2.1. Main Result
Let us consider the conditional location-scale model:
where
and
are measurable functions,
with compact subsets
and
,
and
, where
denotes the true model parameter belonging to
;
is a sequence of iid random variables with zero mean and unit variance. In what follows, we assume that
is strictly stationary and ergodic and that
is independent of past observations
for
. In this section, we consider the entropy-based GOF test proposed by Lee, Vonta and Karagrigoriou [
9] for the location-scale models in (
1). To this end, we set up the hypotheses:
where
denotes the innovation distribution of the model and
can be any family of distributions.
To carry out the test, inspired by Rosenblatt [
15], we check whether the transformed random variables
follow a uniform distribution on
, say,
, where
and
are the true parameters. Since the parameters are unknown, by replacing those with their estimates, we check the departure from
based on
with
, where
and
with
: see Francq and Zakoian [
16], who take this approach of using initial values for GARCH models.
The entropy-based GOF test is constructed based on the Boltzmann–Shannon entropy defined by
for any density function
f. It is noteworthy that the
actually measures the distance between a distribution with density
f and the uniform distribution. Lee, Vonta and Karagrigoriou [
9] construct a GOF test using an approximation form of the integral in (
3). For any distribution
F, we introduce
where the
’s are weights with
and
,
m is the number of disjoint intervals for partitioning the data range, and
are preassigned partition points. Note that the argument in (
4) is a good approximation of that in (
3) when
are all equal to 1—see
Section 2.1 of Lee, Vonta and Karagrigoriou [
9], and also their Remark 1 concerning the role of weights
w.
Further, we define the residual empirical process:
with
, where
is any consistent estimator of
under the null; for example, the maximum likelihood estimator (MLE). We then define the entropy test by
.
To derive the null limiting distribution of the entropy test, we impose the regularity conditions as follows:
- (C1)
(i) For some random variable V and constant , for all ;
(ii) For some random variable V and constant , for all .
- (C2)
(i) For all , and are differentiable in and on some neighborhoods of and of ;
(ii) There exists a random variable V and constant such that for all , and .
- (C3)
(i) For all , and are twice differentiable in and , where and are the ones in (C2)(i);
(ii) and ;
(iii) and
- (C4)
(i) is continuous and has a positive density ;
(ii) and are uniformly continuous on ;
(iii) For some , , , and .
- (C5)
is twice continuously differentiable with respect to and there exists such that and is uniformly continuous on
- (C6)
Under the null, and .
Remark 1. The above conditions can be found in Kim and Lee [2]. They show that a class of GARCH and TGARCH models with ASTD and AEPD innovations satisfy the regularity conditions and the MLE is asymptotically normal. Below is the main result of this section: see the proof in
Section 2.2.
Theorem 1. Under (
C1)∼(
C6),
we havewhere with and Moreover, we are led to the following result, the detailed proof of which is omitted for brevity because it is essentially the same as that of Lee, Vonta and Karagrigoriou [
9] and Lee, Lee and Park [
10].
Theorem 2. Suppose that the assumptions in Theorem 1 hold. Then, under ,
if as ,
we have that for all large m, as ,
where is any finite subset of the class of all weights and is the Brownian bridge on .
Here, the symbol as indicates that the limiting distribution of is approximately the same as the distribution of A as n tends to ∞. More precisely, we can write , where as and for all .
Remark 2. As seen in the proof of Theorem 2 of Lee, Lee and Park [10], one can easily check that owing to Theorem 1, under the null,wherein the term: becomes negligible as n tends to infinity when m is large. This yields Theorem 2. 2.2. Proof of Theorem 1
We reexpress
as follows:
where
Since
owing to Lemma 1 below, we handle the two terms
and
. Let
and let
be a sequence of positive integer numbers with
and
as
. We express
, where
for some
between
x and
. By Taylor’s theorem, we can express
with
for some
between
and
. Then, owing to
(C1)(i) and
(C4)(iii),
and due to the ergodic theorem, Lemma 4 of Amemiya [
17], (
C3)(
iii), (
C4)(
iii), and (
C6), we get
, so that
Similarly, it can be easily seen that
Next, we analyze
. Owing to the ergodic theorem, Lemma 4 of Amemiya [
17],
,
,
, and
, we have
for some
, which is no more than
where
is an intermediate point between
and
.
Meanwhile, since
and
, we can find (large)
, such that on the event
,
, with probability tending to 1,
Hence, owing to the ergodic theorem, Lemma 4 of Amemiya [
17],
,
,
, and
, we can have that on
and for
,
for some
and intermediate vector
between
and
, which is negligible. Because for
,
, it holds that
, which together with (
10) and (
11) indicates
Since owing to and , we establish the theorem.
Lemma 1. Under the assumptions in Theorem 1, we have
Proof of Lemma 1. Due to
, for any
, there exists
such that
, where
is a compact neighborhood of
with
for all
. For a positive real number
, we partition
into a finite number, say,
of subsets
with diameter less than
. Set
Let be an integer such that , where and is the largest integer that does not exceed x. We divide the interval into parts by the points with .
Then, for any points
in
, we have
Putting
, for
, we can express
with
and
and
are the same as
and
, with
and
replaced by
and
,
, respectively.
To show , we verify that , . Below, we only provide the proof for the cases of , since the cases of can be handled similarly.
We first deal with
. By the mean value theorem, we can see that
is no more than
with
Note that the term in (
16) is
due to Lemma 1, and
where
is a real number between
and
. Using an argument similar to that in (12), we can see that
, which can be made arbitrarily small by taking sufficiently small
. Hence, we get
.
Next, because
, we can write
Hence, it remains to show that
Put
and
. Note that
forms an array of martingale differences. Then, we get
and further, applying Rosenthal’s inequality (Hall and Heyde [
18], p. 23),
By the mean value theorem, we can have
for some
and
between
and
, so that
, by using an argument such as that in (
12). Therefore, since
, we have
by (
20). This, together with (
19), validates the lemma. ☐
Lemma 2. Under the assumptions in Theorem 1, for every ,
we havewhere .
Proof of Lemma 2. The lemma can be proven by using (C2)∼(C4) and the second-order Taylor’s expansion theorem centered at x and y. We omit the details for brevity. ☐