Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol25.Issue1-2.Jan2001:
Journal of Economic Dynamics & Control
25 (2001) 149}184
Term structure views of monetary policy under
alternative models of agent expectationsq
Sharon Kozicki!,*, P.A. Tinsley"
!Federal Reserve Bank of Kansas City, 925 Grand Boulevard, Kansas City, MO, 64198 USA
"Faculty of Economics and Politics, University of Cambridge, Cambridge CB3 9DD, UK
Abstract
Term structure models and many descriptions of the transmission of monetary policy
rest on the empirical relevance of the expectations hypothesis. Small di!erences in the
perceived policy reaction function in VAR models of agent expectations strongly in#uence the relevance in the transmission mechanism of the expected short rate component
of bond yields. Mean-reverting or di!erence-stationary characterizations of interest rates
require large and volatile term premiums to match the observable term structure.
However, short rate descriptions that capture shifting perceptions of long-horizon
in#ation evident in survey data support a more substantial term structure role for short
rate expectations. ( 2001 Elsevier Science B.V. All rights reserved.
JEL classixcation: E4
Keywords: Changepoints; Expectations hypothesis; Nonstationary in#ation; Shifting
endpoint
* Corresponding author. Tel.: #1-816-881-2561; fax: #1-816-881-2199.
E-mail addresses: [email protected] (S. Kozicki), [email protected] (P.A. Tinsley)
q
We thank Dennis Ho!man, Vance Roley, Juoko Vilmunen, seminar participants at the University of Washington, session participants at the 1998 annual meetings of the AEA and the Fourth
International Conference of the Society for Computational Economics, and anonymous referees for
comments and suggestions. The views expressed are those of the authors and do not necessarily
represent those of the Federal Reserve Bank of Kansas City or the Federal Reserve System.
0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 7 2 - X
150
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
1. Introduction
The term structure of interest rates is often characterized as a crucial transmission channel of monetary policy where accurate market perceptions of policy
are required for e!ective policy execution. This description rests on three
propositions: First, a short-term interest rate, such as the overnight federal funds
rate, provides an adequate summary of monetary policy actions. Second, longterm bond yields are important determinants of the opportunity cost of investments. Third, under the expectations hypothesis, the anticipated path of the
policy short-term rate is the main determinant of the term structure of bond
rates. Although the empirical validity of each of these three propositions has
been criticized, the accumulation of empirical evidence against the third proposition * the expectations hypothesis * is impressively large.1 If variations in
current bond rates are not well-connected to current and expected movements
in the policy-controlled short-term rate, then the conventional description of
monetary policy transmission is vacuous.
In no-arbitrage formulations of the term structure, variation in bond
rates due to current and expected movements in short rates is determined
by the description of short rate dynamics. Under the standard "nance assumption that short rates are mean-reverting, the average of expected future
short rates is considerably smoother than historical bond rates. Thus, the
expectations hypothesis is empirically rejected by tests which assume constant
term premiums, and postwar shifts in the term structure are often attributed
to sizable movements in &liquidity' and &term' premiums. However, postwar
data are consistent with descriptions of short rate movements other than
mean reversion. This paper shows that small di!erences in the speci"cation of
the stochastic process for the short rate strongly in#uence the relative importance of short rate expectations and residual term premiums in bond rate
movements.
As an alternative to conventional descriptions of short rate dynamics,
a simple class of time-varying descriptions of short rate behavior is examined.
Given well-documented shifts in postwar monetary policy, it seems highly
probable that the short rate expectations of bond traders are heavily in#uenced
by shifting perceptions of monetary policy. Short rate descriptions that capture
shifting perceptions of the long-run objective of monetary policy, formulated as
1 For instance, Campbell and Shiller (1991) compare estimated &theoretical' spreads to actual
spreads and conclude that the spread is too variable to accord with the expectations hypothesis. In
addition, in regressions of long-rate changes on spreads, Shiller (1979), Shiller et al. (1983), and
Campbell and Shiller (1991) "nd that coe$cient estimates on the spread tend to be signi"cantly
di!erent from the hypothesized value of one and frequently negative.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
151
a long-run in#ation target, are supportive of a substantial term structure role for
short rate expectations.2
In addition to rejecting the implication that high bond rates in the 1980s
re#ect large term premiums, the preferred description of short rate behavior also
does not support interpretations that perceived in#ation targets were rapidly
reversed in the 1980s. More likely, the long-run in#ation target perceived by the
market behaved similarly to survey data on long-run expected in#ation. Survey
data suggest agents were cautious in adjusting their perceptions of long-run
in#ation.
This paper illustrates the term structure implications of alternative representations of monetary policy perceptions of agents. Section 2 develops a discrete-time, no-arbitrage model of the term structure where the stochastic
discount factor is related to a vector of macroeconomic variables. Forecasts of
this state vector are generated by a VAR with #exible speci"cations of long-run
behavior. Section 3 discusses three VARs with di!erent characterizations of
agent perceptions of long-run behavior. Owing to the di!erent characterizations
of perceived long-run behavior, the VARs embed perceived policy reaction
functions with di!erent implied in#ation targets. Section 4 indicates how estimates of expected short-rate components of bond yields and conventional
&residual' estimates of term premiums depend critically on assumptions regarding the long-run behavior of policy targets in VAR proxies of agent expectations.
Section 5 concludes.
2. A model of the term structure of interest rates
A discrete-time term structure model is outlined in this section to illustrate the
pivotal role of long-run forecasts by VAR models in representing agent expectations. Bond yields are linked to expectations of monetary policy through a VAR
proxy for expectations formation which contains an implicit policy reaction
function. The implicit policy reaction function relates short term interest rates to
deviations of economic variables from their long-run equilibrium values, including deviations of in#ation from the long-run in#ation goal of monetary policy.
Thus, bond yields depend on monetary policy primarily through the in#uence of
market perceptions of policy on short rate expectations.
2 As developed later, shifting perceptions are represented in this paper as sequential learning,
where agents test for changepoints in parameters of linear time series models. Other approaches to
the apparent nonstationarity of interest rates include the fractional di!erence processes in Backus
and Zin (1993), the multiple regimes models in Evans and Lewis (1995) and Bekaert et al. (1997), and
the ¢ral tendency' factor of Balduzzi et al. (1998).
152
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
The price of an n-period nominal bond, P , satis"es
n,t
P "E [P
M ],
(1)
n,t
t n~1,t`1 t`1
where M
is the nominal stochastic discount factor. Assuming M
and the
t`1
t`1
bond price are conditionally joint lognormally distributed, (1) can be represented as
p "E Mm #p
N#1Var [m #p
],
(2)
2
n,t
t t`1
n~1,t`1
t t`1
n~1,t`1
with lower-case letters denoting the natural logarithms of the corresponding
upper-case letters. The log yield-to-maturity on an n-period bond is equal to
1
r "! p .
n,t
n n,t
(3)
A recursive expression for log yields can be obtained by substituting the yieldprice relationship into Eq. (2):
1
1
N! Var (m !(n!1)r
).
r " E M!m #(n!1)r
t`1
n~1,t`1
t t`1
n~1,t`1
n,t n t
2n
(4)
Noting that for n"1, p
"p
"0, the yield on a one-period bond is
n~1,t`1
0,t`1
r "E (!m )!1Var (m ).
(5)
2
1,t
t
t`1
t t`1
Using the initial condition (5), the di!erence equation (4) can be solved to
express the yield of an n-period bond as
C
D
1
n~1
r " E + r
#h ,
n,t n t
t`i
n,t
i/0
where the last component
(6)
1 n~1
+ Var ((n!i)r
)
h "!
t
n~i,t`i
n,t
2n
i/1
1 n~1
# + Cov (m , (n!i)r
),
(7)
t t`i
n~i,t`i
n
i/1
is the term premium.3 Thus, Eq. (6) explicitly identi"es the roles of expected
future short rates and the term premium in longer maturity bond yields.
To enable use of VAR forecast systems, we use a discrete-time variant of the
&a$ne' class of bond pricing models, such as discussed by Backus et al. (1996)
3 The "rst term in h is a Jensen's inequality term while the second contains the more convenn,t
tional liquidity, risk, or term premium. Generally, we will use the phrase &term premium' to refer
to h .
n,t
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
153
and Campbell et al. (1997), where bond log yields are linear functions of state
variables. Consequently, the stochastic discount factor, m , is speci"ed to be
t`1
a linear function of a vector of state variables, x .
t
!m "/ #/@x #w ,
(8)
t`1
0
t
t`1
where w
is a serially uncorrelated innovation. The discount factor shock is
t`1
a weighted average of shocks to the state vector, u , and an additional
t`1
innovation term, l ,
t`1
w "b@u #l ,
(9)
t`1
t`1
t`1
where u
and l
are uncorrelated at all leads and lags. The shocks have zero
t`1
t`1
means, and the variance of l
is constant, E (l2 ),p2. The conditional
l
t`1
t t`i
, where
covariance of the state innovations is denoted as E (u u@ ),R
u,t`1
t t`1 t`1
the covariances may be time varying,
Var (m )"b@R
b#p2,
l
t t`1
u,t`1
(10)
"b #b@x #p2,
l
0
t
and the second line in (10) imposes the a$ne restriction that any time variations
in the variances of state innovations are due to square-root processes, as in Cox
et al. (1985) and Pearson and Sun (1994).
Because all forecast models of the state vector in this paper contain interest
rate feedback rules to represent predictable policy responses, we assume one of
the state variables, x , is the one-period interest rate, r ,r . This imposes
1,t
1,t
t
restrictions on elements of / and /. To illustrate, substitute Eqs. (8) and (10)
0
into Eq. (5), giving
r "E (!m )!1 Var (m ),
t t`1
2
t
t
t`1
(11)
"/ #/@x !1[b #b@x #p2].
v
0
t 2 0
t
If the covariance matrix of the state innovations is constant, b in Eq. (10) is
a zero vector. In this instance, the one-period interest rate is equal to the "rst
element of the state vector, x "r , if / "(1/2)[b #p2] and / is a selector
v
1,t
t
0
0
vector with one in the "rst element, / "1 (and zeros elsewhere). As an
1
alternative example, suppose the "rst element of b is a positive constant, b '0,
1
so that the variance of the stochastic discount factor is a linear function of the
level of x (and the remaining elements of b are zero). To again equate the
1,t
one-period interest rate to the "rst state variable, the only alteration required of
the restrictions for the constant variance case is to rede"ne the "rst element of
the selector vector: / "1#(1/2)b .
1
1
Returning to the expected short-rate terms in the bond rate equation (6), the
expectations of bond traders are represented by a VAR forecasting system. The
154
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
vector of macroeconomic state variables, x , is a nonexplosive VAR process
t
about the endpoint vector, k(t~1),
=
x "Hx #(I!H)k(t~1)#u ,
=
t
t
t~1
(12)
where the coe$cient matrix, H, has all eigenvalues less than or equal to one in
magnitude. Following Kozicki and Tinsley (1998), the endpoint, k(I), is the
=
limiting conditional expectation of the vector x ,
t
(13)
k(I), lim E x ,
=
I t`k
k?=
with I indexing the time subscript of the information set on which expectations
are conditioned.
