1. INTRODUCTION AND MOTIVATION
Continuous-time diffusion processes arise in many applications in
econometrics, but perhaps nowhere do they play as large a role as in
finance. Following the pathbreaking work of Black and Scholes (1973), the use of continuous-time diffusion
processes has become a common feature of many applications, especially
asset pricing models. This is probably due to the following two
reasons. The first one is that continuous-time diffusion processes are
able to mimic some important macroeconomic and financial phenomena (see
Sundaresan, 2001). The second reason is that
various parametric diffusion processes have already been used nicely to
model financial data. In both theory and practice, however, one needs
to specify whether a parametric diffusion process is appropriate for a
given set of financial data. In other words, one needs to determine
whether it is appropriate to use a diffusion process with both the
drift and the volatility assumed to be parametric for a given set of
financial data. To justify whether the use of parametric diffusion
processes is appropriate or not for a given set of financial data,
empirical researchers have recently shown a preference for
nonparametric alternatives. Aït-Sahalia (1996a, 1996b) was among
the first to pioneer the nonparametric approach. Other related studies
include Jiang and Knight (1997), Stanton
(1997), Chapman and Pearson (2000), Gao and King (2001), Hong and Li (2004), and Fan and Zhang (2003). Aït-Sahalia (1996a) considers testing the marginal density
functions of a class of diffusion processes under the β-mixing
condition. Pritsker (1998) conducts a finite
sample simulation of a nonparametric kernel test proposed in
Aït-Sahalia (1996a). The principal
result of Pritsker (1998) is that the test
rejects true models much too often when asymptotic critical values are
used. This suggests that the use of an asymptotic critical value may
not be suitable in the finite sample analysis of a test power. In
addition, the use of an estimation-based bandwidth in the nonparametric
kernel test may also contribute to the poor performance of the test in
finite sample studies, because an estimation-based optimal bandwidth
may not necessarily imply that the corresponding test is optimal. We
have been motivated by these two aspects to establish a simulation
procedure for the choice of both an appropriate critical value and a
test optimum bandwidth to improve the test proposed in Aït-Sahalia
(1996b).
Recently, Horowitz and Spokoiny (2001) have
developed a new test of a parametric model of a conditional mean
function against a nonparametric alternative. The test adapts to the
unknown smoothness of the alternative model and is uniformly consistent
against alternatives whose distance from the parametric model converges
to zero at the fastest possible rate. This rate is slower than
T−1/2, where T is the number of
observations. To the best of our knowledge, the problem of extending
the approach of Horowitz and Spokoiny (2001)
to construct an adaptive and optimal test for marginal density
functions has not been considered. This paper then proposes an adaptive
test for testing marginal density functions. The proposed test has an
optimal-rate property. In theory, the proposed test is consistent
against some local alternatives with an optimal rate as stated in
Section 3. In practice, we demonstrate how to apply the test in Section
4 through using a simulated example. Our studies show that the proposed
test has some advantages over the test proposed in Aït-Sahalia
(1996a).
The rest of the paper is organized as follows. Section 2 discusses
the testing of the marginal density. An adaptive test procedure is
proposed in Section 3. Section 4 provides an example of implementation.
Section 5 concludes the paper with some remarks on extensions.
Mathematical assumptions and proofs are relegated to Appendixes
A–C.
2. TESTING MARGINAL DENSITY FUNCTIONS
Consider a continuous-time diffusion process of the form
where μ(·) and σ(·) > 0 are, respectively,
the univariate drift and volatility functions of the process indexed by
θ and Bt is standard Brownian
motion.
Let {rt} satisfy model (2.1) and
f (·,θ) be a parametric form of the marginal
density function of {rt}. Within the
diffusion process, f (·,θ) is completely
determined by the corresponding drift μ(·,θ) and the
diffusion σ(·,θ) (see Aït-Sahalia, 1996a, expression (6)) given by
where {rt} is distributed on D
= (xmin,xmax) with
−∞ ≤ xmin <
xmax ≤ ∞, both the lower bound
x0 and ξ(θ) can be chosen to ensure that
f (x,θ) is a probability density, and θ is
an unknown parameter vector. Let Θ denote a parameter space in
Rq and θ0 ∈ Θ
denote the true value of θ.
Let f (x) be a nonparametric form of the density
function. The null and alternative hypotheses are
where 0 ≤ CT ≤ 1,
limT→∞ CT =
0, and ΔT(x) is a continuous function
satisfying ∫ΔT(x) dx = 0
and f (x) ≥ 0 under
.
Theoretically, this requires that under
,
the alternative function is still a probability density. In practice, the
form of ΔT(x) needs to be constructed.
The simple and natural choice of ΔT(x)
is ΔT(x) =
f1(x,θ1) −
f (x,θ1), where
f1(x,θ) is another specified density
function and θ1 ∈ Θ. For example, f
(x,θ) is the marginal density of
{rt} satisfying the CIR model proposed in
Cox, Ingersoll, and Ross (1985), and
f1(x,θ) is the marginal density of
{rt} satisfying the AG model proposed in
Ahn and Gao (1999).
For this case, the hypothesis structure (2.3) can be written as
This is equivalent to
This basically requires us to test whether
{rt} is sampled from f
(x,θ0) or from f
(x,θ1) with probability 1 −
CT and from
f1(x,θ1) with probability
CT. Obviously, such a structure of the
null hypothesis versus a sequence of local alternatives naturally
extends the usual structure of the null hypothesis against a global
alternative of the form
For the diffusion process, we observe the process at dates
{tΔ|t = 0,1,…,}, where Δ > 0
is generally small but fixed. Let Xt =
r(t−1)Δ for t ≥ 1
throughout this section. Let k(·) be a kernel function,
be the standard kernel density estimator of f (x).
Intuitively, it is natural to compare
directly.
In a seminal paper, Aït-Sahalia (1996a) uses a test statistic of the form
where
.
