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Keywords:

  • C13;
  • C32;
  • E30;
  • E43;
  • E52
  • affine term structure model;
  • macro finance;
  • unit root;
  • stochastic volatility

Abstract

  1. Top of page
  2. Abstract
  3. 1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK
  4. 2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK
  5. 3. THE EMPIRICAL MODELS
  6. 4. CONCLUSION
  7. Appendices
  8. LITERATURE CITED

This paper generalizes the standard homoscedastic macro-finance model by allowing for stochastic volatility, using the “square root” specification of the mainstream finance literature. Empirically, this specification dominates the standard model because it is consistent with the square root volatility found in macroeconomic time series. Thus it establishes an important connection between the stochastic volatility of the mainstream finance model and macro-economic volatility of the Okun–Friedman type. This research opens the way to a richer specification of both macro-economic and term structure models, incorporating the best features of both macro-finance and mainstream finance models.

This paper develops a general affine macro-finance model of the U.S. macroeconomy and the Treasury bond market. As the name suggests the macro-finance approach allows bond yields to reflect macro-economic variables as well as latent variables representing financial market factors. It is based on the central bank model (CBM) developed by Svensson (1999), Rudebusch (2002), Smets (1999), Kozicki and Tinsley (2005), and others, which represents the behavior of the macroeconomy in terms of the output gap (gt), inflation t) and the short-term interest rate (rt). The model developed in this paper allows bond yields to reflect changes in macro-economic volatility related to the underlying rate of inflation.

In turn, the behavior of bond yields helps inform the specification of the macroeconomy, yielding new insights into the operation of monetary policy. In particular, early macro-finance studies showed that although macro-economic variables provide a good description of the behavior of short rates they do not provide an adequate description of long-term yields (Kozicki and Tinsley 2001, Ang and Piazzesi 2003). This finding has spawned an important macro-econometric literature that augments the CBM with latent variables, capturing exogenous changes in inflation and interest rates (Kozicki and Tinsley 2005 provide a useful summary). This literature shows that these rates are characterized by a nonstationary common trend (or unit root) that seems to be explained by the underlying rate of inflation. It follows the standard macro-econometric literature in assuming a homoskedastic (fixed) variance structure. This situation is familiar to macro-economic modelers but poses a potential problem for term structure researchers: it is well known that asymptotic (long-maturity) yields are not properly defined if the interest rate is driven by a random walk (a homoskedastic unit root process).

This theoretical problem was first raised as an empirical issue by Dewachter and Lyrio (2006), but with this notable exception, macro-finance modelers have avoided it by assuming that the underlying inflation variable follows a near-unit-root process (Ang and Piazzesi 2003, Rudebusch and Wu 2003, Dewachter, Lyrio, and Maes 2006). However, because this variable is stationary, it mean-reverts to a constant rather than the variable end-point suggested by unit root macro-economic models. As Kozicki and Tinsley (2005), Dewachter and Lyrio (2006), and others note, this means that it cannot be interpreted as a long-run inflation expectation because it is anchored to a constant that cannot be influenced by monetary policy.

Mainstream finance yield curve research avoids these problems by using heteroskedastic (stochastic volatility) interest rate models based on Cox, Ingersoll, and Ross (1985, hereafter CIR). I modify their continuous time specification for use with discrete-time macro-economic data, getting a sensible forward rate asymptote without placing constraints on the roots of system. The stochastic trend is estimated using the extended Kalman filter, which is also standard in the mainstream finance literature. Model restrictions allow the stochastic trend-volatility term to be interpreted as an inflation trend, consistent with the hypothesis that macro-economic volatility is influenced by the underlying rate of inflation (Okun 1971, Friedman 1977, Engel 1982).1 This specification encompasses the standard macro-finance model, which is decisively rejected by the data.

The research described in this paper was initially motivated by my interest in the asymptotic yield problem raised by early drafts of the Dewachter and Lyrio (2006) paper. I expected the new specification to outperform the standard one in explaining long maturity yields and with this in mind I extended the conventional (10-year maximum) maturity data set to include a 15-year yield, the longest available historically. However, the results are surprising in this respect. The new specification does give a dramatic improvement in fit, but the main reason for this is the importance of the Okun–Friedman heteroskedasticity effect found in the macro-data. Once this is allowed for, its more flexible yield curve specification adds very little. This finding suggests that this new macro-finance framework—which uses estimates of macro-economic volatility to inform the stochastic volatility parameters of the term structure model—should be better at discriminating between rival models than the mainstream finance one (Chen and Scott 1993, Dai and Singleton 2002), which does not. It further suggests that CBM-based studies of monetary policy should use the heteroskedastic framework rather than the current homoskedastic one, allowing the effects of the stochastic trend on the second as well as the first moments of the system to be analyzed.

The paper is set out as follows. The next section describes the macro-economic model and its stochastic structure, supported by Appendix A. Section 2 derives the bond pricing model, supported by Appendix B. It discusses the theoretical problems posed by the unit root in the standard specification and shows how these are avoided in the general affine specification. The two respective empirical models are compared in Section 3. Section 4 offers a brief conclusion and suggestions for further research.

1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK

  1. Top of page
  2. Abstract
  3. 1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK
  4. 2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK
  5. 3. THE EMPIRICAL MODELS
  6. 4. CONCLUSION
  7. Appendices
  8. LITERATURE CITED

This section specifies the macro-economic framework. This is “general affine” or “exponential-affine” in the sense of Duffie and Kan (1996) and Duffie, Filipovic, and Schachtermayer (2003, to be explained). It consists of a heteroscedastic macro-economic vector autoregression (VAR) augmented by two latent variables, which is specified under the physical (or observed) probability measure inline image. The yield model is specified under the risk neutral measure inline image in the next section.

1.1 The Macro-Economic Dynamics

The macro-model is based on the CBM. It represents the behavior of the macroeconomy in terms of the inflation rate t), output gap (gt) and the 3-month Treasury Bill rate (r1,t). These are part of an n-vector zt of macro-economic variables driven by the difference equation system:

  • image(1)

where G is a lower triangular matrix, ηt is an n-vector of independent and identically distributed (i.i.d.) orthogonal errors, and yt is a k-vector of latent factors. These follow the first-order process:

  • image(2)

where ɛt is a k-vector of i.i.d orthogonal errors, θ={θ1,…, θk}′ and Ξ=Diag1,…, ξk}.2 It is assumed that zt is observed without measurement error and that yt is unobservable. I estimate yt using the extended Kalman filter (Harvey 1989, Duffee and Stanton 2004) as described in Appendix C.

The specific model developed in this paper defines zt={gt, πt, r1,t}′, inline image, ɛt=inline image, and inline image. In my preferred model, inline image is a martingale driving the inflation asymptote: inline image, where inline image is a shift constant. The central tendency r* of the real interest rate r is inline image plus the constant inline image is assumed to be a mean-reverting Gaussian variable inline image, so inline image plays an identical role to inline image, which it is convenient to set to zero, making inline image the long-run mean. The central tendency of the nominal interest rate is thus inline image, which reverts to the asymptote inline image The output gap is also assumed to be a zero-mean-reverting variable: g*t= 0. These equilibrium conditions are enforced by imposing a set of restrictions on (1):

  • image

where

  • image(3)

to give the equilibrium solution z*t= (I−ΣLl=1Φl)−1Φ0(yt+ϕ) =R(yt+ϕ).

