The Term Structure of Interest Rates and the Monetary Transmission Mechanism

Transcription

1 The Term Structure of Interest Rates and the Monetary Transmission Mechanism Massimiliano Marzo Ulf Söderström Paolo Zagaglia November 6, 7 Preliminary and Incomplete Please do not circulate without the authors permission Abstract We provide empirical evidence that the term structure of interest rates is an integral part of the monetary transmission mechanism. Based on these findings, we amend the standard monetary business cycle model to generate an endogenous term structure of interest rates where movements in the term structure and in term premia) have a direct effect on private agents spending decisions. The model features bond market segmentation through adjustment costs for bond holdings and transaction costs between money and bonds. Our model is able to replicate the main stylized facts concerning the relation between output fluctuations and the yield spread: the negative correlation between the yield spread and the output gap and the positive correlation between the yield spread and future output growth. Furthermore, the model implies that movements in term premia are negatively correlated with future output growth, in line with recent evidence. Keywords: Monetary policy, Yield curve, Term premia. JEL Classification: E43, E44, E5. Marzo: University of Bologna, [email protected]; Söderström: Bocconi University, IGIER and CEPR, [email protected]; Zagaglia: Stockholm University, [email protected]. We are grateful for comments from Diego Rodriguez-Palenzuela, Carlo Rosa, David Small, Paolo Surico, John Williams, Bernhard Winkler and participants at a seminar at Stockholm University, the 1th International Conference on Computing in Economics and Finance of the Society for Computational Economics in Limassol, Cyprus, in June 6, and the Workshop on Monetary and Fiscal Policy, the Exchange Rate and the Term Structure at Bocconi University in March 7. We also acknowledge financial support from the Ministero dell Università e della Ricerca.

2 1 Introduction Many informal accounts of the monetary transmission mechanism assign an important role to the term structure of interest rates. Through open market operations, the central bank is able to affect the level of money market interest rates. Movements in money market rates, in turn, affect longer-term interest rates, which have an impact on private agents spending decisions. The effects of monetary policy on long-term interest rates are partly due to changes in expectations concerning future monetary policy and thus money market rates), and partly through movements in term premia. The standard model framework used in modern monetary economics emphasizes the effects of monetary policy on long-term interest rates through expectations. However, the approach typically used to analyze these models involves a log-linear approximation around the steady state of the model, and does not allow for the existence of time-varying term premia. This framework is therefore not able to capture the effects of term premia on the macroeconomy. More recently, several contributions have combined macroeconomic models with asset pricing models from the finance literature to generate time-varying term premia and study the relationship between monetary policy and the term structure. 1 In a seminal paper, Evans and Marshall 1998) use three popular approaches to identify the response of the yield curve to monetary-policy shocks in the U.S. in a structural vector autoregression, finding that the responses decline along the maturity structure. They conclude that the dynamics of the yield curve is driven by liquidity effects, and propose a limited-participation model capable of replicating that pattern. Other contributions for instance, Ang and Piazzesi, 3) have documented the joint dynamics of the term structure and the macroeconomy through no-arbitrage conditions, where the term structure is priced through the reduced form of a pricing kernel. Yet another strand of the literature has focused on the termstructure implications of the standard monetary business cycle or New-Keynesian ) model, for instance, Rudebusch and Wu 3, 7), Bekaert, Cho and Moreno 3), Hördahl, Tristani, and Vestin 6), and Ravenna and Seppälä 7a,b). However, most of these contributions price the term structure of interest rates using the kernel for one-period bonds extracted from the solution of the dynamic model. Consequently, while these models are able to generate time-varying term premia, movements in term premia do not affect private agents spending decisions: the term premia depend on the macroeconomy, but there is no feedback in the opposite direction. In this paper, we instead propose a unified framework to study the interactions between monetary policy, the macroeconomy, and the term structure of interest rates, with an explicit feedback from the term structure to the economy. Inspired by the portfolio approach of Tobin 1969, 198), recently adopted by Andrés, López-Salido and Nelson 4), we introduce bonds of different maturities in a monetary business cycle model typically used for monetary policy analysis. Although the feedback channel works only through money demand, non-trivial interactions between monetary policy and the term structure arise. We begin in Section by documenting the existence of a feedback effect in the U.S. economy. We augment a Vector Auto-Regression VAR) model that is typically used to study monetary 1 See Diebold, Piazzesi, and Rudebusch 5) for an overview. See also Rudebusch, Sack, and Swanson 7). 1

3 policy shocks to include three measures of government bond yields at short, medium and long maturity. We apply block-exogeneity tests in the spirit of Sims 198) to investigate whether the block of yields Granger-causes the rest of the system, including output and inflation. We find that rejections of the null hypothesis of block exogeneity are large and robust to different definitions of short, medium and long-term rates. Motivated by the empirical evidence, we propose in Section 3 an extension of the New Keynesian model that introduces a feedback from the term structure to the macroeconomy. We introduce several bonds of different maturities in the model, thus characterizing the entire term structure. Since the yields are endogenous variables, they are obtained from the log-linearized solution of the model, and no higher-order approximation method is needed to generate time variation in term premia. Furthermore, movements in bond yields and thus in term premia) affect private returns on bond holdings, and thus the spending decisions of private agents. A prerequisite for any feedback mechanism to exist is that there are positive bond holdings of different maturities at each point in time. We obtain this by generating market segmentation through bond-adjustment costs: households pay a cost each time they change holdings of bonds for each maturity. Since these costs are assumed to be a function of output, they induce a feedback through the resource constraint of the economy. As a second step, we assume that the household s portfolio choice is affected by the money content of bonds. Money and bonds are imperfect substitutes in such a way that consumers pay a transaction cost each time they reallocate wealth between bonds and money. This implies that the degree of imperfect substitutability between money and bonds affects the yields. Our contribution is closely related to Andrés, López-Salido and Nelson 4). Following Tobin 1969), they present a version of the New Keynesian model where the supply of long-term bonds affects both the level of the yields and the term premia. Their estimation results for the U.S. economy indicate that movements in the stocks of financial assets help explaining the deviations of long-term rates from the expectations hypothesis. This key feature is present also in our model. However, three points of difference are worth mentioning. First, in Andrés, López-Salido and Nelson 4) the segmentation of the bond market is due to exogenous transaction shocks. We instead assume that there are bond-adjustment costs. The reason is that we want to shed light on the optimal allocation choice between bonds. Second, the adjustment cost functions of our model are cast in terms of real output. This introduces a clear unit of measure for the costs. Moreover, it enhances the endogeneity of bond premia, since any factor that affects output has a relative impact also on the cost of trading bonds. Third, in contrast to Andrés, López-Salido and Nelson 4), we do not impose adjustment costs on the per-period change of the money stock. Having calibrated our model on U.S. data in Section 4, we show in Section 5 that our framework matches the negative contemporaneous correlation between the yield spread and the output gap in the U.S., as well as the decreasing correlation between the yields and the output gap along the maturity structure. The model also captures the response of the term structure to monetary policy shocks found in U.S. data, and the predictive power of the yield spread and the term premium for future output growth documented in the literature see, for example, Rudebusch, Sack, and Swanson, 7). Finally, we also show that the adjustment costs between money and bonds are essential to induce volatility in the yields and to match the empirical correlations. The larger the weight