It then follows from (13) of Aït-Sahalia (1996a) that as T → ∞
under the β-mixing and some other conditions, where
in which R(k) =
∫k2(u) du < ∞ and
k(j)(0) denotes the j-times
convolution product of k(·) given by
The preceding test statistic is based on
,
which measures directly the difference between
.
It can be shown that under H0,
This implies that it has the same order as the mean square error of
if h is chosen to be O(T−1/5).
Thus, to obtain an asymptotically normal distribution with zero mean,
h has to satisfy limT→∞
Th4.5 = 0 as required in Assumption A5 of Aït-Sahalia
(1996a). This implies undersmoothing.
To reduce the bias and avoid undersmoothing, we propose a
nonparametric estimator,
,
of f (x,θ) of the form
where
is a consistent estimator of θ,
wt(x) =
wt(x,h) =
(1/T)kh(x −
Xt) ×
[(s2(x) −
s1(x)(x −
Xt))/(s2(x)s0(x)
− s12(x))], and
for r = 0,1,2.
We also define
If
is a
-consistent
estimator of θ, then we have
It follows from Fan and Gijbels (1996) that
provided that the first three derivatives of f (x)
exist, where ξ1 and ξ2 lie between
x and h and x,
ck, and dk
are constants depending on functionals of k(·), and
σk2 =
∫x2k(x) dx.
This implies that as T → ∞
As can be seen from (2.7), the use of the difference
can avoid undersmoothing. In other words, we can still assume lim
supT→∞ Th5 <
∞.
Let us now establish our test statistic. We first have a look at the
following distance function:
This naturally suggests estimating D(f,θ) by
We then propose using a test statistic of the form
We now state the main results of this section. Their proofs are
relegated to Appendix A.
THEOREM 2.1. (i) Suppose that Assumptions A.1–A.5 in
Appendix A hold. Then under
in (2.3) we have
where
.
(ii) Assume that the conditions of (i) hold. In addition, assume
that there is a random data-driven
such that
.
Then under
in (2.3) we have
THEOREM 2.2. (i) Suppose that Assumptions A.1–A.5 in
Appendix A hold. Then under
in (2.3) we have
where
are as defined in (2.4).
(ii) Assume that the conditions of (i) hold. In addition, assume
that there is a random data-driven
such that
.
Then under
in (2.3) we have
Remark 2.1. (i) Similar to
of (2.4), one may replace
by
(ii) As can be seen from Theorem 2.2(i), we need to estimate both the
asymptotic mean and variance of
involved in practice. It is possible to avoid estimating this kind of
unknown quantity by introducing a weight function into
.
In both theory and practice, however, the asymptotic power of the test may
depend on the choice of such a weight function. We therefore follow a
suggestion made by two of the referees and use the natural form
to construct an adaptive test in Section 3.
(iii) Theorem 2.2(i) establishes an asymptotic normality test
statistic. Theorem 2.2(ii) shows that the asymptotic normality remains
unchanged when h is replaced with the random data-driven
,
which is known as the plug-in method. Fan and Gijbels (1996, pp. 152–154) have shown
that the plug-in method has some advantages in applications. Whether
the proposed test statistic
is optimal has not been discussed. A modified form of the test statistic
is shown to be optimal, and the detailed discussion is given in Section 3.
3. AN ADAPTIVE TEST PROCEDURE
Section 2 establishes the asymptotic normality of the test statistic
for testing the marginal densities. The test statistic has nontrivial
power only if CT converges more slowly
than T−1/2. To improve the asymptotic
power properties of the test, we consider extending the approach of
Horowitz and Spokoiny (2001) for testing
nonparametric regression functions. It is assumed that a marginal
density function g belongs to a class of s-times
(s ≥ 2) differentiable density functions on
R1, such as a Hölder, Sobolev, or Besov class,
,
which is separated from the null hypothesis by some distance
CT that converges to zero as T →
∞. The objective of this section is to find the fastest rate at which
CT can approach zero while permitting consistent
testing uniformly over
.
This rate is called the optimal rate of testing. A test is consistent
uniformly over
if
Thus, the optimal rate of testing is the fastest rate at which
CT can approach zero while maintaining
(3.1).
3.1. Asymptotic Behavior of the Test Statistic under
the Null Hypothesis
As can be seen in Section 2, the proposed test statistic depends on
the bandwidth. This section then suggests using
where HT = {h =
hmaxak : h
≥ hmin, k = 0,1,2,…}, in which 0
< hmin < hmax and 0 <
a < 1. Let JT denote the
number of elements of HT. In this case,
JT ≤
log1/a(hmax/hmin).
Detailed conditions on hmin and
hmax will be given in Assumption B.3 in Appendix
B.
Simulation Scheme.
We discuss how to obtain a critical value for L*. The exact
α-level critical value, lα* (0 < α
< 1), is the 1 − α quantile of the exact finite sample
distribution of L*. Because θ0 is unknown,
lα* cannot be evaluated in practice. We
therefore suggest choosing a simulated α-level critical value,
lα, by using the following simulation
procedure.
1. For the simulation, we either use resamples of the sampled
data Xt or generate the data
Xt from the marginal density f
(x,θ0) or the corresponding transition density
with an initial value of θ0 under
.
2. The true value θ0 is estimated based on the
simulated {Xt}, and the resulting estimate is
denoted by
.
3. We choose HT as specified
following (3.2) with hmin and hmax
satisfying Assumption B.3 in Appendix B and then compute L* of
(3.2) using the simulated {Xt} and
.
4. Repeat the preceding steps M times and produce
M versions of L*, Lm* for
m = 1,2…,M. The simulated critical value
lα is then the (1 − α)% percentile of
the M values, Lm* for m
= 1,2…,M, of L*.
We now state the following result, and its proof is relegated to
Appendix B.
