Yield Curve in an Estimated Nonlinear Macro Model

Transcription

1 Yield Curve in an Estimated Nonlinear Macro Model Taeyoung Doh Federal Reserve Bank of Kansas City November, 16, 7 Abstract This paper estimates a dynamic stochastic general equilibrium (DSGE) model using macro and yield curve data to identify the macro factors that drive the movements in the yield curve. I propose new closed-form solutions for bond yields to make the estimation of a nonlinear version of the model practically feasible and apply a likelihood-based Bayesian approach. The main findings from the empirical analysis of US data are as follows. First, generating sizeable term premia without highly volatile macro variables is difficult even if nonlinearities are taken into account. Second, however, demeaned term structure variables are well explained by the estimated macro factors. A persistent monetary policy shock accounts for the level of the yield curv, while a markup shock and a transitory monetary policy shock drive the slope and curvature, respectively. Finally, the persistent monetary policy shock is highly correlated with the inflation forecasts from the survey data. JEL CLASSIFICATION: C3, E43, G1 KEY WORDS: Bayesian Econometrics DSGE Model Term Structure of Interest Rates Taeyoung Doh : Research Department, Federal Reserve Bank of Kansas City, 95 Grand Blvd, Kansas City, MO 64198; Taeyoung.Doh@kc.frb.orgThe views expressed herein are solely those of the author and do not necessarily reflect the views of the Federal Reserve Bank of Kansas City or the Federal Reserve System.

2 1 1 Introduction Factor models of the yield curve are empirically successful at explaining movements in the yield curve. These factors are typically extracted from a statistical decomposition of the yield curve. However, the economic interpretation of these factors is not clear. Recent empirical studies on the macroeconomics of the term structure (Ang and Piazzesi (3), Bikbov and Chernov (5), and Diebold et al. (6) among others) show a close link between macroeconomic variables and bond prices. They augment statistical factors of the yield curve with macroeconomic variables. Despite the inclusion of macro variables into their models, they still retain latent term structure factors to explain the yield curve dynamics. These latent factors are often called level, slope, and curvature. In this paper, we set up and estimate a New Keynesian dynamic stochastic general equilibrium (DSGE) model to explain the joint fluctuations of macroeconomic variables and the yield curve. In the model, shocks to technology, markups of firms, and the inflation target of the central bank as well as a transitory monetary policy shock drive economic fluctuation. We do not allow latent term structure factors and stick to restrictions from the fully specified DSGE model in order to maximize the explanatory power of macro factors for the yield curve dynamics. One innovation of this paper is to propose new closed form solutions for bond yields which enable us to estimate a nonlinear macro model compatible with time varying term premia. The closed form expressions for bond yields implied by a log-linearized DSGE model combine a log-linear approximation with the conditional log-normal distribution of the pricing kernel. However, certain terms which can contribute to time varying term premia are completely missing in such an approach. Hence, the model is not able to match the time variation of term premia documented in empirical studies of the US data (Campbell and Shiller (1991), for example). For this reason, Hördahl et al. (5a) and Rudebusch and Wu (4) use an exogenously given pricing kernel, which allows time-varying term premia even though they derive macro dynamics from the log-linearized version of DSGE models. This paper does not take such a short-cut and relies on the pricing kernel fully

3 consistent with the DSGE model. In this sense, my approach is closely related to Bekaert et al. (5), Hördahl et al. (5b), and Ravenna and Seppälä (6). However, none of them estimates a nonlinear version of the macro model as in this paper. Bekaert et al. (5) take the log-linear and log-normal approach, which excludes time varying term premia. Hördahl et al. (5b) and Ravenna and Seppälä (6) consider nonlinear solutions for bond prices but they do not estimate their models. Also term premia are still restricted to be time-invariant in Hördahl et al. (5b) because they rely on a second order approximation to Euler equations for bond prices. Departing from the existing literature, we apply a likelihood-based estimation approach using sequential Monte Carlo methods to approximate the likelihood for a nonlinear macro model as in An and Schorfheide (6), and Fernández-Villaverde and Rubio-Ramírez (6). This paper adopts Bayesian methods which have been widely used in the estimation of a log-linearized DSGE model. The main findings from the empirical analysis of US data are as follows. First, a tension exists in improving macro implications of the model and term structure implications simultaneously. For example, the mean term premium is larger when inflation risk is higher. But amplifying inflation risk to match the mean term premium increases the volatility inflation much higher than we can observe in the sample data. Thus, getting both moments right is difficult. Nonlinear analysis does not resolve this problem because nonlinear terms in the model increase term premia by amplifying the volatilities of macro variables. Second, once average values of term structure variables are taken out as free parameters, demeaned empirical proxies for latent term structure factors are well explained by the estimated macro factors. The finding is supported by high R s in the regressions of empirical proxies for level, slope, and curvature on a constant and the estimated macro factors. A persistent monetary policy shock accounts for the level of the yield curv, while a markup shock and a transitory monetary policy shock drive the slope and curvature, respectively. In addition, fluctuations in the yield term premium are linked with the time variation of the estimated target inflation of the central bank. Third, the estimated target inflation shock, which explains the level of the yield curve, is