The endpoint representation is su$ciently general to encompass more traditional VAR forecasting systems in which elements of x
are classi"ed as either
t`1
I(0) or I(1). In particular, if an element of x
is I(0), then the corresponding
t`1
is I(1), then the
element of k(t) will be a constant; if an element of x
=
t`1
corresponding element of k(t) will be a linear combination of elements of x . The
=
t
endpoint representation also encompasses cases in which elements of x
may
t`1
be subject to sporadic mean shifts with the corresponding elements of k(t) deter=
ministic but subject to occasional shifts. Such speci"cations are convenient for
explicitly accounting for changes in the perceived long-run in#ation goals of
monetary policy within a VAR framework. Three competing VAR speci"cations, di!ering primarily according to endpoint assumptions, are examined in
detail in the next subsection.
As earlier, we continue the convention that the short-term interest rate is the
"rst element of the state vector, r "n@ x , where n is a selector vector with one in
t
1 t
1
the "rst element. Multistep forecasts of short rates can be constructed using the
state-variable VAR. Expression (12) implies
(14)
E r "n@ [Hix #(I!Hi)k(t) ].
=
1
t
t t`i
Substituting into (6), the n-period yield can be rewritten in terms of the vector of
state variables as
1 n~1
(15)
r " + n@ [Hix #(I!Hi)k(t) ]#h .
=
n,t
1
t
n,t n
i/0
Eq. (15) relates bond yields in t to contemporaneous observations on state
variables, including the one-period rate in t. Predictions of yields, conditional on
data through t!1, can be constructed as
1 n~1
E r " + n@ [Hi`1x #(I!Hi`1)k(t~1)]#E h .
=
t~1 n,t
1
t~1
t~1 n,t n
i/0
(16)
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
155
This term structure model has two noteworthy features. First, because the
short rate equation in the VAR approximates agent perceptions of the monetary
policy reaction function, monetary policy in#uences bond yields through a dependence of short rate expectations on expected policy. Second, if the conditional variance of the stochastic discount factor is a linear function of the level of
the state vector, as illustrated earlier, then the model admits also the potential
for time-varying term premiums.
3. VARs with 5xed and time-varying policy goals
This section discusses the use of VARs to approximate agent expectations,
especially longer-horizon expectations. Because expectations models of the term
structure of interest rates embed long-horizon forecasts, the limiting conditional
forecasts, or endpoints, of the VAR play a crucial role in term structure models.
In particular, standard VAR speci"cations frequently used in empirical macroeconomic studies embed an assumption that the market believes the long-run
policy goal for in#ation has not and will not change over time.4 This assumption
is contradicted by survey data on long-horizon in#ation expectations. This
section shows how to generalize VAR speci"cations to permit expectations to be
in#uenced by shifting perceptions of monetary policy goals. Operationally, such
shifts are mapped into time-varying intercepts of the VAR equations.
Typically, forward-looking models, including those of the term structure of
interest rates, contain implicit assumptions on long-horizon behavior. These
long-run conditions are usually determined by the decision to include variables
in level or di!erenced formats. This choice of dynamic formats has minimal
impact on short-horizon VAR forecasts for variables that are highly persistent,
such as interest rates or in#ation. However, the dynamic format can have
a substantial e!ect on long-horizon forecasts, as shown in Kozicki and Tinsley
(1998) for univariate autoregressive models. Dynamic speci"cations are similarly
important when VAR forecasts are used to proxy for long-horizon market
expectations.
Kozicki and Tinsley (1998) indicate that a convenient way to examine longrun assumptions in a time series model is to identify the endpoints of the series
4 In general, the standard approaches include variables such as interest rates and in#ation in levels
in VARs. When, as is usually the case, point estimates of the VAR coe$cients are consistent with
mean-reversion of VAR variables, the VARs implicitly embed an assumption that the VAR
endpoints for interest rates and in#ation are constant. Since the endpoint of the in#ation process
reasonably can be assumed to correspond to the long-run policy goal for in#ation, these VARs
implicitly assume that the long-run policy goal for in#ation is a constant. Clarida et al. (1997, 1998)
estimate policy rules assuming a constant in#ation target since 1979.
156
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
being modeled. As introduced in the previous section, the endpoint of the vector
E x
where the expectation is
of time series, x , is denoted k(I),lim
=
k?= I t`k
t
conditioned on information set I.5 If a series is mean reverting or I(0), then the
endpoint of the series will be a constant. This feature is just a restatement of the
property that long-run forecasts of a mean-reverting series converge to a constant, which is equal to the mean of the series in large samples. By contrast, if
a series is di!erence stationary, I(1), then, in a univariate autoregressive model,
the endpoint of the series will be a moving average of recent observations of the
series. In a multivariate VAR, the endpoint of an I(1) series will be either
a moving average of the series or a linear combination of moving averages of the
series and other series with which it is cointegrated.
VARs can be readily rewritten to explicitly reveal dynamic adjustments of
deviations from endpoints. An advantage of this reformulation is that alternative characterizations of long-run behavior can be easily implemented and
interpreted. For example, the VARs considered in this section contain an
implicit policy reaction function which relates deviations of the short rate from
its endpoint to deviations of in#ation and capacity utilization from their respective endpoints. The in#ation endpoint thus plays the role of the perceived policy
target for in#ation. The implicit endpoint restrictions imposed by I(0) assumptions correspond to an assumption that the perceived policy target for in#ation
is "xed and equal to the sample mean of in#ation. An alternative assumption
that in#ation is I(1) corresponds to a characterization that the perceived policy
target for in#ation is a moving average of in#ation.6 Relaxing the implicit
endpoint restrictions imposed by I(0) or I(1) assumptions allows additional
characterizations of time variation in the perceived policy goal for in#ation.
The remainder of this section discusses three di!erent VAR speci"cations to
illustrate the long-horizon implications of alternative endpoint speci"cations.
The VARs di!er in their assumptions about long-run behavior. In particular,
each incorporates a di!erent characterization of the in#ation endpoint, which
represents the market perception of the long-run policy target for in#ation. The
VARs are similar in that each contains the same three variables: a one-month
nominal interest rate, r, in#ation, n, and the rate of capacity utilization, y. All
interest rates referenced in this paper are end-of-month zero-coupon Treasury
bond yields.7 Monthly in#ation is measured by the BEA chain-weighted
5 Strictly speaking, the in"nite horizon subscript is applicable only for stochastic processes that do
not contain unbounded deterministic trends.
6 This assumes no cointegration between capacity utilization, interest rates, and in#ation. See the
discussion on this point later in the section.
7 Interest rates for 1960m }1991m are from McCulloch and Kwon (1993). For comparability,
1
1
these data were extended to 1997m by applying the cubic spline estimator described in McCulloch
9
(1975) to end-of-month yields. Our thanks to Mark Fisher for his generous assistance.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
157
de#ator for personal consumption expenditures, and monthly capacity utilization by the FRB index for manufacturing.
3.1. The xxed endpoints VAR
The general structure of this VAR resembles the representative small-scale
VARs used by macroeconomists such as Bernanke and Blinder (1992) and
Fuhrer and Moore (1995) to represent the responses and e!ects of monetary
policy. Such VAR speci"cations, with all variables included in levels, are typically "xed endpoint speci"cations as the estimated coe$cients are usually
consistent with mean reversion.
The top rows of Table 1 summarize estimates of univariate twelve-order
autoregressions for the one-month bond rate, r, the in#ation rate, n, and the
manufacturing utilization rate, y. The point estimates of the coe$cients of the
lagged levels of each variable, b, are consistent with mean-reverting behavior.
Table 1
Autoregressions under alternative endpoint speci"cations!
*x "c#bx #bH(¸)*x #a
t
t~1
t~1
t
x
t
r
t
n
t
y
t
r !rt
t
=
n !nt #
t
=
n !nt $
t
=
c
b
bH(1)
R2
SEE
Fixed endpoints"
0.246
(2.4)
0.310
(2.2)
2.65
(3.6)
!0.038
(!2.5)
!0.076
(!2.5)
!0.032
(!3.6)HHH
0.065
(0.3)
!1.85
(!4.1)
0.544
(5.1)
0.07
0.684
0.29
1.64
0.18
0.783
Shifting endpoints"
!0.019
(!0.5)
!0.002
(!0.0)
0.016
(0.2)
!0.087
(!3.5)HHH
!0.228
(!4.2)HHH
!0.105
(!3.0)HH
0.116
(0.4)
!1.12
(!2.1)
!1.40
(!3.1)
0.09
0.689
0.32
1.60
0.28
1.65
!The one-month nominal bond rate is denoted by r, the PCE in#ation rate by n, and the
manufacturing capacity utilization rate by y. Monthly in#ation and utilization rates are seasonally
adjusted, and the sample spans are 1955m }1997m for n and y, and 1966m }1997m for r. Entries
2
9
1
9
in parentheses are t-ratios. Endpoint constructions are discussed in the text, where xt denotes the
=
endpoint or long-horizon expectation of x as conditioned on information available in t.
"A unit root in endpoint deviations is rejected at 90% (H), 95% (HH), or 99% (HHH) signi"cance levels.
#In#ation endpoints (calendar time).
$In#ation endpoints as perceived in &real time'.
158
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Under this speci"cation, a long-horizon forecast will converge to the "xed
endpoint, !c/b. For the sample spans underlying Table 1, the estimated
endpoints are r "6.5, n "4.1, and y "82.8.8
=
=
=
Selected coe$cients of a "xed endpoints VAR for r, n, and y, using a common
sample span of 1966m }1997m ,9 are shown in the top panel of Table 2.10 The
1
9
short rate equation plays the role of a policy reaction function where a shortterm interest rate, controlled by the policy authority, responds to deviations of
in#ation from a "xed long-run policy target for in#ation and to deviations of
output from trend.
In a mean-reverting VAR, the implied policy target for in#ation is equivalent
to the "xed endpoint for in#ation, assuming information on policy targets is
symmetrically available to all public and private sector agents. As shown in the
top panel of Fig. 1, the "xed endpoint for in#ation over the 1966}1997 sample is
estimated to be a bit less than 5% at an annual rate.11 A feature of "xed
endpoints is that forecasts formulated in any period of the sample will eventually
converge to the same endpoint. This property is illustrated in the top panel of
Fig. 2, where two multiperiod forecasts of monthly in#ation rates are shown.
The "rst forecast originates in 1972m and the second in 1979m , when the
1
10
dramatic change in the policy operating procedures of the Fed was announced
by Paul Volcker. Despite the very di!erent history of in#ation preceding each
forecast, the assumption of the "xed endpoints VAR is that the underlying
policy target for in#ation is believed to be constant.
One check on whether the VAR does a reasonable job at replicating long-run
agent expectations is to compare VAR-based forecasts of 10 yr in#ation rates to
survey data on long-horizon in#ation expectations. This is done in Fig. 3. The
top panel of Fig. 3 shows VAR-based predictions of 10 yr in#ation rates,
8 In remaining empirical work, the mean of the utilization rate is removed, implying a "xed
endpoint of zero.
9 Models using a short-term interest rate to approximate the operating policy of the Federal
Reserve often omit observations before 1966 due to changes in the behavior of the interbank market
for overnight funds prior to the mid-1960s; see discussion of the relationship between the federal
funds rate and the Federal Reserve discount rate in Tinsley et al. (1982). The VARs in Table 2
contains six lags of each variable. In the case of the "xed endpoints model, the p-values of
likelihood ratios indicate that a three-lag speci"cation is rejected in favor of six lags, but a six-lag
speci"cation is not rejected against the alternatives of 9 or 12 lags.