THEOREM 3.1. Assume that Assumptions A.1, A.3, and A.4 in
Appendix A and B.1–B.3 listed in Appendix B hold. Then under
The main result on the behavior of the test statistic L* under
is that lα is an asymptotically correct α-level
critical value under any model in
.
3.2. Consistency against a Fixed Alternative
We now show that L* is consistent against a fixed
alternative model. Assume that model (1.1) holds. Let the parameter set
Θ be an open subset of Rq. Let
satisfy Assumption B.1 in Appendix B. For convenience, let
Measure the distance between
by the normalized l2 distance
where ∥·∥ denotes the euclidean norm. If
is false, then
for all sufficiently large T
and some Cρ > 0. A consistent test will
reject a false
with probability approaching one as T → ∞.
The following theorem establishes a consistency result, and its proof
is relegated to Appendix B.
THEOREM 3.2. Assume that the conditions of Theorem 3.1 hold. In
addition, if there is a Cρ > 0 such that
holds then
3.3. Consistency against a Sequence of Local
Alternatives
In this section, we consider the consistency of L* under
local alternatives of the form
with
for some constant C0 > 0 and θ1 ∈
Θ.
Let
We now have that
To ensure that the rate of convergence of
fT to the parametric model
F(θ1) is the same as the rate of convergence
of CT to zero, in view of (3.4), we need
to assume that ΔT(x) is a continuous
function that is normalized so that
for some δ > 0. When ΔT(·) does
not depend on T, condition (3.5) can be replaced by E
[Δ2(X1)] > 0, which
holds automatically when Δ(·) ≠ 0.
We now state the following consistency result, and its proof is
relegated to Appendix B.
THEOREM 3.3. Assume that Assumptions A.1, A.3, and A.4 in
Appendix A and B.1–B.3 with hmax =
cmax(log log T)−1 for
some constant cmax > 0 in Appendix B hold. Let
be a
-consistent
estimator of θ. Let fT satisfy
(3.3) with
for some constant C > 0. In addition, let condition (3.5)
hold. Then
The result shows that the power of the adaptive, rate-optimal test
approaches one as T → ∞ for any function
ΔT(·) and sequence
{CT} that satisfy the conditions of
Theorem 3.3.
3.4. Consistency against a Sequence of Smooth
Alternatives
This section establishes that L* is consistent uniformly
over alternatives in a Hölder smoothness class whose distance from
the parametric model approaches zero at the fastest possible rate. It
can be shown that we can extend the results to Sobolev and Besov
classes under more technical conditions.
Before specifying our smoothness classes, we introduce the following
notation. Define the Hölder norm
where Sf = {x ∈
R1 : f (x) > 0}.
The smoothness classes that we consider consist of functions
f ∈ S(H,s) ≡ {f
: ∥ f ∥H,s ≤
cH} for some (unknown) s ≥ 2
and cH < ∞.
For some s ≥ 2 and all sufficiently large
Cf < ∞, define
where
is as defined in Section 3.2.
We now state the following consistency result, and its proof is
relegated to Appendix B.
THEOREM 3.4. Assume that Assumptions A.1, A.3, and A.4 in
Appendix A and B.1–B.3 in Appendix B hold. Then for 0 < α
< 1 and BH,T as defined in
(3.7)
Remark 3.1. Theorems 3.1–3.4 show that we have established some
consistency results for the proposed test given in (3.2). Such consistency
results correspond to Theorems 1–4 of Horowitz and Spokoiny (2001) for a fixed design regression case. In our
case, we deal with the case where the observations are stationary and
α-mixing time series. In addition, the optimum version L*
is asymptotically consistent as established in Theorem 3.2. This is one
of the advantages of our test over existing ones, such as the natural
competitor proposed in Aït-Sahalia (1996a). In Section 4, we show that our test also
outperforms the natural competitor in the finite sample case.
4. EXAMPLE OF IMPLEMENTATION AND
APPLICATION IN DIFFUSION MODELS
This section illustrates the proposed adaptive test by the following
example. As the bootstrap simulation procedure for selecting both the
bandwidth and simulated critical values is extremely computationally
demanding, especially for large numbers of data, we only consider using
the CIR model proposed by Cox et al. (1985)
and show how to implement the adaptive test statistic L* of
(3.2) in practice through using a simulated example. The main reason
for choosing the model is not only because both the marginal and
transition density functions have closed forms but also because the
model has been studied extensively in the literature. See, for example,
Aït-Sahalia (1999) and Hong and Li
(2004).
Example 4.1
We consider using the CIR model given by
where κ > 0, β > 0, and σ > 0 are unknown
parameters and Bt is standard Brownian
motion. It can be shown that {rt} is
distributed on R+ = (0,∞) if
2κβ/σ2 ≥ 1. Furthermore, it follows from
Lemma 3.1 of Masry and Tjøstheim (1995) that the process
{rt} satisfies Assumption A.1(i).
Alternatively, one may apply Assumption A.3′ of Aït-Sahalia
(1996b, p. 552) to verify that
{rt} is strictly stationary and
α-mixing.
As a result of (2.2), the marginal density function of
{rt} satisfying model (4.1) is
where θ = (β,κ,σ), ν =
2κβ/σ2 − 1, and Γ(·) is the
usual gamma function. Let θ0 be the true value of
θ.
To construct a sequence of local alternatives, we also consider using
a marginal density of the form
where ν1 = 2κ/σ2. It is known
that f1(x,θ) is the marginal density
of {rt} satisfying the AG model proposed
in Ahn and Gao (1999)
with parameter values κ > 0, β > 0, and σ > 0.
The necessary and sufficient conditions for stationarity and
unattainability of 0 and ∞ in finite expected time are the pairs
κ > 0 and β > 0 (see Ahn and Gao,
1999). To show that {rt} is
strictly stationary and α-mixing, as explained in Appendix A of Ahn
and Gao (1999, pp. 755–756), one needs
only to verify Assumption A.3′ of Aït-Sahalia (1996b, p. 552). It is easy to see that such an
assumption holds for the marginal density, drift, and diffusion
functions given in (4.3) and (4.4).
