Accounting for a Shift in Term Structure Behavior with No-Arbitrage and Macro-Finance Models

Accounting for a Shift in Term Structure Behavior
with No -Arbitrage and Macro -Finance Models?
Glenn D. Rudebusch†
Tao Wu‡
November 2004
Revised July 2005
Abstract
This paper examines a shift in the dynamics of the term structure of interest rates
in the U.S. during the mid-1980s. We document this shift using standard interest
rate regressions and using dynamic, a?ne, no-arbitrage models estimated for the
pre- and post-shift subsamples. The term structure shift largely appears to be the
result of changes in the pricing of risk associated with a “level” factor. Using a
macro-?nance model, we suggest a link between this shift in term structure behavior
and changes in the dynamics and risk pricing of the Federal Reserve’s in?ation target
as perceived by investors.
?For helpful comments, we thank seminar and conference participants at Bocconi, the Federal Reserve Board,
and the NBER Summer Institute, in addition to many colleagues in the Federal Reserve System. Golnaz Motiey
and Vuong Nguyen provided excellent research assistance. The views expressed in this paper do not necessarily
re?ect those of the Federal Reserve Bank of San Francisco. The title of an earlier version of this paper was “The
Recent Shift in Term Structure Behavior from a No-Arbitrage Macro-Finance Perspective.”
†Federal Reserve Bank of San Francisco; www.frbsf.org/economists/grudebusch; Glenn.Rudebusch@sf.frb.org.
‡Federal Reserve Bank of San Francisco; Tao.Wu@sf.frb.org.

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1. Introduction
During the past few decades, the U.S. economy has undergone an important transformation that
has likely altered the nature of uncertainty and risk in the economy as well as investors’ attitudes
and pricing of that risk. A key aspect of this transformation is the precipitous decline in overall
macroeconomic volatility: Since the middle of the 1980s, the volatility of real GDP growth has
been about 35 percent lower than earlier in the postwar period (as noted by Kim and Nelson 1999
and McConnell and Perez-Quiros 2000). Several factors may underlie this “Great Moderation”
in economic ?uctuations.1 For example, better economic policy in the later sample may have
helped stabilize the economy; indeed, many have argued that the conduct of U.S. monetary
policy improved dramatically during the mid-1980s, helping to usher in the current period
of diminished output volatility as well as remarkably low and stable in?ation. Alternatively,
the recent quiescence in real activity and in?ation may largely re?ect good luck–that is, a
temporary run of smaller economic shocks. Other potential factors include non-policy changes
in the dynamics of the economy arising from, for example, improved inventory management
or a greater share in aggregate output accounted for by the relatively stable service sector.
Finally, the development of deeper and more integrated ?nancial markets and the introduction
of new ?nancial instruments may also have played a role both in damping the magnitude of
economic ?uctuations and in mitigating their e?ects on investors. Given such dramatic shifts in
the economic environment, a change in the behavior of the term structure of interest rates, and
especially in the size and dynamics of risk premiums, would hardly be surprising.
This paper examines how the dynamics of the term structure and of interest rate risk may
have changed over time. We use a?ne, no-arbitrage, asset pricing models of the type popular
in the ?nance literature to investigate the recent shift in the behavior of the term structure;
however, our investigation is also informed by the above literature on the recent transformation
of the U.S. economy and by consideration of the macroeconomic underpinnings of the term
structure factors in ?nance models.2 The payo? from this joint analysis is bi-directional as
well. The macro-?nance perspective helps illuminate the nature of the shift in the behavior
of the term structure, highlighting in particular the importance of a shift in investors’ views
1 For references to the quickly growing literature on this topic, see Blanchard and Simon (2001) and Stock and
Watson (2003).
2 The connection between the macroeconomic and ?nance views of the term structure has been a very fertile
area for recent research, including, for example, Piazzesi (2005), Diebold, Rudebusch, and Aruoba (2004), Hördahl,
Tristani, and Vestin (2004), Rudebusch and Wu (2004), Wu (2001), Dewachter and Lyrio (2002), Du?ee (2004),
and Kozicki and Tinsley (2001, 2005).
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regarding the risk associated with the in?ation goals of the monetary authority. In addition, the
shift in term structure behavior, as viewed using a no-arbitrage ?nance model, sheds light on the
nature of recent macroeconomic changes. Speci?cally, if one assumes that the factors underlying
recent changes in the macroeconomy also have left their imprint on the yield curve, the ?nance
models suggest that more than just good luck was responsible for the recent macroeconomic
transformation. Instead, a favorable change in economic dynamics, likely linked to a shift in
the monetary policy environment, appears to have been an important element of the Great
Moderation.