4 on the relation of complementarity between bonds and money, the more volatile are the yields. This is because in order to minimize the reallocation of bond holdings across periods, most of the adjustment takes place through changes in bond prices. We end in Section 6 by discussing some possible directions for future research based on our results. The U.S. term structure and the macroeconomy: Some stylized facts We begin by reporting some stylized facts for the U.S. term structure of interest rates over the period from 1987:4 to 6:4. We study four maturities: 3 months, 1 year, 5 years and 1 years, as well as the 1 year 1 year spread and the federal funds rate. The interest rate data are quarterly averages of monthly data on treasury bills for the 3-month rate) and constant maturity rates for the 1-, 5-, and 1-year rates). We also report correlations between the yields and the output gap, the percentage deviation of real GDP from potential, calculated by the Congressional Budget Office. All data were obtained from the FRED database of the Federal Reserve of St. Louis. Figure 1 shows the data used. 3 Table 1 reports the mean and standard deviation of the yields and their correlation with the output gap. The average yield curve is upward-sloping, with yields of shorter maturities more volatile than those of longer maturities. All yields are procyclical and the correlations with output decrease with maturity. Thus, as is also clear from Figure 1, the yield spread is countercyclical. Figure shows the impulse responses of the macroeconomy and the term structure to a monetary policy shock estimated from a Vector Auto-Regression on monthly U.S. data from 1959:1 to 6:1. 4 The VAR includes 1 lags of six variables ordered as follows: the industrial production index in logs), the producer price index finished goods less food and energy, in logs), the federal funds rate, and the 1-year, 5-year, and 1-year yields. We identify the monetary policy shock through a Choleski decomposition. As shown in Figure, a contractionary monetary policy shock leads to a gradual contraction in industrial production and a slow decline in the producer price level. All yields increase significantly for the first 6 8 months after the monetary policy shock, with the response of the 1-year rate being twice as large as that of the 5-year rate and three times as large as the 1-year rate. This is consistent with much of the empirical literature on monetary policy and the term structure of interest rates, for example, Evans and Marshall 1998, 3). A common assumption in the literature is that the term structure has no direct effects on the macroeconomy, see, for instance, Evans and Marshall 1998, 3). We here use the estimated VAR model to study whether this hypothesis is supported by U.S. data. Following Sims 198), we test for block exogeneity of the yields. Let the vector x t collect the variables of the VAR model, that is, industrial production, producer-price inflation, commodity-price inflation, the federal funds 3 We do not use zero-coupon yields, which would be more closely related to the yields in our theoretical model. However, data on zero-coupon constructed by Duffee ) for the sample 195Q1 1998Q4 are essentially indentical to the constant-maturity rates used here. Also, estimating potential output through a Hodrick-Prescott filter gives very similar results. 4 We estimate the VAR on a longer sample of monthly data in order to better identify the effects of monetary policy on the term structure. Evans and Marshall 3) also use monthly data starting in

5 rate, and the 1-, 5- and 1-year yields. The structural form of the VAR can be written as q A x t = k + A i x t i + ε t, 1) i=1 and the resulting reduced form is q x t = k + B i x t i + u t. ) i=1 Testing for block exogeneity of the macro variables with respect to the three yields implies testing the restriction that the last three columns of the B i matrices i = 1,..., q) are all zero. The test involves estimating an unrestricted VAR which includes all variables also the yields), and a restricted VAR which excludes the yields. Given the variance-covariance matrix Ω U of the unrestricted model, and the variance-covarianze matrix Ω R of the restricted model, we compute the likelihood-ratio test statistic LR = T p) log Ω R log Ω U ), 3) where T is the number of observations, p indicates the number of parameters of the unrestricted system, and Ω denotes the determinant of Ω. The null hypothesis is that the block of restricted variables the yields) does not enter the remaining part of the system. The test statistic asymptotically follows a χ -distribution with the number of degrees of freedom equal to the number of restrictions in the system. Table reports the results from block-exogeneity tests with alternative indicators for yields on short, medium and long-term bonds. For instance, in the first panel, the short rate is the three-month rate, the medium rate is the one-year rate, and the yield on long maturities is the ten-year rate. We report the results for the full sample from January 1959:1 to December 6, the pre-greenspan and the Greenspan-Bernanke sample. The first column of Table shows that the null is strongly rejected for the full sample, implying that the macro variables respond to shocks originating in the term structure. The results in the remaining columns reveal that the evidence against block exogeneity is stronger for the pre-greenspan period than for the Greenspan-Bernanke period. During the earlier period, the restrictions are always strongly rejected, whereas in the later period the null of block exogeneity is rejected less often. 5 3 The model In this section, we develop a business cycle model with an endogenous term structure of interest rates, which is an integral part of the transmission of monetary policy. The starting point for our analysis is a New-Keynesian model with sticky prices, habits in consumption, and capital adjustment costs. To this model we add an endogenous term structure of interest rates by assuming 5 The asymptotic distribution of the test statistic is sensitive to the presence of unit roots see Watson, 1994). Failing to account for cointegration in the VAR would lead the test statistic to have a non-standard asymptotic distribution. In order to check the robustness of our results, we also computed block exogeneity tests in VARs with either all variables in first differences, or with the macro variables in first differences and the yields in levels. Firstdifferencing strengthens the rejections of the null for the pre-greenspan sample, while the case for block exogeneity in the Greenspan-Bernanke also becomes more compelling. However, the main impression from Table is confirmed. 4

6 that households allocate their assets among four different types of bonds, which we interpret as being of different maturity: very short-term money market bonds, short-term bonds, medium-term bonds, and long-term bonds. As households are assumed to face costs when adjusting their bond holdings, there is a non-zero demand for each type of bond, and the expectations hypothesis does not hold. Households also face transaction costs for money holdings, so the effect of term structure movements operate through households money demand. 3.1 Households There is a continuum of identical and infinitely-lived households indexed by i [, 1]. For convenience we omit the index i in what follows.) These households obtain utility from consumption of a bundle C t of differentiated goods relative to an endogenous habit level, real money holdings M t /P t, and disutility from labor L t according to the utility function u C t, C t 1, M ) t 1, L t = P t 1 1/σ C t γc t 1 ) 1 1/σ χ Mt P t ) 1 χ where C t is a constant elasticity of substitution aggregator of differentiated goods: Ψ 1 + 1/ψ L1+1/ψ t, 4) [ 1 θ/θ 1) C t = C t j) dj] θ 1)/θ, 5) σ determines the elasticity of intertemporal substitution, γ determines the importance of habits, χ is the elasticity of money demand, ψ is the elasticity of labor supply, and θ is the elasticity of substitution across the different varieties of goods. Households allocate their wealth among money holdings, accumulation of capital, which is rented to firms, and holdings of four types of nominal bonds. We interpret these different bonds as very short-term money market bonds denoted B t ), short-term bonds B S,t ), medium-term bonds B M,t ), and long-term bonds B L,t ), which pay the returns R t, R S,t, R M,t, and R L,t, respectively. The representative household maximizes its life-time utility U t = E t= β t u C t, C t 1, M ) t P t,, L t, 6) where β is a discount factor, subject to the budget constraint B t + B S,t ) 1 + AC S B M,t ) P t P t AC M B L,t ) t P t AC L t P t t + M t 1 + AC m ) t ) + C t + I t 1 + AC I P t t = R t B t 1 P t + R S,t 1 B S,t 1 P t + R M,t 1 B M,t 1 P t + R L,t 1 B L,t 1 P t + M t 1 P t + w t L t + q t K t + Ω t τ t AC P t, 7) where the AC ι t terms are different adjustment costs, to be specified below, P t is the aggregate price level, and I t is investment. The household obtains income from renting capital, K t, to firms at the rental rate q t, labour services, w t L t, where w t is the real wage, and from its share in firms real profits, Ω t. Finally, households pay a real lump-sum tax τ t. 5

