Uncertainty shocks, monetary policy and long-term interest rates

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Uncertainty shocks, monetary policy and long-term interest rates Gianni Amisano and Oreste Tristani 2019-024 Please cite this paper as: Amisano, Gianni, and Oreste Tristani (2019). “Uncertainty shocks, monetary policy and long-term interest rates,” Finance and Economics Discussion Series 2019-024. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2019.024. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Uncertainty shocks, monetary policy and long-term interest rates∗ Gianni Amisano† Federal Reserve Board and Georgetown University Oreste Tristani‡ European Central Bank and CEPR February 8, 2019 Abstract Westudytherelationshipbetweenmonetarypolicyandlong-termratesin a structural, general equilibrium model estimated on both macro and yields data from the United States. Regime shifts in the conditional variance of productivity shocks, or ”uncertainty shocks”, are an important model ingredient. First, they account for countercyclical movements in risk premia. Second, they induce changes in the demand for precautionary saving, which affects expected future real rates. Through changes in both risk-premia and expected future real rates, uncertainty shocks account for about 1/2 of the varianceoflong-termnominalyieldsoverlonghorizons. Theremainingdriver of long-term yields are changes in inflation expectations induced by conventional,autoregressiveshocks. Long-terminflationexpectationsimpliedbyour model are in line with those based on survey data over the 1980s and 1990s, but less strongly anchored in the 2000s. ∗Anolderversionofthispaperwascirculatedwiththetitle”ADSGEmodelofthetermstructure with regime switches”. We thank for useful comments and suggestions: Martin Andreasen, Michael Bauer, Francesco Bianchi, James Bullard, Yunjong Eo, Andrew Figura, John Geweke, Jinill Kim, Gary Koop, James Morley, Haroon Mumtaz, Adrian Pagan, Giorgio Primiceri, Tommaso Proietti, Frank Schorfheide, Rodney Strachan, Eric Swanson, Herman van Dijk, Paolo ZagagliaandTaoZha. Wearealsogratefulforcommentsfromseminar/conferenceparticipantsatthe following institutions: Banks of England, Finland, Italy, Japan, Erasmus University Rotterdam, FederalReserveBoard,2018FRSMacroSystemConference,KoreaUniversity,NBER-NSFDSGE Conference,NBERSummerInstitute,NorgesBank,ReserveBanksofAustraliaandNewZealand, StrathclydeUniversity,SBIESWorkshopinChicago,UniversitiesofBresciaandGenoa,University ofSydney,UniversityofTechnologySydneyandWarwickBusinessSchool. WealsothankMichael Kister and Niko Paulson for excellent research assistance. The opinions expressed are personal and should not be attributed to the Board of Governors of the Federal Reserve System, or to the European Central Bank. †Email: gianni.amisano@frb.gov. ‡Email: oreste.tristani@ecb.int. 1

JEL classification: C11, C34, E40, E43, E52. Keywords: monetary policy rules, uncertainty shocks, term structure of interest rates, regime switches, Bayesian estimation. 1 Introduction Following Smets and Wouters (2007), estimated general equilibrium models with sticky prices have become popular tools for monetary policy analysis in central banks. Partly as a result of the 2008 financial crisis and the ensuing recession, these models have become much richer in a number of directions, notably including financial frictions (e.g. Christiano, Motto and Rostagno, 2014, or Del Negro et al., 2017). Due to linearization, however, most of these models are not ideally suited to describe bond risk premia and they in fact almost always abstract from bond yields in estimation. Nevertheless, long-term rates and risk premia are often an object of interestinmonetarypolicyanalysis—seeforexampleGoodfriend’s(1993)discussion of an ”inflation scare” in bond yields in the early 1990s, or the famous Greenspan (2005) speech on the ”bond markets conundrum”.1 It would be desirable to have models allowing for a joint and coherent explanation of macroeconomic dynamics and bond risk premia. Thispapershowsthattheestimated,nonlinearversionofasimplenew-Keynesian model can provide an internally consistent account of the evolution of macroeconomic and yields data in the United States. To achieve this goal, two model modifications are necessary compared to versions used in standard macro-only applications: non-expected utility preferences and stochastic volatility. Non-expected utility—a model feature which is now standard in calibrated analyses of asset prices in production economies with endogenous labor supply—allows one to increase riskaversion independently of the elasticity of intertemporal substitution.2 Stochastic 1Goodfriend (1993) defines an inflation scare as a significant increase in long term nominal interest rates in the absence of an increase in policy rates. 2See e.g. Tallarini (2000), Rudebusch and Swanson (2012), van Binsbergen et al. (2012), Andreasen (2012a), Colacito and Croce (2013), Swanson (2018). 2

volatility, which in our model takes the form of regime switching, leads to ”uncertainty shocks” that produce variations in bond risk premia and, at the same time, to changes in households’ demand for precautionary saving. Through the latter mechanism, uncertainty shocks contribute to affect macroeconomic fluctuations.3 We show that our simple model specification provides a reasonably good and parsimonious account of U.S. data on aggregate consumption, inflation, short and long-term interest rates. The model also fits well dimensions of the data which were notdirectlyusedinestimation, suchaslongforwardrates. Theconditionalstandard deviations implied by our DSGE model for the macro-variables are broadly in line with those generated by a reduced-form Markov-switching VAR including the same variables. Finally, our model does not suffer from the puzzle identified in Backus, Gregory and Zin (1989), as it can simultaneously account for the positive slope of the term structure and a positive autocorrelation of consumption growth. The serial autocorrelation of yields is strikingly consistent with their sample autocorrelation coefficients. Our results shed new light on the determinants of long-term interest rates over the business cycle. Long-term rates are not roughly constant, as implied by simple, linearized models, but subject to three broad sources of time-variation: risk-premia, expected future real rate and expected future inflation. Fluctuations in bond risk-premia are induced by uncertainty shocks. Our results suggest that such fluctuations are cyclical. Recessions are persistent periods of heightened uncertainty, which leads to a rise in risk premia. During economic recoveries, uncertainty returns to normal levels and risk premia become again small. Ifthisreductioninriskpremiaoccursattheturningpointofthepolicyinterestrates cycle—aswasthecasein2004accordingtoourestimates—observedlong-termyields can remain roughly unchanged masking the expectations of a future policy tightening and producing the apparent ”conundrum” noted in Greenspan (2005). If, by contrast, the fall in uncertainty occurs before the monetary policy tightening phase 3See also Gourio (2012, 2013), Andreasen (2012b) for the case of time variations in disaster risk. 3

begins, as was the case e.g. in 1994 according to our estimates, the response of observed long-term rates is more directly informative of expected changes in future interest rates. The second source of fluctuations in long-term nominal rates are developments in current and expected future real rates. Once again, uncertainty shocks over future realizations of technology play an important role. The higher level of uncertainty which characterizes recessions leads to an increase in households’ demand for precautionary saving. Current and expected future real rates, therefore, tend to fall to clear the savings market. For given long-term inflation expectations, this mechanism also leads to a fall in expected future nominal interest rates. During recoveries, uncertainty over future realizations of technology switches back to lower levels and ”confidence” returns. The demand for precautionary saving falls back to normal levels and expected real yields increase. The ensuing increase in nominal yields is neither a signal of expected inflationary pressure, nor of changes in the anti-inflationary credibility of the central bank. Such movements in real rates also induce macroeconomic fluctuations. The monetary policy reaction function shapes the strenghts of their effects on consumption and inflation. As described above, uncertainty shocks produce changes in the demand for precautionary saving. If monetary policy is conducted according to the standard Taylor rule, a hypothesis we maintain in our analysis, it does not internalize the ensuing changes in equilibrium real interest rates. For example, after an increase in uncertainty, policy rates are cut due to the ensuing weakness in consumption and inflation, but they do not fall enough to offset the drop in equilibrium real rates induced by the stronger demand for precautionary saving. Consequently, consumption, output and inflation remain weak for a prolonged period of time. The third driver of developments in bond yields are long-term inflation expectations. In our model these movements mostly arise as a result of linear shocks and conventional, first-order dynamics. More specifically, conventional technology shocks are responsible for the bulk of the secular change in long-term nominal yields. To affect yields at the 10-year horizon, they must be even more persistent, than typ- 4

ically found in applications solely based on macroeconomic data. To assess the plausibility of our estimates, we compare model-based long-term inflation expectations with measures of expectations available from the Federal Reserve Bank of Philadelphia’s quarterly Survey of Professional Forecasters. There is no reason why the two measures should agree, since they reflect the different information sets of survey participants and of ”financial markets,” as embodied in bond prices and extracted through the lens of our model. Nevertheless, the broad level and the medium term trends in the two series are quite similar. They both point to a progressive and parallel fall in inflation expectations over the 1980s, then hover around the 2.5% level since the end of the 1990s. At the higher frequency, however, our model is consistent with a less extreme anchoring of 10-year inflation expectations compared to surveys.4 After falling towards 2.5 percent over the 1990s, survey measures remain constant at that level thereafter. By contrast, model-implied measures are generally more volatile. For example they temporarily increase again, sharply, during the ”inflation scare” of 1993. They hover closely around 2.5 percent at the turn of the millennium, but then fall to levels close to 1 percent during the recession of the early 2000s and even below 1 percent ahead of the Great recession. In sum, bond prices suggest that 10year inflation expectations over the 2000s are less dogmatically anchored than one would conclude based on survey data. It is noticeable that inflation developments after the Great recession turned out to be more in line with the expectations implied by our model than with survey expectations. We also derive model-based expectations measures for the late 1960s and early 1970s, a period where the survey measures are unavailable. Unsurprisingly, our results suggest an increase in 10-year inflation expectations over this early part of the sample. The rise in expectations is not abrupt, but very gradual and persistent. Our paper is related to the literature exploring the term structure implications of macro-models. Many of these papers are theoretical and look at the asset pricing 4We assume a constant inflation target in the monetary policy rule, so inflation expectations are by assumptions always anchored in the very long run. 5

implicationsofmacromodels–seee.g. PiazzesiandSchneider(2006),Rudebuschand Swanson (2012), Swanson (2014). Amongst the empirical papers, De Graeve, Emiris and Wouters (2007) estimate a standard DSGE model using both macroeconomic and term structure data, but rely on the loglinearized version of that model and must therefore introduce additional parameters to allow for non-zero risk-premia. Christoffel, Jaccard and Kilponen (2011) also estimate the linearized version of a new Keynesian model, and then draw bond pricing implications using a higher order approximation. Bekaert, Cho and Moreno (2010) and Campbell, Pflueger and Viceira (2013) follow an intermediate route and study asset prices in a linearized New Keynesian model assuming a stochastic discount factor that is related to the new Keynesian model’s equations in a reduced-form manner. The papers most similar to ours are Doh (2011, 2012), van Binsbergen et al. (2012) and Andreasen (2012), which estimate nonlinear models with macroeconomic and term structure data. In contrast to all these papers, we allow for regime switches in the variance of shocks and argue that this is an essential model feature to fit bonds and macro data. Moreover, the focus of all these papers is on the model’s ability to fit yields, while we highlight the model’s implications for the transmission of monetary policy to longrunrates. Fromthisperspective,weareclosertoCochrane(2008,2017)andAtkeson and Kehoe (2008). All these papers are related to the huge consumption-based asset pricing literature and build on the results of either Campbell and Cochrane (1999) or Bansal and Yaron (2004). In a recent contributions to this literature, Schorfheide, Song and Yaron (2018) highlight the importance of allowing for measurement error in consumption in a long-run risk model. We also allow for measurement errors in our estimation. Our paper is also related to the literature documenting time variation in macroeconomic volatility in a reduced form setting, including e.g. McDonnell and Perez- Quiros (2000), Sims and Zha (2006), Primiceri (2005). Justiniano and Primiceri (2008) allows for shifts in the volatility of structural shocks in a linearized, mediumscale DSGE model applied to the U.S. economy. Compared to Justiniano and Primiceri’s, we rely on a smaller, but non-linear model, which allows us to explore the 6

