Term structure, inflationand real activity 1

Andrea BerardiUniversità di Verona

June 16, 2004

1Correspondence to: Andrea Berardi, Università di Verona, Department of Eco-nomics, via Giardino Giusti 2, 37129 Verona, Italy. E-mail: andrea.berardi@univr.it.I am particularly indebted to Stephen Schaefer for innumerable insightful discussionsand suggestions. I also thank Krishna Ramaswamy, Eduardo Schwartz, Ken Single-ton, Walter Torous, two anonymous referees and seminar participants at the AmericanFinance Association, Western Finance Association and CEPR European Summer Sym-posium in Financial Markets for useful comments and suggestions. All errors are myresponsibility.

Abstract

This paper presents and estimates a structural model which investigates therelationship between the term structure of interest rates, inflation and GDPgrowth. The model allows us to: (i) study the cross-sectional restrictions whichlink the dynamics of the term structure to output growth and inflation and ob-tain reliable estimates of unobservable variables, such as the real interest rate,the expected inflation rate, their stochastic central tendencies and time-varyingbond risk premia; (ii) fit the actual term structure of nominal interest rates anduse nominal bond prices to derive estimates for the implicit term structures ofreal interest rates and expected inflation rates; (iii) obtain endogenous expecta-tions for the macroeconomic variables and produce accurate forecasts of futureinflation and GDP growth rates.

The relation between the term structure of interest rates, inflation and realactivity has long been recognized as central to both macroeconomic and financetheory and as critical in formulating economic policy and investment decisions.Expectations of inflation and economic growth also play a key role in pricingfinancial instruments as well as in determining the actions of governments andcentral banks in attempting to smooth business cycle fluctuations. This relationis reciprocal, in the sense that changes in asset prices and economic policieschange current levels of inflation and real growth and, therefore, expectationsabout their future path.This paper investigates the relationship between interest rates, inflation and

economic growth in the context of the continuous time theory of the term struc-ture. In particular, the paper develops and estimates a structural model whichallows us to: (i) study the cross-sectional restrictions which link the dynamicsof the term structure to output growth and inflation; (ii) fit the actual termstructure of nominal interest rates and use nominal bond prices to derive esti-mates for the implicit unobservable term structures of real interest rates andexpected inflation rates; (iii) obtain endogenous expectations for future inflationand GDP growth rates and test their forecasting ability.Extensive theoretical and empirical work has been devoted to the study of

the relation between real activity, inflation and term structure of interest rates.So far, however, there is neither a commonly accepted theoretical frameworknor unanimity on empirical regularities.According to C. Harvey (1988 and 1989) and subsequent analyses by, among

others, Estrella and Hardouvelis (1991), Plosser and Rouwenhorst (1994), Chap-man (1997), Kamara (1997) and Roma and Torous (1997), the shape of the termstructure reflects information about future economic growth. In particular, apositive slope of the yield curve predicts an increase in the level of real activity,whereas a flattening or a negative slope of the yield curve is associated with afuture recession. Conversely, Evans and Marshall (2001) and Ang and Piazzesi(2003) show that macroeconomic factors have a significant impact on the termstructure of interest rates.Empirical studies on the information content of the term structure for future

inflation have been undertaken, among the others, by Fama (1990) and Mishkin(1990). They suggest that only the long end of the term structure of interestrates contains information about future inflation, whereas the short end of theterm structure provides almost no information about the future path of inflation.All the studies cited above are related to the theory of interest rates but do

not address the problem of specifying a complete model of the term structurewhich fully analyzes the links between inflation, economic growth and bondprices and might be used for forecasting purposes.The inclusion of these macroeconomic relations in a general equilibrium term

structure framework has its origins in the continuous time model of Cox, In-gersoll and Ross (CIR, 1985a and 1985b), which describes a complete economywith production and a stochastic investment opportunity set and endogenouslydetermines optimal consumption, portfolio choice and equilibrium asset prices.However, most term structure models based on a CIR-type approach do

not explicitly consider the role of macro variables. In this respect, significantexceptions are provided by Breeden (1986), Foresi (1989), Pennacchi (1991), Sun(1992), Pearson and Sun (1994), den Haan (1995), Bakshi and Chen (1996) and,more recently, by Wu (2001), Ang and Bekaert (2003), Ang and Piazzesi (2003),

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Ang, Piazzesi and Wei (2003), Buraschi and Jiltsov (2003) and Dewachter andLyrio (2003).Considering data on macro variables such as inflation and output along with

the data on bonds places more emphasis on the macroeconomic aspects under-lying the dynamics of the yield curve and on the feedback effects which run frominterest rates to the fundamentals. Such a ”macro” approach would be moresuitable for both medium-term investment and economic policy purposes, as italso permits information contained in the term structure of interest rates to beused in predicting inflation and economic growth.In this paper, we propose a structural model of the yield curve based on

an economy whose dynamics are determined by the real interest rate, the ex-pected inflation rate and their time-varying central tendencies. The equilibriumaspects of the model impose nontrivial restrictions on the joint dynamics ofbond yields and macro variables. Real and monetary variables in this economyare interrelated and influence the term structure of interest rates, while bondyields convey information about expectations of economic fundamentals, suchas production, consumption and inflation. Closed form solutions for the equilib-rium value of nominal and real bonds and for expectations of inflation rates andoutput growth rates over any future time interval are derived. These enable usto analyze the cross-equation restrictions which link the real and nominal termstructure of interest rates to the dynamics of production and the price level anduse the model to forecast inflation and GDP growth. The model is estimatedusing U.S. quarterly data for the sample period 1960-2002 by applying a maxi-mum likelihood approach where the Kalman filter algorithm is used to computethe unobservable variables.With respect to other studies in the field, the main contribution of the paper

consists in: (i) building an internally consistent structural model which providesa nontrivial extension of previous term structure models investigating the re-lation between real rates and inflation, such as Pennacchi (1991), Sun (1992)and Pearson and Sun (1994), and represents a nontrivial alternative to recentapproaches advanced in modelling the relation between bond yields and macrovariables, such as Wu (2001), Ang and Piazzesi (2003) and Ang, Piazzesi andWei (2003); (ii) deriving endogenous estimates of time-varying bond risk premiausing a framework which differs substantially from those recently proposed byCochrane and Piazzesi (2002) and Buraschi and Jiltsov (2003); (iii) obtainingthe term structures of real interest rates and expected inflation rates from ob-servations on nominal bond prices using a more structured approach than theAng and Bekaert (2003) regime-switching model; (iv) exploiting the equilibriumcross-sectional restrictions imposed by the model to obtain accurate forecasts offuture inflation and GDP growth rates by making use of term structure data.The plan of the paper is the following. Section I contains a description of

the data used in the empirical work and a preliminary analysis based on theprincipal components technique. Section II illustrates the structural term struc-ture model, whereas section III describes the econometric specification and theestimation method. Section IV gives the empirical results on model estimationand section V analyzes the out-of-sample forecasts of the model. Section VIconcludes.

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I Some preliminary evidence

This section provides a preliminary view on the relation between the term struc-ture of interest rates, inflation and GDP growth based on the principal compo-nents analysis technique.The empirical investigation is based on quarterly data for the United States

from the first quarter of 1960 to the fourth quarter of 2002 (172 observations).Bond yields are end-of-quarter annualized zero coupon yields with maturities 3months, 1 to 10 and 20 years. For the period between the first quarter of 1960and the first quarter of 1991, the data are taken from McCulloch and Kwon(1993). From the second quarter of 1991 to the fourth quarter of 2002, zerocoupon yields calculated from Treasury coupon strip prices are used1.As a measure of economic activity, we use seasonally adjusted data on real

Gross Domestic Product (GDP), expressed in constant dollars, from the U.S.Department of Commerce’s Bureau of Economic Analysis. The seasonally ad-justed GDP deflator is used as a proxy for the price level.A principal components analysis provides insight on the relation between the

term structure and macro variables. Moreover, it provides preliminary evidenceon the number of factors which should be taken into account in order to explaintheir joint dynamics.We apply the principal components analysis to the entire set of variables,

i.e., changes in the annualized quarterly GDP growth rate and inflation rate aswell as changes in the bond yields.The first principal component, which accounts for 78.3% of the total variance

of the dependent variables, is highly correlated (0.95) with changes in a ”level”factor calculated as the average of a short (3-month), a medium (5-year) anda long (20-year) term yield. The second and the third principal componentsexplain, respectively, 7.5% and 6.9% of the total variability and are stronglycorrelated with both changes in the inflation rate (0.80 and 0.44, respectively)and changes in the GDP growth rate (−0.65 and 0.68, respectively). The fourthprincipal component, which explains 5.1% of the total variance, is a ”slope”factor and is correlated (0.72) with changes in the spread between the 20-yearand the 3-month maturity yields. The fifth principal component, which accountsfor 1.3% of the variability, is correlated (0.62) to a ”curvature” factor (sum ofthe 3-month and the 20-year yields minus twice the 5-year yield). The rest of theprincipal components explain in aggregate only about 1% of the total variance.Table 1 shows the sensitivity of the three term structure factors (level, slope

and curvature) and the two macro factors (GDP growth and inflation) to each ofthe five principal components. We can notice that, on the one hand, the secondand the third principal components, which are highly correlated with inflationand GDP growth, have both a positive, statistically significant, impact on thelevel and curvature term structure factors and a negative effect on the slope. Onthe other hand, the first, the fourth and the fifth principal components, whichare related to the term structure factors, significantly affect GDP growth andinflation. In particular, we notice that the level factor negatively affects GDPgrowth, whereas, consistently with C. Harvey (1989) and Mishkin (1990), theslope factor is positively related to GDP growth and inflation rates.In general, this analysis shows that there are strict interrelations between

1We are grateful to Roger Brown and Stephen Schaefer for providing these data.

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macro and term structure variables. In fact, we observe that the factors ex-plaining the variability of GDP growth and inflation have significant impact onnominal yields and, at the same time, the factors driving the dynamics of theyield curve affect both output and inflation. This evidence seems to support astructural model relating all these variables.

II The model

In this section, we develop a structural term structure model in which the dy-namics of bond yields and macro factors are strictly interrelated. The underlyingbasic assumptions are consistent with CIR (1985a): (i) continuous time compet-itive economy composed by a fixed number of identical individuals with rationalexpectations and time-additive logarithmic utility functions; (ii) a single physi-cal good is produced, which may be allocated to consumption or investment; (iii)a single endogenous production process exists, which is affected by stochastictechnological changes; (iv) there are markets for instantaneous borrowing andlending at the same riskless interest rate and markets for default-free nominaland real bonds.We do not explicitly model money and monetary policy and, as in most

term structure models including inflation (such as, for example, CIR (1985b),Pennacchi (1991), Sun (1992), Pearson and Sun (1994), Buraschi and Jiltsov(2003), Ang and Bekaert (2003)), we assume an exogenously given process forthe price level and the expected inflation rate. In other words, we assume thatthere exists an underlying equilibrium in the money market which supports theobserved price level2. Monetary policy will be indirectly taken into account inour framework by modelling time-varying central tendencies for the real interestrate and the expected inflation rate and interpreting these variables as proxiesfor the Fed medium-long term interest rate targets, as in Balduzzi, Das andForesi (1995), Jegadeesh and Pennacchi (1996) and Piazzesi (2003), for example.The solution of the individuals’ intertemporal problem determines the total

consumption and production plan, the price level, the optimal allocation ofwealth among activities and the equilibrium value of the riskless interest rateand nominal and real bond prices. As shown by CIR (1985a), in equilibriumall wealth should be invested in the physical production process. Therefore,equilibrium is characterized by a zero supply of bonds and no borrowing andlending.

A Production, inflation and state variables

In what follows, we outline the structural model3. Following CIR (1985b),Pennacchi (1991) and Sun (1992), we assume that the state variables drivingthe dynamics of the economy are the instantaneous real interest rate r andexpected inflation rate π. However, we depart from those models in allowinga time-varying central tendency in both these variables. We denote by θ the

2Endogenizing the price level and developing a monetary economy requires a more sophis-ticated theoretical framework, which would probably allow us to get a richer structure oftestable implications. However, this is beyond the scope of this paper. For term structuremodels which explicitly deal with money and endogenous inflation, see Stulz (1986), Foresi(1989), Marshall (1992), den Haan (1995) and Bakshi and Chen (1996).

