Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries

Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries Aruoba, S. Boragan, Pablo Cuba-Borda and Frank Schorfheide Please cite paper as: Aruoba, S. Boragan, Pablo Cuba-Borda and Frank Schorfheide (2016). Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries International Finance Discussion Papers 1163. http://dx.doi.org/10.17016/IFDP.2016.1163 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1163 May 2016

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1163 May 2016 Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries S. Bora˘gan Aruoba, Pablo Cuba-Borda and Frank Schorfheide NOTE:InternationalFinanceDiscussionPapersarepreliminarymaterialscirculatedtostimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. Thispapercanbedownloadedwithoutchargefrom the Social Science Research Network electronic library at www.ssrn.com.

Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries ∗ S. Bora˘gan Aruoba Pablo Cuba-Borda Frank Schorfheide University of Maryland Federal Reserve Board University of Pennsylvania CEPR and NBER First Version: September 10, 2012 Current Version: January 22, 2016 Abstract We compute a sunspot equilibrium in an estimated small-scale New Keynesian model with a zero lower bound (ZLB) constraint on nominal interest rates and a full set of stochastic fundamental shocks. In this equilibrium a sunspot shock can move the economy from a regime in which inflation is close to the central bank’s target to a regime in which the central bank misses its target, inflation rates are negative, and interest rates are close to zero with high probability. A nonlinear filter is used to examine whether the U.S. in the aftermath of the Great Recession and Japan in the late 1990s transitioned to a deflation regime. The results are somewhat sensitive to the model specification, but on balance, the answer is affirmative for Japan and negative for the U.S. JEL CLASSIFICATION: C5, E4, E5 KEY WORDS: Deflation, DSGE Models, Japan, Multiple Equilibria, Nonlinear Filtering, Nonlinear Solution Methods, Sunspots, U.S., ZLB ∗ Correspondence: B. Aruoba: Department of Economics, University of Maryland, College Park, MD 20742. Email: aruoba@econ.umd.edu. P. Cuba-Borda: Division of International Finance, Federal Reserve Board, 20th Street & Constitution Ave., NW, Washington, DC 20551. Email: pablo.a.cubaborda@frb.gov. F. Schorfheide: Department of Economics, 3718 Locust Walk, University of Pennsylvania, Philadelphia, PA 19104. Email: schorf@ssc.upenn.edu. This paper previously circulated under the title “Macroeconomic Dynamics Near the ZLB: A Tale of Two Equilibria”. We are thankful for helpful comments and suggestions from the editor, the referees, Saroj Bhattarai, Jeff Campbell, John Cochrane, Hiroshi Fujiki, Hidehiko Matsumoto, Karel Mertens, Morten Ravn, Stephanie Schmitt-Grohe, Mike Woodford, and participants of various conferences and seminars. Much of this paper was written while Aruoba and Schorfheide visited the Federal Reserve Bank of Philadelphia, whose hospitality they are thankful for. Aruoba and Schorfheide gratefullyacknowledgefinancialsupportfromtheNationalScienceFoundationundergrantsSES1061725and 1424843. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

This Version: January 22, 2016 1 1 Introduction Japan has experienced near-zero interest rates and a deflation of about -1% since the late 1990s. In the U.S. the federal funds rate dropped below 20 basis points in December 2008 and has stayed near zero in the aftermath of the Great Recession through the end of 2015. Investors’ access to money, which yields a zero nominal return, prevents interest rates from falling below zero and thereby creates a zero lower bound (ZLB) for nominal interest rates. The recent experiences of the U.S. and Japan have raised concern among policy makers about a long-lasting switch to a regime in which interest rates are zero, inflation is low, and conventional macroeconomic policies are less effective.1 For instance, in the aftermath of the Great recession the president of the Federal Reserve Bank of St. Louis, James Bullard wrote: “During this recovery, the U.S. economy is susceptible to negative shocks that may dampen inflationexpectations. Thiscouldpushtheeconomyintoanunintended,lownominalinterest rate steady state. Escape from such an outcome is problematic. [...] The United States is closer to a Japanese-style outcome today than at any time in recent history.” (Bullard (2010)) The key contribution of this paper is to provide the first formal econometric analysis of the likelihood that the U.S. and Japan have transitioned to a long-lasting zero interest rate and low inflation regime. Starting point is a standard small-scale New Keynesian dynamic stochastic general equilibrium (DSGE) model. We explicitly impose the ZLB constraint on the interest rate feedback rule. We assume that in normal times monetary policy is active in the sense that the central bank changes interest rates more than one-for-one in response to deviations of inflation from the target. Moreover, fiscal policy is assumed to be passive in the sense that the fiscal authority uses lump-sum taxes to balance the government budget constraint in every period. It is well known, that in such an environment there are two steady states. In the targeted-inflation steady state, inflation equals the value targeted by the central bank and nominal interest rates are strictly positive. In the second steady state, 1Ueda(2012)providesaverythoroughreviewofthepoliciesusedintheU.S.andJapanandheconcludes that“theJapaneseeconomyseemstobetrappedinan‘equilibrium’wherebyonlyexogenousforcesgenerate movements to a better equilibrium with a higher rate of inflation.”

This Version: January 22, 2016 2 the deflation steady state, nominal interest rates are zero and inflation rates are negative.2 To assess the likelihood of a transition to a deflation regime, we construct a stochastic equilibrium, in which agents react to fundamental shocks (e.g., technology shocks, demand shocks, monetary policy shocks) as well as to a Markov-switching sunspot shock that can move the economy between a targeted inflation regime and a deflation regime. This equilibrium offers two potential explanations for the recent experience in the U.S. and Japan: the economies may have been pushed to the ZLB either by a sequence of adverse fundamental shocks in the targeted-inflation regime or by a switch to the deflation regime. The type of explanation has important implications, not just for the central bank’s ability to stimulate the economy using conventional interest rate policies, but also for the effectiveness of fiscal policy, as documented in Mertens and Ravn (2014). Aggregating results from different model specifications, we find that Japan shifted from the targeted-inflation regime into the deflation regime in 1999 and remained there until the end of our sample. The U.S., in contrast, remained in the targeted-inflation regime throughout its ZLB episode, with the possible exception of the first part of 2009, where the evidence is more mixed. We use a first-order perturbation approximation of the targeted-inflation regime to estimate the model parameters based on U.S. and Japanese data on output, consumptionoutput ratio, inflation, and interest rates. Our estimation sample is chosen such that the observations pre-date the episodes of zero nominal interest rates and are consistent with the targeted-inflation regime. We estimate four model specifications that differ in terms of observables and interest rate feedback rule. Once the parameter estimates are obtained, we work directly with the nonlinear sunspot equilibrium. At a technical level, our paper is the first paper to use global projection methods to compute a sunspot equilibrium for a DSGE model with a full set of stochastic shocks that can be used to track macroeconomic time series. Identifying the regime (targeted-inflation versus deflation) is not as easy as computing 2Thesecondsteadystateisoftencalledundesirable. However,inthecontextofastandardNewKeynesian DSGE model with an explicit money demand motive the steady state is not necessarily bad in terms of welfare. While negative inflation rates in conjunction with a cost of adjusting nominal prices lead to an output loss, the zero interest rate implies that the welfare losses arising from the opportunity costs of holding real money balances are eliminated. We leave a careful normative analysis to future work.

This Version: January 22, 2016 3 average inflation and interest rates and comparing them to the corresponding steady state values. Thus, for each model specification we use a nonlinear filter to extract the sequence of shocks that can explain the data. Most importantly, we obtain estimates of the probability that the economies were in either the targeted-inflation or the deflation regime. Due to the sequential nature of our filter conditional on parameter estimates, the results of the filter provide a quasi-real-time assessment of the state of the economy. We find that for each country three out of the four specifications agree, but there is considerable uncertainty to reach a definitive conclusion. Finally, we aggregate the results based on quasi posterior model probabilities that we obtain using predictive likelihoods for the set of observables that is common across specifications. Our paper is related to three strands of the literature: sunspots and multiplicity of equilibria in New Keynesian DSGE models; global projection methods for the solution of DSGE models; and the use of particle filters to extract hidden states in nonlinear state-space models. TherelevanceofsunspotsineconomicmodelswasfirstdiscussedinCassandShell(1983), who define sunspots as “extrinsic uncertainty, that is, random phenomena that do not affect tastes, endowments, or production possibilities.” Sunspot shocks can affect economic outcomes in environments in which there does not exist a unique equilibrium. A review of the sunspot literature in macroeconomics is provided by Benhabib and Farmer (1999). Subsequently, there has been extensive research on multiplicity of equilibria in New Keynesian DSGE models generated by so-called passive monetary policy rules that do not respond strongly enough to inflation deviations from target. Such policy rules are associated with undetermined local fluctuations in the neighborhood of the targeted-inflation steady state. An econometric analysis of this type of multiplicity is provided by Lubik and Schorfheide (2004). If monetary policy is active instead of passive then the local dynamics near the targetedinflationsteady stateareunique (subjectto thecaveatsemphasizedin Cochrane(2011)), but a second steady state arises from the kink in the monetary policy rule induced by the ZLB. In this second steady state nominal interest rates are zero and inflation rates are negative.

This Version: January 22, 2016 4 Because in the neighborhood of this second steady state the central bank is unable to lower interest rates in response to a drop in inflation, the local dynamics are indeterminate. Hirose (2014) estimates a linearized New Keynesian DSGE model by imposing that the economy is permanently at the ZLB and parameterizing the multiplicity of solutions as in Lubik and Schorfheide (2004). Benhabib et al. (2001a,b) were the first to construct equilibria in which the economy transitions from the targeted-inflation steady state toward the deflation steady state. Recently, Armenter (2014) generalizes their results to a model in which monetary policy is not represented by a Taylor rule, but it is optimally chosen to maximize social welfare. It should be clear from the above discussion that the model considered in our analysis has many equilibria. This opens the door for two research strategies: (i) characterize as many equilibria as possible and then examine which of these equilibria is consistent with the data. (ii) Choose one particular equilibrium and condition the empirical analysis on that equilibrium. The papers by Lubik and Schorfheide (2004) (studying inflation and interest rate dynamics pre and post Volcker disinflation) and Cochrane (2015) (studying inflation and interest rate dynamics during a ZLB episode and a subsequent exit from the ZLB) consider linearized DSGE models and are examples of the first approach. Our paper pursues the second avenue: we consider a particular equilibrium within which we can address the question whether an economy has transitioned into a long-lasting deflation regime. While thereareotherequilibriathatallowforsimilartransitions, yetmightexhibitdifferentregimeconditional dynamics, at present it is computationally not feasible to enumerate. Thus, we focus our empirical analysis on an interesting equilibrium for which we do have a solution. Conditional on being in the targeted-inflation regime, the dynamics are very similar to the dynamics that arise in the targeted-inflation equilibrium, that is studied in, for instance, Maliar and Maliar (2015), Ferna´ndez-Villaverde et al. (2015), and Gust et al. (2012). A sunspot equilibrium similar to ours has been recently analyzed by Mertens and Ravn (2014), but in a model with a much more restrictive exogenous shock structure. Related, Schmitt-Groh´e and Uribe (2015) study an equilibrium in which confidence shocks, which resemble a change in regimes in our model, combined with downward nominal wage rigidity

