Download Discrete-time affine term structure

Discrete-time affine term structure:
an econometric formulation
Dean Fantazzini∗
Mario A. Maggi†
Abstract
Discrete-time Affine Term Structure Models can be expressed in
AR (1)-ARCH form, but it is not possible to get a non-negative variance equations simply by restricting the parameters. In this paper
we resort to a distribution assumption in order to assure the variance
to be non-negative. We present a complete formulation for one-factor
and multi-factor models with Gamma conditional noise distribution.
This way we get a well defined volatility process that avoids any problem both in generating processes and in computing the conditional
likelihoods of observations. The log-likelihood function is derived
for one- and multi-factor specifications. Moreover, we implement a
one-factor estimation both with simulated and US interest rate data.
Finally, we compare the estimation results with a standard ATSM
with Gaussian disturbances.
Keywords: Discrete-time Affine Term Structure Models, ARCH, VAR,
Maximum Likelihood Estimation
JEL Classification: C32, C51, E43.
∗
PhD student in Economics, Dipartimento di Economia Politica e Metodi
Quantitativi, Università di Pavia, [email protected]
†
Researcher in Mathematical Finance, Dipartimento di Ricerche Aziendali, Università di Pavia, [email protected]
1
1
Introduction
Affine term structure models (ATSM) constitute a wide family
of term structure models. ATSM are characterized by a defaultfree zero coupon bond (ZCB) evaluation function P which is
exponentially linear:
Pt (τ ) = exp {A (τ ) − B (τ ) Xt }
(1)
being: t the current valuation date, τ the time to maturity, A =
A (τ ) a real function, B = B (τ ) ∈ IRn a vector function and
Xt ∈ IRn a vector collecting the n stochastic risk factors. As
Dai and Singleton (2000) clearly show, ATSM constitute a wide
class of models encompassing the majority of the most popular
and standard models (for instance, the Vasiček (1977) and the
Cox, Ingersoll and Ross (1985) models are special cases).
In many works the model chosen is a continuous-time one,
e.g. see Dai and Singleton (2000), Duarte (2004), Duffee (2002),
Duffie and Kan (1996). A strong contribution to the success
of continuous-time ATSM comes from the tractability of their
pricing formulas and their dynamics, together with the actual
possibility to use them empirically. However, multi-factor continuous-time models are quite demanding in terms of estimation
procedures. In fact, out of some special cases, ATSM do not admit general closed forms for the likelihood function and for the
distribution function moments of bond prices, yields and factors. This leads to use estimation techniques such as the quasimaximum likelihood, the efficient and the simulated methods of
moments, for example see Berardi (1995), Chen and Scott (1993),
Dai and Singleton (2000), Duffee and Stanton (2000), Duffie and
Singleton (1997), Fisher and Gilles, (1996), Liu (1997), Liu and
Pan (1997), and Singleton (2001).
Discrete-time models are used as well. For instance, Backus
2
et al. (1998) show some simple discrete-time one- and multifactor affine models estimations. Ang and Piazzesi (2003) use
a discrete-time VAR model for the term structure. Rudenbusch
and Wu (2004) deal with a VAR(1) models that can easily be expressed as a Gaussian ATSM. Ichiue (2003) employs a Gaussian
VAR ATSM. In general, it seems that discrete-time models are
well suited for term structure models that uses macroeconomic
variable as factors. In fact, even if market data can be sampled at different frequencies from low to very high (intraday)
frequency, macroeconomic variable are typically monthly data.
For this reason, a discrete-time setting may be convenient for
macro-finance term structure models. Moreover, their estimation is quite affordable.
As Backus et al. (1998) remark, discrete time version of
ATSM processes can take negative value with a positive probability. In non-Gaussian affine processes, the factors linearly enter
the volatility equation making possible that the model breaks
down when a negative volatility value appears. The probability of this event can be kept small enough, but it still remains
positive. This fact can produce emergency stops in simulating
algorithms or in likelihood computation. In this paper we propose a way out of this stalemate. We model the disturbances
of the volatility factor with non-central Gamma distributions
with a suitable conditional lower bound, that imply the volatility processes to be non-negative with probability one. We begin
from one-factor models where it is simple to set up our model.
We estimate a one-factor model with US interest rates data,
finding that it behaves well, but it does not really improves
over a standard one-factor ATSM. Indeed, our proposal is useful
for multi-factor models, where it allows to exploit much more
flexibility in describing the volatility and the correlations than
Gaussian VAR models or independent factors affine models. In
3
fact, it is possible to adapt to discrete-time the Dai and Singleton (2000) specification that splits the factor vector into volatility factors and other factors.
The paper proceeds as follows. Section 2 presents the discretetime formulation of an ATSM in a general AR(1)-ARCH form.
In section 3 we introduce the Gamma disturbances hypothesis for
one-factor models. These simple models allows to better understand the implications of our assumption. The likelihood function is derived and used in estimation in section 3.1. Section 3.2