3For a full derivation, see Appendix A.

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stochastic long-term mean of the real interest rate and by ξ the stochastic long-term mean of the expected inflation rate. We also introduce an additionalfactor, a zero-mean variable α, which, as will be explained in more detail below,represents a shock to the conditional mean of output growth.We assume that, under the physical probability measure, the state variables

follow a joint elastic random walk process of the form:

ds = (φ+ Γs) dt+Σdz (1)

where:

s ≡

rπθξα

, φ ≡

00φθφξ0

, Γ ≡

γrr γrπ −γrr −γrπ 0γπr γππ −γπr −γππ 00 0 γθθ 0 00 0 0 γξξ 00 0 0 0 γαα

Σ is a diagonal matrix and z a vector of standard Brownian motions.As in Pennacchi (1991) and Bakshi and Chen (1996), we assume that there

is a single technology producing a single physical good and that the produc-tion output follows a gaussian-type stochastic process. According to the CIR(1985a)-type general equilibrium framework, the logarithmic utility hypothesisunderlying the model implies that the expected rate of return on the productionprocess should be equal to the sum of the real interest rate and the variance ofthe production process (see also Breeden (1986)). However, in practice, the re-lation between real yields and expected output growth can significantly changeover time. Chapman (1997) shows that this relation has been quite unstable inthe U.S. over the last fifty years and has also exhibited a negative sign for someperiods. In fact, recursively regressing quarterly GDP growth rates against anaive estimator of the real interest rate4, we observe that the slope coefficientvaries significantly over time.For this reason, in order to better describe the dynamics of output, we relax

the CIR general equilibrium restriction by allowing the expected rate of returnon the production process and the real interest rate to differ by an amount whichvaries over time according to the zero-mean gaussian process α. Therefore, thevariable α plays the role of an exogenous shock to the conditional mean of outputgrowth.In common with most term structure models, we assume that the price level

evolves according to an exogenous gaussian process, which is determined by theprevailing monetary policy.Therefore, the vector of macro variables, composed of the production output

q and the price level p, evolves according to the following process:

dm

m= (φm + Γms) dt+Σmdzm (2)

where:4The regressions use a fixed ten year window of quarterly data beginning in 1960:Q1.

Proceeding recursively, we obtain parameter estimates for the period from 1970:Q1 to 2002:Q4.The estimator of the real interest rate is built by (i) estimating the expected inflation ratefrom the one-step forecast of an ARIMA(1,0,1) model applied to the logarithm of the GDPdeflator and then (ii) subtracting this series from the observed 3-month zero coupon yield.

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dm

m≡µdq/qdp/p

¶, φm ≡

µσ2q0

¶, Γm ≡

µ1 0 0 0 10 1 0 0 0

¶Σm is a diagonal matrix, with elements σq and σp, and zm is a vector of

standard Brownian motions.Innovations in the state variables, output and price level could be mutually

correlated and the correlation matrices Ω ≡ Et dz, dz0, Ωm ≡ Et dzm, dz0mand Ξ ≡ Et dz, dz0m full. However, we assume that innovations in the shockvariable α are uncorrelated with innovations in the structural variables.From equation (2), expressions for logarithmic changes in production and

the price level over a time interval [t, t + τ ] can be obtained. For both theseexpressions and the stochastic differential equations (1) a closed form solutioncan be derived and the dynamics of the economy summarized by the followingset of equations:

Et s(t+ τ) = a(τ) +B(τ)s(t) (3)

Covt s(t+ τ), s(t+ τ)0 = C(τ) (4)

Et ln [m(t+ τ)/m(t)] = h(τ) + J(τ)s(t) (5)

Covt©ln [m(t+ τ)/m(t)] , ln [m(t+ τ)/m(t)]0

ª=M(τ) (6)

Covt©s(t+ τ), ln [m(t+ τ)/m(t)]0

ª= Q(τ) (7)

where τ > 0 and where a(τ), B(τ), C(τ), h(τ), J(τ), M(τ) and Q(τ) are nonlinear transformations of the original set of coefficients in equations (1-2)5.Equations (3-7) completely describe the dynamics of the underlying econ-

omy6. This structure implies that cross-equation restrictions link the dynamicsof all the variables in the economy, i.e., the state variables, output and the pricelevel.

B Risk premia and term structure of interest rates

In the CIR (1985a) framework, the volatility and correlation structure endoge-nously determines the factor risk premia coefficients:

Λ0 ≡ Σ−1µCovt

½ds,

dq

q

¾+Covt

½ds,

dp

p

¾¶= ΞΣmι

with: ι0 ≡ ¡ 1 1¢.

5The functional form of these parameters is derived in Appendix A.6Note that we could obtain a similar structure starting with a model which depends on the

same two unobservable state variables used by Bakshi and Chen (1996) - a monetary shockand a technological shock - and imposing on them a time-varying central tendency. Assumingthat two different linear combinations of the monetary factor and the technological factoraffect the drift term of the production process and the price level, respectively, and usingthe general equilibrium argument, we may show that both the instantaneous real interestrate and the instantaneous expected inflation rate can be written as linear combinations ofthe underlying monetary and technological shocks. Therefore, through a simple change invariables, we could rewrite the model in terms of this ”new” set of state variables, i.e., theinstantaneous real interest rate, the instantaneous expected inflation rate and their centraltendencies, and arrive at a model similar to that in equations (3-7).

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The homoskedasticity assumption underlying the gaussian structure of themodel implies that risk premia are constant. However, following Duffee (2002)(see also Dai and Singleton (2002) and Duarte (2003)), we adopt the ”essentiallyaffine” specification, which allows the market price of risk to vary independentlyof interest rate volatility. In this case, factor risk premia are assumed to be affinein the state variables:

Λ ≡ Λ0 + Λ1s (8)

where Λ0 is specified as above and Λ1 is a matrix of free parameters. Thisspecification implies that the matrix of mean reversion coefficients under therisk-adjusted probability measure is given by eΓ ≡ Γ − ΣΛ1. However, someconstraints on the form of matrix Λ1 must be imposed in order to preserve astructure consistent with θ and ξ being the stochastic central tendencies of rand π under the risk-adjusted probability measure. In addition, we also assumethat there is no risk premium on the shock variable α. Therefore, we have:

Λ1 ≡

λrr λrπ −λrr −λrπ 0λπr λππ −λπr −λππ 00 0 λθθ 0 00 0 0 λξξ 00 0 0 0 0

This model admits a closed form solution for the equilibrium price of a

nominal unit discount bond with maturity τ . The nominal term structure takesthe form:

Y (t; t+ τ) = κ0(τ) + κ(τ)s(t) (9)

where the coefficients κ0(τ) and κ(τ) depend on the underlying model parame-ters.Using a similar argument, we can determine the closed form solution for the

equilibrium price of a real unit discount bond with maturity τ and, therefore,an expression for the real term structure:

y(t; t+ τ) = ω0(τ) + ω(τ)s(t) (10)

where the coefficients ω0(τ) and ω(τ) are also non linear functions of the originalset of parameters7.The solution allows one to assess the influence on bond pricing of the co-

variance between output growth and inflation. In fact, a neat relation betweennominal and real yields can be derived:

Y (t; t+ τ) = y(t; t+ τ) +1

τEt

½lnp(t+ τ)

p(t)

¾− 12τV art

½lnp(t+ τ)

p(t)

¾−1τCovt

½ln

q(t)

q(t+ τ), ln

p(t)

p(t+ τ)

¾(11)

The covariance term in equation (11) enables us to evaluate the role of therelationship between economic growth and inflation in determining the price

7Again, for a full derivation of the term structure equations (9) and (10), see Appendix A.

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of zero coupon bonds. This effect is constant over time, but can differ acrossbonds. The specification implies that shocks to the real side of the economy canaffect both the level and the slope of the nominal term structure through theoutput-inflation correlation.

III Estimation method

The theoretical model above is developed in a continuous time and gives ana-lytical solutions for bond prices, economic growth and inflation. However, themodel is estimated using data available only at discrete observation intervals.The discrete time transformation of the continuous time model can be obtainedby exploiting the solution to the stochastic differential equations which describethe dynamics of the variables. This is given by a discrete time state space form,composed of state transition equations related to the unobserved state variablesand measurement equations related to bond yields, output growth and inflation:

Zt = H +K · st + et (12)

st = A+B · st−τ + υt (13)

where:

Zt ≡ Y (t; t+ τ i)ln (q(t+ τ)/q(t))ln (p(t+ τ)/p(t))

,H ≡ κ0(τ i)

hq(τ)hp(τ)

,K ≡ κ0(τ i)Jq(τ)Jp(τ)

, i = 1, ..., n

s0t ≡¡r(t) π(t) θ(t) ξ(t) α(t)

¢, A ≡ a(τ), B ≡ B(τ)

τ is the time interval, and τ i, i = 1, ..., n, are the maturities of the bondsincluded in the sample.The structure of the coefficients implies that cross-equation restrictions must

be imposed on the model, as the vectors H and A and the matrices K and Bin the state space form (12-13) are nonlinear functions of the original modelparameters in equations (1-2) and (8).The error terms et and υt are assumed to be normally and independently

distributed. For the covariance matrices relating error terms in the state vari-ables, output growth and inflation we impose all the cross-equation restrictionsimplied by the matrices C(τ), M(τ) and Q(τ) of equations (4), (6) and (7).The deviation of actual yields from their theoretical values is assumed to

be due to ”observation error”. This noise in zero coupon bond yields mightarise from the fact that they are typically computed from averages of bid andask prices or from the fact that they are estimated from the prices of couponbonds. In contrast, the errors in the equations for output growth and inflationmeasured over the time interval [t, t + τ ] are ”forecasting errors”, which arisefrom the difference between actual and expected values for output growth andinflation. In the state space form (12-13) both these forecasting errors and theobservation errors on bond yields are referred to as ”measurement errors”.We assume that observation errors in bond yields are mutually uncorrelated

and uncorrelated with forecasting errors in τ -period output growth and inflation.

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Moreover, in order to limit the number of parameters which must be estimated,we assume that the errors on bond yields are cross-sectionally homoskedastic.As stated earlier, the forecasting errors in τ -period output growth and infla-

tion rates and the transition errors υt are correlated, so that additional cross-equation restrictions on model’s parameters are imposed, which might furtherhelp to identify the model.Measurement errors et and transition errors υt are assumed to be uncor-

related at all lags and to be uncorrelated with the initial state vector. Theparameters of the state space model are estimated by the maximum likelihoodmethod with the Kalman filter algorithm used to calculate the values of theunobserved state variables8. As is well known, the Kalman filter algorithm isa recursive procedure which computes the conditional expectation of the statevector st given the information available at time t, assuming that the meanand the covariance matrix of the initial state vector are known. The inclusionof equations for τ-period economic growth and inflation into the system allowsthe unobservable state variables to be uniquely identified. Without these equa-tions, i.e., including only the term structure equations and the state transitionequations, an identification problem would arise from the fact that any bondpricing result may be obtained from different orthogonal transformations of theunderlying state variables and model parameters9.

IV Model estimation

The model is estimated using the dataset described in section I, that is, quarterlydata for the United States over the period 1960:Q1-2002:Q4. For bond yields,we use end-of-quarter annualized zero coupon yield data for six maturities (3months, 1, 3, 5, 10 and 20 years), whereas seasonally adjusted data on realGDP and the GDP deflator are used as proxies for output and the price level,respectively.In this section, we first present some specification tests investigating the

adequacy of the model. Second, we illustrate the estimated parameter values,state variables and implicit risk premia. Then, we evaluate the goodness of fitof the model with respect to bond yields and macro variables by comparing itsperformance with that of some alternative models. Finally, we show the modelestimates of the unobservable term structures of real interest rates and expectedinflation rates.

