Y
i
in the analysis to obtain a vector Y
i
, estimate rk{z}, and
sup
z
rk{z} in the model Y
i
= Z
i
Z
i
1
VX
i
+ U
i
, 1.7
assuming that the covariance matrix of U
i
is nonsingular, and then obtain estimates of rk{
θ z} and sup
z
rk{
θ z} by adding 1.
We shall apply this estimation procedure to estimate local and global ranks in the demand system constructed from the CEX
and the ACCRA data sets. Related estimation of local ranks in a demand system given by a nonparametric model can be found
in Fortuna
2008 .
1.3 Outline of the Article
The rest of the article is organized as follows. In Section 2
, we state the assumptions which are used in connection to the
model 1.1
. In Section 3
, we introduce an estimator for the ma-
trix z based on 1.4
and, in particular, state its asymptotic normality result. Local and global rank tests for the matrices
z
are studied in Sections 4
and 5
, respectively. Simulation experiment is presented in Section
6 . Estimation of local and
global ranks in a demand system can be found in Section 7
. Some proofs are postponed till Appendices
A and
B .
2. ASSUMPTIONS AND OTHER NOTATION
We shall use the following assumptions on the variables X
i
, Z
i
, and U
i
, on the functions θ and v, and on the kernel K.
Assumption 1. The function K is a symmetric kernel on R
q
of order s, that is, K has a compact support, is bounded and satisfies the following conditions: i
R
q
Kz dz = 1 and ii
R
q
z
b
Kz dz = 0 for any b ∈ N ∪ {0}
q
satisfying 1 ≤ |b| s, where z
b
= z
b
1
1
· · · z
b
q
q
with z = z
1
, . . . , z
q
, b = b
1
, . . . , b
q
,
and |b| = b
1
+ · · · + b
q
. There are many possible choices for such kernels K. In the
simulation experiment and the application below, when q = 1, we use the popular Epanechnikov kernel Kz = 31 − z
2
4, for |z| ≤ 1, of the order s = 2. The kernel Kz = 157z
4
− 10z
2
+ 332, for |z| ≤ 1, of the order s = 4, is another pos- sibility. For a given kernel K, we shall also use the following
related kernels throughout the article: K
p
z = |Kz|
p
K
p p
, 2.1
where K
p p
=
R
q
|Kz|
p
dz ,
Kz =
R
q
KvKz − v dv
2.2 the so-called convolution kernel. The notation K
h
z
will stand for a scaled kernel h
−q
Kzh. We shall also use obvi-
ous combinations of the notation above such as K
2,h
and so on. Assumption 2.
Suppose that X
i
, Z
i
∈ R
p
× R
q
, i = 1, . . . , N
, are iid random vectors such that the support H
z
of Z
i
is the Cartesian product of compact intervals H
z
= [a
1
, b
1
] × · · · × [a
p
, b
q
] and Z
i
is continuously distributed with a density pz. Suppose that the density pz has s continuous bounded deriv-
atives on H
z
, and that pz is bounded away from zero in the
interior of H
z
. Assumption 3.
Suppose that the error terms U
i
, i = 1, . . . , N,
are iid random vectors, independent of the sequence X
i
, Z
i
and such that EU
i
= 0 and EU
i
U
′ i
= , 2.3
where is a m × m positive definite matrix. Suppose also that E|U
i
|
u
∞ where u ≥ 4. Local rank tests can also be obtained under a weaker, het-
eroscedasticity assumption on U
i
, that is, EU
i
U
′ i
|X
i
= x, Z
i
=
z = x, z. Under the stronger condition 2.3
, the limit co- variance matrix in the asymptotic normality result for
θ z has a
convenient Kronecker product structure. The proof of the global rank tests uses this Kronecker product structure and hence the
stronger condition 2.3
. Regarding Assumption 2
, the condi-
tion of bounded support of Z
i
can be removed by requiring fi- nite suitable moments of functions ψ
k
defined below. Assumption L4.
In the case of local rank tests, the function θ
: H
z
→ R
mn
is such that each of its component functions has s
continuous derivatives on H
z
. Assumption G4.
In the case of global rank tests, suppose that
the component functions of θ z are real analytic on H
z
see the discussion below.
Assuming smoothness i.e., continuity of derivatives of some
order of the function θ z is standard for varying-coefficient models see the references provided in Section
1.1 . The as-
sumption of analytic θ z for global rank tests is less common and requires further explanation. According to one possible de-
finition, a function f is analytic if its Taylor series converges to the function f at a neighborhood of each point. We assume
analyticity just in order to have smoothness of the eigenvectors of some analytic matrices involving θ z. It is well known that
smoothness of a matrix is not sufficient to have smooth eigen- vectors see, e.g., Kato
1976 ; Bunse-Gerstner et al.
1991 .
To state the next assumptions, let φ
k
z = E vX
i
vX
i ′
k
|Z
i
= z
=: ψ
k
z pz
−1
, k = 1, 2, 4,
2.4
where, for a matrix A, we write A
2
= AA
′
and A
4
= A
2
A
2 ′
. We write φ and ψ for φ
1
and ψ
1
, respectively. Assumption L5.