We begin our analysis in Section 2 with a simple empirical characterization of the recent
shift in the term structure in the U.S.. For this purpose, we use regressions of the change in a
long-term interest rate on the lagged spread between long and short rates. Following Campbell
and Shiller (1991), such regressions have been widely used to test the expectations hypothesis
of the term structure, which assumes that the risk or term premiums embedded in long rates
are constant. We ?nd–as have many others–that these tests often reject the expectations
hypothesis; however, of more interest for our purposes is the apparent signi?cant shift in the
estimated coe?cients from these regressions. Indeed, since the mid-1980s, there is much less
evidence against the expectations hypothesis than before, which suggests a shift in risk pricing
and in the properties of risk premiums.
We use these term structure regression results as a summary statistic for characterizing the
changing empirical behavior of the term structure. Accordingly, the regression evidence is a
useful ?rst step to a more formal modeling perspective on the change in the term structure,
which is provided in Section 3 using an estimated dynamic, a?ne, no-arbitrage model of bond
pricing. The no-arbitrage model provides an obvious setting in which to examine changes in
interest rate behavior and time-varying term premiums. Indeed, as demonstrated by Backus
et al. (2001), Du?ee (2002), and Dai and Singleton (2002), a?ne, no-arbitrage models with
a rich speci?cation of the dynamics of risk premiums are broadly consistent with the usual
full-sample term structure regression results of the type obtained in Section 2. We conduct
a similar consistency check between models and regression results, though from a somewhat
di?erent perspective. Namely, given our evidence of a signi?cant shift in the term structure
regression results, we estimate a?ne, no-arbitrage models for each of the two subsamples that
are associated with the di?erent regression results. We ?nd a statistically signi?cant di?erence
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between the two estimated bond pricing models. In addition, the subsample models are able to
account for much of the disparity between the subsample term structure regression results, thus
supporting the empirical characterization of structural change in Section 2.
Beyond merely documenting the recent change in term structure behavior through regres-
sion analysis and model estimates, we also begin the more di?cult task of understanding and
accounting for such time variation. In Sections 4 and 5, we illuminate the economic changes
that may account for the shift in term structure behavior. We ?rst use the estimated subsample
no-arbitrage models to parse out whether a change in underlying factor dynamics or a change
in risk pricing is more important in accounting for the shift in term structure behavior. In
this regard, we ?nd that changes in pricing the risk associated with a “level” factor are crucial
for accounting for the shift in term structure behavior. We then provide an interpretation of
this shift in terms of possible recent macroeconomic changes using the macro-?nance model of
Rudebusch and Wu (2004). Our results suggest a link between the recent shift in term structure
behavior and changes in the risk and dynamics of the central bank’s in?ation target as perceived
by investors.
At this point, it is perhaps useful to note some recent related research. There has been
little analysis of the potential e?ects on asset pricing induced by the important structural shifts
in the economy documented in the macroeconomics literature. Indeed, the ?nance literature
often treats the entire postwar period as a long homogenous sample. An exception to this
practice is the literature on regime-switching models of interest rates, including, for example,
Hamilton (1988), Ang and Bekaert (2002), Bansal and Zhou (2002), and Dai, Singleton, and
Yang (2003). These papers attempt to capture the postwar dynamics of interest rates with
models that contain a succession of alternating regimes that are often linked informally to
business cycles or interest rate policies. In contrast, we are interested in a single break in the
behavior of the term structure, with our attention focused by the macroeconomic evidence that
suggests the shift occurred during the middle to late 1980s. Also, following the macroeconomic
evidence, we have no expectation that this change will be reversed (and we incorporate no pricing
of further regime change risk). Of course, regime switching at a cyclical frequency could coexist
with a single large shift in risk pricing as well, but our interest here is in the latter. Accordingly,
our analysis is related to other work, including Watson (1999), who examined a shift in the
unconditional volatility of interest rates, and Lange, Sack, and Whitesell (2003), and Swanson
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(2005), who considered a change in the forecastability of short-term interest rates. However,
in contrast to these analyses, we examine a shift in behavior of risk pricing using both simple
regression indicators as well as formal dynamic bond pricing models. Finally, others, notably
Fuhrer (1996) and Kozicki and Tinsley (2005), have also linked the term structure regression
estimates to the behavior of the perceived in?ation target using di?erent methodologies.