The corresponding structure of the test problem (2.3) for this
example can be constructed as
where
in which θ1 ∈ Θ. The reason for choosing
such ΔT(·) as the local shift function
is to ensure that the models under
fluctuate closely around those under
.
The choice of (4.5) and (4.6) ensures that (3.7) holds with s = 2.
This implies that the adaptive test is consistent against the sequence with
an optimal rate. Note that Assumptions B.1 and B.2 hold.
In the following simulation, we consider using a class of
alternatives of the form
where θ1 ∈ Θ and 0 < ψ < 1 is
defined as the truncation parameter to be chosen.
To compute the nonparametric estimators involved, we choose the
normal kernel function given by
throughout the simulation. Observe that Assumptions A.1–A.4
hold. For the CIR and AG models, we simulate the data from their
marginal density and transitional functions, which all have closed
forms.
In the detailed simulation, we simulate the data from (4.2) for the
CIR model, (4.3) for the AG model, and then (4.7) under
.
Using the simulated data, we compute
in which R(k) =
are used after the choice of (4.8) and HT is
as defined following (3.2) with
.
Note that Assumption B.3 holds.
To compare L* with
in (2.4), we construct a test statistic of the form
where h* is chosen by using the following
procedure.
[bull ] We simulate Xt with
probability 1 − ψ from the CIR model and with probability ψ
from the AG model with an initial value of θ1 under
.
[bull ] Use the simulated data {Xt :
t = 1,2,…,T} to estimate
θ1.
[bull ] Compute the resulting function of h given
by
[bull ] Repeat the preceding steps Q = 1,000 times and
produce Q versions of
denoted by
for m = 1,2,…,Q. Use the Q functions of
h,
for m = 1,2,…,Q, to construct their empirical
bootstrap distribution function, that is,
where I(U ≤ u) is the usual
indicator function.
[bull ] For a given asymptotic critical value ecvα
at the level α (e.g., ecv0.005 = 1.645 at the 5% level), we
then calculate the following power function:
[bull ] Find approximately at which h value the power
function ψ(h) is maximized. Denote the maximizer by
h*.
We then consider using the same choice of the parameter values as in
(17) of Pritsker (1998). This means that the
baseline model is model (4.1) with
.
In this example, the same parameter values were also used as
θ1 in computing the power of the tests L* and
L0*. The truncation parameter was chosen as
ψ = 0 under
whereas the truncation parameter was chosen as
under
.
Three different sizes of sample T = 1,000, 2,755, or 5,500 were
then considered. The corresponding simulated critical values,
lα and l0α, of
L* and L0* at the α level are then
found by using the simulation scheme proposed in Section 3.1. The sizes
of the tests were then computed based on the data simulated under
,
and the power values of the tests were calculated based on the data
generated under
.
In implementing the simulation procedure, we used M = 1,000
involved in the simulation scheme proposed in Section 3.1. The number
of simulations in producing Table 1 was also
1,000. Both the size and the power of L* and
L0* are given in Table
1.
Rejection rates for the marginal density tests
Remark 4.1. (i) As can be seen from Table 1,
the power values of both L* and L0* look
reasonable when
,
or about 3%. This may show that both L* and L0*
are practically applicable to the medium sample case, because the difference
between the null hypothesis and its alternative was made deliberately close.
We also computed the power of the tests for the case where
or 5%. Our small sample results showed that the power of L* was
already 100% even when T = 1,000. In general, it is true that the
power increases as ψ increases for each case. Observe that L*
is slightly more powerful than L0*, although
h* involved in
has been chosen based on the assessment of its power. We observe that
the sizes of the two tests are also close to either 5% in the first
half of Table 1 or 1% in the second half of
Table 1.
(ii) We also examined the dependence of the power on the choice of
the initial parameter values. Our experience suggests that the power of
the tests mainly depends on the choice of the truncation parameter
ψ. This is both understandable and expected, because the test
statistics finally depend only on the estimation and reestimation
procedure of the vector of the initial parameters rather than the
initial parameter values themselves. This is probably why artificial
values or parameter values estimated from a set of real data are used
as initial values for starting a simulation procedure. For example,
Hong and Li (2004) use the parameter values
estimated from the U.S. interest rate series for their simulation
procedure.
(iii) Compared with existing results (see Pritsker, 1998), both the size and power of
L0* have been significantly improved. This is
probably because (a) the choice of h involved in
is based on the assessment of the power of
rather than using an estimation-based optimal value and (b) to avoid
using the asymptotic distribution of
and then an asymptotic critical value of 1.645 at the 5% or 2.33 at the 1%
level, we have used the bootstrap-based simulated critical value,
l0α, at the level α. We also computed both
the power and size values for the case where h was chosen by
using a cross-validation criterion, and the resulting sizes and power
values were similar to those obtained by Pritsker (1998), although L* always performed better
than L0*. This further demonstrates that the
asymptotic distribution of either
can only provide some kind of idea about the asymptotic behavior. In practice,
we strongly suggest using the proposed bootstrap simulation procedure
for choosing a simulated critical value rather than an asymptotic
critical value.
5. CONCLUSION
In this paper, we have considered testing the general continuous-time
diffusion model (1.1) under the α-mixing condition. The results for
continuous-time models under the α-mixing condition complement some
existing results under the β-mixing condition. See, for example,
Aït-Sahalia (1996a). Moreover, an
adaptive and optimal test procedure has been established. This
extension corresponds to Horowitz and Spokoiny (2001) for the fixed design nonparametric regression
and then to Chen, Gao, and Li (2001) for a
nonparametric time series regression model. To deal with the
α-mixing condition, we have established some novel results for
moment inequalities (see Lemma C.2) and limit theorems (see Lemma A.1)
for degenerate U-statistics of strongly dependent processes.