effects of uncertainty shocks on households’ demand for precautionary saving and on bond risk premia. Our modelling of second moments is however more parsimonious and less flexible in uncovering trend shifts in volatility. Finally, our paper is related to the literature on uncertainty shocks spawned from Bloom (2009). In Bloom (2009), an increase in uncertainty induces firms to temporarily reduce investment and hiring. In our model, higher uncertainty over future technology shocks induces households to increase their precautionary saving. Consumption demand will tend to fall. Due to monopolistic competition and sticky prices, this will bring down output and inflation. Uncertainty shocks therefore act like demand shocks. This is consistent with the results in Basu and Bundick (2012), which relies on a more comprehensive, calibrated model of the U.S. economy and analyses uncertainty shocks in both technology and preferences. Bianchi, Ilut and Schneider (2014) put forward a model with ambiguity averse investors, where regime shifts generate large low frequency movements in asset prices. The rest of the paper is organized as follows. Section 2 describes the model, focusing on its distinguishing features: the distribution of the shocks and the utility function, which is of the class proposed by Epstein and Zin (1989) and Weil (1990), but extended to allow for habit persistence in consumption. The methods that we adopt to solve and estimate the model are described next, in section 3. Such methods are non-standard, because we need to solve the model to a second order approximation in order to capture precautionary savings effects. We demonstrate that the reduced form of the model is quadratic in the state variables with continuous support and includes regime-switching intercepts, as well as variances. We then estimate the non-linear reduced form using Bayesian methods. Section 4 described the estimation results and presents a few goodness-of-fit measures and Section 5 illustrates the role of uncertainty shocks through the analysis of impulse responses and the variance decomposition of forecast errors. The implications of our estimates for the relationship between monetary policy and risk premia and for the transmission of monetary policy to long-term rates are discussed in Section 6. 7

Section 7 concludes. A technical appendix, available on-line5, describes in details some of the computations carried out in this paper. 2 The model We start from a simple version of the new-Keynesian model that has been shown to account relatively well for the dynamics of key nominal and real macroeconomic variables—see e.g. Smets and Wouters (2007). We thus assume nominal price rigidities, external habit persistence, inflation indexation, and a monetary policy rule with “interestrate smoothing”. Sinceour interestis onthe model’s implications for long-term interest rates, we simplify it by abstracting from capital accumulation and real wage rigidities. Our results suggest that even our simple model can go a long way in explaining the data of interest to us. ComparedtothenewKeynesianbenchmark, weintroducetwokeymodifications. The first is to allow for stochastic regime switching in the variance of structural shocks. The evidence of time variation in the variance of macroeconomic shocks is well-established–see e.g. Justiniano and Primiceri (2008), McDonnell and Perez- Quiros (2000), Primiceri (2005) and Sims and Zha (2006). The novelty in our paper is to explore the implications of time varying variances on bond prices. Our second modification, which is already common in the consumption-based asset pricing literature, is to adopt the non-expected utility specification for preferences proposed by Epstein and Zin (1989) and Weil (1990). Here we extend this specification to a general equilibrium model in which we also allow for habit persistence in consumption and labour-leisure choice. 2.1 Structural shocks A key distinguishing feature of our model are changes in the demand for precautionary saving induced by variations in the conditional variance of the structural shocks. We therefore start the description of our model from the distribution of structural 5https://sites.google.com/site/gianniamisanowebsite/ 8

shocks. In macroeconomic applications, exogenous shocks are almost always assumed to be (log-) normal, partly because models are typically log-linearized and researchers aremainlyinterestedincharacterizingconditionalmeans. However,Hamilton(2008) argues that a correct modelling of conditional variances is always necessary, for example because inference on conditional means can be inappropriately influenced by outliers and high-variance episodes. The need for an appropriate treatment of heteroskedasticity becomes even more compelling when models are solved nonlinearly, because conditional variances have a direct impact on conditional means. Inthispaper, weassumethatvariancesaresubjecttostochasticregimeswitches. We will allow for shocks to the level and growth rates of technology, to markups, to the monetary policy rule and to a non-interest-rate-sensitive component of output G . G will enter GDP like government spending, but we do not model it t t explicitly since our interest is not on fiscal policy. We only use G to allow for t a demand-type shock and we therefore refer to it generically as ”demand shock”. The conditional variance of any of these shocks could in principle be subject to regime switching, but in this paper we adopt a parsimonious specification such that only (level) productivity, monetary policy and demand shocks have regime switching variances.6 More specifically, we will assume that the technology shock z , the monetary t policy shocks η and the demand shock G have standard deviations that can int t dependently switch between a high and a low regime. Denoting the low variance regime by 1 and the high variance regime by 0, we write σ = σ s +σ (1−s ) z,sz,t z,0 z,t z,1 z,t σ = σ s +σ (1−s ) G,sG,t G,0 G,t G,1 G,t σ = σ s +σ (1−s ) η,sη,t η,0 η,t η,1 η,t where the variables s , s and s can assume the discrete values 0 and 1. For z,t G,t η,t 6We have also estimated versions of the model allowing for regime-switching in the variance of mark-up and technology growth shocks. These dimensions of regime switching receive little support from the data. 9

each variable s (j = z,G,η), the probabilities of remaining in states 0 and 1 are j,t constant and equal to p and p , while the probabilities of switching to the other j,0 j,1 state will be 1−p and 1−p , respectively. j,0 j,1 2.2 Households WeassumethateachhouseholdiprovidesN (i)hoursofdifferentiatedlaborservices to firms in exchange for a labour income w (i)N (i). Each household owns an equal t t (cid:82)1 share of all firms j and receives profits Ψ (j)dj. As in Erceg, Henderson and 0 t Levin (2000), an employment agency combines households’ labor hours in the same proportionsasfirmswouldchoose. Theagency’sdemandforeachhousehold’slabour is therefore equal to the sum of firms’ demands. The labor index L has the Dixitt (cid:20) (cid:21) θw,t (cid:82)1 θw,t−1 θw,t−1 Stiglitz form L t = 0 N t (i) θw,t di , where θ w,t > 1 is subject to exogenous shocks. At time t, the employment agency minimizes the cost of producing a given amount of the aggregate labor index, taking each household’s wage rate w (i) as t given,andthensellsunitsofthelaborindextotheproductionsectorattheaggregate (cid:104) (cid:105) 1 wage index w = (cid:82)1 w(i)1−θw,tdi 1−θw,t. The employment agency’s demand for the t 0 labor hours of household i is given by (cid:18) w (i) (cid:19)−θw,t t N (i) = L . (1) t t w t Each household i maximizes its intertemporal utility with respect to consumption, the wage rate and holdings of a complete portfolio of state-contingent assets, subject to the demand for its labour (1) and the budget constraint (cid:90) 1 P C (i)+E Q W (i) ≤ W (i)+w (i)N (i)+ Ψ (j)dj (2) t t t t,t+1 t+1 t t t t 0 where C is a consumption index satisfying t (cid:18)(cid:90) 1 θ−1 (cid:19) θ− θ 1 C = C (z) θ dz . t t 0 In the budget constraint, W (i) denotes the beginning-of-period value of a comt plete portfolio of state contingent assets held by household i, Q is their price t,t+1 10

and Ψ (j) are the profits received from investment in firm j. The price level P is t t defined as the minimal cost of buying one unit of C , hence equal to t (cid:18)(cid:90) 1 (cid:19) 1− 1 θ P = p(z)1−θdz . t 0 Equation (2) states that each household can only consume or hold assets for amounts that must be less than or equal to its salary, the profits received from holding equity in all the existing firms and the revenues from holding a portfolio of state-contingent assets. Households’ preferences are described by the Kreps and Porteus (1978) specification proposed by Epstein and Zin (1989). In that paper, utility is defined recursively through the aggregator U such that (cid:26) (cid:27) 1 U (cid:2) C , (cid:0) E V1−γ(cid:1)(cid:3) = (1−β)C1−ψ +β (cid:0) E V1−γ(cid:1)1 1 − − ψ γ 1−ψ , ψ,γ (cid:54)= 1, (3) t t t+1 t t t+1 where β,ψ and γ are positive constants. Using a specification equivalent to that in equation (3), Weil (1990) shows that β is, under certainty, the subjective discount factor, but time preference is in general endogenous under uncertainty. The parameter γ is the relative risk aversion coefficient for timeless gambles. The parameter 1/ψ measures the elasticity of intertemporal substitution for deterministic consumption paths. The distinguishing feature of the Epstein-Zin-Weil preferences, compared to the standard expected utility specification, is that the coefficient of relative risk aversion can differ from the reciprocal of the intertemporal elasticity of substitution. In addition, Kreps and Porteus (1978) show that, again contrary to the expected utility specification, the timing of uncertainty is relevant in their class of preferences. The specification in equation (3) displays preferences for an early resolution of uncertainty when the aggregator is convex in its second argument, i.e. when γ > ψ. Any source of risk will be reflected in asset prices not only if it makes consumption more volatile, but also if it affects the temporal distribution of consumption volatility. We generalize the utility function in equation (3) by allowing for habit formation and a labour-leisure choice, as in standard, general equilibrium macro-models. 11

The generalization to allow for the labour-leisure choice has already been used, for example, in Rudebusch and Swanson (2012). We additionally allow for habit formation because it has been shown to be important to match the dynamic behavior of aggregate consumption–see e.g. Fuhrer (2000). As a result, time-t utility of household i will not only depend on consumption C but it will be a more general function of consumption and leisure t U (i) = u{C (i)−hΞ C ,1−N (i)} t t t t−1 t where leisure is written as 1 − N because total hours are normalized to 1, the h t parameter represents the force of external habits and Ξ is the rate of growth of t technology.7 With our more general preferences specification, γ is no-longer related one-toone to risk aversion. Swanson (2012) discusses the appropriate measures of risk aversion in a dynamic setting with consumption and leisure entering the utility function. However, 1/ψ continuestomeasurethelong-runelasticityofintertemporal substitution of consumption.8 The first order conditions for household i include (ignoring the i subscript for simplificy) u w N,t t = µ w,t u P c,t t and (cid:34) (cid:35)γ−ψ (cid:18) J (cid:19)1−γ 1−γ (cid:18) J (cid:19)−(γ−ψ)(cid:18) u (cid:19)−ψ u 1 t+1 t+1 t+1 c,t+1 Q = β E , (4) t,t+1 t J J u u Π t t t c,t t+1 where J = J(W (i)) is household i’s maximum value function 9, Π is the inflation t t t ratebetweentandt−1, andthemark-upµ ≡ (θ −1)/θ followsanexogenous w,t w,t w,t autoregressive process µ w,t+1 = µ1 w −ρµ(µ w,t )ρµeεµ t+1, εµ t+1 ∼ N (0,σ µ ). 7GuarigliaandRossi(2002)alsouseexpectedutilitypreferencescombinedwithhabitformation to study precautionary savings in UK consumption. Koskievic (1999) studies an intertemporal consumption-leisure model with non-expected utility. 8See section A.9 of the online appendix. 9For details, see section A.1 of the online appendix. 12

The gross nominal, one-period interest rate, I , equals the conditional expectat tion of the stochastic discount factor, i.e. I−1 = Pb = E Q , (5) t 1,t t t,t+1 where Pb denotes the price of the one- period bond. 1,t BasedonthestochasticdiscountfactorQ , thepriceofan-periodzerocoupon t,t+1 bond Pb can be written as n,t (cid:2) (cid:3) Pb = E Q Pb n,t t t,t+1 n−1,t+1 and the corresponding nominal yield R as n,t 1/I = Pb . n,t n,t Note that we will focus on a symmetric equilibrium in which nominal wage rates are all allowed to change optimally at any point in time, so that individual nominal wages will equal the average w . t Equation (4) highlights how our model nests the standard power utility case, in which ψ = γ and the maximum value function J disappears from the first order t conditions. The same equations also demonstrate that the parameter γ only affects the dynamics of higher order approximations. It is straightforward to see that, to first order, the term (cid:2) E (J /J )1−γ(cid:3)(γ−ψ)/(1−γ) (J /J )−(γ−ψ) cancels out in the t t+1 t t+1 t interest rate equation (5). 2.3 Firms We assume a continuum of monopolistically competitive firms (indexed on the unit interval by j), each of which produces a differentiated good. Demand arises from households’ consumption and from the exogenous component G , which is an aggret gate of differentiated goods of the same form as households’ consumption. It follows (cid:16) (cid:17)−θ that total demand for the output of firm j takes the form YD(j) = Pt(j) YD. t Pt t YD is an index of aggregate demand which satisfies YD = C +G . t t t t 13