This system can be consolidated by defining xt={yt, zt}′; vt={ɛ′t, η′t}′ and combining (1) and (2) to get an Lth order difference system described in Appendix A as (A1). The yield model employs the state space form (Harvey 1989), obtained by arranging this as first-order difference system describing the dynamics of the state vector:

  • image(4)

where Xt={yt, zt, … , zt-1}′ is the state vector, Wt=C⋅{ɛ′t, η′t, 01,Nkn}′, and Θ, Φ, and C are defined in Appendix A. Xt has dimension N=k+nl.

The macro-economic data were provided by Datastream and are shown in Figure 1. πt is the annual CPI inflation rate and r1,t is the 3-month Treasury Bill rate. The output gap series gt is the quarterly OECD measure, derived from a Hodrick–Prescott filter. The yield data were taken from McCulloch and Kwon (1991), updated by the New York Federal Reserve Bank.3 These have been extensively used in the empirical literature on the yield curve. To represent this curve I use 1-, 2-, 3-, 5-, 7-, 10-, and 15-year maturities. These yield data are available on a monthly basis, but the macro-economic data dictated a quarterly time frame (1961Q4–2004Q1, a total of 170 periods). The quarterly yield data are shown in Figure 2. The 15-year yield is typically shown shown at the top of the figure, while the shorter maturity yields are most frequently shown at the bottom; however, reversals in this ordering tend to occur near business cycle peaks.

image

Figure 1. Macroeconomic Variables. Note: CPI inflation and 3-month T-Bill intrest rate are from Datastream. Output gap is from OECD.

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image

Figure 2. U.S. Treasury Discount Yields: United States. Note: U.S. Treasury discount bond equivalent yield data are compiled by McCulloch and Kwon (1990) updated by the New York Federal Reserve Bank.

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These inflation and interest rates all exhibit a high degree of persistence, which could be the effect of slow mean reversion, unit roots, or structural breaks. Table 1 shows the means, standard deviations, and first-order autocorrelation coefficients of these data, as well as KPSS (Kwiatowski et al. 1992) and adjusted Dickey–Fuller (ADF) test results. The ADF tests show that the null hypothesis of nonstationarity for these variables cannot be rejected at the 5% level. The KPSS statistics for inflation and 1- to 10-year interest rates are only significant at the 10% level, suggesting that the null hypothesis of stationarity may just be acceptable. However, the KPSS statistic for the 15-year rate is almost significant at the 5% level. Moreover, Fama (2006) argues persuasively that the long upswing and downswing in rates evident in the figures was the result of a succession of permanent shocks that were on balance positive until 1981 and negative thereafter. In this paper, I follow Kozicki and Tinsley (2005), Dewachter and Lyrio (2006), and Fama (2006) in analyzing a macro-econometric model characterized by a unit root.

Table 1.  Data Summary Statistics: 1961Q4–2004Q1
 gπr1r4r8r12r20r28r40r60
  1. Note: Inflation (π) and 3-month T-Bill rate (r1) are from Datastream. Output gap (g) is from OECD. Yield data are U.S. Treasury discount bond equivalent data compiled by McCulloch and Kwon (1990) updated by the New York Federal Reserve Bank. Mean denotes sample arithmetic mean expressed as percentage per annum; Std. is standard deviation and Auto. is the first-order quarterly autocorrelation coefficient. Skew. and Kurt. are standard measures of skewness (the third moment) and kurtosis (the fourth moment). KPSS is the Kwiatowski et al. (1992) statistic testing the null hypothesis of level stationarity. The 10% and 5% significance levels are 0.347 and 0.463, respectively. ADF is the adjusted Dickey–Fuller statistic testing the null hypothesis of nonstationarity. The 10% and 5% significance levels are 2.575 and 2.877, respectively.

Mean 0.0266 4.4345 5.8051 6.3954 6.6305 6.7849 7.0021 7.1478 7.2513 7.4383
Std. 2.3382 2.9841 2.7750 2.8174 2.7355 2.6508 2.5447 2.4790 2.4194 2.3833
Skew.−0.575  1.4161 1.1907 1.8820 1.8697 1.8793 1.9633 1.9622 1.9215 1.8188
Kurt. 1.0230 1.5178 2.3986 1.1986 1.0622 0.9505 0.9166 0.7550 0.5929 0.3945
Auto. 0.4632 0.9921 0.9815 0.9892 0.9923 0.9944 0.9953 0.9963 0.9969 0.9971
KPSS 0.2151 0.3399 0.3100 0.3307 0.3348 0.3399 0.3475 0.3548 0.3761 0.4298
ADF−4.133 −2.411 −2.110 −2.100 −2.063 −2.031 −2.043 −1.991 −1.951 −2.091 

1.2 The Stochastic Structure

The standard macro-finance model assumes that the volatility structure is homoskedastic and Gaussian: WtN(0N, Ω), while mainstream finance models usually assume that volatility is stochastic, driven by square root processes in one or more of the state variables.4Dai and Singleton (2000) derive “admissibility” conditions to ensure that these state variables remain nonnegative and the variances are well defined. They classify an admissible model with N state variables and m independent square root factors conditioning volatility is classed as inline image Thus the standard macro-finance model (which is homoskedastic) is classified as inline image and the mainstream specification (with a single stochastic volatility term) as inline image.

This paper develops a model that encompasses the inline image and inline image specifications. These models all generate affine yield curves because the probability distributions underpinning them are all “exponential-affine” in the sense of Duffie, Filipovic, and Schachtermayer (2003). They define a process as exponential-affine under any measure inline image if the conditional moment generating function (MGF) (inline image | Xt]) for Xt+1, is an exponential-affine (log-linear) function of Xt. EM denotes the expectation under the measure inline image while E and V denote the mean and variance under the state price density and ν is a vector of Laplace coefficients. The MGF is the Laplace transform of the density of Xt+1. For example inline image assumes that ɛ1,t is normally distributed with mean zero and standard deviation δ01 and we use the formula for the MGF of a normal variable:

  • image(5)

In inline image, this latent variable also drives volatility through a square root process similar to the diffusion for the spot interest rate in CIR. They show that in discrete time, this has a noncentral conditional χ2 distribution. If we normalize the time interval (st) in their equation (18) as unity and replace r by inline image:

  • image(6)

where 2c is the scale factor, inline image is the noncentrality parameter, and 2cμ shows the degrees of freedom.

This process is exponential-affine because its conditional MGF is a log-linear function of inline image:

  • image(7)

(provided that ν < c,Johnson and Kotz 1970). Differentiating (7) with respect to ν once, twice, and then setting ν to zero gives the conditional mean and variance:

  • image

where

  • image(8)

In the limiting case of a unit root, the degree of freedom parameter is zero. This model is studied by Seigel (1979) and his basic results are reported in chapter 29 of Johnson and Kotz (1970). Important results have also been obtained for this case by Gourieroux and Jasiak (2002). In this limit: inline image; δ01=μ= 0; c= 2/δ11, and (8) simplifies to:

  • image(9)

This process is a martingale: the expectation of any future value is equal to the current value. However, unlike the random walk model, the error variance is also proportional to this value. These models can all be represented as: inline image To be consistent with (2) we set the intercept inline image equal to μ for the models (5) and (8) and to zero in (9).