Both Lemmas A.1 and C.2 are applicable to some other estimation and
testing of diffusion processes with the α-mixing condition (for
more about various mixing conditions, see Doukhan,
1995). In addition, we have demonstrated how to implement the
proposed test procedure in practice through using a simulated
example.
The results given in this paper can be extended in a number of
directions. First, it is possible to consider testing for both the
marginal and transition density functions simultaneously, because the
transition density can capture the full dynamics of a diffusion process
and, in particular, can distinguish the diffusion processes that have
the same marginal density but different transition densities. Second,
the results of this paper for the short-range dependent continuous-time
case can be extended to the long-range dependent continuous-time case.
Third, one probably can relax the strict stationarity and the mixing
condition, as the recent work by Aït-Sahalia (1999) and Karlsen and Tjøstheim (2001) indicates that it is possible to do such work
without the stationarity and the mixing condition. This part is
particularly important for two reasons: (i) for the long-range
dependent case one needs to avoid assuming both the long-range
dependence and the mixing condition, as they contradict each other; and
(ii) some important models are nonstationary. These are some issues
left for future research.
APPENDIX A
This Appendix lists the necessary assumptions for the establishment
and the proof of the main results given in Section 2.
A.1. Assumptions. Let the parameter set Θ be an open subset
of Rq. Let
.
Define [dtri ]θ f (x,θ) =
∂f (x,θ)/∂θ,
[dtri ]θ2 f (x,θ) =
∂2f
(x,θ)/∂θ∂θ′, and
[dtri ]θ3 f (x,θ) =
∂3f
(x,θ)/∂θ∂θ′∂θ′′
whenever these derivatives exist. For any q × q
matrix D, define
where
.
Assumption A.1. (i) Assume that the process
{rt} is strictly stationary and α-mixing
with the mixing coefficient α(t) =
Cααt defined by
for all s,t ≥ 1, where 0 <
Cα < ∞ and 0 < α < 1 are
constants and Ωij denotes
the σ-field generated by {rt :
i ≤ t ≤ j}.
(ii) Assume that the univariate kernel function k(·)
is nonnegative, symmetric, and four-times differentiable on
R1 = (−∞,∞). In addition,
.
Assumption A.2. (i) The parameter space Θ ⊂
Rq is compact. In a neighborhood of the true
parameter θ0, f (x,θ) is twice
continuously differentiable in θ; E [(∂f
(x,θ)/∂θ)(∂f
(x,θ)/∂θ)τ] is of full
rank. In addition, assume that G(x) is a positive and
integrable function with E
[G(Xt)] < ∞
uniformly in t ≥ 1 such that
supθ∈Θ| f
(Xt,θ)|2 ≤
G(Xt) and
supθ∈Θ∥[dtri ]θ
j f
(Xt,θ)∥2 ≤
G(Xt) for j = 1,2,3,
where for
.
(ii) Assume that
is a
-consistent
estimator of θ0.
Assumption A.3. For every θ ∈ Θ:
(i) The drift and the diffusion functions are three times
continuously differentiable in x ∈ R+
= (0,∞), and σ > 0 on R+.
(ii) The integral of
converges at both boundaries of D, where v is fixed in
D.
(iii) The integral of
diverges at both boundaries of D.
Assumption A.4. (i) Assume that the first three derivatives of
f (x) are continuous on D and that f
(x) > cf > 0 on the interior of
D for some cf > 0. In addition,
both f (x) and f2(x) are
integrable on D.
(ii) The initial random variable r0 is distributed as
f (x).
(iii) The true drift and diffusion functions satisfy Assumption A.3.
Assumption A.5. The bandwidth parameter h satisfies
that
Remark A.1. Assumptions A.1–A.4 are quite natural in
this kind of problem. Assumptions A.2–A.4 correspond to Assumptions
A0, A1, and A3 of Aït-Sahalia (1996a).
Assumption A.1 is the exception. Assumption A.1(i) assumes the α-mixing
condition, which is weaker than the β-mixing condition. Assumption A.1(ii)
is quite general, allowing the use of the standard normal kernel.
Assumption A.5 ensures that the theoretically optimum value of
hoptimal = CT−1/5 can
be included. This is important, because there may be cases in which
hoptimal is also optimal for testing purposes.
A.2. Technical Lemmas.
The following lemmas are necessary for the proof of the main results
stated in Section 2.
LEMMA A.1. Let ξt be an
r-dimensional strictly stationary and strong mixing (α-mixing) stochastic
process. Let φ(·,·) be a symmetric Borel function
defined on Rr ×
Rr. Assume that for any fixed x ∈
Rr, E
[φ(ξ1,x)] = 0 and E
[φ(ξi,ξj)|Ω0j−1]
= 0 for any i < j, where
Ωij denotes the σ-field
generated by {ξs : i ≤
s ≤ j}. Let
.
For some small constant 0 < δ < 1, let
where the maximization over P in the equation for
MT4 is taken over the four probability
measures
P(ξ1,ξi,ξj,ξk),
P(ξ1)P(ξi,ξj,ξk),
P(ξ1)P(ξi1)P(ξi2,ξi3),
and
P(ξ1)P(ξi)P(ξj)P(ξk),
where
(i1,i2,i3)
is the permutation of (i,j,k) in
ascending order;
Assume that all the MT′s are
finite. Let
If
limT→∞(max{MT,NT}/σT2)
= 0, then
Remark A.2. Lemma A.1 establishes central limit theorems for
degenerate U-statistics of strongly dependent processes. It should
be pointed out that the conclusion of Lemma A.1 remains true when the
martingale assumption that E
[φ(ξi,ξj)|Ω0j−1]
= 0 for any i < j is removed. Such a martingale
assumption is used only for a direct application of an existing central
limit theorem (CLT) for martingales. Without such a condition, one
needs only to decompose
and then apply the martingale CLT to
.