applies to US interest rate data the procedures pointed out in
section 3.1. Section 4 deals with multi-factor models combining
the results obtained for one-factor models with the Dai and Singleton (2000) general affine formulation. The likelihood function
is derived for multi-factor processes too. Section 5 concludes.
2
Discrete-time ATSM
The dynamics of prices and interest rates in financial markets
are driven by micro- and macro-economic variables, such as the
expectations, risk attitude and feelings of economic agents, decisions of political economy. We consider the general case of an
economy driven by n random factors or state variables stacked in
a column-vector Xt ∈ IRn , with t ∈ {0, 1, . . . , T }. We assume the
absence of arbitrage and frictions (no transaction costs, bid-ask
spread or any type of frictions are in place) on the fixed income
market.
An n-factor discrete-time ATSM is described by the relations
Xt = Xt−1 + K (Θ − Xt−1 ) + εt , t = 1, 2, . . . , T,
εt ∼ f [0] , Σ2t , white noise,
Σ2t = diag (αi + βi Xt−1 ) , i = 1, . . . , n,
4
(2)
(3)
(4)
with Xt , Θ, εt ∈ IRn , βi ∈ IRn , K, Σ2t ∈ IRn×n , αi ∈ IR and [0]
the null vector. We denote Ωt the information set available at
time t, i.e. Ωt = {Xz , εz , z = 0, 1, . . . , t}.
The relevance of ATSM both in discrete and in continuous time stems from the computational tractability of the zero
coupon pricing function. In fact, the zero coupon bond (ZCB)
prices are given by the formula
Pt (τ ) = exp (Aτ + Bτ Xt ) ,
(5)
where τ = τ1 , τ2 , . . . , τm is the time to maturity. The yield-tomaturity, or simply the spot rate, between t and (t + τ ) is
rtτ = −
Aτ + Bτ Xt
.
τ
The rate with shortest maturity is the short rate rt = rtτ1 . It
can be shown that formula (5) can be recursively derived from
a simple difference system (e.g., see. Rudebusch and Wu (2004)
for the details).
In this paper we deal with the dynamics (2) of risk factors
under the historical probability measure. In the following section
we present some result for one-factor models and then we extend
them to multi-factor discrete-time ATSM.
3
One-factor ATSM
Owing to the linear relationship among different maturities rates
(e.g., see Duffie and Kan (1996)), in a one-factor ATSM the factor can always be identified with the short rate, so relations (2)–
5
(4) become
rt = rt−1 + k (θ − rt−1 ) + εt ,
εt ∼ f 0, σt2 , white noise,
σt2 = α + βrt−1 .
t = 1, 2, . . . , T
(6)
(7)
(8)
that does not look like a GARCH model. Equation (6) defines
an AR (1) process and can be written in a more common way:
rt = kθ + (1 − k) rt−1 + εt ,
t = 1, 2, . . . , T.
The volatility process depends on lagged interest rates. Provided
that 0 < k < 1, the MA (∞) form of {rt }
rt = θ +
+∞
X
j=0
(1 − k)j εt−j ;
moreover, is convenient to write the volatility process {σt2 } as a
MA (∞):
"
#
+∞
X
(1 − k)j εt−j−1 ,
σt2 = α + β θ +
j=0
with a null weight on the current disturbance. Writing the first
term out of the summation, we get
σt2
= α + βθ + βεt−1 + β (1 − k)
+∞
X
j=0
(1 − k)j εt−j−2 .
Also the lagged volatility has a MA (∞) representation
"
#
+∞
X
j
2
σt−1
=α+β θ+
(1 − k) εt−j−2 .
j=0
6
(9)
Therefore, relation (9) can be written as
β
+∞
X
j=0
2
(1 − k)j εt−j−2 = σt−1
− α − βθ,
so it follows that σt2 has the following AR (1) representation
2
σt2 = k (α + βθ) + (1 − k) σt−1
+ βεt−1 .
(10)
This representation is useful in order to analyze the condition
for the non-negativity of the conditional variance of noise. Although equation (10) is quite similar to a GARCH (1, 1) volatility process, the presence of lagged noise, which is not squared,
makes more difficult to find non-negativity condition. The idea
we present is simple. If the noise process is Gaussian with β 6= 0,
equation (10) has no chance to produce non-negative variances
with probability one (see also Backus et al. (1998)). So, there exists a noise conditional distribution assuring that the conditional
variance is always non-negative? A simple answer is to choose
a distribution for εt−1 |Ωt−2 whose support is suitably bounded
from below. Equation (10) suggests that this bound can depend
on Ωt−2 :
k (α + βθ) 1 − k 2
min {εt−1 |Ωt−2 } = a ≧ −
+
σt−1 ,
β
β
this can be written as
k (α + βθ) 1 − k
min {εt |Ωt−1 } = a ≧ −
+
var (εt |Ωt−1 ) ,
β
β
(11)
for t = 1, 2, . . . T,
7
that is, the lower bound linearly depends on the conditional variance. Moreover, the noise must have mean zero and variance σt2
E [εt |Ωt−1 ] = 0,
var (εt |Ωt−1 ) = σt2 .
(12)
(13)
There are various distribution with support bounded from
below, for example the χ2 , the Beta, the Gamma, the Exponential, the Weibull distributions. Let we suppose that the noise
conditional distribution is a non-central gamma with parameters λt > 0 and qt > 0. We choose this distribution because
it has enough parameters whose value can be chosen to match
conditions (11) to (13), and its moments have easy expressions
that do not introduce computational burdens for estimation.
Consider the random variable z ∼ G (λ, q), with λ > 0, q > 0.
Its density is
fz (z) =
λq q−1 −λz
z e , z ≧ 0,
Γ (q)
and its mean and variance are
E [z] =
q
q
, var (z) = 2 .
λ
λ
In order to get a zero mean distribution, we need that our
variable takes negative values too. However, we need a lower
bound a, so we displace the random variable z by the amount a:
x=z+a
λq
(x − a)q−1 e−λ(x−a) , x ≧ a,
Γ (q)
q
q
E [x] = + a, var (x) = 2 .
λ
λ
fx (x) =
8
We now come back to conditions (11), (12) and (13) which
become (remark that we consider condition (11) with the equality sign)
h
i