In inline image, this stochastic trend also conditions the volatility of the other variables. It is ordered as inline image, the first variable in the (k+n) vector xt. The other contemporaneous variables are put into an (k+n− 1) vector x2,t, so that: xt={x1,t, x2,t}′ and conformably: vt={w1,t, v2,t}′ and wt={w1,t, w2,t}′, where inline image. Similarly, writing Xt={x1,t, X2,t}′ and partitioning Wt, Θ, Φ,C conformably (see Appendix A), (4) becomes:

  • image(10)

where inline image, and inline image. In this paper subscripts 1 and 2 denote partitions of N (or k+n) dimensional vectors and matrices into 1 and N− 1 (or k+n− 1). The stochastic structure for (10) is described in Appendix A. The distribution of x2,t and x2,t conditional upon x1,t−1 is assumed to be Gaussian. The conditional covariance of x2,t is Σ01x1,t−1, where: Σi=C22ΔiC22 and Δi=Diag{{δi2, … , δi(k+n)}, 0′Nkn}; i= 0, 1.C22 is a lower triangular and Δi; i= 0, 1 are deficient diagonal (N− 1)2 matrices.

This model is the discrete time analogue of the “general affine”inline image model developed by Duffie and Kan (1996), which generalizes the CIR model by adding a translation or shift constant to a model variable like an interest rate or stochastic trend when defining the volatility term in the rate diffusion. However, it is convenient to allow for this shift by using inline image in (3) instead, keeping inline image a CIR process but making the inflation asymptote π* a Duffie and Kan (1996) process. Importantly, the Duffie and Kan inline image model encompasses inline image. Similarly, the discrete time inline image model is a special case of my inline image model, which can be obtained by taking the limit as c tends to infinity. Setting δ11 and Σ1 to zero then makes x1,t and x2,t homoskedastic. This also renders the distribution of inline image Gaussian, allowing this to be defined in the inline image model as another zero-mean-reverting latent variable 1= 0).5 These models are “admissible” in the sense of Dai and Singleton (2000, 2002) in that they ensure that the variance structure remains nonnegative definite.6 They can be used independently of the yield specification, as for example by Kozicki and Tinsley (2005). However, my interest is to use them jointly with bond market data, using the macro-yield framework developed in the next section.

2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK

  1. Top of page
  2. Abstract
  3. 1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK
  4. 2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK
  5. 3. THE EMPIRICAL MODELS
  6. 4. CONCLUSION
  7. Appendices
  8. LITERATURE CITED

This section is supported by Appendix B and shows how exponential-affine MGFs can be used to model the yield curve consistently with the macro-models of the previous section. It is based on the fact (Duffie, Filipovic, and Schachtermayer 2003) that the MGF of a distribution that is exponential-affine under the risk neutral measure inline image generates an exponential-affine discount bond price model:

  • image(11)

The natural logarithm of this price is denoted by pτ,t and is linear in Xt. Reversing sign and dividing by maturity τ gives the discount yields: rτ,t=−pτ,t/τ=ατ+β′τXt, where αττ/τ; βττ. The slope coefficients βτ are known as “factor loadings.” Stacking the yield equations for τ= 4, 8, 12, 20, 28, 40, and 60 quarters and adding an error vector et gives the multivariate regression model used in Section 3 for the (7 × 1) vector rt:

  • image

where

  • image(12)

and et is an i.i.d. error vector.7 This is the affine yield curve framework used in this paper. The remainder of this section discusses the measure inline image used for asset pricing and the “essentially affine” yield models inline image and inline image corresponding to the macro-models inline image and inline image of the previous section.

2.1 The Risk Neutral Measure Q

Assets are priced under the risk neutral measure, which adjusts the state probabilities in such a way that they all have the same expected return. These adjustments depend upon “price of risk” parameters that show the effect of model variables on the risk premia. For the yield model to be affine these prices must also be affine in the state variables. So, for example, the variable λ1,t shows the price of risk associated with the stochastic trend, which plays an important role in this analysis:

  • image(13)

If this is zero, then an “asymptotic” or “end-point” portfolio that is constructed so that it is only exposed to shocks in x1,t has a zero risk premium and is expected to earn the spot rate. If it is constant 1,t10), then variations in this risk premium depend only upon variations in volatility, such as those induced by x1,t in inline image. This parameter plays the key role in that model. If λ11 is also nonzero then the trend can influence the risk premia thorough variations in the price of risk, even if volatility is fixed as in inline image, so λ11 plays the key role in that model. Appendix B shows how the prices of risk associated with the other variables are adjusted, following Duffee (2002). After this modification, the models inline image and inline image are classified as the “essentially affine” models inline image and inline image, respectively, represented by the empirical models M0 and M1 in Section 3. Following Dewachter and Lyrio (2006), both models incorporate the restrictions: ξ1= 1; Λ12= 0N−1.

As Appendix B explains, variations in λ1,t are a nuisance in the inline image model M1 and the conventional assumption is that λ1,t10 for that model. On the other hand, it is important to allow λ1,t to reflect variations in x1,t in M0, so in this case λ1,t1011x1,t. This means that M1 does not encompass M0. However, an encompassing specification that nests both models can be constructed by relaxing the usual macro-finance assumption that bond market participants use the true value of ξ1 in pricing. This gives my “baseline” model M2, which uses a new parameter inline image to describe their estimate of the speed of adjustment of x1,t under inline image, which may differ from the parameter ξ1 defined in the macro-model.8 This modification is a technical one, designed to allow encompassing tests, but could detect an error in the market estimate of ξ1. Appendix B derives the MGF of M2 under measure inline image and shows how it can be used as a “model generating function” to derive the other empirical models. It can for instance be used as a moment generating function to give the dynamics of this system under inline image:

  • image(14)

where the time t expectations of the error terms are zero under inline image. Table 2 shows parameter values for models M0-M2, where

  • image(15)

These reduced form parameters show the effects of the coefficients λ10 and λ11 modeling the price of risk associated with x1,t. The vectors Λ20 and Λ21 and the matrix Λ22 are defined in (B6) and model the price of risk associated with x2,t. Obviously, if all of these parameters are set to zero, the parameters revert to those shown (for inline image) in the first column. The parameter values for model M2 are shown in the second column. Those for M1 in the next column assume the market uses the true value of inline imageξ1. The parameters for M0 in the final column are standard and Appendix B shows that they can be derived from those for M2 by setting δ11 and Σ1 to zero and taking the limit as c tends to infinity. In this case we replace inline image by inline image, which is the Duffee (2002) risk-adjusted parameter.

Table 2.  Dynamic Coefficients for Different Yield Models and Measures
  1. Note: This table shows how the shift from the state probability measure inline image underpinning the macro-model to the risk-neutral measures used to price financial assets shift the coefficients of (14). The scalars λ10 and λ11 reflect the price of risk (or risk–reward ratio) associated with variations in x1,t, while Hi and ϒi; i= 0, 1 are matrices of parameters reflecting the price of risk associated with the other state variables and are defined in (15). The baseline model M2 allows inline image to differ from the parameter ξ1 defined in the macro-model. M1 is the EA1(N) model with inline image. M0 specializes M2 by setting δ11 and Σ1 to zero and taking the limit as c tends to infinity. This gives the standard EA0(N) model with the Duffee (2002) risk adjustments shown in the final column.