Using the condition that E
[φ(ξ1,x)] = 0 for each given
x, one can show that the terms involving E
[φ(ξi,ξj)|Ω0j−1]
are negligible (see Roussas and Ioannides, 1987, Theorem 5.5). Thus, as assumed in Lemma 3.2
of Hjellvik, Yao, and Tjøstheim (1996)
and Theorem 2.1 of Fan and Li (1998), the
condition that E
[φ(ξ1,x)] = 0 for each given
x is the key assumption.
Proof. Let
To prove Lemma A.1, it suffices to show that as T →
∞
and
By Lemma C.1 (with η1 = φik,
η2 = φjk, l = 2,
pi = 2(1 + δ), and Q =
1/(1 + δ)),
Therefore,
because
.
Observe that
Let ηijk =
1/3(φikφjk +
φijφkj +
φjiφki) and
ηij =
1/3∫φikφjk
dP(ξk).
Then by Lemma C.2(i) in Appendix C,
Let Cφ =
∫φ122φ342
dP(ξ1) dP(ξ2)
dP(ξ3) dP(ξ4), where
P(ξ) denotes the probability measure of ξ.
Using Lemma C.1 repeatedly, we have that for different
i,j,k,l
where Δ(i,j,k,l) is the
minimum increment in the sequence that is the permutation of
i,j,k,l in ascending order.
Similar to (A.5), one can have for all different
i,j,k,l
Therefore,
It now follows from (A.3)–(A.5) that for any ε > 0
Thus, (A.1) holds.
Note that for 2 ≤ k ≤ T,
It is easy to see that
Similar to (A.5), one can have for any (i,j) ≠
(s,t),
where Δ(·) is as defined in (A.5).
Consequently, the first two terms on the right-hand side of (A.7) are
of order
O(T3MT41/(1+δ)),
because
.
Thus, (A.2) follows from
This finishes the proof. █
Before stating the following lemmas, we define the following notation.
using
where
in which A is the T × T matrix with
{ast} as its (s,t)
element.
We assume without loss of generality throughout the rest of this
paper that
LEMMA A.2. Under the conditions of Theorem 2.1, we have as T
→ ∞
Proof. We now prove (A.9). It follows from Assumptions A.2 and
A.3 that as T → ∞
This completes the proof of the first part of (A.9). For the proof of
the second part of (A.9), let
Then
We first look at the main component of
σT2. We now have
Using Assumptions A.1–A.4, we have as T → ∞
where L(x) = ∫k(x +
y)k(y) dy is as defined in
(2.5).
Therefore, as T → ∞
where k(4)(·) is as defined in (2.5).
Similarly, one can show that as T → ∞
We now deal with the remainder term of
var[N0T(h)] . By Lemma
C.1 (with η1 = φik,
η2 = φjk, l = 2,
pi = 2(1 + δ), and Q =
1/(1 + δ)),
where MT1 is as defined in Lemma A.1.
Therefore, using the fact that
,
whose proof is similar to that of (A.17), which follows.
Equations (A.10)–(A.12) imply
This finishes the proof of the second part of (A.9). █
LEMMA A.3. Under the conditions of Theorem 2.1, we have as T
→ ∞
and
where C1 is a constant and
are as defined in Section 2.
Proof. We now give only the proof of (A.14) in some detail, as
the proofs of (A.13) and (A.14) are similar and quite standard and the
details follow similarly from some existing results. See, for example, Fan
and Gijbels (1996).
In view of the definition of
wt(x) and the second equation of
(A.13), to prove (A.14), it suffices to show that as T →
∞
using a Taylor expansion to f (x) −
f (x − vh). This finishes the proof of
(A.14). █
A.3. Proof of Theorem 2.1.
Proof of Theorem 2.1(i). To prove Theorem 2.1(i), in view of
Remark A.2 and Lemma A.3, it suffices to show that
To apply Lemma A.1, let ξt =
Xt and
φ(ξs,ξt) =
φst defined previously. Let
MT and NT be
defined as in Lemma A.1. We now verify only the following condition
listed in Lemma A.1:
for MT1, MT
21, MT3,
MT51, MT52,
and MT6, where
σh2 =
hσ02. The others follow
similarly.
For the MT part, one justifies only
The others follow similarly.
Let ψst =
(1/Th)∫k((x −
Xs)/h)k((x
−
Xt)/h)p(x)
dx. We now have
where L(·) is as defined previously.
For any given 1 < ζ < 2 and T sufficiently large,
we obtain
using Assumption A.1(ii), where f
(x,y,z) denotes the joint density of
(Xi,Xj,Xk)
and Cp is a constant.
Thus, as T → ∞
Hence, (A.17) shows that (A.15) holds for the first part of
MT1. The proof for the second part of
MT1 follows in a similar way. Similarly,
we have that as T → ∞
using Assumption A.1(ii).
This implies that as T → ∞
Thus, (A.18) now shows that (A.15) holds for
MT3. It follows from the structure of
{ψij} that (A.15) holds automatically for
MT51, MT52,
and MT6, because E
[φst] = 0 for s ≠
t.
We now prove that (A.15) holds for MT
21. For some 0 < δ < 1 and 1 ≤ i <
j < k ≤ T, let
MT 21 = E
[|ψikψjk|2(1+δ)]
. Similar to (A.16) and (A.17), we obtain that as T →
∞
using the fact that limT→∞ Th
= ∞.
This completes the proof of (A.15) for MT
21, and thus (A.15) holds for the first part of
{φst}. Similarly, one can show that (A.15)
holds for the other parts of {φst}. Thus, we
have shown that under
The proof of Theorem 2.1(i) is therefore finished. █
Proof of Theorem 2.1(ii). Note that as T →
∞
using the continuity of
in h. This completes the proof of Theorem 2.1(ii). █
Proof of Theorem 2.2. The proof follows from Theorem 2.1 and the
following standard result:
APPENDIX B
This Appendix lists the necessary assumptions for the establishment
and the proof of the main results given in Section 3.