Firms have the production function Y (j) = A Lα(j) t t t where L is the labour index L defined above and A is a mixture of two shocks t t t A = Z B such that, in logs, t t t b = b +ξ +εξ, εz ∼ N (0,σ ) t t−1 t t+1 ξ z = ρ z +εz, εz ∼ N (cid:0) 0,σ (cid:1) t z t−1 t t+1 z,sz,t where ξ is the long run productivity growth rate. This specification allows for both a standard, stationary technology shock and for a stochastic trend, represented by B . t For the solution and estimation of the model, we will explicitly take the stochastic trend into account. As in Rotemberg (1982), we assume the firms face quadratic costs in adjusting their prices. This assumption is also adopted, for example, by Schmitt-Groh´e and Uribe (2004b) and it is known to yield first-order inflation dynamics around a zero inflation steady state equivalent to those arising from the assumption of Calvo pricing.10 From our viewpoint, it has the advantage of greater computational simplicity, as it allows us to avoid including an additional state variable in the model, i.e. the cross-sectional dispersion of prices across firms. The specific assumption we adopt is that firm j faces a quadratic cost when changing its prices in period t, compared to period t − 1. Consistently with what is typically done in the Calvo pricing literature, we modify the original Rotemberg (1982) formulation for partial indexation of prices to lagged inflation. More specifically, we assume the following specification for the quadratic adjustment cost (cid:32) (cid:33)2 ζ Pj t −(Π∗)1−ιΠι Y 2 Pj t−1 t t−1 where Π∗ is the central bank’s inflation target. In a symmetric equilibrium, firms’ 10The equivalence does not hold exactly around a positive inflation steady state – see Ascari and Rossi (2010). Moreover two pricing models have in general different welfare implications – see Lombardo and Vestin (2008). 14

profits maximization problem leads to (θ−1)Y +ζ (cid:0) Π −(Π∗)1−ιΠι (cid:1) Y Π = t t t−1 t t (cid:18) (cid:19)1 θ w t Y t α +E Q ζ (cid:0) Π −(Π∗)1−ιΠι (cid:1) Y Π αP A t t,t+1 t+1 t t+1 t+1 t t 2.4 Monetary policy and market clearing We close the model with the simple Taylor-type policy rule (cid:18) Π∗(cid:19)1−ρI (cid:18) Π (cid:19)ψ Π (cid:32) Y(cid:101) (cid:33)ψY I = t t IρI eηt+1 t β Π∗ Y(cid:101) t−1 where Y(cid:101) ≡ Y /B is detrended aggregate output, Y(cid:101) its steady state level, Π∗ is the t t t constant inflation target and η is a policy shock such that t+1 η t+1 = eεη t+1, εη t+1 ∼ N (cid:0) 0,σ η,sη,t (cid:1) . Market clearing in the labour market requires labour demand to equal labour supply. In addition, the total demand for hours worked in the economy must equal the sum of the hours worked by all individuals. Taking into account that at any point in time the nominal wage rate is identical across all labor markets because all wages are allowed to change optimally, individual wages will equal the average w . As a result, all households will chose to supply the same amount of labour and t labour market equilibrium will require that (cid:18) (cid:19)1 Y α t L = t A t Market clearing in the goods market requires ζ Y = C +G + (cid:0) Π −(Π∗)1−ιΠι (cid:1)2 Y t t t 2 t t−1 t where G is an exogenous stochastic process which captures additional non-interestt rate-sensitive components of output and which we specify in deviation from the stochastic growth trend B , so that t G (cid:18) gY (cid:19)1−ρg (cid:18) G (cid:19)ρg B t = B B t−1 eεg t εG t+1 ∼ N (cid:0) 0,σ G,sG,t (cid:1) t t−1 15

where the long run level g is specified in percent of output, so that g ≡ G/Y. The variable G is a common way to introduce demand shocks in the model t (see e.g. Rudebusch and Swanson, 2012). It also breaks the theoretical equivalence between GDP (net of price adjustment costs) and consumption. As a result, we will use both these variables in estimation. 3 Solution and estimation methods 3.1 Solution To solve the model, we first approximate the system around a deterministic steady state in which all real variables are detrended by the technological level B . In the t solution, we expand variables around their natural logarithms, which are denoted by lower-case letters. We collect all (detrended) predetermined variables (including both lagged endogenous predetermined variables and exogenous states with continuous support) in a vector x and all the non-predetermined variables in a vector y . t t The reduced form of the model can thus be written in compact form as y = g(x ,σ,s ) t t (cid:101) t x = h(x ,σ,s )+σΣ(s )u t+1 t (cid:101) t (cid:101) t t+1 for matrix functions g(·), h(·), and Σ(·) and a vector of i.i.d. innovations u . t The vector s includes the state variables that index the discrete regimes. σ is a t (cid:101) perturbation parameter. Following Hamilton (1994), we can write the law of motion of the discrete processes s as t s = κ +κ s +ν t+1 0 1 t t+1 for a vector κ and a matrix κ . The law of motion of state s , for example, 0 1 z,t is written as s = (1−p ) + (−1+p +p )s + ν , where ν is an z,t+1 z,0 z,1 z,0 z,t z,t+1 z,t+1 innovation with mean zero and heteroskedastic variance. 16

For the solution, we follow the approach described in Amisano and Tristani (2011), which exploits the model property that regime switches only affect the shock variances. We can therefore apply standard perturbation methods (as in, for example, Schmitt-Groh´e and Uribe, 2004a, or Gomme and Klein, 2011) and approximate the solution as a function of the state vector x and perturbation parameter σ, but t (cid:101) keep it fully nonlinear as a function of the vector s . More specifically, we seek a t second-order approximation to the functions g(x ,σ,s ) and h(x ,σ,s ) around the t (cid:101) t t (cid:101) t non-stochastic steady state, namely the point where x = x and σ = 0. t (cid:101) Due to the presence of the discrete regimes in the system, both the steady state and the coefficients of the second order approximation could potentially depend on s in a nonlinear fashion. Since the discrete states only affect the variance of the t shocks, however, they disappear when σ = 0 so that the non-stochastic steady state (cid:101) is not regime-dependent. Amisano and Tristani (2011) demonstrate that the second order approximation can be written as 1 (cid:0) (cid:1) g(x ,σ,s ) = Fx + I ⊗x(cid:48) Ex +k σ2 (6) t (cid:101) t (cid:98)t 2 ny (cid:98)t (cid:98)t y,st(cid:101) and 1 h(x ,σ,s ) = Px + (I ⊗x(cid:48))Gx +k σ2 (7) t (cid:101) t (cid:98)t 2 nx (cid:98)t (cid:98)t x,st(cid:101) where F, E, P and G are constant vectors and matrices and only the vectors k y,st and k are regime dependent. x,st The second order approximation proposed by Amisano and Tristani (2011) can be interpreted as a particular case of the more general solution method provided by Foerster et al. (2016), namely a situation in which no Markov switching parameter affects the steady state solution of the model. See Foerster et al. et al., p. 639. Note that regime-switching plays no role to a first order approximation. The quadratic terms in the vector of predetermined variables with continuous support arealsoregimeinvariant. Changesinvolatilityonlyhaveanimpactonthequadratic terms in the perturbation parameter σ. Such terms would be constant in a model (cid:101) with homoskedastic shocks. 17

3.2 Estimation Exploiting this feature of the solution, the reduced form system of equations (6) and (7) can be re-written as 1 (cid:16) (cid:17) y = k +Fxˆ + I ⊗xˆ(cid:48) Exˆ +Dv (8) t y,j t+1 2 ny t+1 t+1 t+1 1 (cid:16) (cid:17) x = k +Pxˆ + I ⊗xˆ(cid:48) Gxˆ +σ ×Σ ×u (9) t+1 x,i t 2 nx t t (cid:101) i t+1 s (cid:118) Markov switching with (8×8) transition probability P t P = P ⊗P ⊗P G η z (cid:20) (cid:21) p 1−p P = h,1 h,1 ,h = G,η,z h 1−p p h,0 h,0 where k = k y,j y,st+1=j k = k x,i x,st=i Σ = Σ(s = i),i = 1,2,..,8;j = 1,2,...,8. i t Note that there are, overall, 8 states, since the volatility of 3 shocks can be either low or high, with independent probabilities across different shocks. The vector y includes all observable variables, and v and u are measuret t+1 t+1 ment and structural shocks, respectively. In this representation, the regime switching variables affect the system by changing the intercepts k , k and the loadings y,j x,i of the structural innovations Σ (we indicate here with i the value of the discrete i state variables at t and with j the value of the discrete state variables at t+1). If a linear approximation were used, we would have a linear state space model withMarkovswitchingwherethelikelihoodcouldbeaccuratelycomputedbyafinite mixture Kalman filter based approximation (see Kim, 1994, Kim and Nelson, 1999, and Schorfheide, 2005). In the quadratic case, however, such approximation is not available, and one possible approach is to rely on Sequential Monte Carlo (SMC) techniques for the computation of the likelihood. (see Herbst and Schorfheide, 2016). SMC-based likelihood approximations turn out to be very computationally expensive and not 18

very reliable in the case of our model, given the frequent occurrence of abrupt volatility changes. Basedontheobservationthatquadraticterms1/2 (cid:0) I ⊗xˆ(cid:48) (cid:1) Exˆ and1/2 (cid:0) I ⊗xˆ(cid:48) (cid:1) Gxˆ ny t+1 t+1 nx t t in equations (8) and (9) tend to be small, we therefore proceed as follows. At any point in time, we first linearize the two quadratic terms around the conditional mean of the continuous state variables. In a homoskedastic setting, this would correspond to applying the extended Kalman filter (Durbin and Koopman, 2012, section 10.2). In our model with regime switching, the linearisation must be conditional on the prevailing regime. As a result, at any point in time we can rewrite equations (8) and (9) as y = (cid:101)k(i,j) +F(cid:101) (i,j)xˆ +Dv t y,t+1 t+1 t+1 t+1 x = (cid:101)k(i) +P(cid:101) (i)x +Σ u (cid:98)t+1 x,t t (cid:98)t i t+1 for suitably defined coefficients (cid:101)k(i,j) , F(cid:101) (i,j), (cid:101)k(i) and P(cid:101) (i). Note that, in contrast y,t+1 t+1 x,t t to the original system (6)-(6), in the above equations both the intercepts (cid:101)k(i,j) , y,t+1 (cid:101)k(i) and the slope coefficients F(cid:101) (i,j), P(cid:101) (i) become regime-dependent. Nevertheless, x,t t+1 t we are still in the world of linear state space models with Markov switching. To compute the likelihood, we can therefore apply Kim’s (1994) approximate filter— see the on-line appendix for a description of the likelihood compution procedure. We then combine the likelihood with a prior and sample from the posterior using a tunedMetropolis-Hastingsalgorithm. ThisapproachbasedontheextendedKalman FilterlinearisationiscomputationallymuchfasterthanusingsequentialMonteCarlo methods. 4 Empirical results 4.1 Functional forms In our empirical analysis we need to choose a functional form for the utility aggregator u{C −hΞ C ,1−N }. As shown by King, Plosser and Rebelo (1988), t t t−1 t 19