2. Regression Evidence of a Term Structure Shift
In this section, to help guide our subsequent model-based analysis, we provide a simple empirical
characterization of the recent shift in the behavior of the term structure. This characterization,
which also provides a metric to assess the extent of any such shift, is based on a regression test
of the expectations hypothesis that was popularized by Campbell and Shiller (1991).
To derive this regression test, consider the following decomposition of the yield of a pure
discount bond into average expected future yields and a term premium Et?m,t:
m?1
im,t = (1/m)
Et(it+j) + Et?m,t,
(1)
j=0
where im,t is the continuously compounded yield to maturity at time t of an m-month nominal
zero-coupon bond with the notational simpli?cation for the one-month rate of it ? i1,t. Bekaert,
Hodrick, and Marshall (1997a) derive equation (1) from a modern asset pricing equation and
show that the term premium is a function of second- and higher-order conditional moments of
the stochastic discount factor (or pricing kernel). If these moments vary over time, then so will
the term premium. If not, then term premiums will be constant, the expectations hypothesis will
hold, and changes in long-term rates will result only from changes in expected future short-term
rates. In this special case, we can obtain from equation (1) the pricing equation
m?1
mim,t ? (m ? 1)im?1,t+1 = it + const. +
(Et(it+j) ? Et+1(it+j)),
(2)
j=1
where the left-hand side is the one-month holding period return of a bond of maturity m and
the right-hand side is the one-month short rate plus a constant premium plus an expectational
term.3 This expectational term represents the capital gains or losses resulting from revisions to
expected future short rates made between periods t and t + 1. With rational expectations, these
3 The holding period return is the pro?t or loss from buying an m-period bond at time t and selling the same
(aged) (m ? 1)-period bond at time t + 1. If bm,t is the price of this m-period nominal bond, then the return is
bm?1,t+1/bm,t, the log of which is the left-hand side of the equation.
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revisions are unpredictable at time t, so they can be interpreted as a white noise error term.
Equation (2) then leads naturally to the “long-rate regression” form of Campbell and Shiller
(1991):
im?1,t+1 ? im,t = ?m + ?m(im,t ? it)/(m ? 1) + ?m,t,
(3)
where ?m and ?m are maturity-speci?c regression intercept and slope coe?cients, and ?m,t is
the white noise expectational term (scaled by 1 ? m). Under the expectations hypothesis, the
estimated slope coe?cient ?m will equal unity; that is, the term spread will be an optimal
forecast of future change in the long rate (adjusted for a constant risk premium), so when the
spread between long and short rates widens (narrows), the long rate should rise (fall) in the
following period.
Deviations from the expectation hypothesis will push the slope coe?cient away from one.
In particular, as noted early on by Mankiw and Miron (1986), a time-varying term premium
can drive the estimated ?m to zero or even to negative values as the resulting term spread
re?ects variation in expected risk premiums rather than in future rates. In our analysis below,
we construct models in which the time variation in the term premium (or equivalently the
conditional heteroskedasticity of the discount factor) is su?cient to generate the regression
coe?cients found in the data, which are often signi?cantly less than one. However, we are not
primarily interested in the slope coe?cients as indicators of the expectations hypothesis; instead,
we use them as simple summary statistics of term structure behavior, and we interpret shifts in
these coe?cients as indications that the term structure behavior has changed. Of course, the
fact that so many researchers have focused so much e?ort on estimating these slope coe?cients
makes them of particular interest, but other simple metrics of term structure change could also
be considered (as in Watson 1999 and Lange, Sack, and Whitesell 2003).
Table 1 collects estimates of the slope coe?cient ?m in equation (3) over various samples
for eight di?erent long-rate maturities–each column uses a di?erent maturity m. In each case,
the underlying interest rate data are from end-of-month, zero-coupon U.S. Treasury securities.
The original full-sample (1952-1987) estimates from Campbell and Shiller (1991) are shown at
the top along with coe?cient standard errors in parentheses. Estimates and standard errors
from a more recent sample (1970-1995) from Dai and Singleton (2002) are shown directly below.
These two sets of estimates are similar and representative of the literature. In particular, both
sets of estimates are uniformly negative and decrease steadily as the maturity of the long rate
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increases–falling from about ?0.3 for m = 3 to less than ?4.0 at a long-rate maturity of 10
years.
