We propose a simple, flexible approach to nonparametric estimation and specification testing for a two-factor interest rate model. These methods are illustrated with a Monte Carlo experiment and an empirical example.Nonparametric local linear estimation, Two-factor term structure models, Model specification tests

Brennan S. Thompson

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 3 
  
Nonparametric estimation and specification testing of a two-factor interest rate 
model 
  
 
 
Brennan S. Thompson  
Ryerson University 
Abstract 
We propose a simple, flexible approach to nonparametric estimation and specification testing for a two-factor interest 
rate model. These methods are illustrated with a Monte Carlo experiment and an empirical example.
The author wishes to Thanasis Stengos, Alex Maynard, and Yiguo Sun for their helpful comments. 
Citation: Brennan S. Thompson, (2009) ''Nonparametric estimation and specification testing of a two-factor interest rate model'', Economics 
Bulletin, Vol. 29 no.3 pp. 2343-2349. 
Submitted: Mar 05 2009.   Published: September 16, 2009. 
 
     1. Introduction
Letting rt be the short rate, lt be the consol rate, and st = rt−lt be the spread, the two-factor continuous-time
model of Schaefer and Schwartz (1984) is
dst = µ1(st,lt)dt + σ1(st,lt)dW1,t (1)
dlt = µ2(st,lt)dt + σ2(st,lt)dW2,t, (2)
where µk(·) and σk(·), for k = 1,2, are unknown drift and diﬀusion functions, respectively, and W1,t and
W2,t are independent standard Wiener processes.
Knight et al. (2006) have proposed a semiparametric estimation method for this model, which requires
placing linear identifying restrictions on µ1(·). In this note, we propose a ﬂexible, fully nonparametric
estimation method, as well as a procedure for testing parametric speciﬁcations of this model. Aside from
not placing any restrictive assumptions on the model, our method has the advantage of utilizing local linear
(LL) regression, which avoids the well-known boundary bias of local constant (Nadaraya-Watson) regression
(Chapman and Pearson, 2000; Fan and Zhang, 2003).
2. Estimation
Suppose we have observations {(s∆,l∆),(s2∆,l2∆),...,(sn∆,ln∆)} at dates {t1 = ∆,t2 = 2∆,...,tn = n∆}
on the time interval [0,T], where ∆ = T/n is the discretization step. Let Y1,i = (s(i+1)∆ − si∆), Y2,i =
(l(i+1)∆ −li∆), X1,i = si∆, and X2,i = li∆, for i = 1,...,(n−1). Using a Euler-type approximation, we can
rewrite system (1) – (2) as
Yk,i ≈ µk(X1,i,X2,i)∆ + σk(X1,i,X2,i)k,i
√
∆, k = 1,2,
with E(k,i|X1,i,X2,i) = 0, Var(k,i|X1,i,X2,i) = 1, and E(1,i2,i|X1,i,X2,i) = 0.
Since the errors in these equations are uncorrelated, they can eﬃciently be estimated separately. Here,
we propose doing so by applying the method of Fan and Yao (1998) to each equation.1 This method is
explained as follows. Suppose the drift function, µk(·), is known and let Rk = [Yk/∆−µk(X1,X2)]2∆. Since
E(Rk|X1 = x1,X2 = x2) = σ2
k(x1,x2), an ideal estimate of the squared diﬀusion function can be found by
regressing Rk,i = [Yk,i/∆ − µk(X1,i,X2,i)]2∆ on X1,i and X2,i, using LL estimation (see below).
In practice, the drift function is unknown, but we can use the following two-stage procedure. In the ﬁrst
stage, we repalce µk(X1,i,X2,i) in Rk,i with an LL estimate, ˆ µk(X1,i,X2,i), which is obtained as the value
of a0 solving
min
a0,a1,a2
n−1 X
j=1
[Yk,j/∆ − a0 − a1(X1,j − X1,i) − a2(X2,j − X2,i)]2K

