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BaumBasma Bekdache
Department of EconomicsDepartment of Economics
Boston CollegeWayne State University
Chestnut Hill MA 02167Detroit MI 48202
June 1996
Introduction
In this paper, we test the multivariate model of securities’ excess returns
formulated by
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Engle et al. (1990)
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on an expanded set of maturities. By applying their
methodology to the entire Treasury term structure, we consider the applicability of a
parsimonious common factor approach to the dynamics of short, medium, and
longterm interest rates.
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zero to 12 months’ tenor discount bills, which are readily available from CRSP as
the “Fama files.” In this study, we consider the entire Treasury term structure–for
bills, notes, and bonds–so that both money market and capital market returns may
be modeled. We make use of a set of monthly estimates of Treasury market spot
yields constructed from coupon securities’ quotations by
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Coleman et al. (1993,
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CFI).
Our work is based on the spot yields for the 14 specific tenors analyzed by CFI
for their sample period of 1955 through 1992,1 transforming them into estimated
onemonth holding period returns.2 We model excess return series, created by
subtracting the annualized holding period return on a onemonth Treasury from
the holding period return for each longer tenor.
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term structure suggest that a
workable term structure model should explicitly consider time variation in the
second moments of residual series as well as capture the interaction among tenors.
The following section presents such a model in which we have implemented time
variation in the second moments, as well as asymmetry in the modelled conditional
variances, using the approach of
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Gourieroux and Monfort (1992).
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2. Estimates of FactorGARCH models for the Treasury term structure
Term structure modelling has followed two broad strands of development:
general equilibrium models, such as those pioneered by Cox, Ingersoll and Ross
(1985), and noarbitrage partial equilibrium models.
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Estimates of FactorGARCH models for the Treasury term structure
Term structure modelling has followed two broad strands of development:
general equilibrium models, such as those pioneered by
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Cox, Ingersoll and Ross (1985), and
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noarbitrage partial equilibrium models. In this paper, we consider a
model of the latter genre, developed by Engle et al. (1990), and extend it to the
consideration of the complete Treasury term structure rather than just its short end.
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Estimates of FactorGARCH models for the Treasury term structure
Term structure modelling has followed two broad strands of development:
general equilibrium models, such as those pioneered by Cox, Ingersoll and Ross (1985), and noarbitrage partial equilibrium models. In this paper, we consider a
model of the latter genre, developed by
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Engle et al. (1990), and
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extend it to the
consideration of the complete Treasury term structure rather than just its short end.
In this framework, we consider whether timevarying volatility in asset returns is a
meaningful determinant of excess returns in the medium and longterm sectors of
the Treasury market.
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The term structure literature contains scattered evidence that
conclusions drawn from Treasury bill data do not readily extend to the medium and
long term sectors of the Treasury market. For instance,
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Engsted and Tangaard (1994)
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extend the work of Hall, Anderson and Granger (1992) and study the cointegration
properties of the term structure of interest rates, using 2, 5, and 10year yields from
McCulloch and Kwon’s (1993) data.
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The term structure literature contains scattered evidence that
conclusions drawn from Treasury bill data do not readily extend to the medium and
long term sectors of the Treasury market. For instance, Engsted and Tangaard (1994) extend the work of
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Hall, Anderson and Granger (1992) and
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study the cointegration
properties of the term structure of interest rates, using 2, 5, and 10year yields from
McCulloch and Kwon’s (1993) data. They find that the breakdown in the
cointegrating relationship between short rates that occurs during the 19791982
Federal Reserve operating policy shift does not appear in the longer maturity term
structure.
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This suggests that risk premia behave differently over the maturity
structure, and that the explanation that term premia become nonstationary with a
regime shift may not hold true for longer maturities.