10 Sims (1992) and Christiano et al. (1994) include commodity price indexes as an additional
determinant of interest rate policy responses. Although we explored relative commodity price
indexes in preliminary work, inclusion raised several speci"cation issues, such as the well-known
problem (Prebisch-Singer) of long-run negative trends in relative commodity prices, without contributing essential insights into the issue of time-varying endpoints.
11 Using a similar calculation, Clarida et al. (1998) estimated a constant in#ation target slightly
larger than 4% at an annual rate over 1979:10}1994:12.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
159
Table 2
VARs under alternative endpoint speci"cations!
*x "c#Ax8
#AH(¸)*x #a
t
t~1
t~1
t
x8 "x !x(t)
t
t
=
Regressor coe$cients
n
r
c
*r
*n
*y
*r
*n
*y
*r
*n
*y
A
Fixed endpoints
!0.007
!0.033
(!0.1)
(!1.8)
0.313
0.037
(1.1)
(0.7)
0.379
!0.056
(3.73)
(!3.2)
AH(1)
A
AH(1)
A
AH(1)
SEE
!0.425
(!2.7)
0.202
(0.5)
0.281
(1.9)
0.050
(2.7)
!0.114
(!2.3)
!0.009
(!0.5)
!0.305
(!3.5)
!1.64
(!6.99)
0.084
(1.0)
0.026
(2.9)
0.043
(1.8)
!0.039
(!4.5)
0.285
(3.3)
0.166
(0.7)
0.457
(5.6)
0.663
!0.176
(!2.4)
!1.94
(!9.8)
0.059
(0.8)
0.030
(3.4)
0.042
(1.8)
!0.028
(!3.4)
0.254
(3.1)
0.298
(1.4)
0.562
(7.2)
!0.261
(3.0)
!1.64
(!7.1)
0.064
(0.8)
0.032
(3.3)
0.066
(2.5)
!0.014
(!1.5)
0.196
(2.0)
0.098
(0.4)
0.390
(4.2)
Moving average endpoints
0.018
!0.462
(0.5)
(!3.3)
0.013
0.111
(0.1)
(0.3)
!0.020
!0.000
(!0.6)
(!0.0)
Shifting endpoints
!0.002
!0.069
(!0.7)
(!2.2)
0.044
0.023
(0.5)
(0.3)
!0.036
!0.091
(!1.1)
(!3.1)
y
!0.276
(!1.6)
0.184
(0.4)
0.302
(1.9)
0.029
(1.7)
!0.112
(!2.4)
!0.006
(!0.4)
1.77
0.626
0.668
1.78
0.639
0.664
1.77
0.631
!The 3]1 vector, x, contains the one-month nominal bond rate, r, the PCE in#ation rate, n, and the manufacturing capacity utilization rate, y. Monthly in#ation and utilization rates are seasonally adjusted, and the sample
span is 1966m }1997m . Entries in parentheses are t-ratios. Endpoint constructions are discussed in the text,
1
9
where x(t) denotes the endpoint or long-horizon expectation of x as conditioned on information available in t.
=
historical in#ation, and a series of spliced survey measures of long-horizon
in#ation expectations.12 Whereas the survey data have been gradually trending
down, the VAR-based predictions are relatively #at and barely deviate from the
"xed endpoint. This feature suggests that predictions from the "xed endpoint
VAR may not be very good proxies for agent expectations.
12 In the 1980s, Richard Hoey, an economist at Drexel Burnham Lambert, conducted a survey of
market participants which asked for forecasts of in#ation in the second "ve years of a 10 yr forecast
horizon. This series was used prior to 1990m . The remainder of the spliced series is the long-run
7
expectations series from the Survey of Professional Forecasters (SPF), currently published by the
Federal Reserve Bank of Philadelphia.
160
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Fig. 1. Alternative in#ation endpoints.
3.2. The moving average endpoints VAR
The fundamental di!erence between the "xed endpoints VAR and the moving
average endpoints VAR is that the latter incorporates unit root restrictions
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
161
Fig. 2. VAR multiperiod predictions of 1-month in#ation rate.
on the 1-month interest rate and in#ation rate, r and n. Returning to the top
of Table 1, the format of the autoregressions in r, n, and y, is the same
as that required for an ADF test of unit root behavior. Indeed, the t-ratios
of the coe$cients of the lagged dependent variable indicate that a unit root
hypothesis cannot be rejected for both the short-term interest rate and
162
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Fig. 3. Concatenated VAR predictions of 10 yr in#ation rates.
in#ation rate. The results of estimating the three-variable VAR under the
condition that r and n are di!erence stationary is summarized in the middle
section of Table 2.13
13 Cointegration between r and n has not been imposed. A reasonable assumption might be that
after-tax real rates are stationary. This would imply that r and n are cointegrated with a cointegrating vector that re#ects tax rates. Since tax rates have changed over time, r and n would not be
cointegrated with a constant cointegrating vector. Time-varying cointegrating restrictions are
beyond the scope of this paper.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
163
In the present discussion, the interesting property of this speci"cation is that
the endpoints of r and n are now also I(1). In each period, the conditional
endpoint of a multiperiod in#ation forecast is a moving average of in#ation in
the months just prior to the forecast period.14 This moving average property is
displayed in the middle panel of Fig. 1, where it is seen that the in#ation
endpoint * which functions as the market view of the long-run policy target
* now moves very closely with recent in#ation history. The two multiperiod
forecasts of in#ation generated by the moving endpoints VAR are presented in
the middle panel of Fig. 2. Whereas the long-run in#ation forecast in 1972m is
1
now about 1% lower than the "xed endpoint in the top panel, the moving
average endpoint for in#ation forecasts originating in 1979m is an alarming
10
10%, owing to the large run-up in historical in#ation in preceding months.
An implication of the moving-average property of the endpoint is that
VAR-based 10 yr in#ation forecasts will follow recent in#ation history. The
middle panel of Fig. 3 illustrates this property. VAR-based predictions of 10 yr
in#ation rates follow actual in#ation quite closely. Unlike predictions based on
the "xed endpoints VAR, 10 yr in#ation predictions based on the moving
average endpoints VAR are quite variable. In fact, the long-horizon predictions
are more variable than survey data. Whereas survey data suggest that longhorizon in#ation expectations adjust slowly, with a signi"cant lag compared to
actual in#ation, predictions based on the moving average VAR adjust quite
rapidly.
3.3. A shifting endpoints VAR
The third VAR speci"cation admits endpoints that can di!er from either the
"xed endpoints implied by the mean-reverting speci"cation or the moving
average endpoints provided by the di!erence-stationary speci"cation. The format of an autoregression or VAR can be written to explicitly account for
deviations from endpoints, according to
(17)
*x "c#A(x !x(t~1))#AH(¸)*x #a ,
=
t~1
t
t
t~1
where AH(¸) denotes a polynomial in the lag operator, ¸ix "x . When
t
t~i
expressed in this format, it is clear that endpoints need not be constrained to be
"xed or moving averages.
In some instances, direct measurements of the implied endpoints or longhorizon forecasts of agents are available. For example, under the expectations
hypothesis, bond rates contain ex ante long-horizon forecasts of short rates.
14 As shown in Kozicki and Tinsley (1998), the moving average endpoint is precisely the
Beveridge}Nelson &permanent' component of unit-root stochastic processes. By construction, deviations from moving average endpoints of I(1) series are stationary.
164
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Thus, as suggested in Kozicki and Tinsley (1998), one measure of the short-term
interest rate endpoint is the average of expected short rates from t#n to t#n@,
for n@'n,
n@(r !h )!n(r !h )
n{
n,t
n ,
r( (t) " n{,t
=
n@!n
(18)
where h is an estimate of a constant term premium for an n-period bond. For
n
the current study, estimates of the shifting endpoints of the short-term interest
rate are based on the average of expected short rates between the 5 yr and 10 yr
maturities. One check of the reasonableness of this interest rate endpoint is that
deviations of the one-month bond rate from this endpoint should be stationary
if agents' bond rate forecasts in a given period are internally consistent. The
fourth equation listed in Table 1 indicates that the interest rate deviations from
the constructed series of nominal rate endpoints reject the hypothesis of unit
root behavior at a 99% signi"cance level.
It is more di$cult to obtain direct measurements of long-horizon endpoints
for agents' forecasts of in#ation. Although several surveys of expected 5}10 yr
in#ation emerged in the 1980s, contiguous survey estimates of expected longhorizon in#ation are not available for earlier decades. To generate backcasts of
agent expectations of long-run in#ation, bond traders are assumed to be &boundedly-rational' statistical agents, who sequentially test for intercept shifts in an
autoregressive model of in#ation. The statistical learning model is based on
a univariate autoregressive model of in#ation. The intuition behind the assumed
learning behavior is that agents infer an error in their current estimate of the
long-run policy goal of in#ation when they observe a string of prediction errors
for in#ation of the same sign, and adjust their outlook accordingly. These shifts
in the long-run perceptions induce nonstationarity in the mean of the in#ation
process.
Central bankers of developed economies are generally cautious and slow to
change either operational policies or strategic objectives. Thus, changes in
long-run policy objectives are likely episodic with frequencies that are more
appropriately measured in half-decades rather than months. This inference
seems to be consistent with the small number of policy regimes typically
identi"ed in postwar analyses of policy, such as Huizinga and Mishkin (1986).
A simple representation of episodic policy shifts in the in#ation process as
considered by agents is
(19)
*n "b #+ d(t!q )*b k #b n #b(¸)*n #a .
1 t~1
t~1
t
t
0
k
0,q
k
To represent possible shifts in the intercept, the dummy variable notation, d( . ),
denotes the Heaviside switching function where d(t!q )"0 if t(q . If
k
k
a change in the intercept is detected for period q , the function is switched &on' in
k
all subsequent periods, d(t!q )"1 if t5q .
k
k
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
165
In the absence of intercept shifts, the long-run in#ation rate de"ned by (19) is
n "!b /b . However, if intercept shifts are sequentially detected, the evolu=
0 1
tion of expected long-run in#ation is determined by the current estimate of the
intercept, n(t) "![b #+ k *b k ]/b . By not presuming ex ante knowledge
=
0
q :t 0,q
1
of the number of distinct policy regimes, the changepoint learning model allows
agents to be less informed about possible policy regimes than would be the case
for a shifting-regimes analysis.15
In the learning simulations referenced in this paper, agents test for intercept
shifts in a twelve-order autoregression in in#ation using critical values extrapolated from Andrews (1993). The search by simulated agents for signi"cant
changepoints is in &real time' as new observations are added in expanding
monthly samples over the period 1954m }1997m . To account for hetero1
9
geneous expectations among bond traders, agents are viewed as subscribing to
one of m market forecast &newsletters'. Before testing for a new in#ation endpoint, each newsletter accumulates a di!erent minimum number of monthly
observations since the date of the last estimated changepoint. The minimum
number of observations used by the ith newsletter is denoted s . Results disi
cussed here are based on "fteen newsletters, m"15. The minimum sample
collected since the last changepoint ranges from 12 months for newsletter 1 to 96
months for newsletter 15: s "6(1#i).
i
An important by-product of this simulated model of sequential, &real-time'
learning by agents is that it generates both the date of an estimated changepoint
and the date of the subsequent recognition of that changepoint, when su$cient
observations have been collected to satisfy the relevant critical value of the
changepoint test. The concatenation of shifted endpoints in the estimated
in#ation model is termed the &calendar time' endpoint series, and the concatenation of estimated endpoints when perceived by the learning agents is termed the
&real time' endpoint series.
