the shocks is significantly reduced in the post-Volcker period. These results provide evidence that, according

Table 7: Posterior distribution of unrestricted parameters

		Pre-Volcker (55:4 - 79:2)	Post-Volcker (84:1 - 07:3)
Coeﬃcient	Description	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]

a	Technology shock	0.54	[0.47,0.60]	0.36	[0.31,0.40]
r	Taylor rule inﬂation	0.85	[0.73,0.97]	1.80	[1.37,2.18]
ry	Taylor rule output gap	0.15	[0.09,0.21]	0.05	[0.02,0.10]
ry	Taylor rule (output gap)	0.17	[0.12,0.22]	0.17	[0.12,0.21]
	Taylor rule smoothing	0.86	[0.80,0.91]	0.84	[0.80,0.88]
b	Risk premium shock	0.14	[0.08,0.19]	0.15	[0.08,0.21]
g	Government sp. shock	0.53	[0.46,0.60]	0.41	[0.36,0.46]
I	Investment-speciﬁc shock	0.45	[0.35,0.56]	0.30	[0.23,0.37]
r	Monetary policy shock	0.17	[0.15,0.20]	0.12	[0.10,0.14]
p	Price mark-up shock	0.30	[0.25,0.34]	0.09	[0.07,0.11]
w	Wage mark-up shock	0.26	[0.22,0.30]	0.31	[0.25,0.37]
	Sunspot shock	0.06	[0.01,0.11]	-	-

a	Persistence technology	0.97	[0.96,0.98]	0.93	[0.89,0.96]
b	Persistence risk premium	0.75	[0.55,0.93]	0.38	[0.05,0.71]
g	Persistence government sp.	0.90	[0.86,0.95]	0.97	[0.95,0.98]
I	Persistence investment-speciﬁc	0.68	[0.55,0.81]	0.74	[0.62,0.86]
r	Persistence monetary policy	0.35	[0.20,0.51]	0.32	[0.19,0.45]
p	Persistence price mark-up	0.24	[0.04,0.42]	0.82	[0.73,0.92]
w	Persistence wage mark-up	0.34	[0.12,0.55]	0.69	[0.50,0.88]
p	MA price mark-up	0.77	[0.64,0.92]	0.63	[0.44,0.83]
w	MA wage mark-up	0.38	[0.21,0.57]	0.56	[0.32,0.80]
ga	Cov(a; g)	0.62	[0.46,0.76]	0.40	[0.22,0.57]

The table reports the posterior estimates of policy parameters and the exogenous shocks for the pre- and post-Volcker period.

to the SW model, boththeories contribute to explain the reduction in U.S. macroeconomic

volatility.

6.5 IRFs from Bayesian Vector Autoregressions

In this section, I provide additional empirical evidence in support of the ﬁndings of section

6.2 in which I show that the transmission of fundamental shocks in the SW model diﬀers substantially across regimes. In particular, I estimate a medium-scale BVAR model and doc- ument that the propagation mechanism of productivity and monetary policy shocks depends on the conduct of monetary policy also when adopting a reduced-form approach.

The data consists of eight time series. In addition to the seven quarterly U.S. macroeconomic time series used in SW, I also include the time series for total factor productivity obtained

from the estimation of the SW model. Using the eight observables, I estimate a BVAR in each of the three subsamples described in section 4: the ﬁrst sample from 1955:1 until 1969:4, the second from 1970:1 until 1979:2 and the last sample from 1984:1 until 2007:3.

I estimate a BVAR model of the following form:

yt= C + B1yt 1+ ::: + Bpyt p+ "t

where ytis a n1vector of endogenous variables, "tN(0; )is an n1vector of exogenous shocks, and C; B1; :::; Bpand are matrices of suitable dimensions containing the model's unknown parameters. To estimate the BVAR model, I adopt the hierarchical approach of Giannone et al. (2015). Their estimation strategy ensures a good performance in terms of accuracy of the estimation of impulse response functions in identiﬁed VARs. Giannone et al. (2015) propose a hierarchical modeling approach that combines the conjugate priors most commonly used in the literature. First, they set a "Minnesota" prior assuming that each variable follows a random walk process (Litterman, 1979, 1980). Second, they include a "sum-of-coeﬃcients" prior assuming that a no-change forecast for each variable is a good forecast at the beginning of the sample (Doan et al., 1984). Finally, they set a "dummy- initial-observation" prior such that a no-change forecast for all variables is a good forecast at the beginning of the sample, consistent with cointegration (Sims, 1993).