This Version: January 22, 2016 5 can deliver jobless recoveries near the ZLB in a mostly analytical analysis. More recently, Piazza (2015) shows that different monetary policy rules may change the path from the targeted-inflation steady state to the deflation steady state and demonstrates in a calibrated model that a combination of a perfect-foresight sunspot shock and a shock to the growth rate of the economy can generate dynamics similar to the Japanese experience. Our paper is the first to compute a sunspot equilibrium in a New Keynesian DSGE model that is rich enough to track macroeconomic time series and to use a filter to extract the evolution of the hidden sunspot shock. In terms of solution method, our work is most closely related to the papers by Judd et al. (2010), Maliar and Maliar (2015), Ferna´ndez-Villaverde et al. (2015), and Gust et al. (2012).3 All of these papers use global projection methods to approximate agents’ decision rules in a New Keynesian DSGE model with a ZLB constraint. However, these papers solely consider an equilibrium in which the economy is always in the targeted-inflation regime – what we could call a targeted-inflation equilibrium – and some important details of the implementation of the solution algorithm are different. To improve the accuracy of the model solution, we introduce two novel features. First, we use a piece-wise smooth approximation with two separate functions characterizing the decisions when the ZLB is binding and when it is not. This means all our decision rules allow for kinks at points in the state space where the ZLB becomes binding. Second, when constructing a grid of points in the model’s state space for which the equilibrium conditions are explicitly evaluated by the projection approach, we combine draws from the ergodic distribution of the DSGE model with values of the state variables obtained by applying our filtering procedure. This modification of the ergodic-set method proposed by Judd et al. (2010) ensures that the model solution is accurate in a region of the state space that is unlikely ex ante under the ergodic distribution of the model, but very important ex post to 3Most of the other papers that study DSGE models with a ZLB constraint take various shortcuts to solve the model. In particular, following Eggertsson and Woodford (2003), many authors assume that an exogenous Markov-switching process pushes the economy to the ZLB. The subsequent exit from the ZLB is exogenous and occurs with a prespecified probability. The absence of other shocks makes it impossible to use the model to track actual data. Unfortunately, model properties tend to be very sensitive to the approximation technique and to implicit or explicit assumptions about the probability of leaving the ZLB, see Braun et al. (2012) and Fern´andez-Villaverde et al. (2015).

This Version: January 22, 2016 6 explain the observed data. With respect to the empirical analysis, the only other papers that combine a projection solution with a nonlinear filter to track U.S. data throughout the Great Recession period are Gust et al. (2012) and Cuba-Borda (2014). Both papers restrict their attention to the targeted-inflation equilibrium. The first focuses on parameter estimation using post- 2008 data in a New Keynesian model like ours and examine the extent to which the ZLB constrained the ability of monetary policy to stabilize the economy. The latter extracts fundamental shocks to account for the decline in economic activity during the U.S. Great Recession in a medium-scale model with investment. Our paper is the first to fit a nonlinear DSGE model with an explicit ZLB constraint to Japanese data. The remainder of the paper is organized as follows. Section 2 presents a simple twoequationmodelthatweusetoillustratethemultiplicityofequilibriainmonetarymodelswith ZLBconstraints. Wealsohighlighttheparticularequilibriumstudiedinthispaper. TheNew Keynesian model that is used for the quantitative analysis is presented in Section 3, and the solution of the model is discussed in Section 4. Section 5 describes the parameter estimates for the different model specifications in this paper and illustrates the dynamic properties of one of the estimated model specifications. Section 6 presents our main results regarding the identification of the sunspot regime in each country. Section 7 concludes. Detailed derivations, descriptions of algorithms, and additional quantitative results are summarized in an Online Appendix. 2 A Two-Equation Example We begin with a simple two-equation example to characterize the sunspot equilibrium that we will study in the remainder of this paper in the context of a New Keynesian DSGE model with an interest-rate feedback rule and the ZLB constraint. Suppose that the economy can be described by a consumption Euler equation of the form (cid:20) (cid:21) R 1 = E M t (1) t t+1 π t+1

This Version: January 22, 2016 7 and the monetary policy rule (cid:40) (cid:41) (cid:18) π (cid:19)ψ t R = max 1, rπ , ψ > 1. (2) t ∗ π ∗ In the fully-specified DSGE model introduced in Section 3 below, the stochastic discount factor M that appears in (1) is given by t+1 d (cid:18) C (cid:19)−τ t+1 t+1 M = β , t+1 d C t t where C is consumption, d is a discount factor shock, and 1/τ is the intertemporal elasticity t t of substitution. We define r = 1/M , (3) t t where r can be interpreted as the real rate of return for a one-period asset. To keep the t example simple, we assume that r exogenous and follows a stationary AR(1) process t (cid:16)r (cid:17) (cid:16)r (cid:17) t+1 t log = ρlog +σ(cid:15) , (cid:15) ∼ iidN(0,1) and ρ ∈ (0,1). (4) t+1 t+1 r r The parameter r corresponds to the steady state (σ = 0) of the real interest rate. We assume that the gross nominal interest rate is bounded from below by one, which is captured by the max operator in (2). Throughout the paper we refer to this bound as the ZLB because the net interest rate cannot fall below zero. The parameter π in the monetary policy rule ∗ represents the central bank’s target inflation. Loglinearizing around π = π , r = r and R = rπ and using hats to denote percentage t ∗ t t ∗ deviations from this point, yields the system R ˆ = E [rˆ +πˆ ] (5) t t t+1 t+1 ˆ R = max{−log(rπ ),ψπˆ } (6) t ∗ t rˆ = ρrˆ +σε (7) t+1 t t+1 (5) is a version of the Fisher equation, which relates the nominal interest rate to the expected

This Version: January 22, 2016 8 real interest rate and inflation. A similar system of equations arises from the log-linearized equilibrium conditions of many monetary DSGE models. Combining (5) and (6) and using E [rˆ ] = ρrˆ yields the following expectational difference equation for inflation t t+1 t E [πˆ ] = max (cid:8) −log(rπ )−ρrˆ,ψπˆ −ρrˆ (cid:9) . (8) t t+1 ∗ t t t Just as the original system comprising (1) and (2), the linearized difference equation (8) has two steady states. In the targeted-inflation steady state inflation equals π , and the ∗ ˆ nominal interest rate is R = rπ , so that πˆ = R = 0. In the deflation steady state, ∗ ∗ ˆ πˆ = R = −log(rπ ) and thus inflation equals 1/r, and the nominal interest rate is at the ∗ ZLB. The presence of two steady states suggests that the rational expectations difference equation (8) also has multiple stochastic solutions. We find solutions to this equation using a guess-and-verify approach (see Online Appendix for details). Suppose that we conjecture πˆ = θ +θ rˆ. (9) t 0 1 t It can be verified that a solution that fluctuates around the targeted-inflation steady state (henceforth targeted-inflation equilibrium) is given by4 ρ θ∗ = 0, θ∗ = > 0. (10) 0 1 ψ −ρ Becausearoundthetargeted-inflationsteadystatenominalinterestratesrespondtoinflation more than one-for-one, the local dynamics are unique. Wecanalsoobtainasolutionthatfluctuatesaroundthedeflationsteadystate(henceforth deflation equilibrium): θD = −log(rπ ), θD = −1. (11) 0 ∗ 1 Because around the deflation steady state nominal interest rates do not respond to inflation, the local dynamics are indeterminate and one could construct other solutions (see, 4It is assumed that the exogenous movements in r are sufficiently small such that (ψθ ∗ −ρ)rˆ ≥ (cid:98)t 1 t −log(rπ ) for all t. ∗

This Version: January 22, 2016 9 Figure 1: Inflation Dynamics in the Two-Equation Model Targeted-Inflation and Sunspot Equilibrium Deflation Equilibria 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 100 200 300 400 500 100 200 300 400 500 Notes: The figure shows annualized net inflation rate, 400logπ . In the left panel, the blue line shows t the targeted-inflation equilibrium, and the red line shows the deflation equilibrium. In the right panel, the shaded area corresponds to periods in which the system is in the deflation regime, s =0. t for instance, Lubik and Schorfheide (2004) as well as the Online Appendix) that may involve a sunspot shock ζ with the property that E [ζ ] = 0 or the dependence of inflation t t−1 t on rˆ . However, we will restrict our attention to (11). Note that (10) and (11) have t−1 drastically different dynamics: inflation and real interest rates have a positive correlation in the targeted-inflation equilibrium, while this correlation switches signs in the deflation equilibrium. In the remainder of the paper we will focus on an equilibrium in which a two-state Markov-switching sunspot shock s ∈ {0,1} triggers transitions from a targeted-inflation t regime to a deflation regime and vice versa: πˆ(s) = θ (s )+θ (s )rˆ. (12) t 0 t 1 t t where θ (s ) and θ (s ) denote the regime-specific intercept and slope of the linear decision 0 t 1 t rule. Throughout this paper, we assume that the sunspot process s evolves independently t

This Version: January 22, 2016 10 Table 1: Decision Rule Coefficients Targeted-Inflation Equilibrium θ∗ = 0 θ∗ = 1.5 0 1 Deflation Equilibrium θD = −0.01 θD = −1 0 1 Sunspot Equilibrium θ (1) = −0.0002 θ (1) = 1.4611 0 1 θ (0) = −0.0105 θ (0) = −1.1295 0 1 from the fundamental shocks.5 If the regimes are persistent, then the intercepts and slopes are similar in magnitude (but not identical) to the coefficients in (10) and (11), respectively. The precise values depend on the transition probabilities of the Markov switching process and ensure that (8) holds in every period t. A numerical illustration is provided in Figure 1. We set π = 1.005, ψ = 1.5, r = 1.005, ∗ σ = 0.0007, ρ = 0.9, p = 0.99 and p = 0.95. The implied decision rule coefficients are 11 00 summarizedinTable1. TheleftpanelofFigure1comparesthepathsofannualizednetinflation (400logπ ) under the targeted-inflation equilibrium (10) and the deflation equilibrium t (11). The inflation paths are shifted by the difference between 400logπ and 400log(1/r), ∗ which is 4%, and display perfect negative correlation. The right panel shows the sunspot equilibrium with visible shifts from the targeted-inflation regime to the deflation regime (shaded areas) and back. We close this section with the following remarks: (i) The linearized two-equation model has many stochastic equilibria. (ii) We will focus on a particular sunspot equilibrium that is interesting for our empirical analysis because it can capture long lasting transitions into and out of a regime in which interest rates are zero and inflation rates are low. (iii) Our empirical analysis will be based on a small-scale DSGE model rather than the two-equation model presented in this section. (iv) We will not log-linearize the equilibrium conditions of the DSGE model, instead we will work with the nonlinear equilibrium conditions. (v) Unlike in the simple two-equation example we will not assume that interest rates are always strictly greater than zero in the targeted-inflation regime and always equal to zero in the 5For toy models we were able to construct equilibria in which the Markov transition is triggered by (cid:15) . t But we were unable to numerically construct such solutions for the DSGE model presented in Section 3.