7 According to the traditional asset allocation theory, agents hold different types of assets in their portfolio depending on each asset s risk/return trade-off and expectations about the future path of this trade-off. For government bonds, the risk element is exclusively related to the uncertainty with respect to the future path of returns. The main difficulty when modelling assets with different rates of return in general equilibrium is due to the solution technique employed which, for computational reasons, typically involves Taylor approximations up to first or second order) of the system of equations around the steady state. Of course, this procedure eliminates any role for higher-order terms, making it difficult to allow a full portfolio choice on the basis of the risk/return trade-off. We instead implement an alternative methodology that allows for the simultaneous presence of different rates of return on government bonds. To ensure a non-zero demand for each bond, we follow Andrés, López-Salido and Nelson 4) and generalize the Tobin 1969, 198) model of portfolio allocation by inserting a set of portfolio adjustment frictions, which can be rationalized as transaction costs. We assume that bond trading is costly to each agent, and, in particular, that bond adjustment costs are quadratic and given by AC ι t = φ ι for ι = S, M, L. Bι,t /P t B ι,t 1 /P t 1 ) Y t, 8) The adjustment cost is paid in terms of aggregate output Y t, an assumption that allows us to better quantify the magnitude of these costs in terms of the budget for the representative household, and also implies that spreads between the different bonds returns vary over time. In steady state, as long as φ ι these adjustment costs are non-zero. Furthermore, in order to capture the entire dimension of costs involved in any financial transaction we also assume transaction costs for money holdings. As in Andrés, López-Salido and Nelson 4), the money transaction costs follow AC m t = [ v S ) Mt κ S 1 + v ) M Mt κ M 1 + v ) ] L Mt κ L 1 Y t, 9) B S,t B M,t B L,t where κ i = B i /M is the inverse of the steady-state money/bond ratio for bond ι. Also these costs are paid in terms of real output Y t, and they measure the amount of resources spent in order to shift the portfolio allocation between money and bonds of various maturities. 6 Finally. the money/bond transaction costs are present only during the transition from one steady state to another one, and are zero at the steady state. Consequently, only the bond adjustment costs are present in steady state, and are therefore responsible for the different long-run rates of return. To obtain realistic internal propagation mechanisms in the model and generate fluctuations of the rental rate of capital compatible with the empirical evidence we introduce quadratic investment adjustment costs, following Abel and Blanchard 1983) and Kim ). Thus, AC I t = φ K It K t ), 1) 6 The transaction cost functions between money and bonds can be written in a Cobb-Douglas form with an elasticity of substitution equal to 1, implying that money and bonds are imperfect substitutes. 6

8 where I t is the level of investment. The law of motion of the capital stock is given by K t+1 = I t + 1 δ)k t, 11) where δ is the depreciation rate of the capital stock. Finally, the term ACt P in the budget constraint 7) is due to price adjustment costs, which are specified below. 3. Optimality conditions The optimal intra-temporal consumption choice implies the typical demand function C t j) C t = [ ] θ Pt j), 1) P t where P t is the aggregate price index given by [ 1 1/1 θ) P t = P t j) dj] 1 θ. 13) Maximizing life-time utility in equation 6) subject to the budget constraint 7) implies that the optimal inter-temporal consumption choice satisfies C t γc t 1 ) 1/σ βγe t C t+1 γc t ) 1/σ = λ t, 14) where λ t is the marginal utility of consumption; the optimal labor supply is given by ΨL 1/ψ t = λ t w t ; 15) the optimal money holdings follow m χ t + βe t λ t+1 = λ t [1 + ACt m ] 16) π t+1 ) ) ) ] Yt mt Yt mt Yt + v M κ M κ M 1 + v L κ L κ L 1, b S,t b M,t b M,t b L,t +λ t m t [ v S κ S mt b S,t κ S 1 where m t = M t /P t are real money holdings, b ι,t = B ι,t /P t are real holdings of bond ι, and π t = P t /P t 1 is the gross rate of inflation; holdings of the money market bond follow βe t R t λ t+1 π t+1 = λ t ; 17) and holdings of the remaining three bonds satisfy { ) 3 R ι,t λ t+1 bι,t+1 βe t + βφ ι E t λ t+1 Y t+1} π t+1 b ι,t = λ t [1 + 3 ) ) ) ] φ bι,t mt mt ι Y t v ι κ ι κ ι 1, 18) b ι,t b ι,t 1 b ι,t for ι = S, M, L. Note that in the case without bond adjustment and transaction costs φ ι = v ι = b L,t 7

9 κ ι = ) the optimality conditions for the four bonds are identical, so all bonds give the same return. The different returns of the bonds thus arise from the presence of adjustment and transaction costs. Households also own the capital stock which is rented to firms. The first-order conditions for the capital stock and investment are then given by β 1 δ) E t µ t+1 = µ t λ t [ ) ] 3 It q t + φ K, 19) K t and βe t µ t+1 = λ t [1 + 3 φ K It K t ) ], ) where µ t is the marginal value of capital. 3.3 Firms Firms indexed by j [, 1]) produce and sell differentiated final goods in a monopolistically competitive market. These goods are produced using capital and labor following the Cobb-Douglas production function Y t j) = a t K t j) α L t j) 1 α Φ, 1) where a t is a technology process given by log a t ) = ρ a log a t 1 ) + ε a t, ) Φ is a fixed cost to ensure that profits are zero in steady state, and ε a t is an i.i.d. shock with zero mean and constant variance σa. Firms set prices to maximize the expected future stream of profits subject to a quadratic price adjustment cost, following Rotemberg 198). 7 The price-adjustment cost function ACt P takes the form AC P t = φ P ) Pt π Y t, 3) P t 1 so price changes that deviate from the steady-state rate of inflation π are costly. The presence of price adjustment costs implies that the firm s price-setting problem is dynamic. The expected future profit stream is evaluated through a stochastic pricing kernel for contingent claims ρ t, which plays the role of the firms discount factor. However, assuming that each agent has access to a complete set of markets for contingent claims, the discount factors of firms and households are equal: E t ρ t+1 ρ t = βe t λ t+1 λ t. 4) 7 Alternative ways to include nominal rigidities include the Calvo 1983) or Taylor 198) models of staggered prices. As these schemes have similar implications for the dynamics of aggregate inflation, the choice is not crucial for our purposes. 8

10 Each firm chooses its production inputs to maximize profits subject to the production function 1). The first-order conditions with respect to capital and labor are then given by q t = α 1 1 ) ) Yt + Φ e y, 5) t K t w t = 1 α) 1 1 ) ) Yt + Φ e y, 6) t L t where we have omitted the index j and where e y t denotes the output demand elasticity, determined by 1 e y t = 1 θ { [ ]} λt+1 1 φ P π t π) π t + βφ P E t π t+1 π) π Y t+1 t+1. 7) λ t Y t Equation 7) measures the gross price markup over marginal cost. Without costs of price adjustment φ P = ), this markup is constant and equal to θ/θ 1). With this formulation it is straightforward to see that all supply side shocks affect the magnitude and the cyclical properties of the markup. When price adjustment is costly φ P > ), the log-linearized version of equation 7) is given by a typical New Keynesian Phillips curve. 3.4 The government sector The government determines the level of taxes and bond supply, while the central bank determines the level of the money market interest rate. The government budget constraint is given by B t + B S,t + B M,t + B L,t + M t + τ t 8) P t P t P t P t P t = R t 1 B t 1 P t + R S,t 1 B S,t 1 P t + R M,t 1 B M,t 1 P t + R L,t 1 B L,t 1 P t + M t 1 P t + g t, where g t is government spending. For simplicity, define the government s total liabilities as B t B S,t B M,t B L,t h t = R t + R S,t + R M,t + R L,t + M t. 9) P t P t P t P t P t Then we can rewrite the government budget constraint as h t + R t R S,t ) b S,t + R t R M,t ) b M,t + R t R L,t ) b L,t = R t π t h t 1 + R t g t τ t ) R t 1)m t 3) In order to close the model, we assume that the real supply of short-, medium-, and long-term bonds follow the exogenous processes log b ι,t ) = ρ ι log b ι,t 1 ) + ε ι t, 31) for ι = S, M, L, where ε ι t are i.i.d. shocks with zero mean and constant σ ι. To avoid the emergence of inflation as a fiscal phenomenon, as in Leeper 1991) and Schmitt- Grohé and Uribe 6), we assume a feedback rule for fiscal policy such that the total amount of 9