Model: M2M1M0
Type:  EA1(N)EA0(N)
Measure:PQQ1Q0
θ1inline imageinline imageinline image
inline imageinline imageinline imageinline image
Θ2inline imageinline imageinline image
Φ21inline imageinline imageinline image
Φ22ΦQ2222−Λ22inline imageinline image

2.2 TheEA1(N)Yield Specifications (M1 and M2)

The coefficients of (11) are partitioned Ψτ conformably with (10) as 1,τ, Ψ′2,τ}′. They are recursive in maturity. Since pτ,1=r1,t its coefficients have the starting values γ11,1= 0; and Ψ2,1=J2,r, where J2,r is a selection vector such that J2,rX2,t=r1,t. It is also recursive in the sense that Ψ2,τ does not depend upon ψ1,τ−1 (or γτ−1):

  • image(16)

where inline image (defined in Table 2) adjusts Φ22 to allow for the effect of X2,t on the associated prices of risk. I assume that the roots of this subsystem are stable under inline image, so this has the asymptote:

  • image(17)

Dividing by τ and taking the limit as this goes to infinity gives the limit for β2 shown in Table 3. This Gaussian substructure is common to all models, but the structure of the remaining coefficients depends critically upon the model specification, particularly if the system is nonstationary.

Table 3.  Asymptotic Yield Coefficients for the Limit of a Unit Root
ModelEA0(N)EA1(N)
  1. Note: This table shows the effect on the yield model of imposing unit root restrictions inline image on x1,t under inline image. The maturity limits for its factor loading (β*1) are very different in the two models. This is reflected in the behavior of the forward rate asymptote (f*t). β*2=Ψ*2/τ is common to both specifications where Ψ*2= limτ[RIGHTWARDS ARROW]Ψ2,τ=inline image.

Limit:  
 β*1= limτ[RIGHTWARDS ARROW]β1,τinline image0
 β*2= limτ[RIGHTWARDS ARROW]β2,τ0N−10N−1
 f*t= limτ[RIGHTWARDS ARROW]fτ,t2H0)′Ψ*2+τβ*1x1,tinline image
inline image 

Importantly, as Dai and Singleton (2002) and others note, nonstationarity under inline image is not a problem in the EA1 specification. Indeed their results, like previous mainstream inline image estimates (Chen and Scott 1993) suggest that there is a root that is significantly larger than unity under inline image. The volatility of x1,t+1 is linear in x1,t and so the equation determining its price coefficient includes a nonlinear Jensen effect. For M2:

  • image(18)

For a regular solution:

  • image(19)

As Campbell, Lo, and Mackinlay (1997) note in a similar heteroskedastic yield curve model,9 the price parameter ψ*1 is determined by a quadratic rather than a linear equation and is well defined (with β*1= 0) even if inline image.10 For the intercept:

  • image(20)

Unit roots are not a problem in inline image. Indeed, they simplify the model structure, giving model M1 with inline image. Substituting μ= 0 into (20)“switches off” the logarithmic term and makes γτ and hence the asymptotic forward rate f* independent of ψ*1, as shown in Appendix B. In other words, because the volatility of x1,t is proportional to x1,t−1, the associated Jensen effects are found in (18) but not in (20). This also simplifies the risk premium, derived in Appendix B as (B12).

2.3 The StandardEA0(N)Specification (M0)

The pricing formulae for this model are well known and Appendix B considers them as limits of these baseline M2 formulae. Taking the limit of (18) as c tends to zero gives a quadratic recursion. However, the restriction δ11= 0“switches off” this quadratic effect, reducing (18) to a linear recursion:

  • image(21)

The intercepts do follow a quadratic recursion:

  • image(22)

with the parameters defined in Table 2. In contrast to the M1 model, the intercept exhibits the nonlinear Jensen effects in M0 and not the first slope coefficient.

If inline image, then clearly (21) has a unit root and as Appendix B and Table 3 show, the long forward rates behave like (−)τ2 in the limit, reflecting the well-known asymptotic problem. In the specification of Dewachter and Lyrio (2006), x1 has a unit root under inline image but is mean reverting under inline image: inline image. This provides a neat way of avoiding the asymptotic yield problem while allowing the inflation asymptote in the macro-model to be a variable end-point as in Kozicki and Tinsley (2001). However, this restriction constrains the rate at which ψ1,τ grows in the recursion (21), constraining the effect of the stochastic trend on short-maturity yields. It also means that the associated factor risk premium, the excess return expected for holding the “asymptotic” portfolio (constructed so that the portfolio weights sum to: ψ1= 1, Ψ′2= 0N−1), is negatively related to the inflation trend, as Appendix B demonstrates. Moreover, the EA1(N) model relaxes this restriction and my results, reflecting those of mainstream research (e.g., Chen and Scott 1993, Dai and Singleton 2002), suggest that this root is significantly greater than unity under inline image.

3. THE EMPIRICAL MODELS

  1. Top of page
  2. Abstract
  3. 1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK
  4. 2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK
  5. 3. THE EMPIRICAL MODELS
  6. 4. CONCLUSION
  7. Appendices
  8. LITERATURE CITED

The empirical model consists of a heteroskedastic VAR describing the three macro-economic variables (4) and the associated equations describing the seven representative yields (12). It is estimated by quasi-maximum likelihood and the extended Kalman filter, which gives optimal linear estimates of the latent variables in this situation. The likelihood function is derived in Appendix C. The preliminary tests reported in Section 1.1 indicated the presence of a unit root in the macro-economic and yield data. Further (Akaike information criterion (AIC)) tests suggested a third-order lag structure for (1). With n= 3, k= 2, and l= 3, there are N= 11 state variables. Consequently this research focussed on the inline image and inline image specifications. The baseline model M2 uses 66 parameters11 and has a log-likelihood L(2) = 747.5 as shown in Table 5.

Table 5.  Model Evaluation 1961Q4–2004Q1
Model MSpecification* macro/financeRestrictions (w.r.t M4)ParametersLog-likelihood
k(M)k(4)-k(M)k(3)-k(M)k(2)-k(M)L(M)L(4)-L(M)L(3)-L(M)L(2)-L(M)
  1. Notes: (*) Model specification Am(N), EAm(N), where m= number of variables conditioning volatility and N= number of state variables.

 Standardinline image        
0A0(N)/EA0(N)inline image6016116678.2142.4 135.6 138.6 
χ295δ11= 0, Δ1= 0      26.30 19.68 12.59
pξ1= 1       0.0   0.0   0.0 
1Stochastic trend         
A1(N)/EA1(N)inline image6313 3747.0  5.40   1.00
χ295inline image      22.36   7.81
pinline image       0.97   0.80
2Baseline         
A1(N)/EA1(N)inline image6610      
χ295inline image    747.5  4.4   
p       18.31  
         0.93  
3Hybrid         
A1(N)/EA0(N)inline image71 5  745.4  8.60  
χ295       11.07  
p        0.13  
4Encompassing 76       
A1(N)/EA1(N)     749.7   
χ295
p

The inline image model M1 specializes this by assuming a unit root and maintaining the standard identity between macro- and yield parameters under inline image: inline image.12 These 3 restrictions are easily accepted by the data: the χ2(3) likelihood ratio test gives an acceptance value of p= 0.97. The unit root inline image specification M0 is also nested in M2, employing six restrictions: δ11 and Δ1(4) are set to zero and again ξ1= 1. However, its log-likelihood of L(0) = 678.2 is much lower than for the other models and it is overwhelmingly rejected against M2. This rejection is largely due to the restriction δ11= 0. The effect of this is twofold: (i) it makes the stochastic trend homoskedastic and (ii) it removes the nonlinearity from (18), reducing it to the linear recursion (21). Theoretically, these two effects are inextricably related because the parameters of the stochastic structure structure (δ01, δ11, Δ0, Δ1, and C22) are not affected by the change of measure and the conventional macro-finance assumption is that they are the same in the macro- and yield models. However further relaxing this assumption allows (i) and (ii) to be separated.