This Version: January 22, 2016 11 deflation regime. Instead, our nonlinear decision rules imply that interest rates could be zero in the targeted-inflation regime and they could be strictly positive in the deflation regime. Likewise, in both regimes it is possible to observe both positive and negative inflation rates. 3 A Prototypical New Keynesian DSGE Model Our quantitative analysis will be based on a small-scale New Keynesian DSGE model. Variantsofthismodelhavebeenwidelystudiedintheliteratureanditspropertiesarediscussedin detail in Woodford (2003). The model economy consists of perfectly competitive final-goodsproducing firms, a continuum of monopolistically competitive intermediate goods producers, a continuum of identical households, and a government that engages in monetary and fiscal policy. To keep the dimension of the state space manageable, we abstract from capital accumulation and wage rigidities. We describe the preferences and technologies of the agents in Section 3.1, and summarize the equilibrium conditions in Section 3.2. 3.1 Preferences and Technologies Households. Households derive utility from consumption C relative to an exogenous habit t stock and disutility from hours worked H . We assume that the habit stock is given by the t leveloftechnologyA , whichensuresthattheeconomyevolvesalongabalancedgrowthpath. t We also assume that the households value transaction services from real money balances, detrended by A , and include them in the utility function. The households maximize t (cid:34) (cid:32) (cid:33)(cid:35) (cid:88) ∞ (C /A )1−τ −1 H1+1/η (cid:18) M (cid:19) E βsd t+s t+s −χ t+s +χ V t+s , (13) t t+s H M 1−τ 1+1/η P A t+s t+s s=0 subject to the budget constraint P C +T +M +B = P W H +M +R B +P D +P SC . t t t t t t t t t−1 t−1 t−1 t t t t

This Version: January 22, 2016 12 Here β is the discount factor, d is a shock to the discount factor, 1/τ is the intertemporal t elasticity of substitution, η is the Frisch labor supply elasticity, and P is the price of the final t good. The shock d captures frictions that affect intertemporal preferences in a reduced-form t way. Fluctuations in d affect households patience and their desire to postpone consumption. t As we demonstrate below, and as is commonly exploited in the literature, a sufficiently large shock to d makes the central bank cut interest rates all the way to the ZLB. The households t supply labor services to the firms in a perfectly competitive labor market, taking the real wage W as given. At the end of period t, households hold money in the amount of M . They t t have access to a bond market where nominal government bonds B that pay gross interest R t t are traded. Furthermore, the households receive profits D from the firms and pay lump-sum t taxes T . SC is the net cash inflow from trading a full set of state-contingent securities. t t Detrended real money balances M /(P A ) enter the utility function in an additively t t t separablefashion. AnempiricaljustificationofthisassumptionisprovidedbyIreland(2004). As a consequence, the equilibrium has a block diagonal structure under the interest-rate feedback rule that we will specify below: the level of output, inflation, and interest rates can be determined independently of the money stock. We assume that the marginal utility V(cid:48)(m) is decreasing in real money balances m and reaches zero for m = m¯, which is the amount of money held in steady state by households if the net nominal interest rate is zero. Since the return on holding money is zero, it provides the rationale for the ZLB on nominal rates. More specifically since households can hold as well as issue debt at the market rate R , their problem does not have a solution when R < 1. The ZLB ensures the existence of t t a monetary equilibrium. Firms. The final-goods producers aggregate intermediate goods, indexed by j ∈ [0,1], using the technology: (cid:18)(cid:90) 1 (cid:19) 1− 1 ν Y = Y (j)1−νdj . t t 0 The firms take input prices P (j) and output prices P as given. Profit maximization implies t t that the demand for inputs is given by (cid:18) P (j) (cid:19)−1/ν t Y (j) = Y . t t P t

This Version: January 22, 2016 13 Undertheassumptionoffreeentryintothefinal-goodsmarket,profitsarezeroinequilibrium, and the price of the aggregate good is given by (cid:18)(cid:90) 1 (cid:19) ν− ν 1 ν−1 P = P (j) dj . (14) t t ν 0 We define inflation as π = P /P . t t t−1 Intermediate good j is produced by a monopolist who has access to the following production technology: Y (j) = A H (j), (15) t t t where A is an exogenous productivity process that is common to all firms and H (j) is the t t firm-specific labor input. Intermediate-goods-producing firms face quadratic price adjustment costs of the form φ (cid:18) P (j) (cid:19)2 t AC (j) = −π¯ Y (j), t t 2 P (j) t−1 where φ governs the price stickiness in the economy and π¯ is a baseline rate of price change that does not require the payment of any adjustment costs. In our quantitative analysis, we set π¯ = π , where π is the target inflation rate of the central bank, which in turn is ∗ ∗ the steady state inflation rate in the targeted-inflation equilibrium. Firm j chooses its labor input H (j) and the price P (j) to maximize the present value of future profits t t (cid:34) (cid:35) ∞ (cid:18) (cid:19) (cid:88) P (j) E βsQ t+s Y (j)−W H (j)−AC (j) . (16) t t+s|t t+s t+s t+s t+s P t+s s=0 Here, Q is the time t value to the household of a unit of the consumption good in period t+s|t t+s, which is treated as exogenous by the firm. Government Policies. Monetary policy is described by an interest rate feedback rule. Because the ZLB constraint is an important part of our analysis we introduce it explicitly as follows: R = max{1, R∗e(cid:15)R,t}. (17) t t Here R∗ is the systematic part of monetary policy which reacts to the current state of the t

This Version: January 22, 2016 14 economy and (cid:15) is a monetary policy shock. We consider two specifications for R∗, which R,t t we refer to as growth and gap specifications. The growth specification takes the form (cid:34) (cid:18) π (cid:19)ψ1 (cid:18) Y (cid:19)ψ2 (cid:35)1−ρR Growth : R∗ = rπ t t RρR . (18) t ∗ π γY t−1 ∗ t−1 Here r is the steady-state real interest rate and π is the target-inflation rate. Provided that ∗ the ZLB is not binding, the central bank reacts to deviations of inflation from the target rate π and deviations of output growth from its long-run value γ. ∗ Under the gap specification, the central bank reacts to a measure of the output gap in addition to inflation deviations from target: (cid:34) (cid:18) π (cid:19)ψ1 (cid:18) Y (cid:19)ψ2 (cid:35)1−ρR Gap : R∗ = rπ t t RρR , (19) t ∗ π Y∗ t−1 ∗ t where Y∗ is the target level of output. In theoretical studies the targeted level of output t often corresponds to the level of output in the absence of nominal rigidities and mark-up shocks, because from an optimal policy perspective, this is the level of output around which the central bank should stabilize fluctuations. However, historically, at least in the U.S., the central bank has tried to keep output close to the official measure of potential output, which is well approximated by a slow-moving trend. Thus we use exponential smoothing to construct Y∗ directly from historical output data. It is given by t logY∗ = αlogY∗ +(1−α)logY +αlogγ. (20) t t−1 t where α is a parameter we calibrate such that logY∗ tracks an official measure of potential t output. We use two alternative policy rules in an attempt to capture the dynamics of R∗, which t is in principle latent when the economy is at the ZLB. For instance, the U.S. experienced a large negative rate of output growth in 2008:Q4. Under the growth rule, this creates a large drop in R∗, but the drop is short-lived because output growth subsequently recovers. t Under the gap rule, the reduction in R∗ is more persistent, because the level of output stays t

This Version: January 22, 2016 15 below its historical average for a long period of time. Our analysis is sensitive to the desired interest rate, because R∗ determines how constrained the central bank is by the ZLB and t how likely it is that it will leave the ZLB in the subsequent quarters. The government consumes a stochastic fraction of aggregate output. We assume that government spending evolves according to (cid:18) (cid:19) 1 G = 1− Y . (21) t t g t The government levies a lump-sum tax T (or provides a subsidy if T is negative) to finance t t any shortfalls in government revenues (or to rebate any surplus). Its budget constraint is given by P G +M +R B = T +M +B . (22) t t t−1 t−1 t−1 t t t Exogenous shocks. The model economy is perturbed by four (fundamental) exogenous processes. Aggregate productivity evolves according to logA = logγ +logA +logz , where logz = ρ logz +σ (cid:15) . (23) t t−1 t t z t−1 z z,t Thus, on average, the economy grows at the rate γ, and z generates exogenous stationary t fluctuations of the technology growth rate around this long-run trend. We assume that the government spending shock follows the AR(1) law of motion logg = (1−ρ )logg +ρ logg +σ (cid:15) . (24) t g ∗ g t−1 g g,t While we formally introduce the exogenous process g as a government spending shock, we t interpret it more broadly as an exogenous demand shock that contributes to fluctuations in output. (21), (23) and (24) imply that log output and government spending are cointegrated and that the log government spending-output ratio is stationary. The shock to the discount factor evolves according to logd = ρ logd +σ (cid:15) (25) t d t−1 d d,t The monetary policy shock (cid:15) is assumed to be serially uncorrelated. We stack the four R,t

This Version: January 22, 2016 16 innovations into the vector (cid:15) = [(cid:15) ,(cid:15) ,(cid:15) ,(cid:15) ](cid:48) and assume that (cid:15) ∼ iidN(0,I). t z,t g,t d,t r,t t In addition to the fundamental shock processes, agents in the model economy observe an exogenous sunspot shock s , which follows a two-state Markov-switching process t   (1−p ) if s = 0 P{s = 1} = 00 t−1 . (26) t  p if s = 1 11 t−1 3.2 Equilibrium Conditions Because the exogenous productivity process has a stochastic trend, it is convenient to characterize the equilibrium conditions of the model economy in terms of detrended consumption c ≡ C /A and detrended output y ≡ Y /A . We write the consumption Euler equation t t t t t t (sometimes called the IS equation) as c−τ = βR E , (27) t t t where (cid:20) d c−τ (cid:21) E = E t+1 t+1 . (28) t t d γz π t t+1 t+1 The solution algorithm approximates the conditional expectation E using a Chebychev polyt nomial in terms of the state variables. In a symmetric equilibrium, in which all firms set the same price P (j), the price-setting decision of the firms leads to the condition t (cid:20) (cid:21) d φβE t+1 c−τ y (π −π¯)π (29) t d t+1 t+1 t+1 t+1 t (cid:26) (cid:20)(cid:18) (cid:19) (cid:21) (cid:27) 1 (cid:16) (cid:17) 1 π¯ = c−τy 1−χ cτy1/η +φ(π −π¯) 1− π + −1 . t t ν h t t t 2ν t 2ν A log-linearization of (29) leads to the standard New Keynesian Phillips curve. We show in the Online Appendix that the aggregate resource constraint can be expressed as (cid:20) (cid:21) 1 φ c = − (π −π¯)2 y . (30) t t t g 2 t

This Version: January 22, 2016 17 It reflects both government spending as well as the resource cost (in terms of output) caused by price changes. Finally, we reproduce the monetary policy rule    (cid:34) (cid:18) π (cid:19)ψ1 (cid:18) y (cid:19)ψ2 (cid:35)1−ρR  Growth: R = max 1, rπ t t z RρR eσR(cid:15)R,t , t ∗ π y t t−1  ∗ t−1  (31)    (cid:34) (cid:18) π (cid:19)ψ1 (cid:18) y (cid:19)αψ2 (cid:35)1−ρR  Gap: R = max 1, rπ t t z RρR eσR(cid:15)R,t . t ∗ π y∗ t t−1  ∗ t−1  wherey∗ ≡ Y∗/A . Wedonotuseameasureofmoneyinourempiricalanalysisandtherefore t t t drop the equilibrium condition that determines money demand. As the two-equation model in Section 2, the New Keynesian model with the ZLB constraint has two steady states, which we refer to as the targeted-inflation and the deflation steady states. In the targeted-inflation steady state, inflation equals π and the gross inter- ∗ est rate equals rπ , while in the deflation steady state, inflation equals 1/r and the interest ∗ rate is at the ZLB. Subsequently, we will focus on a stochastic sunspot equilibrium with a targeted-inflation regime (s = 1) and a deflation regime (s = 0). t t 4 Solution Algorithm We now discuss some key features of the algorithm that is used to solve the nonlinear DSGE model presented in the previous section. Additional details can be found in the Online Appendix. We utilize a global approximation method following Judd (1992) where the decision rules are approximated by combinations of Chebyshev polynomials. The minimum set of state variables associated with our DSGE model is S = (R ,y ,d ,g ,z ,(cid:15) ,s ) (32) t t−1 t−1 t t t R,t t for the growth specifications and S = (R ,y∗ ,d ,g ,z ,(cid:15) ,s ). (33) t t−1 t−1 t t t R,t t