11 tax collection is a function of the total government s liabilities outstanding in the economy: [ ] b S,t 1 b S T t = ψ + ψ 1 h t 1 h) + ψ R S,t 1 R S π t π [ ] [ ] b M,t 1 b M b L,t 1 b L +ψ R M,t 1 R M + ψ R L,t 1 R L, 3) π t π π t π where T t is nominal lump-sum taxes. In order to rule out indeterminacy we restrict the parameter ψ 1 to the range β 1 1, β 1 + 1). This configuration defines fiscal policy as passive in the sense of Leeper 1991), or Ricardian in the sense of Woodford 3), and implies that fiscal policy is not allowed to act independently from the outstanding stock of government s liabilities, but taxes are set in order to avoid an explosive path for public debt. We assume that government expenditure follows the exogenous AR1) process log g t ) = ρ g log g t 1 ) + ε g t, 33) where ε g t is an i.i.d. disturbance with zero mean and variance σ g. The central bank is assumed to set the money market interest rate its policy rate) R t according to the Taylor 1993) rule log ) Rt = α R log R Rt 1 R ) [ + 1 α R ) α π log πt ) + α Y log π )] Yt + ε R t. 34) Y This formulation assumes that the money market rate is determined by the deviation of inflation and output from the steady state with a gradual adjustment. We also include an exogenous monetary-policy shock ε R t, which is normally distributed with zero mean and variance σ R. 3.5 Resource constraint Finally, the model is completed by the resource constraint Y t = C t + I t [ + [ φ S 1 + φ K BS,t /P t B S,t 1 /P t 1 [ v S + It K t ) ] ) + φ M Mt B S,t κ S 1 + g t + φ P BM,t /P t B M,t 1 /P t 1 ) + v M ) Pt π Y t P t 1 ) + φ L Mt B M,t κ M 1 BL,t /P t ) ] Y t B L,t 1 /P t 1 ) + v ) ] L Mt κ L 1 Y t. 35) B L,t Thus, total output is allocated to consumption, investment including the capital adjustment cost), government spending, the price adjustment cost, and the sum of adjustment costs for bond and money holdings. 3.6 Model summary and log-linear approximation The complete model consists of equations for the endogenous variables: the household budget constraint 7), the capital accumulation equation 11), the households optimality conditions 14) ), the production function 1), the firms optimality conditions in 5) 7), the government s 1

12 total liabilities in 9), the government budget constraint 3), the fiscal policy rule 3), the monetary policy rule 34), and the resource constraint in 35). In addition, the model includes six exogenous processes: a technology shock in ), the three bond supply equation in 31), a government spending shock in 33) and a monetary policy shock in 34). We log-linearize the model around its steady state. 8 In order to gain intuition about the interaction between the term structure and the macroeconomy, it is useful to study the log-linearized equations for money demand and bond pricing. Denote by z t the log deviation of the variable z t from its steady-state level z. The log-linearized approximation of the Euler equation 17) delivers the standard pricing equation for money market bonds E t λt+1 E t π t+1 = λ t R t. 36) This expression can be substituted into the log-linearized version of the money demand equation 16) to obtain m t = A 1 E t c t+1 + A c t 1 A 3 c t A 4 Rt + A 5 bs,t + A 6 bm,t + A 7 bl,t, 37) where the A coefficients are convolutions of the structural parameters. Given the imperfect substitution between money and bond holdings, money demand is affected by the quantities of bonds at short, medium and long maturities. However, differently from the framework of Andrés, López- Salido and Nelson 4), the bond yields do not affect money demand directly, only the money market rate enters the money demand equation. The log-linearized version of optimality condition for bond holdings in equation 18) is given by R ι,t = B 1 λt B E t λt+1 + B 3 ỹ t B 4 E t ỹ t+1 + B 5 E t π t+1 ] [B 6 m t B 7 bι,t B 8 bι,t 1 B 9 E t bι,t+1 38) for ι = S, M, L, where the term in brackets underlines the role played by the adjustment costs between bonds and money. As the bond yields enter only the optimality conditions embedded in equations 36) 38), the transmission channel from the term structure to consumption and the rest of the economy works through money demand and its relation with bonds at different maturities. 4 Calibration The model is calibrated to match the behavior of quarterly U.S. data over the period from 1987:4 to 6:4, as reported in Tables 1 and 3. A subset of the parameters are chosen based on other studies or empirical evidence. For instance, the elasticity of substitution across differentiated goods θ has been set equal to 6, in order to generate a markup of 1., as in Schmitt-Grohé and Uribe 6). The habit formation parameter γ is set equal to.7, following the empirical evidence reported by, for instance, Christiano, Eichenbaum, and Evans 5), Smets and Wouters 7), or Altig, Christiano, Eichenbaum, and Lindé 5). The elasticity of substitution σ is calibrated to.5, which is within the range of values used in the real business cycle literature. The depreciation rate of capital and the capital share of production α are set to.5 and.36, respectively, following 8 The steady-state relationships are shown in Appendix A and the complete log-linear system of equations in Appendix B. 11

13 Christiano, Eichenbaum, and Evans 5). Finally, the price adjustment cost parameter φ P is set to 1, as suggested by empirical evidence reported by Ireland 4). We calibrate several parameters so that the steady state of our model matches the stylized facts reported in Table 3. The discount factor β is set to.9975 in order to match the average real interest rate over the sample. The elasticity of real money balances χ has been chosen to match the money-to-output ratio implied by the steady-state relation of the first-order condition on money. This gives χ = 7. Similarly, the elasticity of labor supply ψ has been chosen from the steady-state version of the first-order condition for labor by matching the ratio of market to non-market activities for the U.S. economy, yielding ψ = 1. The capital adjustment cost φ K is calibrated through the first-order conditions with respect to capital and investment in steady state, in order to match the real interest rate of the U.S. economy and the relevant share of investment to GDP. This gives φ K = 1, 15. In the calibration of the bond-adjustment costs, we use the optimality conditions for the three types of bonds. For each bond, we choose the adjustment cost to match average yields reported in Table 1, using the 1-year rate as the measure for the yield on short-term bonds, the 5-year rate as the medium-term yield, and the 1-year rate as the yield on the long-term bond. This gives φ S =.6, φ M =.5, and φ L =.7. The parameters of the cost for money versus bond holdings are not easily calibrated, as these costs are zero in steady state. We set these parameters such that the pattern of contemporaneous correlations between the different interest rates and the output gap is broadly consistent with the empirical evidence, yielding v S = 5., v M = 4., and v L =.5. The debt to GDP ratio in steady state is set to 45 percent, while the ratio of debt at various maturities to total debt are taken from Missale 1999), reported in Table 3. The parameters of the processes for the technology, government spending and monetary policy shocks are calibrated to have values similar to those of Altig, Christiano, Eichenbaum, and Lindé 5), Andrés, López-Salido and Nelson 4) and Kim ). The autoregressive coefficients are set to ρ a =.95 and ρ g =.9, while the standard deviations are σ a =.1 and σ R =.5. The volatility of government spending shocks is set to the value used by Christiano and Eichenbaum 199), σ g =.1. To set the autoregressive parameters for the bond supply processes we estimate a first-order autoregression on yearly data taken from Missale 1999), which gives ρ S =.7, ρ M =.73, and ρ L =.75. The volatility of the bond-supply shocks are set to match the contemporaneous correlation between the yield spread and the output gap over the sample. The resulting values are σ S =.9, σ M =.3, and σ L =.95. Finally, the parameters in fiscal policy rule are set to ψ 1 = ψ =.3 and ψ is set to match the ratio of government spending to output in the U.S. economy, yielding ψ =.158. The parameters in the monetary policy rule are set to α π = 1.5, α Y =, and α R =.7. Table 4 reports all calibrated parameters. The calibration delivers an inflation rate equal to.53 percent per year. The capital-output ratio generated by the model is 1.49 on a quarterly basis), which is close to the value of 11.3 suggested by Christiano 1991). The ratio of public spending to GDP in steady state is percent, while the tax to GDP ratio is 19.7 percent. Both values are very close to both to those used in empirical studies and to those of theoretical RBC studies. 1