4 3 highly correlated with inflation forecasts form the survey data. This high correlation is obtained only when yield curve data are included in the estimation of the macro model. This is another evidence that the yield curve can provide additional information about market s expectations of the long run inflation as much emphasized by Kozicki and Tinsley (5). The above findings can be compared to the empirical results reported in the literature on term structure implications of a New Keynesian DSGE model. The tension between the magnitude of the mean term premium and the volatility of inflation seems to be in conflict with Hördahl et al. (5b) and Ravenna and Seppälä (6) who show that a New Keynesian DSGE model can replicate the size of average term premia well. As mentioned by Rudebusch et al. (6), both papers use extremely high persistence and volatilities of shocks to generate sizeable term premia. Our estimation results provide evidence that the historical data are not consistent with those parameter values. Regarding economic interpretation of term structure factors, my interpretation of the level and curvature of the yield curve is in line with Bekaert et al. (5). However, unlike Bekaert et al. (5) who attribute the slope to the transitory monetary policy shock, I associate the slope with a markup shock which is a real disturbance. The reason for the difference is that the estimate of the monetary policy reaction to real disturbances is very close to zero in Bekaert et al. (5) while our estimate is positive. The paper is organized as follows. Section describes the model economy and presents a second order approximation to model solutions, which includes some potentially important nonlinearities. Section 3 proposes a new method to construct measurement equations for bond yields under the second order approximation. Section 4 describes the econometric methodology used. Section 5 discusses the estimation results of the model with US dataset. Section 6 discusses the economic interpretation of term structure variables. Section 7 concludes. Technical details are provided in the appendix.

5 4 Model Economy We consider a small scale New Keynesian model that has been widely used for business cycle and monetary policy analysis. The model economy consists of a representative final good producing firm in a competitive market, a continuum of intermediate goods producing firms under imperfect competition, a representative household, a monetary as well as a fiscal authority, and complete markets for statecontingent claims. Our model is a variant of the model with habit persistence, which is discussed in Woodford (3)..1 Private Agents The production sector in the economy consists of two parts. One is the perfectly competitive final good sector and the other is the intermediate goods sector made of a continuum of monopolistically competitive firms. The final good sector combines each intermediate good indexed by j [, 1] using the technology ( 1 Y t = ) ζ t Y t (j) ζ t 1 ζ t 1 ζ t dj (1) The representative firm in the final good sector maximizes its profit given output prices P t and input prices P t (j). The resulting input demand is given by Y t (j) = ( P t(j) P t ) ζ t Y t () Hence, ζ t > 1 represents the elasticity of demand for each intermediate good. Production technology for intermediate good j is linear with respect to labor. Y t (j) = A t N t (j) (3) where A t is an exogenously given common technology and N t (j) is the labor input of firm j. We assume labor market is perfectly competitive and denote the real wage

6 5 as W t. Firms in the intermediate goods sector face nominal rigidities in the form of quadratic price adjustment costs. AC t (j) = φ ( P t(j) P t 1 (j) π t ) Y t (j) (4) Here φ is a parameter governing the degree of price stickiness in this economy and π t is the target inflation of central bank in terms of the price of the final good. 1 Firm j decides its labor input N t (j) and the price P t (j) to maximize the present value of profit stream E t [ β s λ t+s ( P t+s(j) Y t+s (j) W t+s N t+s (j) AC t+s (j))] (5) λ t P t+s s= where λ t+s is the marginal utility of a final good to the representative household at time t + s, which is exogenous from the viewpoint of the firm. The representative household maximizes its utility by choosing consumption (C t ), and labor supply(h t ). 3 We deflate consumption by the current technology level to ensure a balanced growth path for the economy. Also, we introduce a kind of internal habit formation into the utility function. The specification corresponds to the closed economy version of Lubik and Schorfheide (5). The objective function of the household is given by E t [ β s ( (Ca t+s/a t+s ) 1 τ 1 1 τ s= 1 ν H1+ t+s )] (6) ν 1 Alternatively, we can introduce Calvo-type price stickiness in which only a small fraction of firms can reoptimize their prices. If we additionally assume firms, which cannot reoptimize their prices, adjust prices by the current period target inflation of the central bank, our specification would be equivalent to such a Calvo-type sticky price model up to the first order. ( 1 The price of the final good is given by P t = P t(j) 1 ζ t 1 ζ dj t. 3 We do not model money explicitly. If we introduce real money balance which is separable from other arguments in the utility function, all the following arguments would not be affected. ) 1

7 6 where Ct+s a = C t+s he u a Ct+s 1 is the consumption relative to the habit level which is determined by the previous period consumption, h is the parameter governing the magnitude of habit persistence, τ the curvature of utility function, ν the short-run (Frisch) labor supply elasticity. Under the assumption of complete asset markets, the household s budget constraint is of the form P t C t + P n,t (B n,t B n+1,t 1 ) + T t = P t W t H t + B 1,t 1 + Q t + Π t (7) n=1 where P n,t the price of an n quarter bond, B n,t bond holding, T t lump-sum tax or subsidy, Q t the net cash flow from participating in state-contingent security markets and Π t the aggregate profit, respectively.. Monetary Policy The monetary policy of the central bank is assumed to follow a forward-looking Taylor rule with interest rate inertia. The nominal gross interest rate reacts to expected inflation, and output gap in the following way: (1 + i t ) = (1 + i t ) 1 ρ i (1 + i t 1 ) ρ i e η iɛ i,t 1 + i t = ((1 + r )(π ))( Et(π t+1) πt ) γ p ( Y t ) γ A ty f y t (8) where r is the steady state real interest rate which is eu a β defined by P t P t 1, π the steady state inflation and y f t perturbed by a markup shock. 4 1, π t the actual inflation the steady state value of Y t A t In the model, the time varying target inflation is assumed to be exogenously given. ρ i captures the degree of interest rate inertia. 4 The concept is comparable to the natural rate of output in Bekaert et al. (5). They define the natural rate of output as the steady state output in a flexible price economy and interpret a markup shock as a shock to the natural rate of output. However, this is different from the usual definition of the natural rate of output in which nominal rigidities are assumed away but no restriction on exogenous disturbances is imposed. Still, the use of y f t rather than the steady state value of Y t A t may seem to be odd. Here, the purpose is to enhance the central bank s ability to adjust the short term target interest rate in response to a markup shock. An alternative specification for this purpose is to assume the target inflation is reacting to real disturbances as in Ireland (7).