To identify the productivity and monetary policy shocks, I adopt a relatively standard iden- tiﬁcation scheme. First, I order (the logarithm of) total factor productivity (tfpt) as ﬁrst variable, therefore assuming that it does not react contemporaneously to any other shock. Second, I order the inﬂation rate as second variable, assuming that it can react on impact to unexpected changes in productivity while reacting with a lag to any other shock. Fi- nally, the federal funds rate is ordered last, thus assuming that all the other variables of the BVAR model react with a lag to a monetary policy shock. This identiﬁcation scheme results in ordering the variables of the model as ytfltfpt; dlPt; dlGDPt; dlCONSt; dlINVt; dlW AGt; lHourst; F EDF UNDStg, where dldenotes the percentage change measured as

log diﬀerence and ldenotes the log. Finally, due to the small size of the ﬁrst two samples, I estimate the model with one lag, while for the last sample period, I choose ﬁve lags in line with Giannone et al. (2015).41

Burns-MillerPost-Volcker

0.6		Productivity
			0.1

0.4
0.2				0

0				-0.1

-0.2
0	5	10	15	20
		Output		0.2
0.5

0				0.1

-0.5				0
-1				-0.1
0	5	10	15	20

		Inflation		0.6		Productivity				Inflation
									0.1

					0.4
					0.2					0

			50th quantile	0
			16th quantile						-0.1

			84th quantile		-0.2
0	5	10	15	20
0	5	10	15	20	0	5	10	15	20
		Interest rate				Output			0.2		Interest rate
					0.5							50th	quantile

16th quantile

					0					0.1	84th quantile


					-0.5					0
0	5	10	15	20	-1					-0.1
0	5	10	15	20	0	5	10	15	20

Figure 7: Productivity shock from BVAR

The ﬁgure reports the 50th (solid) as well as the 16th and 84th percentiles (dashed) of the distribution of the impulse response functions of the BVAR to a one standard deviation productivity shock.

Figure 7 reports the 50th (solid line) as well as the 16th and 84th percentiles (dashed lines) of the distribution of the IRFs to a one standard deviation exogenous shock to total factor productivity in two sample periods. The four panels on the left report the IRFs of selected variables during the Burns-Miller period identiﬁed as a period of passive monetary policy from the estimation of the SW model.42The four panels to the right report the IRFs of the same variables for the post-Volcker period associated with a more active stance of the Federal Reserve. The variables reported in ﬁgure 7 are total factor productivity, the inﬂation rate, economic activity and the federal funds rate, while appendix F reports the IRFs of all variables included in the BVAR speciﬁcation.

The size of the estimated one standard deviation shock during the Burns-Miller period (ap- proximately 50 basis points) almost halves during the post-Volcker period (approximately 30 basis points). However, these estimates are fully in line with the size of the productivity

While the choice of one lag for the ﬁrst two subsample ensures a precise estimate of the model parameters, the results are qualitatively unchanged to the number of lags.
The results are roughly unchanged when considering the Martin subsample that is also associated with the conduct of a passive monetary policy.

shock estimated from the SW model and reported in table 2. Moreover, the comparison of the IRFs between the two subsamples shows the same diﬀerences in the propagation of the productivity shock highlighted in section 6.2. During the Burns-Miller period, productivity shocks led not only to an expansion of economic activity, but also to a persistent inﬂationaryeﬀect and a corresponding rise in the federal funds rate. On the contrary, a productivity shock under a more active monetary regime is associated with a rise in output, a temporary deﬂationaryeﬀect, and an initial drop in the federal funds rate. Strikingly similar patterns and magnitudes are reported in ﬁgure 1 that presented the Bayesian IRFs to a productivity shock in the context of the SW model.

Burns-MillerPost-Volcker

0.1		Productivity	0.2

0				0.1

				0
-0.1
				-0.1
-0.2				-0.2

0	5	10	15	20
0		Output		0.3

-0.5				0.2

-1
-1.5				0.1

-2				0

-2.5				-0.1
0	5	10	15	20

		Inflation				Productivity
					0.1			0.02
					0			0.01

								0
					-0.1
								-0.01
					-0.2			-0.02
0	5	10	15	20
0	5	10	15	20

		Interest rate		0		Output	0.3
			50th quantile

			16th quantile		-0.5
			84th quantile				0.2

					-1			0.1
					-1.5

					-2			0

0	5	10	15	20	-2.5			-0.1
0	5	10	15	20

		Inflation
0	5	10	15	20

Interest rate

		50th	quantile
		16th quantile
		84th quantile
0	5	10	15	20

Figure 8: Monetary policy shock from BVAR

The ﬁgure reports the 50th (solid) as well as the 16th and 84th percentiles (dashed) of the distribution of the impulse response functions of the BVAR to a one standard deviation monetary policy shock.

Figure 8 reports the IRFs of the same variables to a one standard deviation shock to the federal funds rate in the two subsamples. Even in this case, the size of the monetary policy shock is in line with the estimated standard deviation of the monetary policy shock in the SW model in the two samples. As reported in table 2, the mean estimate was 22 basis points for the Burns-Miller period and 12 basis points for the post-Volcker period. Moreover, the transmission of an exogenous tightening in monetary policy diﬀers substantially in terms of both magnitudes and direction between the two subsamples. During the post-Volcker period, the increase in federal funds leads to a lagged decline in both output and productivity. The dynamics of the inﬂation rate are in line with the "price puzzle" (Sims, 1992), for which an

increase in the federal funds rate predicts an initial rise in inﬂation and a lagged decline. However, the magnitudes are small and not signiﬁcant. The propagation substantially diﬀers for the Burns-Miller sample during which the exogenous monetary policy tightening leads to a persistent drop in economic activity and productivity, while inducing a signiﬁcant inﬂationary eﬀect. Even in this case, the IRFs are similar to the estimated Bayesian IRFs from the SW model and plotted in ﬁgure 3.