To explore this idea, I constructed two new models: M3 and M4. The “encompassing” specification M4 nests all of the other models. It is based on M2 but uses 10 new parameters (inline image, and inline image) to replace their macro-equivalents in the yield model. It has the log-likelihood value L(4) = 749.7. As Table 5 shows, model M1 is acceptable against this alternative, suggesting that the conventional macro-finance assumption is valid. Model M3 then specializes M4 by using the restrictions inline image and inline image. This is a hybrid in which the true macro-model is the heteroscedastic inline image model, but the bond market mistakenly uses a best-fit inline image specification instead of inline image Comparing its log-likelihood L(3) with that L(0) of M0 which it nests, gives an estimate of effect (i) of heteroskedasticity in the macroeconomy. This is highly significant. Indeed, M3 is accepted against the alternative of M4 (p= 0.13), with a likelihood almost as high as that for M2 and my preferred model M1. Although these models are not directly comparable, this observation tells us that (ii) the increase in fit due to the use of a macro-consistent yield model, is in practice relatively small. In other words, if the yield model parameters (inline image, and inline image) are estimated separately they not very well determined statistically. In practice, these parameters are determined by the macro-finance restriction, which equates them with their macro-model analogues and thus uses macro data to pin them down.

Since these L(M) values are the sum of log-likelihood values at each period (Appendix C) they can be analyzed as time series. Figure 3 shows the effect of disaggregating the differences [L(3)–L(0)] due to effect (i) of heteroskedasticity in the macroeconomy. This is very marked in the early 1980s when the stochastic trend peaks, but also noticeable when the trend is low, at the beginning and the end of the estimation period. The difference [L(4)–L(3)] due to (ii), the use of the more general inline image yield curve specification, is much smaller and reflecting this, the residuals from the heteroscedastic macro-based specifications M1, M2, M3, and M4 are all very similar. Despite the theoretical superiority of the inline image-based yield specification, it is hard to see any systematic improvement over maturity or over time. The impulse responses and factor loadings of these models are also similar. For that reason I now focus on the results for the preferred model, M1.

image

Figure 3. Model Performance (Log-Likelihood by Period). Note: M0 is the standard heteroskedastic A0(N)/EA0(N) model. M3 is a hybrid in which the true macro-model is the heteroskedastic A1(N) model, but the bond market mistakenly uses a best-fit EA0(N) specification instead of EA0(N). This comparison gives an estimate of effect of heteroskedasticity in the macroeconomy.

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3.1 The Empirical Macro-Model

At the core of this model there is a macro VAR with a steady state solution dictated by the restrictions (3). The novelty here is the introduction of the square root volatility effects implied by the CIR-based term structure model. The model is driven by a nominal factor x1,t and a real factor x2,t. Model estimates of these factors are shown in Figure 4, along with their 95% confidence intervals. Most of the work is done by the nominal factor, which has a unit root. Since x1,t has a noncentral chi-square distribution, the downside variance is smaller than the upside, but this asymmetry is only apparent at the beginning of the estimation period when the underlying inflation rate is low. This variable drives the conditional heteroskedasticity in the macro-variables. Their one-quarter-ahead forecasts values and 95% confidence intervals are shown in Figure 5. The effect of heteroskedasticity is particularly pronounced in the case of the spot rate, consistent with the finding in univariate models (Chen et al. 1992, Ait-Sahalia 1996, Stanton 1997, and others). Its variance is low over the first 4 years of the estimation period, consistent with the ex post stability of interest rates over this period (Figure 5c). The behavior of the spot rate over the medium term is also influenced by the real factor x2,t, as is clear from a comparison with Figure 4(b). As expected, this real interest series reveals a marked tightening of monetary policy in the late 1970s, with a very relaxed stance in the early 1990s and again post-millennium. The model attributes the ultra-low interest rates seen over the early years of the millennium to a relaxation of monetary policy, coming against a background of a low underlying inflation rate.

imageimage

Figure 4. (a) Inflation Factor (with 95% Confidence Band) and (b) Real Rate Factor (with 95% Confidence Band).

imageimageimage

Figure 5. (a) Output Gap Variability, (b) Inflation Variability, and (c) Spot Rate Variability (One Step Ahead Estimate Plus 95% Confidence Interval).

How firmly do these factors anchor inflation and interest rates? This question depends upon whether output, inflation, and interest rates and the real factor x2,t are contintegrated with the nonstationary nominal factor x1,t. This was checked by running ADF tests on the residuals of the output, inflation, and interest rate equations, which decisively reject nonstationarity (Table 6b). These variables adjust quickly and smoothly to their equilibrium values. This mean-reversion effect can be summarized in terms of the model's eigenvalues. The autoregressive coefficient associated with x1,t is unity, but the other roots are stable and are reported in Table 7. Four pairs of roots are sinusoidal, reflecting the cyclical nature of the macro-economic data.13

Table 6b.  Residual Error Statistics M1 1961Q4–2004Q1
 gπr1r4r8r12r20r28r40r60
  1. Notes: The first row reports the unadjusted R2; the second reports the root mean square error (RMSE). ADF is the adjusted Dickey–Fuller statistic testing the null hypothesis of nonstationarity. The 10% and 5% significance levels are 2.575 and 2.877, respectively. Auto. is the first order quarterly autocorrelation coefficient (with t-value in parentheses).

R20.90200.96640.90990.86160.87970.89010.91580.92970.93410.9355
RMSE0.18240.54520.83211.0450.94580.85630.73590.65500.61880.6034
ADF−9.4633−6.3951−6.3239−6.6522−6.7983−7.0042−7.1833−7.2131−7.0550−6.9903
Auto.−0.1218−0.1283−0.0933−0.02610.00490.09480.10810.08320.0153−0.0021
(1.57)(1.59)(1.21)(0.34)(0.06)(1.23)(1.98)(1.08)(0.19)(0.02)
Table 7.  Eigenvalues of the Dynamic Responses in M1 (in Order of Absolute Value)
M0M1
11
0.92415 ± 0.08084i 0.93103 ± 0.10677
0.8577084000
0.55264 ± 0.27966i0.61927
−0.00649 ± 0.51542i 0.51854 ± 0.18382
0.51152 0.03600 ± 0.50129
−0.24485 ± 0.012387−0.26430 ± 0.09198

These cyclical effects are seen more clearly in the impulse responses, which show the dynamic effects of innovations in the macro-economic variables on the system. Because these innovations are correlated empirically, we work with orthogonalized innovations using the triangular factorization defined in (A1). The orthogonalized impulse responses show the effect on the macro-economic system of increasing each of these shocks by 1 percentage point for just one period using the Wold representation of the system. This arrangement is affected by the ordering of the macro-economic variables in the vector xt. Like Kozicki and Tinsley (2005), I adopt the standard ordering: inline image. The first shock 1) reflects permanent policy or expectational changes in the inflation asymptote while the second 2) reflects structural shocks to the real interest rate. Conventionally, υ3 is interpreted as a positive demand shock and 4) as a negative supply shock. Finally, 5) represents transitory changes in monetary policy.