8 7 The fiscal side of government policy is passive. The government does not make any independent expenditure and finances the debt maturing on the current period by new issues of bonds and lump-sum tax. Its period by period budget constraint is given by n=1 P n,t(b n,t B n+1,t 1 ) + T t = B 1,t 1..3 Exogenous Processes The model economy is subject to four structural disturbances. Aggregate productivity evolves according to u a,t = A t A t 1, ln u a,t+1 = (1 ρ a )u a + ρ a ln u a,t + η a ɛ a,t+1 (9) If there were no nominal rigidities, firms in the intermediate goods sector would choose their price-marginal cost ratios equal to markups of firms. ζ t ζ t 1. These are called desired We assume they are affected by a persistent shock which follows an AR(1) process with ARCH(1) structure. f t = ζ t ζ t 1, ln f t+1 = (1 ρ f ) ln f + ρ f ln f t + η,f + η 1,f (ln f t ln f ) ɛ f,t+1 (1) Monetary policy is exposed to a serially uncorrelated shock(ɛ i,t ) and a persistent target inflation shock. ln π t+1 = (1 ρ π ) ln π + ρ π ln π t + η π ɛ π,t+1 (11) In principle, we can introduce ARCH effects to other structural shocks. However, adding ARCH effects to other disturbances turns out to increase the volatility of inflation too much with little impact on the magnitude of the term premium. Alternatively, one can introduce independent volatility shocks as in Justiniano and Primiceri (5) and Fernández-Villaverde and Rubio-Ramírez (6). Constructing closed-form solutions for bond yields in such cases is more complicated. These considerations narrow down our choice.

9 8 All the four serially uncorrelated innovations(ɛ a,t, ɛ f,t, ɛ i,t, ɛ π,t) are independent of each other at all leads and lags. Each innovation is assumed to follow a standard normal distribution..4 Equilibrium Conditions Market clearing conditions for the final good market and labor market are given by Y t = C t + AC t, H t = N t (1) The first order conditions for firms and the represent household can be expressed as follows: λ t A t = λ a t = (C a t /A t ) τ βhe u ae t ((C a t+1/a t+1 ) τ A t /A t+1 ) (13) 1 = βe t [( λa t+1 λ a ) A t 1 + i t ] (14) t A t+1 π t 1 = ζ t [1 ( Yt A t ) 1 ν λ a t ] + φπ t (π t π t ) φ ζ t(π t π t ) φβe t [( λa t+1 λ a ) Y t+1/a t+1 π t+1 (π t+1 π t Y t /A t+1)] (15) t The Euler equation for the short term interest rate can be rewritten by using the one-period ahead nominal stochastic discount factor(m t,t+1 ). M t,t+1 = β λ t+1 λ t 1 π t+1, E t (M t,t+1 (1 + i t )) = 1 (16).5 Model Solutions To solve the model around the steady state, we need to eliminate the non-stationary trend of technology and make all the variables stationary. For example, consumption is detrended by the technology level (c t = C t /A t ). In our model economy, the steady state values of detrended output(y t /A t ) and consumption(c ) are given by

10 9 (Y t /A t ) = c 1 1 βh = ( f (1 h) τ ) τ+ 1 ν (17) For notational convenience, the percentage deviation from the steady state value of a variable d t from its steady state d is denoted by ˆd t = ln(d t /d ). Equations (8) - (15), which consist of equilibrium conditions and specifications for exogenous processes form a rational expectations system with respect to: y t = [ Y t /A ˆ t, C t ˆ/A t, ˆπ t, (1 + it ), C a ˆ t /A t, ˆλ a t ](control variable) and x t = [û a,t, ˆf t, ɛ i,t, ˆπ t, (1 + it 1 ), C t 1 ˆ/A t 1 ](state variable). E t f(y t+1, y t, x t+1, x t, σɛ t+1 ) = (18) Here σ [, 1] is a perturbation parameter which determines the distance from the deterministic steady state. σ = corresponds to the non-stochastic steady state. The approximate economy is associated with σ = 1. Since σɛ t+1 is the only source of uncertainty, the approximation order in the perturbed system is determined by the the degree of powers of σ in the approximated system. The exact solution of the model is given by y t = g(x t, x t 1, σ), x t+1 = h(x t, σ) + ση(x t )ɛ t+1 (19) x t 1 appears in y t because the previous period s markup affects the conditional volatility of the current markup. Since the time varying volatility is not captured in the log-linear approximation, the first order accurate approximate solution considers only the constant volatility as follows. y t g x x t, x t+1 h x x t + ση()ɛ t+1 () While the log-linear approximation can provide insights on macro dynamics, asset pricing implications of the model cannot be studied in the pure log-linear approximation because there is no consideration of risk in the log-linearized macro dynamics.