B.1. Assumptions.
Assumption B.1. The parameter set Θ is an open subset of
Rq for some q ≥ 1. The parametric
family
satisfies the following conditions.
holds with probability one (almost surely).
Assumption B.2. (i) Let
be true. Then θ0 ∈ Θ and
for any ε > 0 and all sufficiently large
CL.
(ii) Let
be false. Then there is a θ* ∈ Θ such that
for any ε > 0 and all sufficiently large
CL.
Assumption B.3. (i) Assume that the set
HT has the structure of (3.2) with
cminT−γ =
hmin < hmax =
cmax(log log T)−1, where
γ, cmin, and cmax are some
constants satisfying 0 < γ < 1 and 0 <
cmin,cmax < ∞.
(ii) Assume that ΔT(x) is continuous
in x ∈ D and satisfies
for all T ≥ 1.
Remark B.1. Assumptions B.1(i) and B.1(ii) are quite standard
in this kind of problem. See Assumptions 1(i) and (ii) of Horowitz and
Spokoiny (2001). Assumption B.1(iii) is required
to ensure that the marginal density function is identifiable. A similar
condition is used in Assumption 1(iii) of Horowitz and Spokoiny (2001). It can be shown that Assumption B.1(iii)
holds when f (x,θ) belongs to classes of simple
linear and certain nonlinear functions in θ. The identifiability
assumption is imposed to exclude the case where f
(x,θ) is flat as a function of θ over certain range
of θ and some value of x, because such a function may be
neither identifiable nor a probability density. Assumption B.2 is
needed to ensure that the true version of θ under
can be estimated by a
-consistent
estimator. Assumption B.3(i) imposes some conditions on both
hmin and hmax. The theoretical
condition on hmin is quite general. In practice, we
would suggest using
to include the estimation-based optimal bandwidth hoptimal
= Cn−[1/(2s+1)], because
the estimation-based optimal value may also be optimal for testing
purposes in some cases. The restriction on hmax is
required only for the proof of Theorem 3.3. It should be noted that
hmax is not necessarily the optimal bandwidth such
that the power of the resulting test is maximized. As explained at the
beginning of Section 2, both the existence and reasonableness of
Assumption B.3(ii) can be justified. Unlike the regression setting
discussed in Horowitz and Spokoiny (2001), we
need to assume
to ensure that the alternative is also a probability density. As the main
results in Section 2 are only concerned with the null hypothesis, we do
not need to assume such a rigorous condition for the main results.
This paper considers using only a set of discrete bandwidths for
constructing the adaptive test. It is believed that some corresponding
results of Theorems 3.1–3.4 can be established for the case where
HT is an interval of continuous bandwidth
values. As HT is always chosen as a set of
discrete bandwidths in practice, we therefore think that such an
extension from a set of discrete bandwidths to an interval of
continuous bandwidth values may just be for theoretical and technical
consideration. As such an extension also involves much more tedious and
technical details, we do not discuss this issue in detail in this
paper.
B.2. Technical Lemmas. Before stating the necessary lemmas for
the proof of the results given in Section 3, we introduce the following
notation.
LEMMA B.1. Suppose that the conditions of Theorem 2.1 hold.
in probability, where C > 0 is a
constant.
(ii) For each θ ∈ Θ and sufficiently large
T
Proof. (i) It follows from the definition of
QT(θ) that
To prove Lemma B.1(i), one first needs to show that
in probability for some constant C > 0.
Using the conditions of Lemma B.1, we now have
in probability.
In view of (B.2), to prove Lemma B.1(i), it suffices to show that
in probability.
A Taylor series expansion to f
(Xt,θ) − f
(Xt,θ0) and an application
of Assumption B.1(i) imply (B.3). This finishes the proof of Lemma
B.1(i).
(ii) Let λmin(A) and
λmax(A) denote the smallest and largest
eigenvalues of A, respectively. In view of
to prove Lemma B.1(ii), it suffices to show that for n large
enough
for some C > 0. Similar to the proof of Lemma A.2 of Gao,
Tong, and Wolff (2002), one can easily finish
the proof of (B.5). █
Without loss of generality, we consider the case of q = 1 in
the following lemmas and their proofs. Define
LEMMA B.2. Under the conditions of Theorem 3.1, we have for any
given θ ∈ Θ and i = 1,2
Proof. It suffices to show that for any large constant
C0 > 0
where
Similar to the proof of (A.1), one can show that as T →
∞
for some function C(θ).
Using Lemmas C.1 and C.2 in Appendix C and the fact that E
[εt(x)] = 0 for x
∈ D, one can show that as T → ∞
Thus, equations (B.7)–(B.9) complete the proof. █
LEMMA B.3. Under the conditions of Theorem 3.1, we have as T
→ ∞
Proof. Similar to (B.7), we have for large constant
C0 > 0
Similar to (B.8), we can have as T → ∞
Analogous to (B.9), one can show that as T → ∞
Thus, equations (B.11)–(B.13) complete the proof of (B.10).
█
LEMMA B.4. Under the conditions of Theorem 3.1, we have for each
u > 0,
under
.
Proof. We now prove (B.14). Using a Taylor series expansion to
f (Xt,θ) − f
(Xt,θ0) and Assumption
B.1, we have for θ′ between θ and θ0
Hence, (B.4), (B.10), (B.15), and Assumption B.1(i) imply
The proof of (B.14) follows from (B.15) and (B.16). █
LEMMA B.5. Suppose that the conditions of Theorem 3.1 hold. Then
for every u > 0, some h ∈
HT, and as T → ∞
under
.
Proof. In view of the definition of
Qn(θ), to prove (B.17), it suffices to
show that as T → ∞
where qT = E
[QT(θ*)] .
Note that
where θ′ lies between θ and θ*.
In view of (B.6), (B.10), (B.18), and Assumptions B.1(i) and B.2(ii),
to prove (B.17), it suffices to show that for any δ > 0,
as T → ∞.
