Unspanned macroeconomic factors in the yield curve

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Unspanned macroeconomic factors in the yield curve Laura Coroneo, Domenico Giannone, and Michele Modugno 2014-57 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Unspanned macroeconomic factors in the yield curve Laura Coroneo Domenico Giannone University of York Luiss University of Rome, ECARES, EIEF and CEPR Michele Modugno Board of Governors of the Federal Reserve System July 30, 2014 Abstract Inthispaper,weextractcommonfactorsfromacross-sectionofU.S.macro-variablesandTreasury zero-coupon yields. We find that two macroeconomic factors have an important predictive content for government bond yields and excess returns. These factors are not spanned by the cross-section of yields and are well proxied by economic growth and real interest rates. JEL classification codes: C33, C53, E43, E44, G12. Keywords: Yield Curve; Government Bonds; Factor models; Forecasting. We thank Carlo Altavilla, Andrea Carriero, Valentina Corradi, Rachel Griffith, Matteo Luciani, Denise Osborn, Jean-Charles Wijnandts and two anonymous referees for useful comments. We also thank seminar participants at HEC Montreal, Federal Reserve Bank of Saint Louis, the 2012 International Conference on Computing in Economics and Finance, the 2012 European Meetings of the Econometric Society, the University of York, the cemmap, UCL and Bank of England workshop on Frontiers of Macroeconometrics andthe2013ViennaWorkshoponHighDimensionalTimeSeries. Anyremainingerrorsareourown. Laura Coroneo gratefully acknowledges the support of the ESRC grant ES/K001345/1 and Domenico Giannone was supported by the “Action de recherche concert´e” contract ARC-AUWB/2010-15/ULB-11 and by the IAP research network grant nr. P7/06 of the Belgian government (Belgian Science Policy). The opinions in this paper are those of the authors and do not necessarily reflect the views of the Board of Governors of the Federal Reserve System.

1 Introduction Governmentbond yieldswith different maturitiesand macroeconomicvariablesare bothcharacterized by a high degree of comovement, indicating that the bulk of their dynamics is driven by a few common forces. Three common factors, usually interpreted as the level, slope and curvature of the yield curve, can explain changes and shifts of the entire cross-section of yields, see Litterman and Scheinkman (1991). Although there is less consensus on the number and nature of macroeconomic factors, two factors, one nominal and one real, summarize well the dynamics of a large variety of macroeconomic indicators for the United States, see Sargent and Sims (1977), Giannone, Reichlin and Sala (2005) and Watson (2005). Macroeconomic factors and yield curve factors are also characterized by a strong interaction. The short end of the yield curve moves closely to the policy instrument under the direct control of the central bank, which responds to changes in inflation, economic activity, or other economic conditions, see Taylor (1993). The average level of the yield curve is usually associated with the inflation rate and the spread between long and short rates with temporary business cycles conditions, see Diebold, Rudebusch and Aruoba (2006). For these reasons, macroeconomic information have been shown to help forecasting future interest rates and excess bond returns, see Ang and Piazzesi (2003), M¨onch (2008), De Pooter, Ravazzolo and van Dijk (2007), Favero, Niu and Sala (2012) and Ludvigson and Ng (2009). In this paper, we aim at identifying the factors summarizing macroeconomic information that is notspannedbythetraditionalyieldcurvefactors. Theeconomicliteraturesofarhasnotaddressed this problem since in existing studies macroeconomic factors are either proxied by preselected observable variables, see Bianchi, Mumtaz and Surico (2009), Dewachter and Lyrio (2006), Diebold et al. (2006), Joslin, Priebsch and Singleton (2014) and Wright (2011), or extracted from a large set of macroeconomic indicators and treated separately from the yield curve factors, see Ang and Piazzesi (2003), Favero et al. (2012), Ludvigson and Ng (2009), M¨onch (2008) and M¨onch (2012). We estimate a macro-yield model that treats macroeconomic factors as unobservable com- 1

ponents that we extract simultaneously with the traditional yield curve factors. The latter are identified by constraining the loadings to follow the smooth pattern proposed by Nelson and Siegel (1987). More specifically, our empirical model is a Dynamic Factor Model (DFM) for Treasury zero-coupon yields and a representative set of macroeconomic variables with restrictions on the factor loadings. Estimation is performed using a Quasi-Maximum Likelihood approach, as proposed by Doz, Giannone and Reichlin (2012). This procedure is easily implementable using the Kalman smoother and the Expectation Maximization algorithm. The estimator has been shown to be feasible when the number of variables is large, and robust to non Gaussianity and to the presence of weak cross-sectional correlation among the idiosyncratic terms. We validate the model by assessing the forecasting ability for yields and excess returns of US government bonds. Using monthly U.S. data from January 1970 to December 2008, we find that a significant component of macroeconomic information is not captured by the yield curve factors and, at the same time, is unspanned by the yield curve, in the sense that it does not affect contemporaneously the cross-section of yields. The unspanned macroeconomic information is driven by two factors that are well proxied by economic growth and real interest rates. These factors have substantial predictive information for bond yields and excess bond returns, in spite of the fact that they do not affect contemporaneously the shape of the yield curve. The macro-yields model explains up to 55% of the variation in excess bond returns and outperforms all existing models in forecasting bond yields and excess returns. Thepaperisorganizedasfollows. Section2presentsthemacro-yieldsmodel. Section3describes the data, the estimation procedure and the information criteria used for model selection. Section 4 describes the empirical results and in-sample validation of the model. Section 5 reports out of sample results for yields and excess bond returns and Section 6 concludes. 2

2 The macro-yields model The macro-yields model that we propose is a dynamic factor model for the joint behavior of government bond yields and macroeconomic indicators. The cross-section of yields is described by the traditionallevel, slopeandcurvaturefactors. Macroeconomicvariablesloadonboththeyieldcurve factors as well as on some additional macro factors, that capture the information in macroeconomic variables over and above the yield curve factors. In addition, these additional macro factors are assumed to not provide any information about the contemporaneous shape of the yield curve. In practice, the level, slope and curvature implied by the Nelson and Siegel (1987) model are assumed to be spanned by both the bond yields and macroeconomic variables. The additional macro factors, instead, are contemporaneously loaded only by the macroeconomic variables and, thus, are unspanned by the cross-section of yield. The joint dynamics of the factors is an unrestricted vector autoregression and the idiosyncratic components follow independent univariate autoregressions. In what follows we detail on each of the points. More specifically, we assume that yields on bonds with different maturities are driven by three common factors. Denoting by y the N ×1 vector of yields with N different maturities at time t, t y y we have y = a + Γ Fy +vy, (1) t y yy t t where Fy is a 3 × 1 vector containing the latent yield-curve factors at time t, Γ is a N × 3 t yy y matrix of factor loadings, and vy is an N ×1 vector of idiosyncratic components. The yield curve t y factors Fy are identified by constraining the factor loadings to follow the smooth pattern proposed t by Nelson and Siegel (1987) (hereafter NS) (cid:20) 1−e−λτ 1−e−λτ (cid:21) a = 0; Γ(τ) = 1 −e−λτ ≡ Γ (τ) , (2) y yy λτ λτ NS (τ) where Γ is the row of the matrix of factor loadings corresponding to the yield with maturity τ yy months and λ is a decay parameter of the factor loadings. Diebold and Li (2006) show that this 3

functional form of the factor loadings, implies that the three yield curve factors can be interpreted as the level, slope and curvature of the yield curve. Indeed, the loading equal to one on the first factor, for all maturities, implies that an increase in this factor increases all yields equally, shifting the level of the yield curve. The loadings on the second factor are large for short maturities, decaying to zero for the long ones. Accordingly, an increase in the second factor increases the slope of the yield curve. Loadings on the third factor are zero for the shortest and the longest maturities, reaching the maximum for medium maturities. Therefore, an increase in this factor augments the curvature of the yield curve. The specific shape of the loadings depends on the decay parameter λ, which we calibrate to the value that maximizes the loading on the curvature factor for the yields with maturity 30 months, as in Diebold and Li (2006). Given these particular functional forms for the loadings on the three yield curve factors, one can summarize movements in the term structure of interest rates into three factors which have a clear-cut interpretation. The NS factors are just linear combinations of yields. The level factor can beproxiedbythelongtermyield,theslopebythespreadbetweenthelongandshortmaturityyield (first derivative) and the curvature by sum of the spreads between a medium and a long term yield, and between a medium and the short term yield (second derivative), see Diebold and Li (2006).1 Due to its flexibility and parsimony, the NS model accurately fits the yield curve and performs well in out-of-sample forecasting exercises, as shown by Diebold and Li (2006) and De Pooter et al. (2007). For these reasons, fixed-income wealth managers in public organizations, investment banks and central banks rely heavily on NS type of models to fit and forecast yield curves, see BIS (2005), ECB(2008), Gu¨rkaynak, SackandWright(2007)andCoroneo, NyholmandVidova-Koleva(2011). Macroeconomic variables, are assumed to be potentially driven by two sources of co-movement, the yield curve factors Fy and macro specific factors. Denoting by x the N ×1 vector of macroet t x conomic variables at time t, we have x = a +Γ Fy +Γ Fx+vx, (3) t x xy t xx t t 1Similar proxies are used by Ang, Piazzesi and Wei (2006) and Duffee (2011a). 4