X1,j − X1,i
h1

K

X2,j − X2,i
h2

,
where K(·) is a nonnegative bounded kernel function with
R
K(v)dv = 1, and h1 and h2 are bandwidths,
with h1 → 0, h2 → 0, and nh1h2 → ∞, as n → ∞. Popular methods of bandwidth selection include plug-in
methods and cross-validation; see Fan and Gijbels (1996).
In the second stage, the residual-based estimator of the squared diﬀusion function is found by regressing
ˆ Rk,i = [Yk,i/∆ − ˆ µk(X1,i,X2,i)]2∆ on X1,i and X2,i using LL estimation. That is, at arbitrary evaluation
points x1 and x2, the estimate ˆ σ2
k(x1,x2) is found as the value of b0 which solves2
min
b0,b1,b2
n−1 X
i=1
[ ˆ Rk,i − b0 − b1(X1,i − x1) − b2(X2,i − x2)]2K

X1,i − x1
g1

K

X2,i − x2
g2

,
where, for k = 1,2, gk is deﬁned analogously to hk.
Fan and Yao (1998) refer to this estimator as adaptive, as it will estimate the squared diﬀusion function
asymptotically as well as the ideal estimator. We compare the ﬁnite-sample performance of these estimators
in Section 4.
1Fan and Yao (1998) consider only the case where there is a single regressor. Thus, the LL estimator here will have a slower
rate of convergence. However, it is still consistent and asymptotically eﬃcient (Ruppert and Wand, 1994; Fan and Gijbels,
1996).
2Note that the ideal estimator discussed above is identical to the residual-based estimator, except that Rk,i is used in place
of ˆ Rk,i.
13. Speciﬁcation testing
We now consider speciﬁcation tests of the general parametric model,
dst = m1(st,lt;θ1)dt + q1(st,lt;φ1)dW1,t
dlt = m2(st,lt;θ2)dt + q2(st,lt;φ2)dW2,t,
where, for k = 1,2, mk(·), and qk(·) are known functions (which may be linear or non-linear), and θk and
φk are unknown parameter vectors.
To test the null hypothesis of correct speciﬁcation, we use the test of Li (1994) and Zheng (1996),
henceforth referred to as the Jn test. Unlike other speciﬁcation tests used in the literature on continuous-
time models, such as the generalized likelihood ratio (GLR) test of Fan et al. (2001), the Jn test has been
shown to be applicable to time-series data (Li, 1999) and insensitive to the choice of bandwidth (Li and
Wang, 1998; Kim and Wang, 2006). The Jn test also beneﬁts from use of the wild bootstrap (Li and Wang,
1998), which is more reliable than the ordinary bootstrap when the model errors are heteroskedastic (Liu,
1988).3 Application of the Jn test to the present framework is described below.
3.1. Testing speciﬁcations of the drift functions
We ﬁrst consider testing speciﬁcations of the drift functions. That is, for k = 1,2, we wish to test
H
(k)
0 : µk(st,lt) = mk(st,lt;θk) vs. H
(k)
1 : µk(st,lt) 6= mk(st,lt;θk).
Note that, in testing speciﬁcations of the drift functions, we place no restrictions on the diﬀusion functions.
Our proposed testing procedure, analogous to that of Kim and Wang (2006), who consider speciﬁcation
tests of the drift function in a single-factor model, is as follows:
Step 1: Use least squares to estimate
Yk,i/∆ = mk(X1,i,X2,i;θk) + uk,i.
Call the resulting estimate ˆ θk, and let ˆ uk,i = Yk,i/∆ − mk(X1,i,X2,i; ˆ θk).
Step 2: Obtain the test statistic,
J(k)
n = (n − 1)
p
h1h2I(k)
n /ˆ Ω(k),
where
I(k)
n =
1
(n − 1)
n−1 X
i=1
ˆ uk,i