Similar to (B.8) and (B.9), one can show that as T →
∞
Thus, equations (B.19) and (B.20) imply that as T →
∞
using qT = CTh(1 +
o(1)) given in the proof of Lemma B.1(ii), where C is
a constant independent of T. Lemma B.5 therefore follows from
(B.21). █
Recall the notation introduced in (A.9). We assume without loss of
generality that k(4)(0) = 1 in Lemma A.2. Define
LEMMA B.6. Suppose that the conditions of Theorem 3.1 hold. Then
as T → ∞
uniformly over h ∈ HT.
Proof. The proof of (B.23) follows from (2.7) and (2.8)
immediately. █
LEMMA B.7. Suppose that the conditions of Theorem 3.1 hold.
Then maxh∈HT
L0(h) and
maxh∈HT
LT(h) have identical asymptotic
distributions under
.
Proof. Note that
QT(θ0) = 0 under
and that Lemmas A.3 and B.1–B.5 imply as T → ∞
Therefore, equations (B.21), (B.22), and (B.24) complete the proof of
Lemma B.7. █
LEMMA B.8. Suppose that the conditions of Theorem 3.1 hold. Then
for any x ≥ 0, h ∈
HT, and all sufficiently large
T
Proof. It follows from the beginning of the proof of Theorem
2.1(i) that for any small δ > 0 there exists a large integer
T0 ≥ 1 such that for T ≥
T0
where
.
This implies for any T ≥ T0 and
x ≥ 0
using
.
The proof follows by letting
for any x ≥ 0. █
For 0 < α < 1, define
to be the 1 − α quantile of
maxh∈HT
L0(h).
LEMMA B.9. Suppose that the conditions of Theorem 3.1 hold. Then
for large enough T
Proof. The proof is trivial.
LEMMA B.10. Suppose that the conditions of Theorem 3.1 hold.
Suppose that
for some h ∈ HT,
where
Then
Proof. To prove Lemma B.10, in view of Lemmas B.6 and B.7, it
suffices to show that
which holds if
for some h ∈ HT. For any
h ∈ HT, using (B.21) and
then (B.17) we have
On the other hand, condition (B.25) implies that as T →
∞
Observe that
Thus, it follows from (B.26) that as T → ∞
because L0(h) is asymptotically normal
and therefore bounded in probability and
.
Because of (B.27), as T → ∞
This finishes the proof. █
B.3. Proofs of Theorems 3.1–3.4.
Proof of Theorem 3.1. The proof follows from Lemmas B.6 and
B.7.
Proof of Theorem 3.2. This proof is similar to that of Theorem
3.3, which follows, using Lemma B.1(ii). Alternatively, one can follow the
corresponding proof of Theorem 2 of Horowitz and Spokoiny (2001) by using Lemma B.1(ii) and the condition that
to verify (B.25). █
Proof of Theorem 3.3. Condition (3.5) ensures that the rate of
convergence of fT to the parametric model
F(θ1) is the same as the rate of convergence
of CT to zero. In particular, when (3.5)
holds,
In view of Lemma B.10, to complete the proof of Theorem 3.3, it
suffices to verify (B.25). This verification follows from Lemma B.1(ii)
and (B.28). █
Proof of Theorem 3.4. For the proof of Theorem 3.4, one needs
to use the conditions of Theorem 3.4 to finish the proof. In our proof, we
mainly use Lemma B.1(ii) and the condition of Theorem 3.4 that
to verify (B.25). █
APPENDIX C
The following two technical lemmas have already been used in the
proofs of Lemma A.1 and Theorem 2.1. The two lemmas are of general
interest in themselves and can be used for other nonparametric
estimation and testing problems associated with the α-mixing
condition.
LEMMA C.1. Suppose that
Mmn are the
σ-fields generated by a stationary α-mixing process
ξi with the mixing coefficient
α(i). For some positive integers m let
ηi ∈
Msiti
where s1 < t1 <
s2 < t2 <
··· < tm and
suppose ti − si
> τ for all i. Assume further that
for some pi > 1 for which
Then
Proof. See Roussas and Ioannides (1987).
LEMMA C.2. (i) Let ψ(·,·,·) be a
symmetric Borel function defined on Rr ×
Rr ×
Rr. Let the
processξi be defined as in Lemma A.1.
Assume that for any fixed x,y ∈
Rr,
E [ψ(ξ1,x,y)]
= 0. Then
where 0 < δ < 1 is a small constant, C > 0
is a constant independent of T and the function ψ, M =
max{M1,M2,M3},
and
(ii) Let φ(·,·) be a symmetric Borel
function defined on Rr ×
Rr. Let the process
ξi be defined as in Lemma A.1. Assume that for
any fixed x ∈ Rr, E
[φ(ξ1,x)] = 0. Then
where δ > 0 is a constant, C > 0 is a
constant independent of T and the function φ, and
Proof. As the proof of (ii) is similar to that of (i), one
proves only (i). Let i1,…,i6
be distinct integers and 1 ≤ ij ≤
T, let 1 ≤ k1 < ···
< k6 ≤ T be the permutation of
i1,…,i6 in ascending
order, and let dc be the cth
largest difference among kj+1 −
kj, j = 1,…,5. Let
By Lemma C.1 (with η1 =
ψ(ξi1,ξi2,ξi3),
η2 =
ψ(ξi4,ξi5,ξi6),
l = 2, pi = 2(1 + δ) and
Q = 1/(1 + δ)),
Thus,
Similarly,
Analogously, it can be shown in a similar way that
On the other hand, if {k6 −
k5,k2 −
k1} =
{d4,d5}, by using Lemma C.1
three times we have the inequality
Hence,
It follows from (C.3)–(C.7) that
Similar to (C.8), one can show that
Finally, it is easy to see that
The conclusion of Lemma C.2(i) follows immediately from
(C.8)–(C.11). █