where Fx is an r × 1 vector of macroeconomic latent factors, Γ is a N × 3 matrix of factor t xy x loadings on the yield curve factors, Γ is a N ×r matrix of factor loadings on the macro factors, xx x and vx is an N ×1 vector of idiosyncratic components. t x The yield curve and the macroeconomic factors are extracted by estimating (1) and (3) simultaneously           y 0 Γ Γ Fy vy  t  =  + yy yx   t  + t, Γ = Γ , Γ = 0, (4)           yy NS yx x a Γ Γ Fx vx t x xy xx t t where Γ is defined according to (2). NS The joint dynamics of the yield curve and the macroeconomic factors follow a VAR(1)               Fy µ A A Fy uy uy Q Q  t  =  y + yy yx  t−1 + t,  t ∼ N 0, yy yx . (5)               Fx µ A A Fx ux ux Q Q t x xy xx t−1 t t xy xx The idiosyncratic components collected in v = [vy vx](cid:48) are modelled to follow independent t t t autoregressive processes v = Bv +ξ , ξ ∼ N(0,R) (6) t t−1 t t where B and R are diagonal matrices, implying that the common factors fully account for the joint correlation of the observations. The residuals to the idiosyncratic components of the individual variables, ξ , and the innovations driving the common factors, u , are assumed to be normally t t distributed and mutually independent. This assumptions implies that the common factors are not allowed to react to variable specific shocks. The assumptions of Gaussianity and of independence among idiosyncratic components might be sources of miss-specification. It is hard to relax these restrictions since they are necessary to retain parsimony, insure identification of the common and idiosyncratic components and limit computational complexity. However, Doz et al. (2012) have shown that, if the factor structure is strong, the Maximum Likelihood estimates are robust not only to non Gaussianity but also to the 5

presence of limited correlations among idiosyncratic components. Allowing Γ to be different from zero is crucial to insure that the macroeconomic factors Fx xy t capture only those source of co-movement in the macroeconomic variables that are not already spanned by the yield curve factors. Existing studies, instead, have imposed a block-diagonal structure of the factor loadings (Γ = 0 and Γ = 0), either explicitly, as in M¨onch (2012),2 either xy yx implicitly by extracting the macro factors exclusively from macroeconomic variables, as in Ludvigson and Ng (2009). Assumingthatmacroeconomicfactorsdonotprovideanyinformationaboutthecontemporaneous shape of the yield curve (Γ = 0) restricts the macroeconomic factors Fx to be unspanned not yx t only by the yield factors but also by the entire cross-section of yields. This restriction is expected to be immaterial since, as stressed above, the yield factors Fy are notoriously effective at fitting t the entire yield curve. In this remainder of the paper we will maintain the restriction Γ = 0 and leave Γ unyx xy restricted, unless otherwise mentioned. The block-diagonal model (Γ = 0, Γ = 0) and the xy yx unrestricted model (Γ (cid:54)= 0) are considered in Appendix D. xy 3 Estimation and preliminary results 3.1 Data We use monthly U.S. Treasury zero-coupon yield curve data spanning the period January 1970 to December 2008. The bond yield data are taken from the Fama-Bliss dataset available from the Center for Research in Securities Prices (CRSP) and contain observations on three months and one through five-year zero coupon bond yields. The macroeconomic dataset consists of 14 macroeconomic variables, which include five inflation measures, seven real variables, the federal funds rate and a money indicator. Appendix B contains a complete list of the macroeconomic 2Mo¨nch (2012) estimates his model with Bayesian methods since he claims that “Estimation of the model via maximumlikelihoodtechniquesisinfeasibleduetothelargenumberofmodelparameters”(pp. 578)onthecontrary of what is shown in Doz et al. (2012). 6

variables along with the transformation applied to ensure stationarity. Following Ang and Piazzesi (2003), De Pooter et al. (2007), Diebold et al. (2006) and M¨onch (2008), we use annual growth rates for all variables, except for capacity utilization, the federal funds rate, the unemployment rate and the manufacturing index which we keep in levels.3 3.2 Estimation Equations (4)–(6) describe a restricted state-space model with autocorrelated idiosyncratic components for which maximum likelihood estimators of the parameters are not available in closed form. Conditionally on the factors, the model reduces in a set of linear regressions. As consequence, the Maximum Likelihood estimates can be easily computed using the Expectation Maximization (EM) algorithm, as described in detail in Appendix A.4 We initialize the yield curve factors with the NS factors using the two-steps OLS procedure introduced by Diebold and Li (2006). We then project the macroeconomic variables on the NS factors and use the principal components of the residuals of this regression to initialize the unspanned macroeconomic factors. Γ is restricted to be equal to the NS loadings. All the other yy parameters are initialized with the OLS estimates obtained using the initial guesses of yield and macro factors described above. Given the initial parameters, a new set of factors is obtained using the Kalman smoother. If we stop at this stage, we have the two-step procedure of Doz, Giannone and Reichlin (2011). Roughly speaking, Maximum Likelihood estimates are obtained by iterating these two steps until convergence, see Doz et al. (2012).5 For comparison, we also estimate an only-yields model, which uses only the information contained in the yields. This is a restricted version of the macro-yields model in equations (4)–(6) with 3Since the selection of variables has an element of arbitrariness, we have performed robustness checks with an alternativedatabasesconstructedbyBanbura,Giannone,ModugnoandReichlin(2012)thatincludesallthevariables thatareconstantlymonitoredbymarketparticipants. Results,reportedinaseparateAppendix,showthatthemain findings are confirmed. 4Using the Expectation Conditional Restricted Maximization (ECRM) algorithm is also possible to estimate λ, but, despite the increase in the computation burden, the empirical results remain qualitatively similar to those obtained by setting λ to the value that maximizes the loading of the the yields with maturity 30 months on the curvature factor. 5In Appendix E, we compare the initial estimates and find that the Maximum Likelihood provide significant improvements. 7

Q = A = Γ = 0 and can hence be estimated using the same procedure. yx yx xy 3.3 Model selection The macro-yields model decomposes variations in yields and macroeconomic variables into yield curve factors, unspanned macroeconomic factors and idiosyncratic noises. The yield curve factors are identified as the NS factors which have a clear interpretation as level, slope and curvature. However, the true number of unspanned macroeconomic factors is unknown. We select the optimal number of factors using an information criteria approach. The idea is to choose the number of factors that maximizes the general fit of the model using a penalty function to account for the loss in parsimony. BaiandNg(2002)deriveinformationcriteriatodeterminethenumberoffactorsinapproximate factor models when the factors are estimated by principal components. They also show that their IC informationcriterioncanbeappliedtoanyconsistentestimatorofthefactorsprovidedthatthe 3 penalty function is derived from the correct convergence rate. For the quasi-maximum likelihood estimator, Doz et al. (2012) show that it converges to the true value at a rate equal to (cid:26)√ N (cid:27) C∗2 = min T, (7) NT logN where N and T denote the cross-section and the time dimension, respectively. Thus, a modified Bai and Ng (2002) information criterion that can be used to select the optimal number of factors when estimation is performed by quasi-maximum likelihood is as follows logC∗2 IC∗(s) = log(V(s,Fˆ∗ ))+s g(N,T), g(N,T) = NT (8) (s) C∗2 NT where s denotes the number of factors, Fˆ are the estimated factors and V(s,Fˆ∗ ) is the sum (s) (s) of squared idiosyncratic components (divided by NT) when s factors are estimated. The penalty function g(N,T) is a function of both N and T and depends on C∗2 , the convergence rate of the NT estimator, in our case given by (7). 8

Table 1: Model selection Number of factors IC∗ V 3 0.02 0.44 4 -0.03 0.31 5 -0.11 0.22 6 0.01 0.18 7 0.23 0.17 8 0.43 0.16 This table reports the information criterionIC∗,asshownin(8)and(7),and the sum of the variance of the idiosyncraticcomponents(dividedbyNT),V, when different numbers of factors are estimated. To select the number of factors in the macro-yields model, we estimate the macro-yields model in equations (4)–(6) allowing from three up to a total of eight factors, where the first three are identified as the yield curve factors and the others are unspanned macro factors. Table 1 reports theinformationcriterion, asshowninEquation(8), andthesumofthevarianceoftheidiosyncratic components for these different specifications of the macro-yields model. The information criterion selects the model with five factors, i.e. three yield curve factors plus two unspanned factors. This is also confirmed by the fact that the strongest reduction in the sum of the variances of the idiosyncratic components is obtained passing from the four to the five factors specification. Thus our macro-yields model is a latent factor model with three factors that explain the cross-section of yields and two unspanned macroeconomics factors. 4 In sample results 4.1 Model fit Table 2 reports the share of variance of the macroeconomic variables explained by the macro-yields factors. Results show that, as expected, the yield curve factors explain most of the variance of the federal funds rate and the yields at different maturities. They also explain the part of the 9

Table 2: Cumulative variance of yields and macro variables explained by the macro-yields factors Level Slope Curv UM1 UM2 Government bond yield with maturity 3 months 0.59 0.94 1.00 1.00 1.00 Government bond yield with maturity 1 year 0.61 0.83 0.99 0.99 0.99 Government bond yield with maturity 2 years 0.65 0.78 0.99 0.99 0.99 Government bond yield with maturity 3 years 0.70 0.79 1.00 1.00 1.00 Government bond yield with maturity 4 years 0.74 0.80 0.99 0.99 0.99 Government bond yield with maturity 5 years 0.78 0.82 0.99 0.99 0.99 Average Hourly Earnings: Total Private 0.07 0.29 0.33 0.33 0.67 Consumer Price Index: All Items 0.19 0.48 0.48 0.50 0.85 Real Disposable Personal Income 0.00 0.02 0.03 0.34 0.36 Effective Federal Funds Rate 0.53 0.93 0.96 0.96 0.97 House Sales - New One Family Houses 0.00 0.19 0.19 0.23 0.23 Industrial Production Index 0.02 0.02 0.03 0.69 0.69 M1 Money Stock 0.17 0.25 0.25 0.25 0.31 ISM Manufacturing: PMI Composite Index (NAPM) 0.03 0.05 0.05 0.61 0.65 Payments All Employees: Total nonfarm 0.00 0.02 0.10 0.70 0.70 Personal Consumption Expenditures 0.16 0.23 0.33 0.46 0.78 Producer Price Index: Crude Materials 0.03 0.14 0.14 0.20 0.43 Producer Price Index: Finished Goods 0.03 0.32 0.32 0.33 0.80 Capacity Utilization: Total Industry 0.02 0.16 0.21 0.63 0.64 Civilian Unemployment Rate 0.44 0.54 0.55 0.65 0.68 Thistablereportsthecumulativeshareofvarianceofyieldsandmacrovariablesexplainedbythemacroyields factors. The first three columns refer to the yield curve factors (level, slope and curvature) and the last two to the unspanned macroeconomic factors (UM1 and UM2). 10