A Specification tests

The model imposes several moment restrictions and encompasses more parsi-monious models, such as, for example, the Pennacchi (1991) model. In order to

8See Anderson and Moore (1979) and A. Harvey (1989). Recently, the Kalman filter hasbeen widely applied in the estimation of term structure models. See, among the others, Pen-nacchi (1991), Chen and Scott (1994), Gong and Remolona (1996), Jegadeesh and Pennacchi(1996), Dewachter and Lyrio (2003).

9 In a similar way, Pennacchi (1991) is able to identify the real interest rate and the expectedinflation rate in the Kalman filter estimation of his model as he uses data on survey inflationforecasts along with data on bond yields. Instead, the factors cannot be identified in modelsestimated using no other data than bond yields (see, for example, Chen and Scott (1994) andGong and Remolona (1996)).

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test the adequacy of the model, we perform some preliminary specification testsfollowing Duffee (2002) and Ang and Bekaert (2003).First, we apply a GMM-type test to assess the closeness of the estimated

unconditional moments to the sample moments. This test is based on the pointstatistic H = (h−h)0Σ−1h (h−h) , where h are the estimates of the unconditionalmoments obtained from the sample, h are the unconditional moments estimatedby the model, Σh is the covariance matrix of the sample estimates of the un-conditional moments, which is estimated using GMM with the Newey-West cor-rection for heteroskedasticity and autocorrelation. Under the null hypothesis,this statistic is distributed as χ2(n), where n is the number of overidentifyingrestrictions. We test for first and second moments of bond yields and yieldchanges, for maturities of 3 months, 1, 3, 5, 10 and 20 years, and GDP growthand inflation rates with time horizon 3, 6, 9 and 12 months.Second, we apply a GMM-based test for first order serial correlation in the

estimated scaled residuals of equation (12). We test the following null hypoth-esis: E(etet−1) = 0.Third, using a Wald statistic10, we test a null hypothesis involving restric-

tions on the model parameter vector. The restricted models are represented by:(i) a two-factor model, in which central tendencies are constant and there is noshock to the conditional mean of output growth (the variable α is fixed at itszero long-term mean); (ii) a three-factor model in which central tendencies areconstant; (iii) a four-factor model in which the shock to the conditional mean ofoutput growth is fixed at zero; (iv) a five-factor completely affine specificationin which risk premia are constant (the coefficients of matrix Λ1 in equation (8)are all equal to 0).The results in table 2 show that the model performs relatively well. In Panel

A we observe that the unconditional moments implied by the model estimatesare within one standard error of data moments, with the only exception of thevariance of the GDP growth rate. The goodness of fit of the model is reflectedalso in Panel B, where the χ2 test allows us to reject the null hypothesis thatthe first and second moments of yields, yield changes and inflation rates overdifferent time horizons estimated by the model are significantly different fromthose estimated from the sample. In the case of GDP growth rates, we noticethat the model tends to underestimate the volatility and the null hypothesiscannot be rejected.Panel C shows that, as predicted by the model, the estimated residuals

for bond yields and quarterly GDP growth and inflation rates are not seriallycorrelated. Moreover, we observe that the different restrictions on the model’sparameters are rejected, indicating that the time-varying central tendencies,the shock on the conditional mean of output growth and the essentially affinespecification for the risk premia all play a significant role in fitting the relationbetween the term structure and macro variables.

B Parameter estimates

Table 3 reports the estimated parameter values under the physical probabilitymeasure.We notice that the real interest rate r exhibits significant mean reversion

10 See, for example, Hamilton (1994, ch.14).

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(0.57) towards its time-varying central tendency θ, which is much more persis-tent. Instead, both the expected inflation rate π and its central tendency ξ areclose to a random walk.Figure 1 shows the estimated dynamics of the state variables in response to

a one standard deviation innovation. We observe that for the real interest ratethe half-life (the time it takes for the variable to return half way towards itsinitial value) is approximately 2 years (panel A), whereas shocks to the expectedinflation rate are very persistent (panel B). A one standard deviation shock tothe central tendency of the expected inflation rate also has persistent effectson itself and the expected inflation rate (panel D). The mean reversion of thecentral tendency of the real interest rate (panel C) is relatively slow, with ahalf-life of about 6 years.The cross-effects coefficients relating the real interest rate and the expected

inflation rate are negative. Moreover, we observe in figure 1 that a one stan-dard deviation innovation in the real interest rate has a negative impact on theexpected inflation rate (panel A), and vice versa (panel B). In this respect, ourestimated drift term coefficients differ substantially from those originally esti-mated by Pennacchi (1991), who found positive cross-effects between the levelof the real interest rate and the expected inflation rate11.The estimated volatility coefficients indicate that the volatility of the real

interest rate is generally higher than that of (i) the expected inflation rate (249vs. 149 basis points) and (ii) its stochastic central tendency (186 basis points).The first result is consistent with that of Pennacchi (1991), where the estimatedvolatilities were 266 and 106 basis points, respectively. Sun (1992) also findsthat the real interest rate is more volatile than the expected inflation rate. Ingeneral, all these findings sharply contrast with the Fama (1975) hypothesis ofa constant real interest rate.Table 4 shows the correlations between innovations in the real and nominal

variables calculated for finite time intervals.The correlation between innovations in the real interest rate and the expected

inflation rate is negative and increases in size with the interval over which theyare calculated. This evidence, along with the estimated negative cross-effectscoefficients seen above, supports the so-called ”Mundell (1963) - Tobin (1965)effect”, which has extensively been documented in the empirical literature (see,among the others, Mishkin (1992) and Boudoukh (1993)).Both the real interest rate and the expected inflation rate are positively

related to innovations in their respective central tendencies, whereas the twocentral tendencies are negatively correlated.Consistent with the predictions of Fama and Gibbons (1982), we find that

innovations in output growth appear to be positively correlated with instan-taneous innovations in both the real interest rate and its time-varying centraltendency and slightly negatively correlated with innovations in expected infla-tion.The correlation between innovations in output growth and actual inflation is

positive at the short end and becomes negative for longer time horizons. Equa-tion (11) above shows that the covariance between output growth and inflationhas a direct impact on nominal yields. Using the estimated model coefficients,11However, estimating the essentially affine version of the Pennacchi model for the sample

period 1960-2002 (see Appendix B) we find that this relation is actually negative for ourdataset.

11

we observe that this covariance affects nominal yields positively and the size ofthe effect, which is almost negligible for short-term maturities, tends to increasewith bonds maturity. In fact, the impact of the GDP - inflation covarianceon the unconditional mean of nominal yields is only about 3 basis points forthe 3-year maturity and increases substantially in the case of longer maturitybonds: 24 and 71 basis points, respectively, for the 10-year and the 20-yearmaturity. This is partially in contrast with one of the theoretical specificationsproposed by Bakshi and Chen (1996), which relates the real term structure onlyto output shocks and the nominal term structure only to inflationary shocks.The interrelation between real and monetary variables, instead, seems to playa key role in determining the nominal term structure.

C State variables

Figure 2 shows the estimated series of the unobservable state variables, thatis, the instantaneous real interest rate and expected inflation rate and theirtime-varying central tendencies, calculated applying the fixed-interval smooth-ing algorithm to the Kalman filter estimates12. For comparison purposes, weinclude a proxy for the corresponding ex-post measure. In particular, we includethe annualized quarterly inflation rate realized a quarter ahead and the realizedshort-term real interest rate, calculated as the difference between the 3-monthnominal yield and the 3-month realized inflation rate for the same period.Consistent with previous estimates (see, for example, Pennacchi (1991), Sun

(1992) and Ang and Bekaert (2003)), the real interest rate becomes negative inthe Seventies and exhibits a sharp increase in level and volatility in the 1979-82 period, when the Federal Reserve changed its monetary policy procedures.Thereafter, especially in the second part of the sample, the real interest rateseems to revert quickly towards its central tendency. The expected inflationrate shows a positive trend for the first part of the sample and declines consid-erably after the Fed experiment. The series estimated for the long-term centraltendencies look reasonable13 .The unobservable shock variable α, which affects the conditional mean of

output growth, displays a significant positive correlation with GDP growthrates. The correlation decreases as the time horizon lenghtens, from about0.8 at the 3 month horizon to 0.3 at the 2 year horizon. By regressing theestimated series against a constant using GMM with the Newey-West correc-tion, we cannot reject at the 10% significance level the null hypothesis thatthe unconditional mean of the variable is zero, as predicted by the theoreticalmodel.Figure 3 shows the impact of the unobservable factors on the shape of the

term structure, i.e, the effect on nominal yields of a one standard deviation

12 See, for example, A. Harvey (1989, ch.3) and Hamilton (1994, ch.13).13The theoretical framework implies that the sum of the real interest rate and the expected

inflation rate is approximately equal to the instantaneous nominal interest rate. FollowingJegadeesh and Pennacchi (1996), we might interpret the sum of the central tendencies as aproxy for the unobservable nominal target rate of the Fed monetary policy. Long-run targetsreflect individuals’ expectations about medium-long term inflation and economic growth and,implicitly, individuals’ beliefs regarding future monetary policy. Our estimates imply that thehalf-life of the nominal rate is about 2 years, whereas its central tendency is very persistent.This result appears to be in contrast with the evidence presented in Jegadeesh and Pennacchi(1996), where the central tendency displays higher mean reversion than the short rate.

12

innovation in the variables. We notice that the sensitivity of yield changes toinnovations in the expected inflation rate and its central tendency is almostconstant across maturity. Therefore, these persistent factors can be associatedto the level of yields and account for parallel shifts in the yield curve. The realinterest rate, instead, appears to be mainly related to the slope of the termstructure, as a positive shock in this variable has a significant positive impacton short-term yields which declines with maturity. Finally, the central tendencyof the real interest rate plays the role of the factor affecting the curvature of theyield curve, as its innovations have a hump-shaped influence on yields, with apeak around the 4-year maturity. Intuitively, this may be explained as a sortof ”business cycle” effect. In fact, a positive impulse to the central tendencyof the real rate is generally associated with a growing real activity and with asubsequent increase in short term interest rates during the expansion period,which lasts on average 4-5 years.

D Risk premia

The model’s essentially affine specification endogenously determines instanta-neous risk premia, given by the expression: RP (τ) = −τ · κ(τ) · Λ, where Λand κ(τ) are defined as in equation (8) and (9), respectively. Panel A of table5 shows summary statistics on the estimated risk premia. We observe that riskpremia on the real interest rate are almost constant with maturity (around 20basis points), whereas risk premia on expected inflation slowly increase withbond maturity and are around 25 basis points, on average. This evidence isin part consistent with that presented by Buraschi and Jiltsov (2003), in thatwe also obtain a positive slope of the average term structure of inflation riskpremium. However, the estimated average level of the inflation risk premium isonly about half of that estimated by Buraschi and Jiltsov (2003)14.As in Cochrane and Piazzesi (2002), we find that the estimated risk premia

are linked to the business cycle and tend to be higher during recessions thanduring expansions. In fact, the average (across maturities) risk premia on thereal interest rate and the expected inflation rate are significantly negativelycorrelated, −0.45 and −0.35, respectively, with the cyclical component of GDP,which is calculated as the difference between the level of real GDP and its trendcomponent estimated through the Hodrick-Prescott filter.The estimated expected excess returns on the central tendency of the real

interest rate and the expected inflation rate increase with maturity and areequal, respectively, to 45 and 30 basis points, on average. Therefore, the totalrisk premium also has an increasing structure and ranges approximately between30 (1-year) and 200 (10-year) basis points.The estimated expected risk premia are positively correlated with realized

1-year excess bond returns, although the model predictions, as expected, exhibitless variability than the ex-post values.

14The estimated risk premia are also consistent with those of Gong and Remolona (1996)for the sub-period 1984:Q1-1996:Q4. In the case of the 5-year maturity, we get similar values,with the risk premium on the real interest rate which is, on average, about half the riskpremium on the expected inflation rate (110 basis points). However, for the 10-year maturity,Gong and Remolona (1996) estimate risk premia of approximately the same size for both thereal and the inflation side, whereas our model produces much higher risk premia on inflation(140 vs. 50 basis points, on average).

