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working paper
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economics
OPTIMAL INSTRUMENTAL VARIABLES
ESTIMATION FOR ARMA MODELS
Guido Kuersteiner
No.
99-07
March, 1999
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
OPTIMAL INSTRUMENTAL VARIABLES
ESTIMATION FOR ARMA MODELS
Guido Kuersteiner
No.
99-07
March, 1999
MASSACHUSETTS
INSTITUTE OF
TECHNOLOGY
MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
50
riTUTE
10GY
R 1 4 1999
—
Optimal Instrumental Variables Estimation
for
ARMA
Models
By Guido M.
In this paper a
new
class of
Kuersteiner 1
Instrumental Variables estimators for linear
ARMA models is developed. Previously, IV estimators based on lagged observations as instruments have been used to account
for unmodelled MA(q) errors in the estimation of the AR parameters. Here it
is shown that these IV methods can be used to improve efficiency of linear time
series estimators in the presence of unmodelled conditional heteroskedasticity.
processes and in particular
Moreover an IV estimator for both the AR and MA parts is developed. One
consequence of these results is that Gaussian estimators for linear time series
models are inefficient members of this IV class. A leading example of an inefficient member is the OLS estimator for AR(p) models which is known to be
efficient under homoskedasticity.
Keywords:
ARMA,
conditional heteroskedasticity, instrumental variables, efficiency lower-
bound, frequency domain.
1
Address:
MIT
gkuerste@mit.edu.
Dept.
of Economics,
50 Memorial Drive,
Cambridge,
MA
02142,
USA. Email:
1.
Introduction 2
This paper considers instrumental variables (IV) estimators
for linear
time series models.
framework has been studied by Hayashi and Sims (1983), Stoica,
Soderstrom and Friedlander (1985) and Hansen and Singleton (1991, 1996). In these papers
efficient estimation of autoregressive roots under the presence of moving average errors has
been analyzed. The moving average part of the model is not estimated but rather treated
Efficient estimation in this
as a nuisance parameter.
observations.
It is also
The
class of instruments
assumed
restricted to linear functions of past
is
in this literature that the innovations are conditionally
homoskedastic.
Here
it is
shown that the same
class of
IV estimators based en
linear functions of past
observations can be used to improve efficiency of estimators for linear time series models in
the presence of unmodelled conditional heteroskedasticity.
this
paper
is
consequence of the results of
that standard estimators of linear process models based on Gaussian Pseudo
Likelihood functions are inefficient
heteroskedastic.
inefficient
A
GMM
estimators
This means in particular that
GMM estimators
if
if
OLS
the innovations are conditionally
estimators for AR(p) models are
the innovations are heteroskedastic.
In addition the paper extends the current literature in two directions.
estimator for general linear models, including
MA(q)
parts of
under the assumption of conditionally heteroskedastic innovations. Second,
IV estimators with
filters
First,
ARMA models,
is
an IV
introduced
for the class of
paper derives exact functional forms of optimal
Hansen and Singleton (1991) for a simpler estimation
linear instruments the
of the type developed in
shown how the filters depend on fourth order cumulants of the innovation
and the impulse response function of the underlying process. This formulation
allows to give exact conditions on the distribution of the error process under which optimal
problem.
It is
distribution
instrumental variables estimators are feasible.
optimal weight matrix
The
results in this
is
A
detailed analysis of the properties of the
provided.
paper are presented
tions driving the linear process.
for the case of
martingale difference innova-
Alternatively similar formulas with the
same
efficiency
implications could be obtained under the weaker assumption of white noise innovations.
In this case the space of permissible instruments
is
generated by
all
linear combinations of
past observations and the efficiency bounds developed here are identical to the bounds of
Hansen (1985) and Hansen, Heaton and Ogaki (1988). In the case of martingale difference
innovations Hansen's bounds are based on a larger class of instruments and are therefore
tighter than the bounds obtained here.