variance of price indices, unemployment, nominal earnings, nominal consumption and money, in line with previous studies, see Diebold et al. (2006). The first unspanned macro factor captures the dynamics of industrial production and other real variables, while the second unspanned factor mainly explains inflation and other nominal variables.6 Figure 1 displays the estimated factors of the macro-yields model. The top three plots report the yield curve factors, while the bottom two refer to the unspanned factors. The estimated yield curvefactorsofthemacro-yieldsmodelarehighlycorrelatedwiththeNSfactors,whichweestimate by ordinary least squares as in Diebold and Li (2006) and report in dashed red lines in the top plots. The differences between the NS factors and the first three macro-yields factors are due to the fact that, in the macro-yields model, the yield curve factors are common to both yield curve and macroeconomicvariables. Infact,inthemacro-yieldsmodel,weextracttheyieldcurvefactorsfrom both yields and macroeconomic variables and impose the NS restrictions on the factors loadings of the yields to identify them as yield curve factors. The two bottom plots of Figure 1 show the unspanned macro factors. The bottom left plot reports the first unspanned macro factor along with the industrial production index, while the bottom right plot reports the second unspanned macroeconomic factor along with the real interest rate (computed as the difference between the federal funds rate and the consumer price index). As it is clear from the plots, the first unspanned macroeconomicfactorcloselytrackstheindustrialproductionindex, withacorrelationof90%, and the second unspanned macroeconomic factor proxies the real interest, with a correlation of 74%. This is in line with the fact that, as reported in Table 2, the first unspanned macroeconomic factor explains mainly measures of real economic activity, while nominal variables are explained partly by the yield curve factors and partly by the second unspanned factor. We can thus conclude that the macro-yields models identifies two unspanned macroeconomic factors: real economic activity and real interest rate. In the next Section we assess the quantitative importance of the unspanned 6The two macroeconomic factors are not identified since any transformation HFx, with H non-singular, gives an t observationally equivalent model. In order to achieve identification additional restrictions are required. We do not imposesuchrestrictionsandtheEMalgorithmconvergestotheMaximumLikelihoodsolutionthatis”close”tothe initialisation. Identificationcanbeachievedbyassumingthatthesecondfactorisnotloadedbyoneoftheindicators of real economic activity. Once we impose this restriction, results do not change. 11

Figure 1: Macro-yields factors Levelvs.NS Slopevs.NS Curvaturevs.NS 10 6 14 8 4 12 6 2 4 10 2 0 8 0 -2 -2 6 -4 -4 4 -6 -6 80 90 00 80 90 00 80 90 00 UM1vs.IP UM2vs.r Model 2 Proxy 2 1 1 0 0 -1 -2 -1 -3 -2 80 90 00 80 90 00 This figure displays the estimated factors of the macro-yields model. The dashed red lines in the three top graphs refer to the NS yield curve factors estimated by ordinary least squares as in Diebold and Li (2006). The red dashed lineinthebottomleftplotreferstotheindustrialproductionindex(IP),whilethereddashedlineinthebottomplot refers to the real interest rate (FFR-CPI). The grey-shaded areas indicate the recessions as defined by the NBER. 12

macroeconomic factors in explaining bond risk premia. 4.2 Bond risk premia The bond risk premium measures the compensation required by risk averse investors to hold longterm government bonds for facing capital loss risk, if the bond is sold before maturity. Long-term yields are determined by market expectations for the short rates over the holding period of the long-term asset plus a yield risk premium. Assuming a minimum investment horizon of one year, we have (cid:16) τ (cid:17)−1 (cid:88) (τ) (12) (τ) y = E [y ]+yrp . (9) t 12 t t+i t i=0,12,...,τ−12 An alternative measure for the bond risk premium can be obtained by looking at bond returns. Theone-yearholdingperiodbondreturnforabondwithmaturityτ monthsisthereturnofbuying a bond with τ months to maturity at time t, selling it one year later, at time t+12, as a bond with τ −12 months to maturity, i.e. (τ) (τ−12) (τ) r = −(τ −12)y +τy . (10) t+12 t+12 t The expected one-year holding period return on long term bonds equals the expected return on short term bond plus the return risk premium (τ) (12) (τ) E [r ] = y +rrp , (11) t t+12 t t accordingly the return risk premium is the one-year expected return in excess of the one-year rate (τ) (τ) (12) (τ) rrp = E [r ]−y ≡ E [rx ]. (12) t t t+12 t t t+12 13

The relation between the return risk premium and the yield risk premium is as follows 1 (cid:104) (cid:105) (τ) (τ) (τ−12) (24) yrp = E rrp +rrp +...+rrp , (13) t τ t t t+12 t+τ−24 which means that the yield risk premium is the average of expected future return risk premia of declining maturity. This implies that the statements in Equations (9) and (11) are equivalent, if one equation holds with zero (constant) bond risk premium, the other equation holds with zero (constant) bond risk premium as well. The expectations hypothesis of the term structure of interest rates states that the yield risk premium is constant. This implies that expected excess returns are time invariant and, thus, excess bondreturnsshouldnotbepredictablewithvariablesintheinformationsetattimet. However,the expectations hypothesis has been empirically rejected since Fama and Bliss (1987) and Campbell and Shiller (1991). They find that excess returns can be predicted by forward rate spreads and by yield spreads, respectively. More recent evidence by Cochrane and Piazzesi (2005) shows that a linear combination of forward rates (the CP factor) explains between 30% and 35% of the variation in expected excess bond returns. Moreover, Ludvigson and Ng (2009) find that macroeconomic factors constructed as linear and non-linear combinations of principal components extracted from a large data-set of macroeconomic variables (the LN factor) have important forecasting power for future excess returns on U.S. government bonds, above and beyond the predictive power contained in forward rates and yield spreads. Cooper and Priestley (2009) also find that the output gap has in-sample and out-of-sample predictive power for U.S. excess bond returns. The top panel of Figure 2 shows the 5 years to maturity yield along with the corresponding components as in Equation (9), where the sum of expectations is the sum of forecasts produced withourmacro-yieldsmodelandtheriskpremiumisthedifferencebetweenthe5yearstomaturity yield and the sum of the forecasts of the 1 year to maturity yields. The expectation component is larger than the risk premium but the graph shows that there is substantial variation of the risk premium over time, which is not compatible with the expectations hypothesis. The middle 14

Figure 2: Yield risk premium, 5-year bond 15 10 5 0 72 75 77 80 82 85 87 90 92 95 97 00 02 05 07 Yield ExpectationsMY RiskPremiumMY 4 2 0 -2 72 75 77 80 82 85 87 90 92 95 97 00 02 05 07 RiskPremiumMY IPGrowth 4 3 2 1 0 -1 72 75 77 80 82 85 87 90 92 95 97 00 02 05 07 RiskPremiumMY RiskPremiumOY This figure displays the yield risk premium using the 5 years to maturity bond. The top panel shows the 5 years to maturity yield (red dashed line) along with the corresponding expectation (green dot-dashed line) and the yield risk premium (blue line) components, computed as in Equation (9) using the macro-yields model. The middle panelreportstheyieldriskpremiumaccordingtothemacro-yieldsmodel(blueline)andthestandardizedindustrial productiongrowth(reddashedline). Thebottomplotshowstheyieldriskpremiumobtainedfromthemacro-yields model (blue line) and the only-yields model (red dashed line). The grey-shaded areas indicate the recessions as defined by the NBER. 15

graph plots the risk premium against the industrial production index growth and it reveals that the yield risk premium obtained from the macro-yields model displays a clear counter-cyclical pattern. Its correlation with the industrial production index growth is -0.33. This is consistent with the fact that investors want to be compensated for bearing risks related to recessions. Conversely, the bottomgraphinFigure2,showstheriskpremiumobtainedfromtheonly-yieldsmodel. Thismodel delivers an acyclical risk premium, with a correlation of only -0.07 with the industrial production index growth. This indicates that using macro variables greatly improves the estimates of the risk premium. Given that, as shown in Equation (13), the yield risk premium is the average of expected future returnriskpremiaofdecliningmaturity, weanalyzethepredictiveabilityofthemacro-yieldsmodel for excess returns and compare it with the predictions of the only-yields model. We also compare ourresultswithpredictionsobtainedusingtheCPfactor, theLNfactorandtheCPandLNfactors combined. We implement predictive regressions for the CP and LN factors by regressing excess bond returns on the predictive factors X = {CP ,LN } , as follows t t t (τ) (τ) rx = βX +ε . (14) t+12 t t+12 We construct the predictive factors X by pooling the predictive regression for the individual mat turities rx = γx +ε , (15) t+12 t t+12 where rx = 1 (cid:80) rx (τ) and x contains the predictor variables. To construct the t+12 4 τ=24,36,48,60 t+12 t CP factor we use the following predictor variables xCP = [1,y (12) ,f (24) ,...,f (60) ], where f (τ) t t t t t denotes the τ-month forward rate.7 We estimate equation (15) using xCP as predictor variables t 7The τ-month forward rate for loans between time t+τ −12 and t+τ is defined as f(τ) =−(τ −12)y(τ−12)+τy(τ). t t t 16