A sampling of shifting in#ation endpoints generated by competing forecast
newsletters is shown in Fig. 4. The top panel indicates the in#ation endpoints
generated by the newsletter that collects a minimum of 30 months of observations after detecting a changepoint before testing for a subsequent shift. The
middle and bottom panels correspond to newsletters with minimum observation
collections of 60 and 90 months respectively. As indicated, newsletters with
a smaller number of minimum observations have a shorter lag in detecting
changepoints, as indicated by the shorter horizontal displacements between the
calendar time endpoints series (bold line) and the &real time' endpoints series
(dashed line). Newsletters using smaller minimum samples also can generate
15 Regime-shifting speci"cations, such as proposed by Hamilton (1989), typically presume ex ante
knowledge of the number of policy regimes, a constraint that is di$cult to verify empirically and to
implement in real time policy analysis.
166
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Fig. 4. Shifting in#ation endpoints used by competing forecast newsletters.
more volatile long-run forecasts with frequent adjustments of estimated endpoints. This di!erence depends on the underlying behavior of historical in#ation, as can seen by comparing the frequency and size of estimated changepoints
in the 1970s and early 1980s with the infrequent endpoint adjustments by all
newsletters after the mid-1980s.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
167
An aggregate in#ation endpoint series is constructed by weighted averages of
the di!erent newsletter estimates of long-run in#ation,
m
m
+ n "1,
(20)
n( (t) " + n n( (t) ,
i,t
=
i,t i,=
i/1
i/1
where n 50 is a nonnegative weight indicating the proportion of bond traders
i,t
subscribing to the ith newsletter in month t.
The relative contribution by each forecast newsletter, n , to the aggregate
i,t
in#ation endpoint is estimated by relating movements in the after-tax endpoint
of the nominal interest rate to the movements of the aggregate of newsletter
in#ation endpoints. That is, the long-run, after-tax nominal interest rate is
assumed equal to the sum of the long-run real rate and the long-run in#ation
rate,
(21)
(1!t )r(t) "o #n(t) ,
=
=
x,t =
where t is the tax rate on bond earnings and o is the long-horizon expectation
x
=
of the after-tax real rate. Notice that this speci"cation assumes after-tax real
rates are stationary. Eq. (21) has two important properties. First, unlike the
traditional Fisher equation, nominal rates will move more than one-for-one
with expected in#ation under positive tax rates.16 Second, shifting agent perceptions of the long-run policy goal for in#ation, as captured by n , are a likely
=
source of shifts in the perceived endpoint of the nominal rate, r .
=
The selection of forecast newsletters by agents is described by a stochastic
discrete choice model. The utility of the ith newsletter to its subscriber subpopulation in period t is represented by
u(i) "u #e , i"1,2, 15,
(22)
t
i,t
i,t
where u denotes that portion of utility that is related to observable variables.
i,t
This component is assumed to consist of a constant taste preference for the ith
newsletter, v , and a quadratic penalty on deviations from the expected conseni
sus forecast of long-run in#ation.17
u "v #v (n(i)(t) !n(t) )2, i"1,2, 15,
=
=
i,t
i
0
(23)
16 Crowder and Ho!man (1996) illustrate the historical relevance of the nonzero tax rate.
Estimated tax rates are drawn from McCulloch and Kwon (1993). McCulloch (1975) found that the
e!ective tax rate that best explains the prices of U.S. Treasury securities lies in the range, 0.22 to 0.30.
As noted by Green and Odegaard (1997), tax rates fell substantially after 1986.
17 Traders are assumed to have disparate priors about the frequency of shifts in the long-run
in#ation objectives of policy but place some weight also on the long-run views of other agents. For
more extensive discussion of the dynamic implications of discrete choice models with social
interactions, see Brock and Durlauf (1995) and Brock and Hommes (1996).
168
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
where n(i)(t) is the in#ation endpoint selected by the ith newsletter.18 To simplify
=
notation, time subscripts will be dropped in some subsequent equations if the
omission is not misleading.
The residual portion of utility associated with the ith forecast newsletter, e , is
i,t
assumed to follow an extreme value distribution with a zero mean and variance
proportional to c~2. As shown in Anderson et al. (1992), the assumptions of
discrete choice utility maximization and the extreme value distribution of
residual utility imply that the relevant choice probabilities can be characterized
by a multinomial logit model. Thus, the probability of selecting the ith forecast
newsletter is
A
B
n "Pr u(i)"max u( j) ,
i
j
"Pr(e !e 4u !u ,2, e !e 4u !u ),
1
i
i
1
n
i
i
n
i
ecu
.
(24)
"
+m ecuj
j/1
Estimates of the parameters (o , c, v , i"0,2, m) de"ned by the system of
=
i
equations (20)}(24) over the sample 1955m }1997m are shown in Table 3. The
7
9
estimate of the long-run after-tax real rate, o , is 2.21. The frequency distribu=
tion of agents over newsletters is smoothed by constraining the constant taste
preferences, v (i"1,2, m), to lie on a third-degree polynomial. Given that the
i
choice probabilities are invariant to a constant addition to utility, the constant
taste parameter associated with the newsletter using the largest minimum
observation sample is normalized to zero, v "0. Whereas the estimated
15
constant taste parameters are all positive, the estimate of the coe$cient of
squared deviations of newsletter endpoints from the consensus endpoint, v , is
0
signi"cantly negative, indicating forecasts nearer the consensus estimate tend to
attract more subscribers. Finally, the nonnegative precision parameter, c, is
associated with the normalization, (+m v2)1@2"1.19 The c parameter indicates
j/0 j
the concentration of agent preferences. That is, if cP#R, the newsletter
with the highest observable utility characteristics will attract all subscribers.
Conversely, if cP0, each newsletter will attract an equal market share of
subscribers, on average.
As indicated in Table 4, the sample mean estimates of the newsletter subscriber shares, g6 , indicate that more than 40% of the simulated agents are
i,t
18 To reduce computations in the estimation stage, the contraction estimate of the aggregate
in#ation endpoint in each month, n(t) , is proxied by the aggregate in#ation endpoint generated in the
=
preceding month, t!1.
19 This normalization is similar to that suggested in McFadden (1981).
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
169
Table 3
Discrete choice selection of forecast newsletters!
15
(1!t )r(t) "o # + n n( (t)
x,t =
=
i,t i,=
i/1
15
n "ecui, t + ecuj, t
i,t
j/1
u "v #v (n( (t) !n( (t~1))2
i,t
i
0 i,=
=
N
o
=
v }v
1 5
v }v
6 10
v }v
11 15
v
2.21
(27.1)
*
*
*
*
*
*
*
*
0.318
(1.9)
0.298
(2.4)
0.297
(2.7)
0.284
(2.9)
0.285
(2.9)
0.289
(2.7)
0.292
(2.5)
0.293
(2.3)
0.288
(2.1)
0.275
(1.9)
0.251
(1.8)
0.214
(1.6)
0.162
(1.5)
0.091
(1.5)
0.0
*
!0.101
(!8.2)
*
*
*
*
*
*
*
*
0
R2
0.86
*
*
*
*
*
*
*
*
*
!The endpoint of the after-tax nominal rate, (1!t )r(t) , is de"ned as the sum of the long-run real
x,t =
interest rate, o and of a weighted average of in#ation endpoints used by competing forecast
=
newsletters, n(t) . The weights are the probabilities of choosing a forecast newsletter and, as noted in
i,=
the text, the choice probabilities, n , are generated by a multinomial logit model. Estimates are
i,t
based on the monthly sample, 1955m }1997m . The estimated precision is: c"3.28. Estimates of
7
9
remaining parameters are listed in the table, along with t-ratios in parentheses.
clustered, on average, around newsletters 7}11, which use minimum observation
samples from 48 to 72 months. Thus, the implied aggregate &real time' lag in
detecting past shifts in in#ation endpoints appears to be relatively lengthy with
a minimum detection lag of around "ve years. However, the sizeable standard
deviations of the relative subscription shares of each newsletter suggest substantial variation over time in the aggregate detection lag. To illustrate this time
variation, Fig. 5 plots the aggregate minimum detection lag, de"ned as the
weighted average of minimum observation periods used by each newsletter,
+m n s . The estimated aggregate minimum detection lag ranges from lows of
i/1 i,t i
about three years (36 months) to highs of around six years (72 months). The
adjustment pattern in Fig. 5 suggests that a sizeable and sustained change in the
level of recent in#ation leads to an initial increase in the aggregate minimum
detection lag followed by a relatively rapid decrease. This reversing movement
in the aggregate detection lag is driven by the penalty on deviations of newsletter
in#ation endpoints from the consensus aggregate endpoint. That is, agents
initially cancel subscriptions to the newsletters with short detection lags whose
long-run forecasts appear to be overreacting to recent in#ation observations.
170
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Table 4
Distribution of agents over in#ation forecast newsletters!
15
15
n( (t) " + n n( (t) , n 50, + n "1
=
i,t i,=
i,t
i,t
i/1
i/1
i
s"
i
n6 #
i,t
p(n )$
i,t
i
s"
i
n6
i,t
p(n )
i,t
1
2
3
4
5
6
7
8
12
18
24
30
36
42
48
54
0.058
0.068
0.064
0.058
0.077
0.072
0.075
0.084
0.052
0.044
0.044
0.044
0.044
0.030
0.040
0.032
9
10
11
12
13
14
15
*
60
66
72
78
84
90
96
*
0.081
0.085
0.085
0.060
0.073
0.044
0.034
*
0.033
0.038
0.039
0.034
0.032
0.022
0.030
*
!The aggregate in#ation endpoint, n(t) , is a weighted average of in#ation endpoints used by
=
simulated forecast newsletters, n(t) , where the weights are the estimated choice probabilities of
i,=
newsletters, n , over the 1955}1997 sample.
i,t
"Minimum number of monthly observations since last changepoint, s "6(1#i).
i
#Average share of agent subscriptions to the ith newsletter.
$Standard deviation of subscription shares.
Fig. 5. Average minimum detection lag of in#ation endpoint shifts.
However, if additional forecast newsletters also revise their endpoint estimates
in the same direction, the aggregate endpoint is eventually dominated by revised
endpoint estimates, and subscribers abruptly move away from the most conservative newsletters whose endpoints have not yet been revised.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
171
The last two autoregressions shown in Table 1 provide evidence against the
hypothesis that deviations of monthly in#ation from either the aggregated
calendar time endpoints series or the aggregated &real time' endpoints series are
I(1).20 Because the purpose of the VAR is to generate agent expectations that
approximate information available to markets when the historical data are
recorded, the assumption of symmetric access to policy target information
implicit in the "xed and moving average endpoints models is dropped, and the
&real time' in#ation endpoint is selected as a more realistic estimate of historical
market perceptions of the long-run policy target for in#ation. This selection is
supported in the bottom panel of Fig. 3, which shows that the predictions of
10 yr in#ation rates based on the shifting endpoints VAR resemble surveys of
expected long-run in#ation.
Estimated VAR coe$cients are provided in the bottom panel of Table 2.