Conclusions

The conventional interpretation of the post-war data through the lens of a medium-scale NK model depends on the standard assumption of active monetary policy. In this framework, the run-up in inﬂation from the early 1960s until the late 1970s is exclusively attributed to mark-up shocks.

I have argued that the assumption of active monetary policy over the entire post-war period cannot be easily reconciled with the data and substantially aﬀects the identiﬁcation of the sources of U.S. business cycles. To understand the behavior of the data prior to the Volcker disinﬂation, one must allow for the possibility that the central bank operated under passive monetary policy.

I have found that the data strongly support the evidence of passive monetary policy before 1979, even when adopting the rich structure of the SW model. I have shown that non-fundamental sunspot shocks are not quantitatively relevant to explain the U.S. macroeconomic volatility prior to 1979. On the contrary, the conduct of passive monetary policy alters the transmission of fundamental shocks that propagate via the de-anchoring of self- fulﬁlling inﬂation expectations. Under this monetary regime, a sequence of positive productivity shocks in the 1960s caused inﬂationary pressures that the central bank accommodated. Once the assumption of active monetary policy is relaxed, mark-up shocks only account for the sudden rise in inﬂation during the oil crisis of the mid-1970s.

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A Appendix A

Appendix A reports the prior distributions of the structural parameters and volatility of the shocks used for the estimation of the SW model.

Table 8: Prior distribution of structural parameters

Coeﬃcient	Description	Distr.	Mean	Std. Dev
	Adjustment cost	Normal	4.00	1.50
c	IES	Normal	1.50	0.37
h	Habit Persistence	Beta	0.70	0.10
l	Labor supply elasticity	Normal	2.00	0.75
w	Wage stickiness	Beta	0.50	0.10
p	Price Stickiness	Beta	0.50	0.10
w	Wage Indexation	Beta	0.50	0.15
p	Price Indexation	Beta	0.50	0.15
	Capacity utilization elasticity	Beta	0.50	0.15
	Share of ﬁxed costs	Normal	1.25	0.12
	Share of capital	Normal	0.30	0.05
	S.S. inﬂation rate (quart.)	Gamma	0.62	0.10
100( 11)	Discount factor	Gamma	0.25	0.10
	S.S. hours worked	Normal	0.00	2.00
l
	Trend growth rate	Normal	0.40	0.10
r	Taylor rule inﬂation	Normal	1.00	0.35
ry	Taylor rule output gap	Normal	0.12	0.05
ry	Taylor rule (output gap)	Normal	0.12	0.05
	Taylor rule smoothing	Beta	0.75	0.10

The table reports the prior distribution for the structural parameters of the model.

Table 9: Prior distribution of exogenous processes

Coeﬃcient	Description	Distr.	Mean	Std. Dev
a	Technology shock	Invgamma	0.10	2.00
b	Risk premium shock	Invgamma	0.10	2.00
g	Government sp. shock	Invgamma	0.10	2.00
I	Investment-speciﬁc shock	Invgamma	0.10	2.00
r	Monetary policy shock	Invgamma	0.10	2.00
p	Price mark-up shock	Invgamma	0.10	2.00
w	Wage mark-up shock	Invgamma	0.10	2.00
	Sunspot shock	Uniform[0,1]	0.50	0.29
a	Persistence technology	Beta	0.50	0.20
b	Persistence risk premium	Beta	0.50	0.20
g	Persistence government sp.	Beta	0.50	0.20
I	Persistence investment-speciﬁc	Beta	0.50	0.20
r	Persistence monetary policy	Beta	0.50	0.20
p	Persistence price mark-up	Beta	0.50	0.20
w	Persistence wage mark-up	Beta	0.50	0.20
p	Mov. Avg. term, price mark-up	Beta	0.50	0.20
w	Mov. Avg. term, wage mark-up	Beta	0.50	0.20
ga	Cov(a; g)	Normal	0.50	0.25
p	Corr( ; p)	Uniform[-1,1]	0	0.57

The table reports the prior distribution for the exogenous processes of the model.