A detailed analysis of the linear class of instruments is justified by the fact that the
Gaussian estimators are a member of this class. Any IV procedure dominating the Gaussian
estimators therefore has to contain these linear instruments in the set of
all
instruments
2
This paper is partly based on results in my Ph.D. dissertation at Yale University, 1997. I wish to
thank Peter C.B. Phillips for continued encouragement and support. I have also benefited from comments
by Donald Andrews, Jinyong Hahn, Jerry Hausman, Whitney Newey, Oliver Linton, Chris Sims and
participants of the econometrics workshop at Yale and the University of Pennsylvania. I benefited from
helpful comments of three anonymous referees on an earlier paper that inspired the current extensions. All
remaining errors are my own. Financial support from an Alfred P. Sloan Doctoral Dissertation Fellowship
is gratefully acknowledged.
used.
The main
technical difficulty in extending previous procedures to the estimation of
the moving average case
lies
in the consistency proof.
We
give a general characterization
We
of instrument processes that lead to consistent estimators.
optimal instrument
it
is
satisfies these criteria.
we do not
In this paper
focus on implementation issues. For most parts of the analysis
assumed that the optimal instrument
is
known a
a procedure for estimation of the weight matrix
feasible
procedure
then establish that the
is
priori.
It is clear
that in practice
needed. In Kuersteiner (1997) such a
developed under stronger assumptions about the joint distribution
is
these assumptions are satisfied then the procedures developed
of the error process.
If
in Kuersteiner (1997)
can be directly applied to the present context.
are provided for this case.
We
also give
an exact formula
for
Explicit formulas
a feasible version of the
optimal procedure under the more general conditions analyzed in this paper. In this case
the feasible estimator depends on a bandwidth parameter.
of this parameter for the estimator to maintain
A
maximal
rate of expansion
order asymptotic properties
its first
is
provided. However, optimal bandwidth selection procedures are beyond the scope of the
paper.
The paper
is
organized as follows.
Section 2 introduces the assumptions about the
innovation sequence and specifies the inference problem.
Section 3 develops an instru-
and proves consistency
In Section 4 it is shown
mental variables estimator for estimation of linear process models
and asymptotic normality of estimators
how
for the
ARMA
class.
to factorize the asymptotic covariance matrix of this class of instrumental variables
estimators in a
way
to obtain a lower bound. Section 5 uses the lowerbound to obtain an
IV estimator depending on the data periodogram and
an optimal frequency domain filter. Proofs of some important lemmas are contained in
Appendix A while the proofs of the results in the paper are contained in Appendix B.
explicit formulation of the optimal
2.
Model
Specification
The econometrician observes a
following mechanism
finite stretch of
data {yt}"=1 which
is
generated by the
oo
yt
= J2c(p,j)e
t
-J
(2.1)
3=0
for a
given
(3
=
functions c(.,j)
Rd
and
are known.
(3
<E
c(/3,j)
We
impose the identifying restriction
The innovations
:
Rd
x
N—
>
R.
The parameter
define the lag polynomial
c(/3, 0)
=
C(0,z)
/?
=
is
unknown but
Yl'jLo C (P d)
z^
the
an d
1.
assumed to be a martingale difference sequence. The martingale
difference property imposes restrictions on the fourth order cumulants. These restrictions
can be conveniently summarized by defining the following function
e<
are
It
should be emphasized that a (s,r)
=
«..-
equal to the fourth order cumulant for s,r
is
" (a r)
( Q r7 = a
[
We
u+ a
'
.
,
(2.3)
v
;
r
if s
Let
assume that we have a probability space (Cl,T,P) with a
Q T+i C
such that Tt
(j- fields
T Vi.
The doubly
infinite
=
{e J }^_ 00 generates the filtration Tt such that jFt
{e 4 }^._ 00 are
summarized
Assumption Al.
E (e 2
surely, (Hi)
°°>
0.