Figure 6 shows the results of this exercise. This gives a plausible description of the macro-economic dynamics. As in the model of Kozicki and Tinsley (2005), the use of Kalman filters to pick up the effect of unobservable expectational influences seems to solve the notorious price puzzle—the tendency (noted originally by Sims 1992) for increases in policy interest rates to anticipate inflationary developments and apparently cause inflation. The nominal filter dictates the long-run equilibrium of the macroeconomy (and its volatility). These effects are persistent, but the responses of the macro-economic variables to surprises in inflation, output, and interest rates are rapid. They are largely exponential in nature, suggesting that monetary policy has been effective in securing its objectives quickly, without significant policy reversals or cycles.

image

Figure 6. Model M1 Macro-Economic Impulse Responses. Notes: Each panel shows the effect of a shock to one the five orthogonal innovations (ɛ, η) shown in (1) and (2). These shocks increase the each of the five driving variables in turn by 1 percentage point compared to its historical value for just one period. Since inline image is a martingale, the first shock 1) has a permanent effect on inflation and interest rates, while other shocks are transient. The dashed line shows the effect on output, the dotted line the effect on inflation, the continuous line the effect on the spot rate and the dotted–dash line the effect on the 10 year yield. Elapsed time is measured in quarters.

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These results are reflected in Figure 7, which report the results of the analysis of variance (ANOVA) exercise. These figures show the share of the total variance attributable to the innovations at different lag lengths and are also obtained using the Wold representation of the system, as described in Cochrane (1997). They indicate the contribution each innovation would make to the volatility of each model variable if the error process was suddenly started (having been dormant previously). So, for example, we see that the output surprise η1,t accounts for nearly all of the short-run volatility in output, with similar results for the responses of inflation and spot rates to their own innovations. However, the effect of other innovations builds up over time.

image

Figure 7. Model M1 Analysis of Variance. Notes: Each panel shows the contribution to total variance of innovations in each of the orthogonal shocks representing innovations in each of the five driving variables. Elapsed time is measured in quarters.

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3.2 The Empirical Yield Model

The behavior of the yield curve is dictated by the factor loadings. These are depicted in Figure 8, as a function of maturity (expressed in quarters). The first panel shows the loadings on x1,t (broken line), and x2,t (continuous line). The second panel shows the loadings on π (dotted line), g (broken line), and the spot rate (continuous line). The spot rate provides the link between the macro-economic model and the term structure. Since it is the 3-month yield, this variable has a unit coefficient at a maturity of one quarter and other factors have a zero loading. The spot rate loadings decline over the next few years, reflecting the adjustment of the spot rate toward x1,t and x2,t. The spot rate thus determines the slope of the short-term yield curve. Three- to 5-year maturity yields are strongly influenced by the behavior of the real rate factor x2,t. The loading on this factor then fades gradually over the longer maturities, allowing this to act as a “curvature” factor. In contrast, the loading upon x1,t moves up to unity and then increases gradually with maturity over the 2- to 15-year maturity range, so that it acts as a “level” factor. The loadings for output and inflation have a humped shape, but are relatively small.

image

Figure 8. Factor Loadings in Model M1. Notes: Panels (i) and (ii) show the effect of orthogonal shocks to the financial factors (η) and macro-variables (ɛ), respectively. These shocks increase each of these driving variables in turn by 1 percentage point compared to its historical value for one period. Maturity is measured in quarters.

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Figure 9 shows the risk premia implied by models M0 and M1 for the 15-year yield. Although the loadings for these models are similar, the risk premia differ because they also depend upon the way that the stochastic trend affects volatility (via λ10 in M1) and the price of risk (λ11 in M0). The parameter λ10 is significantly negative in M1, so the stochastic trend has a positive effect on the asymptotic risk premium in that model (Appendix B). In M0, λ11 adopts a positive value in order to keep the model dynamics stable under inline image, so the stochastic trend has a negative effect on the asymptotic premium. This means that although the stochastic trend has a positive effect on the 15-year premia in both models, this is more powerful in M1 than it is in M0. The real factor also has a strong positive effect in both models, as is clear from the figure. The effect of the macro-economic variables on the risk premia in these models is relatively small in the 15-year area. The impulse responses for the 5-year yield and ANOVA results for the 10-year yield are shown in Figures 6 and 7. These reflect the combined effect of the factor loadings and the dynamic characteristics of the model variables discussed in the previous section. The behavior of the 5-year yield depends upon the spot rate and the financial factors. The variance of the 10-year yield (Figure 7) is dominated by the shocks to the two financial factors, reflecting the “level” and “curvature” effects. The effect of the spot rate and other macro-economic variables is negligible at this maturity. Table 6 shows that the joint macro-yield model closely replicates the first three moments of the data shown in Table 1.

image

Figure 9. Risk Premia (on 15-Year Maturity).

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Table 6a.  Summary Statistics for Estimated Values, M1 1961Q4–2004Q1
 gπr1r4r8r12r20r28r40r60
  1. Notes: Mean denotes sample arithmetic mean expressed as percentage per annum, and Std. is standard deviation. Skew. and Kurt. are standard measures of skewness and kurtosis.

Mean 0.1433 4.4933 5.91036.58046.78466.90407.16377.26767.36267.4621
Std. 2.239922.963082.55552.62222.57822.52772.45822.42042.38892.3193
Skew.−0.553  1.3980 0.91570.74110.81960.87490.92120.92980.92380.8994
Kurt. 3.9787 4.5119 4.42883.85373.83523.80213.72203.65123.57513.4752

4. CONCLUSION

  1. Top of page
  2. Abstract
  3. 1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK
  4. 2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK
  5. 3. THE EMPIRICAL MODELS
  6. 4. CONCLUSION
  7. Appendices
  8. LITERATURE CITED

This research aligns the new macro-finance model with the mainstream finance literature, using a latent variable with stochastic first and second moments to model the unit root. Because volatility depends upon the stochastic trend in this model, the Jensen effects induced by the convexity of the bond price function affect the associated slope parameter and not the intercept. This means that the trend affects the steady state inflation and spot rates without disturbing the asymptotic forward rate. The model was initially designed to tackle the asymptotic problem posed by the unit root, but in practice it seems that its superior performance stems from its ability to handle the heteroscedasticity of the macro-economic data rather than the asymptotic yield problem. Unfortunately, it is not possible to test these models on longer term yields using this historical data set because there have been long periods when the U.S. Treasury did not fund in the 20- to 30-year area. However, 30 year issuance has now resumed and the growing demand from pension providers is likely to keep this funding window open. Moreover, the increasing number of 50-year issues in the U.K. and French Treasury markets should generate data better suited to an empirical test of asymptotic model behavior.

In the meantime, the significance of the inflation-driven conditional heteroskedasticity found in U.S. macro-data motivates the use of the general affine model to study both the macroeconomy and the bond market. In contrast to the volatility-clustering effects implied by GARCH macro-models, this conditional heteroskedasticity is persistent, exhibiting a unit root. Mathematically, it is more tractable than the GARCH model, generating linear structures that could lend themselves not just to research on the term structure but to optimal control and similar intertemporal optimization problems. Empirically, this finding helps to explain the “Great Moderation”—the fall in output, interest rate, and inflation volatility seen since the mid 1980s (Bernanke 2004, Kim, Nelson, and Piger 2004)—attributing it to the fall in the inflation trend associated with the recession of the early 1980s. It reminds us that this so-called moderation is actually a return to the low-inflation, low-volatility epoch that characterized the early postwar years. The “general affine” macro-model inline image helps to explain both the rise and subsequent fall in volatility.

Compared to the mainstream finance model of the bond market, the macro-finance inline image model can use a relatively large number of factors (11) because the parameters of the model are informed by macroeconomic as well as yield data (with a total of 1,700 data points). It can also use an unrestricted specification of the price of risk, with a large number of parameters. It is particularly informative about the stochastic volatility parameters, identifying these with the volatility parameters of the macro-model, which are well determined. The relative adjustment speeds mean that the behavior of the yield curve is largely dictated by three factors: the inflation end-point, the real interest rate factor, and the spot rate. The model is consistent with the traditional three-factor finance specification in this sense but links these factors to the behavior of the macroeconomy. This research opens the way to new CBM-based studies of monetary policy and a much richer term structure specification, incorporating the best features of both macro-finance and mainstream finance models.