Table 3: In-sample fit of excess bond returns Maturity MY OY CP LN LN+CP 2y 0.55 0.12 0.22 0.33 0.41 3y 0.53 0.12 0.24 0.33 0.43 4y 0.50 0.14 0.27 0.32 0.43 5y 0.46 0.15 0.24 0.30 0.40 This table reports the R2 for one-year ahead one year holdingperiodexcessbondreturnsfromdifferentmodels. The columns MY and OY refer to the modelimpliedexpectedexcessbondreturnsfromthemacroyieldsmodel(MY)andtheonly-yieldsmodel(OY)respectively. The columns CP, LN and CP+LN refer to the predictive regression using the Cochrane and Piazzesi (2005) factor (CP), the Ludvigson and Ng (2009) factor (LN), and both the Cochrane and Piazzesi(2005)andtheLudvigsonandNg(2009)factors jointly. and construct the CP factor as CP = γˆCPxCP. To construct the LN factor, we use as predictor t t variables xLN = [1,PC1 ,...,PC8 ,PC13], where PC denotes principal components extracted t t t t from a large dataset of 131 macroeconomic data series.8 We then estimate equation (15) using xLN t as predictor variables and construct the LN factor as LN = γˆLNxLN. t t Notice that the LN factors are constructed aggregating principal components extracted from a setofmacroeconomicandfinancialvariableswithoutimposingthattheyareunspannedbythecross section of the yields similarly to the factors extracted by assuming a block-diagonal structure on the factor loadings. As a consequence, those factors duplicate information that is already spanned by the yield factors.9 Results in Table 3 show that the macro-yields model explains about 46-55% of the variation of one-year ahead excess returns, while the only-yields model can explain only the 12-15% of the variation of the one-year ahead excess returns. Table 3 reports also the R-squared from the predictive regressions of excess bond returns on the CP and the LN factors. Results show that the CP factor explains 22-27% of the variation in one-year ahead excess returns, slightly lower than 8The 131 macroeconomic data series used to construct the LN factor have been downloaded from Sydney C. Ludvigson’s website at http://www.econ.nyu.edu/user/ludvigsons/Data&ReplicationFiles.zip. 9See Appendix D.2. 17

the value reported in Cochrane and Piazzesi (2005). This is due to the fact that our predictive regressions are estimated on a more updated sample, and the performance of the CP factor has deteriorated over time, as also shown by Thornton and Valente (2012). The LN factors explain a third of the variation of future excess bond returns, while the CP and LN factors jointly explain 40-43% of the variation in one-year ahead excess bond returns, lower than what is explained by our macro-yields model. We can thus conclude that, in-sample, the macro-yields model outperforms the CP and the LN factors even combined. Figure 3 shows the predicted and realized average excess bond returns from the macro-yields andtheonly-yieldsmodel,andalsofromthepredictiveregressionsusingtheCPandtheLNfactors. The figure shows that the predicted excess bond returns from the only-yields model are quite flat, indicating that the yield curve factors poorly predict excess bond returns. The CP factor seems doing a better job than the only-yields model, but does not improve over the macro-yields model. The macro-yields model is able to better predict the average excess return, also with respect to the LN factor. 4.3 Unspanning conditions Results in the previous section show that the unspanned macro factors play an important role in explaining the term premium, despite being constrained to not affect current yields. In the context of Equation (9), this can only happen if the unspanned macro factors have offsetting effects on average expected future short rates and term premia, see Duffee (2011b). To understand whether our macro factors are truly unspanned by the yield curve, we compute the risk premium of an unrestricted macro-yields model which does not impose the zero restriction on the factor loadings of the yields on the macro factors, i.e. Γ (cid:54)= 0 in Equation (4).10 The yx estimates of the bond premium delivered by this model are practically indistinguishable from the estimates obtained using the macro-yields model which instead imposes the restriction Γ = 0 yx (the correlation between the estimates is 0.99). The fact that imposing the unspanning restrictions 10In Appendix D we report more extensive results for the unrestricted macro-yields model. 18

Figure 3: Average 1-year holding period excess return: realized and predicted 10 10 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −8 −8 −10 rx MY −10 rx OY 75 80 85 90 95 00 05 75 80 85 90 95 00 05 10 10 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −8 −8 −10 rx CP −10 rx LN 75 80 85 90 95 00 05 75 80 85 90 95 00 05 This figure displays the average excess return rx (blue continuous line) and the corresponding predicted values t+12 from different models (dashed red line). The dashed red line in the top plots refer to the model-implied predicted values from the macro-yields MY model (top right) and only-yields OY model (top left). The dashed red line in the bottomplotsrefertothepredictedvaluesfromthepredictiveregressionsusingtheCPfactors(bottomleft)andthe LN factor (bottom right). The grey-shaded areas indicate the recessions as defined by the NBER. 19

Table 4: Likelihood ratio test statistic for the unspanning restrictions H Test statistic p-value 0 Γ = 0 12.85 0.38 yx A = 0 79.03 0.00 yx This table reports the likelihood ratio test statistic for the unspanning restrictions and the corresponding p-values, computed using a chi-squared distribution with degrees of freedom equal to the number of restrictions tested. The first line refers to the null hypothesesΓ =0inEquation(4)whilethesecond yx line refers to the null hypotheses A = 0 in Equayx tion (5). has no effect on the yield risk premium indicates that the macro factors are unspanned by the yield curve. In practice, this means that, in periods of recession, the unspanned macro factors increase the risk premium and decrease the expected future short rates by the same amount, without contributing to a steepening of the current yield curve. Conversely, in periods of economic expansion, the unspanned macro factors decrease the bond premium and increase the expected future short rates by the same amount, without contributing to a flattening of the current yield curve. Changes in the current shape of the yield curve can only be determined by changes in the yield curve factors. To formally test for the unspanning properties in the context of our state-space macro-yields model, wedefineafactorasunspannedbytheyieldcurveifitsatisfiesthefollowingtwoconditions. First, it doesn’t affect the current cross-section of yields, i.e. it is not loaded contemporaneously by the yields (Γ = 0 in Equation (4)). Second, it has predictive ability for the yield curve factors yx (A (cid:54)= 0 in Equation (5)), see also Joslin et al. (2014). yx The unspanning conditions can be tested performing likelihood ratio tests, as follows LR = 2×(L −L ) (16) u r where LR has a chi-squared probability distribution with degrees of freedom equal to the number of restrictions imposed. To compute the likelihood ratio test for for the zero restrictions on the 20

factor loadings, L denotes the loglikelihood of an unrestricted macro-yield model that does not u impose the restriction Γ = 0 and L is the loglikelihood of our macro-yields model. The test yx r statistic in Table 4 shows that we cannot reject the null of zero factor loadings of the yields on the macro factors. This implies that, indeed, the macro factors do not affect the current shape of the yield curve.11 To test the predictive ability of the macro factors obtained from macro-yields model in Equations (4)–(6) for the yield factors, and therefore the yield curve of interest rates, we perform the likelihood ratio test statistics in Equation (16), where, in this case, L is the loglikelihood of our u macro-yield model and L is the restricted loglikelihood obtained imposing A = 0 in Equation r yx (5). Results in Table 4 show that we can reject the null hypothesis of no Granger causality from the macro factors to the yield curve factors. These testing results show that the macroeconomic factors identified by the macro-yields model do not explain the cross-section of yields but have predictive ability for the future evolution of the yield curve. As a consequence, they satisfy both conditions for being truly unspanned macroeconomic factors. Moreover, looking at the coefficients and their relative standard errors, available in Appendix F, we can infer that the first unspanned factor, proxied by economic growth, Granger causes the slope and the curvature, while the second unspanned factor, proxied by the real interest rate, Granger causes the level. 5 Out of sample forecast To evaluate the predictive ability of the macro-yields model, we generate out-of-sample iterative forecasts of the factors, as follows E (F∗ ) ≡ Fˆ∗ = (Aˆ∗)hFˆ∗ , t t+h t+h|t |t t|t 11ThisresultisduetothefactthatalmostthetotalityofthefluctuationsofbondyieldsisexplainedbytheNelson andSiegelfactors. Thesameresultmaynotholdwhentheyieldfactorsprovideapoorerfitoftheyields,asinJoslin, Le and Singleton (2013). 21

where h denotes the forecast horizon and Aˆ∗ is estimated using the information available till time |t t.12 We then compute out-of-sample forecasts of the yields given the projected factors, in this way E (z ) ≡ zˆ = Γˆ∗Fˆ∗ . t t+h t+h|t |t t+h|t where Γˆ∗ is estimated using data up to time t. |t Collecting the excess returns for bonds with maturities from two to five years in the vector rx . t We compute the out-of-sample predictions of excess bond returns as follows E (rx ) ≡ rx = Π y +Π y = Π (Γˆ∗F∗ )+Π y , (17) t t+12 t+12|t 1 t+12|t 2 t 1 |t t+12|t 2 t (cid:20) (cid:21) (cid:20) (cid:21) where Π 1 = D [−1:−K] 0 [K×1] , Π 2 = −1 [K×1] D [2:K+1] , D [−1:−K] denotes a diagonal matrix with elements −1,−2,...,−K in the diagonal and K + 1 denotes the total number of maturities. Notice that Equation (17) implies that the forecast errors made in forecasting the excess returns are proportional to the ones made in forecasting the yields, i.e. rx −rx = t+12|t t+12 Π (y −y ), see Carriero, Kapetanios and Marcellino (2012). 1 t+12|t t+12 We forecast yields and excess returns recursively using data from January 1970 until the time that the forecast is made, beginning in January 1990 to December 2008. 5.1 Yields To evaluate the prediction accuracy of the macro yields model for out of sample forecasts of yields, we use the Mean Squared Forecast Error (MSFE), i.e. the average squared error in the evaluation period for the h-months ahead forecast of the yield (or excess return) with maturity τ MSFEt1(τ,h,M) = 1 (cid:88) t1 (cid:16) yˆ (τ) (M)−y (τ) (cid:17)2 , (18) t0 t −t +1 t+h|t t+h 1 0 t=t0 12See Appendix A for the definitions of F∗, Γ∗ and A∗. t 22