^^
=
4
(r, r)
>
M Y1T=
Remark
E~
l
l
a
=
stationary and ergodic,
(s,r)\
\
o\ almost surely where a 2
=B <
co, (vi)
E (s 2 e 2_
Assumption Al(ii) could be relaxed
1.
a(et,et-i,
...)•
The assumptions on
E (e
t
as follows:
e^ is strictly
('ij
Tt-\)
\
filtration Tt of increasing
sequence of random variables
>a some a>
s)
to
(ii)
not constant,
is
Ee
t
£s
=
for
\
E
for all
t
=
Tt-\)
(iv)
^
(ef)
s.
s at
the cost of
more complicated expressions for the optimal instruments. Assumption
that the second moments are conditionally heterogeneous. A consequence
slightly
states
E (s 2 et-
almost
— a2 <
Al(iii)
is
that
s^r^O
and depend on s for s — r ^ 0.
Assumption (v) limits the dependence in higher moments by imposing a summability
condition on the fourth cumulants. The assumption is needed to prove invertibility of the
infinite dimensional weight matrix of the optimal
estimator. Assumption (vi) is not
restrictive. Its only purpose is to guarantee that the innovation distribution does not have
all its mass concentrated at zero.
terms of the form
s £t-r)
are nonzero for
GMM
Remark
2.
It
can be checked that processes in the
ARCH, GARCH, EG ARCH and
stochastic volatility class satisfy the assumptions, provided the parametrization implies
that
is
Eel <
known from Milhoj
oo. It is well
(1985) or Nelson (1990) that this condition
satisfied only if additional restrictions limiting the
temporal dependence of conditional
variances and/or the innovation distribution are imposed on the parameter space.
By
definition of the conditional expectation operator,
<r t
Tt-\ measurable.
is
As-
sumption (Al) implies that e 2 is strictly stationary and ergodic and therefore covariance
stationary. It should be emphasized that no assumptions about third moments are made.
In particular this allows for skewness in the error process.
For the special case of an
ARMA(p, q)
process, the lag polynomial has the familiar
rational form
C(/M =
with 9(z)
=
Let gyy (P,X)
1-01-z-..
=
|C(/3, e
.
-0 q zi and
lA
)
|
where
4>{z)
\z\
=
^
= \-faz-.
(zz*)
1
'
.
(2.4)
=
.~4> zP and/?'
p
for z
G
C
and
z*
is
{^,...,^,6^
...,6
q
).
the complex conjugate
2
of
z.
of y t is given by fyy ((3,\) = %^9yy{P, A).
are needed to insure identification of the model and for
Under Assumption (Al), the spectrum
Further restrictions on C(0, e lX )
The necessary assumptions are
(1976), and Deistler, Dunsmuir and
consistency and asymptotic normality of the estimators.
discussed in
Hannan
Hannan
(1978).
(1973),
As shown
Dunsmuir and Hannan
in these articles, a careful distinction
between convergence of
the parameters in c(/3,j) and the structural form parameters
needed. Consistency proofs
is
An
typically establish convergence in the pointwise topology.
identification condition
is
then needed to obtain convergence in the quotient topology.
Some
of the results of this paper are presented for the general formulation C((3,z).
At some points however a specialization
sharper results. This
is
to the
ARM A
case
made
is
in order to obtain
especially the case for the consistency proof. In that case abstract
high level assumptions can be
made
precise for the specific functional form of the
ARM A
model.
C([R d xN],R)
In the general case the functions c(0,j) G
are restricted to satisfy the
following additional constraints.
Assumption Bl. Let C(0, z) =
=
defined by
that
(3
open
is
G 0. The
c(/?,0)
=
1.
Rd
g
{(5
in Mr.
zJ
Yl'jLo c {PiJ)
\C(0,z)\~
2
±
for
'
\z\
The parameter space
<
\C{p,z)\
1,
Let the compact closure of
in
Rd
2
^
for
that
£~
|j|
\c(0,j)\
<
oo
<
\z\
1}.
Assume
be denoted by 0. Assume
coefficients c([3,j) are twice continuously differentiate in
We require for (5 G
a subset ofM. d
is
and££o
G
\j\
and
for all j
\-^c(0,j)\
<
oo.