k(α+βθ)
1−k 2

a
=
−
+
σ
 t
t
β
β
qt
(14)
+ at = 0

 λqtt = σ 2
t
λ2
t
From system (14) it is possible to solve for the values of a, λ and
q:

h
i
k(α+βθ)
1−k 2

a
=
−
+
σ

t
β
β
 t
at
λt = − σ 2
t


 q = a2t2
t
σt
that is, the parameters of the noise distribution at time t are
known functions of the model parameters and of σt2 ∈ Ωt .
Summing up, we can formulate the one-factor discrete-time
ATSM model as an “affine Γ-ARCH (1, 1)” model:

rt = kθ + (1 − k) rt−1 + εt



2

var (ε

t ) = σt


2
 2
k (α + βθ) + (1 − k) σt−1
+ βεt−1 , if β 6= 0,
σt =
α,
otherwise



εt − at ∼


h G (λt , qt )
i


 at = − k(α+βθ) + 1−k σ 2 , λt = − a2t , qt = a2t2 .
t
β
β
σt
σt
(15)
For example, a set of parameter values that can mimic a short
rate dynamics is
k = 0.025, θ = 5, α = 0.002, β = 0.1.
(16)
A (little) flaw is that the probability of having a negative rate
is positive. Some MC simulations with parameter values (16)
9
have shown that this probability is extremely low: with Gaussian
disturbances P [rt < 0] ≃ 1.76 × 10−4 , whereas with mean zero
Gamma disturbances P [rt < 0] ≃ 0, 1 . However, this problem
can be mended by means of the same idea exploited before:
choose at suitably as follows
k (α + βθ) 1 − k 2
at = − min
+
σt , rt−1
(17)
β
β
and the short rate will be non-negative with probability one.
Equation (17) will be useful in section 4 in order to ensure nonnegative volatility equation for some factor.
Given a set of parameter values Ξ = (k, θ, α, β), the conditional distribution of an observation is
[(rt |rt−1 , Ξ) − kθ − (1 − k) rt−1 − at ] ∼ G (λt , qt ) ,
so its conditional likelihood is
f (rt |rt−1 , Ξ) =
λq t
[rt − kθ − (1 − k) rt−1 − at ]qt −1
Γ (qt )
· e−λt [rt −kθ−(1−k)rt−1 −at ] .
The sample likelihood and log-likelihood are, respectively,
T
Y
λq t
[rt − kθ − (1 − k) rt−1 − at ]qt −1
L (r̄, Ξ) = f (r0 )
Γ (qt )
t=1
· e−λt [rt −kθ−(1−k)rt−1 −at ] ,
ln L (r̄, Ξ) = ln f (r0 ) +
T
X
t=1
qt ln λt − ln Γ (qt ) + (qt − 1)
·ln [rt − kθ − (1 − k) rt−1 − at ]−λt [rt − kθ − (1 − k) rt−1 − at ] ,
1
In different 106 time-steps simulated processes a negative rt never occurred.
10
model
ATSM
k̂
θ̂
0.0291181 4.8496
(0.0000)
Γ-ARCH
true model: Γ-ARCH 0.025
β̂
0.0255651 0.09082
(0.0000) (0.0654)
0.0290948 4.8497
(0.0000)
α̂
(0.0000)
0.0247342 0.09130
(0.0000) (0.0469)
(0.0000)
5
0.1
0.002
Table 1: Maximum likelihood parameters’ estimates, (p-values).
Sample: T = 2000 time-steps generation of a Γ−GARCH
porcess.
where [rt − kθ − (1 − k) rt−1 ] is the conditional innovation
(εt |Ωt−1 , Ξ). In order to calculate it recursively we need the
initial value of the short rate r0 and its volatility σ02 .2
3.1
A test estimation
We perform some preliminary test of the ML estimation. We estimate the parameters from some MC simulation of the Γ-ARCH
process. We compare the results obtained from two different
specifications: the Γ-ARCH and the ATSM ones. Tables 1, and 2
display the results.
2
In our algorithms we equal r0 to the first data and σ02 to the variance
unconditional mean given the parameter set.