Table 5: Out-of-sample performance for yields Macro-Yields Maturity 3m 1y 2y 3y 4y 5y h=1 1.17 1.05 1.06 1.00 1.05 1.14 h=3 0.79* 0.93 0.99 0.96 0.99 1.02 h=6 0.78** 0.89 0.94 0.93 0.93 0.94 h=12 0.69** 0.74** 0.79** 0.80*** 0.80*** 0.80*** h=24 0.62*** 0.66*** 0.74** 0.82** 0.88* 0.97 Only-Yields Maturity 3m 1y 2y 3y 4y 5y h=1 0.93 1.09 1.17 1.11 1.07 1.11 h=3 0.96 1.13 1.20 1.14 1.10 1.13 h=6 0.99 1.18 1.25 1.21 1.15 1.16 h=12 1.04 1.16 1.26 1.27 1.25 1.26 h=24 1.06 1.12 1.27 1.39 1.49 1.62 ThistablereportstherelativeMSFEofthemacro-yieldsmodelandtheonly-yields model over the MSFE of the random walk for multi-step predictions of the yields. Thefirstcolumnreportstheforecasthorizonh. ThesamplestartsonJanuary1970 andtheevaluationperiodisJanuary1990toDecember2008. *,**and***denote significantoutperformanceat10%,5%and1%levelwithrespecttotherandomwalk according to the White (2000) reality check test with 1,000 bootstrap replications using an average block size of 12 observations. (τ) where t and t denote, respectively, the start and the end of the evaluation period, y is the 0 1 t+h (τ) realized yield with maturity τ at time t+h and yˆ (M) is the h-step ahead forecast of the yield t+h|t with maturity τ from model M using the information available up to t. Forecast results for yields are usually expressed as relative performance with respect to the random walk, which is a na¨ıve benchmark for yield curve forecasting very difficult to outperform, given the high persistency of the yields. The random walk h-steps ahead prediction at time t of the yield with maturity τ is (τ) (τ) (τ) E (y ) ≡ yˆ = y , t t+h t+h|t t where the optimal predictor does not change regardless of the forecast horizon. To measure the relativeperformanceofthemacro-yieldsmodelwithrespecttotherandomwalk,weusetherelative 23

Figure 4: 12-months ahead smoothed squared forecast errors for yields Maturity 3 months Maturity 3 months 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 MY RW OY RW 0 0 95 97 00 02 05 07 95 97 00 02 05 07 Maturity 36 months Maturity 36 months 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 MY RW OY RW 0 0 95 97 00 02 05 07 95 97 00 02 05 07 Maturity 60 months Maturity 60 months 2 2 1.5 1.5 1 1 0.5 0.5 MY RW OY RW 0 0 95 97 00 02 05 07 95 97 00 02 05 07 This figure displays the 5-years rolling 12-months ahead squared forecast error for the yields with 3, 36 and 60 months to maturity. The blue continuous line refers to the 5-years rolling squared forecast error of the macro-yields MY model (left plots) and of the only-yields OY model (right plots). The dashed red line refers to 5-years rolling squared forecast error of the random walk. The dates on the horizontal axis refer to the end of the rolling window period. The grey-shaded areas indicate the recessions as defined by the NBER. 24

MSFE computed as MSFEt1(τ,h,M) rMSFEt1(τ,h,M) = t0 . t0 MSFEt1(τ,h,RW) t0 Table 5 reports the rMSFE with respect to the random walk for the macro-yields model the only-yields model. Results in Table 5 show that the only-yields model is outperformed by the macro-yields model for all but the 1-month horizon. Moreover, the macro-yields model outperforms the random walk at 3-, 6-, 12- and 24-month ahead for all the maturities, with significant outperformance according to the White (2000) reality check test for the 12- and 24-month ahead forecasts.13 ThisevidenceiscorroboratedbyFigure4, whichreportsthe12-monthaheadsmoothed squared forecast errors of the macro-yields, the only-yields and the random walk models for yields with 3-, 36- and 60-month to maturity. The figure highlights how the macro-yields model has been systematically outperforming the random walk, especially in the last part of the evaluation sample for the short maturity and the first part of the sample for long maturities. The only-yield model, instead has been performing as the random walk in the first part of the evaluation sample but its performance deteriorated in the last part of the evaluation sample, significantly underperforming the random walk. This indicates that the unspanned macroeconomic factors, while not important for explaining the contemporaneous yields curve, contain useful information to predict the future yield curve factors and, thus, the future evolution of the yield curve. 5.2 Excess bond returns Out of sample forecast results for excess bond returns are reported in Table 6, which contains the relative MSFE of the macro-yields model (MY) with respect to the constant excess return benchmark, where one-year holding period excess returns are unforecastable atone year horizon, as in the expectation hypothesis. We use the expectation hypothesis since, because of its simplicity, represents a benchmark of unpredictability. The macro-yields model outperforms the constant excess return benchmark for all maturities and the outperformance is significant for all maturities according to the White (2000) reality check test. 13For more details about the reality check test see Appendix C. 25

Figure 5: Smoothed mean squared forecast errors for excess bond returns 20 20 15 15 10 10 5 5 EH MY EH OY 0 0 95 97 00 02 05 07 95 97 00 02 05 07 20 20 15 15 10 10 5 5 EH CP EH LN 0 0 95 97 00 02 05 07 95 97 00 02 05 07 This figure displays the 5-years rolling mean squared forecast error for one-year holding period excess bond returns fromtheexpectationhypothesisEH(bluecontinuousline)andthecorrespondingvaluesfromdifferentmodels(dashed redline). Thedashedredlineinthetopplotsreferto5-yearsrollingmeansquaredforecasterrorofthemacro-yields MY model (top right) and only-yields OY model (top left). The dashed red line in the bottom plots refer to the 5-years rolling mean squared forecast error from the predictive regressions using the CP factors (bottom left) and the LN factor (bottom right). The dates on the horizontal axis refer to the end of the rolling window period. The grey-shaded areas indicate the recessions as defined by the NBER. 26

Table 6: Out-of-sample predictive performance for excess returns Maturity MY OY CP LN LN+CP 2y 0.76** 1.20 1.17 0.80 0.80 3y 0.75** 1.20 1.21 0.79 0.83 4y 0.74** 1.18 1.21 0.78 0.83 5y 0.75** 1.18 1.18 0.81 0.83 This table reports the relative MSFE of the macroyields model (MY), the only-yields model (OY), the Cochrane and Piazzesi (2005) factor (CP), the Ludvigson and Ng (2009) (LN) factor, the Cochrane and Piazzesi (2005) and the Ludvigson and Ng (2009) factors combined(LN+CP)withrespecttotheexpectationhypothesisforexcessreturns. ThesamplestartsonJanuary 1970 and the evaluation period is January 1990 to December2008. *and**denotesignificantoutperformance at10%and5%levelwithrespecttotheexpectationhypothesis according the White (2000) reality check test with1,000bootstrapreplicationsusinganaverageblock size of 12 observations. Table6alsoreportstheout-of-samplerelativeMSFEsoftheexcessbondreturnsforecastsusing the CP factor, the LN factor, and the CP and LN factors combined obtained from the predictive regressions in (14). The worst performing models are the ones that do not use macroeconomic variable, i.e. the only-yield model and the CP factors. In line with the predictive regressions of excess bond returns and with the 12-month ahead out-of-sample forecast performance of the macro-yields model for the yields, results in Table 6 show that the macro-yields model is the best performing model for the prediction of the 1-year excess bond returns for all maturities followed by the combination of the CP and LN factors. However, although the unspanned model significantly outperforms the na¨ıve benchmark while the CP+LN does not, we cannot reject the hypothesis that the forecasts of these two models are statistically equally accurate. To further understand the performance of the macro-yields model to predict 1-year holding period excess bond returns, Figure 5 plots the 5-year rolling mean squared forecast error of the macro-yields model, the only-yields model, the CP and LN factors along with the 5-year rolling mean squared forecast error under the expectation hypothesis (EH). The figure shows that the performance of the only-yield model and the CP factors are similar: both models outperform the 27

expectation hypothesis in the first part of the evaluation sample but display large forecast errors in the second part. Also the performance of the macro-yields model and the LN factors are similar, they both provide more accurate predictions than the expectation hypothesis, in particular in the last part of the evaluation period. However, it is clear from the figure that the macro-yields model, apart from being the best performing model on average, as shown in Table 6, it is the best performing model for the whole evaluation period. This is a clear evidence that the unspanned macroeconomic factors identified by the proposed macro-yields model as related with economic growth and real interest rates have predictive ability for the yield curve factors and, thus, for excess bond returns. 6 Conclusions In this paper we analyze the predictive content of macroeconomic information for the yield curve of interestratesandexcessbondreturnsintheUnitedStates. Wefindthattwomacroeconomicfactors characterizing economic growth and real interest are unspanned by the cross-section of government bond yields and have significant predictive power for the bond yields and excess returns. In future research, we plan to extend our empirical specification to allow for the zero lower bound of interest rates, non-synchronicity of macroeconomic data releases and mixed frequencies. The macro-yields model presented in this paper cannot be estimated on a sample that includes the great recession, as it does not ensure a zero lower bound on interest rates. Our model model can, however, be easily extended to deal with this issue by anchoring the shorter end of the yield curve using market expectation, along the lines of Altavilla, Costantini, Giacomini and Ragusa (2012). Data revisions and jagged edges due to the non-synchronicity of macroeconomic data releases areimportantcharacteristicstobetakenintoaccountwhenextractingmacroeconomicinformation, seeGiannone,ReichlinandSmall(2008). Inaddition,bondyieldsareavailableathigherfrequencies than macroeconomic variables. These features can be easily incorporated into our empirical model along the line described in Banbura et al. (2012). 28

A Estimation procedure We can rewrite the macro-yields model in equations (4)–(6) in compact form as z = a+ΓF +v , (19) t t t F = µ+AF +u , u ∼ N(0,Q) (20) t t−1 t t v = Bv +ξ , ξ ∼ N(0,R) (21) t t−1 t t             y Fy 0 Γ Γ A A Q Q wherez =  t , F =  t , a =  , Γ =  yy yx , A =  yy yx , Q =  yy yx , t   t           x Fx a Γ Γ A A Q Q t t x xy xx xy xx xy xx   µ y µ =   and Γ = Γ is the matrix whose rows correspond to the smooth patterns proposed   yy NS µ x by Nelson and Siegel (1987) and shown in equation (2). In addition Γ = 0, as the macroeconomic yx factors Fx are unspanned by the cross-section of yields Γ = 0. We also estimate the only-yields t yx model using the same procedure, as it implies the following restrictions in (19)–(20): z = y , F = t t t Fy, a = 0, Γ = Γ , µ = µ . t NS y The macro-yields model in (19)–(20) can be put in a state-space form augmenting the states F t with the idiosyncratic components v and a constant c as follows t t z = Γ∗F∗+v∗, v∗ ∼ N(0,R∗) t t t t F∗ = A∗F∗ +u∗, u∗ ∼ N(0,Q∗) t t−1 t t         F A µ ... 0 u Q ... 0 t t (cid:20) (cid:21)         whereΓ∗ = Γ a I N ,F t ∗ =    c t    ,A∗ =    . . . . . . 1 . . .    ,u∗ t =    ν t    ,Q∗ =    . . . ε . . .            v 0 ... ... B ξ 0 ... R t t and R = εI , with ε a very small fixed coefficient. In this state-space form, c an additional state n t variable restricted to one at every time t. 29