B Appendix B

This appendix studies the implications of the a priorirestriction about equilibrium uniqueness imposed in SW for the model estimation and presents the results of predictive checks indicating that the indeterminate speciﬁcation better ﬁts the data. As shown in table 1, the assumption is validated by the data exclusively for the post-Volcker period. On the contrary, the restriction is rejected when considering the sample prior to 1979. Table 10 reports the posterior distribution of the structural parameters estimated for each of the two local maxima found by the Metropolis-Hastings algorithm for the ﬁrst sample period (1955:4-1969:4). The table allows for a comparison with the estimation results that would be obtained by imposing the same a prioriassumption as in SW.43While most of the estimates are unchanged, relaxing the restriction implies that the Taylor rule coeﬃcient on inﬂation is estimated to be associated with a weak response of the monetary authority, therefore rejecting the assumption imposed in SW. As shown in section 6.2 and 6.3, this ﬁnding has crucial implications for the propagation of the shocks and to explain U.S. business cycle ﬂuctuations.

43Similar diﬀerences arise when using the second subsample (1970:1-1979:2) to study how the imposition of the assumption in SW would impact the results.

The comparison of the posterior estimates also highlights a higher degree of both the wage and inﬂation indexation, as well as more persistence of the price mark-up shock. This ﬁnding is in line with the intuition provided in section 3.1. A characteristic feature of indeterminate models lies in the richer endogenous persistence. Hence, when imposing the assumption of an active monetary policy, the model incurs a diﬃculty in matching the observed persistence in the data and mistakenly suggests a higher persistence than in the representation favored by the data.

Table 10: Posterior distribution of structural parameters (1955-1969)

Period: 1955:4-1969:4		Indeterminacy	Determinacy
Coeﬃcient	Description	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]
	Adjustment cost	4.75	[2.90,6.63]	5.07	[3.25,6.95]
c	IES	1.11	[0.86,1.37]	1.13	[0.78,1.46]
h	Habit Persistence	0.62	[0.49,0.74]	0.58	[0.43,0.74]
l	Labor supply elasticity	1.98	[0.89,3.07]	1.34	[0.28,2.20]
w	Wage stickiness	0.74	[0.63,0.85]	0.78	[0.69,0.86]
p	Price Stickiness	0.60	[0.50,0.67]	0.62	[0.50,0.74]
w	Wage Indexation	0.33	[0.13,0.53]	0.44	[0.21,0.67]
p	Price Indexation	0.30	[0.12,0.47]	0.45	[0.17,0.72]
	Capacity utiliz. elasticity	0.52	[0.31,0.74]	0.42	[0.20,0.63]
	Share of ﬁxed costs	1.59	[1.46,1.71]	1.65	[1.49,1.82]
	Share of capital	0.24	[0.20,0.29]	0.25	[0.19,0.29]
	S.S. inﬂation rate (quart.)	0.60	[0.44,0.76]	0.63	[0.49,0.77]
100( 11)	Discount factor	0.17	[0.06,0.26]	0.21	[0.07,0.34]
	S.S. hours worked	1.33	[0.17,2.54]	1.83	[0.60,3.01]
l
	Trend growth rate (quart.)	0.57	[0.38,0.76]	0.47	[0.33,0.59]
r	Taylor rule inﬂation	0.64	[0.35,0.99]	1.35	[0.98,1.69]
ry	Taylor rule output gap	0.13	[0.05,0.20]	0.14	[0.06,0.22]
ry	Taylor rule (output gap)	0.11	[0.07,0.15]	0.12	[0.08,0.17]
	Taylor rule smoothing	0.87	[0.81,0.94]	0.88	[0.83,0.93]

The table compares the posterior estimates of structural parameters for the pre-Volcker period under indeterminacy and determinacy.

To evaluate in more detail the ﬁt of the SW model under the determinate and indeterminate speciﬁcations, I also conduct predictive checks to examine whether the model captures relevant features of the data. The role of predictive checks in Bayesian analysis has been highlighted in Lancaster (2004), Geweke (2005), Geweke (2007), Del Negro and Schorfheide (2011), and Herbst and Schorfheide (2012), among others. In particular, I run posterior predictive checks in which the distribution of the model parameters is conditioned on the observed data. I follow four steps. First, I make 500 parameter draws from the posterior predictive distribution. Second, I simulate a trajectory for each observable in the SW model

Table 11: Posterior distribution of exogenous processes (1955-1969)

Period: 1955:4-1969:4		Indeterminacy	Determinacy
Coeﬃcient	Description	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]
a	Technology shock	0.52	[0.44,0.61]	0.52	[0.43,0.61]
b	Risk premium shock	0.19	[0.11,0.27]	0.12	[0.04,0.26]
g	Government sp. shock	0.50	[0.42,0.58]	0.50	[0.42,0.58]
I	Investment-speciﬁc shock	0.61	[0.42,0.79]	0.57	[0.39,0.75]
r	Monetary policy shock	0.11	[0.09,0.12]	0.11	[0.09,0.14]
p	Price mark-up shock	0.24	[0.20,0.28]	0.24	[0.18,0.30]
w	Wage mark-up shock	0.24	[0.18,0.28]	0.26	[0.19,0.33]
	Sunspot shock	0.14	[0.07,0.21]	-	-
a	Persistence technology	0.95	[0.92,0.99]	0.92	[0.86,0.99]
b	Persistence risk premium	0.58	[0.34,0.83]	0.79	[0.31,0.98]
g	Persistence government sp.	0.86	[0.77,0.94]	0.84	[0.73,0.95]
I	Persistence investment-speciﬁc	0.50	[0.30,0.70]	0.54	[0.33,0.74]
r	Persistence monetary policy	0.50	[0.31,0.69]	0.44	[0.27,0.60]
p	Persistence price mark-up	0.24	[0.04,0.41]	0.47	[0.12,0.93]
w	Persistence wage mark-up	0.62	[0.34,0.90]	0.47	[0.18,0.76]
p	MA price mark-up	0.64	[0.43,0.85]	0.71	[0.35,0.99]
w	MA wage mark-up	0.50	[0.25,0.74]	0.56	[0.26,0.91]
ga	Cov(a; g)	0.57	[0.38,0.76]	0.57	[0.38,0.78]
p	Corr( ; p)	0.92	[0.82,0.99]	-	-