This Version: January 22, 2016 18 for gap specifications, and d is only relevant in the version with the discount factor shock. t We included the regime-switching process s into the state vector because our goal is to t characterize a sunspot equilibrium. An (approximate) solution of the DSGE model is a set of decision rules π = π(S ;Θ), E = E(S ;Θ), c = c(S ;Θ), y = y(S ;Θ), and R = R(S ;Θ) t t t t t t t t t t that solve the nonlinear rational expectations system given by (27) to (31), and the laws of motionoftheexogenousprocesses. Notethatconditionalonπ(S ;Θ)andE(S ;Θ),Equations t t (27), (30) and (31) directly determine c(S ;Θ), y(S ;Θ), and R(S ;Θ). Thus, we only use t t t Chebyshev polynomials to approximate π(S ;Θ) and E(S ;Θ). In our notation the coefficient t t vectorΘ ≡ {θ },i = 1,...,N,parameterizesallofthedecisionrulesandN isthetotalnumber i of coefficients. The solution algorithm amounts to specifying a grid of points G = {S ,...,S } in the 1 M model’s state space and determining Θ by minimizing the (unweighted) sum of squared residuals associated with (28) and (29). Because (28) and (29) are functions of S , we are t evaluating the residuals for each S ∈ G and then sum the M squared residuals. There are t two non-standard aspects of our solution method that we will now discuss in more detail: (i) the piecewise smooth representation of the functions π(·;Θ) and E(·;Θ) and (ii) our iterative procedure of choosing grid points G. Piece-wise Smooth Decision Rules. The max operator in the monetary policy rule potentially introduces kinks in the decision rules π(S ) and E(S ). While Chebyshev polyt t nomials, which are smooth functions of the states, can in principle approximate functions with a kink, such approximations are quite inaccurate for low-order polynomials. Thus, unlike Judd et al. (2010), Fern´andez-Villaverde et al. (2015), and Gust et al. (2012), we use a piece-wise smooth approximation of the functions π(S ) and E(S ) by postulating t t   f1(S ;Θ) if s = 1 and R(S ) > 1   π t t t        f π 2(S t ;Θ) if s t = 1 and R(S t ) = 1 π(S ;Θ) = (34) t  f3(S ;Θ) if s = 0 and R(S ) > 1   π t t t       f4(S ;Θ) if s = 0 and R(S ) = 1  π t t t

This Version: January 22, 2016 19 Figure 2: Sample Decision Rules Interest Rate In.ation 3 4 2 2 1 0 0 -6 -4 -2 0 -6 -4 -2 0 g^ g^ Output Consumption 1 2.5 0 2 -1 1.5 -2 1 -3 -4 0.5 -6 -4 -2 0 -6 -4 -2 0 g^ g^ Note: This figure depicts the decision rules for 3vGrowth using parameter values estimated for the U.S. as described in Section 5.2. The x-axis corresponds to the state variable g , in percentage deviations from its t steady state. The other state variables are fixed: s =1, R =1, and y , z , and (cid:15) set to their means t t−1 t−1 t R,t conditional on s =1. t and similarly for E(S ,Θ), where the functions fi(·) are linear combinations of a complete t j set of Chebyshev polynomials up to fourth order. In our experience, the flexibility of the piece-wise smooth approximation yields more accurate decision rules, especially for inflation. Figure 2 shows a slice of the decision rules. We vary g over a wide range where the ZLB is both slack and binding. To generate the t figure we condition on s = 1, R = 1 and set y , z , and (cid:15) to their means conditional t t−1 t−1 t R,t

This Version: January 22, 2016 20 on s = 1. The monetary policy rule has kink due to the ZLB, while the decision rule for t inflation has an apparent kink due to the piece-wise smooth approximation in (34). The decision rules for output and consumption inherit the kinks in the decision rule for inflation (and E(S )) and in the monetary policy rule. The kinks, especially the ones in the decision t rules for inflation and consumption, are very severe. For instance, if the ZLB is binding, consumption is increasing in gˆ. If the ZLB is non-binding consumption falls as gˆ rises. As a consequence, a smooth approximation obtained from a single Chebyshev polynomial would do a very poor job capturing the actual decision rules.6 Choice of Grid Points. With regard to the choice of grid points, projection methods that require the solution to be accurate on a fixed grid, e.g., a tensor product grid, become exceedingly difficult to implement as the number of state variables increases above three. While the Smolyak grid proposed by Kru¨ger and Kubler (2004) can alleviate the curse of dimensionalitytosomeextent, webuildonrecentworkbyJuddetal.(2010), whichproposed to simulate the model to be solved, to distinguish clusters on the simulated series, and to use the clusters’ centers as a grid for projections.7 We modify their methodology significantly by combining simulated grid points with states obtained from the data using a nonlinear filter. Doing so is necessary to capture the behavior of the model in low probability regions of the state space that are important for our analysis. For example, when s = 1 (s = 0), negative t t (positive) inflation is typically outside the ergodic distribution of the model. Because the set of simulated grid points that represent the ergodic distribution and the filtered states both depend on the solution of the model, some iteration of solution, on the one hand, and simulation and filtering, on the other hand, is required. For a given solution we simulate the model and get a set of points that characterize the ergodic distribution. We then run a particle filter, details of which are provided in the Online Appendix, to obtain 6Inanearlierversionofthepaperweindeedsolvedthemodelbothwaysandillustratedthatthesmooth approximationleadstoapproximationerrorsthatareanorderofmagnitudelargerrelativetothepiece-wise smooth approximation. 7The work by Judd, Maliar, and Maliar evolved considerably over time. We initially built on the working paper version, Judd et al. (2010). In the published version of the paper, Maliar and Maliar (2015), also consider (cid:15)-distinguishable (EDS) grids and locally-adaptive EDS grids. Their locally-adaptive grids are similar in spirit to our approach, which tries to control accuracy in a region of the state space that is important for the substantive analysis, even if it is far in the tails of the ergodic distribution.

This Version: January 22, 2016 21 the grid points which are consistent with data. We repeat this until we obtain a stable grid, which typically happens after three to five iterations. We parameterize each fi(·) in (34) for i = 1,...,4 and j = π,E with 210 parameters for j a total of 1,680 elements in Θ and use M = 880 including the grid points from the ergodic distribution and the filtered states. For a given set of filtered states and simulated grid, the solution takes about six minutes on a single-core Windows-based computer using MATLAB where some computationally-intensive parts of the code are run using Fortran via mex files. The approximation errors are in the order of 10−4 on average, expressed in consumption units. 5 Model Estimation and Dynamics The data sets used in the empirical analysis are described in Section 5.1. In Section 5.2, we estimate the parameters of the DSGE model for the U.S. and Japan using data from before the economies reached the ZLB. These parameter estimates are the starting point for the subsequent analysis. We consider four different specifications for each country that differ in terms of number of observable variables used and the details of the monetary policy rule. We solve the model using the nonlinear methods outlined in the previous section and in Section 5.3, we illustrate the dynamic properties of one of the estimated models by focusing on the economy’s ergodic distributions and by presenting regime-specific impulse responses. 5.1 Data Thesubsequentempiricalanalysisisbasedonlogofrealper-capitaGDP,thelogconsumptionoutput ratio, GDP deflator inflation, and interest rates for the U.S. and Japan. The U.S. interest rate is the federal funds rate and for Japan we use the Bank of Japan’s uncollateralized call rate. Consumption for the U.S. is the real personal consumption expenditures and real private consumption for Japan, where we normalize by an appropriate population measure to convert to per-capita terms. Further details about the data are provided in the Online Appendix.

This Version: January 22, 2016 22 The time series are plotted in Figure 3. The U.S. sample starts in 1984:Q1, after the Great Moderation and ends in 2015:Q2. The time series for Japan range from 1981:Q1 to 2015:Q1. The vertical lines denote the end of the estimation sample for each country, 2007:Q4 for the U.S. and 1994:Q4 for Japan. For the U.S. the fourth quarter of 2007 marks the beginning of the Great Recession, which was followed with a long-lasting spell of zero interest rate starting in 2009. In Japan, short-term interest rates dropped below 50 basis points in 1995:Q4 and have stayed at or near zero ever since. An important feature of the ZLB episode for Japan is consistently negative inflation rates – average inflation for Japan from 1999:Q1 to the end of the sample is nearly −1%.8 This is in stark contrast with the U.S., which experienced only two quarters of mildly negative inflation (2009:Q2 and Q3) and two quarters of inflation less than 0.5% at the very end of the sample. These features of the data are important (though not sufficient) for the identification of the sunspot regimes. 5.2 Model Estimation For both the U.S. and Japan we estimate four versions of the DSGE model that differ in terms the monetary policy rules (growth vs. gap as in (31)) and the variables included in the estimation. For both countries, the first data set (three variables, henceforth 3v) comprises thelogofoutput,inflation,andinterestrates. Theseconddataset(fourvariables,henceforth 4v) also includes the log consumption-output ratio. For the U.S. the 3v data set is the standard data set for the estimation of small-scale DSGE models in the literature before the Great Recession, with the minor difference that we use the level of output instead of output growth. In this version we treat consumption as a latent variable and switch off the discount factor shock. Thus, the model is driven by technology growth, government spending (aggregate demand) and monetary policy shocks, which, again, is the typical specification for these models before the Great Recession. To estimate the DSGE model on the 4v data set, we activate the discount factor shock, which 8The three positive spikes for Japanese GDP deflator inflation in 1997:Q2, 2008:Q4 and 2014:Q2 are unusual events that are not visible in, for instance, CPI inflation. The first and the third spike are due to increases in the value-added tax and the second is when a large decline in oil prices leads to a decrease in the import deflator which in turn generated a large jump in the GDP deflator. In our subsequent analysis we treat these observations as missing observations.