The matrix φz is positive definite, and has
s continuous derivatives on H
z
. The matrices φ
2
z
, φ
4
z
are continuous on H
z
. Assumption G5.
In the case of global rank tests, suppose in
addition that the components of ψz are real analytic. If vx = v
1
x, . . . , v
n
x
′
, the conditions on φ
2
z and
φ
4
z in Assumption
L5 are effectively those on
E
4 k=1
v
i
k
X
i
Z
i
= z
and E
8 k=1
v
i
k
X
i
Z
i
= z ,
Downloaded by [Universitas Maritim Raja Ali Haji] at 23:08 11 January 2016
where i
1
, . . . , i
8
are arbitrary indices from 1, . . . , n. In addition to
2.4 , we shall also use the d × d matrix z such that
z
−1
= α
′
ψ z
−1
α, 2.5
where α is defined in 1.2
. Observe that, if the matrix ψz is positive definite, then z is positive definite and, if ψz is
real analytic, then z is also real analytic.
The global rank tests will be constructed over a subset H of H
z
satisfying the following assumption. Assumption G6.
Suppose that H is a proper, compact subset of H
z
. The condition on properness in Assumption
G6 is neces-
sary to avoid boundary effects when using kernel smoothing. A number of boundary corrections are available conveniently
summarized in Karunamuni and Alberts 2005
but using them here would unnecessarily complicate the technical level of the
article. Moreover, in simulations reported below, our global rank test appears to work quite well even ignoring the boundary
issue.
Assumption G7. Suppose that q = 1, 2, or 3.
As seen from the proofs for global rank tests, the global rank test statistic has a leading bias term of the order h
2−q2
. The bias term is thus negligible only under Assumption
G7 , and would
have to be corrected properly in higher dimensions q. 3.
KERNEL–BASED ESTIMATOR Let K be a kernel defined in Assumption
1 of Section
2 . We
shall use throughout a kernel-based estimator θ z
for the ma-
trix θ z defined by 1.4
. The estimator θ z
can also be ex- pressed as
θ z = YD
z
v
′
vD
z
v
′ −1
=: 1
N YD
z
v
′
ψ z
−1
, 3.1
where Y = Y
1
· · · Y
N
, v = vX
1
· · · vX
N
, and
D
z
= diag{K
h
z − Z
1
, . . . , K
h
z − Z
N
}. A more general esti-
mator for θ z can be defined based on the idea of local polyno- mial regression Fan and Gijbels
1996 ; Fan and Zhang
1999 .
We work with the estimator 3.1
for proof simplicity, espe- cially in the context of global rank tests.
Define the estimator
z for the submatrix z of θ z by
using 1.2
as
z = θ zα
. The following result establishes the asymptotic normality of
z
. The proof is standard and can be found in Donald, Fortuna, and Pipiras
2010 .
Theorem 1. Under Assumptions
1 –
3 ,
L4 –
L5 of Section
2 ,
we have, for fixed z in the interior of H
z
, √
Nh
q
vec z − z →
d
N 0,
z, 3.2
as N → ∞, h → 0,
Nh
q
→ ∞ and
Nh
q+2s
→ 0, 3.3
with
z = z
−1
⊗ K
2 2
, 3.4
where the matrix z is defined by 2.5
. Suppose in addition that N
1−2u
h
q
ln N → ∞ where u appears in Assumption 3
.
Then, the limiting covariance matrix
z in 3.4
can be esti- mated consistently by
z = z
−1
⊗ K
2 2
, 3.5
where
z
−1
= α
′
ψ z
−1
α with
ψ z
−1
appearing in 3.1
, and
= 1
N p
H N
i=1
Y
i
− θ Z
i
vX
i
× Y
i
− θ Z
i
vX
i ′
1
{Z
i
∈H}
3.6 with H satisfying Assumption
G6 ,
p
H
= N
−1 N
i=1
1
{Z
i
∈H}
and 1
A
denoting an indicator function. Remark 1.
Under slightly different assumptions, includ-
ing the heteroscedasticity assumption EU
i
U
′ i
|X
i
= x, Z
i
=
z = x, z, the asymptotic normality result for θ z
is also proved in Li et al.
2002 see also theorem 9.3 in
Li and Racine 2007
. The resulting covariance matrix is
ψ z
−1
EvX
i
vX
i ′
X
i
, Z
i
|Z
i
= zψz
−1
K
2 2
in the case m = 1.
The restriction of Z
i
to H appears in 3.6
for simplicity in or- der not to deal with the boundary effects in the proofs. It is quite
likely that the restriction can be removed but at the expense of a significant expansion of the proof. At least from a practical
perspective, we do not see much difference in our simulations between using and not using the restriction to H.
4. LOCAL RANK TESTS
We consider here the hypothesis testing problem of H :
rk{z} ≤ r against H
1
: rk{z} r, where z and r are fixed.