The long-rate regression slope estimates from our full data sample, which runs from 1970 to
2002, are shown in the middle rows of Table 1. Despite di?erences in the sample ranges, our
full-sample estimates match the earlier results of Campbell and Shiller and Dai and Singleton
quite closely.4 In particular, our full-sample estimates of the slope coe?cients are uniformly
negative and decline with maturity to almost -4.0 at the long end. The numbers in brackets
below the standard errors are p-values of the null expectations hypothesis that ?m = 1. These
p-values indicate that for each of the nine regressions over our full sample the expectations
null hypothesis can be rejected at the 5 percent signi?cance level and often at the 1 percent
level. It should be noted that–as in the remainder of this section–the reported standard errors
and p-values are based on the usual asymptotic distributions (with a standard correction for
heteroskedasticity). Questions have been raised in the literature about the appropriateness of
such asymptotic distributions for inference in small samples; therefore, in the Appendix, we
report monte carlo simulations that indicate that in this application the small-sample biases are
not leading us astray.
We are primarily interested in regression results from shorter samples, and our strong prior–
based on the shifts in the economy described in the introduction–is that the most likely po-
tential breakdate for term structure behavior would occur around the middle or late 1980s. In
particular, econometric evidence (e.g., Kim and Nelson 1999 and McConnell and Perez-Quiros
2000) suggests that a likely date for the start of reduced volatility in economic activity is 1984.
In addition, there appears to have been an important shift in the conduct of monetary policy
during the 1980s, perhaps triggered or reinforced by the appointment of Fed Chairman Alan
Greenspan in late 1987. Of course, for pricing risk in real time, investors may have needed some
time to learn about and assess the importance of these changes, which makes the choice of a
breakdate somewhat indeterminate. We will examine a variety of potential breakdates below;
however, for an initial look at the data with an a priori choice of a breakdate, the lower half of
Table 1 provides estimates when the sample is split into an earlier “subsample A” that runs from
4 These estimates may also di?er because of variations in the methods used to create the zero-coupon yields
data–particularly in interpolating missing maturities and smoothing out idiosyncratic observations (e.g., Bliss
1997). Our data are unsmoothed Fama-Bliss yields data, kindly supplied by Robert Bliss, but we obtained
qualitatively similar breakpoint results with smoothed Fama-Bliss data (the type of data used in Dai and Singleton
2002). A ?nal di?erence is in approximating im?1,t+1 by im,t+1. We have the entire maturity set of yields, so we
do not employ this approximation, but the other authors in Table 1 do apply it.
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1970 through 1987 and a later “subsample B” that goes from 1988 to 2002. (This is the split
suggested by the change in Fed Chairmen and conveniently supplies two subsamples of nearly
equal size.)
The long-rate regression results in the lower half of Table 1 show an interesting di?erence
across the two subsamples. The slope estimates from the nine long-rate regressions are all neg-
ative in subsample A, as in the full sample, while they are predominately positive in the later
subsample B. Furthermore, the expectations hypothesis is rejected in every subsample A regres-
sion, while it is rejected in only one subsample B regression (at the 3-month horizon). Note
that this lack of rejection does not re?ect in?ated standard errors from a short sample. In fact,
for each maturity, the standard errors from the subsample B regressions are smaller than the
full-sample ones.
Evidence from a formal break test is given in the bottom line in Table 1, which shows
the p-value at each maturity for a Chow-type F -test that the slope coe?cient has not shifted
between subsamples A and B.5 Taken one maturity at a time, the evidence of a shift in the
slope coe?cient is decidedly mixed. For the three regressions using 6-, 9-, and 12-month long
rates, the evidence suggests a clear break, while at other maturities, the p-values are typically
in the 15 to 20 percent range. The Table 1 coe?cients and standard errors from the A and
B subsamples are also displayed in Figure 1. It is clear that the ±2 standard error bands
overlap considerably except at fairly short horizons, which is consistent with the predominance
of insigni?cant individual breakdate p-values.
Still, the fact that all of the slope coe?cients, taken as a group, have shifted in the same
direction in the later subsample is highly suggestive of a structural break in the behavior of the
term structure. Rigorous statistical evidence on this point requires the formulation of a joint
test. The next section will develop closely related evidence in the context of an empirical no-
arbitrage model of the entire term structure. However, in the spirit of the regression analysis of
this section, we also examine evidence on the joint signi?cance of simultaneous changes in several
of the slope coe?cients by stacking several long-rate regressions for di?erent maturities into one
system regression. Although none of these long-rate regressions share a common regressor or
regressand, it is highly likely that their error terms are correlated, so the system Seemingly Un-
related Regression (SUR) technique will generate more precise estimates.6 Speci?cally, we stack
5 The speci?c test used adds two variables to the long-rate regression: a dummy variable that is non-zero only
during subsample B and a spread times that dummy. The break test is an F-test of the signi?cance of the latter.