Appendices

  1. Top of page
  2. Abstract
  3. 1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK
  4. 2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK
  5. 3. THE EMPIRICAL MODELS
  6. 4. CONCLUSION
  7. Appendices
  8. LITERATURE CITED

APPENDIX A: THE STATE-SPACE REPRESENTATION OF THE MODEL

Define xt={yt, zt}′; vt={ɛ′t, η′t}′ and combine (1) and (2) to get:

  • image((A1))

where

  • image

Stacking (A1) puts the system into state space form (4), where Xt={yt, zt, … , zt-1}′, Wt=C.{ɛ′t, η′t, 01,Nkn}′ and:

  • image((A2))

The second matrix repartitions Φ conformably with (10), so that Φ21 is (N− 1) × 1 and Φ22 is (N− 1)2. Similarly:

  • image

where C21 is (N− 1) × 1 and C22 is (N− 1)2. Comparing this with the partition of (A2), note that:

  • image((A3))

Similarly for xt={x1,t, x2,t}′ repartition vt={w1,t, v2,t}′, wt={w1,t, w2,t}′ and (A1) conformably and write Γ as:

  • image

where Γ22 is a (k+n− 1)2 lower triangular matrix with unit diagonals and Γ21 is a (k+n− 1) column vector. The errors in x2,t+1 are decomposed into orthogonal components that are related to w1,t+1 and v2,t+1=stu2,t+1:

  • image((A4))

where u2,t+1 is an (n+k− 1) vector of standard normal variables, inline image; δmj≥ 0, m= 0, 1; j= 1, … , k+n, and E[u2,t+1w1,t+1]= 0(k+n−1); u2,t+1N[0(k+n−1), I(k+n−1)]. The error structure of (10) follows from (A4) as:

  • image((A5))
  • image((A6))

where U2,t+1={(u2,t+ 1, 0′N-k-n}′; inline image; D=Diag{1′k+n−1, 0′Nkn}, so that StD=St. This implies the Gaussian conditional MGFs:

  • image((A7))

where

  • image((A8))

Finally, the conditional value X2,t+1 | x1,t+1 can be represented using (10), (A3), and (A5) as:

  • image((A9))

APPENDIX B: AFFINE YIELD STRUCTURES

This appendix derives the MGF of the distribution under the risk neutral measure inline image and shows how it can be employed as a “model generating function” to derive the yield model, forward rates, and risk premia as well as the moments of the macro-system under inline image.

B.1 The Risk-Neutral Probability Measure

Measure inline image adjusts the state probabilities using a multiplicative state-dependent subjective-utility weight Nt+1 (with the logarithm nt+1) so that the time t conditional risk neutral expectation inline image of a scalar random variable Zt+1:

  • image((B1))

Bond (and other asset) prices are discounted expectations of future payoffs and prices defined under this measure:

  • image((B2))

(Campbell, Lo, and MacKinlay 1997, Cochrane 2000). Recall that the MGF for measure inline image is:

  • image((B3))

Using (11) to replace Pτ−1,t+1 in (B2) gives a similar form with ν=−Ψτ−1:

  • image((B4))

where Jr is a selection vector such that JrXt=r1,t.

The discount factor is naturally exponential and if the MGF under the risk-neutral measure inline image is exponential-affine in Xt, then so is the expectation. Thus, (B4) is of the form (11), with coefficients that are obtained recursively by matching the coefficients (for maturity τ− 1) of the state variables Xt in the exponents of (B4) with those (for τ) in (11). For this to be the case, Nt+1 must be an exponential-affine function of the state variables or error terms:

  • image((B5))

where λ1,t is a scalar, and Λ2,t=[λ′2,t, 0′N−(k+n)]′ is a (N− 1) × 1 deficient vector containing λ2,t, which is a (k+n− 1) × 1 vector of coefficients related to the prices of risk associated with shocks to x2,t+1. In the basic affine model class these coefficients are constant and variations in the risk premia only depend upon those in volatility. However, in the “essentially affine” specification of Duffee (2002) they are linear in xt, allowing an addition source of variation in the premia. I adopt his specification using (13) with:

  • image((B6))

where Λ′12, Λ20 and Λ21 are (N− 1) × 1 vectors and Λ22 is an (N− 1)2 matrix of parameters to be estimated. The parameter λ11 allows x1 to influence the asymptotic risk premium through the price of risk. However, since x1 affects this through volatility it is redundant in inline image and is set to zero. For the EA1 specifications M1 and M2 to be admissible it is also necessary that Λ12= 0N−1 and to facilitate the encompassing tests I follow Dewachter and Lyrio (2006) and use this restriction for M0.

B.2  The MGF under the Risk-Neutral Measure

Using (B1) and (B3), the MGF of the distribution under the risk-neutral measure inline image can be represented as: inline image=E[exp [nt+1+ν′Xt+1] | Xt]. Substituting (B5) and (A9) and noting that x1,t+1 and U2,t+1 are independent allows this to be factorized as:

  • image((B7))

For the baseline model M2, these expectations are evaluated using (7) and (A7), substituting inline image for inline image:

  • image

This probability density is normalized to unity using: inline image. Substituting this back and using (15), (B6), and setting λ1,t10 gives:

  • image((B8))

Using this as a moment-generating function (differentiating w.r.t. 1, ν2} and setting these parameters to zero) gives (14). The formulae (16), (18), and (20) follow by substituting ν=−Ψτ−1, into (B8), substituting this into (B4) and equating the coefficients of Xt in the exponent with those in (11). M1 follows immediately from the restrictions noted in the main text.

B.3 The Standard EA0 Model as a Special Case

The standard way to obtain the moments and yield structure for the EA0 specification is to use (5) instead of (7) to evaluate the second expectation in (B7). However it is more instructive to derive these from the formulae for M2, taking the limit as c tends to infinity and setting Σ1 and δ11 to zero. This specializes the baseline parameters as shown in the final column of Table 2. We expand the denominator in inline image as the geometric series [1 −λ10/c+ (λ10/c)2−⋯] and use δ01=μ/c to get the second-order approximation: inline image, which is approximated arbitrarily closely by inline image for large values of c (o(c−1) denotes terms of order c−1  or smaller). inline image follows from inline image in the same way. Similarly, inline image may be written as the second-order approximation: inline image, which reduces to the first-order expansion inline image upon the substitution of zero for the limiting value of inline image as c tends to infinity. inline image follows from inline image in the same way.

Now consider the EA0 yield curve formulae. First, note that the recursive nature of the coefficient system means that inline image, and c do not affect (16). Next, expand the denominators in (18) as above to give the quadratic approximation:

  • image((B9))

Substituting the limit δ11= 0 reduces this to a linear difference equation and then substituting (15), (A3), Σ1= 0 and the coefficients of Table 2 gives:

  • image((B10))

which reproduces (21) upon substitution of inline image. Finally, to specialize the intercept (22) for EA0, take a Taylor approximation of the logarithmic term in (20) around ln [1] use δ01=μ/c and then expand the denominator terms.