11 1 Jermann (1998) obtains the risk premium term by combining the conditional lognormality of the model-implied pricing kernel with the log-linear approximation of macro dynamics. He applies this method in the context of studying asset pricing implications of a real business cycle model. Wu (5) takes the same approach to study term structure implications of a New Keynesian model. He computes equilibrium bond prices in an affine term structure model derived from the log-linear and log-normal approach. The basic idea of this can be illustrated by using the short term interest rate as an example. If we apply the log-linear approximation to the stochastic discount factor, ˆMt,t+1 is represented by a linear function of (x t, ɛ t+1 ). ˆM t,t+1 = (ˆλ a t+1 ˆλ a t ) û a,t+1 ˆπ t+1 = m l xx t + m l ɛɛ t+1 (1) ˆ Euler equation implies E t (e ˆM t,t i t ) = 1. Using the normality of ɛ t+1, we can express the short term interest rate implied by the log-linear and log-normal framework as follows: 1 + i l t = E t( ˆM t,t+1 ) V ar t( ˆM t,t+1 ) = m l xx t ml ɛm l ɛ () The above expression is only a partial solution. To evaluate the impacts of macroeconomic risks on the short term interest rate more accurately, we need to compute second order solutions. We follow Schmitt-Grohé and Uribe (4) to obtain the second order approximate solution of the DSGE model. The exact solutions for control variables include x t 1 as well as x t. However, up to the second order approximation of g and h around the deterministic steady state (x t, σ) = (, ), it can be shown there is no need to add x t 1 as a vector of additional state variables to approximate policy functions. Result 1 : Up to the second order, the coefficients in the approximate solutions of g and h are not affected by time varying volatility. Indeed they are the same as those under the homoskedastic case where η(x t ) is replaced by η(). Proof In Appendix.

12 11 The resulting approximate solutions are: 5 y t 1 g σσσ + g x x t + 1 (I n y x t ) (g xx )x t (3) x t+1 1 h σσσ + h x x t + 1 (I n x x t ) (h xx )x t + η(x t )σɛ t+1 (4) The second order accurate solution of the short term interest rate is given by: 1 + i q t = 1 gi σσ + g i xx t + 1 x tg i xxx t (5) Here gσσ i denotes the second order derivative of the policy function of the short term interest rate with respect to σ. As emphasized by Schmitt-Grohé and Uribe (4), the first order coefficients in the second order approximation must be the same as coefficients in the first order approximation. This implies gx i = m l x. The difference between two approximate solutions for the short term rate is difficult to characterize analytically because x t in the second order approximation follows different law of motion from that in the first order approximation. Due to the inclusion of the past consumption and the past interest rate, there are nonlinear terms in the transition equation of x t under the second order approximation. For a special case in which x t consists of variables whose transition equations do not contain any nonlinear terms, gxx i t and m l xx t are canceled out. Then the difference can be expressed as follows: q,l i = 1 + i q t ( 1 + i l t ) = 1 gi σσ + x tgxxx i t + ml ɛm l ɛ (6) gσσ i is the precautionary saving term which is negative while ml ɛm l ɛ is always positive. Therefore, the sign of q,l i is ambiguous even in this simple case. 5 Here, the representation of solutions follows Klein (5).

13 1 3 Measurement Equations for Macro Variables and Bond Yields Approximate model solutions give us the dynamics of detrended macro variables. To estimate the model, we need to define measurement equations that relate macro state variables to observations of macro and term structure variables. Given approximate solutions for macro dynamics, we can relate observed macro variables such as real per capita GDP (GDP t ), quarter to quarter inflation rates (INF t ), and 3-month Treasury bill rates (INT t ) with model implied values plus measurement errors whose standard deviations are σ my, σ mπ, σ mi. GDP t = ln A t + ln( Y t ) + Y ˆ t /A t + ξ y,t A t INF t = ln π + ˆπ t + ξ π,t INT t = ln(1 + i ) + ˆ 1 + i t + ξ i,t (7) Since bond yields are not included in a vector of policy variables y t, measurement equations for bond yields are not as straightforward as those for macro variables. Furthermore, model implied bond yields must satisfy no-arbitrage conditions. In a log-linearized economy, analytical solutions for arbitrage-free bond yields can be easily derived by using the log-normality of the stochastic discount factor. Bekaert et al. (5) and Wu (5) explain the derivation in detail. However, when a second order approximation is used, the log stochastic discount factor implied by approximate solutions does not follow a normal distribution. The challenge here is to solve the following Euler equations for bond prices when ˆM t,t+1 and ˆp n 1,t+1 are not normally distributed. eˆp n,t = E t (e ˆM t,t+1 +ˆp n 1,t+1 ) eˆp n 1 n,t = E t (e ˆM j= t+j,t+j+1 ), ˆp 1,t = 1 + i t (8) One solution is to approximate the conditional expectations numerically (Ravenna and Seppälä (6)) or analytically (Hördahl et al. (5b)). By contrast, a novel

14 13 way of computing the exact conditional expectations based on the normality of structural shocks is proposed in this paper. The literature using the approximation of conditional expectations also assumes the normality of shocks but they do not use that assumption to derive arbitrage-free bond prices. Hence, the method suggested here is more accurate than their methods in the sense that no approximation to conditional expectations are involved. 6 The detailed derivation of closed-form solutions is explained in appendix but the main idea can be described as follows. The short term interest rate is obtained as a quadratic function of state variables in the approximate solution. Also, the log stochastic discount factor is a quadratic function of (x t, x t+1 ). ˆM t,t+1 = m + m 1 x t + x tm x t + m 3 x t+1 + x t+1m 4 x t+1 (9) One complication that we face in computing conditional expectations is that the log stochastic discount factor M t,t+1 is quadratic with respect to x t+1 as well as x t. Mechanical forward iterations of the second order approximation for x t+1 generate extra higher order terms such as x 4 t and x 3 t. As discussed in Kim et.al. (5), these extra higher order terms do not necessarily increase the accuracy of the approximation because they do not correspond to higher order terms of the Taylor series expansion. Following their suggestion, we use the pruning scheme which effectively ignores extra higher order terms. Therefore, we consider terms up to the second order of (x t, ɛ t+1 ) in the equation (9). It needs to be emphasized that terms like x t ɛ t+1 in the approximate pricing kernel can generate a time varying risk premium if we compute the conditional expectations in Euler equations exactly. The detailed discussion is provided in the appendix. Now we compute conditional expectations appearing in equation (8) by using the properties of quadratic forms of a multi-variate normal random variables as ex- 6 Rudebusch et al. (6) solve the macro model by a third-order approximation to equilibrium conditions and obtain numerical solutions for bond prices as third order polynomials of macro state variables. Whether or not their solutions are more accurate than those in this paper is not clear because of the tradeoff between the accuracy of approximate macro solutions and that of approximate Euler equations for bond prices.