consistency with long run growth requires a functional form of the following type u = (C −hΞ C )v(N ) t t t−1 t where v(N ) is a decreasing function. Various options are available for v(N ). We t t rely on the particular specification proposed by Trabandt and Uhlig (2011), which implies a constant Frisch elasticity of labour supply in the absence of habits and with standard, expected-utility preferences. The specification implies (cid:18) (cid:19) ψ 1+1 1−ψ v(N ) = 1−η(1−ψ)N φ . t t 4.2 Data description We estimate the model on quarterly US data over the sample period from 1966Q1 to 2009Q1. Ourestimationsamplestartsin1966,becausethisisthedatewhenaTaylor rule begins providing a reasonable characterization of Federal Reserve policy.11 We end in 2009Q1 when the zero bound constraint, which we do not explicitly include in our model, becomes binding. Concerning the macro data, we use per capita total real personal consumption, per capita GDP and inflation. We need to use both GDP and consumption because the former enters the Taylor rule and the latter the Euler equation. Given that we abstract from investment, G will capture all not-interest-sensitive components of t GDP. Inflation is measured as the logarithmic first-difference in the consumption deflator (all macro variables are from the FRED database of the St. Louis Fed). We use continuously compounded yields on 3-month, 3-year and 10-year zero-coupon bonds from Gu¨rkaynak, Sack and Wright (2007).12 Prior to the analysis, we take logarithmic first differences for consumption and GDP,whichareassumedtofollowastochastictrend. Nootherdatatransformations 11According to Fuhrer (1996), ”since 1966, understanding the behaviour of the short rate has beenequivalenttounderstandingthebehaviouroftheFed,whichhassincethattimeessentiallyset the federal Funds rate at a target level, in response to movements in inflation and real activity”. Goodfriend (1991) argues that even under the period of official reserves targeting, the Federal Reserve had in mind an implicit target for the Funds rate. 12The data are available at http://www.federalreserve.gov/pubs/feds/2006. 20

are applied. All variables are expressed as quarterly rates, so that 0.0025 represents an annualized interest rate, inflation rate, or growth rate equal to 1 percent. 4.3 Prior distributions Prior and posterior distributions for our model are presented in Table 1. Concerning regime switching processes, we assume beta priors for transition probabilities. We expect the states to be relatively persistent, so we centre all distributions around a value of 0.9, which implies a persistence of 2.5 years for each state. Since it is well know that in mixture models the likelihood is invariant to label permutations of the discrete states, for each of the three discrete volatility states (government spending, technology and monetary policy shocks), we achieve identification by calling ”state 1” the low variance state, and we assign a symmetric prior across the two states. Prior and posterior draws not complying with the inequality constraint are therefore suitably permuted (see Geweke 2007). We use inverse gamma priors for the standard deviations of the shocks. With the exception of the technology growth shock, which has a tighter prior centred around a small value because the process is a random walk, we keep the prior distribution relatively dispersed around a mean value of 0.003. The regime-switching standard deviations also have the same prior distribution in the high and low regimes. To ensure identification, however, all draws from the prior are first ordered and then assigned to the high or low state. Table 1 reports the resulting empirical distribution for the prior of regimeswitching standard deviations. Concerning the persistence of the shocks, we use beta priors centred around the value of 0.85. Forthepolicyrule, weuserelativelyloosepriorscentredaroundparametervalues estimated from quarterly data over a pre-sample period running from 1953 to 1965, namely ρ = 0.85, ψ = 0.2 and ψ = 0.02. I Π Y The priors for all utility parameters are specified broadly in line with the rest of the literature. For the φ parameter we rely on a normal prior centred around 1.0, a value in between macro estimates and micro estimates of the Frisch elasticity 21

of labour supply (see e.g. the evidence reviewed in Chetty et al., 2011). We use a shifted Gamma distribution for ψ and γ, to ensure that ψ,γ > 1. We centre the distribution of ψ around a value above but close to 1. For the γ parameter, which contributes to shape risk aversion, we use a very large standard deviation whose prior 95 percent confidence set goes from 2 to 30. The habit parameter has a beta prior centred around 0.5. Finally, for β we use a relatively tight prior with a mean of 0.9985. This is consistent with assumptions made in models with growth–see e.g. Christiano, Motto and Rostagno (2014). For the long run parameters Ξ and Π∗ we rely on more dogmatic priors. For Ξ, which determines the growth rate of the economy in the non-stochastic steady state, we use a tight prior centred around 0.005. This implies an annualized growth rate of 2 percent, which is consistent with the average per-capita U.S. GDP/GNP growth from the 1870s to the 1950s–see Maddison (2013). For the inflation target, we choose a prior centred around 1.0063 that gives most mass to annualized values between 2 and 3 percent. Thepriceadjustmentcostζ istypicallycalibratedbasedontheimpliedfrequency of adjustment of prices in linearized models. In our model, however, the relationship ismorecomplexduetoboththenonlinearityofthemodelandthepresenceofsteady state inflation. We therefore centre the prior around 15, which is roughly consistent, for example, with the value used in Schmitt-Groh´e and Uribe (2004b), but allow for a relatively large standard deviation. For inflation indexation, we rely on a beta prior centred around 0.5. The elasticity of intratemporal substitution θ, which is weakly identified, is set dogmatically at 6. Similarly, we set the gross wage mark-up µ to 1.2. w 4.4 Posterior distributions The posterior distributions of structural parameters in Table 1 suggest that the data are informative about the estimation of most parameters, as witnessed by the typically smaller standard deviation of the posterior distribution compared to the prior distribution. 22

Table 1(a): Structural parameter estimates post mean post sd prior mean prior sd p 0.8760 0.0556 0.8997 0.0657 G,11 p 0.9413 0.0351 0.8994 0.0662 G,00 p 0.9595 0.0196 0.8996 0.0657 η,11 p 0.9079 0.0447 0.8998 0.0658 η,00 p 0.9728 0.0091 0.9013 0.0651 z,11 p 0.9317 0.0190 0.8993 0.0662 z,00 σ 0.0033 0.0021 0.0021 0.0008 G,1 σ 0.0316 0.0037 0.0039 0.0028 G,0 σ 0.0013 0.0001 0.0021 0.0008 η,1 σ 0.0039 0.0005 0.0039 0.0027 η,0 σ 0.0109 0.0021 0.0022 0.0008 z,1 σ 0.0271 0.0050 0.0039 0.0029 z,0 σ 0.1768 0.0217 0.0031 0.0026 µ σ 0.0062 0.0003 0.0012 0.0005 ξ ρ 0.5487 0.0581 0.8552 0.0916 µ ρ 0.9889 0.0018 0.8582 0.0899 z ρ 0.9091 0.0298 0.8559 0.0906 G Π 1.0061 0.0007 1.0063 0.0007 ψ 0.2676 0.0241 0.1990 0.1001 π ψ 0.0497 0.0075 0.0200 0.0100 y ρ 0.9135 0.0169 0.8494 0.1002 i Ξ 1.0045 0.0004 1.0050 0.0010 ι 0.7334 0.1116 0.5003 0.1899 φ 0.6156 0.0846 1.0023 0.5049 γ 11.5185 3.6747 10.9537 6.9730 ψ 1.3075 0.0868 1.2035 0.2900 ζ 33.8071 3.1344 14.9744 6.9819 h 0.8619 0.0261 0.4996 0.1886 β 0.9984 0.0006 0.9986 0.0014 Table1(b): Measurementerrors(annualized) post mean post sd prior mean prior sd σ 0.0006 0.0007 0.0006 0.0005 me,π σ 0.0005 0.0003 0.0006 0.0004 me,∆c σ 1.4429 0.2470 0.2018 0.1079 me,∆y σ 0.0005 0.0003 0.0005 0.0004 me,i σ 0.2878 0.0305 0.5512 0.4147 me,i12 σ 0.1748 0.0199 0.5525 0.3994 me,i40 Legend: ”sd” denotes the standard deviation. Priors: Beta distributionforβ,h,ι,ζ,ρ ,ρ ,ρ ; Gammadistributionfor ξ g π ψ ,ψ andallstandarddeviations; shiftedGammadistribu- π y tion (domain from 1 to ∞) for γ, φ, Ξ, Π∗; Gaussian distribution for ρ . Posterior distributions are based on 200,000 i draws. 23

More specifically, the different regimes in the volatilities of monetary policy, technology and government spending shocks are clearly identified. For monetary policy, the standard deviations in the low and high regimes are equal to 0.13 percent and 0.39 percent respectively. These values straddle the constant standard deviation of 0.24 percent estimated in Smets and Wouters (2007). The standard deviation of technology shocks change between 1.1 percent in the low volatility regime and 2.7 percent in the high volatility regime. The difference between the two volatility regimes is largest for demand shocks: their standard deviation shifts between 0.33 and 3.2 percent. The posterior mode of the transition probabilities suggests that the low-volatility states are more persistent for monetary policy and technology shocks. For policy shocks, the ergodic probability of being in the low-volatility state is approximately 0.69, which is consistent with the idea that policy shocks were small over most of the sample, except for the Volcker disinflation period. Both the low and the high volatility states are more persistent for technology shocks. These states are countercyclical, being persistently high during recessions and low over expansions. The ergodic probability of the low volatility state for technology shocks is 0.71. By contrast, the volatility of demand shocks is more persistent in the high state, whose ergodic probability is 0.68. Based on these results, we refer to low-volatility regimes for policy and technology shocks as ”normal regimes”. As in estimates solely based on macro data, shock processes tend be highly serially correlated. At 0.99, the correlation of the level technology shock process is especially high. Together with the features of the monetary policy rule, this implies that technology shocks have very persistent effects. The estimated parameter values of the coefficients of the monetary policy rule are of particular interest. Ceteris paribus different parameters of the policy rule will be associated with different expectations of the future path of short-term interest rates, thus different configurations of the yield curve. Since we explicitly use yields data when estimating the model, our estimates of the policy rule parameters should be more informative than those obtained without including yields in the 24

econometrician’s information set. Given the well-known problems of general equilibrium models to match the unconditional volatility of long-term yields (see e.g. Den Haan, 1995), one would expect the degree of interest rate smoothing to be higher than in estimates ignoring yields data. A higher smoothing coefficient would impart persistence to any movements in the short-term rate. Its variability would thus be transmitted to longer rates (for a discussion of this point, see Ho¨rdahl, Tristani and Vestin, 2008). To compare our estimates to those in the literature, it is useful to rewrite the rule (2.4) in partial adjustment form. In deviation from the non-stochastic steady state, our parameter estimates imply under a first order approximation13: (cid:104) (cid:16) (cid:17)(cid:105) (cid:98)i t = 0.09 3.09 (π t −π∗)+0.57 (cid:98)y (cid:101)t −(cid:98)y (cid:101) +0.91(cid:98)i t−1 +η t+1 . (10) where hats denote deviations from the non-stochastic steady state and tildes denote variables in deviation from the stochastic growth trend. Equation (10) confirms the above intuition. Compared to the estimates in Smets and Wouters (2007), our parameters imply a somewhat higher, but not exceedingly high, inflation response coefficient.14 The more striking feature of our estimates, however, is the increase in the degree of interest rate smoothing (0.91 vs. 0.81 in Smets and Wouters). More inertial movements in short-term rates imply that longer-term yields can be systematically affected by monetary policy. This feature is important for the model to be able to generate variation at longer maturities in the term-structure of interest rates.15 The estimates of the other structural parameters are roughly consistent with the existing literature. Concerning long-run means, the mode of the quarterly trend growth rate of 13Second and higher order terms would be zero, because the policy rule is log-linear. 14The parameter estimates are not fully comparable, because the policy rule used in Smets and Wouters (2007) includes additional arguments. 15DeGraeve,EmirisandWouters(2009)alsousesyieldsdatainestimation,butobtainsinterest ratesmoothingestimatessimilartoSmetsandWouters(2007). InDeGraeve,EmirisandWouters (2009),however,persistentmovementsinpolicyinterestratesaredrivenbychangesinastochastic inflation target, which is almost a random walk. 25

technologyis0.45andthequarterlyinflationtargetis0.61, bothwithintheposterior distribution of estimates obtained in Smets and Wouters (2007). Amongst preference parameters, the posterior mean of φ is 0.6. Our estimate of ψ implies a long-run elasticity of intertemporal substitution of consumption of 0.76, which is in line with other available estimates (see e.g. Basu and Kimball, 2002). The γ parameter is equal to 11.5 and the habit parameter h = 0.86. Together, these two parameters are suggestive of a high level of risk aversion, which is in line with the results in Piazzesi and Schneider (2006), or in Rudebusch and Swanson (2012). 4.5 Goodness of fit measures The model fit is good for both macroeconomic and yields data. This claim is supported by various pieces of evidence. First, measurement errors on all variables are small. This is perhaps not surprising for macro variables and for the short-term interest rate, given the results in Smets and Wouters (2007). For longer-term yields, however, one could expect a worse performance. Nevertheless, both 3-year and 10-year rates are fit rather well. The measurement errors on these two variables are equal to 29 and 18 basis points, respectively. This is a comparable fit to the results in more empirically flexible models such as Ang and Piazzesi (2003).16 Second, wechecktheimplicationsofourmodelforthedynamicauto-correlations of all observable variables–see Figure (1). The auto-correlations of consumption growth are particularly interesting, because of the difficulty for simple asset pricing models to simultaneously generate positive serial correlation in consumption growth and a positive slope of the term structure–see Backus, Gregory and Zin (1989). We compare model-implied auto-correlations at lags up to 20 quarters to sample autocorrelations. Model-based correlations are computed for all posterior draws. Figure (1) shows the mean and error bands corresponding to a 95 percent confidence set. 16Ang and Piazzesi (2003) is however estimated on more volatile, monthly data. 26