The restrictions on the factor loadings Γ∗ and on the transition matrix A∗ can be written as H vec(Γ∗) = q , H vec(A∗) = q , 1 1 2 2 where H and H are selection matrices, and q and q contain the restrictions. 1 2 1 2 We assume that F∗ ∼ N(π ,V ), and define y = [y ,...,y ] and F∗ = [F∗,...,F∗]. Then 1 1 1 1 T 1 T denoting the parameters by θ = {Γ∗,A∗,Q∗,π ,V }, we can write the joint loglikelihood of z and 1 1 t F , for t = 1,...,T, as t T (cid:18) (cid:19) (cid:88) 1 L(z,F∗;θ) = − [z −Γ∗F∗](cid:48)(R∗)−1[z −Γ∗F∗] + 2 t t t t t=1 T (cid:18) (cid:19) T (cid:88) 1 − log|R∗|− [F∗−A∗F∗ ](cid:48)(Q∗)−1[F∗−A∗F∗ ] + 2 2 t t−1 t t−1 t=2 T −1 1 − log|Q∗|+ [F∗−π ](cid:48)V−1[F∗−π ]+ 2 2 1 1 1 1 1 1 T(p+k) − log|V |− log2π+λ(cid:48) (H vec(Γ∗)−q )+λ(cid:48) (H vec(A∗)−q ) 2 1 2 1 1 1 2 2 2 where λ contains the lagrangian multipliers associate with the constraints on the factor loadings 1 Γ∗ and λ contains the lagrangian multipliers associated with the constraints on the transition 2 matrix A∗. The computation of the Maximum Likelihood estimates is performed using the EM algorithm. Broadly speaking, the algorithm consists in a sequence of simple steps, each of which uses the Kalman smoother to extract the common factors for a given set of parameters and multivariate regressionstoestimatetheparametersgiventhefactors. Inpractice,weusetherestrictedversionof the EM algorithm, the Expectation Restricted Maximization, since we need to impose the smooth pattern on the factor loadings of the yields on the NS factors. The ERM algorithm alternates Kalman filter extraction of the factors to the restricted maximization of the likelihood. At the j-th iteration the ERM algorithm performs two steps: 1. In the Expectation-step, we compute the expected log-likelihood conditional on the data and 30

the estimates from the previous iteration, i.e. L(θ) = E[L(z,F∗;θ(j−1))|z] which depends on three expectations Fˆ∗ ≡ E[F∗;θ(j−1)|z] t t P ≡ E[F∗(F∗)(cid:48);θ(j−1)|z] t t t P ≡ E[F∗(F∗ )(cid:48);θ(j−1)|z] t,t−1 t t−1 These expectations can be computed, for given parameters of the model, using the Kalman filter. 2. In the Restricted Maximization-step, we update the parameters maximizing the expected log-likelihood with respect to θ: θ(j) = argmaxL(θ) θ This can be implemented taking the corresponding partial derivative of the expected log likelihood, setting to zero, and solving. The procedure outlined above can be extended to estimate also the decay parameter λ controlling for the shape of the loadings of the yields on the slope and curvature factors. Since the factor loadings are a non-linear function λ, an additional step consisting in the numerical maximization of the conditional likelihood with respect to λ is required. The procedure is know as Expectation Conditional Restricted Maximization (ECRM) algorithm. 31

B Data Table 7: Macroeconomic variables Series N. Mnemonic Description Transformation 1 AHE Average Hourly Earnings: Total Private 1 2 CPI Consumer Price Index: All Items 1 3 INC Real Disposable Personal Income 1 4 FFR Effective Federal Funds Rate 0 5 HSal House Sales - New One Family Houses 1 6 IP Industrial Production Index 1 7 M1 M1 Money Stock 1 8 Manf ISM Manufacturing: PMI Composite Index (NAPM) 0 9 Paym All Employees: Total nonfarm 1 10 PCE Personal Consumption Expenditures 1 11 PPIc Producer Price Index: Crude Materials 1 12 PPIf Producer Price Index: Finished Goods 1 13 CU Capacity Utilization: Total Industry 0 14 Unem Civilian Unemployment Rate 0 This table lists the 14 macro variables used to estimate the macro-yields. Most series have been transformed prior to the estimation, as reported in the last column of the table. The transformation codes are: 0 = no transformation and 1= annual growth rate. C Reality check test To compare the out of sample predictive ability of a model with respect to the benchmark, we use the reality check test of White (2000), as we compare only non-nested models. If we denote by e (b) the forecast errors of the benchmark and by e (M) the forecast errors of t t the model under consideration. Then we can define the null hypothesis of no predictive superiority over the benchmark as H : f = E(f ) ≡ E(e (b)2−e (M)2) ≤ 0 (22) 0 t t t The test is then based on the statistic 1 (cid:88) t1 f = fˆ (23) t t −t 1 0 t=t0 32

wheret andt denote, respectively, thestartandtheendoftheevaluationperiod, andhatsdenote 0 1 estimated statistics. To approximate the asymptotic distribution of the test statistic, we use block-bootstrap as follows: 1. Wegeneratebootstrappedforecasterrorseˆ∗(b)andeˆ∗(M)usingthestationaryblock-bootstrap t t of Politis and Romano (1994) with average block size of 12. This procedure is analogous to the moving blocks bootstrap, but, instead of using blocks of fixed length uses blocks of random length, distributed according to the geometric distribution with mean block length 12. Also to give the same probability of resampling to all observations, we use a circular scheme. 2. Construct the bootstrapped test statistic as f ∗ = 1 (cid:88) t1 (eˆ∗(b)2−eˆ∗(M)2) t −t t t 1 0 t=t0 3. Repeat steps 1 and 2 for 1,000 times to obtain an estimate of the distribution of the test ∗ ∗ ∗ statistic f = [f ,...,f ]. (1) (1,000) 4. Compare V = (t − t )1/2f with the quantiles of V∗ = (t − t )1/2(f ∗ − f) to obtain the 1 0 1 0 p-value. 33

D Unrestricted vs block-diagonal macro-yields model In this Appendix we report results for two alternative macro-yields models that use different assumptions about the matrix of factor loadings Γ in Equation (4). The first model that we consider is an unrestricted macro-yields model, which does not impose the zero restrictions on the factor loadings of yields on macro factors, i.e. Γ is allowed to be different from zero. This model alyx lows the macro factors Fx to directly affect the cross-section of yields. The second model imposes t block-diagonality of the matrix of factor loadings Γ, i.e. Γ = 0 and Γ = 0. This implies that yx xy the yield curve factors and the macro factors are treated separately, as in M¨onch (2012). Notice that when estimating the block-diagonal macro-yields model, we omit the federal funds rate from the macro variables used to estimate the model, as in M¨onch (2012). The reason is that we cannot reliably impose a zero restriction on the factor loadings of the federal funds rate on the yield curve factors. Results in Tables 8–11, show that the in-sample and out-of-sample performances of the unrestricted macro-yields model are equal to the ones of the macro-yields model with unspanned macro factors in Tables 2–6. This provides evidence that the zero restrictions on the factor loadings are satisfied by the data, implying that the macro factors Fx are unspanned by the cross-section of t yields. On the contrary, imposing a block-diagonal structure on the matrix of factor loadings has substantialimplicationsforthetheidentificationofthefactors. Table 8showsthattheinformation criterion selects the model with six factors, i.e. three yield curve factors and three macro factors. This indicates that this model is less parsimonious than our macro-yields model. In addition, as shown in Table 9, the macro factors in the block-diagonal macro-yield model are highly correlated with the yield curve factors and are, thus, spanned by the cross-section of yields. This happens because the macro variables have a factor in common with the yields which the block-diagonal model treats as an additional macro factor. Table 9 also shows that the correlation of the first macro factor of the block-diagonal macro-yields model with the industrial production index is 34

Table 8: Model selection Unrestricted Block-diagonal Number of factors IC* V IC* V 3 0.02 0.44 1.17 1.36 4 -0.02 0.31 1.30 1.16 5 -0.11 0.22 1.50 1.05 6 0.01 0.18 0.89 0.43 7 0.25 0.17 1.06 0.38 8 0.43 0.16 1.42 0.41 This table reports the information criterion IC*, as shown in Equation (7) and in Equation (8), and the sum of the variance of the idiosyncratic components (divided by NT), V , whendifferentnumbersoffactorsareestimated. Thefirsttwo columnsrefertotheunrestrictedmacro-yieldsmodel. Thelast two columns refer to the block-diagonal macro-yields model. weaker than in our model. More importantly, the second and third macro factors do not have a clear-cut interpretation. Despite the issues with the interpretation of the factors, the fit of the block-diagonal macroyields model is comparable to the fit of our macro-yields, as shown in Table 10. However, Table 11 reveals that the lack of parsimony of the block-diagonal macro-yields model leads to out-of-sample underperformances. 35

Table 9: Factor identification Unrestricted L S C M1 M2 NS L 0.98 0.00 0.32 -0.08 0.01 NS S 0.03 0.97 0.28 0.11 0.08 NS C 0.25 0.37 0.86 0.12 0.06 IP -0.08 0.03 0.16 0.91 -0.09 R 0.47 0.05 0.10 0.20 -0.76 Block-diagonal L S C M1 M2 M3 NS L 0.99 -0.03 0.21 -0.07 0.46 0.57 NS S -0.03 0.99 0.38 0.51 0.44 -0.38 NS C 0.18 0.36 0.98 0.36 0.22 0.13 IP -0.10 0.04 0.20 0.69 -0.49 0.20 R 0.44 0.08 0.15 -0.05 -0.35 0.20 This table reports the correlation of the estimated factors with their corresponding empirical proxies–the Nelson and Siegel level (NS L), slope (NS S) and curvature (NS C), the industrial production index (IP) and the real interest rate (R). The top panel refers to the unrestricted macro-yieldsmodel. Thebottompanelreferstotheblockdiagonal macro-yields model. 36