This Version: January 22, 2016 23 Figure 3: Data U.S. 1984-2015 Japan 1981-2015 Output Output 11 10.6 10.8 10.4 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 Consumption/Output Consumption/Output −42 −52 −44 −54 −56 −46 −58 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 Inflation(%) Inflation(%) 6 4 4 2 2 0 −2 0 −4 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 NominalInterestRate(%) NominalInterestRate(%) 10 5 5 0 0 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 Note: Output is the natural logarithm of per-capita output, consumption-output ratio is also in natural logarithm, scaled by 100, and inflation and nominal interest rate are in annualized percentage units. The vertical red line in each figure show the end of the estimation sample. is widely used in the ZLB literature to drive model economies to the ZLB. Using a variant of consumption as an observable is natural, because the discount rate shock influences consumption directly. During the estimation periods, it is clear that the data favor the targeted-inflation regime because both inflation and the nominal interest rate are positive. Moreover, we verify that the values of the state variables that are needed to rationalize the observations fall into a region of the state space in which the decision rules of the nonlinear model are well approximated by the decision rules obtained from a first-order perturbation solution of the DSGE model that ignores the ZLB. The first-order perturbation solution can be computed much faster and is numerically more stable than the global approximation to the sunspot equilibrium discussed in Section 4. We use a standard random walk Metropolis-Hastings

This Version: January 22, 2016 24 (RWMH) algorithm to estimate the log-linearized DSGE models over the pre-ZLB sample periods. The implementation of the posterior sampler follows An and Schorfheide (2007) and is described in the Online Appendix.9 We fix a subset of the parameters prior to the estimation. First, we want our model’s average inflation conditional on being in the targeted-inflation regime to equal the average inflation in the estimation sample in each country. The former depends not only on π ∗ but also on the values for the sunspot transition parameters p and p . The latter two 11 00 determine the expected durations of staying in each regime and therefore influence the longruninflationexpectations. Welooselycalibratethesethreeparameterstomatchthefollowing three observations: (i) average inflation conditional on s = 1 equals average inflation in the t estimation sample; (ii) long-run inflation expectations when s = 1 are only slightly lower t than average inflation; and (iii) when s transitions from one to zero, inflation expectations t fall by about 1% in Japan and about 20 basis points in the U. S. Observations (ii) and (iii) are somewhat crude, obtained from long-run inflation expectations for Japan and the U.S. at the start of their ZLB experiences.10 This procedure yields p = 0.95 and p = 0.99 00 11 for the U.S. and p = 0.92 and p = 0.99 for Japan. These values make the deflation 00 11 regime (s = 0) less persistent than the targeted-inflation regime (s = 1) and imply that the t t unconditional probability of being in the deflation regime (s = 0) is 0.17 for the U.S. and t 0.11 for Japan. Note that we identify the regime probabilities from the change in inflation expectations instead of the relative duration of the ZLB spell, which would be very sensitive to the start date of the estimation sample. The π values we use for each specification / ∗ country are tabulated in the Online Appendix. We choose values for γ and β such that the steady state of the model matches the average output growth, and interest rates over the estimation sample period. The steady state government expenditure-to-output ratio is determined from national accounts data. 9The only somewhat nonstandard aspect of our methodology is the initialization of the Kalman filter to handle the nonstationarity in the log level of output. 10This calibration involves fixing these three parameters, estimating the linear model to get values for the remaining parameters, solving the full nonlinear model to calculate the long-run averages of inflation andinflationexpectationsviasimulationsanditeratinguntilwefindareasonablefit. Thelong-runinflation expectations are computed using the Consensus Forecast in Japan and taken from Aruoba (2014) for the U.S. (see Online Appendix).

This Version: January 22, 2016 25 Becauseoursampledoesnotincludeobservationsonlabormarketvariables, wefixtheFrisch labor supply elasticity. Based on R´ıos-Rull et al. (2012), who provide a detailed discussion of parameter values that are appropriate for DSGE models of U.S. data, we set η = 0.72 for the U.S. Our value for Japan is based on Kuroda and Yamamoto (2008) who use micro-level data to estimate labor supply elasticities along the intensive and extensive margin for males and females. The authors report a range of values which we aggregate into η = 0.85. The parameter ν, which captures the elasticity of substitution between intermediate goods, is set to 0.1. It is not separately identifiable from the price adjustment cost parameter φ. Finally, we calibrate the smoothing parameter α for trend output in (20) to make the implied trend output close to a measure from the data. For the U.S. we use the output gap measure produced by the Congressional Budget Office, and for Japan we use the potential growth rate from the Bank of Japan to construct an output gap measure. For each country we estimate four DSGE model specifications: 3vGrowth, 4vGrowth, 3vGap, and 4vGap. The marginal prior distributions for τ, κ, ψ , ψ and the parameters of 1 2 the exogenous shock processes are tabulated in the Online Appendix. For the inverse IES τ we use Gamma distributions with mean 2 and standard deviations of 0.25 (U.S.) and 0.5 (Japan). We re-parametrize the price adjustment cost parameter φ in terms of the implied slope of the linearized New Keynesian Phillips curve: κ = τ(1−ν)/(νπ2φ). Our prior for κ ∗ has a mean of 0.3 and a standard deviation of 0.1, encompassing fairly flat and fairly steep Phillips curves. Our benchmark priors for the policy rule coefficients ψ and ψ are centered 1 2 at 1.5 and 0.5, respectively, with standard deviations of 0.3 and 0.25, respectively. For the 3variable specifications, the likelihood function was fairly uninformative about the policy rule coefficients.11 Thus, wereplacedthebenchmarkpriordistributionsforψ andψ withtighter 1 2 prior distributions. For the 3vGap specifications the modified priors are centered at the parameter values obtained from the estimation of the corresponding 4-variable specification. We truncate the prior distribution at the boundary of the determinacy region associated with the linearized version of the DSGE model. Thus, we are essentially imposing the existence of a second steady state (which requires that ψ > 1) when we are estimating 1 11This problem is well recognized in the literature; see, for instance, Cochrane (2011) for a theoretical appraisal and Mavroeidis (2010) for identification-robust inference in single-equation estimation settings.

This Version: January 22, 2016 26 the model. In view of the empirical results in Lubik and Schorfheide (2004) who estimate a similar model without imposing determinacy on post-1982 data and find no evidence in favor of ψ < 1, the ψ > 1 restriction strikes us as reasonable. 1 1 The resulting posterior estimates reported in Table 2 are in line with the estimates reported elsewhere in the literature. Most notable are the implicit estimates of the slope of the New Keynesian Phillips curve, which are around 0.3 for the U.S. and 0.5 for Japan, implying fairly flexible prices and relatively small real effects of unanticipated interest rate changes.12 The posterior distributions for most of the estimated parameters move somewhat significantly away from their priors, or at least they get much tighter. A notable exception is the elasticity of intertemporal substitution parameter τ for the U.S., which remains near the prior mean of 2. 5.3 Equilibrium Dynamics In this section we discuss the dynamics of the estimated specifications. Since we have a total of eight estimated specifications, we focus on the 4vGrowth specification for the U.S. All models behave qualitatively similarly, though sometimes there are quantitative differences, which we point out when relevant. We start with an illustration of the ergodic distribution by simulating a long sequence of observations. Given our choices of p and p for the U.S., 00 11 approximately 17% of the observations are associated with the deflation regime, whereas the remaining 83% are associated with the targeted-inflation regime. Figure 4 depicts contour plots for the joint probability density function of inflation and interest rates conditional on the regimes s = 0 and s = 1, respectively. Formally, we show p(R ,π |s = j) for j = 0,1, t t t t t whichmeansthatthetwosetsofcontoursarenotweightedbytheunconditionalprobabilities P{s = j}. In the contour plots each line represents one percentile with the outermost line t showing the 99th percentile. Under the deflation regime there is a high probability that the interest rate is equal to zero, which leads to a point mass on the x-axis and is not reflected in the contour plot. 12A survey of DSGE-model-based New Keynesian Phillips curve is provided in Schorfheide (2008). Our estimates fall within the range of the estimates obtained in the literature.

This Version: January 22, 2016 27 Figure 4: Regime-Conditional Ergodic Distribution: 4vGrowth, U.S. Data Inflation (%) )%( etaR lanimoN 0.99 0.95 0.7 0.5 0.3 0.1 0.2 0.4 0.6 0.8 0 0.9 .9 9 −10 −5 0 5 10 21 01 8 6 4 2 0 0.99 0.95 0.9 0.6 0.7 0.8 −10 −5 0 5 10 21 01 8 6 4 2 0 Notes: Figure depicts the joint probability density function (kernel density estimate) of annualized net inflationandinterestratesconditionalonthetargeted-inflationregimeandthedeflationregime,respectively. Formally, the two sets of contours correspond to p(R ,π |s =j) for j =0,1. t t t

This Version: January 22, 2016 28 sretemaraP ledoM EGSD fo roiretsoP :2 elbaT napaJ SU paGv4 paGv3 htworGv4 htworGv3 paGv4 paGv3 htworGv4 htworGv3 retemaraP )84.2,01.1( 37.1 )97.1,97.0( 32.1 )48.1,17.0( 12.1 )54.1,95.0( 69.0 )86.2,09.1( 82.2 )74.2,66.1( 30.2 )15.2,27.1( 01.2 )33.2,65.1( 29.1 τ )85.0,32.0( 93.0 )87.0,73.0( 55.0 )96.0,03.0( 84.0 )07.0,92.0( 84.0 )44.0,41.0( 72.0 )05.0,81.0( 23.0 )44.0,02.0( 13.0 )83.0,71.0( 62.0 κ )01.2,63.1( 96.1 ∗)57.1,95.1( ∗76.1 )22.2,34.1( 08.1 ∗)66.1,91.1( ∗34.1 )99.2,71.2( 55.2 ∗)85.2,24.2( ∗05.2 )71.3,12.2( 76.2 ∗)65.1,83.1( ∗74.1 ψ 1 )22.0,90.0( 51.0 ∗)71.0,41.0( ∗51.0 )36.0,42.0( 24.0 )46.0,82.0( 44.0 )44.0,62.0( 53.0 ∗)73.0,43.0( ∗63.0 )42.1,16.0( 09.0 ∗)28.0,97.0( ∗08.0 ψ 2 )58.0,07.0( 87.0 )08.0,46.0( 37.0 )28.0,16.0( 37.0 )57.0,94.0( 46.0 )48.0,77.0( 18.0 )87.0,76.0( 37.0 )48.0,67.0( 08.0 )37.0,16.0( 76.0 ρ r )79.0,68.0( 29.0 )09.0,18.0( 68.0 )89.0,09.0( 49.0 )19.0,08.0( 68.0 )39.0,28.0( 78.0 )28.0,57.0( 97.0 )79.0,88.0( 39.0 )09.0,48.0( 78.0 ρ g )92.0,40.0( 41.0 )31.0,20.0( 70.0 )21.0,20.0( 60.0 )80.0,10.0( 30.0 )65.0,71.0( 63.0 )71.0,30.0( 90.0 )92.0,50.0( 61.0 )02.0,30.0( 01.0 ρ z )59.0,58.0( 09.0 )59.0,48.0( 09.0 )79.0,29.0( 49.0 )69.0,09.0( 39.0 ρ d )32.0,51.0( 91.0 )62.0,71.0( 12.0 )82.0,61.0( 12.0 )23.0,81.0( 42.0 )61.0,21.0( 41.0 )32.0,61.0( 91.0 )91.0,41.0( 61.0 )62.0,81.0( 22.0 σ001 r )09.0,66.0( 77.0 )78.1,01.1( 34.1 )88.0,46.0( 57.0 )43.1,86.0( 69.0 )35.0,24.0( 74.0 )62.1,49.0( 90.1 )25.0,14.0( 64.0 )37.0,15.0( 16.0 σ001 g )82.1,39.0( 90.1 )32.1,38.0( 10.1 )92.1,49.0( 01.1 )54.1,30.1( 22.1 )54.0,33.0( 93.0 )86.0,84.0( 75.0 )25.0,14.0( 64.0 )47.0,65.0( 46.0 σ001 z )59.1,39.0( 73.1 )85.1,17.0( 01.1 )48.3,66.1( 74.2 )06.2,03.1( 48.1 σ001 d elbiderc %09 dna snaem roiretsop troper eW .napaJ rof 4Q:4991-1Q:1891 dna .S.U eht rof 4Q:7002-1Q:4891 era selpmas noitamitse ehT :setoN HMWR a morf sward 000,05 tsal eht no desab era stluser llA .sesehtnerap ni )noitubirtsid roiretsop eht fo elitnecrep ht59 dna ht5( slavretni rethgit a htiw retemarap siht rof roirp kramhcneb eht decalper ew taht etacidni ∗ a htiw seirtnE .sward 000,05 tsrfi eht gnidracsid retfa ,mhtirogla .sliated rehtruf rof xidneppA enilnO eeS .roirp