Similar evidence is found in a study by
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Froot (1989),
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where survey data on
interest rate expectations are used to study the relative importance of timevarying
term premia and expectational errors in explaining rejections of the pure
3 The moving correlations are computed annually from 36 monthly observations; the date on the
horizontal axis is the left endpoint of the threeyear spa
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They conclude that there are “nontrivial differences in the risk characteristics in
agents investing at different maturities” (p. 64) and suggest that a segmented
markets approach may be warranted.
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Engle et al. (1990,
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henceforth ENR) argue that a multivariate approach to the
modelling of asset returns is clearly justified, since even in static asset pricing
models, the full covariance matrix of asset returns is required to derive estimates of
a single asset’s risk premium.
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weighted bond portfolio
and a pure stock portfolio–as sufficient, and apply a recursive representation where
4the stockmarket portfolio’s excess returns are generated in a univariate model, but
the bond portfolio’s excess returns depend as well on the stockmarket portfolio’s
behavior.
2.1 Excess returns for Treasury and stockmarket index portfolios
We use the
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Coleman et al. (1993,
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henceforth CFI) yields to construct estimates
of onemonth holding period returns. Comparable returns on a diversified stock
portfolio are derived from the CRSP valueweighted index for the NYSE/AMEX,
which is first available in July 1962.
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Although
the number of significant eigenvalues does not specifically indicate the number of
dynamic factors appropriate for our model, it would appear that a twofactor model
might be able to capture the behavior of the excess returns series.
2.2 Asymmetric GARCH models of portfolio excess returns
Following
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Engle et al. (1990,
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p.223), we do not attempt to determine portfolio
weights within the model, but rather specify weights for two factorrepresenting
portfolios: an equallyweighted bond portfolio and a portfolio containing only the
stockmarket index.
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As noted above, the set of tenors for Treasuries in our data imply
that the equallyweighted bond portfolio will have risk characteristics approximately
equal to a 5.5year tenor security. In fitting a GARCH model to the portfolio excess
returns series, we considered various asymmetric forms of the basic GARCH model.
Other researchers (cf.
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Gourieroux and Monfort (1992))
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have found support for
asymmetries in either the mean equation or the conditional variance equation of
the GARCH formulation. In the context of the factorrepresenting portfolios’ excess
returns, we might expect the conditional variance to respond differently to increases
and decreases in risk.
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In a first
application, we model the security excess returns as a linear function of the two
portfolios’ excess return series, with a conditional variance given by a linear
function of the two portfolios’ conditional variances. This model (analogous to
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Engle et al., (1990,
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p.226)) may be written as:
Ri,t=iλ+iBondβtBondˆR+iStockβtStockˆR+i,tυ
υi,tt−1ℑ~N0,i,th()
hi,t=iσ+iBond
2
β
Bondt
θˆ+
Stocki
2
β
Stockt
θˆ
(4)
This model incorporates a constant, λ, in the mean equation, to capture nontimevarying components of the individual security’s risk premium (for instance,
the “ontherun” effect that newly auctioned Treasury securities exhibit, in which
their
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While the β coefficients are highly significant for
all tenors, examination of their asymptotic standard errors indicate that the direct
effect of the stockmarket portfolio’s excess returns and conditional variance is of
considerable importance for shortterm Treasury returns, but never meaningful for
tenors greater than two years. In contrast to
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Engle et al. (1990,
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p.226), we found that
the stockmarket index has significant direct effects on the 312 month segment of the
yield curve.
Since the stockmarket factor does not appear to play a direct role in the
individual securities’ excess returns and conditional variances for medium or long
tenors, we respecify the model as a singlefactor model in which the estimated excess
return and con
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We are investigating the feasibility of this approach, which is computationally
burdensome.
10the model should be contrasted with the general equilibrium models of the term
structure, such as that of
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Cox, Ingersoll and Ross (1985),
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which generate perfect
correlations among rates at all tenors. The correlations of these predicted values are
quite consistent with the original excess return series to which they are fit; for
instance, the correlations between 3month securities’ excess returns and those of 1,
2, 5 and 20year securities are 0.768, 0.685, 0.550, and 0.381 over the full sample,
wh
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