Figure 1: Model implied and sample dynamic correlations D p D c y 1.00 1.00 0.8 0.75 0.75 0.50 0.50 0.4 0.25 0.25 0.0 0.00 0.00 −0.25 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 i i i 12 40 1.00 1.00 1.00 0.75 0.75 0.75 0.50 0.50 0.50 0.25 0.25 0.25 0.00 0.00 0.00 −0.25 −0.25 −0.25 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 90% bands Posterior mean Sample Note: dotted lines display auto-correlation coefficients from the data; solid lines and shaded areas denote the mean and the 5th and 95th percentiles of the distribution across parameter draws of the theoretical correlation coefficients implied by the model. The figure indicates that the distribution of model-implied auto-correlations always includes its empirical counterpart. The fit is especially good for short and long-term interest rates, whose mean model-implied auto-correlation almost perfectly matches the sample autocorrelation at all lags. Third, we test the implications of our model for dimensions of the data which were not directly used in estimation, notably for forward rates at various horizons. Model-implied and actual 3-month forward rates in 1, 3 and 10 years are reported in Figure (2). Note that the 1-year rate was not used in estimation. Nevertheless, the model tracks well the evolution of all forward rates. 27

Figure 2: Model based and actual forward rates at different maturities, % p.a. 1−year 15 10 5 0 1970 1980 1990 2000 2010 3−year 16 12 8 4 1970 1980 1990 2000 2010 10−year 15 12 9 6 1970 1980 1990 2000 2010 Actual Model Note: forward rates are taken from Gu¨rkaynak, Sack and Wright (2007). 4.6 Conditional standard deviations and volatility regimes As a final validation test, we compare model-implied conditional volatilities of the macro variables to a few reduced form counterparts. Since there is no unique way of computing conditional standard deviations, we try two alternative approaches. The first one is a Markov switching VAR including the same observable variables as our DSGE model. It is a natural benchmark for comparison, because it allows for a similar type of parameter variation as the structural model, but without theory-based cross-equation restrictions. To maximize comparability, we allow for 8 parameter regimes for means and variances. Mean and variance parameters are all assumed to 28

switch simultaneously. In the second approach, we use for each series the simple, univariate unobserved component model proposed by Stock and Watson (2007). Annualized conditional standard deviations for inflation, the short-term interest rate and the rates of growth of consumption and GDP are reported in Figure (3). Forallfourvariables, conditionalstandarddeviationsarecomputedaccordingtoour DSGEmodelandtothetworeduced-formmodels. Thefigurealsoshowsconditional standard deviations implied by the Smets and Wouters (2007) model, which are constant over time given that the model is conditionally Gaussian. Figure 3: Conditional one-step- ahead standard deviations, % p.a. p D c 6 4 4 2 2 0 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 D y i 10.0 4 7.5 3 2 5.0 1 2.5 0 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 MS−DSGE SW−DSGE SW−UC MS−VAR Note: MS-DSGE indicates the model used in this paper, SW-DSGE is the Smets and Wouters (2007) linear DSGE model, SW-UC is the Stock and Watson (2007) univariate unobserved component model with stochastic volatility model, and MS- VAR denotes an 8-state Markov Switching VAR where only the intercepts and the shock covariance matrix is allowed to vary across regimes. 29

Estimates from the two reduced form models concur on detecting some degree of time-variation in conditional standard deviations. Compared to the estimates in the Smets and Wouters (2007) model, they tend to find high variances in the seventies and lower variances in the nineties. However, the estimates also show that estimates of conditional standard deviations are somewhat model dependent. The unobserved component model finds very large time-variation over the sample. For inflation, for example, the standard deviation climbs to high peaks in the seventies and then falls to low levels in the nineties. By contrast, the Markov switching VAR finds smaller variations over time. Taking into account heterogeneity across models, the estimates obtained from our MS-DSGE are broadly in line with the other benchmarks. Our model also tends to find high standard deviations in the seventies and lower in the nineties. The range of variation, however, is smaller than in the reduced-form specifications for consumption and the short-term interest rates. The exception is inflation, which is found to have occasional volatility bursts also in the nineties. Figure (4) displays filtered and smoothed estimates of the probability of being in a high-variance regime for the three heteroskedastic shocks. 30

Figure 4: Probabilities of high volatility states G 1.00 0.75 0.50 0.25 1970 1980 1990 2000 2010 Monetary Policy 1.00 0.75 0.50 0.25 0.00 1970 1980 1990 2000 2010 Technology 1.00 0.75 0.50 0.25 0.00 1970 1980 1990 2000 2010 Filtered Smoothed Note: filtered and smoothed probabilities are computed by averaging across draws from the full sample posterior probability of the parameters. Thedemandshockhashighvarianceinthefirstpartofthesampleandlowervariance during the Great moderation period. Concerning the monetary policy shock, our results are consistent with those in Justiniano and Primiceri (2008), where heteroskedasticity takes the form of stochastic volatility, rather than regime switching. The policy shock has a high variance during the mid-1970s and again during the so-called ”Volcker disinflation” period in 1979-83. One marginally different feature of our results, is that the increase in volatility in 1979 is estimated to be very rapid in real time. This is arguably consistent with the spikes which can be observed in the short term interest rate over this period. Such sudden increases in volatility can more easily be captured by a regime-switching model than by a stochastic volatility model. The most striking feature of the regimes for the variance of technology shocks in Figure (4) is that they are strongly cyclical. Starting in 1980, the standard deviation of these shocks tends to increase at the beginning of each recessions and 31

to fall again after a few quarters. This pattern is quite systematic, especially over the 1990s and the 2000s. The period of the Volcker disinflation is therefore unique in being characterized by high variance of government spending, policy and technology shocks. 5 The role of uncertainty shocks We next focus on the impulse responses to uncertainty shocks. To understand the impact of these shocks, consider the second-order approximation to the Euler equation (cid:18) (cid:19) 1 h 1 (cid:16) (cid:17) 1−h 1 n (cid:98) (cid:101) c t = E t ((cid:98) (cid:101) c t+1 )+ (cid:98) (cid:101) c t−1 − (cid:98)i t −E t [π (cid:98)t+1 ] + 1+ E t ∆(cid:98)l t+1 1+h 1+h ψ 1+h φ 1−n (cid:20) (cid:18) (cid:19) (cid:21) 1−h 1 (cid:16) (cid:17) 1 n −(γ −ψ) 1+h Cov t 1−h ∆(cid:98) (cid:101) c t+1 −h∆(cid:98) (cid:101) c t + 1+ φ 1−n ∆(cid:98)l t+1 ,ξ(cid:98) t+1 +(cid:98) (cid:101)j t+1 γ −ψ (cid:104) (cid:105) 1 ψ −1 (cid:104) (cid:105) − ψ Cov t ψξ(cid:98) t+1 +π (cid:98)t+1 ,ξ(cid:98) t+1 +(cid:98) (cid:101)j t+1 + 2 (γ −ψ) ψ Var t ξ(cid:98) t+1 +(cid:98) (cid:101)j t+1 1 − Var Ω (11) t t+1 2 where all variables are expressed in deviation from the non-stochastic steady state. More specifically, (cid:98)c denotes consumption in deviation from the stochastic trend, (cid:101)t (cid:98)i t denotes the nominal interest rate, π (cid:98)t the inflation rate, (cid:98)l t hours worked, ξ(cid:98) t the productivity growth rate and (cid:98) (cid:101)j is the conditional present discounted stream of t future consumption and hours (cid:98) (cid:101)j t+1 +ξ(cid:98) t+1 (cid:34) (cid:32) (cid:33)(cid:35) ∞ (cid:18) (cid:19) = (cid:88)(cid:0) βΞ1−ψ (cid:1)i E ξ(cid:98) + (cid:0) 1−βΞ1−ψ (cid:1) (cid:98) (cid:101) c t+1+i −h(cid:98) (cid:101) c t+1+i−1 − ψ 1+ 1 n (cid:98)l t+1 t+1+i t+1+i 1−h 1−ψ φ 1−n i=0 Finally ψ ≡ ψ1+h is the inverse of the short-run elasticity of intertemporal 1−h substitution, n ≡ η(1−ψ)N1+1/φ and Var Ω denotes other second order terms t t+1 such as Var(cid:98)c , or Var ξ(cid:98) which tend to be quantitatively small and which are t(cid:101)t+1 t t+1 are made explicitly in the appendix. The first row on the right-hand-side of equation (11) includes first-order terms. As in the standard new-Keynesian model with habits (see e.g. Woodford, 2003), 32

consumption is partly backward-looking, partly forward looking, and it is negatively related to the real interest rate. The following terms in the equation arise under Epstein-Zin-Weilutilityandinvolveconditionalvarianceandcovariancesofexpected future utility news, (cid:98) (cid:101)j t+1 +ξ(cid:98) t+1 . The covariance term in the second row is positive when news about expected future utility growth tend to be associated with high expected growth in consumption and hours worked at t+1. When this covariance increases, i.e. expected future utility growth becomes riskier, households increase their precautionary saving and reduce their consumption. This trasmission channel is stronger, the more agents are unwilling to adjust their level of utility across states (i.e. the higher γ). Note, however, that this channel is dampenend by the adjustment in the real interest rate, which falls, thus stimulating the demand for consumption goods, to restore the equilibium in the savings market. A similar effect is produced by increases in the first covariance on the last row of the equation, which is positive when news about expected future utility growth are associated with high inflation and/or high productivity growth at t + 1. When ψ (cid:54)= 1 increases in the variance of revisions of expected future utility also play a role and tend to boost consumption, according to the following term in the last row of the equation. This effect is, however, small for our estimated parameter values. To summarize, increases in the volatility of structural shocks which bring about an increase in the conditional covariance terms in equation (11) tend to produce an increase in precautionary saving and a fall in consumption. For our estimated parameter values, these effects are small in reaction to changes in the conditional variance of demand and policy shocks, but non-negligible after switches in the standard deviation of (level) technology shocks. We report in Figure 5 the impulse responses to an increase in the variance of technology from the low to the high regime. For illustrative purposes, in this figures we assume that no further changes in the variance regime occur after the shock. 33