Coe$cients in columns labeled &A' correspond to the sum of coe$cients on the
lagged levels of the relevant regressor variable. Interestingly, in most instances,
the sum of coe$cients in the shifting endpoints VAR is insigni"cantly di!erent
from the corresponding su
25 (2001) 149}184
Term structure views of monetary policy under
alternative models of agent expectationsq
Sharon Kozicki!,*, P.A. Tinsley"
!Federal Reserve Bank of Kansas City, 925 Grand Boulevard, Kansas City, MO, 64198 USA
"Faculty of Economics and Politics, University of Cambridge, Cambridge CB3 9DD, UK
Abstract
Term structure models and many descriptions of the transmission of monetary policy
rest on the empirical relevance of the expectations hypothesis. Small di!erences in the
perceived policy reaction function in VAR models of agent expectations strongly in#uence the relevance in the transmission mechanism of the expected short rate component
of bond yields. Mean-reverting or di!erence-stationary characterizations of interest rates
require large and volatile term premiums to match the observable term structure.
However, short rate descriptions that capture shifting perceptions of long-horizon
in#ation evident in survey data support a more substantial term structure role for short
rate expectations. ( 2001 Elsevier Science B.V. All rights reserved.
JEL classixcation: E4
Keywords: Changepoints; Expectations hypothesis; Nonstationary in#ation; Shifting
endpoint
* Corresponding author. Tel.: #1-816-881-2561; fax: #1-816-881-2199.
E-mail addresses: [email protected] (S. Kozicki), [email protected] (P.A. Tinsley)
q
We thank Dennis Ho!man, Vance Roley, Juoko Vilmunen, seminar participants at the University of Washington, session participants at the 1998 annual meetings of the AEA and the Fourth
International Conference of the Society for Computational Economics, and anonymous referees for
comments and suggestions. The views expressed are those of the authors and do not necessarily
represent those of the Federal Reserve Bank of Kansas City or the Federal Reserve System.
0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 7 2 - X
150
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
1. Introduction
The term structure of interest rates is often characterized as a crucial transmission channel of monetary policy where accurate market perceptions of policy
are required for e!ective policy execution. This description rests on three
propositions: First, a short-term interest rate, such as the overnight federal funds
rate, provides an adequate summary of monetary policy actions. Second, longterm bond yields are important determinants of the opportunity cost of investments. Third, under the expectations hypothesis, the anticipated path of the
policy short-term rate is the main determinant of the term structure of bond
rates. Although the empirical validity of each of these three propositions has
been criticized, the accumulation of empirical evidence against the third proposition * the expectations hypothesis * is impressively large.1 If variations in
current bond rates are not well-connected to current and expected movements
in the policy-controlled short-term rate, then the conventional description of
monetary policy transmission is vacuous.
In no-arbitrage formulations of the term structure, variation in bond
rates due to current and expected movements in short rates is determined
by the description of short rate dynamics. Under the standard "nance assumption that short rates are mean-reverting, the average of expected future
short rates is considerably smoother than historical bond rates. Thus, the
expectations hypothesis is empirically rejected by tests which assume constant
term premiums, and postwar shifts in the term structure are often attributed
to sizable movements in &liquidity' and &term' premiums. However, postwar
data are consistent with descriptions of short rate movements other than
mean reversion. This paper shows that small di!erences in the speci"cation of
the stochastic process for the short rate strongly in#uence the relative importance of short rate expectations and residual term premiums in bond rate
movements.
As an alternative to conventional descriptions of short rate dynamics,
a simple class of time-varying descriptions of short rate behavior is examined.
Given well-documented shifts in postwar monetary policy, it seems highly
probable that the short rate expectations of bond traders are heavily in#uenced
by shifting perceptions of monetary policy. Short rate descriptions that capture
shifting perceptions of the long-run objective of monetary policy, formulated as
1 For instance, Campbell and Shiller (1991) compare estimated &theoretical' spreads to actual
spreads and conclude that the spread is too variable to accord with the expectations hypothesis. In
addition, in regressions of long-rate changes on spreads, Shiller (1979), Shiller et al. (1983), and
Campbell and Shiller (1991) "nd that coe$cient estimates on the spread tend to be signi"cantly
di!erent from the hypothesized value of one and frequently negative.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
151
a long-run in#ation target, are supportive of a substantial term structure role for
short rate expectations.2
In addition to rejecting the implication that high bond rates in the 1980s
re#ect large term premiums, the preferred description of short rate behavior also
does not support interpretations that perceived in#ation targets were rapidly
reversed in the 1980s. More likely, the long-run in#ation target perceived by the
market behaved similarly to survey data on long-run expected in#ation. Survey
data suggest agents were cautious in adjusting their perceptions of long-run
in#ation.
This paper illustrates the term structure implications of alternative representations of monetary policy perceptions of agents. Section 2 develops a discrete-time, no-arbitrage model of the term structure where the stochastic
discount factor is related to a vector of macroeconomic variables. Forecasts of
this state vector are generated by a VAR with #exible speci"cations of long-run
behavior. Section 3 discusses three VARs with di!erent characterizations of
agent perceptions of long-run behavior. Owing to the di!erent characterizations
of perceived long-run behavior, the VARs embed perceived policy reaction
functions with di!erent implied in#ation targets. Section 4 indicates how estimates of expected short-rate components of bond yields and conventional
&residual' estimates of term premiums depend critically on assumptions regarding the long-run behavior of policy targets in VAR proxies of agent expectations.
Section 5 concludes.
2. A model of the term structure of interest rates
A discrete-time term structure model is outlined in this section to illustrate the
pivotal role of long-run forecasts by VAR models in representing agent expectations. Bond yields are linked to expectations of monetary policy through a VAR
proxy for expectations formation which contains an implicit policy reaction
function. The implicit policy reaction function relates short term interest rates to
deviations of economic variables from their long-run equilibrium values, including deviations of in#ation from the long-run in#ation goal of monetary policy.
Thus, bond yields depend on monetary policy primarily through the in#uence of
market perceptions of policy on short rate expectations.
2 As developed later, shifting perceptions are represented in this paper as sequential learning,
where agents test for changepoints in parameters of linear time series models. Other approaches to
the apparent nonstationarity of interest rates include the fractional di!erence processes in Backus
and Zin (1993), the multiple regimes models in Evans and Lewis (1995) and Bekaert et al. (1997), and
the ¢ral tendency' factor of Balduzzi et al. (1998).
152
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
The price of an n-period nominal bond, P , satis"es
n,t
P "E [P
M ],
(1)
n,t
t n~1,t`1 t`1
where M
is the nominal stochastic discount factor. Assuming M
and the
t`1
t`1
bond price are conditionally joint lognormally distributed, (1) can be represented as
p "E Mm #p
N#1Var [m #p
],
(2)
2
n,t
t t`1
n~1,t`1
t t`1
n~1,t`1
with lower-case letters denoting the natural logarithms of the corresponding
upper-case letters. The log yield-to-maturity on an n-period bond is equal to
1
r "! p .
n,t
n n,t
(3)
A recursive expression for log yields can be obtained by substituting the yieldprice relationship into Eq. (2):
1
1
N! Var (m !(n!1)r
).
r " E M!m #(n!1)r
t`1
n~1,t`1
t t`1
n~1,t`1
n,t n t
2n
(4)
Noting that for n"1, p
"p
"0, the yield on a one-period bond is
n~1,t`1
0,t`1
r "E (!m )!1Var (m ).
(5)
2
1,t
t
t`1
t t`1
Using the initial condition (5), the di!erence equation (4) can be solved to
express the yield of an n-period bond as
C
D
1
n~1
r " E + r
#h ,
n,t n t
t`i
n,t
i/0
where the last component
(6)
1 n~1
+ Var ((n!i)r
)
h "!
t
n~i,t`i
n,t
2n
i/1
1 n~1
# + Cov (m , (n!i)r
),
(7)
t t`i
n~i,t`i
n
i/1
is the term premium.3 Thus, Eq. (6) explicitly identi"es the roles of expected
future short rates and the term premium in longer maturity bond yields.
To enable use of VAR forecast systems, we use a discrete-time variant of the
&a$ne' class of bond pricing models, such as discussed by Backus et al. (1996)
3 The "rst term in h is a Jensen's inequality term while the second contains the more convenn,t
tional liquidity, risk, or term premium. Generally, we will use the phrase &term premium' to refer
to h .
n,t
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
153
and Campbell et al. (1997), where bond log yields are linear functions of state
variables. Consequently, the stochastic discount factor, m , is speci"ed to be
t`1
a linear function of a vector of state variables, x .
t
!m "/ #/@x #w ,
(8)
t`1
0
t
t`1
where w
is a serially uncorrelated innovation. The discount factor shock is
t`1
a weighted average of shocks to the state vector, u , and an additional
t`1
innovation term, l ,
t`1
w "b@u #l ,
(9)
t`1
t`1
t`1
where u
and l
are uncorrelated at all leads and lags. The shocks have zero
t`1
t`1
means, and the variance of l
is constant, E (l2 ),p2. The conditional
l
t`1
t t`i
, where
covariance of the state innovations is denoted as E (u u@ ),R
u,t`1
t t`1 t`1
the covariances may be time varying,
Var (m )"b@R
b#p2,
l
t t`1
u,t`1
(10)
"b #b@x #p2,
l
0
t
and the second line in (10) imposes the a$ne restriction that any time variations
in the variances of state innovations are due to square-root processes, as in Cox
et al. (1985) and Pearson and Sun (1994).
Because all forecast models of the state vector in this paper contain interest
rate feedback rules to represent predictable policy responses, we assume one of
the state variables, x , is the one-period interest rate, r ,r . This imposes
1,t
1,t
t
restrictions on elements of / and /. To illustrate, substitute Eqs. (8) and (10)
0
into Eq. (5), giving
r "E (!m )!1 Var (m ),
t t`1
2
t
t
t`1
(11)
"/ #/@x !1[b #b@x #p2].
v
0
t 2 0
t
If the covariance matrix of the state innovations is constant, b in Eq. (10) is
a zero vector. In this instance, the one-period interest rate is equal to the "rst
element of the state vector, x "r , if / "(1/2)[b #p2] and / is a selector
v
1,t
t
0
0
vector with one in the "rst element, / "1 (and zeros elsewhere). As an
1
alternative example, suppose the "rst element of b is a positive constant, b '0,
1
so that the variance of the stochastic discount factor is a linear function of the
level of x (and the remaining elements of b are zero). To again equate the
1,t
one-period interest rate to the "rst state variable, the only alteration required of
the restrictions for the constant variance case is to rede"ne the "rst element of
the selector vector: / "1#(1/2)b .
1
1
Returning to the expected short-rate terms in the bond rate equation (6), the
expectations of bond traders are represented by a VAR forecasting system. The
154
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
vector of macroeconomic state variables, x , is a nonexplosive VAR process
t
about the endpoint vector, k(t~1),
=
x "Hx #(I!H)k(t~1)#u ,
=
t
t
t~1
(12)
where the coe$cient matrix, H, has all eigenvalues less than or equal to one in
magnitude. Following Kozicki and Tinsley (1998), the endpoint, k(I), is the
=
limiting conditional expectation of the vector x ,
t
(13)
k(I), lim E x ,
=
I t`k
k?=
with I indexing the time subscript of the information set on which expectations
are conditioned.