13

In panel B of table 5 we compare the predictive power of the model estimatesfor 1-year excess bond returns with that of the Cochrane and Piazzesi (2002)single-factor model. The single factor in the Cochrane-Piazzesi model is a linearfunction of forward rates which is obtained by running the following regression:

ER(t+ 1) = γ0+5Xi=1

γi · fw(t; 0, i) + u(t+ 1) = (γ0 · fw(t)) + u(t+ 1)

where ER(t + 1;T ) ≡ F (t + 1;T − 1) / F (t;T ) − 1 / F (t; 1) is the 1-yearexcess return on a T -maturity bond and ER(t+1) is the average of excess bondreturns for T = 2, 3, 4, 5. F (t;T ) represents the price at time t of a T -maturityunit zero coupon bond and fw(t;T −1, T ) is the forward rate for loans betweentime t+ T − 1 and t+ T .Then, in a second step, the following forecasting regressions are estimated:

ER(t+ 1;T ) = bT · (γ0 · fw(t)) + u(t+ 1;T ) , T = 2, 3, 4, 5

We run the same two regressions by using as a predictor the estimated modelrisk premia, RP (t;T ), T = 2, 3, 4, 5, instead of the forward rates.We observe, consistent with Cochrane and Piazzesi (2002), that the return-

forecasting factor is a tent-shaped linear function of the regressors, i.e., forwardrates or estimated model risk premia, and that the normalized coefficients bTtend to increase with the maturity of the bonds. The results show that the modelpredictive power, measured in terms of R

2, is also comparable to that produced

by the Cochrane-Piazzesi framework and explains about 36% of time variationin 1-year excess bond returns. As pointed out by Cochrane and Piazzesi (2002),this is considerably higher than the value obtained in the ”benchmark” stud-ies of Fama and Bliss (1987) and Campbell and Shiller (1991)15. According tothis evidence, we might argue that including macro variables in a term struc-ture model does indeed provide relevant information for the prediction of bondreturns.

E Goodness of fit

In what follows, we analyze the in-sample adequacy of the structural model inpredicting bond yields, inflation and GDP growth and compare its performanceswith those of some competing models. In particular, as competing models, weconsider an essentially affine version of the Pennacchi (1991) constant-meanequilibrium term structure model (PCM) and the Ang, Piazzesi and Wei (2003)VAR model (APW). Moreover, as a benchmark in evaluating bond yields pre-dictions, we estimate a 5-factor latent variables essentially affine term structuremodel (ATSM) of the Dai and Singleton (2000) - Duffee (2002) type, whereasour benchmarks for the predictions of GDP growth rates and inflation rates are,respectively, the C. Harvey (1989) term spread regression model (HTS) and therandom walk model (RW). All these models and their parameter estimates areillustrated in Appendix B.

15We estimate the Fama-Bliss regressions for our dataset by regressing the 1-year excessreturn on a T -maturity bond, T = 2, 3, 4, 5, against the forward spread of the correspondingmaturity. The estimated R

2’s range between 11% and 16%. For brevity, we do not include

these regressions in Table 5.

14

Table 6 shows the results of the following regression, in which realized in-flation rates and GDP growth rates are regressed against the correspondingexpectations generated by the model estimates:

X (t; t+ τ) = β0 + β1 · E X (t; t+ τ)+ e(t)where:

X (t; t+ τ) ≡ ln (x(t+ τ) / x(t)) ; x = q, p

We run these regressions for both the structural model and the competingmodels. The inflation expectations produced by the model are accurate predic-tions of future inflation, as the adjusted R2’s of the regressions are around 95%for all time horizons up to two years (panel A). Moreover, the inflation estimatorappears to be unbiased. The model inflation predictions are more precise thanthose provided by the alternative models, as the corresponding adjusted R2’sare about 75-80% for the RW process and the APW model and only around25% for the PCM model.The output growth expectations implicit in the model estimates are also

relatively accurate predictions of actual GDP growth rates (panel B), althoughthe accuracy of the fit tends to sharply decrease as the time horizon lengthens.In fact, the adjusted R2 of the regression is 94% at the 3-month horizon andonly 40% at the 1-year horizon. The competing models, instead, are very poorin predicting short term GDP growth rates and improve their performancesas the time horizon lenghtens. The HTS and APW estimators look unbiased,whereas in the case of the PCM model we observe that the sign of the regressioncoefficient is wrong. This is an effect of the restriction imposed by the Pennacchi(1991) framework (and, in general, by general equilibrium term structure modelsof the CIR (1985a and 1985b) type) on the relation between expected outputgrowth and real yields. In particular, it implies that output growth expectationsare determined one-to-one by the real interest rate. However, as shown byChapman (1997), the sign of this relation can change over time because of theeffect of exogenous factors, such as, for example, the monetary policy. Thisevidence underlines the relevance of the shock variable α introduced in thestructural model to capture the dynamics of output growth, especially in theshort term.The ability of the structural model to fit the macro variables does not involve

a significant trade-off in terms of the fit of bond yields. In order to evaluatethe performances of the competing models in fitting bond yields, we use as abenchmark the maximum likelihood - Kalman filter estimates of a 5-factor latentvariable model (ATSM), specified as in the Dai and Singleton (2000) canonicalform and extended to include the Duffee (2002) essentially affine generalization(for details, see Appendix B). This model is also estimated using end-of-quarterannualized zero coupon yields for the maturities 3 months, 1, 3, 5, 10 and 20years.Table 7 reports summary statistics on in-sample estimation errors. Bond

yield errors generated by the structural model are relatively low, with no ev-idence of systematic under/over pricing, as the errors have both positive andnegative mean values at the various maturities. Root mean squared errors rangebetween 7 and 21 basis points. Only at short term maturities are these errorssignificantly larger than those obtained by the ATSM. At medium-long term

15

maturities they are not significantly different. It is worth noting that the modelalso fits well the yields not included in the estimation sample (2, 4, 6 to 9 yearsmaturity) and, in general, provides a more accurate fit than the PCM and theAPW16 models.

F Real and expected inflation term structures

An interesting feature of the structural model is represented by the fact thatit allows one to extrapolate the unobservable real and expected inflation termstructures from nominal bond prices. A natural benchmark to evaluate themodel estimates would be represented by market values extracted from inflation-indexed bonds. However, a liquid market with a relatively long series of obser-vations exists only in the U.K.17, whereas the U.S. Treasury has only recentlyinitiated the issue of inflation-indexed bonds. Estimates of the U.S. real termstructure calculated from these bonds start only in 1997 and a different bench-mark must be used in order to analyze the properties of the real and expectedinflation term structures estimated by the model for the sample period 1960-2002. In particular, we use ex-post realized real yields and inflation rates. Theτ -maturity real interest rate realized at time t is calculated as the differencebetween the time t observed τ-maturity nominal yield and the annualized infla-tion rate realized between time t and t+ τ . The latter is used as a benchmarkto test the predictive ability of the model term structure estimates for futureinflation.Table 8 contains summary statistics on the estimated real and expected

inflation term structures. On average, real yield curves are humped, as they areupward sloping for maturities up to 5-6 years and downward sloping at the longend, with volatility which tends to decrease with the time horizon. Instead, theterm structure of expected inflation rates is, on average, upward sloping withdecreasing volatility. As in Ang and Bekaert (2003), we find that real rates areprocyclical and expected inflation is countercyclical. In fact, the estimated realrates are generally higher during expansions than recessions18, whereas expectedinflation rates behave in the opposite way, with the volatilities of both variablesincreasing during recessions. For example, we estimate that the 2-year real rateis on average 2.86% with a standard deviation of 1.62% during expansions and2.40% with a standard deviation of 2.78% during recessions. The corresponding2-year expected inflation rate is on average 3.42% with a standard deviation of2.01% during expansions and 4.72% with a standard deviation of 3.07% duringrecessions.We compare the moments estimated by the model with the sample moments

of ex-post realized real yields and inflation rates with maturity from 1 to 5 yearsusing the χ2 test based on the point statistic H = (h − h)0Σ−1h (h − h), whichhas been described in section IV.A above.We observe that the unconditional moments implied by the model predic-

tions for real interest rates and inflation rates are within one standard error16The APW model fits particularly well the 3-month, 5-year and 20-year maturity yields

because they are the inputs of the observable factors (short rate, spread and curvature) usedin the estimation (see Appendix B).17Empirical evidence on this market has been provided by Brown and Schaefer (1994a),

Barr and Campbell (1997), Evans (1998).18Expansion and recession periods as defined by the NBER.

16

of realized data moments, with the exception of the variance of real yields. Infact, the volatility of estimated real yields is too low with respect to the oneobserved ex-post, especially for long term maturities. In general, however, themodel seems to provide relatively accurate predictions on the dynamics of thelevel of both the real and the inflation term structures. The correlation betweenthe series estimated by the model and the corresponding ex-post realized valuesis very high.The model estimates of real rates can be compared with the corresponding

values calculated from quoted U.S. inflation-indexed bonds using the McCul-loch and Kochin (1998) procedure19 . The sample period for this comparisonis limited, as it ranges only between 1997:Q3 and 2002:Q4 (22 observations).The estimated series appear to behave in a very similar way and to be highlycorrelated at all maturities (between 0.83 and 0.94), both in levels and quarterlychanges, although the real rates estimated by the structural model tend to belower in level than those extracted using the McCulloch-Kochin method.

V Out-of-sample forecastsIn this section, we evaluate the out-of-sample predictive accuracy of the struc-tural model for inflation and GDP growth rates and compare its performancewith that of the competing models, i.e., the Pennacchi (1991) constant-meanmodel (PCM) and the Ang, Piazzesi and Wei (2003) VAR model (APW). Wealso consider as benchmarks the inflation forecasts of a random walk model(RW), the GDP growth forecasts of the C. Harvey (1989) term spread regres-sion model (HTS) and the forecasts on inflation and GDP growth rates providedby the Survey of Professional Forecasters published by the Federal Reserve Bankof Philadelphia20.Predictions of inflation and output growth are endogenously determined by

the structural model under cross-equation restrictions which involve parametersand unobservable state variables that also determine the dynamics of the termstructure of interest rates (equations (3-9)). Imposing these restrictions meansthat forecasts of real growth rates and inflation rates over different horizons arelinked through the structural parameters of the model.The forecasts are obtained by using the preceeding thirty years of quar-

terly data to estimate the parameters needed to predict subsequent inflationand GDP growth rates over the different horizons. Proceeding recursively, wecompute forecasts for all the competing models from 1990:Q1 to 2002:Q4 (52observations) and compare these to realized inflation and GDP growth rates.The estimation of the state space form (12-13) at time t involves the use of

GDP growth and inflation rates over the next quarter [t, t+1]. As these variablesare not observed at time t, we build a proxy for them by recursively estimatingone-step-ahead forecasts generated by an ARIMA(1,0,1) process applied to thetime series of the logarithm of the price level and real GDP. These estimatorsare used as inputs for the estimation of the state space form (12-13) at time t.

19These values are published and upgraded quarterly on McCulloch’s website:http://economics.sbs.ohio-state.edu/jhm/jhm.html20This survey is formed using the macroeconomic forecasts supplied by several main

economic research centers and financial institutions. Before 1990, it was known as theASA/NBER survey.