The table compares the posterior estimates of the parameters associated with the exogenous processes for the pre-Volcker period under indeterminacy and determinacy.

of the same length as the sample period over which the model is estimated, while I drop the ﬁrst 1,000 observations. Third, I convert the simulated trajectories into sample means and standard deviations for which I then have an approximated predictive distribution. Finally, I compute the same moments from the actual data and evaluate if they are located in the tails of the predictive distributions. The rationale for such exercise is that if the moments based on the actual data lie in the tails of the distribution, then the model shows diﬃculties in explaining the observed features of the data.

In table 12, I consider the period from 1955:1 to 1969:4.44The table compares the means and standard deviations based on the data with the corresponding moments simulated from the predictive distributions under determinacy and indeterminacy. For each model speciﬁca- tion, I report the mean and the 90 percent probability intervals obtained from the predictive posterior distributions for both the mean and standard deviation of each observable of the SW model. The table conﬁrms the ﬁnding in the previous section for which the conduct of

44The results for the period from 1970:1 to 1979:2 are generally in line with the ﬁndings provided in this appendix, while the diﬀerences between model speciﬁcations are more moderate due to the shorter sample size.

Table 12: Posterior predictive checks (1955-1969)

1955:4-1969:4		Data	Determinacy	Indeterminacy
			Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]
Mean	dlGDPt	0.57	0.46	[0.25,0.68]	0.51	[0.32,0.69]
	dlPt	0.61	0.64	[-0.13,1.30]	0.48	[-0.13,1.17]
	F F Rt	0.93	1.41	[-0.28,2.92]	1.09	[0.53,1.74]
	dlCONSt	0.55	0.47	[0.33,0.63]	0.51	[0.39,0.63]
	dlINVt	0.53	0.46	[-0.10,0.99]	0.51	[0.00,0.99]
	lHourst	2.29	2.60	[0.24,4.89]	1.31	[0.14,2.44]
	dlW AGt	0.57	0.47	[0.40,0.54]	0.50	[0.43,0.58]
Standard dev.	dlGDPt	1.02	1.51	[1.00,2.71]	1.12	[0.87,1.38]
	dlPt	0.38	0.58	[0.33,1.18]	0.40	[0.25,0.63]
	F F Rt	0.42	0.90	[0.32,2.36]	0.34	[0.20,0.52]
	dlCONSt	0.76	1.14	[0.61,2.50]	0.69	[0.51,0.93]
	dlINVt	2.24	2.67	[1.74,4.18]	2.74	[1.91,3.73]
	lHourst	1.72	2.45	[1.16,5.32]	1.60	[1.01,2.30]
	dlW AGt	0.48	0.50	[0.40,0.62]	0.53	[0.42,0.65]

The table compares the mean and standard deviation based on the data with the corresponding moments simulated from the predictive distribution under determinacy and indetermi- nacy.

monetary policy was more likely to be passive over this sample period. While both model speciﬁcations generally capture the mean of the observables, the indeterminate speciﬁca- tion substantially outperforms the determinate model in explaining the standard deviations. Under the indeterminate speciﬁcation, the mean of the predictive distribution for the ob- servables' standard deviation captures the corresponding moment obtained from the data. On the contrary, for almost all the observables, the standard deviations from the data are located on the tails of the predictive posterior distributions for the determinate speciﬁcation.

Table 13 reports similar statistics for the sample period from 1984:1 to 2007:3. Over this period, the estimation of the SW model provided evidence of an active monetary policy consistent with a determinate speciﬁcation. The predictive checks presented in the table conﬁrm this ﬁnding. The mean of the observed data are generally captured by both the determinate and indeterminate speciﬁcation. However, the determinate model substantially outperforms the indeterminate version in explaining the standard deviation of the observed data. The indeterminate speciﬁcation predicts a signiﬁcantly higher volatility in the inﬂation rate, as well as the growth rate of output, investment, and wages.