This Version: January 22, 2016 29 As expected, the two regime-conditional distributions are approximately centered near the respective steady state values. Average inflation when s = 1 is slightly above π (2.5% t ∗ versus 2.4%) and average inflation conditional on s = 0 is below inflation at the deflation t steady state (−4.2% versus −2.8%). Under the targeted-inflation regime, inflation is positive with probability 99.7%. The probability of reaching the ZLB given s = 1 is virtually zero t given the shock processes estimated based on the pre-Great-Recession sample. This means that rationalizing the post-2008 U.S. experience with the targeted-inflation regime requires large shocks that are unlikely in view of the pre-2008 data. Under the deflation regime, on the other hand, interest rates are zero with 89% probability – even in the absence of extreme shocks – and inflation rates are negative with 97.6% probability.13 To better understand how the economy evolves in each regime, we compute impulse response functions (IRFs) to one standard deviation shocks conditional on remaining in the same regime throughout the response. Prior to the shock the economy is assumed to be at the mean of the regime-conditional ergodic distribution. The IRFs are plotted in Figure 5. Each column corresponds to one of the variables of interest (output, consumption, inflation, and interest rate) and each row corresponds to one of the structural innovations ((cid:15) , (cid:15) , z,t g,t (cid:15) , and (cid:15) ). d,t R,t The responses conditional on s = 1 are standard. The shock to technology raises output t and consumption permanently. Because it is a supply shock, prices and quantities move in opposite directions. The reaction to the positive output growth dominates in the monetary policy rule and therefore the interest rate rises. The government spending shock acts like an aggregate demand shock, increasing output and inflation temporarily. In response the central bank raises interest rates. Because nominal interest rates rise more strongly than inflation, the real interest rate increases, which reduces consumption. To understand the response to the discount factor shock innovation (cid:15) , recall that the d,t stochastic discount factor M is a function of βd /d . In log-linear terms, an unantict+1 t+1 t ipated rise in d ˆ implies that E [d ˆ − d ˆ ] = (ρ − 1)d ˆ is negative, because d ˆ follows an t t t+1 t d t t ˆ AR(1) process. Thus, a positive d shock makes the households less patient. This induces an t 13Because average inflation in the estimation sample and hence π is lower in Japan, it is more likely to ∗ observe deflation when s =1 but virtually impossible to observe positive inflation when s =0. t t

This Version: January 22, 2016 30 Figure 5: Impulse Response Functions: 4vGrowth, U.S. Data Growth Model: 4US4, 1std Output (%) Consumption (%) In.ation (%) Interest Rate (%) 0.55 0.55 0.2 0 0 z 0.5 0.5 -0.1 0.1 0.45 0.45 -0.2 -0.3 0 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 0.2 0.2 0 0.2 0 g 0 0 -0.1 0.1 -0.2 0 -0.2 -0.2 -0.3 -0.1 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 0.1 0.1 1 0 0 0.4 0 d -0.1 -0.1 0 0.2 -0.2 -0.2 0 -0.3 -0.3 -1 -0.2 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 0 0 0.3 0 0 0.2 R-0.05 -0.05 -0.2 0.1 -0.4 -0.1 -0.1 -0.6 0 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 Notes: The figure depicts impulse response functions to one-standard deviation shocks conditional. The economy is assumed to be at the mean of the regime-conditional distribution when the shocks hit and to stay in the regime in the remaining periods. Solid lines depict the responses for s = 1 and dashed lines t show the responses for s =0. For output and consumption the figure shows percentage deviations from the t baseline path. For the interest rate and inflation it shows differences in annualized percentages relative to the baseline path. increase in consumption and output, and an associated rise in inflation. The central bank reacts to these by increasing the interest rate, dampening the effect of the discount factor shock. The discount factor can be interpreted as an aggregate demand shock in the sense that it generates positive comovement between output and inflation. Unlike an expansionary g shock, however, the d shock raises consumption. t t A shock to monetary policy that increases the interest rate has the usual effects: output and inflation fall and, because the real interest rate rises, consumption falls as well. According to our estimates, the degree of price stickiness is relatively small and therefore the

This Version: January 22, 2016 31 New Keynesian Phillips curve is relatively steep. Thus, the real effect of an unanticipated monetary policy shock is small (output and consumption drop by about 10 basis points) in comparison to the inflation response (annualized inflation falls by about 40 basis points). The IRFs conditonal on the s = 0 regime display some important differences. In this t case, a positive technology shock increases inflation slightly. On the other hand, positive government spending and discount factor shocks reduce inflation. Thus, the signs of the inflationresponsesswitch, comparedtothes = 1regime. Thisresultislinkedtothefindings t of Eggertsson (2011) and Mertens and Ravn (2014), who show that positive demand shocks may lead to a negative comovement of prices and output in the deflation regime.14 The sign switchingfortheinflationresponsetothediscountfactorshockisthesamephenomenonthat we demonstrated in Section 2 for the simple model, in which inflation responds positively to a real-rate shock in the s = 1 regime but negatively in the s = 0 regime. We also t t observe in Figure 5 that consumption falls instead of rises in response to a discount factor shock because of the decline in the real interest rate. Finally, monetary policy is much less effective in the deflation regime. 6 Evidence of a Sunspot Switch We are now ready to address our main empirical question: did the U.S. and Japan experience a change in regimes due to a switch in the sunpot variable at or near the beginning of their ZLB episodes? We do this in multiple steps. First, we examine the evidence that individual pairs of inflation and interest rate observations provide about the prevailing sunspot regime in Section 6.1. Next, we use a nonlinear filter in Section 6.2 to track the sunspot regime over time, which brings in information from other variables and allows for dynamics to matter. The analysis up to this point is using all four specifications for each country. Finally, we aggregate the filtering results across the different model specifications in Section 6.3. 14Morespecifically,MertensandRavn(2014)showthattheEEcurve,whichplotsinflationversusoutput using the relationship in (27) with necessary substitutions, has two segments, one downward sloping and one upward sloping. If the equilibrium is in the upward-sloping portion, then a positive demand shock may generate a decrease in inflation while increasing output.

This Version: January 22, 2016 32 6.1 Static Analysis: Evidence from Inflation and Interest Rate Observations Our goal in this section is to conduct inference on the hidden process s . In Figure 4 t we showed the bivariate ergodic distribution of (π ,R ) for the targeted-inflation and the t t deflation regimes. Glancing at Figure 4, it seems clear that an observation of a 3% inflation and a 6% interest rate is strong evidence in favor of s = 1. Conversely, zero interest rates t combined with an inflation rate of -5% provides evidence for s = 0. However, if the interest t rate is zero and inflation is low, as it has been the case for the U.S. since 2009, it is more difficult to determine by visual inspection which regime is favored by the data. The heatmap in Figure 6 shows p(R ,π |s = 1)P{s = 1} P{s = 1|R ,π } = t t t t . (35) t t t p(R ,π |s = 1)P{s = 1}+p(R ,π |s = 0)P{s = 0} t t t t t t t t for a section of the (π ,R ) space, for which the evidence about the sunspot shock based t t on the contour plot in Figure 4 is ambiguous.15 Suppose interest rates are around 25 basis points. Then an inflation rate of more than 1.5% would be interpreted as some evidence for s = 1 (indicated by the warm colors), whereas an inflation rate below 0.75% would be t evidence in favor of s = 0 (indicated by dark blue). t Because our main interest is to infer the sunspot regime during the respective ZLB episodes of the U.S. and Japan, we want to zoom in to the bottom part of the heatmap figure. Thus, we now compute P{s = 1|ZLB,π }, where we interpret interest rates in the t t range from 0% to 0.25% as the ZLB being binding. Results are depicted in Figure 7. Unlike in the heatmap we now apply a kernel smoother to approximate the probabilities. In this figure vertical lines correspond to the inflation value of a ZLB observation for the country. Not surprisingly, all probabilities start at zero for low inflation observations and are increasing as inflation increases. The probabilities can be compared to three thresholds: 0.5, P{s = 1}, and P{s = 1|ZLB}. The first threshold is natural to construct a point t t 15To generate the heatmap we define bins for inflation and interest rates and count the number of realizations within each bin based on a long simulation from the model. The probability that s = 1 in a bin t simply is the fraction of s=1 observations in that bin.

This Version: January 22, 2016 33 Figure 6: P{s = 1|π ,R }: 4vGrowth, U.S. Data t t t Notes: The legend for the colors is to the right of the heatmap. estimator of s that is restricted to the set {0,1}. As soon as the posterior probability of s t t exceeds 0.5, the point estimate (under a 0-1 loss function) is sˆ = 1. The second threshold t is the prior probability of being in the targeted-inflation regime, which is 0.83 for the U.S. and 0.89 for Japan. If the inflation and interest rate pair exceeds the second threshold, then the data provide additional evidence in favor of the targeted-inflation regime. If the third threshold is exceeded, then the inflation observation increases the evidence against the deflation regime conditional on the economy being at the ZLB. The first two thresholds are shown by horizontal lines in Figure 7. The third threshold conditions on being at the ZLB. The probabilities P{s = 1|ZLB} are close to zero for all specifications and are not shown. t As we saw in Figure 3, inflation rates in the U.S. were mostly positive and inflation rates in Japan were mostly negative during the ZLB period. The plots in Figure 7 suggest, ignoring the fact that evidence from multiple observations should be aggregated, that the Japanese inflation and interest rate data imply evidence in favor of the deflation regime: at the observed π ’s the posterior probabilities of s = 1 are very close to zero, substantially t t

This Version: January 22, 2016 34 Figure 7: P{s = 1|ZLB,π } t t U.S. - Gap Japan - Gap 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -1 0 1 2 -1 0 1 2 In.ation (%) In.ation (%) U.S. - Growth Japan - Growth 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -1 0 1 2 -1 0 1 2 In.ation (%) In.ation (%) Notes: For the purposes of this figure ZLB is defined as interest rate being between 0% and 0.25%. In each panel, solid lines show the 4-variable specification and dashed lines show the 3-variable specification. Each vertical line shows the inflation value for a data point in the ZLB sample for the country. The horizontal dashed line shows the country-specific threshold P{s = 1}, which is 0.83 for the U.S. and 0.89 for Japan. t The horizontal solid line is at 0.5. below any of the three thresholds, with the exception of the 4vGap specification for Japan, which clears the third threshold but not the other two. For the U.S. the conclusion depends on the model specification and the threshold used. Using 0.5 as the cutoff for a point estimate of s that is restricted to zero or one, the growth t specifications imply that most of the observations favor the targeted-inflation regime, while the gap specifications favor the deflation regime. Relative to the prior distribution P{s = 1} t the evidence in almost all of the inflation and interest rate observations leads to a downward