14 5 Results Table 5 summarizes the behavior of the model, reporting the yield curve standard deviations and correlations with output from simulated data. 9 The standard deviations of the short rate and the spread are similar to those of the data reported in Table 1, while the policy rate, the medium-term rate and the long rate are much more stable in the model than in the data. The contemporaneous correlations of all yields with output are positive and decreasing with maturity, as in the data. The yield spread is countercyclical, and its correlation with output is similar to that found in U.S. data. 5.1 Model responses to shocks Figures 3 5 show the model impulse responses to one standard deviation innovations to the productivity, monetary policy and government spending shocks, respectively. A positive productivity shock in Figure 3 triggers a negative response of inflation, which drives the monetary policy rate below its long-run value. All yields fall, but with the response decreasing with maturity. This causes the term spread defined as the difference between the long and the short-term rate to respond positively. The yields at short, medium and long maturities increase because because the decrease in inflation increases the real value of bonds, raising the demand for bonds. Given the fixed supply of bonds, the increase in the price of bonds results lower in yields. Money demand increases, due to the reduction in the money market rate as well as the positive demand for bonds off the steady state. The movements in these two components compensate for the negative impact on money demand coming an increase in consumption, which is due entirely to the increase in household welfare. We also report the reaction of the term premium, which is computed as the deviation of the long-term yield in the model from the level consistent with the expectations hypothesis. 1 The lower-right panel of Figure 3 shows that the term premium fall on impact, as the long-term yield falls more than that consistent with the expectations hypothesis. Figure 4 reports the impulse responses following a monetary policy shock. An exogenous increase in the monetary policy rate reduces output and inflation, and the fall in inflation increases the real value of outstanding debt. However, the reduction in output and consumption requires the yields to rise, and the price of bonds to decline for the financial markets to clear. The fall in the demand of bonds and the initial increase in the opportunity cost of money drives money demand below its long-run value. The response of the yields is decreasing along the term structure, so the term spread falls. The term premium increases on impact. Finally, an exogenous shock to government spending in Figure 5 generates an inflationary waste of resources that crowds out consumption. Since optimal consumption plans imply a reduction in consumer spending throughout the transition, the response of the term structure is consistent with a reduction in the demand for bonds. We note that the upsurge in inflation reduces the real 9 We simulate 5 samples of 1, observations, discarding the first simulated observations. 1 Given a yield of maturity N, denoted Rt N, the term premium ξn t is defined by the relation R N t = 1 N N 1 j= E tr t+j + ξ N t, where R t is the short-term interest rate. Thus, the term premium is the deviation of the long-term yield Rt N the level consistent with the expectations hypothesis, 1/N) N 1 j= E tr t+j. from 13

15 outstanding value of bonds. Differently from the cases of productivity and monetary policy shocks, the term spread is now negatively correlated with output. The term premium rises on impact. We note that the responses of the term premium are entirely consistent with those reported by Rudebusch, Sack, and Swanson 7), who derive a time-varying term premium using a thirdorder approximation of a small business cycle model. 5. Sensitivity analysis In order to gain some intuition for the mechanisms in the model, we now study the model behavior when we vary the parameters of the Taylor rule 34) and the parameters in the adjustment cost function between money and bonds in equation 9). 11 The baseline calibration sets the monetary policy response to output α y ) to zero. Table 6 reports the standard deviations of the term structure and the contemporaneous correlations with output as this coefficient is increased to.5 and.9. As monetary policy becomes more responsive to output, the policy rate becomes more volatile and less procyclical, while the medium- and long-term rates become more volatile. Also, the medium-term rate is now countercyclical. The volatility of the term spread falls, and the correlation with output increases. These effects can be explained by the fact that the Euler equations can be re-written as a function of the spreads, and the policy rate is determined by the Taylor rule. Therefore, the term spread is less countercyclical when monetary policy is more responsive to output. Figure 6 shows the impulse responses to a productivity shock for three values of the policy response to output. 1 The more accomodative is the policy rule with respect to output, the larger is the response of inflation to a productivity shock see Schmitt-Grohé and Uribe, 6), and the higher is the resulting increase in the real value of debt, leading to larger movements in the term structure. Together with the larger decrease in the money market rate, this implies that money demand increases more when α y is large. Table 7 and Figure 7 study the case when the policy rule coefficient for inflation α π ) varies between 1.1 and 1.9 the baseline value is 1.5). A larger α π makes all the yields less volatile. Figure 7 shows that all yields respond less to a productivity shock, but output responds more, when α π is large. Table 8 reports the simulated moments for various values of the parameters in the adjustment cost functions between money and bonds. Panel b) shows that when all these adjustment costs are set to zero, the volatility of the yields falls to almost zero, all yields become countercyclical and the term spread procyclical. The remaining panels show that the standard deviation of each yield is increasing in its own adjustment cost, but to some extent also in the adjustment costs for the other maturities. Figure 8 reports the impulse responses to a productivity shock for different values of the adjustment cost between money and short-term bond holdings, v S. As v S increases, the response of the short-term rate becomes larger, whereas the reaction of money demand gets smaller. The intuition is straightforward. Although the elasticity of substitution does not change as a function of the 11 We do not perform any sensitivity analysis concerning the parameters φ ι in the bond adjustment cost function, as these parameters are obtained from the steady-state relations of the model. 1 The responses to monetary policy and government spending shocks do not change much with respect to the baseline calibration, and are therefore not reported. 14

16 adjustment cost, the weight on the relation of complementarity between money and bonds in the household s budget constraint increases. Since there is a desired ratio between money and bonds, the lower the adjustment cost, the higher the propensity to reallocate income between money and short-term bonds. Hence, in order to minimize the changes of holdings of short-term bonds, most of the adjustment takes place through their prices. 5.3 Output growth, the yield spread and the term premium Finally, we study the implications of our model for the predictive power of yield spread and the term premium for output growth. After Estrella and Hardouvelis 1991), a large body of research has documented the predictive power of the yield spread for future output growth. 13 Traditional explanations of the predictive power of the yield spread emphasize the fact that asset prices incorporate market views on the future stance of monetary policy. tightens, the yield curve flattens and future output subsequently falls. As monetary policy We here study the predictive power of the term spread in our model by estimating the predictive regression 4/h Y t+h Y t ) = β + β 1 [R L,t R S,t ] + ε t 39) on simulated data from the model. 14 Table 9 reports the average of the estimated β 1 coefficients over the simulated samples for horizons of h = 1,..., 8 quarters. For horizons of two quarters and longer, our model captures the positive correlation between the yield spread and future output growth. Although not reported here for brevity, various experiments show that this result is robust to changes in the parameter configuration. Rudebusch, Sack, and Swanson 7) study the empirical properties of the term premium in the U.S. term structure and its relation with output growth. They show that the level of term premium has no statistically significant relationship with future output growth, whereas the first difference of the term premium is countercyclical, that is, a declining term premium is significantly correlated with high output growth. In order to verify whether our model fits these stylized facts, we decompose the yield spread into an expectations-related component ER t and a term premium T P t. For any maturity N, this decomposition is given by R N t R t = 1 N N 1 j= E t R t+j R t + R N t 1 N N 1 j= E t R t+j = ER t + T P t, 4) where R t is the short-term interest rate. The expectations-related component is the part of the yield spread that is consistent with the expectations hypothesis of the term structure, while the term premium is the deviation of the spread from the expectations hypothesis. Applying this decomposition to the long-short spread in equation 39), we obtain the predictive regression Y t+4 Y t = β + β 1 ER t + β T P t + ε t. 41) 13 See Watson and Stock 3) for an extensive overview of the field. 14 We here simulate 1, samples of 1, observations, discarding the first observations. 15