15 14 plained in Johnson and Kotz (197). In the end, our closed-form solutions resemble bond prices in the discrete time counterpart of a continuous time quadratic term structure model studied in Ahn et al. (). The difference is that our solutions are derived from the approximate stochastic discount factor, which is consistent with the underlying macro model. Result : The log bond price of maturity of n quarter, p n,t, has the following representation under the no-arbitrage assumption: ˆp n,t = a n + b n x t + x tc n x t Proof : In Appendix. The yield of n-quarter bond ŷ n,t can be computed as ˆp n,t n. Therefore, measurement equations for approximate bond yields are given by y n,t = ln(1 + i ) a n n b n n x t x c n t n x t + ξ n,t (3) where ξ n,t is the measurement error whose standard deviation is equal to σ m,i. 4 Econometric Methodology We estimate the underlying DSGE model by Bayesian methods. To construct the likelihood function, we use particle filtering that approximates the unknown density of a latent macro state vector x t given observations up to time t by simulationbased methods. Since particle filtering assumes neither the linearity nor Gaussian distributions, it is suitable for a general nonlinear and non-gaussian model, which nests our quadratic model as a special case. Doucet et al. (1) provide a comprehensive review of theoretical and practical issues. Recently, An and Schorfheide (6) and Fernàndez-Villaverde and Rubio-Ramírez (6) apply particle filtering to the estimation of nonlinear macro models. Our econometric methodology is built upon theirs. We combine the likelihood function with the prior density to compute the posterior density kernel. Posterior inferences on parameters are made

16 15 by running multiple MCMC chains and selecting the highest posterior region. Unobserved macro factors are backed out based on the entire observations by Monte Carlo smoothing applied in Fernàndez-Villaverde and Rubio-Ramírez (6). See the appendix for detailed descriptions of all the procedures used in the estimation. 4.1 Construction of the likelihood by particle filtering Previous sections provide the laws of motion of state variables and the measurement equations for observed macro variables and bond yields under the second order approximation. The resulting state transition equation (4) and measurement equations (7) and (3) can be casted into the following nonlinear state space model. x t = Γ (ϑ) + Γ 1 (ϑ)x t 1 + (I nx x t 1 ) Γ (ϑ)x t 1 + η(x t 1 )σɛ t (31) z t = α (ϑ) + α 1 (ϑ)x t + (I nz x t ) α (ϑ)x t + ξ t where ξ t N (, H) (3) z t = [ln Y t, ln(1 + π t ), ln(1 + i t ), y 4,t, y 8,t, y 1,t, y 16,t, y,t ] ϑ = [τ, β, ν, ln f, φ, u a, γ p, γ y, ρ a, ρ f, ρ i, ρ π, η a, η,f, η i, η π, ln A, ln π, h, η 1,f ] z t and ξ t are a set of observed variables and a vector of measurement errors, respectively. H is assumed to be diagonal implying measurement errors are independent. Standard deviations of measurement errors are fixed because estimating them is much more complicated in the nonlinear model. 7 We allow measurement errors for all the observed variables. If the above state space model is linear and Gaussian, Kalman filter is optimal and we can compute the prediction error likelihood by recursively applying it. However, once we get out of the linear Gaussian case, Kalman filter is no longer optimal. Simulation-based particle filtering is found to work well in the estimation of a DSGE model solved with nonlinear methods. (An (5), Fernández-Villaverde and Rubio-Ramírez (6)) The basic idea of particle filtering is to approximate unknown filtering density p(x t z t, ϑ) by a large swarm of particles x i t (i = 1, N). 8 Arulampalam et al. () provide an excellent survey 7 The detailed explanation is given in appendix. 8 Here, z t denotes observations up to time t, [z 1,, z t ].

17 16 of various algorithms. We use the sequential importance resampling filter suggested by Kitagawa (1996). The same algorithm is used and discussed in An (5) in the context of the constructing the likelihood for a DSGE model solved with a second order approximation to equilibrium conditions. 4. Posterior simulation Once the likelihood function is obtained, we can combine that information with the prior information on structural parameters to compute the posterior density. Prior information on structural parameters can be represented by the following prior density p(ϑ). All the parameters are assumed to be independent a priori. The posterior kernel is the product of prior density and likelihood: p(ϑ z T ) p(ϑ)l(ϑ z T ) (33) The analytical form of the posterior density is not known but we can use simulation methods to generate draws from the posterior density. Markov Chain Monte Carlo (MCMC) methods can be used for this purpose as explained in An and Schorfheide (6), for example. The idea of MCMC methods is to construct a Markov chain of parameters whose stationary distribution would be the posterior densities of parameters. The Markov chain is typically initialized at the posterior mode that is found by the numerical maximization of the posterior density kernel. However, the numerical optimization step rarely converges to the point with small standard errors of parameters and is a bit sensitive to starting points in our case. Alternatively, we initialize a MCMC chain at the point with the highest loglikelihood among 1 draws of parameters from prior distributions. And multiple MCMC chains starting from different points are implemented. If different chains are wandering around deeply separated regions, we pick up the highest posterior area. In the linear analysis, two separated regions have the similar magnitude of the posterior density and the selection of the highest posterior region is difficult. Unless it is mentioned otherwise, we report results of the linear analysis from the posterior region, which has a higher posterior density in the nonlinear analysis.