This Version: January 22, 2016 35 revision of the probability that the economy is in the targeted-inflation regime. However, this downward revision is not as strong as in the case of Japan because U.S. inflation rates remained mostly positive. While individual inflation and interest rate observation provide some evidence about the regime, this evidence does not suffice to determine whether the U.S. or Japan did transition to the deflation regime. First, the economy evolves dynamically and the probability of being in one regime or another depends not only on the observed variables but also on the state of the economy, including the history of s . Second, variables other than inflation may contain t key information that may help distinguish the two regimes – this is evident from Figure 6 by the wide yellow-colored region where the probability of being in the two regimes are about the same. Third, the four different specifications may and do disagree. In the next two sections we tackle these issues to obtain a single and clear answer to the question of which regime the two countries were in their ZLB episodes. 6.2 Dynamic Analysis: Evidence from a Nonlinear Filter We now use a nonlinear filter to conduct inference about the hidden state s . The filter t addresses two of the above-mentioned shortcomings of the static analysis: it accounts for the state of the economy in period t − 1 and it also uses information from output and the consumption-output ratio (4-variable specifications). The DSGE model has a nonlinear state-space representation of the form yo = Ψ(x )+ν t t t x = F (x ,(cid:15) ) (36) t st t−1 t   (1−p ) if s = 0 P{s = 1} = 00 t−1 t  p if s = 1 11 t−1 Here yo is the vector of observables. We use the o superscript to distinguish the vector t of observables from detrended output in our DSGE model. For the 3-variable specifications it consists of log of output, inflation, and nominal interest rates. For the 4-variable

This Version: January 22, 2016 36 specifications the vector also includes the log consumption-output ratio. Yo is the se- 1:t quence {yo,...,yo}. The vector x stacks the continuous state variables, which are given 1 t t by x = [R ,y ,y∗,y ,d ,z ,g ,A ](cid:48), and s ∈ {0,1} is the Markov-switching process, where t t t t t−1 t t t t t y is only necessary in the growth specifications, y∗ is only necessary in the gap specifit−1 t cations, and d is only relevant for the 4-variable specifications.16 The lower case output t variables in the state vector are detrended by the level of technology A . The first equation t in (36) is the measurement equation, where ν ∼ N(0,Σ ) is a vector of measurement errors. t ν The second equation corresponds to the law of motion of the continuous state variables. The vector (cid:15) ∼ N(0,I) stacks the innovations (cid:15) , (cid:15) , (cid:15) , and (cid:15) , where once again (cid:15) is t d,t z,t g,t R,t d,t used only in the 4-variable specifications. The functions F (·) and F (·) are generated by 0 1 the model solution procedure described in Section 4. The third equation represents the law of motion of the Markov-switching process. Given the system in (36) and conditioning on the posterior mean estimates obtained in Section 5.2, we use a sequential Monte Carlo filter (also known as the particle filter) to extract estimates of the sunspot shock process s , and the latent state x .17 Because the filter t t is sequential, the results of this filter can be thought of as a quasi-real-time assessment of the probability of a sunspot switch.18 Figure 8 depicts the filtered probabilities P{s = 1|Yo }. t 1:t As in Figure 7, we plot the prior P{s = 1} as a dashed horizontal line in each panel. t Using the simple rule by which P{s = 1|Yo } > P{s = 1} is interpreted as evidence in t 1:t t favor of s = 1, we find that three out of four specifications for the U.S. indicate that the t economy stayed in the targeted-inflation regime after 2008, although the conclusions for 4vGrowth is somewhat less strong. For Japan, we draw the opposite conclusions. Three out of four specifications suggest that Japan transitioned to the deflation regime in the late 1990s. The exceptions are the 4vGap specification for the U.S. and 4vGrowth specification for Japan. It is interesting to note that across the four specifications for the U.S. there is some uncertainty which vindicates Bullard (2010)’s concern of the possibility of a shift to 16The econometric state variables x of the state-space representation are slightly different from the t economic state variables S that appear in the solution. t 17This filter is described in the Online Appendix. A more detailed exposition is provided in Herbst and Schorfheide (2015). 18We use the qualifier “quasi” because the data we use is not the real-time data but what is available as of the date we write the paper.

This Version: January 22, 2016 37 Figure 8: Filtered Probability of Targeted-Inflation Regime U.S. - Gap Japan - Gap 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2004 2006 2008 2010 2012 2014 1995 2000 2005 2010 2015 U.S. - Growth Japan - Growth 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2004 2006 2008 2010 2012 2014 1995 2000 2005 2010 2015 Notes: The figure shows the filtered probabilities P{s =1|Yo } for each specification and country, starting t 1:t five years prior to the start of the ZLB episode for the country. In each panel, solid lines show the 4variable specification and dashed lines show the 3-variable specification. The dashed horizontal line shows the country-specific threshold P{s =1}, which is 0.83 for the U.S. and 0.89 for Japan. The solid horizontal t line shows 0.5. the deflationary regime. Thefilteralsogeneratesestimatesoftheexogenousshockprocessesandtheirinnovations. The subsequent discussion focuses on the 4vGrowth specification for the U.S. and the 4vGap specification for Japan. We will see in Section 6.3 that the inference about s from these t two specifications is by and large consistent with the conclusions drawn after aggregation across the four specifications for each country. Time series plots for the filtered innovations (cid:15) are provided in Figure 9. Recall that in our model logC /Y ≈ −logg (in a first-order t t t t approximation of the aggregate resource constraint the relationship holds exactly). Thus,

This Version: January 22, 2016 38 the government spending shock by construction tracks the consumption-output ratio. In the last quarter of 2008, the U.S. experienced a large drop in output, which turned out to be permanent. In our model, this is captured by a negative technology growth shock of roughly 3.5 standard deviations. In addition, the aggregate demand shock g dropped by about 2 t standard deviations and the discount factor innovation is also negative in 2008:Q4. All three adverse shocks generate a drop in interest rates (see Figure 5) and push the economy toward the ZLB. While the two adverse demand shocks are deflationary, the adverse technology shock is inflationary. This is consistent with the modest decline in inflation.19 After 2008:Q4, the technology growth shocks stay slightly negative on average, depressing output growth and preventing a quick and full recovery. The discount factor shock innovations also remain on average negative, delaying the mean reversion of the d process t and keeping the economy near the ZLB. Moreover, the filtered sequence of monetary policy shocks is mostly negative. In the absence of these shocks interest rates would have been between 0.75% and 1%. Thus, from the perspective of the DSGE model, U.S. monetary policy is more expansionary in the aftermath of the Great Recession than what is implied by the systematic part of the interest rate feedback rule. Given the long-lasting drop in output, one might expect the Phillips curve relationship in the DSGE model to imply a significant deflation, which did not occur in the U.S.. In this regard, our simple DSGE models works similarly as the richer DSGE model studied in Del Negro et al. (2015). One important reason for why the New Keynesian Phillips curve embedded in the DSGE model does not predict deflation is that the Phillips curve is forward looking. Inflation depends on the sum of discounted expected future marginal costs. Because the model has a fairly strong mean reversion, it predicts marginal costs to rise in the medium run, whichallowsthemodeltoexplainwhattheliteraturehastermedthe“missingdeflation” in the U.S. Moreover, the expansionary monetary policy contributed to the positive inflation rates. The 4vGap specification for Japan implies that the economy transitioned to the deflation 19Due to the simplicity of the DSGE model, the shock decomposition is not refined enough to generate a more detailed narrative of the recent U.S. experience that emphasizes the disruption in financial intermediation. Shocks to the financial system and nonlinearities generated by its disruption, are interpreted as large technology or discount factor shocks by our model.

This Version: January 22, 2016 39 Figure 9: Filtered Shock Innovations (cid:15) t 2 0 −2 −4 1985 1990 1995 2000 2005 2010 2015 z 0htworGygolonhceT U.S. 4vGrowth 2 0 −2 1985 1990 1995 2000 2005 2010 2015 g 0gnidnepStnemnrevoG 3 2 1 0 −1 −2 −3 1985 1990 1995 2000 2005 2010 2015 d 0rotcaFtnuocsiD 2 1 0 −1 −2 1985 1990 1995 2000 2005 2010 2015 R 0yciloPyratenoM Japan 4vGap 2 0 −2 −4 1985 1990 1995 2000 2005 2010 2015 4 2 0 −2 −4 1985 1990 1995 2000 2005 2010 2015 4 2 0 −2 −4 1985 1990 1995 2000 2005 2010 2015 2 0 −2 −4 1985 1990 1995 2000 2005 2010 2015 Note: Innovations are shown in multiples of their standard deviations. The solid vertical line shows the end of the estimation sample and the dashed vertical line shows the beginning of the ZLB episode. regime in the late 1990s. In the deflation regime the negative inflation rates generate a nonnegligible resource cost and the approximation logC /Y ≈ −logg is no longer accurate. t t t The discount factor shock also affects the consumption-output ratio. As is apparent from the impulseresponsesinFigure5noneoftheshockshasasignificantimpactontheinterestrates, which are with high probability zero. The filter essentially inverts these relationships. Most notably, the slow growth of the Japanese economy since the late 1990s maps into technology growth innovations that are on average negative. An inspection of the regime-conditional ergodic distributions drawn in Figure 4 indicates that inflation rates in the deflation regime

This Version: January 22, 2016 40 are with high probability less than -4%.20 Actual Japanese inflation, while being negative, has always been above -4%, which is translated by the filter in a sequence of discount factor innovations that are fairly volatile and on average below zero. 6.3 Aggregating the Results We now formally aggregate the results from the four different specifications for each country by computing weights for each specification that are related to the goodness of fit. The obvious difficulty here is that the four specifications do not share a common dataset. In order to compare 3-variable and 4-variable specifications, we follow the approach in Del Negro et al. (2016) and construct one-step-ahead predictive densities for the subset of common observations. Let zo be the 3×1 vector of output, inflation, and interest rates. These three t variables are the core variables that most New-Keynesian models aim to capture. Moreover, let p(zo|Yo ,M ) be the predictive density for zo given specification M and the t − 1 t 1:t−1 j t j information set Yo .21 Based on the predictive densities, we can define the quasi model 1:t−1 probabilities (cid:81)T p(zo|Yo ,M ) p˜(M ) = t=T0 t 1:t−1 j (37) t j (cid:80)4 (cid:81)T p(zo|Yo ,M ) j=1 t=T0 t 1:t−1 j and use them to create weighted averages of P{s = 1|Yo ,M }. t 1:t j The results are presented in Figure 10. For each specification, we plot the log of the numerator of (37) in the top two panels. We take T to be the beginnings of the respective 0 ZLB periods. Each line can be interpreted as running predictive score of a model specification. For the U.S. the difference in fit between the four specifications is relatively small. The performance differential between the best and the worst specification does not significantly widen over time, though the relative rankings change. Until the end of 2013 the 3vGrowth specification dominates, whereas after 2014, the 4vGrowth specification attains the highest predictive score. The 3vGap specification is the least preferred. For Japan the 3vGrowth specification also is best, though by a much larger margin. Unlike for the U.S., in the case of 20The contours for 4vGap-Japan look similar to the contours for 4vGrowth-U.S., which are shown in the figure. 21The conditioning information set here differs across the 3-variable and 4-variable specifications. To avoid an overly tedious notation, we did not introduce a j index for the information set.