17 Equation 41) can then be differenced to obtain a specification where the changes in the two components predict output growth one year ahead: Y t+4 Y t = β + β 1 ER t + β ER t 4 + β 3 T P t + β 4 T P t 4 + ε t, 4) or Y t+4 Y t = β + β 1 [ER t ER t 4 ] + β [T P t T P t 4 ] + ε t. 43) Table 1 reports the estimates of these three predictive regressions on simulated model data. 15 The first column shows that the relation between future output growth is related to the term premium in a different way than the expectations-related component: the while a large expectations-related component is positively correlated with future output growth, a large term premium is instead negatively correlated. The third column of Table 1 instead shows that our model predicts that declining term premia are followed by faster output growth, although this result is not statistically significant. These properties of the term premium are again consistent with those reported by Rudebusch, Sack, and Swanson 7). 6 Concluding remarks We set out to study the feedback effect from the term structure of interest rates to the macroeconomy. We used an estimated Vector Auto-Regressive model to show that government bond yields in the U.S. have a direct impact on the macroeconomic variables such as output and inflation: thus the macroeconomy is not exogenous with respect to the term structure. Encouraged by on this evidence, we proposed a modification of the standard New Keynesian model that generates endogenous term premia and a feedback from the term structure to the economy. In order to allow for differences in the returns on government bonds, we introduced transaction costs for government bonds of two types: costly reallocation between bonds of different maturities themselves; and costly reallocation between each type of bonds and money. These transaction costs generate a wedge between the returns on bonds with different maturities, which, in turn, creates a feedback from the term structure to private agents spending decisions. Our empirical results suggest that the feedback was more significant in the period before 1987 than in the post-1987 period. This weakening of the feedback effect seems to coincide with a weaker predictive power of the yield spread for output growth see Dotsey, 1998) and a decline in the level and volatility of the term premium see, for instance, Rudebusch, Sack, and Swanson, 7). Future work could study more carefully the role of the term premium in generating these results. We take our model as a starting point for a more thorough discussion on the role of the term structure in the New Keynesian framework. An attractive feature of our setup is that it involves no higher-order approximation methods. Thus, the premia do not arise from time-varying risk. A useful extension would therefore be to apply a second-order solution method that would induce time-variation in risk. 15 Adding lagged output growth as a regressor does not change the results reported here. 16

18 A The steady state The computation of the deterministic steady state takes as given the values for θ, δ, α, φ P, γ, I/Y, C/Y, b/b, b S /B, b M /B, b L /B, Y, L, χ, ψ, σ, π, R, R S, R M, and R L. We normalize all variables with respect to aggregate income Y, whose steady-state value is normalized to Y = 1. From the first-order condition for consumption in equation 14), we recover the steady-state value of λ as λ = 1 γβ) [1 γ) C] 1 σ. A1) With the steady-state inflation rate π set to 3.7% per year, and the federal funds rate R given by 4.8% per year, we recover the value of the discount rate from equation 17). From the optimality condition for labor in 15), we calibrate the parameters ψ and Ψ in order to match the steady-state ratio of market to non-market activities for the U.S., which is given by.3. The steady-state level of wage is directly obtained from the firm s first-order condition for labor in equation 6). The level of the capital stock is recovered from the accumulation equation 11), consistently with a share of investment to GDP of.3 on a quarterly basis. Combining the firm s optimality conditions for labor and capital with the product exhaustion theorem, we obtain Y = QK + wl = 1 1 ) Y + Φ) θ = 1 1 ) AK α L 1 α. θ A) From this equation, we can see the presence of a positive wedge due to monopolistic competition. To ensure zero profits, we set the fixed cost Φ to Φ = AKα L 1 α. θ A3) Thus, given L, K and the steady state level of Y, we can recover the steady-state level of the technology shock A from equation A). We also note that at the steady state the dynamic markup is equal to the elasticity of substitution among differentiated goods, θ. We set the values of κ S, κ M, κ L equal to the steady-state ratios of, respectively, short-term, medium-term, and long-term debt to money. Combining the Euler Equation 17) with the first-order condition for money 16) at the steady state, we get m χ = λ 1 1 ). A4) R From equation A4), we calibrate the parameter χ in order to match the steady-state value of the federal funds rate R, and the money to output ratio m/y, given λ. We calibrate the bond adjustment costs to preserve the differences between the yields at the deterministic steady state. Indexing the first-order conditions with respect to short-term bonds 17

19 B S, B M and B L with ι = S, M, L, we have βλ R [ ι π + βλφ ιy = λ ] φ ιy. A5) From A5) we observe that bond adjustment costs are non-zero at the steady state, and they are a function of aggregate output. To understand the role of transaction costs, we substitute equation A5) into the steady-state version of 17). After rearranging, we get ) R ι 3 R = 1 + φ ιy β, A6) and the spread between yield R ι and R can be rewritten as R ι R R = φ ι Y ) 3 β. A7) Equation A7) identifies the spread existing between the yield on bond ι and the federal funds rate as a function of each bond adjustment cost. From A6), we can calibrate φ ι according to φ ι = βr ι/π 1 Y 3/ β). A8) To calibrate the cost of capital installment φ k, we lead equation 19) one period and combine with ): βe t λ t+1 λ t { 1 δ) [ φ k which can be rearranged to give E t It+1 K t+1 ) ] 1 + 3/)φ k I t /K t ) ) } 3 It+1 + Q t+1 + φ k = ) K t+1 φ It k, A9) K t [ 1 δ) 1 + 3/)φ k I t+1 /K t+1 ) ] + Q t+1 + φ k I t+1 /K t+1 ) 3 = E t { Rt π t+1 }. A1) From A1) we see that if φ k =, the rental rate Q t net of depreciation) is equal to the real interest rate R t /π t+1. Thus, the investment adjustment cost creates a wedge between the real interest rate and the rental rate of capital. We calibrate φ k to match the values of R, K, π, I, λ, Q, by setting φ k = 1/β) λq + δ 1 I/K) [3/) 1 δ 1/β) + λi/k)]. A11) 18