Table 13: Posterior predictive checks (1984-2007)

1984:1-2007:3		Data	Determinacy	Indeterminacy
			Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]
Mean	dlGDPt	0.50	0.45	[0.40,0.51]	0.43	[0.33,0.54]
	dlPt	0.62	0.80	[0.64,0.96]	0.68	[-0.83,2.19]
	F F Rt	1.33	1.70	[1.48,1.91]	1.58	[0.17,3.04]
	dlCONSt	0.57	0.46	[0.41,0.50]	0.43	[0.34,0.53]
	dlINVt	0.48	0.46	[0.31,0.59]	0.43	[0.18,0.68]
	lHourst	1.20	1.73	[0.89,2.51]	1.02	[-1.18,3.04]
	dlW AGt	0.40	0.45	[0.39,0.52]	0.43	[0.28,0.60]
Standard dev.	dlGDPt	0.54	0.63	[0.53,0.73]	0.74	[0.61,0.87]
	dlPt	0.23	0.29	[0.22,0.37]	0.57	[0.29,1.02]
	F F Rt	0.60	0.29	[0.22,0.38]	0.60	[0.32,1.03]
	dlCONSt	0.50	0.45	[0.36,0.54]	0.55	[0.41,0.69]
	dlINVt	1.47	1.40	[1.07,1.82]	1.89	[1.44,2.44]
	lHourst	2.00	1.11	[0.77,1.55]	2.18	[1.27,3.23]
	dlW AGt	0.72	0.72	[0.58,0.89]	1.31	[0.98,1.68]

The table compares the mean and standard deviation based on the data with the corresponding moments simulated from the predictive distribution under determinacy and indetermi- nacy.

C Appendix C

Del Negro and Schorfheide (2004) modify the model in Lubik and Schorfheide (2004) to

describe an economy that evolves along a balanced growth path. In particular, total factor

productivity is assumed to follow an exogenous unit root process of the form lnAt= ln+

lnAt 1+ zt, where zt= zzt 1+ "z;tand "z;tcan be interpreted as a technology shock. The

model is represented by equations (6)(9) and consists of a dynamic IS curve

xt= Et(xt+1) (RtEt(t+1)) + (1 g)gt+ zzt;

(6)

a NK Phillips curve

t=	Et(t+1) + (xtgt) ;	(7)
r

and a Taylor rule,

Rt= RRt 1+ (1 R) [ 1t+ 2(xtzt)] + "R;t;

(8)

where r= =is the steady-state real interest rate and the demand shock, gtfollows a

univariate AR(1) processes gt=ggt 1+"g;t. The standard deviations of the fundamental shocks "g;t, "z;tand "R;tare denoted by g, zand R. The rational expectation forecast errors are deﬁned as

1;t =xt Et 1 [xt];2;t =t Et 1 [t];

(9)

and we deﬁne the vector of endogenous variables as Xt(xt; t; Rt; Et(xt+1), Et(t+1); gt; zt)0. To estimate the model, I use Bayesian techniques and consider three of the seven quarterly U.S. macroeconomic time series used in SW. Relative to the measurement equation (4) in SW, I estimate the model in (6)(9) using the following measurement equation,

2	dlGDPt	3		2	ln	3		21	0	0	0	0	0	03	(10)
6	dlPt	7	=	6	ln	7	+	60	1	0	0	0	0	07 Xt;
6		7		6		7		6						7
6F EDF UNDSt	7		6lnr +ln	7		60	0	1	0	0	0	07
6		7		6		7		6						7
4		5		4		5		4						5

where dl denotes the percentage change measured as log diﬀerence. The series considered are: the growth rate in real GDP, the inﬂation rate measured by the GDP deﬂator, and the federal funds rate. The parameter ln is the quarterly trend growth rate of real GDP, and ln and lnr are quarterly steady-state inﬂation and real interest rates expressed in percentages. I deﬁne the vectors of fundamental shocks and non-fundamental errors as,

0	0
"t = ("R;t; "g;t; "z;t);	t = (1;t;2;t)

and the vector of parameters as

= (ln; ln; lnr; 1; 2; R; ; ; g; z; g; z; R)0:

Finally, while Del Negro and Schorfheide (2004) restrict the prior distributions to impose the determinate solution, table 14 reports the same prior distributions for the model parameters that are in common with Lubik and Schorfheide (2004) who tested for indeterminacy in U.S.

monetary policy.45However, the prior distributions for the model parameters that deﬁne the balanced growth path of the model, fln; ln; lnrg, are the same as in Del Negro and Schorfheide (2004).