11
corresponding parameters
GARCH(1, 1) parameters
in Γ-ARCH
ĉ = 0.11756
(0.028222)
kθ = 0.125
φ̂ = 0.96885
(0.0053405)
(1 − k) = 0.975
ω̂ = 0.011499
ϕ̂ = 0.89524
(0.0033932) k (α + βθ) = 0.1255
(0.01579)
ψ̂ = 0.081588
(0.012067)
(1 − k) = 0.975
β = 0.1
log lik. = −1978.1
Table 2: GARCH (1, 1) estimation (standard errors). Sample:
T = 2000 time-steps generation of a Γ-GARCH porcess.
We estimate also a standard AR (1)-GARCH (1, 1) model
rt = c + φrt−1 + ǫt ,
ǫt ∼ N (0, σt2 ) ,
var (ǫt |Ωt−1 ) = σt2 ,
(18)
2
σt2 = ω + ϕσt−1
+ ψǫ2t−1 .
In table 2 the ML estimate are compared to parameters values
(remark that in this model the innovation is squared, so its coefficient ψ can not match β).
12
model
ATSM
Γ-ARCH
k̂
θ̂
α̂
β̂
0.0039782
5.6421
−0.038377
0.042987
(0.3699)
(0.2522)
(0.0000)
(0.0000)
0.0044149
5.6459
−0.039794
0.044950
(0.3406)
(0.2226)
(0.0000)
(0.0000)
Table 3: Maximum likelihood parameters’ estimates, (p-values).
Sample: monthly US three months T-bill rates, from 1962:1 to
2005:2.
3.2
An application to US interest rates
The maximum likelihood algorithm previously tuned up with
the artificial data presented in section 3.1 is now applied to actual interest rate data. The data-set contains monthly sampled 3-months T-bill rates from 1962:1 to 2005:2. Table 3 reports the maximum likelihood estimate of a standard ATSM
with Gaussian innovation and our Γ-ARCH specification. The
p-values of the AR(1) coefficient are quite high, so we performed
the likelihood ratio tests on each parameter for the two specifications. All tests reject the null of zero parameter values3 .
The results obtained show that the Γ-ARCH specification
works slightly better than a standard ATSM. However, the improvement is so weak that can not justify the use of a Γ-ARCH.
Anyway, this application allows us to get confidence on our model
so we can build up the multi-factor specification we will present
next.
3
The results are not reported but they are available from the authors’.
13
4
Multi-factor discrete-time ATSM
It is well documented that term structure dynamics might be
modelled accounting for more than one factor (e.g., see Litterman and Scheinkman (1991)). Principal component analysis show that two or three factors are enough to describe the
term structure dynamics. For this reason, in this section we
turn to multi-factor models. Our assumption on the innovation
distribution allows us to fully exploit in a discrete-time setting
the potential flexibility of different specification of multi-factor
ATSM.
We come back to the general multi-factor ATSM presented
in section 1. The Dai and Singleton (2000) and Rudebusch and
Wu (2004) parameters structure can help getting a tractable
specification in discrete-time too. The n factors of the model (2)(4) should be divided into two groups: the first m ≦ n that
we call volatility factors XtB ∈ IRm , and the others n − m
factors XtD ∈ IRn−m . The parameters are partitioned accordingly so that (2)-(4) can be written in the following form: for
t = 1, 2, . . . , T,
 