20 B The log-linearized model Let z t = log z t log z be the log deviation of any variable z t from its steady-state value z. The log-linearized system of equations is then given by: η cc c t + η cc1 c t 1 + βη cc1 E t c t+1 λ λ t = 1 ψ l t + w t + λ t = w w t = 1 α) 1 1 ) [ Y θ q q t = α 1 1 ) Y θ K ỹt α Y ỹ t = η y ã t + αη y kt + 1 α) η y lt h h t = Rb R t + Rb b t + m m t ] Y + Φ) 1 α) Y + Φ L ỹt lt + L θ L 1 1 ) Y + Φ) θ K k t + α ) Y + Φ ẽ y t θ K ) ẽ y t B1) B) B3) B4) B5) B6) C c t + η cy ỹ t + η cbs bs,t + η cbm bm,t + η cbl bl,t η ck kt + T τ t + b b t + m m t + η ci ĩ t + η cπ π t = η cr Rt 1 + b t 1 ) + η crs RS,t 1 + η crm RM,t 1 + η crs + η cxs ) b S,t 1 + η crm + η cxm ) b M,t 1 + η crl RL,t 1 + η cxl + η cxl ) b L,t 1 + m π m t 1 R R t = 1 φ R ) φ π π π t + 1 φ R ) φ y yỹ t + φ R R R t 1 + ε t ) kt = 1 δ) k i t 1 + ĩ t 1 k T τ t + η τπ π t = ψ 1 h h t 1 + η τs bs,t 1 + R S,t 1 ) + η τm bm,t 1 + R M,t 1 ) +η τl bl,t 1 + R L,t 1 ) B7) B8) B9) b b t + b s bs,t + b m bm,t + b L bl,t + T τ t G g t + m m t + η bπ π t = η b bt 1 + R ) t 1 + η s bs,t 1 + R ) S,t 1 + η m bm,t 1 + R ) M,t 1 + η L bl,t 1 + R ) L,t 1 + m π m t 1 B1) βλ ) π E t λt+1 π t+1 = λ λ t + η m1 m t η ms bs,t η mm bm,t η ml bl,t B11) β 1 δ) µe t µ t+1 = η k kt ĩ t ) + µ µ t λq q t η λ λt B1) βµe t µ t+1 = η µλ λt + η µk ĩt k t ) B13) η π1 E t π t+1 = η π π t θe y t B14) E t [ η sλ1 λt+1 + η sy1 ỹ t+1 ) + η sb1 bs,t+1 η sπ π t+1 ] = η sλ λt + η sy ỹ t + η sb bs,t η bsm m t η sy bs,t 1 η sπ RS,t B15) 19

21 E t [ η mλ1 λt+1 + η my1 ỹ t+1 ) + η mb1 bm,t+1 η mπ π t+1 ] = η mλ λt + η my ỹ t + η mb bm,t η bmm m t η my bm,t 1 η mπ RM,t B16) E t [ η Lλ1 λt+1 + η Ly1 ỹ t+1 ) + η Lb1 bl,t+1 η Lπ π t+1 ] = η Lλ λt + η Ly ỹ t + η Lb bl,t η Lm m t η Ly bl,t 1 η Lπ RL,t B17) E t λt+1 E t π t+1 = λ t R t ã t = ρ A ã t 1 + ε A t g t = ρ G g t 1 + ε G t bs,t = ρ bs bs,t 1 + ε bs t bm,t = ρ bm bm,t 1 + ε bm t bl,t = ρ bl bl,t 1 + ε b L t B18) B19) B) B1) B) B3) The η-coefficients are convolutions of the structural parameters, and are available from the authors upon request.

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23 Ireland, Peter, N. 4), Money s Role in the Monetary Business Cycle, Journal of Money, Credit, and Banking 36 6), Jovanovic, Boyan, and Peter L. Rousseau. 1), Liquidity Effects in the Bond Market, Economic Perspectives Q4/1, Federal Reserve Bank of Chicago, Kim, Jinill ), Constructing and Estimating a Realistic Optimizing Model of Monetary Policy, Journal of Monetary Economics 45, Leeper, Eric, M. 1991), Equilibria under Active and Passive Monetary and Fiscal Policies, Journal of Monetary Economics 7, McCulloch, J. Huston and Heon-Chul Kwon 1993), U.S. Term Structure Data, , Working Paper No. 93-6, Ohio State University. Missale, Alessandro 1999), Public Debt Management, Oxford University Press Ravenna, Federico, and Juha Seppälä 7a), Monetary Policy and Rejections of the Expectations Hypothesis, Manuscript, University of California, Santa Cruz. Ravenna, Federico, and Juha Seppälä 7b), Monetary Policy, Expected Inflation, and Inflation Risk Premia, Manuscript, University of California, Santa Cruz. Rotemberg, Julio J. 198), Monopolistic Price Adjustment and Aggregate Output, Review of Economic Studies 49 4), Rudebusch, Glenn D., Brian P. Sack, and Eric T. Swanson, Macroeconomic Implications of Changes in the Term Premium, Federal Reserve Bank of St. Louis Review 89 4), Rudebusch, Glenn D. and Tao Wu 3), A Macro-Finance Model of the Term Structure, Monetary Policy and the Economy, Federal Reserve Bank of San Francisco Working Paper No Forthcoming, Economic Journal. Rudebusch, Glenn D. and Tao Wu 7), Accounting for a Shift in Term Structure Behavior with No-Arbitrage and Macro-Finance Models, Journal of Money, Credit, and Banking 39 3), Schmitt-Grohé, Stephanie and Martin Uribe 6), Optimal Simple and Implementable Monetary and Fiscal Rules, Journal of Monetary Economics, 546), Sims, Christopher A. 198), Macroeconomics and Reality, Econometrica, 48 1), 1 48 Smets, Frank and Rafael Wouters 6), Shocks and Frictions in U.S. Business Cycles: Bayesian DSGE Approach, American Economic Review 97 3), Taylor, John B. 198), Aggregate Demand Dynamics and Staggered Contracts, Journal of Political Economy 88 1), 1 3. Taylor, John B. 1993), Discretion Versus Policy Rules in Practice, Carnegie-Rochester Conference Series on Public Policy 39, Tobin, James 1969), A General Equilibrium Approach to Monetary Theory, Journal of Money, Credit, and Banking 1, Tobin, James 198), Money and Finance in the Macroeconomic Process, Journal of Money, Credit, and Banking 14, Watson, Mark 1994), Vector Autoregressions and Cointegration, in Handbook of Econometrics, R. F. Engle and D. L. McFadden, Elsevier, Watson, Mark and James H. Stock 3), Forecasting Output and Inflation: The Role of Asset Prices, Journal of Economic Literature, 413), Woodford, Michael 3), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. A

24 Table 1: U.S. term structure statistics, 1987:4 6:4 Federal funds rate 3-month 1-year 5-year 1-year Spread Mean Standard deviation Correlation with output gap This table shows standard deviations and contemporaneous correlations with the output gap for U.S. yields. The yields are constant-maturity interest rates, in percent per year, and the spread is between the 1-year and 1- year yields. The output gap is defined as the percent deviation of real GDP from potential as calculated by the Congressional Budget Office. 3

25 Table : Results from block-exogeneity tests Restrictions Full sample Pre-Greenspan Greenspan-Bernanke a) Unrestricted model: [y; p; pcom; f f r; r3m; r1y; r1y] ) [r3m; r1y; r1y] ) [r1y; r1y] ) r1y e 4) [r3m; r5y; r1y] ) 1.78.) ) b) Unrestricted model: [y; p; pcom; f f r; r3m; r5y; r1y] ) [r5y; r1y] ) r1y e 8) [r3m; r1y; 5y] ) ) ) c) Unrestricted model: [y; p; pcom; f f r; r3m; r1y; r5y] 74.7.) [r1y; r5y] ) r5y ) 4.46 e 8) e 4) d) Unrestricted model: [y; p; pcom; f f r; r1y; r5y; r1y] ) [r1y; r5y; r1y] ) [r5y; r1y] e 7) r1y e 7) e 5) ) ) ) ) ) ) ) ) ) e 5) e 5) ) ) This table reports test statistics and p-values in parentheses) from tests of block exogeneity from VARs with alternative indicators for yields on short, medium and long-term bonds, estimated with 1 lags on monthly U.S. data from 1959:1 to 6:1. The VARs include 1 lags. The variables are as follows: y industrial production index, in logs), p consumer price index, in logs), pcom producer price index: finished goods less food and energy, in logs), ffr Federal funds rate), r3m yield on 3-month bonds), r1y yield on 1-year bonds), r5y yield on 5-year bonds), r1y yield on 1-year bonds). 4