18 Monte Carlo smoothing In empirical finance literature of the yield curve, bond yields are assumed to be functions of a few latent factors. With the specification of the factor dynamics, they build dynamic term structure models that can be taken to data. Filtered estimates of latent factors are obtained from the estimation. However these factors are just outcomes of a statistical decomposition of the yield curve and do not have clear economic interpretation. They are often regressed on the estimated macro factors in order to clarify their economic meaning. Since we do not use latent term structure factors in the macro model, our estimation does not provide direct estimates of latent term structure factors. Ang and Piazzesi (3) and Diebold et al. (6) mention that empirical proxies for latent factors are highly correlated with latent factors extracted from a statistical decomposition. These empirical proxies are obtained by linear transformations of the observed yield curve. For example, the level of the yield curve, which is the most persistent latent factor, is related to the simple average of different bond yields. We regress empirical proxies for latent variables on the estimated macro factors in order to identify macroeconomic fundamentals associated with latent factors. Since macro factors in the model are latent variables, estimates of them conditional on all the observations E(x t z T, ϑ) would be more efficient than filtered estimates E(x t z t, ϑ). In the linear model, we can use the classical fixed interval Kalman smoothing without resorting to simulation. However, in the nonlinear model, we have to rely on simulation methods to back out latent factors. We will use an efficient way of smoothing via backward simulation as proposed in Godsill et.al. (4) and applied in Fernández-Villaverde and Rubio-Ramírez (6). The key idea of the simulation scheme is to regenerate a lot trajectories of resampled state variables stored in the forward filtering. Information from all the observations is used to select particular values of state variables at each time period. The more trajectories we generate, the more accurate smoothed estimates would be. However, the computational cost of generating one trajectory is non-negligible. We choose the number of trajectories used in the smoothing based on the performance for the

19 18 simulated dataset. 5 Estimation Results The econometric methods outlined in the previous section are applied to US data. We briefly describe our dataset in section 5.1. Section 5. describes prior distributions for parameters and tests the model s ability to account for empirical regularities mentioned in section 3. by computing predictive moments from prior draws of parameters. We consider three different versions of the model in order to identify gains in term structure implications from the nonlinear analysis and the ARCH effect of a markup shock. Section 5.3 compares posterior predictive moments to prior predictive moments and discusses the posterior distributions of parameters both in linear and nonlinear analysis. 5.1 Data We estimate the model based on US macro and treasury bond yields data. Macro variables are taken from the Federal Reserve Database (FRED) Saint Louis. The measure of output is per-capita real GDP, obtained by dividing real GDP (GDPC1) by total population (POP). The inflation rate is the log difference of the GDP price index (GDPCTPI). The nominal interest rate is extracted from the Fama CRSP risk free rate file. We select the average quote of 3-month treasury bill rate. Five bond yields (1,,3,4,5 year) are obtained from Fama-Bliss CRSP discount bond yields files. Observations from 1983:QI to :QIV are used for the estimation. To match the frequency of the yields with that of macro data, the monthly observations of the treasury bill rate and bond yields are transformed into quarterly data by arithmetic averaging.

20 19 5. Prior information 5..1 Prior distributions of parameters Prior distributions play an important role in our estimation. For example, they might place less emphasis on regions of the parameter space that are at odds with observations not included in the estimation sample. We tune prior distributions based on pre-sample data. The specification of prior distributions is shown in Table 1. Prior means of parameters are set to match implied steady state values with the average observations much like calibration exercises as in Cooley and Prescott (1995) in a pre-sample. The prior mean of the average technological progress (u a) is set to be.5% at the quarterly frequency to imply the steady state growth rate of % per year that is roughly the average growth rate of per capita real GDP during the pre-sample period (196:QI to 198:QIV). The prior mean of the discount factor (β) is chosen to set the steady state real interest rate to.8% in the annualized percentage. 9 Since we do not use data for hours worked in the estimation, the estimate of the elasticity of labor supply(ν) may not be reliable. We fix the parameter at.5 which is roughly the posterior mean in Chang et al. (6) who estimate a DSGE model with the time series for hours worked. Regarding the steady state log markup, we rely on various sources. Depending on the treatment of fixed cost, the average log markup ranges from nearly zero to.6. Rotemberg and Woodford (1999) suggest that somewhere between.15 and. would be appropriate. However, based on micro data of US manufacturing industries, Chang and Hong (6) argue that.1 would be more reasonable than. if we consider small profit rates observed in data. We use.1 as the prior mean of the steady state log markup. The prior mean of ln A is set in order to match the detrended output of 198:QIII with the steady state level implied by prior mean values of parameters. The prior distribution of ARCH parameter η 1,f is assumed to be a uniform 9 Since the average real interest rate during the pre-sample period was less than %, we cannot set u a and β to match the real interest rate and output growth at the same time. The real interest rate is about 3.% during the sample period and we target the steady state real interest rate which is slightly lower than that.