The endpoint representation is su$ciently general to encompass more traditional VAR forecasting systems in which elements of x
are classi"ed as either
t`1
I(0) or I(1). In particular, if an element of x
is I(0), then the corresponding
t`1
is I(1), then the
element of k(t) will be a constant; if an element of x
=
t`1
corresponding element of k(t) will be a linear combination of elements of x . The
=
t
endpoint representation also encompasses cases in which elements of x
may
t`1
be subject to sporadic mean shifts with the corresponding elements of k(t) deter=
ministic but subject to occasional shifts. Such speci"cations are convenient for
explicitly accounting for changes in the perceived long-run in#ation goals of
monetary policy within a VAR framework. Three competing VAR speci"cations, di!ering primarily according to endpoint assumptions, are examined in
detail in the next subsection.
As earlier, we continue the convention that the short-term interest rate is the
"rst element of the state vector, r "n@ x , where n is a selector vector with one in
t
1 t
1
the "rst element. Multistep forecasts of short rates can be constructed using the
state-variable VAR. Expression (12) implies
(14)
E r "n@ [Hix #(I!Hi)k(t) ].
=
1
t
t t`i
Substituting into (6), the n-period yield can be rewritten in terms of the vector of
state variables as
1 n~1
(15)
r " + n@ [Hix #(I!Hi)k(t) ]#h .
=
n,t
1
t
n,t n
i/0
Eq. (15) relates bond yields in t to contemporaneous observations on state
variables, including the one-period rate in t. Predictions of yields, conditional on
data through t!1, can be constructed as
1 n~1
E r " + n@ [Hi`1x #(I!Hi`1)k(t~1)]#E h .
=
t~1 n,t
1
t~1
t~1 n,t n
i/0
(16)
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
155
This term structure model has two noteworthy features. First, because the
short rate equation in the VAR approximates agent perceptions of the monetary
policy reaction function, monetary policy in#uences bond yields through a dependence of short rate expectations on expected policy. Second, if the conditional variance of the stochastic discount factor is a linear function of the level of
the state vector, as illustrated earlier, then the model admits also the potential
for time-varying term premiums.
3. VARs with 5xed and time-varying policy goals
This section discusses the use of VARs to approximate agent expectations,
especially longer-horizon expectations. Because expectations models of the term
structure of interest rates embed long-horizon forecasts, the limiting conditional
forecasts, or endpoints, of the VAR play a crucial role in term structure models.
In particular, standard VAR speci"cations frequently used in empirical macroeconomic studies embed an assumption that the market believes the long-run
policy goal for in#ation has not and will not change over time.4 This assumption
is contradicted by survey data on long-horizon in#ation expectations. This
section shows how to generalize VAR speci"cations to permit expectations to be
in#uenced by shifting perceptions of monetary policy goals. Operationally, such
shifts are mapped into time-varying intercepts of the VAR equations.
Typically, forward-looking models, including those of the term structure of
interest rates, contain implicit assumptions on long-horizon behavior. These
long-run conditions are usually determined by the decision to include variables
in level or di!erenced formats. This choice of dynamic formats has minimal
impact on short-horizon VAR forecasts for variables that are highly persistent,
such as interest rates or in#ation. However, the dynamic format can have
a substantial e!ect on long-horizon forecasts, as shown in Kozicki and Tinsley
(1998) for univariate autoregressive models. Dynamic speci"cations are similarly
important when VAR forecasts are used to proxy for long-horizon market
expectations.
Kozicki and Tinsley (1998) indicate that a convenient way to examine longrun assumptions in a time series model is to identify the endpoints of the series
4 In general, the standard approaches include variables such as interest rates and in#ation in levels
in VARs. When, as is usually the case, point estimates of the VAR coe$cients are consistent with
mean-reversion of VAR variables, the VARs implicitly embed an assumption that the VAR
endpoints for interest rates and in#ation are constant. Since the endpoint of the in#ation process
reasonably can be assumed to correspond to the long-run policy goal for in#ation, these VARs
implicitly assume that the long-run policy goal for in#ation is a constant. Clarida et al. (1997, 1998)
estimate policy rules assuming a constant in#ation target since 1979.
156
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
being modeled. As introduced in the previous section, the endpoint of the vector
E x
where the expectation is
of time series, x , is denoted k(I),lim
=
k?= I t`k
t
conditioned on information set I.5 If a series is mean reverting or I(0), then the
endpoint of the series will be a constant. This feature is just a restatement of the
property that long-run forecasts of a mean-reverting series converge to a constant, which is equal to the mean of the series in large samples. By contrast, if
a series is di!erence stationary, I(1), then, in a univariate autoregressive model,
the endpoint of the series will be a moving average of recent observations of the
series. In a multivariate VAR, the endpoint of an I(1) series will be either
a moving average of the series or a linear combination of moving averages of the
series and other series with which it is cointegrated.
VARs can be readily rewritten to explicitly reveal dynamic adjustments of
deviations from endpoints. An advantage of this reformulation is that alternative characterizations of long-run behavior can be easily implemented and
interpreted. For example, the VARs considered in this section contain an
implicit policy reaction function which relates deviations of the short rate from
its endpoint to deviations of in#ation and capacity utilization from their respective endpoints. The in#ation endpoint thus plays the role of the perceived policy
target for in#ation. The implicit endpoint restrictions imposed by I(0) assumptions correspond to an assumption that the perceived policy target for in#ation
is "xed and equal to the sample mean of in#ation. An alternative assumption
that in#ation is I(1) corresponds to a characterization that the perceived policy
target for in#ation is a moving average of in#ation.6 Relaxing the implicit
endpoint restrictions imposed by I(0) or I(1) assumptions allows additional
characterizations of time variation in the perceived policy goal for in#ation.
The remainder of this section discusses three di!erent VAR speci"cations to
illustrate the long-horizon implications of alternative endpoint speci"cations.
The VARs di!er in their assumptions about long-run behavior. In particular,
each incorporates a di!erent characterization of the in#ation endpoint, which
represents the market perception of the long-run policy target for in#ation. The
VARs are similar in that each contains the same three variables: a one-month
nominal interest rate, r, in#ation, n, and the rate of capacity utilization, y. All
interest rates referenced in this paper are end-of-month zero-coupon Treasury
bond yields.7 Monthly in#ation is measured by the BEA chain-weighted
5 Strictly speaking, the in"nite horizon subscript is applicable only for stochastic processes that do
not contain unbounded deterministic trends.
6 This assumes no cointegration between capacity utilization, interest rates, and in#ation. See the
discussion on this point later in the section.
7 Interest rates for 1960m }1991m are from McCulloch and Kwon (1993). For comparability,
1
1
these data were extended to 1997m by applying the cubic spline estimator described in McCulloch
9
(1975) to end-of-month yields. Our thanks to Mark Fisher for his generous assistance.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
157
de#ator for personal consumption expenditures, and monthly capacity utilization by the FRB index for manufacturing.
3.1. The xxed endpoints VAR
The general structure of this VAR resembles the representative small-scale
VARs used by macroeconomists such as Bernanke and Blinder (1992) and
Fuhrer and Moore (1995) to represent the responses and e!ects of monetary
policy. Such VAR speci"cations, with all variables included in levels, are typically "xed endpoint speci"cations as the estimated coe$cients are usually
consistent with mean reversion.
The top rows of Table 1 summarize estimates of univariate twelve-order
autoregressions for the one-month bond rate, r, the in#ation rate, n, and the
manufacturing utilization rate, y. The point estimates of the coe$cients of the
lagged levels of each variable, b, are consistent with mean-reverting behavior.
Table 1
Autoregressions under alternative endpoint speci"cations!
*x "c#bx #bH(¸)*x #a
t
t~1
t~1
t
x
t
r
t
n
t
y
t
r !rt
t
=
n !nt #
t
=
n !nt $
t
=
c
b
bH(1)
R2
SEE
Fixed endpoints"
0.246
(2.4)
0.310
(2.2)
2.65
(3.6)
!0.038
(!2.5)
!0.076
(!2.5)
!0.032
(!3.6)HHH
0.065
(0.3)
!1.85
(!4.1)
0.544
(5.1)
0.07
0.684
0.29
1.64
0.18
0.783
Shifting endpoints"
!0.019
(!0.5)
!0.002
(!0.0)
0.016
(0.2)
!0.087
(!3.5)HHH
!0.228
(!4.2)HHH
!0.105
(!3.0)HH
0.116
(0.4)
!1.12
(!2.1)
!1.40
(!3.1)
0.09
0.689
0.32
1.60
0.28
1.65
!The one-month nominal bond rate is denoted by r, the PCE in#ation rate by n, and the
manufacturing capacity utilization rate by y. Monthly in#ation and utilization rates are seasonally
adjusted, and the sample spans are 1955m }1997m for n and y, and 1966m }1997m for r. Entries
2
9
1
9
in parentheses are t-ratios. Endpoint constructions are discussed in the text, where xt denotes the
=
endpoint or long-horizon expectation of x as conditioned on information available in t.
"A unit root in endpoint deviations is rejected at 90% (H), 95% (HH), or 99% (HHH) signi"cance levels.
#In#ation endpoints (calendar time).
$In#ation endpoints as perceived in &real time'.
158
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Under this speci"cation, a long-horizon forecast will converge to the "xed
endpoint, !c/b. For the sample spans underlying Table 1, the estimated
endpoints are r "6.5, n "4.1, and y "82.8.8
=
=
=
Selected coe$cients of a "xed endpoints VAR for r, n, and y, using a common
sample span of 1966m }1997m ,9 are shown in the top panel of Table 2.10 The
1
9
short rate equation plays the role of a policy reaction function where a shortterm interest rate, controlled by the policy authority, responds to deviations of
in#ation from a "xed long-run policy target for in#ation and to deviations of
output from trend.
In a mean-reverting VAR, the implied policy target for in#ation is equivalent
to the "xed endpoint for in#ation, assuming information on policy targets is
symmetrically available to all public and private sector agents. As shown in the
top panel of Fig. 1, the "xed endpoint for in#ation over the 1966}1997 sample is
estimated to be a bit less than 5% at an annual rate.11 A feature of "xed
endpoints is that forecasts formulated in any period of the sample will eventually
converge to the same endpoint. This property is illustrated in the top panel of
Fig. 2, where two multiperiod forecasts of monthly in#ation rates are shown.
The "rst forecast originates in 1972m and the second in 1979m , when the
1
10
dramatic change in the policy operating procedures of the Fed was announced
by Paul Volcker. Despite the very di!erent history of in#ation preceding each
forecast, the assumption of the "xed endpoints VAR is that the underlying
policy target for in#ation is believed to be constant.
One check on whether the VAR does a reasonable job at replicating long-run
agent expectations is to compare VAR-based forecasts of 10 yr in#ation rates to
survey data on long-horizon in#ation expectations. This is done in Fig. 3. The
top panel of Fig. 3 shows VAR-based predictions of 10 yr in#ation rates,
8 In remaining empirical work, the mean of the utilization rate is removed, implying a "xed
endpoint of zero.
9 Models using a short-term interest rate to approximate the operating policy of the Federal
Reserve often omit observations before 1966 due to changes in the behavior of the interbank market
for overnight funds prior to the mid-1960s; see discussion of the relationship between the federal
funds rate and the Federal Reserve discount rate in Tinsley et al. (1982). The VARs in Table 2
contains six lags of each variable. In the case of the "xed endpoints model, the p-values of
likelihood ratios indicate that a three-lag speci"cation is rejected in favor of six lags, but a six-lag
speci"cation is not rejected against the alternatives of 9 or 12 lags.