Figure 5: Impulse responses to a one-off volatility shift for the technology shock p y c 0.0 0.00 0.0 −0.1 −0.05 −0.1 −0.2 −0.10 −0.2 −0.3 −0.15 −0.3 −0.20 −0.4 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 i i i 12 40 0.10 −0.1 −0.04 0.08 −0.2 −0.08 0.06 −0.3 −0.12 0.04 −0.4 0.02 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Mean Median 90% bands Note: the impulse response functions describe the reaction to a one-off regime shift in volatility, from low to high state. The responses reported here are scaled by the difference between the standard deviations of the technology shocks in the two regimes. The increase in the variance of technology shocks generates an increase in the demand for precautionary saving. As a result, the demand for consumption goods falls. Given that prices are sticky and output is demand determined, lower demand for consumption goods generates a fall in output and inflation. The policy rate also falls but, due to the high smoothing coefficient in the Taylor rule, very slowly over time. Afteronetotwoyears,consumption,outputandinflationreturnclosertotheir initial level, but the policy rate remains low for much longer, because the persistent nature of the uncertainty shock keeps the demand for precautionary saving high and it drives down real rates. The impulse response of 3-year interest rates also falls, but more mutedly, while 10-yearratesincreaseslighly. Theseimpuseresponsescompoundtheaforementioned expectations of lower future policy rates with a generalized increase in risk premia 34

duetothehightenedtechnologicaluncertainty. Wedescribevariationsinrisk-premia in more detail in the next section. All in all, an uncertainty shock in technology looks like a demand shock, in the sense of being associated with a fall in output, consumption and prices at the same time. Our results corroborate, in the context of an estimated model, Basu and Bundick’s (2012) finding that a persistent fall in nominal interest rates is an important part of the macroeconomic adjustment mechanism, following an uncertainty shock. If the fall in the nominal interest rate were prevented by the zero lower bound, the macro-economic effects of the shock would be even larger. To gauge the role of uncertainty shocks in driving economic dynamics, Table 2 reports the forecast error variance decompositions of all endogenous variables over horizons of 1, 12 and 40 quarters ahead. Consistently with standard results based on linearized models, uncertainty shocks play a negligible role for most macro variables both at short (1 quarter) and business cycle (3-year) horizons. The only exception are changes in the standard deviation of monetary policy shocks, which account for 50% of the 1-quarter ahead variance of short-term rates. This result is in line with the high volatility of policy rates during the high-inflation years at the beginning of the sample and during the Volcker disinflation. Uncertainty shocks to future productivity play no role in the short-term, but they start playing some role at business-cycle frequency and they become the main driver of the variance of yields over long-term horizons. More specifically, over a forecast horizon of 40 quarters, technological uncertainty shocks account for 53% of the variance of 10-year yields, 52% of the variance of 3-year yields, 45% of the variance of the policy rate, and 33% of the variance of inflation. 35

Table 2a: Variance decomposition, one step ahead π ∆c ∆y i i i 12 40 σ 2[0-4] 0[0-0] 39[24-51] 3[2-6] 9[5-14] 2[1-4] G σ 16[7-26] 2[1-5] 1[0-1] 54[31-70] 1[0-2] 0[0-0] η σ 2[0-7] 0[0-0] 0[0-0] 0[0-2] 5[0-21] 6[0-26] z εG 0[0-0] 0[0-0] 1[0-2] 0[0-0] 0[0-1] 0[0-0] εη 4[3-6] 1[0-1] 0[0-0] 16[8-27] 0[0-0] 0[0-0] εz 15[10-20] 0[0-0] 0[0-0] 4[2-7] 56[41-68] 70[50-81] εξ 1[1-2] 96[93-98] 43[36-49] 5[2-8] 0[0-0] 0[0-0] εµ 59[48-69] 1[0-1] 0[0-0] 16[11-24] 7[4-11] 1[0-2] meas. 0[0-0] 0[0-0] 16[9-27] 0[0-0] 16[11-22] 13[8-18] Table 2b: Variance decomposition, 12-steps-ahead π ∆c ∆y i i i 12 40 σ 3[2-5] 1[1-2] 35[23-45] 6[3-10] 3[1-5] 1[0-1] G σ 14[7-24] 2[1-5] 1[0-2] 7[3-12] 0[0-0] 0[0-0] η σ 19[9-30] 3[1-7] 1[0-2] 25[13-40] 37[20-55] 38[21-56] z εG 0[0-0] 0[0-0] 1[0-3] 0[0-1] 0[0-0] 0[0-0] εη 5[3-7] 1[0-1] 0[0-0] 2[2-4] 0[0-0] 0[0-0] εz 22[16-29] 4[2-6] 1[1-2] 40[29-50] 56[39-71] 59[42-76] εξ 2[1-3] 83[75-90] 43[37-50] 1[0-1] 0[0-0] 0[0-0] εµ 34[27-43] 5[3-9] 2[1-3] 18[13-24] 2[1-3] 0[0-0] meas. 0[0-0] 0[0-0] 16[9-26] 0[0-0] 2[1-2] 1[1-2] Table 2c: Variance decomposition, 40-steps-ahead π ∆c ∆y i i i 12 40 σ 2[1-4] 1[1-2] 33[21-44] 3[1-5] 1[0-2] 0[0-0] G σ 9[4-17] 2[1-5] 1[0-2] 3[1-6] 0[0-0] 0[0-0] η σ 33[19-47] 5[2-10] 2[1-4] 45[28-60] 52[34-68] 53[34-68] z εG 0[0-0] 0[0-0] 1[0-3] 0[0-0] 0[0-0] 0[0-0] εη 4[2-5] 1[0-1] 0[0-0] 1[1-2] 0[0-0] 0[0-0] εz 27[20-35] 4[2-6] 1[1-2] 38[26-52] 45[30-62] 46[31-64] εξ 1[1-2] 81[72-89] 44[37-51] 0[0-1] 0[0-0] 0[0-0] εµ 24[18-31] 5[3-9] 2[1-3] 9[6-13] 1[0-1] 0[0-0] meas. 0[0-0] 0[0-0] 16[9-26] 0[0-0] 1[0-1] 0[0-1] Legend: i denotes the 10-year rate; i is the 3-year rate; i is the short-term rate; 40 12 π is the inflation rate; ∆c denotes the rate of growth of consumption; ∆y is the rate of growth of GDP. The shocks indicated in the first column are the three variance shifts and the five continuous shocks: on government spending, monetary policy shock, temporary and permanent technology components, and mark-up. meas. is shorthand for measurement error. The variance decomposition is reported 1, 12 and 40 quarters ahead. 36

To summarize, uncertainty shocks over the evolution of future productivity play a central role in driving nominal yields over long-term horizons. In the next section, we will demonstrate that this result is related to variations in risk-premia and in expected future real interest rates. The remaining half of the variance of yields over long horizons is explained by standard technology shocks, which produce effects on yields partly because they are extremely persistent, partly due to the endogenous persistence induce by the high interest rate smoothing coefficient. 6 Monetary policy and long term rates We have shown that uncertainty shocks have macroeconomic effects. We now investigate their effects on bond prices. 6.1 Monetary policy and risk premia Nominal bonds reflect risk premia associated with both consumption risk and with inflation risk. Ho¨rdahl, Tristani and Vestin (2008) demonstrate that models with homoskedastic shocks solved to a second order approximation can only generate constant risk premia. Consistently with this result, our model can produce changes inriskpremiaonlywhenthereisachangeinthestandarddeviationofthestructural shocks. In other words, time variation in risk premia is associated with switches in the variance regimes. Atypicallyusedmeasureofriskpremiawhichisindependentofexpectedchanges in the future path of short term interest rate is the expected excess holding period return on a bond of maturity n. This corresponds to the expected return that can be earned by holding an n-maturity bond for one quarter in excess of the one quarter interest rate. To second order, the expected excess holding period return can be written as (cid:2) (cid:3) (cid:98)h n,t −(cid:98)i t = F Bn−1 E t u t+1 u(cid:48) t+1 × (cid:20) (cid:18) (cid:19) (cid:21) 1 1 n ψ F(cid:48) +ψ 1+ F(cid:48) +ψF(cid:48) +F(cid:48) +(γ −ψ)F(cid:48) (12) 1−h c φ 1−n l ξ π j 37

where F denotes the vector of parameters of the first order approximation to the z law of motion of any variable z. This equation highlights that all terms in the excess holding period return are constant, except for the variace-covariance matrix (cid:2) (cid:3) of the structural shocks E u u(cid:48) . Hence, changes in expected excess holding t t+1 t+1 period returns occur as a result of regime switches in conditional variances. Our model approximation generates differences in the market prices of risk across variance regimes, but regime-switching risk is not priced. The first term on the right-hand-side of equation (12), F , represents the first- Bn−1 order impact of each of the structural shocks in vector u on the price of a bond t+1 with maturity n−1. This term captures the degree of riskiness of the bond, namely the typical fluctuation in its price which the representative household can expect. The expression in brackets on the right-hand-side of equation (12) has to do with the household’s willingness to bear a risky bond, or price of risk. It suggests that the price of risk is a function of all dimensions which affect household utility, not just the curvature of the utility function with respect to consumption.17 Swanson (2012, 2018) derive general relationships between the excess holding period return (simply referred to as ”risk premium” in those papers) and an appropriate measure of relative risk aversion in models with endogenous labor supply. The (filtered) expected excess holding period return generated by our model for the 3-year and 10-year bonds is displayed in Figure (6). 17The parameters h, ψ, φ and γ may enter equation (12) also through the endogenous vectors F , F , F and F . c l π j 38

Figure6: Modelimpliedpremia(expectedexcessholdingperiodreturns)atdifferent maturities, % p.a. 15 10 5 0 1970 1980 1990 2000 2010 .a.p % 3−year 10−year Note: filtered and smoothed premia are computed by averaging across draws from the full sample posterior probability of the parameters. Excess holding period returns are relatively large. At the 10-year maturity, they average about 8 percentage points in annualized terms over our estimation sample. This value is in the same order of magnitude as some estimates from the finance literature–see e.g. Figure 1 in Duffee (2002). Also consistently with the finding in that literature (see e.g. Fama and French, 1989), risk premia tend to be countercyclical. After remaining around 3 percentage points over the first decade of our sample, they spike to 12 percentage points in annualized terms around most recessions from 1980 to 2008. In contrast to the finance literature, however, we find variations in risk premia to be much more infrequent. Our results suggest that periods of either high or 39

low risk premia are very persistent, since they go along with stochastic switches in conditional variance regimes. We therefore miss the higher frequency fluctuations which are typically uncovered by finance estimates. One would expect that regime switches in the volatility of all shocks lead to variation in risk premia. In the model, however, variations in risk premia must be associatedwithuncertaintyaboutrevisionsintherateofgrowthoffutureutilityand with their correlations with inflation and with the marginal utility of consumption– see also Restoy and Weil (2011) and Piazzesi and Schneider (2006). From a quantitative perspective, monetary policy and demand shocks have a small impact on the rate of growth of utility over long future horizons. Changes in their variance have therefore a small impact on the size of risk premia. This can be observed through a comparison of figures 4 and 6. Thekeysourceofquantitativelysizabletime-variationinriskpremiaareswitches in the variance of technology shocks. Since these variance regimes are estimated very precisely, also in real time, risk premia oscillate mostly between a high and a low value. Consistently with the cyclicality of technological uncertainty shocks, risk premia increase during every NBER-dated recession, then fall again after a few years. It goes without saying that considerable uncertainty characterizes any estimates of risk premia, because of estimation and model uncertainty. Figure 6 shows that filtering uncertainty is around 5 percentage points at the 10-year maturity. In a classicaleconometricsetting, thesmallsamplebiasinmaximumlikelihoodestimates also plays a role–see e.g. Bauer, Rudebusch and Wu (2014), and Wright (2014). 6.2 Yields and the monetary policy transmission mechanism We have shown that changes in the conditional variances of technology shocks play an important cyclical role. At the beginning of recessions, the increase in volatility is tantamount to a persistent fall in confidence: precautionary saving increases, con- 40