This Version: January 22, 2016 41 Figure 10: Combined Filtered Probabilities Cumulative Predictive Log Likelihood Score U.S. Japan 30 20 Gap 3 25 Gap 4 Growth 3 0 20 Growth 4 H H 15 L L-20 g g o L 10 o L d d e 5 e-40 rP rP 0 Gap 3 -60 Gap 4 -5 Growth 3 Growth 4 -10 -80 2009 2010 2011 2012 2013 2014 2015 2000 2005 2010 2015 Average P{s = 1|Yo } Across Specifications t 1:t U.S. Japan 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2009 2010 2011 2012 2013 2014 2015 2000 2005 2010 2015 Notes: Top panels: the log predictive score is cumulative. T is the beginning of the ZLB episodes. Bottom 0 panels: weconvertthecumulativelogpredictivescoreintoquasimodelprobabilities(see(37))andusethem to compute a weighted average of P{s = 1|Yo } across the four specifications. We consider two choices t 1:t of T : the beginning of the ZLB episodes (red) and the beginning of the estimation sample (black). The 0 horizontal dashed line shows the country-specific threshold P{s = 1}, which is 0.83 for the U.S. and 0.89 t for Japan. The solid line indicates the 0.5 threshold. Japan the gap between the 3vGrowth specification and the three other specifications widens toward the end of the sample. From an ex ante perspective, the relative ranking of the 3- and 4-variable specifications basedonthepredictivelikelihoodforisunclear. The3-variablemodelsareoptimizedtotrack the variables included in the zo vector. The 4-variable models, on the one hand, have an t additional degree of freedom, namely, the latent discount factor shock d , which can improve t

This Version: January 22, 2016 42 the tracking of the trivariate vector zo. On the other hand, the 4-variable specifications t also have to track the consumption-output ratio. Doing so may lead to a deterioration of the one-step-ahead forecast performance for zo. Ex post, it turns out that one of the 3t variable specifications, namely 3vGrowth, dominates the 4-variable specifications. However, for both the U.S. and Japan, the 4-variable specifications are competitive with the 3vGap specification. Even though we are considering a period in which interest rates are zero, there seems to be information about the policy-rule specification (gap versus growth). This information arises from the fact that even if interest rates are currently zero, beliefs about the future conduct of monetary policy affect current output and inflation. The bottom panels of Figure 10 show the quasi model probabilities computed for two choices of T : the beginning of the ZLB episodes (red) and the beginning of the estimation 0 sample (black). The first choice is consistent with the predictive scores depicted in the top panels of the figure. The second choice of T also factors in the fit of the model specifications 0 prior to the ZLB episodes and thereby places more weight on the 3-variable specifications. Using horizontal lines, we depict two of the thresholds discussed in Section 6.1: 0.5 and P{s = 1}. After aggregating the information from the four specifications, we conclude t that the U.S. has remained in the targeted-inflation regime in the aftermath of the Great Recession and that Japan’s ZLB experience is best described by a switch to the deflation regime. For the U.S. there is significant uncertainty about the regime at the beginning of 2009. However, subsequently, there is only a single quarter, 2011:Q4, in which the probability of being in the targeted-inflation regime falls below 0.5. This quarter exhibits an unusually low inflation rate. In 2014, using the weights based on T =2009:Q1 the probability of the 0 targeted-inflation regime falls toward 0.5, because the 4vGrowth specification starts to dominate the weighted average. Recall from the bottom right panel of Figure 8 that the filtered probability of s = 1 drops from 1 to about 0.5 between 2012 and the end of the sample. For t Japan there is only one quarter in which the probability of being in the targeted-inflation regime clears all thresholds. This happens in 2002:Q1, when inflation is positive and seems like an outlier relative to the period before and after. Except in 2000 and in 2002:Q1 the

This Version: January 22, 2016 43 black and the red lines are on top of each other, implying that the inference is not sensitive to the choice of T . 0 7 Conclusion The recent experiences of the U.S. and Japan have raised concern among policy makers about a long-lasting switch to a regime in which interest rates are zero, inflation is low, and conventional macroeconomic policies are less effective. We solve a small-scale New Keynesian DSGE model imposing the ZLB constraint and introducing a non-fundamental Markov sunspot shock that can move an economy between a targeted-inflation regime and a deflation regime. An economy may be pushed to the ZLB either by successive fundamental shocks (e.g., an adverse discount factor shock) in the targeted-inflation regime or by a switch to the deflation regime. We develop a quantitative framework that can distinguish these two possibilities. Our empirical analysis focuses on the U.S. and Japan and utilizes four different DSGE model specifications for each country that differ in terms of the observables used and the monetary policy rule. Using a nonlinear filter, we find that for each country three of the four specifications agree: the U.S. remained in the targeted-inflation regime during its ZLB episode, with the possible exception of the early part of 2009 where evidence is more mixed. Japan switched to the deflation regime in 1999 and remained there until the end of our sample. Weaggregateourresultsusingquasimodelprobabilitiesandthefinalresultsconfirm the above conclusions. Our model is silent as to why the two experiences are different because the sunspot process in our model is purely exogenous. In a richer, but computationally much more challenging specification, the coordination of beliefs may be correlated with fundamentals and be affected by central bank actions. Perhaps one key difference between Japan in 1999 and the U.S. in 2009 is in their conduct of monetary policy. Ito and Mishkin (2006), who provide a summary of the actions taken by the Bank of Japan and the Japanese government conclude that “(...) mistakes in the management of expectations [by the Bank of Japan]

This Version: January 22, 2016 44 are a key reason why Japan found itself in a deflation that it is finding very difficult to get out of”. The actions of U.S. policymakers contrast greatly with those of the Bank of Japan. The Federal Reserve and in general policy makers in the U.S. enacted unconventional policies such as quantitative easing and forward guidance and evidently did a good job in coordinating inflation expectations near its target. We leave a formal quantitative analysis for future research. References An, S. and F. Schorfheide (2007): “Bayesian Analysis of DSGE Models,” Econometric Reviews, 26, 113–172. Armenter, R.(2014): “The Perils of NominalTargets,” Manuscript, Federal Reserve Bank of Philadelphia. Aruoba, S. B. (2014): “Term Structure of Inflation Expectations and Real Interest Rates: The Effects of Unconventional Monetary Policy,” Manuscript, University of Maryland. Benhabib, J. and R. E. Farmer (1999): “Indeterminacy and sunspots in macroeconomics,” in Handbook of Macroeconomics, ed. by J. B. Taylor and M. Woodford, Elsevier, vol. 1, Part A, 387 – 448. Benhabib, J., S. Schmitt-Grohe´, and M. Uribe (2001a): “Monetary Policy and Multiple Equilibria,” The American Economic Review, 91, 167–186. ——— (2001b): “The Perils of Taylor Rules,” Journal of Economic Theory, 96, 40–69. Braun, R. A., L. M. Ko¨rber, and Y. Waki (2012): “Some Unpleasant Properties of Log-Linearized Solutions When the Nominal Rate Is Zero,” Federal Reserve Bank of Atlanta Working Paper Series. Bullard, J.(2010): “SevenFacesof‘ThePeril’,” Federal Reserve Bank of St. Louis Review, 92, 339–352.

This Version: January 22, 2016 45 Cass, D. and K. Shell (1983): “Do Sunspots Matter?” Journal of Political Economy, 91, 193–227. Cochrane, J. H. (2011): “Determinacy and Identification with Taylor Rules,” Journal of Political Economy, 119, 565 – 615. ——— (2015): “The New-Keynesian Liquidity Trap,” Manuscript, Chicago Booth School of Business. Cuba-Borda, P. (2014): “What Explains the Great Recession and the Slow Recovery?” Manuscript, Federal Reserve Board. Del Negro, M., M. P. Giannoni, and F. Schorfheide (2015): “Inflation in the Great Recession and New Keynesian Models,” American Economic Journal: Macroeconomics, 7, 168–196. Del Negro, M., R. Hasegawa, and F. Schorfheide (2016): “Dynamic Prediction Pools: An Investigation of Financial Frictions and Forecasting Performance,” Journal of Econometrics, forthcoming. Eggertsson, G. B. (2011): “What fiscal policy is effective at zero interest rates?” in NBER Macroeconomics Annual 2010, ed. by D. Acemoglu and M. Woodford, University of Chicago Press, vol. 25, 59–112. Eggertsson, G. B. and M. Woodford (2003): “The Zero Bound on Interest Rates and Optimal Monetary Policy,” Brookings Papers on Economic Activity, 2003, 139–211. Ferna´ndez-Villaverde, J., G. Gordon, P. Guerro´n-Quintana, and J. F. Rubio- Ram´ırez (2015): “Nonlinear Adventures at the Zero Lower Bound,” Journal of Economic Dynamics and Control, 57, 182 – 204. Gust, C., D. Lopez-Salido, and M. E. Smith (2012): “The Empirical Implications of the Interest-Rate Lower Bound,” Manuscript, Federal Reserve Board. Herbst, E. and F. Schorfheide (2015): Bayesian Estimation of DSGE Models, Princeton University Press.

This Version: January 22, 2016 46 Hirose, Y. (2014): “An Estimated DSGE Model with a Deflation Steady State,” CAMA Working Paper Series. Ireland, P. N.(2004): “Money’sRoleintheMonetaryBusinessCycle,” Journal of Money, Credit and Banking, 36, 969–983. Ito, T. and F. S. Mishkin (2006): “Two Decades of Japanese Monetary Policy and the Deflation Problem,” in Monetary Policy under Very Low Inflation in the Pacific Rim, ed. by T. Ito and A. Rose, University of Chicago Press, vol. NBER-EASE, Volume 15, 131–202. Judd, K. L. (1992): “Projection Methods for Solving Aggregate Growth Models,” Journal of Economic Theory, 58, 410 – 452. Judd, K. L., L. Maliar, and S. Maliar (2010): “A Cluster-Grid Projection Method: Solving Problems with High Dimensionality,” NBER Working Paper, 15965. Kru¨ger, D. and F. Kubler (2004): “Computing Equilibrium in OLG Models with Production,” Journal of Economic Dynamics and Control, 28, 1411–1436. Kuroda, S. and I. Yamamoto (2008): “Estimating The Frisch Labor Supply Elasticity in Japan,” Journal of the Japanese and International Economies, 22, 566–585. Lubik, T. A. and F. Schorfheide (2004): “Testing for Indeterminancy: An Application to U.S. Monetary Policy,” American Economic Review, 94, 190–217. Maliar, L. and S. Maliar (2015): “Merging Simulation and Projection Approaches to Solve High-Dimensional Problems with an Application to a New Keynesian Model,” Quantitative Economics, 6, 1–47. Mavroeidis, S. (2010): “Monetary Policy Rules and Macroeconomic Stability: Some New Evidence,” American Economic Review, 100, 491–503. Mertens, K. and M. O. Ravn (2014): “Fiscal Policy in an Expectations Driven Liquidity Trap,” The Review of Economic Studies, 81, 1637–1667.

This Version: January 22, 2016 47 Piazza, R. (2015): “Self-Fulfilling Deflations,” Manuscript, Bank of Italy. R´ıos-Rull, J.-V., F. Schorfheide, C. Fuentes-Albero, M. Kryshko, and R. Santaeulalia-Llopis (2012): “Methods versus Substance: Measuring the Effects of Technology Shocks,” Journal of Monetary Economics, 59, 826–846. Schmitt-Grohe´, S. and M. Uribe (2015): “Liquidity Traps and Jobless Recoveries,” Manuscript, Columbia University. Schorfheide, F. (2008): “DSGE Model-Based Estimation of the New Keynesian Phillips Curve,” Federal Reserve Bank of Richmond Economic Quarterly, Fall Issue, 397–433. Ueda, K. (2012): “The Effectiveness of Non-Traditional Monetary Policy Measures: The Case of the Bank of Japan,” The Japanese Economic Review, 63(1), 1–22. Woodford, M. (2003): Interest and Prices, Princeton University Press.