26 Table 3: Stylized facts in U.S. data, 1987:4 6:4 Description Value Investment/GDP ratio.3 Consumption/GDP ratio.57 Real money/output ratio.17 1/4 Ratio of market to non-market activities.3 Debt/GDP ratio.45 Fraction of very short-term debt in total debt.48 Fraction of short-term debt in total debt.1 Fraction of medium-term debt in total debt.556 Fraction of long-term debt in total debt.19 5

27 Table 4: Parameter calibration Description Notation Value Preferences and technology Discount factor β.9975 Elasticity of intertemporal substitution σ.5 Habit formation γ.7 Elasticity of money demand χ 7 Labor supply elasticity ψ 1 Elasticity of substitution across goods θ 6 Capital depreciation rate δ.5 Share of capital in production α.36 Capital adjustment cost φ K 115 Price adjustment cost φ P 1 Bond adjustment costs Short-term bond adjustment cost φ S.6 Medium-term bond adjustment cost φ M.5 Long-term bond adjustment cost φ L.7 Money/short-term bond transaction cost v S 5. Money/medium-term bond transaction cost v M 4. Money/long-term bond transaction cost v L.5 Fiscal and monetary policy Fiscal policy constant ψ.158 Fiscal policy response to nominal liabilities ψ 1.3 Fiscal policy response to short, medium and long-term debt ψ.3 Monetary policy response to inflation α π 1.5 Monetary policy response to output α Y Monetary policy inertia α R.7 Autoregressive parameters Technology ρ a.95 Government spending ρ g.9 Short-term bond supply ρ S.7 Medium-term bond supply ρ M.73 Long-term bond supply ρ L.75 Standard deviations Technology shock σ a.1 Government spending shock σ g.1 Monetary policy shock σ R.5 Short-term bond supply shock σ S.9 Medium-term bond supply shock σ M.3 Long-term bond supply shock σ L.95 6

28 Table 5: Term structure moments in the baseline calibration Policy rate Short-term Medium-term Long-term Spread Standard deviations Contemporaneous correlations with output gap This table shows the simulated standard deviations and contemporaneous correlations with the output gap for yields of different maturity in the baseline calibration of the model. 7

29 Table 6: Model moments when varying the coefficient for output in the monetary policy rule Policy rate Short-term Medium-term Long-term Spread Standard deviations Baseline: α y = α y = α y = Contemporaneous correlations with output gap Baseline: α y = α y = α y = This table shows the simulated standard deviations and contemporaneous correlations with the output gap for yields of different maturity as the coefficient α y for the output gap in the monetary policy rule varies from its baseline value of α y =. The remaining parameters are fixed at the baseline values. 8

30 Table 7: Model moments when varying the coefficient for inflation in the monetary policy rule Policy rate Short-term Medium-term Long-term Spread Standard deviations Baseline: α π = α π = α π = Contemporaneous correlations with output gap Baseline: α π = α π = α π = This table shows the simulated standard deviations and contemporaneous correlations with the output gap for yields of different maturity as the coefficient α π for inflation in the monetary policy rule varies from its baseline value of α π = 1.5. The remaining parameters are fixed at the baseline values. 9

31 Table 8: Model moments when varying the adjustment cost between money and bonds of different maturity Policy rate Short-term Medium-term Long-term Spread a) Baseline: v S = 5., v M = 4., v L =.5 Standard deviations Contemporaneous correlations with output gap b) No adjustment costs: v S = v M = v L = Standard deviations Contemporaneous correlations with output gap c) Adjustment cost for the short-term bond Standard deviations v S = v S = Contemporaneous correlations with output gap v S = v S = d) Adjustment cost for the medium-term bond Standard deviations v M = v M = Contemporaneous correlations with output gap v M = v M = e) Adjustment cost for the long-term bond Standard deviations v L = v L = Contemporaneous correlations with output gap v L = v L = This table shows the simulated standard deviations and contemporaneous correlations with the output gap for yields of different maturity as the parameters determining the adjustment cost between money and bonds vary from their baseline values. The remaining parameters are fixed at the baseline values. 3

32 Table 9: Estimated slope coefficients from predictive regressions for output growth using the yield spread Horizon quarters) This table reports the slope coefficient from the predictive regression 39) estimated on simulated data from the model. The dependent variable is output growth at different horizons, the independent variable is the yield spread. The p-values of all estimated coefficients are zero. 31

33 Table 1: Predictive regressions with the yield spread decomposition Predictive regression 41) 4) 43) ER t.6.6.).) ER t ) T P t.9.8.).4) T P t 4..33) ER t ER t 4.1.) T P t T P t 4.4.7) This table reports the slope coefficients from the predictive regressions 41) 43) estimated on simulated data from the model. The dependent variable is output growth at different horizons, the independent variable is the expectations-related component ER) and the term premium component T P ) of the yield spread. All regressions include constant terms that are not reported. Numbers in parentheses are p-values. 3

34 Figure 1: Term structure yields and output gap in U.S. data, 1987:4 6:4 1 8 a) Federal funds rate Rate Output gap 1 8 b) 3 month rate 1 8 c) 1 year rate d) 5 year rate 1 d) 1 year rate f) 1 year 3 month spread

35 Figure : Estimated impulse responses to a monetary-policy shock Response to One S.D. Monetary Policy Schock. Response of Output to FFR.4 Response of Commodity Prices to FFR.6 Response of FFR to FFR Response of One Year-TB Yield to FFR Response of 5-Years TB Yield to FFR Response of 1 - Years TB Yield to FFR This figures shows impulse responses to a monetary policy shock from an estimated VAR1) on monthly U.S. data from 1959:1 to 6:1, using a Choleski decomposition. The variables are ordered as follows: industrial production index in logs); producer price index finished goods less food and energy, in logs); federal funds rate; 1-year yield; 5-year yield; 1-year yield. 34

36 Figure 3: Model impulse responses to a productivity shock.1 Output.1 Consumption Labor effort x Capital x Money market rate Long term rate.1. x Inflation Short term rate.1. Term spread x Money demand Medium term rate. x Term premia

37 Figure 4: Model impulse responses to a monetary policy shock Output x 1 3 Consumption.1 Labor x Capital x Money market rate 1 3 Long term rate.4. 5 x Inflation Short term rate.4.. Term spread.4.1 x Money demand Medium term rate.4. Term premia

38 Figure 5: Model impulse responses to a government spending shock x 1 3 Output x Capital x Money market rate x Long term rate x 1 4 Consumption 4 x Inflation x Short term rate x Term spread x 1 3 Labor 1 x Money demand x Medium term rate x Term premia

39 Figure 6: Impulse responses to a productivity shock when varying the coefficient for the output gap in the monetary policy rule.1 Output.1 Consumption Labor x Capital Money market rate.5.1 Long term rate.1..5 Inflation.1 Short term rate.5.1 Term spread.5. 1 x Money demand Medium term rate.4 Term premia.5.1 α y = α y =.5 α y =.9 38

40 Figure 7: Impulse responses to a productivity shock when varying the coefficient for inflation in the monetary policy rule.1 Output.1 Consumption Labor x Capital x Money market rate x Long term rate x Inflation Short term rate..4 Term spread x Money demand Medium term rate.1 x Term premia α π =1.5 α π =1.1 α π =1.9 39

41 Figure 8: Impulse responses to a productivity shock when varying the adjustment cost between money and short-term bond holdings.1 Output.1 Consumption Labor x Capital x Money market rate x Long term rate x Inflation Short term rate.1. Term spread..1.1 x Money demand Medium term rate.1 x Term premia v S = v S =6 v S =8 4