6 As the term structure literature has stressed (e.g., Litterman and Scheinkman 1991, Du?e and Kan 1996),
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the individual long-rate regressions for the 3-, 24-, and 60-month maturities, which are three
representative maturities for which the stability null hypotheses of unchanged slope coe?cients
were not rejected in the individual regressions. The system regression for these three maturities
is
?
?
?
? ?
? ?
? ?
?
i2,t+1 ? i3,t
?3
?3 0
0
(i3,t ? it)/2
?3,t
? i23,t+1 ? i24,t ? = ? ?24 ? + ? 0 ?
? ?
? ?
?
24
0
(i24,t ? it)/23 + ?24,t .
(4)
i59,t+1 ? i60,t
?60
0
0
?60
(i60,t ? it)/59
?60,t
The estimation results for this SUR regression are shown in Table 2 for the full sample and
for subsamples A and B. The slope coe?cient estimates in subsamples A and B continue to
show the same stark quantitative di?erences apparent in the individual regressions in Table 1;
however, the coe?cient standard errors are, on average, about half as large in magnitude. This
greater precision sharpens inference, and for these three maturities (which again were chosen for
their individual non-rejection of stability null), the p-value of .007 clearly rejects the joint null
hypothesis of no change in the three slope coe?cients between the A and B subsamples. These
system break test results are representative of other combinations of three or more yields.7
Finally, while we have considered a speci?c breakdate based on a prior view of the timing of
changes in the behavior of aggregate output, in?ation, and monetary policy, it is also useful to
consider testing more generally the null of parameter stability without such a prior. To do this,
we consider all possible breakdates in the middle 70 percent of the full sample for the system
regression, and calculate a Chow-type test statistic at each of these breakdates. Figure 2 shows
this set of test statistics as well as two 10 percent critical values. The less stringent one–the
lower dashed line–is the usual ?2 critical value (6.25) for the hypothesis that a speci?c (a
priori) known breakdate is signi?cant. The more stringent one–the upper dashed line (12.27)–
is based on a test that does not assume any prior knowledge about potential breakdates. It tests
the signi?cance of the maximum value of all Chow-type test statistics calculated at all possible
breakdates in the middle 70 percent of the sample, as given in Andrews (1993).8 Applied to all
possible breakdates for the system regression, the break test statistic does exceed the Andrews
critical value during the late 1980s. This evidence supports our earlier selection of a breakdate,
almost all movements in the yield curve can be captured by a few factors; thus, the errors in individual long-rate
regressions are likely correlated across the regressions. On the other hand, the term spreads used in the regressions
at di?erent maturities are also likely correlated for the same reason. The e?ciency gains from running SUR will
depend on which correlation dominates, and the Appendix provides some evidence on this issue.
7 The expectations hypothesis, namely, that all three slope coe?cients equal unity, is also rejected in each
system regression in Table 2. For subsample B, this rejection re?ects the low value of ?3.
8 For our application, in which the yield spread and change variables are not highly persistent, it appears from
various small-sample simulation studies that this asymptotic distribution is appropriate (see Diebold and Chen
1996 and O’Reilly and Whelan 2004).
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though, not surprisingly, the test is not sensitive enough to single out just one date.
In summary, we take the regression results as indicative of a break in term structure behavior
in the 1980s. Determining the nature of that break in terms of changes in the dynamics of the
short rate or the pricing of interest rate risk is the subject of the remainder of our analysis.
3. Estimating Subsample No-Arbitrage Models
In the preceding section, we provided regression evidence of a signi?cant shift in the behavior of
the term structure during the 1980s. In this section, we estimate dynamic term structure models
that can capture that shift in behavior. The framework we use is a standard representation from
the empirical bond pricing literature that assumes no opportunities for ?nancial arbitrage across
bonds of di?erent maturities.9
We focus on a two-factor, Gaussian, a?ne, no-arbitrage term structure model, or an A0(2)
model as de?ned in Dai and Singleton (2000). The model features a constant volatility of
term structure factors but the risk pricing is state-dependent, which implies conditionally het-
eroskedastic risk premiums. Dai and Singleton (2002) compare the performance of di?erent
dynamic term structure models and ?nd that this type of speci?cation performs best in match-
ing the full-sample long-rate regression coe?cients.10
The model is formulated in discrete time. The state vector relevant for pricing bonds is
assumed to be summarized by two latent term structure factors, Lt and St. These are stacked
in the vector Ft = (Lt,