Table 14: Prior distributions for Del Negro and Schorfheide (2004)

Name	Range	Density	Mean	Std. Dev.	90% interval
1	R+	Gamma	1.1	0.50	[0.43,2.03]
2	R+	Gamma	0.25	0.15	[0.06,0.54]
R	[0;1)	Beta	0.50	0.20	[0.17,0.83]
ln	R+	Normal	0.5	0.25	[0.09,0.91]
ln	R+	Normal	1.00	0.50	[0.17,1.82]
lnr	R+	Gamma	0.50	0.25	[0.17,0.97]
	R+	Gamma	0.50	0.20	[0.22,0.87]
1	R+	Gamma	2.00	0.50	[1.25,2.88]
g	[0;1)	Beta	0.70	0.10	[0.52,0.85]
z	[0;1)	Beta	0.70	0.10	[0.52,0.85]
R	R+	Inverse	Gamma	0.31	0.16	[0.14,0.60]
g	R+	Inverse	Gamma	0.38	0.20	[0.17,0.74]
z	R+	Inverse	Gamma	1.00	0.52	[0.47,1.95]
	R+	Uniform	1.00	0.58	[0.10,1.90]
R	[-1,1]	Uniform	0.00	0.58	[-0.90,0.90]
g	[-1,1]	Uniform	0.00	0.58	[-0.90,0.90]
z	[-1,1]	Uniform	0.00	0.58	[-0.90,0.90]

The table reports the distributions used as priors for the model parameters.

Finally, table 15 presents the posterior distributions for the Del Negro and Schorfheide's (2004) model.

45The only diﬀerence with respect to Lubik and Schorfheide (2004) is that we use a ﬂatter prior for the parameter . While the authors set a gamma distribution with mean 0.5 and standard deviation 0.2, I use a prior that sets the standard deviation to 0.4, leaving the mean unchanged. Choosing a ﬂatter prior avoids facing an issue in the convergence of the parameter.

Table 15: Posterior distributions for Del Negro and Schorfheide (2004)

Period: 1984:1-2007:3	Determinacy	Indeterminacy
Coeﬃcient	Mean	[ 5 , 95 ]	Mean	[ 5 , 95 ]
1	1.58	[1.01,2.12]	0.55	[0.12,1.00]
2	0.44	[0.17,0.70]	0.64	[0.37,0.93]
R	0.87	[0.84,0.91]	0.63	[0.47,0.79]
ln	0.55	[0.39,0.74]	0.52	[0.35,0.68]
ln	1.01	[0.53,1.48]	0.87	[0.20,1.54]
lnr	0.58	[0.43,0.72]	0.60	[0.33,0.87]
	0.10	[0.04,0.16]	0.17	[0.03,0.28]
1	3.96	[3.02,4.90]	3.20	[2.88,4.09]
g	0.97	[0.95,0.98]	0.84	[0.77,0.91]
z	0.67	[0.53,0.82]	0.59	[0.45,0.73]
R	0.16	[0.14,0.18]	0.10	[0.08,0.12]
g	0.37	[0.20,0.53]	0.58	[0.40,0.77]
z	0.41	[0.32,0.49]	0.39	[0.31,0.47]
	-	-	0.23	[0.19,0.26]
R	-	-	-0.433	[-0.68,-0.18]
g	-	-	-0.29	[-0.60,0.01]
z	-	-	-0.44	[-0.72,-0.18]

The table compares the posterior estimates of structural parameters of the Del Negro and Schorfheide (2004) model for the period between 1984:1 and 2007:3 under determinacy and indeterminacy.

D Appendix D

Figure 9 plots the historical decomposition of the output gap for two alternative speciﬁca- tions. The left panel decomposes the output gap for the case of a failure to stabilize the economy, as shown in section 6.1. The right panel reports the decomposition that results from the assumption of equilibrium uniqueness as conducted in SW. The two plots indicate minor diﬀerences and attribute the recessions of the late 1950s to demand shocks and the contractions of the early 1970s to a combination of mark-up and demand shocks. Also, non- fundamental disturbances had almost no eﬀect on the observed ﬂuctuations in the output gap. The similarity of the decomposition should not come as a surprise. Indeed, the analysis conducted in section 6.2 shows that the transmission of the structural shocks on the output gap is roughly invariant to the of monetary policy stance, given that the diﬀerences in the magnitudes are due to the larger size of the estimated standard deviations of the shocks for the pre

Historical"Decomposi0on"2"GDP"growth Historical"Decomposi0on"2"GDP"growth"

!1#

!2#

!1#

!2#

Sunspot#

Monetary#

policy#

Mark!up#

Demand#

Produc:vity#

FigureHistorical"Decomposi0on"2"Inﬂa0on9: decomposition of the output gap (1955-1969)

Historical"Decomposi0on"2"Inﬂa0on"

Historical2.5#decomposition of the output (right). 2#

	1.5#
%"	1#

0.5# 0#

!0.5#

!1#

gap3#under indeterminacy (left) andSunspot#determinacy

2.5#

		2#			Monetary#

%"	1.5#			policy#

			Mark!up#
	1#


		0.5#			Demand#

	0#

	!0.5#			Produc:vity#


	!1#

Historical"Decomposi0on"2"GDP"growth Historical"Decomposi0on"2"GDP"growth"

!1#

!2#

!1#

!2#

Sunspot#

Monetary#

policy#

Mark!up#

Demand#

Produc:vity#

Figure 10: Historical decomposition of the output gap (1970-1979)

Historical"Decomposi0on"2"Inﬂa0on Historical"Decomposi0on"2"Inﬂa0on"