Since z
is an asymptotically normal estimator of z and the related covariance matrix
z can be consistently esti-
mated by
z in
3.5 Theorem
1 , this problem can be read-
ily addressed by one of the rank tests available in the literature. Four such tests, mentioned in Section
1.1 , are the LDU-based
test, the minimum-χ
2
, the SVD-based test and the ALS-based test. We recall below the minimum-χ
2
test only because its in- tegrated version will be used for global rank tests.
Minimum-χ
2
rank test. Applied to the matrix
z with the
covariance matrix z
, the minimum-χ
2
test is based on the statistic
Tr, z = Nh
q
min
rk{}≤r
vec z −
′
z
−1
vec z −
= Nh
q
K
−2 2
m−r i=1
λ
i
z,
4.1 where 0 ≤
λ
1
z ≤ · · · ≤ λ
m
z are the ordered eigenvalues of
the matrix
Ŵz =
−1
z z
z
′
. 4.2
The last equality in 4.1
is standard for the covariance matrix
z having a Kronecker product structure, and can be proved
as, for example, theorem 3 in Cragg and Donald 1993
. For the same covariance matrix
z
, the test statistic 4.1
also coincides with that used in the SVD-based rank test.
Downloaded by [Universitas Maritim Raja Ali Haji] at 23:08 11 January 2016
The next result follows from Cragg and Donald 1997
and Robin and Smith
2000 . In the context of linear regression,
it goes back to Anderson 1951
. Let Y
a×b
be a a × b matrix with independent N 0, 1 entries and set X
a,b
= Y
a×b
Y
′ a×b
. Let also λ
1
X
a,b
≤ · · · ≤ λ
a
X
a,b
be the ordered eigenvalues of the matrix X
a,b
. The notation χ
2
k below stands for a χ
2
- distribution with k degrees of freedom, and a stochastic domi-
nance ξ ≤
d
η means that Pξ x ≤ Pη x for all x ∈ R.
Theorem 2. Under the assumptions of Theorem
1 , we have:
i when r rk{z},
limp
Tr, z = +∞,
4.3
ii when r ≥ rk{z} =: lz,
limd
Tr, z =
m−r i=1
λ
i
X
m−lz,d−lz
≤
d
χ
2
m − rd − r, 4.4
where the inequality ≤
d
becomes the equality =
d
for r =
rk{z}.
Theorem 2
can be used to test for H : rk{z} ≤ r against
H
1
: rk{z} r in a standard way. The resulting local rank tests can be used to estimate rk{z}, for example, by using a
sequential testing procedure see, e.g., Donald 1997
; Robin and Smith
2000 .
5. GLOBAL RANK TESTS
We are interested here in global rank tests, that is, the hy- pothesis testing problem of H
: sup
z∈H
rk{z} ≤ r against
H
1
: sup
z∈H
rk{z} r, where r is fixed and H is as in As-
sumption G6
. Let Tr, z
be the minimum-χ
2
statistic defined by
4.1 and used in the local rank tests. In view of Theorem
2 ,
for global rank tests, it is natural to consider a test statistic based on
H
Tr, z dz.
5.1 The idea here is that, under H
, the term 5.1
is expected to be O
p
1, and under H
1
, it is expected to converge to +∞ in prob- ability. [Instead of
5.1 , other global characteristics could be
considered such as max
z∈H
Tr, z
and, in fact, could be dealt with in the same way as
5.1 by a “linearization” argument em-
ployed in the proof of global rank tests below.] A more precise asymptotics of
5.1 needs to take into account proper centering
and normalization. Hence, consider T
glb
r =
H
Tr, z dz − |H|m − rd − r
h
q 2
|H|
12
√ 2m − rd − r
K
2 2
K
2
5.2 with |H| =
H
dz
, which will play the role of the global rank test statistic.
The next result establishes the asymptotics of the statistic T
glb
r . We write lim supd
ξ ≤
d
η if lim sup P
ξ x ≤ Pη
x for all x ∈ R.
Theorem 3. Under the Assumptions
1 –
3 ,
G4 –
G7 of Sec-
tion 2
, and when Nh
2q
ln
3
N → ∞, N
1−2u
h
q
ln N → ∞, 5.3
Nh
q 2+s
→ 0, we have:
i when r sup
z∈H
rk{z},
limp T
glb
r = +∞, 5.4
ii when r ≥ sup
z∈H
rk{z},
lim supd T
glb
r ≤
d
N 0, 1, 5.5
where lim sup becomes lim and the inequality ≤
d
becomes the equality =
d
when r = rk{z} for all z ∈ H.
The proof of Theorem 3
is outlined in Appendix A
through a sequence of results, their application being summarized to the
end of the appendix and with their detailed proofs being moved to Donald, Fortuna, and Pipiras
2010 . It can be used to test for
H : sup
z∈H
rk{z} ≤ r against H
1
: sup
z∈H
rk{z} r in a
standard way. Several remarks regarding global rank test and its statistic are in place.
Remark 2. Observe that the term |H| appearing in
5.2 twice
is consistent with a linear transformation of the data Z
i
. For example, when m = 1 and H = [a, b], we have
b a
Tr, z dz = b − a
1
T
∗
r, w dw, 5.6