 1
(n − 2)h1h2
n−1 X
j=1,j6=i
ˆ uk,jK

X1,j − X1,i
h1

K

X2,j − X2,i
h2



=
1
(n − 1)(n − 2)h1h2
n−1 X
i=1
n−1 X
j=1,j6=i
ˆ uk,iˆ uk,jK

X1,j − X1,i
h1

K

X2,j − X2,i
h2

,
and
ˆ Ω(k) =



2
(n − 1)(n − 2)h1h2
n−1 X
i=1
n−1 X
j=1,j6=i
ˆ u2
k,iˆ u2
k,j

K

X1,j − X1,i
h1

K

X2,j − X2,i
h2
2



1/2
.
Step 3: Construct a bootstrap sample as
Y ∗
k,i = mk(X1,i,X2,i; ˆ θk)∆ + ˆ u∗
k,i,
3As shown by Li and Wang (1998), even with moderately large samples, the Jn test statistic (given below) does not come
close to attaining its asymptotic distribution (standard normal) under the null.
2where {ˆ u∗
k,i}
n−1
i=1 are obtained through a wild bootstrap resampling scheme (see Li and Wang, 1998) from
{ˆ uk,i}
n−1
i=1 .
Step 4: Using Y ∗
k,i in place of Yk,i, repeat Steps 1 and 2, calling the test statistic obtained in Step 2 J
∗(k)
n .
Finally, repeating Steps 3 – 4 some large number of times, the p-value for the test can be approximated by
computing the proportion of times that J
∗(k)
n exceeds J
(k)
n .
3.2. Testing speciﬁcations of the diﬀusion functions
Analogous to the above, we leave the drift functions unspeciﬁed when testing speciﬁcations of the diﬀusion
functions. That is, for k = 1,2, we test
H
(k)
0 : σ2
k(st,lt) = qk(st,lt;φk) vs. H
(k)
1 : σ2
k(st,lt) 6= qk(st,lt;φk),
while leaving µk(·) unspeciﬁed. This is done as follows:
Step 1: For i = 1,...,(n−1), obtain LL estimates ˆ µk(X1,i,X2,i), and let ˆ Rk,i = [Yk,i/∆−ˆ µk(X1,i,X2,i)]2∆.
Step 2: Use least squares to estimate
ˆ Rk,i = [qk(X1,i,X2,i;φk)]2 + vk,i,
Call the resulting estimate ˆ φk, and let ˆ vk,i = ˆ Rk,i − [qk(X1,i,X2,i; ˆ φk)]2.
Step 3: Obtain the test statistic, J
(k)
n , as given in Step 2 of Section 3.1, but with ˆ vk,i in place of ˆ uk,i, and
g1 and g2 in place of h1 and h2, respectively.
Step 4: Construct a bootstrap sample as
ˆ R∗
k,i = [qk(X1,i,X2,i; ˆ φk)]2 + ˆ v∗
k,i,
where {ˆ v∗
k,i}
n−1
i=1 are obtained through a wild bootstrap resampling scheme from {ˆ vk,i}
n−1
i=1 .
Step 5: Using ˆ R∗
k,i in place of ˆ Rk,i, repeat Steps 2 and 3, calling the test statistic obtained in Step 3 J
∗(k)
n .
Here, the p-value for the test is approximated by repeating Steps 4 – 5 some large number of times, and
computing the proportion of times that J
∗(k)
n exceeds J
(k)
n .
4. Monte Carlo experiment
In this section, we perform a small Monte Carlo experiment, generating 10,000 samples with n = 1,000 from
the parametric model of Schaefer and Schwartz (1984),
dst = β1(α1 − st)dt + γ1dW1,t (3)
dlt = β2(α2 − lt)dt + γ2
p
ltdW2,t. (4)
with (β1,α1,γ1) = (0.72,−0.01,0.007) and (β2,α2,γ2) = (0.8587,0.05,0.03). Here, for k = 1,2 we set
hk = gk = cˆ σXkn−1/6, where c is a constant (speciﬁed below) and ˆ σXk is the sample standard deviation of
Xk. We employ Gaussian kernel functions.
Table I shows the root mean squared error (RMSE) and mean absolute error (MAE) of the ideal and
residual-based estimators of the diﬀusion functions for ∆ = 1/52 and ∆ = 1/250 (i.e., weekly and daily
observations, respectively) and c = 1 (other choices of c were also used, and the results were qualitatively
the same). Overall, the two estimators oﬀer virtually identical performance; the RMSE of the residual-based
estimator is consistently just slightly lower than that of the ideal estimator, while the converse is true for
the MAE. As expected, daily sampling leads to slightly more accurate estimates than weekly sampling.
3Table I: Root mean squared error (RMSE) and mean absolute error (MAE) for the ideal and residual-based
estimators of the diﬀusion functions.
Ideal Residual-based
∆ k RMSE MAE RMSE MAE
1/52 1 1.157 × 10−5 7.686 × 10−6 1.091 × 10−5 7.972 × 10−6
2 1.087 × 10−5 7.106 × 10−6 1.024 × 10−5 7.374 × 10−6
1/250 1 1.019 × 10−5 7.144 × 10−6 0.975 × 10−5 7.322 × 10−6
2 0.943 × 10−5 6.570 × 10−6 0.902 × 10−5 6.744 × 10−6