17

Table 9 summarizes the error properties of the estimated models in fore-casting inflation and GDP growth rates over different time horizons and showsthe values of the Diebold and Mariano (DM, 1995) statistic21 testing the nullhypothesis of no difference in the accuracy of the forecasts provided by thecompeting models with respect to those of the structural model.The striking result that emerges from panel A of the table is the accuracy of

the model in forecasting inflation rates at time horizons up to two years. Thisfinding contrasts with recent results presented by Atkeson and Ohanian (2001)showing that most modern macroeconometric models (such as, for example,Stock andWatson (1999) and related models) fail in producing inflation forecastsmore accurate than a random walk model22. In this respect, we may argue thata structural model which in an internally consistent framework links inflation tothe dynamics of the real side of the economy and to the yield curve and whichincorporates current individuals’ expectations about future inflation and outputgrowth, represents a powerful forecasting tool.At short term horizons, the model forecasting errors, measured in terms of

RMSE’s, are almost half those of the RW model. The DM test rejects the nullhypothesis of no difference in the accuracy of the model’s forecasts with respectto the competing models at any time horizon. In general, the other models areoutperformed by the RW process at all time horizons. We notice that all themodels, including the structural model, tend to over-estimate future inflationrates, with the exception of the survey forecasts at horizons longer than threemonths.Panel B of table 9 shows that the structural model also outperforms the

competing models with regard to GDP growth forecasts. The RMSE’s gener-ated by the models tend to decrease as the time horizon lenghtens. The DMtest allows us to conclude that, at the 5% significance level, the GDP growthforecasts produced by the model are more accurate than those of the alternativeapproaches at any time horizon. In this case, the model does not systematicallyover/under-estimate future GDP growth rates and the mean of the forecasterrors is very close to zero.To sum up, the empirical evidence presented in this section shows that the

cross-equation restrictions imposed by the structural model on the dynamics ofinflation, output, term structure of interest rates and underlying state variablesdo actually provide accurate macro-variables forecasts, which clearly outperformalternative techniques. Therefore, a model which is able to take simultaneouslyinto account information on all these variables seems to improve considerably ourability to predict future inflation and output growth rates. In this respect, themodel may represent a powerful tool for both monetary policy and investmentdecisions.The fact that the model performs well in forecasting the structural variables

does not prevent it from providing relatively accurate out-of-sample predictionsof bond yields. Table 10 shows the predictive accuracy of the model in fore-casting bond yields one, two and four quarters ahead and compares it with theperformance of the benchmark 5-factor latent variables essentially affine termstructure model (ATSM). We observe that, with the exception of the 3-month

21The Diebold-Mariano statistic can be calculated as the t -statistic obtained by regressingthe difference in squared errors between the competing forecasts against a constant, with theNewey-West correction for heteroskedasticity and serial correlation.22On this point, see also Fisher, Liu and Zhou (2002).

18

and 20-year maturity yields for short term forecasting horizons, the DM testdoes not allow us to reject the null hypothesis that the forecasts produced bythe two models are not significantly different. In the case of medium-long termmaturities (7 and 10 years), the RMSE’s produced by the model forecasts areeven lower than those estimated for the benchmark 5-factor model. This indi-cates that using data on macro variables might help in better forecasting thefuture dynamics of bond yields by providing some useful additional informationwhich is not embedded in current bond yields. In fact, as shown by Duffee(2002), the out-of-sample forecasting properties of yields-only models are gen-erally relatively poor in comparison to a simple random walk.

VI Conclusion

In this paper we build and estimate an internally consistent structural modelwhich analyzes the relationship between the term structure of interest rates,inflation and GDP growth. The equilibrium aspects of the model impose non-trivial restrictions on the joint dynamics of bond yields and macro variables andimply that real and monetary variables of the economy are interrelated and in-fluence the shape of the yield curve, while bond yields convey information thatis useful for forecasting economic fundamentals.The results indicate that by considering data on inflation and GDP along

with the data on yields in an affine term structure setting allows us to obtain re-liable estimates of the implicit term structures of real interest rates and expectedinflation rates and to derive sensible predictions for bond returns.Moreover, the empirical evidence shows that by placing more emphasis on

the macroeconomic aspects underlying the dynamics of the yield curve and onthe feedback effects which run from interest rates to the fundamentals, we canobtain forecasts of future inflation and GDP growth rates which are considerablymore accurate than those provided by some well known alternative approaches.

19

Appendix A. Full derivation of the structural modelIn this appendix, we provide a full derivation of the structural term struc-

ture model. We consider a continuous time competitive economy of the CIR(1985a) type composed of a fixed number of identical individuals with rationalexpectations and time-additive logarithmic utility functions of the form:

Et

Z ∞t

exp(−δu) ln(c(u))du

where c(t) is the real consumption flow at time t and δ the utility discountfactor.

State variables, output and inflationWe assume that the state variables driving the dynamics of the economy are

the instantaneous real interest rate r and expected inflation rate π and theirtime-varying central tendencies, θ and ξ, respectively. Moreover, we introduceas state variable a zero-mean variable, α, which represents a shock to the con-ditional mean of output growth.Under the physical probability measure, the state variables follow a joint

elastic random walk process:

ds = (φ+ Γs) dt+Σdz (A.1)

where:

s ≡

rπθξα

, φ ≡

00φθφξ0

, Γ ≡

γrr γrπ −γrr −γrπ 0γπr γππ −γπr −γππ 00 0 γθθ 0 00 0 0 γξξ 00 0 0 0 γαα

Σ is a diagonal matrix and z a vector of standard Brownian motions.Production output, q, and the price level, p, also evolve according to gaussian-

type processes:

dm

m= (φm + Γms) dt+Σmdzm (A.2)

where:

dm

m≡µdq/qdp/p

¶, φm ≡

µσ2q0

¶, Γm ≡

µ1 0 0 0 10 1 0 0 0

¶Σm is a diagonal matrix, with elements σq and σp, and zm is a vector of

standard Brownian motions.In the correlation matrices Ω ≡ Et dz, dz0, Ωm ≡ Et dzm, dz0m, Ξ ≡

Et dz, dz0m, the shock variable α is assumed to be uncorrelated with the otherstructural state variables.Using Ito’s Lemma, from equation (A.2) we derive expressions for logarith-

mic changes in production and the price level over a time interval [t, t+τ ]. Boththese expressions and the stochastic differential equations (A.1) have a closedform solution:

20

Et s(t+ τ) = a(τ) +B(τ)s(t) (A.3)

Covt s(t+ τ), s(t+ τ)0 = C(τ) (A.4)

Et ln [m(t+ τ)/m(t)] = h(τ) + J(τ)s(t) (A.5)

Covt©ln [m(t+ τ)/m(t)] , ln [m(t+ τ)/m(t)]0

ª=M(τ) (A.6)

Covt©s(t+ τ), ln [m(t+ τ)/m(t)]0

ª= Q(τ) (A.7)

where:

a(τ) ≡ Γ−1 (B(τ)− I)φ, B(τ) ≡ exp(Γτ)C(τ) ≡ − (Ψ−B(τ)ΨB(τ)0)

h(τ) =

µhq(τ)hp(τ)

¶≡ ντ − τΓmΓ

−1φ+ J(τ)Γ−1φ

J(τ) =

µJq(τ)Jp(τ)

¶≡ ΓmΓ−1 (B(τ)− I)

ν ≡µ

σ2q/2−σ2p/2

¶M(τ) =M1(τ) +M2(τ) +M

02(τ) +M3(τ)

M1(τ) ≡ ΓmΓ−1C(τ)(Γ−1)0Γ0m + ΓmΓ

−1ΣΩΣ(Γ−1)0Γ0mτ−ΓmΓ−1

©Γ−1 (B(τ)− I)ΣΩΣ+ΣΩΣ (B(τ)− I)0 (Γ−1)0ª (Γ−1)0Γ0m

M2(τ) ≡ ΓmΓ−1©Γ−1 (B(τ)− I)− τ I

ªΣΞΣm

M3(τ) ≡ ΣmΞΣmτQ(τ) = Γ−1 (B(τ)− I)ΣΞΣm +C(τ)(Γ−1)0Γ0m − Γ−1 (B(τ)− I)ΣΩΣ(Γ−1)0Γ0mMatrix Ψ is a function of the coefficients in matrices Γ, Σ and Ω 23.Equation (A.5) holds for any time interval τ and we can use it to derive

expectations for both economic growth and inflation over any time horizon.According to the CIR (1985a) general equilibrium framework, in equilibriumconsumption must be proportional to optimally invested wealth, c = δW , wherewealth is related to output and consumption as follows: dW = W (dq/q) −23Matrix Ψ is obtained as follows. Define with V the matrix formed using as columns the

eigenvectors of Γ. Consider the following transformation:

Ω∗ = V −1ΣΩΣ¡V −1

¢0Then divide each element of this matrix, ω∗ij , by (wi + wj), i.e., the sum of the correspondingeigenvalues of Γ. The elements ω∗∗ij ≡ ω∗ij/(wi + wj), form a transformed matrix Ω∗∗,whichallows us to derive the following matrix:

Ψ = V Ω∗∗V 0

The coefficients of this matrix are functions of the eigenvalues of Γ and the volatility coefficientsin Σ. As the matrix ΣΩΣ is positive definite and the stationarity condition of the modelrequires negative eigenvalues, then the matrix Ψ must be negative definite. On this derivation,see Langetieg (1980).

21

cdt. Therefore, by applying Ito’s Lemma, we can obtain an expression for theexpected value of logarithmic changes in consumption:

Et ln [c(t+ τ)/c(t)] = hq(τ)− δτ + Jq(τ)s(t)

Term structure of interest ratesIn equilibrium, the price at current time t of a nominal unit discount bond

with maturity date T , T = t+ τ , is given by:

F (t;T ) = Et

½exp(−δτ) c(t)

c(t+ τ)

p(t)

p(t+ τ)

¾(A.8)

It is well known (see, for example, Duffie (2001)) that the equilibrium priceof the nominal bond may also be written as:

F (t;T ) = bEtexp

− TZt

R (u) du

(A.9)

where R denotes the instantaneous nominal interest rate and bE the expectationunder the risk-adjusted probability measure. Using Girsanov’s theorem, we canlink equations (A.8) and (A.9) and demonstrate that, consistently with CIR(1985b), the instantaneous nominal interest rate is given by24:

R = r + π −µCovt

½dq

q,dp

p

¾+ V art

½dp

p

¾¶= ε+ β0s (A.10)

with: β0 ≡ ¡ 1 1 0 0 0¢and ε ≡ −σp

¡σqρqp + σp

¢. σq and σp denote

the volatility of the production process and the price level, respectively, in ma-trix Σm above, whereas ρqp is the correlation between the standard Brownianmotions of the production and the price level processes.In this gaussian setup the market prices of risk should be constant over time:

Λ0 ≡ Σ−1µCovt

½ds,

dq

q

¾+Covt

½ds,

dp

p

¾¶= ΞΣmι

with: ι0 ≡ ¡ 1 1¢.

However, following Duffee (2002), we relax this constraint by allowing marketprices of risk to be affine in the state variables:

Λ ≡ Λ0 + Λ1s (A.11)

24As shown by Fischer (1975) and, more extensively, by Benninga and Protopapadakis(1983), under uncertainty the traditional Fisher identity does not hold. In fact, in this casenominal interest rates must contain also two risk terms, which are related to the variabilityof money prices and to the purchasing power riskiness of nominal bonds. The first termderives from the uncertainty about the level of prices in the future and is essentially due tothe application of Jensen’s inequality (see Fischer (1975)). The second term is a risk premiumwhich reflects the covariance between future consumption and nominal bonds in real terms.This point is also developed in Breeden (1986). In the context of our model, the two risk termsof the Benninga-Protopapadakis formulation are represented by the sum of the variance ofstochastic inflation and the covariance between inflation and real growth.