10 Sims (1992) and Christiano et al. (1994) include commodity price indexes as an additional
determinant of interest rate policy responses. Although we explored relative commodity price
indexes in preliminary work, inclusion raised several speci"cation issues, such as the well-known
problem (Prebisch-Singer) of long-run negative trends in relative commodity prices, without contributing essential insights into the issue of time-varying endpoints.
11 Using a similar calculation, Clarida et al. (1998) estimated a constant in#ation target slightly
larger than 4% at an annual rate over 1979:10}1994:12.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
159
Table 2
VARs under alternative endpoint speci"cations!
*x "c#Ax8
#AH(¸)*x #a
t
t~1
t~1
t
x8 "x !x(t)
t
t
=
Regressor coe$cients
n
r
c
*r
*n
*y
*r
*n
*y
*r
*n
*y
A
Fixed endpoints
!0.007
!0.033
(!0.1)
(!1.8)
0.313
0.037
(1.1)
(0.7)
0.379
!0.056
(3.73)
(!3.2)
AH(1)
A
AH(1)
A
AH(1)
SEE
!0.425
(!2.7)
0.202
(0.5)
0.281
(1.9)
0.050
(2.7)
!0.114
(!2.3)
!0.009
(!0.5)
!0.305
(!3.5)
!1.64
(!6.99)
0.084
(1.0)
0.026
(2.9)
0.043
(1.8)
!0.039
(!4.5)
0.285
(3.3)
0.166
(0.7)
0.457
(5.6)
0.663
!0.176
(!2.4)
!1.94
(!9.8)
0.059
(0.8)
0.030
(3.4)
0.042
(1.8)
!0.028
(!3.4)
0.254
(3.1)
0.298
(1.4)
0.562
(7.2)
!0.261
(3.0)
!1.64
(!7.1)
0.064
(0.8)
0.032
(3.3)
0.066
(2.5)
!0.014
(!1.5)
0.196
(2.0)
0.098
(0.4)
0.390
(4.2)
Moving average endpoints
0.018
!0.462
(0.5)
(!3.3)
0.013
0.111
(0.1)
(0.3)
!0.020
!0.000
(!0.6)
(!0.0)
Shifting endpoints
!0.002
!0.069
(!0.7)
(!2.2)
0.044
0.023
(0.5)
(0.3)
!0.036
!0.091
(!1.1)
(!3.1)
y
!0.276
(!1.6)
0.184
(0.4)
0.302
(1.9)
0.029
(1.7)
!0.112
(!2.4)
!0.006
(!0.4)
1.77
0.626
0.668
1.78
0.639
0.664
1.77
0.631
!The 3]1 vector, x, contains the one-month nominal bond rate, r, the PCE in#ation rate, n, and the manufacturing capacity utilization rate, y. Monthly in#ation and utilization rates are seasonally adjusted, and the sample
span is 1966m }1997m . Entries in parentheses are t-ratios. Endpoint constructions are discussed in the text,
1
9
where x(t) denotes the endpoint or long-horizon expectation of x as conditioned on information available in t.
=
historical in#ation, and a series of spliced survey measures of long-horizon
in#ation expectations.12 Whereas the survey data have been gradually trending
down, the VAR-based predictions are relatively #at and barely deviate from the
"xed endpoint. This feature suggests that predictions from the "xed endpoint
VAR may not be very good proxies for agent expectations.
12 In the 1980s, Richard Hoey, an economist at Drexel Burnham Lambert, conducted a survey of
market participants which asked for forecasts of in#ation in the second "ve years of a 10 yr forecast
horizon. This series was used prior to 1990m . The remainder of the spliced series is the long-run
7
expectations series from the Survey of Professional Forecasters (SPF), currently published by the
Federal Reserve Bank of Philadelphia.
160
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Fig. 1. Alternative in#ation endpoints.
3.2. The moving average endpoints VAR
The fundamental di!erence between the "xed endpoints VAR and the moving
average endpoints VAR is that the latter incorporates unit root restrictions
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
161
Fig. 2. VAR multiperiod predictions of 1-month in#ation rate.
on the 1-month interest rate and in#ation rate, r and n. Returning to the top
of Table 1, the format of the autoregressions in r, n, and y, is the same
as that required for an ADF test of unit root behavior. Indeed, the t-ratios
of the coe$cients of the lagged dependent variable indicate that a unit root
hypothesis cannot be rejected for both the short-term interest rate and
162
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Fig. 3. Concatenated VAR predictions of 10 yr in#ation rates.
in#ation rate. The results of estimating the three-variable VAR under the
condition that r and n are di!erence stationary is summarized in the middle
section of Table 2.13
13 Cointegration between r and n has not been imposed. A reasonable assumption might be that
after-tax real rates are stationary. This would imply that r and n are cointegrated with a cointegrating vector that re#ects tax rates. Since tax rates have changed over time, r and n would not be
cointegrated with a constant cointegrating vector. Time-varying cointegrating restrictions are
beyond the scope of this paper.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
163
In the present discussion, the interesting property of this speci"cation is that
the endpoints of r and n are now also I(1). In each period, the conditional
endpoint of a multiperiod in#ation forecast is a moving average of in#ation in
the months just prior to the forecast period.14 This moving average property is
displayed in the middle panel of Fig. 1, where it is seen that the in#ation
endpoint * which functions as the market view of the long-run policy target
* now moves very closely with recent in#ation history. The two multiperiod
forecasts of in#ation generated by the moving endpoints VAR are presented in
the middle panel of Fig. 2. Whereas the long-run in#ation forecast in 1972m is
1
now about 1% lower than the "xed endpoint in the top panel, the moving
average endpoint for in#ation forecasts originating in 1979m is an alarming
10
10%, owing to the large run-up in historical in#ation in preceding months.
An implication of the moving-average property of the endpoint is that
VAR-based 10 yr in#ation forecasts will follow recent in#ation history. The
middle panel of Fig. 3 illustrates this property. VAR-based predictions of 10 yr
in#ation rates follow actual in#ation quite closely. Unlike predictions based on
the "xed endpoints VAR, 10 yr in#ation predictions based on the moving
average endpoints VAR are quite variable. In fact, the long-horizon predictions
are more variable than survey data. Whereas survey data suggest that longhorizon in#ation expectations adjust slowly, with a signi"cant lag compared to
actual in#ation, predictions based on the moving average VAR adjust quite
rapidly.
3.3. A shifting endpoints VAR
The third VAR speci"cation admits endpoints that can di!er from either the
"xed endpoints implied by the mean-reverting speci"cation or the moving
average endpoints provided by the di!erence-stationary speci"cation. The format of an autoregression or VAR can be written to explicitly account for
deviations from endpoints, according to
(17)
*x "c#A(x !x(t~1))#AH(¸)*x #a ,
=
t~1
t
t
t~1
where AH(¸) denotes a polynomial in the lag operator, ¸ix "x . When
t
t~i
expressed in this format, it is clear that endpoints need not be constrained to be
"xed or moving averages.
In some instances, direct measurements of the implied endpoints or longhorizon forecasts of agents are available. For example, under the expectations
hypothesis, bond rates contain ex ante long-horizon forecasts of short rates.
14 As shown in Kozicki and Tinsley (1998), the moving average endpoint is precisely the
Beveridge}Nelson &permanent' component of unit-root stochastic processes. By construction, deviations from moving average endpoints of I(1) series are stationary.
164
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Thus, as suggested in Kozicki and Tinsley (1998), one measure of the short-term
interest rate endpoint is the average of expected short rates from t#n to t#n@,
for n@'n,
n@(r !h )!n(r !h )
n{
n,t
n ,
r( (t) " n{,t
=
n@!n
(18)
where h is an estimate of a constant term premium for an n-period bond. For
n
the current study, estimates of the shifting endpoints of the short-term interest
rate are based on the average of expected short rates between the 5 yr and 10 yr
maturities. One check of the reasonableness of this interest rate endpoint is that
deviations of the one-month bond rate from this endpoint should be stationary
if agents' bond rate forecasts in a given period are internally consistent. The
fourth equation listed in Table 1 indicates that the interest rate deviations from
the constructed series of nominal rate endpoints reject the hypothesis of unit
root behavior at a 99% signi"cance level.
It is more di$cult to obtain direct measurements of long-horizon endpoints
for agents' forecasts of in#ation. Although several surveys of expected 5}10 yr
in#ation emerged in the 1980s, contiguous survey estimates of expected longhorizon in#ation are not available for earlier decades. To generate backcasts of
agent expectations of long-run in#ation, bond traders are assumed to be &boundedly-rational' statistical agents, who sequentially test for intercept shifts in an
autoregressive model of in#ation. The statistical learning model is based on
a univariate autoregressive model of in#ation. The intuition behind the assumed
learning behavior is that agents infer an error in their current estimate of the
long-run policy goal of in#ation when they observe a string of prediction errors
for in#ation of the same sign, and adjust their outlook accordingly. These shifts
in the long-run perceptions induce nonstationarity in the mean of the in#ation
process.
Central bankers of developed economies are generally cautious and slow to
change either operational policies or strategic objectives. Thus, changes in
long-run policy objectives are likely episodic with frequencies that are more
appropriately measured in half-decades rather than months. This inference
seems to be consistent with the small number of policy regimes typically
identi"ed in postwar analyses of policy, such as Huizinga and Mishkin (1986).
A simple representation of episodic policy shifts in the in#ation process as
considered by agents is
(19)
*n "b #+ d(t!q )*b k #b n #b(¸)*n #a .
1 t~1
t~1
t
t
0
k
0,q
k
To represent possible shifts in the intercept, the dummy variable notation, d( . ),
denotes the Heaviside switching function where d(t!q )"0 if t(q . If
k
k
a change in the intercept is detected for period q , the function is switched &on' in
k
all subsequent periods, d(t!q )"1 if t5q .
k
k
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
165
In the absence of intercept shifts, the long-run in#ation rate de"ned by (19) is
n "!b /b . However, if intercept shifts are sequentially detected, the evolu=
0 1
tion of expected long-run in#ation is determined by the current estimate of the
intercept, n(t) "![b #+ k *b k ]/b . By not presuming ex ante knowledge
=
0
q :t 0,q
1
of the number of distinct policy regimes, the changepoint learning model allows
agents to be less informed about possible policy regimes than would be the case
for a shifting-regimes analysis.15
In the learning simulations referenced in this paper, agents test for intercept
shifts in a twelve-order autoregression in in#ation using critical values extrapolated from Andrews (1993). The search by simulated agents for signi"cant
changepoints is in &real time' as new observations are added in expanding
monthly samples over the period 1954m }1997m . To account for hetero1
9
geneous expectations among bond traders, agents are viewed as subscribing to
one of m market forecast &newsletters'. Before testing for a new in#ation endpoint, each newsletter accumulates a di!erent minimum number of monthly
observations since the date of the last estimated changepoint. The minimum
number of observations used by the ith newsletter is denoted s . Results disi
cussed here are based on "fteen newsletters, m"15. The minimum sample
collected since the last changepoint ranges from 12 months for newsletter 1 to 96
months for newsletter 15: s "6(1#i).
i
An important by-product of this simulated model of sequential, &real-time'
learning by agents is that it generates both the date of an estimated changepoint
and the date of the subsequent recognition of that changepoint, when su$cient
observations have been collected to satisfy the relevant critical value of the
changepoint test. The concatenation of shifted endpoints in the estimated
in#ation model is termed the &calendar time' endpoint series, and the concatenation of estimated endpoints when perceived by the learning agents is termed the
&real time' endpoint series.