To asses the size of the Jn test, we test the null hypotheses that the drift and diﬀusion functions are
of the form speciﬁed in (3) – (4). In order to reduce the computational cost, we skip Step 1 in testing the
diﬀusion functions, and assume that the drift functions are known (i.e., we use the ideal squared residuals,
Rk,i). Here, we focus on the weekly case (i.e., ∆ = 1/52), but, to better understand the consequences of
bandwidth choice, we use c = 0.5, c = 1.0, and c = 1.5. Rejection rates at the 5% level, based on 199
bootstrap replications, are shown in Table II.
Table II: Rejection rates for speciﬁcation tests of the drift and diﬀusion functions.
c k Drift Diﬀusion
0.5 1 0.0487 0.0514
2 0.0514 0.0512
1.0 1 0.0527 0.0499
2 0.0511 0.0515
1.5 1 0.0529 0.0499
2 0.0524 0.0496
From Table II, it is clear that the actual rejection rates, for both the drift and diﬀusion functions, are
quite close to the nominal level for all values of c. This conﬁrms the results of Li and Wang (1998) and Fan
and Wang (2006), who ﬁnd that the Jn test is not sensitive to the choice of bandwidth.
5. Empirical example
We close by applying our proposed methodology to some U.S. Treasury security data, which consists of
n = 1,444 weekly observations, covering the period January 2, 1981 to August 28, 2009. The 3-month
secondary market rate is used as the short rate, while the 10-year constant maturity rate is used as the
consol rate. Bandwidths are chosen using least-squares cross-validation, and, as above, Gaussian kernel
functions are used.
Nonparametric estimates of the two-factor model in (1) – (2) are plotted in Figure 1. For the parametric
speciﬁcation in (3) - (4), the Jn test statistics are 1.3205, 2.9309, 13.2746, and 10.4073, for the spread
drift, consol drift, spread diﬀusion, and consol diﬀusion, respectively. Using 999 bootstrap replications, we
estimate p-values for these statistics of 0.0561, 0.0050, 0, and 0, respectively. Hence, we can reject all of
the restrictions at the 1% level, except that on the spread drift function, which can only be rejected at
the 10% level. Thus, this ﬁnding can be seen to oﬀer some limited support for the method of Knight et
al. (2006), which places a linear restriction on the spread drift function. That being said, our proposed
estimation method does oﬀer some increased ﬂexibility, as such a restriction may be more clearly rejected in
other datasets.
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5Figure 1: Nonparametric estimates of the drift and diﬀusion functions
6

A consistent test for linearity in partially linear regression models”

A consistent test of functional form via nonparametric estimation technique”

A reexamination of diﬀusion estimators with applications to ﬁnancial model validation”

A semiparametric two-factor term structure model”

A simple consistent bootstrap test for a parametric regression function”

A two-factor model of the term structure: An approximate analytical solution”

Bootstrap procedures under some non-I.I.D. models”

Consistent model speciﬁcation tests for time series econometric models”

Eﬃcient estimation of conditional variance functions in stochastic regression”

Generalized likelihood ratio statistics and Wilks phenomenon”

Is the short rate drift actually non-linear?”

Local Polynomial Modelling and Its Applications,

Multivariate weighted least squares regression”

Sizes of two bootstrap-based nonparametric speciﬁcation tests for the drift function in continuous time models”

Nonparametric estimation and specification testing of a two-factor interest rate model

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Nonparametric estimation and specification testing of a two-factor interest rate model

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