BB
B
B
[0]   Θ 
 Xt   K
=
+


ΘD
K DB K DD
XtD


 
 

BB
B
B
[0]   Xt−1   εt 
 K

+ I − 
 
+
 εt ,
DB
DD
D
D
K
K
Xt−1
εt
εt ∼ g [0] , Σ2t , white noise,
Σ2t = diag (αi + βi Xt−1 ) , i = 1, . . . , n,
14
with



K BB diagonal,






 βi = βii 1i , βii ∈ IR, 1 ≦ i ≦ m,


βi = βi1 · · · βim [0] ≧ [0] ,






 α ≧ [0] , m + 1 ≦ i ≦ n,
i
(19)
m + 1 ≦ i ≦ n,
where 1i is the i-th basis (row) vector. This way, the factor
dynamics are



Xit = Kii Θi + (1 − Kii ) Xit−1 + εit



1≦i≦m:
εit ∼ fiB (0, σit2 )




 σ2 = α + β X
i
ii it−1
it



Xit = Ki Θ + (1i − Ki ) Xt−1 + εit




m+1≦i≦n:
εit ∼ fiD (0, σit2 )




2
B

 σit = αi + βi1 · · · βim Xit−1
where Ki stands for the i-th row of K. Thus, the volatility factors are m independent univariate processes like (6)-(8), whereas
the other factors are correlated among them and with the volatility factors, moreover, the volatility of the non-volatility factors
linearly depends on the volatility factors. If m = 0, i.e. the
model is a Gaussian VAR(1), then K has to be lower triangular
and the volatility functions become constant.
For volatility factors, we recall the univariate non-negativity
argument, so we assume that εit |Ωt−1 has a non central Gamma
15
distribution of the kind (15) with