Assumption B2. For all (3 G 0, gyy {Po, A) ^ 9yy(P, A) whenever j3 ^ fi Q for some subsets
L C [— 7r,7r] with nonzero Lebesgue measure. Let 8Q = 0\0 and consider any convergent
sequence /3n € 0, n -» P G 90. Then liminf n J\ C-\p n ,e~ lX )f(e iX )d\ >
for some
complex valued f(z) such that f(z)
=
EfcL-oo
Assumption B3. For a neighborhood U
in A
G
[-7r,7r]
Remark
and
€
of /3
h zk with Y2T=-oc IAI
,
UC
<
°°-
2
©o, d g yy (P, X)/dpdp
is
continuous
U.
Assumption (Bl) implies that the functions g yy (P,X) and dgyy (p,\)/dp are
The Lipschitz condition also implies that g~y (P,X) is Lipschitz
continuous on closed subsets ofQ and therefore that -^ \ng yy (P, A) is Lipschitz continuous
on closed subsets ofQ.
3.
Lipschitz continuous.
Remark
4.
Assumption (Bl)
is
stronger than C2.2 in
Dunsmuir (1979) where on the other
is assumed. The stronger summability restrictions are
needed to justify approximations based on the innovation sequence.
hand conditional homoskedasticity
The assumptions
specified here are sufficient to identify the parameters
P
in
C(P,
e
lX
).
lX
For specific functional forms of C(P, e ) the assumptions can be made more explicit. A
leading example is the
A model where the identifiable subset of R d can be described
ARM
more
The
accurately.
the case of an
following
Assumption
Assumption B4. Let C(P,z) =
defined by
common
is
equivalent to the previous assumptions for
ARM A model.
=
{0 G
zeros, 6 q
^
0,
R d \4>{z) ^
4>
p
^
0}.
9 (z) /4>(z).
for
\z\
<
1,
The parameter space
9(z)
+
Let the compact closure
for
ofQ
\z\
in
<
Rd
is
1
,
6 (z)
a subset of
,
<j>(z)
Rd
have no
be denoted by 0.
Remark
5.
sumption
(B4) satisfy the topological properties required in
to
show
that
Dunsmuir and Hannan (1978) show
Deistler,
ARMA models in 6
all
satisfy
ments of (Bl). The only new condition
C(P,
e
lX
)
/^ C~
liminf„
is
Q
and
defined in AsAssumption (Bl). It is easy
the summability and differentiability requirethat
e~ iX ) f(e lX )d\
1
(/3 n
,
>
0.
Since
can be zero on the boundary of the parameter space we can not expect the inteon the boundary in general. The condition requires that the behavior of
gral to be defined
C~
l
(f3 n
,e
lX
)
not too irregular as
is
>
6 dO. For the
/?
enough to consider the MA(q)
satisfied. It is
diverges to infinity as
to a constant if
iAfc
The
case.
unity.
To see
class this condition is
lX
1
lX
J*n 8 n (e~ )~ f(e )d\
integral
this let £- n
lX
n?=i(l - Z jn e )
=
ARMA
more than one of the roots of 6 n (e lX ) approach unity and converges
one root approaches
x
such that B n (e* )
£r=-ooAe
—
j3 n
which leads to
denote the roots of 6 n (e lX )
E£=o^
™th ^(e^)" = nj=i
iX
iX
f_J n {e- )^f(e )d\ = E£=o
1
Afc
'
and
f^) =
••££=o^--&/V
IV estimator results will first be obtained for the general
will then be shown that high level assumptions needed for these
the case when Assumptions (B1-B3) are specialized to (B4).
In the following analysis of the
linear process case.
It
results are satisfied for
3.
Instrumental Variables Estimators
In this section a class of instrumental variables estimators
introduced.
is
are constructed from linear filters of lagged innovations e t
formulation would be to allow for linear
filters
An
.