sumption, output and inflation fall, and so do current and expected future nominal interest rates. After the recovery sets in, the conditional variance of technology switches back to lower levels and confidence returns. The demand for precautionary saving becomes low again, expected future policy interest rates increase. Thedynamicsinducedbyuncertaintyshocksalsohaveimplicationsforlong-term yields. Their movements are the results of three different forces. First, as we have already discussed in the previous section, uncertainty shocks generate fluctuations in risk premia. Cyclical increases in uncertainty lead to larger risk premia, thus higher yields levels. Second, uncertainty shocks produce an increase in precautionary saving, which exerts downward pressure on current and expected future real rates. As a result, nominal yields also tend to fall. These two effects are not present in a model with constant premia, which therefore does not provide a good description of long-term yields during cyclical turning points. A famous example of this shortcoming occurred in 2004, when long-term yields did not increase in the face of an increase in expected future policy rates. Such behavior of long term yields appeared to be an anomaly compared to previous cyclical developments. In his semiannual Monetary Policy Report to the Congress, ChairmanGreenspanstatedthat”thebroadlyunanticipatedbehaviorofworldbond markets remains a conundrum”–see Greenspan (2005). From the perspective of our model, the conundrum is accounted for by the timing of the fall of uncertainty back to normal levels after the recession. In 2004 it overlapped with the beginning of the monetary policy tightening cycle. The combination of lower risk premia and higher expected future policy rates left observed yields essentially unchanged. The third determinant of long-term yields is more conventional. Over periods in which volatility stays constant, long-term rates react to changes in monetary policy rates according to the expectations hypothesis. Changes in long term rates reflect variations in long-term inflation expectations, which are due to standard, Gaussian shocks. This result may explain the acceptable forecasting performance of linearized 41

models over specific periods of time. For example, De Graeve, Emiris and Wouters (2009) finds that a linearized model is competitive with the random walk in forecasting 1-year yields up to 3-year ahead over the 1990:Q1-2007Q1 period, but less successful in forecasting longer maturity yields. This is not so surprising given that, according to our estimates, risk premia tend to be smaller at short horizons and they only increased and fell four times over the 1990:Q1-2007Q1 period. To produce changes in long-term rates, Gaussian shocks must be extremely persistent and they need to be coupled with a high degree of inertia in the monetary policy rule. A single shock plays this role in our model: the level technology shock z . t Figure(7)showsanimpulseresponsetothetechnologyshock. Theshockhasthe usual opposite effects on output and inflation: real variables increase, while inflation falls. The shock also generates extremely drawn out responses of macroeconomic variables. This is due, first, to the high persistence of the autoregressive process for z , whose half-life is of about 15 years.18 Second, it is due to the high interest rate t smoothingcoefficientoftheTaylorrule, whichkeepstheshort-termrealinterestrate positive over many quarters after the shock. The increase in the real interest rates reinforces the initial fall in inflation and requires an increasingly loose monetary policy stance over the first year after the shock. It is only after two years that all variables slowly start returning towards their long-run value in a monotonic fashion. 18The half-life is defined as the number of periods over which the effect of a unit shock remains above 0.5. For an autoregressive process with serial correlation coefficient ρ, the half-life is hl = ln(0.5)/ln(ρ). 42

Figure 7: Generalized impulse responses to a continuous technology shock p y c 2.0 −0.25 0.3 1.5 −0.50 0.2 1.0 −0.75 0.1 0.5 −1.00 0.0 0.0 −1.25 −0.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 i i i 12 40 −0.3 −0.25 −0.3 −0.30 −0.4 −0.35 −0.5 −0.5 −0.40 −0.7 −0.6 −0.45 −0.9 −0.50 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 90% bands mean median Note: responses to a unit shock on the transitory technology component, with variablesexpressedin%p.a.,andtheresponsestoconsumptionandoutputareobtained by cumulating responses to the respective changes. 43

6.3 Implications for 10-year inflation expectations Both uncertainty shocks and level technology shocks affect inflation over a prolonged period. It is therefore instructive to analyze the overall implications of our estimates for long-term inflation expectations–i.e. expected inflation over the next 10-year. These expectations are important as their stability, or ”anchoring”, is often interpreted as a measure of central banks’ anti-inflationary credibility. As a benchmark for comparison, we use expectations by the Federal Reserve Bank of Philadelphia’s quarterly Survey of Professional Forecasters combined with the Blue Chip Economic Indicators, which is available since 1979:Q4.19 These expectations relate to inflation as measured by the consumer price index, which tends to be a bit higher than inflation measured by the index for personal consumption expenditures, the inflation rate that we use in our estimation and that since January 2012 represents the Feds primary measure of inflation. From a secular perspective, a downward trend can clearly be identified in longterm inflation expectations over the 1980s. Over this period, model-implied 10-year inflation expectations are roughly consistent with the survey data–see Figure (8). 19Both surveys report forecasts for the average rate of CPI inflation over the next 10 years. The Blue Chip survey reports long-term inflation forecasts taken twice a year (March and October). Priorto1983,andin1983:4,thevariablewastheGNPdeflatorratherthantheCPI.Asof1991:Q4, we rely on the Philadelphia Survey of Professional Forecasters. 44

Figure 8: Model based and survey long-run inflation expectations, % p.a. 8 6 4 2 0 1970 1980 1990 2000 2010 Model based Survey Note: survey inflation expectations reported here taken from the Federal Reserve Bank of Philadelphia’s quarterly Survey of Professional Forecasters combined with the Blue Chip Economic Indicators, available since 1979:Q4. The high volatility of the early 1980s kept risk premia and yields high, even as expected future inflation and expected future policy rates were coming down. From this long-term perspective, the improved anchoring of inflation expectations in the U.S. is undoubtable. From a more cyclical perspective, however, survey and model-implied results differ. Survey expectations fall steadily towards 2.5 percent over the 1990s and then remain constant at that level through the 2000s. In contrast, yields dynamics interpreted through the lens of our model suggest a much less tight anchoring of inflation expectations. 45

Model-implied measures fall faster than surveys during the policy tightening phase which started in spring 1988 and was followed by the 1990 Gulf War and the ensuing recession. The fall in long-term inflation expectations is smaller than the fall in 10-year yields, because it is accompanied by a surge in volatility and a fall in real rates. Model-implied inflation expectations increase again sharply in 1993. An increase in long-term inflation expectations is in line with the idea of an ”inflation scare”, which was put forward by some commentators in this period. For example, Goodfriend (2002) states: ”Starting from a level of 5.9 percent [in October 1993], the 30-year bond rate rose through 1994 to peak at 8.2 percent just before election day in November. The nearly 2 1/2 percentage point increase in the bond rate indicated that the Fed’s credibility for low inflation was far from secure in 1994.” Following this period, model-implied inflation expectations remain roughly close to the survey measures. However, model-implied expectations diverge again during the recession of the early 2000s, when they fall sharply to levels around 1 percent. These dynamics are arguably consistent with the views of Federal Reserve officials, who expressed concerns about the, albeit remote, possibility of deflation from late 2002 through 2003. In November 2002, the then Governor Bernanke (2002) judged that”thechanceofsignificantdeflationintheUnitedStatesintheforeseeablefuture is extremely small”, but added that ”having said that deflation in the United States is highly unlikely, I would be imprudent to rule out the possibility altogether.” After a return towards 2.5 percent, model-implied long-term inflation expectations fall again ahead of the Great recession, i.e. a period when the possibility of a protracted, low-inflation period was difficult to rule out. To summarize, our model-implied estimate of long-term inflation expectations implicit in bond prices complements comparable information available from survey data. It suggests that long-term inflation expectations are less rigidly anchored than one would conclude, based on survey data. Since our estimates are model-based, they may of course be affected by model misspecification. For example, it is possible that either the transition probability 46

of technology shocks, or the value of their standard deviation, changed in the mid- 2000. As a result, the low-volatility shock could have become an absorbing state. Alternatively, the standard deviation of the high-volatility state could have become much closer to that of the low-volatility state. In both cases, estimated risk premia would have been smaller and, for given level of the nominal yield, inflation expectations would have been higher. We cannot exclude this possibility, but it would only reveal itself slowly over time, as more and more data are accumulated. It is nevertheless noticeable that inflation developments after the Great recession turned out to be more in line with the expectations implied by our model than with survey expectations. With the exception of 2012Q1, the personal consumption expenditure deflator remained persistently below 2 percent after 2008. Over the 9 years between 2009Q1 and 2017Q4 it only averaged 1.5%. By contrast, average inflation over the 10 years starting in 2019Q1 is equal to 1 percent according to model-implied expectations and to 2.5 percent according to the survey. Our model also provides us with measures of long-term inflation expectations over the late 1960s and 1970s, a period for which survey measures are not available. Levin and Taylor (2013) compute far forward inflation expectations over this period based on a simple methodology and suggest that they started drifting up steadily as of 1965 reaching an estimated peak of about 4.5 percent in 1970 and then remained between 3.5 and 4.5 percent over the next several years. Our estimates are broadly consistent with the results in Levin and Taylor (2013). Average 10-year inflation expectations increase to 4 percent in 1971 and then hover between 3 and 3.5 percent in the early 1970s. In the second half of the 1970s they increase again to levels around 4.5 percent and remain at levels above 4 percent until 1980. These estimates are also consistent with the results in Ang, Bekaert and Wei (2008), which are based on a no-arbitrage factor model of the term structure. Compared to the aforementioned two papers, ours has the additional feature of providing consistent estimates of the structural shocks which led to the model-implied developments in inflation expectations. 47

7 Conclusions This paper presents the results of the estimation of a nonlinear macro-yield curve model with Epstein-Zin-Weil preferences, in which the variance of structural shocks is subject to changes of regime. It shows that the model fits the data well: measurement errors are small; the dynamic cross-correlation matrix of the data is closely replicated; long-term forward rates are matched. An important role to account for this performance is played by changes in variance regimes, or uncertainty shocks, which tends to occur during recessions. On the one hand, uncertainty shocks induce changes in the demand for precautionary savings. Expected real and nominal yields also fall, which is consistent with the empirical evidence. On the other hand, the increase in volatility during recessions also boosts uncertainty over future consumption growth. Risk premia increase in a countercyclical fashion, which is consistent with results from the finance literature. Compared to survey evidence, model-based measures of long-term inflation expectations are more variable over the economic cycle. They fall to low levels over the 1980s, but are subject to cyclical ”scares”–either upwards or downwards. They suggest that the Federal Reserve’s credibility for low inflation is less dogmatically established than one would conclude, based on survey data. 48

References [1] Amisano, Gianni and Oreste Tristani (2011), “Exact likelihood computation for nonlinear DSGE models with heteroskedastic innovations,” Journal of Economic Dynamics and Control 35, 2167-2185. [2] Andreasen, Martin M. (2012a), ”An Estimated DSGE Model: Explaining Variation in Nominal Term Premia, Real Term Premia, and Inflation Risk Premia”, European Economic Review 56, 1656–1674. [3] Andreasen, Martin (2012b), ”On the effects of rare disasters and uncertainty shocks for risk premia in nonlinear DSGE models,” Review of Economic Dynamics 15, 295–316. [4] Ang, Andrew, Geert Bekaert and Min Wei (2008), ”The term structure of real rates and inflation expectations”, Journal of Finance 63, 797-849. [5] Ang, Andrew and Monika Piazzesi (2003), ”A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables”, Journal of Monetary Economics 50, 745–787. [6] Atkeson, A.andP.Kehoe(2008), ”Ontheneedforanewapproachtoanalyzing monetary policy”, in D. Acemoglu, K. Rogoff and M. Woodford, eds., NBER Macroeconomics Annual 23, 389-425. [7] Backus, David K., Allan W. Gregory and Stanley E. Zin (1989), ”Risk premiums in the term structure”, Journal of Monetary Economics 24, 371–99. [8] Bansal, Ravi and Amir Sharon (2004), ”Risks for the Long Run: A Potential Resolutionof Asset Pricing Puzzles”, Journal of Finance 54, 1481-509. [9] Basu, Susanto and Brent Bundick (2012), ”Uncertainty Shocks in a Model of Effective Demand”, Boston College Working Paper 774. [10] Basu, Susanto and Miles S. Kimball (2002), ”Long-run labour supply and the elasticity of intertemporal substitution for consumption”, mimeo, University of Michigan. [11] Bauer, Michael D., Glenn D. Rudebusch and Jing Cynthia Wu (2014), “Term Premia and Inflation Uncertainty: Empirical Evidence from an International Panel Dataset: Comment,” American Economic Review 104, 323–337. [12] Bekaert, G., Seonghoon Cho and Antonio Moreno (2010). “New Keynesian Macroeconomics andthe Term Structure,”Journal of Money, Credit and Banking 42, 33–62. [13] Bernanke, Ben S. (2002), ”Deflation: Making sure ”it” doesn’t happen here”, Remarks before the National Economist Club, Washington, 21 November. 49