A sampling of shifting in#ation endpoints generated by competing forecast
newsletters is shown in Fig. 4. The top panel indicates the in#ation endpoints
generated by the newsletter that collects a minimum of 30 months of observations after detecting a changepoint before testing for a subsequent shift. The
middle and bottom panels correspond to newsletters with minimum observation
collections of 60 and 90 months respectively. As indicated, newsletters with
a smaller number of minimum observations have a shorter lag in detecting
changepoints, as indicated by the shorter horizontal displacements between the
calendar time endpoints series (bold line) and the &real time' endpoints series
(dashed line). Newsletters using smaller minimum samples also can generate
15 Regime-shifting speci"cations, such as proposed by Hamilton (1989), typically presume ex ante
knowledge of the number of policy regimes, a constraint that is di$cult to verify empirically and to
implement in real time policy analysis.
166
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Fig. 4. Shifting in#ation endpoints used by competing forecast newsletters.
more volatile long-run forecasts with frequent adjustments of estimated endpoints. This di!erence depends on the underlying behavior of historical in#ation, as can seen by comparing the frequency and size of estimated changepoints
in the 1970s and early 1980s with the infrequent endpoint adjustments by all
newsletters after the mid-1980s.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
167
An aggregate in#ation endpoint series is constructed by weighted averages of
the di!erent newsletter estimates of long-run in#ation,
m
m
+ n "1,
(20)
n( (t) " + n n( (t) ,
i,t
=
i,t i,=
i/1
i/1
where n 50 is a nonnegative weight indicating the proportion of bond traders
i,t
subscribing to the ith newsletter in month t.
The relative contribution by each forecast newsletter, n , to the aggregate
i,t
in#ation endpoint is estimated by relating movements in the after-tax endpoint
of the nominal interest rate to the movements of the aggregate of newsletter
in#ation endpoints. That is, the long-run, after-tax nominal interest rate is
assumed equal to the sum of the long-run real rate and the long-run in#ation
rate,
(21)
(1!t )r(t) "o #n(t) ,
=
=
x,t =
where t is the tax rate on bond earnings and o is the long-horizon expectation
x
=
of the after-tax real rate. Notice that this speci"cation assumes after-tax real
rates are stationary. Eq. (21) has two important properties. First, unlike the
traditional Fisher equation, nominal rates will move more than one-for-one
with expected in#ation under positive tax rates.16 Second, shifting agent perceptions of the long-run policy goal for in#ation, as captured by n , are a likely
=
source of shifts in the perceived endpoint of the nominal rate, r .
=
The selection of forecast newsletters by agents is described by a stochastic
discrete choice model. The utility of the ith newsletter to its subscriber subpopulation in period t is represented by
u(i) "u #e , i"1,2, 15,
(22)
t
i,t
i,t
where u denotes that portion of utility that is related to observable variables.
i,t
This component is assumed to consist of a constant taste preference for the ith
newsletter, v , and a quadratic penalty on deviations from the expected conseni
sus forecast of long-run in#ation.17
u "v #v (n(i)(t) !n(t) )2, i"1,2, 15,
=
=
i,t
i
0
(23)
16 Crowder and Ho!man (1996) illustrate the historical relevance of the nonzero tax rate.
Estimated tax rates are drawn from McCulloch and Kwon (1993). McCulloch (1975) found that the
e!ective tax rate that best explains the prices of U.S. Treasury securities lies in the range, 0.22 to 0.30.
As noted by Green and Odegaard (1997), tax rates fell substantially after 1986.
17 Traders are assumed to have disparate priors about the frequency of shifts in the long-run
in#ation objectives of policy but place some weight also on the long-run views of other agents. For
more extensive discussion of the dynamic implications of discrete choice models with social
interactions, see Brock and Durlauf (1995) and Brock and Hommes (1996).
168
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
where n(i)(t) is the in#ation endpoint selected by the ith newsletter.18 To simplify
=
notation, time subscripts will be dropped in some subsequent equations if the
omission is not misleading.
The residual portion of utility associated with the ith forecast newsletter, e , is
i,t
assumed to follow an extreme value distribution with a zero mean and variance
proportional to c~2. As shown in Anderson et al. (1992), the assumptions of
discrete choice utility maximization and the extreme value distribution of
residual utility imply that the relevant choice probabilities can be characterized
by a multinomial logit model. Thus, the probability of selecting the ith forecast
newsletter is
A
B
n "Pr u(i)"max u( j) ,
i
j
"Pr(e !e 4u !u ,2, e !e 4u !u ),
1
i
i
1
n
i
i
n
i
ecu
.
(24)
"
+m ecuj
j/1
Estimates of the parameters (o , c, v , i"0,2, m) de"ned by the system of
=
i
equations (20)}(24) over the sample 1955m }1997m are shown in Table 3. The
7
9
estimate of the long-run after-tax real rate, o , is 2.21. The frequency distribu=
tion of agents over newsletters is smoothed by constraining the constant taste
preferences, v (i"1,2, m), to lie on a third-degree polynomial. Given that the
i
choice probabilities are invariant to a constant addition to utility, the constant
taste parameter associated with the newsletter using the largest minimum
observation sample is normalized to zero, v "0. Whereas the estimated
15
constant taste parameters are all positive, the estimate of the coe$cient of
squared deviations of newsletter endpoints from the consensus endpoint, v , is
0
signi"cantly negative, indicating forecasts nearer the consensus estimate tend to
attract more subscribers. Finally, the nonnegative precision parameter, c, is
associated with the normalization, (+m v2)1@2"1.19 The c parameter indicates
j/0 j
the concentration of agent preferences. That is, if cP#R, the newsletter
with the highest observable utility characteristics will attract all subscribers.
Conversely, if cP0, each newsletter will attract an equal market share of
subscribers, on average.
As indicated in Table 4, the sample mean estimates of the newsletter subscriber shares, g6 , indicate that more than 40% of the simulated agents are
i,t
18 To reduce computations in the estimation stage, the contraction estimate of the aggregate
in#ation endpoint in each month, n(t) , is proxied by the aggregate in#ation endpoint generated in the
=
preceding month, t!1.
19 This normalization is similar to that suggested in McFadden (1981).
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
169
Table 3
Discrete choice selection of forecast newsletters!
15
(1!t )r(t) "o # + n n( (t)
x,t =
=
i,t i,=
i/1
15
n "ecui, t + ecuj, t
i,t
j/1
u "v #v (n( (t) !n( (t~1))2
i,t
i
0 i,=
=
N
o
=
v }v
1 5
v }v
6 10
v }v
11 15
v
2.21
(27.1)
*
*
*
*
*
*
*
*
0.318
(1.9)
0.298
(2.4)
0.297
(2.7)
0.284
(2.9)
0.285
(2.9)
0.289
(2.7)
0.292
(2.5)
0.293
(2.3)
0.288
(2.1)
0.275
(1.9)
0.251
(1.8)
0.214
(1.6)
0.162
(1.5)
0.091
(1.5)
0.0
*
!0.101
(!8.2)
*
*
*
*
*
*
*
*
0
R2
0.86
*
*
*
*
*
*
*
*
*
!The endpoint of the after-tax nominal rate, (1!t )r(t) , is de"ned as the sum of the long-run real
x,t =
interest rate, o and of a weighted average of in#ation endpoints used by competing forecast
=
newsletters, n(t) . The weights are the probabilities of choosing a forecast newsletter and, as noted in
i,=
the text, the choice probabilities, n , are generated by a multinomial logit model. Estimates are
i,t
based on the monthly sample, 1955m }1997m . The estimated precision is: c"3.28. Estimates of
7
9
remaining parameters are listed in the table, along with t-ratios in parentheses.
clustered, on average, around newsletters 7}11, which use minimum observation
samples from 48 to 72 months. Thus, the implied aggregate &real time' lag in
detecting past shifts in in#ation endpoints appears to be relatively lengthy with
a minimum detection lag of around "ve years. However, the sizeable standard
deviations of the relative subscription shares of each newsletter suggest substantial variation over time in the aggregate detection lag. To illustrate this time
variation, Fig. 5 plots the aggregate minimum detection lag, de"ned as the
weighted average of minimum observation periods used by each newsletter,
+m n s . The estimated aggregate minimum detection lag ranges from lows of
i/1 i,t i
about three years (36 months) to highs of around six years (72 months). The
adjustment pattern in Fig. 5 suggests that a sizeable and sustained change in the
level of recent in#ation leads to an initial increase in the aggregate minimum
detection lag followed by a relatively rapid decrease. This reversing movement
in the aggregate detection lag is driven by the penalty on deviations of newsletter
in#ation endpoints from the consensus aggregate endpoint. That is, agents
initially cancel subscriptions to the newsletters with short detection lags whose
long-run forecasts appear to be overreacting to recent in#ation observations.
170
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
Table 4
Distribution of agents over in#ation forecast newsletters!
15
15
n( (t) " + n n( (t) , n 50, + n "1
=
i,t i,=
i,t
i,t
i/1
i/1
i
s"
i
n6 #
i,t
p(n )$
i,t
i
s"
i
n6
i,t
p(n )
i,t
1
2
3
4
5
6
7
8
12
18
24
30
36
42
48
54
0.058
0.068
0.064
0.058
0.077
0.072
0.075
0.084
0.052
0.044
0.044
0.044
0.044
0.030
0.040
0.032
9
10
11
12
13
14
15
*
60
66
72
78
84
90
96
*
0.081
0.085
0.085
0.060
0.073
0.044
0.034
*
0.033
0.038
0.039
0.034
0.032
0.022
0.030
*
!The aggregate in#ation endpoint, n(t) , is a weighted average of in#ation endpoints used by
=
simulated forecast newsletters, n(t) , where the weights are the estimated choice probabilities of
i,=
newsletters, n , over the 1955}1997 sample.
i,t
"Minimum number of monthly observations since last changepoint, s "6(1#i).
i
#Average share of agent subscriptions to the ith newsletter.
$Standard deviation of subscription shares.
Fig. 5. Average minimum detection lag of in#ation endpoint shifts.
However, if additional forecast newsletters also revise their endpoint estimates
in the same direction, the aggregate endpoint is eventually dominated by revised
endpoint estimates, and subscribers abruptly move away from the most conservative newsletters whose endpoints have not yet been revised.
S. Kozicki, P.A. Tinsley / Journal of Economic Dynamics & Control 25 (2001) 149}184
171
The last two autoregressions shown in Table 1 provide evidence against the
hypothesis that deviations of monthly in#ation from either the aggregated
calendar time endpoints series or the aggregated &real time' endpoints series are
I(1).20 Because the purpose of the VAR is to generate agent expectations that
approximate information available to markets when the historical data are
recorded, the assumption of symmetric access to policy target information
implicit in the "xed and moving average endpoints models is dropped, and the
&real time' in#ation endpoint is selected as a more realistic estimate of historical
market perceptions of the long-run policy target for in#ation. This selection is
supported in the bottom panel of Fig. 3, which shows that the predictions of
10 yr in#ation rates based on the shifting endpoints VAR resemble surveys of
expected long-run in#ation.
Estimated VAR coe$cients are provided in the bottom panel of Table 2.
Coe$cients in columns labeled &A' correspond to the sum of coe$cients on the
lagged levels of the relevant regressor variable. Interestingly, in most instances,
the sum of coe$cients in the shifting endpoints VAR is insigni"cantly di!erent
from the corresponding su