21 distribution over the interval where the existence of the stationary distribution of ln f t is guaranteed. 1 The prior standard deviation of the monetary policy reaction to expected inflation is set to be a bit smaller than the value used in related literature. This is because a loose prior generates a low value of the reaction coefficient, which creates a numerical problem in the computation of the likelihood. It turns out the likelihood is extremely sensitive to the reaction coefficient when the parameter gets closer to 1. For other parameters, we set relatively loose priors to enhance the model s ex-ante explanatory power. Standard deviations of measurement errors of output, inflation rate, and bond yields are fixed at about % of the sample standard deviations of output growth, inflation rate, and the nominal interest rate. Table 1 summarizes the prior information for all the parameters. 5.. Prior predictive checks Before taking the model to data, it would be useful to examine the properties of the term structure of interest rates implied by the macro model. The purpose is to check whether or not the model can replicate the stylized facts of data. According to Den Haan (1995), there are several empirical regularities in the post-war US term structure data. 11 On average, term premium is positive. Consumption (or output) growth is positively autocorrelated. The term structure of bond yield volatility is nearly flat. Nominal interest rates are very persistent. 1 Borkovec and Klüppelberg (1) derive the interval under the condition that ρ f belongs to [, 1]. As we decrease the upper bound of ρ f, the interval of η 1,f in which the strict stationarity of ln f t is satisfied widens. 11 We replace Den Haan (1995) s statement on term spread by term premium because the unconditional means of both of them are same as long as bond yields are stationary.

22 1 A standard one-sector real business cycle model is known to fail to explain these stylized facts. Consider the log real stochastic discount factor implied by a standard real business cycle model as illustrated in Den Haan (1995). m r t,t+1 = ln β τ(ln C t+1 ln C t ) (34) In this case, the autocorrelation of the log real stochastic discount factor is tightly related to that of consumption growth. Therefore, if consumption growth is positively autocorrelated, the uncertainty in h-period ahead stochastic discount factor (Var t (m r t,t+h )) is greater than h Var t(m r t,t+1 ). This means a long-term bond is a better hedge against unforeseen movements in consumption and,therefore, on average commands a negative term premium. 1 A similar argument can be applied to a nominal stochastic discount factor (m t,t+1 = m r t,t+1 π t+1) when inflation and consumption growth are uncorrelated. Because inflation is more positively autocorrelated than consumption growth, the same problem remains for the nominal stochastic discount factor. It seems that positive autocorrelations of consumption growth and inflation make one of stylized facts of the yield curve -a positive term premium, on average- hard to match by a standard equilibrium business cycle model. We can imagine several ways out of this deficiency as pointed out by Hördahl et.al. (5b). First, one can sever the linear relationship between log marginal utility of consumption and log consumption growth by introducing habit formation. Second, a negative correlation between inflation and consumption growth can decrease the variance of the long horizon stochastic discount factor and so increase the long-term bond yield because the intertemporal insurance service value of a long term bond is reduced. 13 Our model incorporates both features. Habit formation is already introduced in the utility function of the representative household. In addition, shocks 1 Indeed, this can be analytically proven when consumption growth is log-normally distributed. In this case, the unconditional mean of the term premium defined by the difference between the long term rate and the average of expected short rates y n,t E t( n 1 j= i t+j ) is nj=1 Var t (ln(y t+j /y t+j 1 ))) τ ( EVar t( n j=1 ln(y t+j /y t+j 1 )) E( ). 13 n n Piazzesi and Schneider (6) and Wachter (6) emphasize this channel in order to get a positive inflation risk premium. However, they also introduce recursive utility (Piazzesi and Schneider (6)) or a nonlinear habit (Wachter (6)) to amplify that impact. n

23 to desired markups of firms can create a negative correlation between inflation and output growth because they depress labor supply, therefore, output by lowering real wage and increase inflation rate at the same time. Regarding the flat term structure of volatility and high persistence, an easy solution is to introduce a highly persistent macro shock which can affect all the bond yields. The problem with the suggestion is that macro implications of the model may deteriorate by doing so. For example, suppose that inflation follows a random walk process. Then, all the nominal yields are non-stationary by construction. High persistence and flat term structure of volatility can be explained by a common trend of bond yields. Even though the non-stationary inflation can improve term structure implications of the macro model in this dimension, it may lead to counterfactual macro implications. The volatility of inflation relative to those of bond yields should be close to one according to the assumption of non-stationarity. However, the volatility of inflation barely exceeds /3 of those of bond yields in US data during our sample period. Therefore, matching the volatility of long term rate without inducing too volatile inflation remains a challenging task in the model considered here. Now we check that the model can replicate the quantitative aspects of the stylized facts of US term structure. Using simulated data from prior draws of parameters, we examine how much the model can replicate the stylized facts. 14 Since the model implied moments of macro and term structure variables rarely have analytical forms, we use simulated data to identify implications of the model. We generate 8 artificial observations of output, inflation, short term interest rate, and 1,, 3, 4, 5 year bond yields from 1 parameters drawn from the prior distributions. 15 Then we compute sample moments of macro variables and term structure variables such as the standard deviation of inflation, mean term premium, and the standard deviation of term premium. It is well known that three latent factors such as level, slope, and curvature can explain most of the time variation of the yield curve (e.g., 14 An introduction to prior predictive checks is found in Geweke (5). 15 In practice, we generate 18 observations and discard the initial 1 observations to attenuate the dependence on the starting point.