[14] Bianchi, Francesco, Cosmin Ilut and Martin Schneider (2014), Uncertainty shocks, asset supply and pricing over the business cycle, NBER Working Paper No. 20081. [15] Bloom, Nicholas (2009), ”The impact of uncertainty shocks”, Econometrica 77, 623–685 [16] Campbell, John Y., and John H. Cochrane, 1999, By force of habit: A consumption-based explanationof aggregate stock market behavior, Journal of Political Economy 107, 205–251. [17] Campbell, John Y., Carolin Pflueger and Luis M. Viceira (2013), ”Monetary Policy Drivers of Bond and Equity Risks ”, mimeo. [18] Chetty, Raj, Adam Guren, Day Manoli and Andrea Weber (2011), ”Are Micro and Macro Labor Supply Elasticities Consistent? A Review of Evidence on the Intensive and Extensive Margins”, American Economic Review Papers and Proceedings 101, 471-75. [19] Christiano, Lawrence J., Roberto Motto and Massimo Rostagno (2014), ”Risk shocks”, American Economic Review 104, 27–65. [20] Christoffel, Kai, Ivan Jaccard and Juha Kilponen (2011), ”Government bond risk premia and the cyclicality of fiscal policy”, ECB Working Paper No. 1411. [21] Clarida, R., J. Gal´ı and M. Gertler (2000), ”Monetary policy rules and macroeconomic stability: evidence and some theory”, Quarterly Journal of Economics, 147-180. [22] Cochrane, John H. (2008), ”Comment on ”On the need for a new approach to analyzing monetary policy””, in D. Acemoglu, K. Rogoff and M. Woodford, eds., NBER Macroeconomics Annual 23, 427-448. [23] Cochrane, John H. (2017), ”Macro-Finance”, Review of Finance, forthcoming. [24] Cochrane, John H. and Monika Piazzesi (2005), “Bond risk premia”, American Economic Review 94, 138-160. [25] Colacito, Riccardo and Mariano Croce (2013), ”International asset pricing with recursive preferences”, Journal of Finance 68, 2651–2686. [26] Dai, G. and K. Singleton (2002), “Expectations puzzles, time-varying risk premia, and affine models of the term structure,” Journal of Financial Economics 63, 415-441. [27] De Graeve, Ferre, Marina Emiris and Raf Wouters (2009), ”A Structural Decomposition of the US Yield Curve”, Journal of Monetary Economics 56, pp. 545-559. 50

[28] Del Negro, M., G. Eggertsson, A. Ferrero and N. Kiyotaki (2017), ”The Great Escape? A Quantitative Evaluation of the Fed’s Liquidity Facilities,” American Economic Review 107, 824-57 [29] Den Haan, W.J. (1995), ”The term structure of interest rates in real and monetary economies”, Journal of Economic Dynamics and Control 19, pp. 909-40. [30] Doh, Taeyoung (2011), ”Yield Curve in an Estimated Nonlinear Macro Model”, Journal of Economic Dynamics and Control 35, 1229-1244. [31] Doh, Taeyoung (2012), ”What Does the Yield Curve Tell Us About the Federal Reserve’s Implicit Inflation Target?”, Journal of Money, Credit, and Banking 44, 469-486. [32] Durbin, J. and S.J. Koopman (2012): Time series analysis with state space methods, second edition, Oxford University Press, Oxford, UK. [33] Duffee, Gregory R. (2002), ”Term premia and interest rate forecasts in affine models”, Journal of Finance 57, 405-443. [34] Herbst, E. and F. Schorfheide (2016), Bayesian estimation of DSGE models, Princeton University Press, Princeton NJ. [35] Fama, Eugene F. and Kenneth R. French (1989), ”Business conditions and expected returns on stocks and bonds”, Journal of Financial Economics 25, 23-49. [36] Fuhrer, JeffreyC.(1996), “Monetarypolicyshiftsandlong-terminterestrates,” Quarterly Journal of Economics 111, pp. 1183-1209. [37] Foerster, A., Rubio-Ramirez J., Waggoner D. and T. Zha (2016) ”Perturbation methods forMarkov-switching dynamic stochastic general equilibrium models”, Quantitative Economics 7, 637-669. [38] Fuhrer, Jeffrey C. (2000), ”Habit Formation in Consumption and Its Implications for Monetary-Policy Models,” American Economic Review 90, 367-390. [39] Geweke, John (2007), ”Interpretation and inference in mixture models: Simple MCMC works”, Computational Statistics and Data Analysis, 51, 3529-3550. [40] Giannoni, Marc P. and Michael Woodford (2005), ”Optimal Inflation Targeting Rules,” in: B.S. Bernanke and M. Woodford, eds., Inflation Targeting, University of Chicago Press. [41] Gomme, Paul and Paul Klein (2011), ”Second-order approximation of dynamic modelswithouttheuseoftensors”, Journal of Economic Dynamics and Control 35, 604-615. [42] Goodfriend, M. (1991), “Interest Rates and the Conduct of Monetary Policy,” Carnegie-Rochester Conference Series on Public Policy 34, 7–30. 51

[43] Goodfriend, Marvin (1993), “Interest Rate Policy and the Inflation Scare Problem: 1979-1992”,Federal ReserveBank of RichmondEconomic Quarterly 79(1), 1–24. [44] Goodfriend, Marvin (2002), “The phases of U.S. monetary policy: 1987 to 2001”, Federal Reserve Bank of Richmond Economic Quarterly 88(4), 1–17. [45] Gourio, Franc¸ois (2012), ”Disaster Risk and Business Cycles,” American Economic Review 102, 2734-66. [46] Gourio, Fran¸cois (2013), ”Credit risk and disaster risk,” American Economic Journal: Macroeconomics 5, 1–34. [47] Greenspan, Alan (2005), Federal Reserve Board semiannual Monetary Policy Report to the Congress Before the Committee on Banking, Housing, and Urban Affairs, U.S. Senate, February 16. [48] Gu¨rkaynak, Refet S., Brian Sack and Eric Swanson (2005), ”The Sensitivity of Long-Term Interest Rates to Economic News: Evidence and Implications for Macroeconomic Models,” American Economic Review 95, 425-436. [49] Gu¨rkaynak, Refet S., Brian Sack and Jonathan H. Wright (2007), ”The U.S. Treasury yield curve: 1961 to the present,” Journal of Monetary Economics 54, 2291-2304. [50] Ho¨rdahl, P., O. Tristani and D. Vestin (2008), ”The yield curve and macroeconomic dynamics”, Economic Journal 118, 1937-70. [51] Justiniano,AlejandroandGiorgioPrimiceri(2008),”TheTimeVaryingVolatility of Macroeconomic Fluctuations”, American Economic Review 98, 604-641. [52] Kim,Chang-Jin(1994),”DynamiclinearmodelswithMarkov-switching”,Journal of Econometrics 60, 1-22. [53] Kim,Chang-JinandCharlesR.Nelson(1999),State-Space Models with Regime- Switching: Classical and Gibbs-Sampling Approaches with Applications, MIT Press. [54] King, Robert G., Charles I. Plosser and Sergio T. Rebelo (1988), ”Production, Growth and Business Cycles I. The Basic Neoclassical Model”, Journal of Monetary Economics 21, 195–232 [55] Levin, Andrew and John B. Taylor (2013), ”Falling Behind the Curve: A Positive Analysis of Stop-Start Monetary Policies and the Great Inflation,” in The Great Inflation: The Rebirth of Modern Central Banking, by Michael Bordo and Athanasios Orphanides. [56] Lubik, Thomas A. and Frank Schorfheide (2004), ”Testing for Indeterminacy: An Application to U.S. Monetary Policy”, American Economic ReviewA 94, pp. 190-217. 52

[57] Maddison, Angus (2003), The World Economy: Historical Statistics, Paris: OECD Publishing. [58] McDonnell, M. M. and Perez-Quiros, G. (2000) ”Output Fluctuations in the UnitedStates: WhathasChangedSincetheEarly1980s?”,American Economic Review 90, 1464-1467 [59] Piazzesi, M. and M. Schneider (2006), ”Equilibrium yield curves”, in D. Acemoglu, K. Rogoff and M. Woodford, eds., NBER Macroeconomics Annual 21, 389 - 472. [60] Primiceri, Giorgio (2005), ”Time Varying Structural Vector Autoregressions and Monetary Policy”, Review of Economic Studies 72, 821-852. [61] Restoy, Fernando and Philippe Weil (2011), “Approximate Equilibrium Asset Prices”, Review of Finance 15, 1-28. [62] Rudebusch, Glenn D. and Eric Swanson (2012), “The Bond Premium in a DSGE Model with Long-Run Real and Nominal Risks”, American Economic Journal: Macroeconomics 4, 105-43. [63] Schmitt-Groh´e, Stephanie and Mart´ın Uribe (2004a), ”Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function”, Journal of Economic Dynamics and Control 28, 755-775. [64] Schmitt-Groh´e, Stephanie and Mart´ın Uribe (2004b), ”Optimal fiscal and monetary policy under sticky prices”, Journal of Economic Theory 114, 198–230. [65] Schorfheide, F. (2005), “Learning and Monetary Policy Shifts,”Review of Economic Dynamics 8, 392-419. [66] Sims, Christopher A. and Tao Zha (2006), ”Were There Regime Switches in U.S. Monetary Policy?”, American Economic Review 96, 54 - 81. [67] Smets, F. and and R. Wouters (2007), “Shocks and Frictions in US Business Cycles: ABayesianDSGEApproach,”American Economic Review 97,586-606. [68] Stock, J. and M. Watson, ”Why Has U.S. Inflation Become Harder to Forecast?”, Journal of Money, Banking and Credit 39, 3-330. [69] Swanson, Eric T. (2012), “Risk Aversion and the Labor Margin in Dynamic Equilibium Models,” American Economic Review 102, 1663-1691. [70] Swanson, Eric T. (2014), ”A macroeconomic model of equities and real, nominal, and defaultable debt”, mimeo. [71] Swanson, EricT.(2018), ”Riskaversion, riskpremia, andthelabormarginwith generalized recursive preferences,” Review of Economic Dynamics 28, 290–321. 53

[72] Tallarini, Thomas D. (2000), “Risk-sensitive real business cycles”, Journal of Monetary Economics 45, 507-532. [73] Trabandt, M. and H. Uhlig (2011), “The Laffer Curve Revisited”, Journal of Monetary Economics 58, 305-327. [74] van Binsbergen, Jules H., Jesu´s Fern´andez-Villaverde, Ralph S.J. Koijen and Juan Rubio-Ram´ırez (2012), ”The term structure of interest rates in a DSGE model with recursive preferences”, Journal of Monetary Economics 59, 634– 648. [75] Woodford, Michael (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. [76] Wright, Jonathan H. (2014), ”Term Premia and Inflation Uncertainty: Empirical Evidence from an International Panel Dataset: Reply”, American Economic Review 104, 338–341. 54