  • image

Neglecting the terms o(c−1) gives a second-order approximation. Substituting back into (20) using (15) and Table 2 gives (22):

  • image((B11))
B.4  Risk Premia

The risk premia depend entirely upon the difference between the two measures inline image and inline image. To see this, note that the premium on a τ-period bond is the expected return, less the spot rate. The gross expected rate of return is the expected payoff E[Pτ−1,t+1 |Xt] divided by its current price Pτ,t=inline image. Taking the natural logarithm expresses this as a percentage return and subtracting the spot rate r1,t then gives the risk premium: inline image This is affine provided that the MGF is exponential-affine under both measures. Substituting (11), (7), and (B8) with ν=Ψ′τ−1 gives the risk premium. In the case of the EA1 unit root model this gives:

  • image((B12))

The linear term on the first line is the compensation for the bond's exposure to shifts in X2,t, which is negligible for a portfolio or security like an ultra-long bond, with a yield that mimics the asymptotic portfolio. The nonlinear term on the second line shows the premium on the asymptotic portfolio and is zero if λ10= 0. It is positively related to x1,t if λ10 < 0, and negatively related if λ10 > 0 since:

  • image

given (19). The risk premium in the EA0(N) model is:

  • image((B13))

Recall that this model requires the restriction λ11 > 0. This means that the associated factor risk premium (the asymptotic premium) is negatively related to the inflation trend. This premium is shown by δ011011x1,t) in the second term. In practice, this effect has the effect of offsetting the effect of the component Ψ′2,τ−1ϒ0x1,t shown in the first term (which is positive in the empirical model). Thus, the influence of the trend on the premia shown in Figure 9 is more pronounced in model M1.

B.5  Forward Rates and Asymptotic Behavior

Taking logs of (11) and maturity-differencing gives the affine forward rate structure:

  • image((B14))

This shows that the asymptotic behavior of the forward rate depends critically upon whether the slope coefficients converge to constants. If so, the last term vanishes and Δγτ+1 and hence the forward rate asymptote (f*t) is constant. Since Ψ*2=limτ[RIGHTWARDS ARROW]Ψ2,τ=inline image is constant, this just depends upon the behavior ψ1,τ.Table 3 shows the asymptotic effect on the yield model of imposing unit root restrictions inline image on x1,t under inline image. In EA1, ψ1,τ asymptotes to a constant, so β*1 and f* are zero. However, (21) shows that in EA0: limτ[RIGHTWARDS ARROW]1,τ+1−ψ1,τ)=inline image=inline image Consequently, inline image. This expression is equal to the asymptotic effect of x1,t on r1,t under inline image and as such it should be close to unity. Substituting these coefficients into (22) and (B14) gives the asymptotic behavior of the forward rate reported in Table 3. The final term in this expression is the Jensen effect associated with ψ1,τx1,t, which behaves like inline image in the limit. These results illustrate the basic theorem of Dybvig, Ingersoll, and Ross (1996), which says that in an arbitrage-free framework “the limiting forward interest rate, if it exists, can never fall.” In other words the forward rate must either fail to converge with maturity (as in the unit root inline image model), or must asymptote to a constant inline image.

APPENDIX C: THE KALMAN FILTER AND THE LIKELIHOOD FUNCTION

These models were estimated and tested using the fminsearch algorithm on Matlab.14 In this model, the unobservable variables are modeled using the extended Kalman filter (Harvey 1989, Duffee and Stanton 2004). This method assumes that the revisions in (2) are approximately normally distributed:

  • image

where

  • image

I represent expectations conditional upon the available information with a “hat” (so that inline imageys; st) and define the covariance matrices:

  • image((C1))

where using (2):

  • image((C2))

Similarly, using (1): inline image, where

  • image((C3))

and using (12): inline image, where

  • image((C4))

The t-conditional covariance matrix for this (t+ 1)-dated system is:

  • image

where inline image; Mt=G[S0+S1yt,t]G′; Si=Diag2i1 … , δ2in}; i= 0, 1 and inline image is defined in (12). This allows the expectations to be updated as:

  • image((C5))
  • image((C6))

The (log) likelihood for period t+ 1 is thus:

  • image((C7))

The log-likelihood for the full sample follows by iterating (C1), (C2), (C5), and (C6) forward given suitable starting values; substituting (C3) and (C4) then summing (C7) over t= 1, … ,T.

Footnotes
  • 1

    Ball (1992) offers a theoretical analysis of this phenomenon and the empirical evidence is examined by Brunner and Hess (1993), Holland (1995), Caporale and McKiernan (1997), and others. There is also an extensive literature on the effect of inflation and macro-economic volatility on the equity risk premium (Brandt and Wang 2003, Lettau, Ludvigson, and Wachter Forthcoming).

  • 2

    In this paper, Diag{δ} represents a matrix with the elements of the row vector δ in the main diagonal and zeros elsewhere. 0a is the (a× 1) × 1 zero vector, 1a is the (a× 1) × 1 summation vector, 0a,b is the (a×b) zero matrix, and Ia is the a2 identity matrix.

  • 3

    I am grateful to Tony Rodrigues of the New York Federal Reserve Bank for supplying a copy of this yield data set.

  • 4

    Preliminary tests showed no significant evidence of autoregressive conditional heteroskedasticity (ARCH) in this data set.

  • 5

    This parameter is equivalent to ϕ1 in (3) in this case.

  • 6

    In inline image the variable driving volatility x1,t has a noncentral χ2 distribution and is nonnegative, keeping the variance structure Σ01x1,t−1 for X2,t nonnegative definite.

  • 7

    The usual convention in macro-finance models is that this represents measurement error.

  • 8

    A shift in the parameters from (4) to (14) could occur either because of the risk adjustment implied by (B5) or because the bond market does not use the true values of the parameters in (4). However, it is not possible to distinguish these two effects without imposing restrictions on (B5).

  • 9

    Their model uses a Gaussian approximation to the CIR process describing the spot rate, due originally to Sun (1992).

  • 10

    Substituting (17) into (18) gives ψ*1 as the solution to: ψ*1=inline imageinline imageΨ*′2ϒ1inline image This may be arranged as: 0 =ϑ2+ϑ (c(1 −ξ1) −ζ) −cζ, where ϑ=ψ*110+Ψ*′2C21, and inline image The intercept term cζ shows the product of the roots and is a very large negative number. Consequently, one root is a large negative and the other a large positive number. Phase analysis reveals that the recursion (18) selects the positive root.

  • 11

    These are inline image, δ01, δ11,Δ0(4), Δ1(4), H1(4), ϒ1(3), Λ22(13), G(3), Φ(27), and ϕ(2). Estimates are reported in Table 4. It was found that although inline image was significant (Table 4c) the remaining elements of the first rows of Λ22 and ϒ1 (or ϒ0) were very poorly determined and could be eliminated without significantly reducing the likelihood. The structural parameters μ and c follow from (8) given ξ1,δ01, and δ11.

  • 12

    This restriction is imposed via (8) by setting δ01= 0.

  • 13

    However, the imaginary components of the first root is small, meaning that the macro-model is dominated by exponential adjustment effects.

  • 14

    I am very grateful to Zhuoshi Liu for suggesting this algorithm and converting my basic files from Mathematica to Matlab.

LITERATURE CITED

  1. Top of page
  2. Abstract
  3. 1. THE GENERAL AFFINE MACRO-MODEL FRAMEWORK
  4. 2. THE GENERAL AFFINE YIELD CURVE FRAMEWORK
  5. 3. THE EMPIRICAL MODELS
  6. 4. CONCLUSION
  7. Appendices
  8. LITERATURE CITED
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