22

where Λ0 is specified as above and Λ1 is a matrix of free parameters. There-fore, the matrix of mean reversion coefficients under the risk-adjusted probabil-ity measure is given by eΓ ≡ Γ − ΣΛ1. In order to preserve a structure of themodel in which θ and ξ are the stochastic central tendencies of r and π underthe risk-adjusted probability measure, we need to impose some constraints onthe form of matrix Λ1. Moreover, we make the assumption that there is no riskpremium on the shock variable α:

Λ1 ≡

λrr λrπ −λrr −λrπ 0λπr λππ −λπr −λππ 00 0 λθθ 0 00 0 0 λξξ 00 0 0 0 0

Given these assumptions on the factor risk premia, a closed form solution

for equilibrium nominal bond prices can be derived:

F (t;T ) = exp [f0(τ)− f(τ)s(t)] (A.12)

where25 : eΓ ≡ Γ−ΣΛ1f0(τ) ≡ −τε− β0eΓ−1 heΓ−1 (B(τ)− I)− τ I

i(φ−ΣΛ0)

+1

2f(τ)eΨf 0(τ) + 1

2β0eΓ−1ΣΩΣ³eΓ−1´0 βτ

−12β0·eΓ−1eΓ−1 (B(τ)− I) eΨ+ eΨ (B(τ)− I)0 ³eΓ−1´0 ³eΓ−1´0¸β

f(τ) ≡ β0eΓ−1 (B(τ)− I)The yield on the unit discount bond is an affine function of the state vari-

ables26 :

Y (t;T ) ≡ − lnF (t;T )T − t = κ0(τ) + κ(τ)s(t) (A.13)

where:

κ0(τ) ≡ −f0(τ)/τ , κ(τ) ≡ f(τ)/τSimilarly, we can derive the closed form solution for the price at current time

t of a real unit discount bond with maturity date T , T = t+ τ :

G(t;T ) = Et

½exp(−δτ) · c(t)

c(t+ τ)

¾= exp [g0(τ)− g(τ)s(t)] (A.14)

25Matrix eΨ is calculated as matrix Ψ above by substituting matrix Γ with matrix eΓ.26Given the specification of the factor risk premia, the model belongs to the class of the

”essentially affine” models introduced by Duffee (2002) (see also Dai and Singleton (2002) andDuarte (2003)), rather than the ”completely affine” models analyzed by Brown and Schaefer(1994b), Duffie and Kan (1996) and Dai and Singleton (2000).

23

where:

g0(τ) ≡ −β0eΓ−1 heΓ−1 (B(τ)− I)− τ Ii(φ−Σλ0)

+1

2g(τ)eΨg0(τ) + 1

2β0eΓ−1ΣΩΣ(eΓ−1)0βτ

−12β0·eΓ−1eΓ−1 (B(τ)− I) eΨ+ eΨ (B(τ)− I)0 ³eΓ−1´0 ³eΓ−1´0¸β

g(τ) ≡ β0eΓ−1 (B(τ)− I)

λ0 ≡ ΞΣmι, β0 ≡ ¡ 1 0 0 0 0

¢, ι0 ≡ ¡ 1 0

¢Real yields are also affine in the state variables:

y(t;T ) ≡ − lnG(t;T )T − t = ω0(τ) + ω(τ)s(t) (A.15)

where:

ω0(τ) ≡ −g0(τ)/τ , ω(τ) ≡ g(τ)/τ

Relation between real and nominal term structureIn order to work out the relation which links real and nominal bonds, we

can rewrite expression (A.8) for the nominal bond price in the following form:

F (t;T ) = exp(−δτ)·Covt

½c(t)

c(t+ τ),p(t)

p(t+ τ)

¾+Et

½c(t)

c(t+ τ)· p(t)

p(t+ τ)

¾¸(A.16)

which, from the definition of real bond price in equation (A.14) and from theimplicit relation between consumption and production, becomes:

F (t;T ) = Covt

½q(t)

q(t+ τ),p(t)

p(t+ τ)

¾+G(t;T ) · Et

½p(t)

p(t+ τ)

¾(A.17)

This equation represents the equilibrium relation between real and nominalbond prices and can be used to assess the influence on bond pricing of the covari-ance between real growth and inflation. Exploiting the fact that the variables inbraces are log-Normally distributed, we can transform equation (A.17) into thefollowing expression, which allows us to derive a neat relation between nominaland real bond prices:

F (t;T ) = G(t;T ) ·Et½

p(t)

p(t+ τ)

¾· exp

·Covt

½ln

q(t)

q(t+ τ), ln

p(t)

p(t+ τ)

¾¸(A.18)

Applying a logarithmic transformation to this expression, we obtain:

24

Y (t;T ) = y(t;T ) +1

τEt

½lnp(t+ τ)

p(t)

¾− 1

2τV art

½lnp(t+ τ)

p(t)

¾−1τCovt

½ln

q(t)

q(t+ τ), ln

p(t)

p(t+ τ)

¾(A.19)

which establishes a direct link between nominal and real yields.

B. Competing forecasting modelsIn this appendix, we report the parameter estimates of some models used

as a benchmark to evaluate the performance of the structural model in pre-dicting bond yields, inflation and GDP growth. These are represented by: (i)an essentially affine version of the Pennacchi (1991) constant-mean equilibriumterm structure model; (ii) the Ang, Piazzesi and Wei (2003) structural VARmodel; (iii) a 5-factor latent variables essentially affine term structure model ofthe Dai and Singleton (2000) - Duffee (2002) type; (iv) the C. Harvey (1989)term spread regression model; (v) a random walk model.

Pennacchi (1991) constant-mean equilibrium model (PCM)The Pennacchi (1991) two-factor model with constant long-term means and

no adjustment for the conditional mean of output growth is nested within theterm structure equilibrium framework developed in Appendix A. In fact, Pen-nacchi’s model assumes that the state variables driving the dynamics of theeconomy are the instantaneous real interest rate r and expected inflation rateπ following a joint elastic random walk process with constant long-term means.The dynamics of the price level and the production output is described bygaussian-type processes, as in equation (A.2), but with no shocks on the condi-tional mean of output growth.With respect to the original Pennacchi’s framework, we introduce more flexi-

bility in the model’s ability in fitting bond risk premia by allowing market pricesof risk to be affine in the state variables (see equation (A.11)). The rest of themodel can be derived following the same steps as in equations (A.3 - A.13)above.The model is estimated using a maximum likelihood - Kalman filter method.

The following tables report the estimated model parameters (in parentheses,asymptotic standard errors):

φ Γ Σ Λ ΩVariable r π (diag) r π π

r 0.0110 −0.4852 0.0084 0.0137 7.7182 −16.8697 0.8089(0.0016) (0.0710) (0.0163) (0.0008 (0.0231) (0.0282) (0.0283)

π 0.0064 −0.2970 −0.0027 0.0107 0.5325 −10.5024(0.0009) (0.0272) (0.0213) (0.0008) (0.0446) (0.0996)

25

Σm Ωm ΞVariable (diag) p q p

r 0.9999 −0.2647(0.0112) (0.0503)

π −0.3798 −0.5900(0.0496) (0.0771)

q 0.0196 −0.0985(0.0005) (0.0468)

p 0.0119(0.0004)

Ang, Piazzesi and Wei (2003) VAR model (APW)Ang, Piazzesi and Wei (APW, 2003) estimate the relation between the term

structure and macro variables in the context of a structural Vector Autore-gression (VAR) model, in which no-arbitrage restrictions are imposed on thedynamics of bond yields.In the original APW specification, the vector of state variables include the

short rate (3 month yield), the term spread (20-year minus 3-month yield) andlagged annualized quarterly GDP growth rate. With respect to their model,we add two more factors, which the principal components analysis in section Ihas shown to be relevant to jointly explain the dynamics of the term structure,inflation and output growth: the ”curvature” factor (3-month plus 20-year yieldminus twice the 5-year yield) and lagged annualized quarterly inflation rate.The five factors are assumed to follow a gaussian VAR with one lag:

st = a+Bst−1 +Φet (A.4)

Cross-equation restrictions stemming from the no-arbitrage condition areimposed and allow the model to endogenously produce expectations of futureinflation and GDP growth rates over any time horizon. Market prices of riskare assumed to be affine in the state variables.

Λt ≡ Λ0 + Λ1stThe model has a no-arbitrage closed form solution in which bond yields are

affine in the underlying state variables, with coefficients which are nonlinearfunction of the original model parameters. In estimating the parameters of themodel, we adopt the consistent two-step procedure used by APW27. In the firststep, we estimate the VAR parameters (the equivalent of equations (A.3) and(A.4) above) using a standard seemingly unrelated regression technique. In thesecond step, we fix these coefficients and maximize a log-likelihood functionto estimate the risk premia coefficients (vector Λ0 and matrix Λ1, with no re-strictions on the parameters). We obtain the following parameter estimates (inparentheses, asymptotic standard errors):

27 In order to check that we were correctly applying the APW two-step estimation procedure,we replicated their three-factor model estimation using the CRSP Fama-Bliss data set for thesample period 1964:Q1 - 2001:Q4. The estimated VAR coefficients are almost equal to thoseestimated by APW, whereas the estimated risk premia coefficients, which exhibit relativelyhigh standard errors in APW estimates, are slightly different in size, but similar in sign.

26

Variable a Bshort rate 0.0006 0.867 0.058 −0.051 0.028 0.137

(0.0024) (0.052) (0.056) (0.099) (0.030) (0.065)spread 0.0009 0.090 0.909 0.153 −0.022 −0.084

(0.0019) (0.040) (0.048) (0.080) (0.022) (0.051)curvature 0.0016 −0.077 −0.036 0.608 −0.001 0.036

(0.0019) (0.037) (0.048) (0.066) (0.018) (0.041)GDP growth 0.0372 −0.226 0.116 −0.172 0.251 −0.046

(0.0113) (0.160) (0.224) (0.371) (0.082) (0.161)inflation 0.0038 0.069 −0.054 0.150 0.016 0.822

(0.0024) (0.033) (0.045) (0.070) (0.026) (0.047)

Variable Φshort rate 0.0206

(0.0001)spread −0.0151 0.0080

(0.0001) (0.0009)curvature 0.0062 −0.0074 0.0112

(0.0002) (0.0009) (0.0001)GDP growth 0.0147 0.0066 −0.0129 0.0610

(0.0093) (0.0440) (0.0167) (0.0044)inflation 0.0020 0.0009 0.0005 −0.0026 0.0225

(0.0082) (0.0038) (0.0059) (0.0024) (0.0047)

Variable Λ0 Λ1short rate 0.086 −6.683 −0.236 11.541 0.358 4.132

(0.002) (0.050) (0.065) (0.315) (0.021) (0.054)spread −0.418 2.305 −6.121 2.387 4.228 6.062

(0.005) (0.270) (0.210) (1.465) (0.171) (0.145)curvature −0.191 0.353 −6.096 −16.964 3.165 1.509

(0.006) (0.293) (0.225) (1.555) (0.198) (0.141)GDP growth 0.889 11.372 −2.980 −12.357 6.817 6.287

(0.025) (1.395) (0.968) (6.687) (0.969) (0.736)inflation −0.731 9.063 −3.297 −1.830 2.965 −2.008

(0.019) (0.203) (0.331) (1.802) (0.071) (0.371)

5-factor latent variables affine term structure model (ATSM)This 5-factor latent variables model is specified as in the Dai and Singleton

(2000) canonical form extended by the Duffee (2002) essentially affine gener-alization. The dynamics of the zero-mean state variables is described by thefollowing process:

ds = −Γsdt+ dzThe instantaneous nominal interest rate and the market prices of risk are

affine in the state variables:

R = δ0 + δ0s

27

Λ ≡ Λ0 + Λ1s

The model provides an affine closed form solution for bond yields and canbe estimated using a maximum likelihood - Kalman filter approach. In order tolimit the number of parameters which must be estimated, we assume that bothmatrix Γ and Λ1 are diagonal28. Moreover, to better identify model parameters(see, for example, Dai and Singleton (2000) and Ang and Piazzesi (2003)), thecoefficient δ0 in the equation of the nominal interest rate is set to the sampleunconditional mean of the short rate (the 3-month yield): 0.0593. The follow-ing table reports the estimated model parameters (in parentheses, asymptoticstandard errors):

δ Γ Λ0 Λ1Variable (diag) (diag)s1 0.0421 −3.0120 −0.7002 3.9248

(0.0184) (0.2351) (0.4674) (0.2529)s2 0.0098 −0.0023 −0.1075 −0.0013

(0.0034) (0.1837) (0.1222) (0.1806)s3 −0.0008 −4.9693 −8.2461 −4.7750

(0.0011) (0.0786) (0.5733) (0.0887)s4 −0.0173 −0.0001 0.2108 0.1639

(0.0050) (0.0043) (0.3448) (0.0483)s5 0.0229 −0.0740 −2.9305 0.6761

(0.0104) (0.2004) (0.3608) (0.1727)

Harvey term spread regression model (HTS)The C. Harvey (1989) model is based on the following regression:

Q (t; t+ τ) = β0 + β1 (Yt(τL)− Yt(τS)) + etwhere Q (t; t+ τ) is the annualized real GDP rate of growth between time tand t + τ , Yt(τL) and Yt(τS) are a long-term yield and a short-term yield,respectively, observed at time t. We estimate the model using the 5-year nominalyield as long-term yield and the 1-year nominal yield as short-term yield29.