Table 10: In sample performance Unrestricted Model L S C M1 M2 Average Hourly Earnings:Total Private 0.07 0.29 0.33 0.33 0.67 Consumer Price Index: All Items 0.19 0.48 0.48 0.50 0.86 Real Disposable Personal Income 0.00 0.02 0.03 0.34 0.36 Effective Federal Funds Rate 0.54 0.93 0.96 0.96 0.97 Hose Sales - New One Family Houses 0.00 0.19 0.19 0.22 0.22 Industrial Production Index 0.02 0.02 0.03 0.70 0.70 M1 Money Stock 0.17 0.25 0.25 0.25 0.31 ISM Manufacturing: PMI Composite Index (NAPM) 0.03 0.05 0.05 0.61 0.65 Payments All Employees: Total nonfarm 0.00 0.02 0.10 0.71 0.71 Personal Consumption Expenditures Price Index 0.16 0.23 0.33 0.47 0.79 Producer Price Index: Crude Materials 0.03 0.13 0.13 0.20 0.43 Producer Price Index: Finished Goods 0.03 0.31 0.31 0.32 0.81 Capacity Utilization: Total Industry 0.02 0.16 0.20 0.62 0.63 Civilian Unemployment Rate 0.44 0.53 0.55 0.64 0.67 Block-diagonal Model L S C M1 M2 M3 Average Hourly Earnings: Total Private 0.00 0.00 0.00 0.09 0.57 0.68 Consumer Price Index: All Items 0.00 0.00 0.00 0.07 0.76 0.76 Real Disposable Personal Income 0.00 0.00 0.00 0.13 0.37 0.42 House Sales - New One Family Houses 0.00 0.00 0.00 0.00 0.07 0.60 Industrial Production Index 0.00 0.00 0.00 0.63 0.80 0.80 M1 Money Stock 0.00 0.00 0.00 0.00 0.05 0.48 ISM Manufacturing: PMI Composite Index (NAPM) 0.00 0.00 0.00 0.59 0.61 0.68 Payments All Employees: Total nonfarm 0.00 0.00 0.00 0.68 0.74 0.74 Personal Consumption Expenditures Price Index 0.00 0.00 0.00 0.32 0.53 0.79 Producer Price Index: Crude Materials 0.00 0.00 0.00 0.42 0.65 0.75 Producer Price Index: Finished Goods 0.00 0.00 0.00 0.15 1.00 1.00 Capacity Utilization: Total Industry 0.00 0.00 0.00 0.64 0.64 0.73 Civilian Unemployment Rate 0.00 0.00 0.00 0.08 0.18 0.42 This table reports the cumulative share of variance of macro variables explained by the yield curve factors (level, slope and curvature) and the macroeconomic factors. The top panel refers to the unrestricted macroyields model. The bottom panel refers to the block-diagonal macro-yields model. 37

Table 11: Out-of-sample performance Unrestricted Maturity 3m 1y 2y 3y 4y 5y h=1 1.21 1.04 1.05 1.01 1.05 1.12 h=3 0.81* 0.93 0.99 0.97 0.99 1.02 h=6 0.79** 0.89 0.95 0.93 0.93 0.94 h=12 0.67** 0.73** 0.78** 0.79*** 0.79*** 0.79*** h=24 0.59*** 0.63*** 0.71*** 0.78*** 0.84** 0.92 Block-diagonal Maturity 3m 1y 2y 3y 4y 5y h=1 0.84 1.10 1.09 1.04 1.04 1.06 h=3 0.80 1.01 1.07 1.03 1.03 1.04 h=6 0.76 0.97 1.04 1.03 1.02 1.02 h=12 0.68*** 0.82 0.92 0.94 0.95 0.95 h=24 0.77 0.83 0.94 1.00 1.05 1.10 ThistablereportstherelativeMSFEoftheunrestrictedmacro-yieldsmodel(top panel) and of the block-diagonal macro-yields model (bottom panel) over the MSFE of the random walk for multi-step predictions of the yields. The first column reports the forecast horizon h. The sample starts on January 1970 and the evaluationperiodisJanuary1990toDecember2008. *,**and***denotesignificant outperformance at 10%, 5% and 1% level with respect to the random walk according the White (2000) reality check test with 1,000 bootstrap replications using an average block size of 12 observations. 38

D.1 The block-diagonal model and principal components This Appendix compares the principal components extracted exclusively from the macroeconomic variables used in this paper with our unspanned macro factors (top panel), and with the factors extracted from a macro-yield model with block-diagonal structure (bottom panel). Table 12 reports the pairwise correlations between the macro-yields factors and the first eight principal components extracted from our dataset of macro variables. The factors extracted from theblock-diagonalmodelareverysimilartothefirstthreeprincipalcomponents. Asstressedinthe paper, this is the consequence of the fact that the block-diagonal model treats the macroeconomic factors separately from the bond yield factors. Instead, the unspanned macroeconomic factors are significantly correlated with the first four principal components. This is not surprising since principal components contains also information that is already spanned by the yield curve. Table 12: Correlation of principal components with other factors Unspanned macro-yields factors PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 UM1 0.32 -0.87 0.19 0.04 0.14 0.09 0.04 0.02 UM2 0.63 0.22 0.19 0.44 -0.07 0.17 -0.40 0.16 Block-diagonal macro-yields factors PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 M1 0.96 -0.08 -0.06 -0.03 -0.08 0.02 0.00 0.01 M2 0.04 -0.97 0.07 -0.14 0.04 -0.01 0.05 -0.06 M3 0.07 -0.01 0.93 -0.07 0.07 0.03 -0.19 0.13 This table reports the correlation of principal components extracted from macro variables with the unspanned macro-yields factors (top panel) and the block-diagonal macro-yields factors (bottom panel). 39

D.2 LN factor and principal components This Appendix compares our macro-yields factors with the Ludvigson and Ng (2009) factor and the principal components used to construct it. Table 13 reports the pairwise correlations between the our macro-yields factors and the first eight principal components extracted from a large dataset of 131 variables.14 Results in Table 13 show that the principal components are highly correlated with the yield curve factors extracted fromourmacro-yieldsmodel. ThisimpliesthatalsotheLNfactorishighlycorrelatedwiththeyield curve factors as it just aggregates information from the principal components without separating the information already spanned by the cross-section of yields. Table 13: Correlation PC and LN factors with our macro-yield factors PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 LN L -0.18 0.04 -0.06 -0.26 0.30 -0.12 0.08 0.10 0.05 S -0.20 0.70 -0.23 0.01 0.33 0.08 0.13 0.06 -0.39 C 0.00 0.18 -0.08 0.01 0.08 -0.10 -0.08 0.15 -0.12 UM1 0.71 0.35 -0.06 0.39 0.05 -0.01 -0.09 0.10 -0.58 UM2 0.04 0.18 -0.04 0.04 -0.03 0.07 0.19 -0.10 -0.34 Thistablereportsthecorrelationofthemacro-yieldsfactorswiththeprincipal components extracted from a large dataset of macro variables and used to construct the Ludvigson and Ng (2009) factor. 14The 131 macroeconomic data series used to construct the LN factor have been downloaded from Sydney C. Ludvigson’s website at http://www.econ.nyu.edu/user/ludvigsons/Data&ReplicationFiles.zip. 40

E Comparison between initial and final estimates In this Appendix we compare the performance of the initial and the final estimates of the macroyields model. The initial estimates of the yield curve factors are computed using the two-steps OLS procedure introduced by Diebold and Li (2006). We then project the macroeconomic variables on the NS factors and use the principal components of the residuals of this regression as the initial estimates of the unspanned macroeconomic factors. These estimated factor are then treated as if they were the true observed factors. The initial parameters are hence estimated by OLS. The final estimates are obtained using the EM algorithm where the initial estimates of the factors are used in order to initialize the algorithm, as described in Section 3 and Appendix A. Table 14 reports the cumulative variance of yields and macro variables explained by the initial estimates of the macro-yields factors. Comparing these results with the ones of Table 2, it is clear that the fit of the initial estimates is at least as good as the fit of the final estimates of the model. Figure 6 shows the initial estimates of the macro-yields factors are much more volatile than the final estimates. However, the correlation between the initial estimates and the final estimates of the macro-yields factors is very high, as shown in Table 15. The out-of-sample performance of the macro-yields model improves when using the QML estimator compared to the initial estimates obtained by OLS and principal components, see Table 16. This is due to the fact that the QML estimator take appropriately into account the dynamics of the common factors and the cross-sectional heteroscedasticity of the idiosyncratic component. 41

Table 14: Cumulative variance explained by the initial estimates of the macro-yields factors Level Slope Curv UM1 UM2 Government bond yield with maturity 3 months 0.63 0.95 1.00 1.00 1.00 Government bond yield with maturity 1 year 0.64 0.83 0.99 0.99 0.99 Government bond yield with maturity 2 years 0.69 0.79 1.00 1.00 1.00 Government bond yield with maturity 3 years 0.74 0.80 1.00 1.00 1.00 Government bond yield with maturity 4 years 0.78 0.82 1.00 1.00 1.00 Government bond yield with maturity 5 years 0.82 0.84 1.00 1.00 1.00 Average Hourly Earnings: Total Private 0.07 0.30 0.35 0.35 0.71 Consumer Price Index: All Items 0.20 0.50 0.50 0.52 0.92 Real Disposable Personal Income 0.00 0.02 0.05 0.42 0.47 Effective Federal Funds Rate 0.57 0.94 0.97 0.97 0.97 House Sales - New One Family Houses 0.00 0.21 0.21 0.28 0.28 Industrial Production Index 0.02 0.02 0.05 0.84 0.86 M1 Money Stock 0.18 0.29 0.31 0.33 0.44 ISM Manufacturing: PMI Composite Index (NAPM) 0.03 0.06 0.06 0.74 0.77 Payments All Employees: Total nonfarm 0.00 0.02 0.15 0.83 0.83 Personal Consumption Expenditures Price Index 0.17 0.23 0.34 0.50 0.83 Producer Price Index: Crude Materials 0.03 0.16 0.16 0.21 0.45 Producer Price Index: Finished Goods 0.03 0.33 0.33 0.36 0.91 Capacity Utilization: Total Industry 0.03 0.19 0.26 0.69 0.69 Civilian Unemployment Rate 0.47 0.61 0.61 0.68 0.76 This table reports the cumulative share of variance of yields and macro variables explained by the initial estimatesofthemacro-yieldsfactors. ThefirstthreecolumnsrefertotheNelson-Siegelyieldcurvefactors (level,slopeandcurvature)estimatedbyOLS,thelasttwocolumnrefertotheunspannedmacroeconomic factors (UM1 and UM2) estimated by principal components. Table 15: Correlation of the final factor estimates with the initial factor estimates L S C UM1 UM2 Initial 0.97 0.96 0.85 0.94 0.96 This table reports the correlation between the finalfactorestimateswiththecorrespondinginitial factor estimates. 42