24 3 Litterman and Scheinkman (1991)). Even though our model does not incorporate latent term structure factors, it has implications for empirical proxies for them. We measure the level as the arithmetic average of three month rate, year bond yield, and five bond yield, the slope as the three month rate minus the five year bond yield, and the curvature as two times of the two year bond yield minus the sum of the three month rate and the five year bond yield. It turns out that these empirical proxies are highly correlated with three latent factors extracted by principal component analysis. 16 To understand the importance of nonlinearities, we generate data from (i)the linear model, (ii)the nonlinear model without the ARCH effect, and (iii)the benchmark nonlinear model with the ARCH effect. 17 The resulting 1 sample moments for three models are plotted in Figure 1 -. Several things are noticeable. First, the ARCH effect gets mean term premium closer to the actual sample moment. The ARCH effect increases mean term premium by 3 basis points on average. Nonlinear terms without the ARCH effect do not change mean term premium much and sometimes even decrease mean term premium compared to the linear model. This can be understood by the precautionary saving term in equation (6) that decreases long term bond yields more than the short term interest rate. 18 Regarding the term structure of volatility, the ARCH effect does not make much difference and all the models do not generate a flat shape of the term structure of volatility. Volatilities of bond yields decrease rapidly over maturities. 19 In addition, inflation is 16 Three latent factors account for 99.98% of the variance of sample yield curve data. The correlation coefficients between latent factors and empirical proxies are.999 (level),.993 (slope), and.774 (curvature). 17 Unlike the estimation, we do not allow measurement errors in the simulation because they can obscure the essential features of the model. For example, the difficulty in generating a positive term premium and a positive autocorrelation of consumption growth in a simple RBC model is hard to detect in the simulation once we introduce measurement errors. 18 In Hördahl et al. (5b), the precautionary saving term is found to increase the term premium when the consumption growth is negatively autocorrelated. Again, this is at odds with a positive autocorrelation of consumption growth in data. 19 The underprediction of the volatility of the level and the overprediction of volatilities of the slope and the curvature result from this mismatching of the term structure of volatility.

25 4 too volatile. The same problem is documented in other literature using equilibrium bond yields implied by a macro model (Den Haan (1995), Piazzesi and Schneider (6), Ravenna and Seppälä (6) etc.). Second, the three models tend to overpredict the volatility of inflation. Assuming less persistent or less volatile shocks may help us in getting the moment closer to the actual sample moment. However, by doing so, we decrease the average term premium which is already lower than the actual sample moment. Figure 1 clarifies this trade-off. Our prior distribution is a compromise between correcting macro implications of the model and preventing worse term structure implications. Third, consistent with the previous discussion of term structure implications of habit formation, all the models can account for a positive autocorrelation of output growth and a positive mean term premium at the same time. 1 Finally, the markup shock generates a negative correlation of output growth and inflation. Since the ARCH effect amplifies the volatility of a markup shock, the impact is more pronounced in the case of the quadratic model with the ARCH effect. 5.3 Posterior analysis Posterior distribution Additional information from data about the model also can be obtained from contrasting prior distributions of parameters with the posterior counterparts. Figures 3 and 4 depict draws from prior and posterior distribution of each parameter. For most parameters, the posterior draws are more concentrated. Also, there are clear shifts of the mean values for some parameters. The curvature of utility function parameter (τ) and the degree of habit formation (h) are higher in the posterior The exception is Wachter (6). She introduces a complicated nonlinear habit shock, which creates a time-varying risk aversion by construction. That mechanism seems to generate the flat term structure of volatility. It is an interesting research agenda if we can replicate the result in the model in which inflation is endogenously determined unlike Wachter (6). 1 Although we do not report here the results, we generate data from the linear model without habit formation too. As expected, moments from simulated data cannot match the positive autocorrelation of output growth and the positive mean term premium at the same time.

26 5 distribution for both linear and nonlinear models. The steady state value of target inflation (ln π ) is higher in the nonlinear model. The steady state target inflation is not the same as the unconditional average inflation in the nonlinear model due to the impacts of quadratic terms while the two concepts coincide in the linear model. The difference can be illustrated by the following equation. E(ln π t ) ln π = 1 gπ σσ + g π i t 1 E(î t 1 ) + g π c t 1 E(ĉ t 1 ) + tr(g π xxe(x t x t)) (35) We can call this term the quadratic adjustment term. The magnitude of this term can be quantified by simulating a long time series and replacing expected values by sample averages. In our case, the term is negative, reflecting the fact the estimated steady state target inflation is higher than the average inflation. The above finding is consistent with An and Schorfheide (6) who call this term stochastic steady state effect. The reaction of the central bank to expected inflation is stronger in the posterior distribution. Interestingly, the nonlinear model indicates a bit more stronger response of the central bank, since the likelihood in the quadratic model decreases rapidly when the reaction parameter gets smaller and approaches the indeterminacy region. The ARCH effect parameter η 1,f is much smaller in the posterior distribution. This suggests that time varying volatility induced by the ARCH effect would have a limited role in the data. Gains from the nonlinear analysis are clear in the better identification of the highest posterior region of parameters. Figure 5 provides the posterior contours of persistence parameters (ρ f, ρ π ) around posterior mean values of them computed from different MCMC chains. It turns out that the nonlinear analysis clearly identifies the highest posterior region of parameters while the linear analysis cannot do that. In the linear analysis, two deeply separated regions of the parameter space are associated with the posterior density of the similar magnitude. Even if we select one region from the linear analysis as the highest posterior area, the nonlinear If η 1,f is greater than 1 ρ f, the second moment of ln f t does not exist. This would make the unconditional distributions of macro variables and bond yields fairly heavy-tailed.