Random walk model (RW)The random walk model uses the current value of a variable as its forecast

for future periods.

28We also estimated the model by allowing these matrices to be lower triangular or full.However, the performance of the model in terms of goodness of fit does not change significantly.29Using yields with different maturities (10 years for the long-term yield and 3 or 6 months

for the short-term yield), the results do not change significantly.

28

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31

Table 1Sensitivity of term structure and macro factors to estimated

principal components

This table shows the sensitivity of changes in the level, the slope and thecurvature term structure factors and in the GDP growth and inflation rates tothe first five principal components estimated from a set of variables composedby changes in yields with maturities 3 months, 1 to 10 and 20 years and changesin the annualized quarterly GDP growth rate and inflation rate. The ”level”factor is the average of the 3-month, 5-year and 20-year maturity yields; the”slope” factor is the spread between the 20-year and the 3-month maturityyields; the ”curvature” factor is given by the sum of the 3-month and the 20-year maturity yields minus twice the 5-year maturity yield. The sample containsend-of-quarter observations for the period 1960:Q1 - 2002:Q4. In parentheses,Newey-West HAC standard errors. Coefficients and standard errors multipliedby 100.

Variable I PC II PC III PC IV PC V PC Const. R2

Level 0.201 0.018 0.115 −0.146 0.255 −0.004 0.99(0.002) (0.002) (0.003) (0.006) (0.012) (0.002)

Slope −0.103 −0.130 −0.498 0.753 −0.178 0.018 0.95(0.009) (0.017) (0.026) (0.027) (0.049) (0.011)

Curvature −0.060 0.127 0.412 −0.522 1.196 0.011 0.99(0.002) (0.005) (0.008) (0.008) (0.018) (0.004)

GDP growth −0.221 −2.578 2.931 1.365 −0.126 0.023 0.99(0.001) (0.002) (0.003) (0.004) (0.006) (0.001)

Inflation 0.028 0.917 0.553 0.547 −0.045 0.005 0.99(0.001) (0.001) (0.001) (0.001) (0.002) (0.001)

32

Table 2Specification tests

This table shows summary statistics on specification tests based on the max-imum likelihood - Kalman filter estimates of the state-space form (12-13). Thesample contains end-of-quarter observations for the period 1960:Q1 - 2002:Q4.Panel A compares the moments estimated by the model with the sample mo-ments of bond yields and annualized quarterly GDP growth and inflation rates.Data are expressed in basis points. The standard errors (S.E.) are calculatedusing GMM with the Newey-West correction. Panel B reports the p-values of aχ2 test based on the following point statistic: H = (h − h)0Σ−1h (h − h), whereh and h are the estimates of the unconditional moments provided by the sam-ple and the model, respectively, and Σh is the covariance matrix of the sampleestimates of the unconditional moments, estimated using GMMwith the Newey-West correction. We test for first and second moments of bond yields and yieldchanges, with maturity 3 months, 1, 3, 5, 10 and 20 years, and GDP growth andinflation rates, with time horizon 3, 6, 9 and 12 months. Panel C contains thep-value of a GMM-based test for the null hypothesis of first order serial corre-lation in the model scaled residuals and the p-values of a Wald statistic testingthe null hypothesis that n restrictions can be placed on the model parametervector: (i) constant central tendencies and no shock to the conditional mean ofoutput growth (the α factor is fixed at its zero long-term mean); (ii) constantcentral tendencies; (iii) no shock to the conditional mean of output growth; (iv)completely affine specification implying constant risk premia.

Panel A: model-estimated sample moments

Variable Mean VarianceModel Sample S.E. Model Sample S.E.

3-month yield 600 593 60 7.8 7.6 2.71-year yield 629 634 62 7.3 7.7 2.73-year yield 674 676 59 6.8 6.8 2.45-year yield 696 696 59 6.5 6.5 2.210 year yield 721 723 57 6.1 5.9 2.020-year yield 743 740 59 5.5 5.6 1.7

GDP growth rate 307 325 36 6.0 12.2 2.2inflation rate 364 375 55 5.6 6.2 1.7

Panel B: χ2 test on unconditional moments

Yields Changes in GDP growth Inflationyields rates rates

n = 6 n = 6 n = 4 n = 4Mean Variance Mean Variance Mean Variance Mean Variance

p-value 0.00 0.00 0.00 0.01 0.09 0.99 0.00 0.00

Panel C: χ2 test on residual correlation and Wald test on parameter restrictions

Residuals Constant means Constant No α Constantcorrelation and No α factor means factor risk premian = 8 n = 19 n = 17 n = 2 n = 6

p-value 0.00 0.00 0.00 0.00 0.00

33

Table 3Estimated parameter values

This table shows the parameter values of the state-space form (12-13) esti-mated by the maximum likelihood - Kalman filter method. Asymptotic standarderrors are in parentheses. The sample contains end-of-quarter observations forthe period 1960:Q1 - 2002:Q4.

φ Γ ΣVariable r π θ ξ α (diag)

r −0.5711 −0.1302 0.5711 0.1302 0.0249(0.0084) (0.0111) (0.0030)

π −0.0404 −0.0542 0.0404 0.0542 0.0149(0.0143) (0.0087) (0.0046)

θ 0.0026 −0.1234 0.0186(0.0008) (0.0077) (0.0059)

ξ 0.0015 −0.0205 0.0069(0.0010) (0.0088) (0.0066)

α −0.5403 0.0666(0.1831) (0.0049)

Λ ΩVariable r π θ ξ π θ ξ

r 4.7517 5.9388 −4.7517 −5.9388 −0.2629 0.4378 −0.0642(0.6778) (0.9407) (0.0158) (0.0174) (0.0205)

π 2.6679 12.6706 −2.6679 −12.6706 −0.1553 0.3334(0.6351) (0.4626) (0.0128) (0.0201)

θ 0 0 −2.2292 0 −0.3602(0.2207) (0.0119)

ξ 0 0 0 −2.9979(0.1691)

Σm Ωm ΞVariable (diag) p q p

r 0.4570 −0.4237(0.0681) (0.0476)

π −0.1130 −0.0455(0.0479) (0.0506)

θ 0.6704 0.2500(0.0201) (0.0587)

ξ 0.2931 0.4919(0.0600) (0.0570)

q 0.0034 0.6288(0.0025) (0.0634)

p 0.0028(0.0042)

34

Table 4Correlation between real and nominal variables

This table shows the correlation between innovations in real and nominalvariables - the instantaneous real interest rate r and expected inflation rate πand their time-varying central tendencies, θ and ξ, respectively, actual GDPgrowth rates, actual inflation rates - calculated for finite time intervals T − tusing the parameter values of the state-space form (12-13) estimated by themaximum likelihood - Kalman filter method. The sample period is 1960:Q1 -2002:Q4.

T − t πT θT ξT ln(qT /qt) ln(pT/pt)3 months rT −0.28 0.48 −0.08 0.43 −0.47

πT −0.15 0.33 −0.11 0.50θT −0.35 0.34 0.12ξT 0.07 0.58

ln(qT /qt) −0.01

T − t πT θT ξT ln(qT/qt) ln(pT/pt)1 year rT −0.31 0.58 −0.12 0.36 −0.35

πT −0.15 0.34 −0.11 0.82θT −0.36 0.23 −0.04ξT 0.00 0.43

ln(qT/qt) −0.13

T − t πT θT ξT ln(qT/qt) ln(pT /pt)3 years rT −0.34 0.73 −0.20 0.36 −0.29

πT −0.15 0.35 −0.14 0.85θT −0.36 0.27 −0.09ξT −0.05 0.35

ln(qT /qt) −0.16

35

Table 5Estimated risk premia

This table analyzes the properties of the instantaneous expected excess re-turns on τ-period zero coupon bonds implicit in the estimated model parametersfor the sample period 1960:Q1 - 2002:Q4. Panel A contains summary statisticson estimated instantaneous risk premia on each state variable. Panel B comparesthe predictive power of the model estimates for 1-year excess bond returns withthose provided by the Cochrane and Piazzesi (CP, 2002) model. The followingregressions are estimated:

ER(t+ 1) = γ0+5Xi=1

γi · fw(t; 0, i) + u(t+ 1) = (γ0 · fw(t)) + u(t+ 1)

ER(t+ 1;T ) = bT · (γ0 · fw(t)) + u(t+ 1;T ) , T = 2, 3, 4, 5

where ER(t + 1;T ) ≡ F (t+ 1;T − 1) / F (t;T )− 1 / F (t; 1) and ER(t + 1) isthe average of excess bond returns for T = 2, 3, 4, 5. F (t;T ) is the price of aT -maturity unit zero coupon bond and fw(t;T − 1, T ) is the forward rate. Thesame two regressions are also estimated by using as a predictor the estimatedmodel risk premia, RP (t;T ), T = 2, 3, 4, 5, instead of the forward rates. Inparentheses, Newey-West HAC standard errors correcting for overlapping ob-servations. The χ2 test is the Wald statistic for the null hypothesis that theslope coefficients are zero (p-value in parenthesis).

Panel A: Summary statistics on estimated instantaneous risk premia

r π θ ξ TotalMaturity Mean Mean Mean Mean Mean St. Dev.1 year 13 10 4 2 29 532-years 19 16 12 6 53 853 years 22 21 22 11 76 1064 years 23 24 32 18 96 1215 years 23 26 42 24 115 1336 years 23 28 51 31 133 1437 years 23 30 60 39 150 1518 years 22 31 68 46 167 1589 years 22 32 75 54 183 16510 years 22 32 82 61 198 171

Panel B: Excess bond returns forecasting regressions

γ0 γ1 γ2 γ3 γ4 γ5 R2

χ2(5)CP −0.040 −2.64 0.82 4.24 1.93 −3.83 0.32 62.5

(0.016) (0.58) (1.23) (1.38) (1.75) (1.50) (0.00)Model −0.017 −148.89 −42.80 214.95 −18.32 −66.38 0.35 35.3

(0.013) (94.03) (92.25) (85.23) (70.80) (50.88) (0.00)

Maturity (years) CP ModelT bT R

2bT R

2

2 0.49 (0.07) 0.34 0.49 (0.08) 0.373 0.86 (0.14) 0.34 0.85 (0.15) 0.354 1.20 (0.19) 0.35 1.19 (0.21) 0.365 1.45 (0.26) 0.33 1.47 (0.27) 0.35

36

Table 6In-sample prediction of inflation and GDP growth

This table compares the in-sample predictive accuracy for inflation and GDPgrowth rates of the macro model and some competing models, i.e., the Pennacchi(1991) constant-mean equilibrium model (PCM), the Ang, Piazzesi and Wei(2003) VAR model (APW), a random walk model (RW) and the C. Harvey(1989) term spread regression model (HTS) (see Appendix B).Panel A and panel B of the table show, respectively, the estimation results

of the following regressions:

X (t; t+ τ) = β0 + β1 ·E X (t; t+ τ)+ e(t), X ≡ P,Q

where: P (t; t+ τ) and Q (t; t+ τ) are the annualized logarithmic change inthe GDP deflator and the real GDP level, respectively, between time t andt + τ ; E P (t; t+ τ) and E Q (t; t+ τ) are the corresponding expectationsgenerated by the model estimates. The sample period is 1960:Q1 - 2002:Q4. Inparentheses, Newey-West HAC standard errors.