Figure 6: Comparison between the initial and the final estimates of the macro-yields factors Level Slope Curvature 10 6 14 8 4 12 6 2 4 10 2 0 8 0 -2 -2 6 -4 -4 4 -6 -6 80 90 00 80 90 00 80 90 00 UnspannedMacro1 UnspannedMacro2 2 2 MY MY0 1 1 0 -1 0 -2 -1 -3 -2 80 90 00 80 90 00 This figure displays initial (MY0) and the final (MY) estimates of the macro-yields factors. The top panel refers to the yield curve factors and the bottom panel to the unspanned macro factors. 43

Table 16: Out-of-sample performance of the initial and the final estimates Initial Estimates Maturity 3m 1y 2y 3y 4y 5y h=1 1.00 1.10 1.10 1.04 1.03 1.01 h=3 1.09 1.18 1.14 1.07 1.06 1.04 h=6 1.13 1.22 1.19 1.14 1.12 1.07 h=12 0.86 0.92 0.95 0.95 0.96 0.94 h=24 0.62** 0.65** 0.72** 0.77* 0.81 0.87 Final Estimates Maturity 3m 1y 2y 3y 4y 5y h=1 1.17 1.05 1.06 1.00 1.05 1.14 h=3 0.79* 0.93 0.99 0.96 0.99 1.02 h=6 0.78** 0.89 0.94 0.93 0.93 0.94 h=12 0.69** 0.74** 0.79** 0.80*** 0.80*** 0.80*** h=24 0.62*** 0.66*** 0.74** 0.82** 0.88* 0.97 This table reports the relative MSFE of the macro-yields model initial and final estimatesovertheMSFEoftherandomwalkformulti-steppredictionsoftheyields. Thefirstcolumnreportstheforecasthorizonh. ThesamplestartsonJanuary1970 andtheevaluationperiodisJanuary1990toDecember2008. *,**and***denote significantoutperformanceat10%,5%and1%levelwithrespecttotherandomwalk according to the White (2000) reality check test with 1,000 bootstrap replications using an average block size of 12 observations. 44

F Estimated parameters In Tables 17–18 we report the estimated parameters of the macro-yields model described in Equations(4)–(6). StandarderrorsarecomputedusingaMonteCarloprocedure,asfollows: weestimate the macro-yields model in Equations (4)–(6) and save the idiosyncratic innovations and the state innovations. We then bootstrap the state innovations and simulate the state variables. In the same way, we bootstrap the idiosyncratic innovations and simulate the idiosyncratic components. We then obtain a sample of artificial yields and macro variables by adding the simulated idiosyncratic components to the simulated state variables multiplied by the estimated factor loadings. We generate a 1000 simulated samples of yields and macro variables and for each simulated sample we estimate the macro-yields model. The standard deviations of parameters reported in Tables 17–18 are the standard deviations of the empirical distribution of each model parameter. Results in Table 17 show that most macro variables have statistically significant factor loadings on the level and slope of the yield curve, while only house sales and industrial production have significant factor loadings on the curvature. The autocorrelation coefficient in the idiosyncratic components is highly significant for all variables, except for one yield. Results in Table 18 show that the level is mainly Granger caused by the second unspanned macro factor, i.e. the real interest rate. The slope and the curvature are mainly Granger caused by the first factor, i.e. industrial production. 45

Table 17: Estimated parameters: measurement equation Γ B R L S C UM1 UM2 a t t t t t x Govt bond yield maturity 3 m 1.000 0.914 0.081 0.000 0.000 0.000 0.425 0.050 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.166) (0.018) Govt bond yield maturity 1 year 1.000 0.709 0.228 0.000 0.000 0.000 0.591 0.021 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.057) (0.003) Govt bond yield maturity 2 years 1.000 0.526 0.294 0.000 0.000 0.000 0.549 0.007 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.061) (0.001) Govt bond yield maturity 3 years 1.000 0.405 0.294 0.000 0.000 0.000 -0.194 0.002 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.198) (0.001) Govt bond yield maturity 4 years 1.000 0.324 0.270 0.000 0.000 0.000 0.548 0.008 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.051) (0.001) Govt bond yield maturity 5 years 1.000 0.267 0.241 0.000 0.000 0.000 0.739 0.009 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.050) (0.001) Av Hourly Earnings: Tot Private -0.055 -0.031 0.087 0.898 -0.054 0.350 0.941 0.021 (0.096) (0.066) (0.073) (0.369) (0.242) (0.625) (0.032) (0.003) CPI: All Items -0.080 -0.089 0.073 0.819 0.211 0.437 0.846 0.103 (0.089) (0.064) (0.067) (0.302) (0.208) (0.591) (0.032) (0.010) Real Disposable Personal Inc -0.016 -0.085 0.087 0.599 -0.287 -0.029 0.778 0.262 (0.082) (0.066) (0.063) (0.375) (0.314) (0.526) (0.034) (0.041) Effective Federal Funds Rate 0.290 -0.174 0.029 -0.294 0.256 -2.441 0.947 0.017 (0.057) (0.040) (0.035) (0.177) (0.140) (0.382) (0.021) (0.002) House Sales -0.017 0.073 0.122 0.833 0.035 0.228 0.963 0.011 (0.104) (0.058) (0.066) (0.308) (0.191) (0.728) (0.021) (0.002) Industrial Production Index 0.174 0.127 0.085 0.363 0.578 -1.092 0.797 0.086 (0.091) (0.060) (0.047) (0.174) (0.171) (0.687) (0.033) (0.008) M1 Money Stock -0.249 -0.065 -0.036 0.267 -0.158 1.750 0.952 0.054 (0.100) (0.069) (0.049) (0.220) (0.237) (0.663) (0.021) (0.006) ISM Man. PMI Composite Index 0.069 0.296 -0.023 -0.191 0.732 -0.028 0.907 0.024 (0.084) (0.068) (0.047) (0.340) (0.284) (0.563) (0.041) (0.006) Paym All Emp: Total nonfarm 0.189 0.292 -0.010 -0.217 0.622 -0.921 0.943 0.009 (0.076) (0.061) (0.041) (0.293) (0.246) (0.530) (0.032) (0.003) PCE 0.320 0.328 0.017 0.001 0.049 -1.843 0.654 0.014 (0.013) (0.012) (0.010) (0.028) (0.027) (0.094) (0.043) (0.003) PPI: Crude Materials -0.068 0.199 0.069 0.642 0.117 0.823 0.982 0.011 (0.104) (0.055) (0.059) (0.245) (0.191) (0.714) (0.022) (0.002) PPI: Finished Goods 0.182 -0.157 -0.041 0.101 0.343 -1.603 0.976 0.044 (0.134) (0.079) (0.056) (0.216) (0.339) (0.914) (0.019) (0.009) CU: Total Industry 0.113 0.250 0.035 -0.048 0.592 -0.432 0.963 0.022 (0.095) (0.064) (0.043) (0.215) (0.223) (0.665) (0.019) (0.004) Civilian Unemp Rate -0.077 0.178 -0.043 0.238 0.505 0.871 0.897 0.144 (0.098) (0.083) (0.058) (0.306) (0.310) (0.675) (0.030) (0.021) Thistablereportstheestimatedparametersandcorrespondingstandarderrorsforthemeasurement equationofthemacro-yieldsmodelinEquations(4)and(6). Thestandarderrorsarecomputedby 1000 Monte Carlo simulations. 46

Table 18: Estimated parameters: state equation A µ Q L S C UM1 UM2 t−1 t−1 t−1 t−1 t−1 L 0.998 0.013 -0.004 -0.020 0.051 0.029 0.055 0.050 0.034 0.004 -0.032 t (0.010) (0.010) (0.014) (0.024) (0.023) (0.064) (0.019) (0.024) (0.050) (0.016) (0.021) S -0.016 0.966 0.025 0.144 0.032 0.049 0.050 0.102 -0.054 0.017 -0.040 t (0.022) (0.017) (0.022) (0.050) (0.039) (0.159) (0.024) (0.049) (0.076) (0.032) (0.041) C 0.033 0.030 0.889 0.155 0.078 -0.221 0.034 -0.054 0.750 -0.079 -0.007 t (0.037) (0.038) (0.048) (0.079) (0.071) (0.248) (0.050) (0.076) (0.241) (0.077) (0.070) UM1 0.005 -0.038 0.006 0.996 0.007 -0.106 0.004 0.017 -0.079 0.032 0.000 t (0.009) (0.023) (0.019) (0.014) (0.016) (0.074) (0.016) (0.032) (0.077) (0.020) (0.015) UM2 0.002 0.009 -0.012 0.009 0.979 -0.004 -0.032 -0.040 -0.007 0.000 0.035 t (0.011) (0.024) (0.020) (0.016) (0.021) (0.079) (0.021) (0.041) (0.070) (0.015) (0.034) Thistablereportstheestimatedparametersandcorrespondingstandarderrorsforthestateequationofthemacro-yields model in Equation (5). The standard errors are computed by 1000 Monte Carlo simulations. 47