Historical2.5#decomposition of the output

(right). 2#

	1.5#
%"	1#

0.5# 0#

!0.5#

!1#

gap under indeterminacy (left) andSunspot#determinacy

2.5#

		2#		Monetary#



%"	1.5#		policy#

	1#		Mark!up#


	0.5#		Demand#

		0#

	!0.5#		Produc:vity#


	!1#

E Appendix E

Historical"Decomposi0on"2"GDP"growth"

		0.8%
I focus on the post-Volcker period and		0.6%
	0.4%
		0.2%
output gap (left panel) and the inﬂation	%"	0%
!0.2%

under an active monetary policy, as		!0.4%
	!0.6%
are in line with those in SW. The		!0.8%
	!1%
		!1.2%

dot com bubble are mostly explained

the

Historical"Decomposi0on"2"GDP"growth

Sunspot%

for the

Monetary%

policy%conducted

Mark!up%

Demand% results

Produc>vity%burst of

kept

Historical"Decomposi0on"2"Inﬂa0on"

0.8%

0.6%

0.4%

0.2% 0%

!0.2%

!0.4%

!0.6%

!0.8%

!1%

!1.2%

0.6%

0.4%

0.2%

!0.2%

!0.4%

!0.6%

Sunspot%

Monetary%

policy%

Mark!up%

Demand%

Produc>vity%

Figure 11: Historical decomposition of output gap and inﬂation rate (1984-2007)

Historical"Decomposi0on"2"Inﬂa0on"

0.6%

0.4%

0.2%

!0.2%

F Appendix F

!0.4%

!0.6%

Sunspot%

Monetary%

policy%

Mark!up%

Demand%

Produc>vity%

0.6		productivity			0.15		inflation			0.5		consumption					investment

0.5															1
				0.1

0.4					0.05					0					0

0.3															-1
					0

0.2															-2
					-0.05					-0.5
0.1

0					-0.1										-3

-0.1					-0.15					-1					-4
0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20
		output			0.4		hours			0.4		real wage			0.2		interest rate
0.5															50th	quantile

59					0.2										0.15			16th quantile
															84th quantile
									0.2

0					0										0.1

					-0.2					0					0.05
-0.5					-0.4										0
										-0.2
					-0.6										-0.05
-1					-0.8					-0.4					-0.1
0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20

Figure 12: Productivity shock from BVAR (1970-1979)

The ﬁgure reports the 50th (solid line) as well as the 16th and 84th percentiles (dashed lines) of the distribution of the impulse response functions of the BVAR to a one standard deviation productivity shock.

	0.1		productivity			0.2		inflation			0		consumption			0		investment

	0.05
	0					0.1					-0.5					-2

	-0.05
						0					-1					-4
	-0.1
	-0.15					-0.1					-1.5					-6

	-0.2
	-0.25					-0.2					-2					-8
	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20
	0		output			0		hours			0.4		real wage			0.3		interest rate

																		50th	quantile
60	-0.5										0.2							16th quantile
						-0.5										0.2		84th quantile
	-1										0

						-1					-0.2					0.1
	-1.5

						-1.5					-0.4					0
	-2

											-0.6
	-2.5					-2										-0.1
	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20

Figure 13: Monetary policy shock from BVAR (1970-1979)

0.6		productivity			0.15		inflation			0.5		consumption					investment

0.5															1
				0.1

0.4					0.05					0					0

0.3															-1
					0

0.2															-2
					-0.05					-0.5
0.1

0					-0.1										-3

-0.1					-0.15					-1					-4
0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20
		output			0.4		hours			0.4		real wage			0.2		interest rate
0.5
																50th	quantile
										0.3
61					0.2									0.15		16th quantile

									0.2							84th quantile
0					0					0.1					0.1

					-0.2					0					0.05
-0.5					-0.4					-0.1					0

										-0.2
-1					-0.6					-0.3					-0.05

					-0.8					-0.4					-0.1
0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20

Figure 14: Productivity shock from BVAR (1984-2007)

	0.1		productivity			0.02		inflation			0		consumption			0		investment

	0.05
	0					0.01					-0.1					-0.5

	-0.05
						0					-0.2					-1
	-0.1
	-0.15					-0.01					-0.3					-1.5

	-0.2
	-0.25					-0.02					-0.4					-2
	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20
	0		output			0.1		hours			0.4		real wage			0.3		interest rate

62																		50th	quantile
					0					0.2							16th quantile
	-0.1														0.2		84th quantile

						-0.1					0

	-0.2										-0.2					0.1
						-0.2

	-0.3										-0.4					0
					-0.3

											-0.6
	-0.4					-0.4										-0.1
	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20	0	5	10	15	20

Figure 15: Monetary policy shock from BVAR (1984-2007)

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Board of Governors of the Federal Reserve System published this content on 05 May 2020 and is solely responsible for the information contained therein. Distributed by Public, unedited and unaltered, on 05 May 2020 15:43:03 UTC

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