Panel A: in-sample predictions for inflation rates

Model PCM APW RWTime horizon β1 R

2β1 R

2β1 R

2β1 R

2

3 months 1.03 0.96 0.73 0.28 1.00 0.79 1 0.77(0.02) (0.17) (0.04)

6 months 1.01 0.97 0.69 0.27 1.03 0.80 1 0.77(0.02) (0.17) (0.06)

9 months 0.99 0.96 0.66 0.25 1.06 0.80 1 0.76(0.03) (0.17) (0.07)

1 year 0.97 0.94 0.63 0.22 1.10 0.79 1 0.75(0.04) (0.18) (0.09)

2 years 0.89 0.82 0.47 0.12 1.19 0.67 1 0.61(0.08) (0.18) (0.14)

Panel B: in-sample predictions for GDP growth rates

Model PCM APW HTSTime horizon β1 R

2β1 R

2β1 R

2β1 R

2

3 months 1.38 0.94 −0.78 0.06 0.99 0.14 0.95 0.05(0.05) (0.27) (0.21) (0.36)

6 months 1.00 0.69 −0.98 0.14 1.22 0.24 1.12 0.11(0.05) (0.28) (0.21) (0.36)

9 months 0.83 0.53 −1.04 0.17 1.24 0.27 1.08 0.13(0.06) (0.27) (0.22) (0.34)

1 year 0.68 0.40 −1.06 0.20 1.25 0.29 1.03 0.15(0.07) (0.27) (0.23) (0.32)

2 years 0.39 0.16 −1.12 0.25 1.12 0.27 0.94 0.20(0.08) (0.26) (0.22) (0.24)

37

Table 7In-sample bond pricing estimation errors

This table shows summary statistics on in-sample estimation errors on bondyields for the structural model and the competing models, i.e., the Pennacchi(1991) constant-mean equilibrium model (PCM) and the Ang, Piazzesi and Wei(2003) VAR model (APW). As a benchmark, the predictions provided by a 5-factor latent variables essentially affine term structure model (ATSM) of the Daiand Singleton (2000) - Duffee (2002) type are used. An error is defined as thedifference between the fitted variable and its observed value and is measuredin basis points. Mean of the errors and root mean squared errors (RMSE)are calculated. The sample contains end-of-quarter observations for the period1960:Q1 - 2002:Q4.

Model PCM APW ATSMMaturity Mean RMSE Mean RMSE Mean RMSE Mean RMSE

3 months 7.20 18.44 8.58 30.11 0.00 0.00 0.07 1.931 year −5.66 20.92 −8.85 22.22 0.28 27.08 −0.07 5.882 years −4.46 15.52 −9.00 23.79 2.07 20.66 −0.68 7.953 years −2.10 10.34 −5.38 22.72 2.66 14.43 −0.48 6.444 years −1.20 7.52 −2.18 20.57 0.51 8.13 −0.78 5.295 years 0.17 6.90 1.73 19.27 −1.10 5.22 0.17 4.716 years −0.90 8.14 3.06 17.06 −4.23 8.39 −0.76 5.387 years −1.34 8.93 4.68 15.82 −5.40 10.85 −0.72 6.088 years −1.80 9.51 5.87 14.67 −5.24 12.17 −0.51 6.659 years −1.98 10.15 6.88 14.18 −3.66 12.76 0.03 7.4710 years −2.25 10.62 7.33 13.85 −1.34 13.05 0.44 7.9820 years 3.33 11.19 −3.23 25.04 0.67 9.86 0.25 7.26

38

Table 8Real and expected inflation term structures

This table shows summary statistics on the implicit unobservable real andexpected inflation term structures estimated by the structural model for thesample period 1960:Q1 - 2002:Q4. We compare the moments estimated by themodel with the sample moments of ex-post realized real yields and inflationrates with maturity from 1 to 5 years. Data are expressed in basis points.The standard errors (S.E.) are calculated using GMM with the Newey-Westcorrection. The table also reports the p-values of a χ2 test based on the followingpoint statistic: H = (h − h)0Σ−1h (h − h), where h and h are the estimates ofthe unconditional moments provided by the sample and the model, respectively,and Σh is the covariance matrix of the sample estimates of the unconditionalmoments, estimated using GMM with the Newey-West correction. The lastcolumn of the table contains the correlation between the series estimated by themodel and the corresponding ex-post realized values.

Mean Variance CorrelationModel Sample S.E. Model Sample S.E.

Real interest rates1 year 257 260 52 3.9 5.7 1.7 0.972 years 282 290 60 3.6 7.1 2.1 0.933 years 297 306 66 3.3 8.1 2.3 0.914 years 303 314 72 3.2 8.9 2.5 0.885 years 305 319 76 3.0 9.4 2.5 0.86

p-value H statistic 0.00 0.99Expected inflation rates

1 year 366 375 55 5.5 5.6 1.6 0.972 years 374 381 55 5.4 5.2 1.4 0.913 years 380 387 54 5.3 4.8 1.2 0.844 years 385 393 53 5.2 4.5 1.1 0.795 years 391 398 53 5.2 4.3 1.0 0.74

p-value H statistic 0.00 0.08

39

Table 9Out-of-sample forecasts of inflation and GDP growth rates

This table summarizes the error properties in forecasting inflation and GDPgrowth rates over different time horizons of the structural model and some alter-native models: the Pennacchi (1991) constant-mean equilibrium model (PCM);the Ang, Piazzesi and Wei (2003) VAR model (APW); a random walk model(RW); the C. Harvey (1989) term spread regression model (HTS); the Surveyof Professional Forecasters published by the Federal Reserve Bank of Philadel-phia. The forecasts are obtained by using the preceeding thirty years of quar-terly data to estimate the parameters needed to forecast subsequent inflationand GDP growth rates over the different horizons. Proceeding recursively, wecompute forecasts for all the competing models from 1990:Q1 to 2002:Q4 (52observations) and compare them to realized inflation and GDP growth rates.An error is defined as the difference between the forecast and the realized value.Root mean squared errors (RMSE) are expressed in basis points. In parenthe-ses the asymptotic p-values of the Diebold-Mariano statistic, which tests thenull hypothesis of no difference in the accuracy of the forecasts provided by thecompeting models with respect to those of the structural model.

Panel A: forecast errors for inflation rates

Model PCM APW RW SurveyHorizon Mean RMSE Mean RMSE Mean RMSE Mean RMSE Mean RMSE3 months 9 47 68 121 49 97 5 88 44 80

(0.00) (0.00) (0.00) (0.00)6 months 14 42 74 113 69 100 8 77 −75 91

(0.00) (0.00) (0.00) (0.00)9 months 18 42 77 111 86 110 11 75 −113 124

(0.00) (0.00) (0.00) (0.00)1 year 21 43 81 112 100 120 13 71 −133 141

(0.00) (0.00) (0.00) (0.00)2 years 37 60 104 124 156 167 23 80

(0.00) (0.00) (0.00)

Panel B: forecast errors for GDP growth rates

Model PCM APW HTS SurveyHorizon Mean RMSE Mean RMSE Mean RMSE Mean RMSE Mean RMSE3 months 1 178 −105 276 115 249 97 271 −39 214

(0.00) (0.00) (0.00) (0.03)6 months 0 153 −105 243 123 220 107 253 −154 236

(0.00) (0.01) (0.00) (0.02)9 months −6 154 −108 234 120 208 100 236 −200 261

(0.00) (0.02) (0.00) (0.01)1 year −12 152 −113 221 113 192 91 219 −227 273

(0.00) (0.05) (0.01) (0.00)2 years −16 117 −114 181 85 143 65 169

(0.01) (0.05) (0.02)

40

Table 10Out-of-sample forecasts of bond yields

This table compares the predictive accuracy in forecasting bond yields one,two and four quarters ahead of the structural model with respect to the bench-mark 5-factor latent variables essentially affine term structure model (ATSM) ofthe Dai and Singleton (2000) - Duffee (2002) type. The forecasts are obtained byusing the preceeding thirty years of quarterly data to estimate the parametersneeded to forecast bond yields. Proceeding recursively, we compute forecastsfrom 1990:Q1 to 2002:Q4 (52 observations) and compare them to realized yields.Forecast errors are measured by a model’s corresponding root mean squared er-rors (RMSE) and are expressed in basis points. In parentheses the asymptoticp-values of the Diebold-Mariano statistic, which tests the null hypothesis of nodifference in the accuracy of the forecasts provided by the structural model withrespect to those of the benchmark ATSM.

Forecast horizon 3 months 6 months 1 yearMaturity ATSM Model ATSM Model ATSM Model3 months 57 95 90 141 160 189

(0.03) (0.03) (0.19)3 years 73 79 103 115 145 149

(0.40) (0.34) (0.86)5 years 68 68 97 99 128 128

(0.88) (0.84) (0.99)7 years 61 58 88 86 116 114

(0.36) (0.72) (0.86)10 years 54 50 78 75 104 105

(0.17) (0.53) (0.87)20 years 40 48 59 70 79 100

(0.02) (0.06) (0.07)

41

Figure 1Response function to innovations in the state variables

This figure shows the estimated dynamics of the state variables in responseto one standard deviation innovation in one of them. Impulse response functionsare calculated using the parameter values of the state-space form (12-13) esti-mated by the maximum likelihood - Kalman filter method. The sample periodis 1960:Q1 - 2002:Q4.

Panel A: innovation in the real interest rate

-0.01

0.00

0.01

0.02

0.03

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20years

real interest rate expected inflation rate central tendency real interest rate central tendency expected inflation ra

Panel B: innovation in the expected inflation interest rate

-0.01

0.00

0.01

0.02

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20years

real interest rate expected inflation rate central tendency real interest rate central tendency expected inflation rat

Panel C: innovation in the central tendency of the real interest rate

-0.01

0.00

0.01

0.02

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20years

real interest rate expected inflation rate central tendency real interest rate central tendency expected inflation ra

Panel D: innovation in the central tendency of the expected inflation rate

-0.01

0.00

0.01

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20years

real interest rate expected inflation rate central tendency real interest rate central tendency expected inflation ra

42

Figure 2Kalman filter estimates of the unobservable state variables

This figure shows the Kalman filter estimated series of the unobservablestate variables for the sample period 1960:Q1 - 2002:Q4. Panel A illustratesthe estimated instantaneous real interest rate, its time-varying central tendencyand the realized short-term real interest rate, which is obtained by subtractingthe realized annualized 3-month inflation rate from the nominal 3-month yield.Panel B shows the estimated instantaneous expected inflation rate, its time-varying central tendency and the annualized 3-month inflation rate realized aquarter ahead.

Panel A: real interest rate

-0.06

-0.03

0

0.03

0.06

0.09

0.12

Q1 60

Q1 62

Q1 64

Q1 66

Q1 68

Q1 70

Q1 72

Q1 74

Q1 76

Q1 78

Q1 80

Q1 82

Q1 84

Q1 86

Q1 88

Q1 90

Q1 92

Q1 94

Q1 96

Q1 98

Q1 00

Q1 02

real interest rate central tendency real interest rate ex-post realized real interest rate

Panel B: expected inflation interest rate

0

0.02

0.04

0.06

0.08

0.1

0.12

Q1 60

Q1 62

Q1 64

Q1 66

Q1 68

Q1 70

Q1 72

Q1 74

Q1 76

Q1 78

Q1 80

Q1 82

Q1 84

Q1 86

Q1 88

Q1 90

Q1 92

Q1 94

Q1 96

Q1 98

Q1 00

Q1 02

expected inflation rate central tendency expected inflation rateex-post realized inflation rate

43

Figure 3Impact on the term structure of innovations in the state variables

This figure shows the effect on nominal yields of one standard deviationinnovation in the unobservable state variables. Impulse response functions arecalculated using the parameter values of the state-space form (12-13) estimatedby the maximum likelihood - Kalman filter method. The sample period is1960:Q1 - 2002:Q4.

0

0.005

0.01

0.015

0.02

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20years

real interest rate expected inflation rate central tendency real interest rate central tendency expected inflation rate

44