Xit = Kii Θi + (1 − Kii ) Xit−1 + εit







var (εit ) = σit2






 σ 2 = Kii (αi + βii Θi ) + (1 − Kii ) σ 2
it
it−1


εit − ait ∼ G (λit , qit )




nh


Kii (αi +βii Θi )

a
=
−
min
+

it

βii




 λit = − ait2 , qit = a2it2 .
σ
σ
it
(1−Kii ) 2
σit
βii
+ βii εit−1
i
o
, Xit−1 ,
it
In a multi-factor model, the non-negativity condition (17) for
the volatility factors XtB become necessary in order to guarantee
that restrictions (19) produce volatilities that are well defined
positive processes for the non-volatility factor XtD as well. No
restriction on the distribution of innovations of non-volatility factors innovations are needed. In the rest of the paper we assume
that εit |Ωt−1 is Gaussian for m + 1 ≦ i ≦ n.
It is straightforward to compute the likelihood function of
the multi-factor discrete-time ATSM presented in this paper. In
fact the conditional innovation
(εt |Ωt−1 ) = Xt − KΘ − (I − K) Xt−1 ,
has independent components whose distribution is Gamma for
the first m components, and Gaussian for the others n − m. The
likelihood of an observation, given the set of parameter values
Ξ, is
n
Y
L (Xt |Ξ) =
Li (εit |Ωt−1 , Ξ) ,
i=1
16
where
λqit
[Xit − Kii Θi − (1 − Kii ) Xit−1 − ait ]qit −1 ·
Γ (qit )
−λit [Xit −Kii Θi −(1−Kii )Xit−1 −ait ]
·e
, for 1 ≦ i ≦ m,
Li (εit |Ωt−1 , Ξ) =
and
Li (εit |Ωt−1 , Ξ) =
σit
1
√
(Xit − Ki Θ − (1i − Ki ) Xt−1 )
exp −
2σit2
2π
2
!
,
for m + 1 ≦ i ≦ n.
Therefore the sample log-likelihood is
T X
n
X
ln L X̄|Ξ = ln f (X0 ) +
Li (εit |Ωt−1 , Ξ) .
t=1 i=1
Obviously, in order to recursively calculate the initial values X0
and σ02 are needed.
A common issue in multi-factor models is the growing number of parameters to estimate. So it is worth to remark that the
number of free parameters in the multi-factor specification we
have just presented is:
4m
(n − m)
(n − m)
n2 − m2 + 2 (n + m)
volatility factors (like m one-factor models)


 n Ki




for each non1 αi

volatility factor: 



 m
βi1 · · · βim
ΘD
total parameters
17
In most fixed income analysis (see also Litterman and Sheinkman (1991)) the number of risk factors does not exceed 4. Therefore, for discrete-time ATSM up to 4 factors the number of parameters is:
n\m
0
1
2
3
4
1
3
4
−
−
−
2
8
9
8
−
−
3
15 16 12 12
−
4
24 25 24 21 16
This means that the models we present are affordable both from
the computational and from the identification point of view. In
fact, the estimation time can be kept reasonable.
5
Conclusions
The model we have presented in this paper is intended to be
an operative model which is developed paying attention to its
econometric implementation. We have extended to discrete-time
the idea of splitting the state vector in order to differently deal
with volatility and non-volatility factors. This idea was successfully introduced in continuous-time models by Dai and Singleton (2000) and has been extensively applied for financial market
study and asset pricing. We think, however, that discrete-time
models may be a better choice for analyzing the low-frequency
data typical in macro-finance literature, where the macroeconomic variables can not be sampled more than once a month.
Simulations and empirical analysis with US interest rate data
18
show that our affine Γ-ARCH model can be a valuable tool
for financial applications, since the volatility processes are nonnegative with probability one. Therefore, the main benefits are
to avoid any emergency stops in simulating algorithms or in likelihood function computations. Moreover, our model can also
preserve the flexibility allowed by multi-factor ATSM.
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