The instruments
alternative, equivalent
of the observable process yt- Estimators
by Hayashi and Sims (1983), Stoica, Soderstrom and
of this form have been proposed
Friedlander (1985) and Hansen and Singleton (1991).
Restricting the instruments to the linear class has implications for the efficiency properties
of the estimators.
It rules
out conditional
GLS
transformations and
ML estimators for
parametric cases. Linearity, on the other hand, leads to a tractable theory. Introduce the
space of absolutely summable sequences l l such that x G I 1 if ]P \xj\ < oo for x = {xj}°°_ 1
.
Define the set
A
of sequences of vectors
A = ia=
where
[.] fc
{aj}f=l
:
a; G
G
clj
Rd
,
Rd
such that
{[oj]*.}^ G
denotes the k-th element of a vector.
We
1
for all
I
define
zj
G
1
Rd
<
k
< d\
as
oo
zt
= ^akCt-k
a.s.
fc=l
for
o G A, a
fixed.
The instruments
satisfy the orthogonality condition
E[(Csince
C _1 (/3
in the
,L)yt
=
time domain.
e t from (2.1).
-1
If
C
(/?
,L)
1
((3
,L)y t )z t }=0
(3.1)
The estimator based on
is
of infinite order as
is
this condition
the case for
is
constructed
MA(q) models
a
sample analog to
needs to be based on an approximation. Such an approximation
(3.1)
can be conveniently analyzed in the frequency domain. It should be stressed however that
_1
(/3,z) be
the estimator is set up in time domain. Let the expansion of the polynomial C
C _1 (/3, z) =
'
The sample analog
j
Y1T=qCj z
'
moment
of the
n
restriction
is
then given by
t-1
CUM = ~I>5^Vt-j
a € A. From
for all
we
(3-2)
j=0
(=1
approximated as well. Discussion of this
where an optimal instrument is considered. For the time
therefore assumed that z t is known.
(3.1)
see that zt has to be
issue will be delayed to Section 5
being
it is
In the frequency
domain the analog of
P C-
(3.1)
iX
l
(l3o,e-
J — IT
where fyz (X)
=
Yi'jL-oo
lyzHY* 3 and
G(P,a)
=
(2ir)-
lyziJ)
=
)fyZ (X)dX
= Ey
f C"
1
is
1
t
We
z t -j.
set
x
^, e^ )fyz (X)dX.
J —IT
Note that fyz (X) typically
lX
a complex vector valued function fyz (X)
(X)dX
is real valued.
)fy Z
is
note that f*n C~ ((3,e
We introduce discrete Fourier transforms of the data defined as
1
and
for the
instrument as
=
oJ n ,z(X)
^>n,y(X)co n ,z(—X). It is easy to
-4=
Y^t=i Zte~
G n {(3,a)
check that
G n ((3,a) =
(27T)"
r— Cr
1
J
We
follow
Hansen (1982)
-
The
cross
u nt y(X) =
lX
{f3,e-
is
— Cd
>
-4=
periodogram
defined in (3.2)
l
[— 7r,7r]
is
.
Also
^"=i Vte~
In ,yz(X)
ltX
=
identical to
)In yz {X)dX.
,
IT
in defining the
/3 n
ltX
:
estimator
f3 n
=argmin||G n (M||
as the solution to
2
(3.3)
.
/3e0
Consistency arguments are complicated by the fact that the parameter space for linear
time series models usually
is
only locally compact. Standard consistency proofs relying on
compactness can therefore not be applied. Hosoya and Taniguchi (1982), Kabaila (1980),
Taniguchi (1983) are assuming compactness of the parameter space to avoid consistency
problems. Such an assumption
imply that
is
an open subset
is
not valid in the
in M.
d
as
ARMA
was shown by
Stationarity restrictions
case.
Deistler,
Dunsmuir and Hannan
(1978).
Huber (1967) probably
when the parameter space
level
is
is
the
first
reference to discuss consistency of A/-estimators
not compact.
The formulation
there
is
in
terms of low
assumptions on the criterion function and the data generating process which are
not readily adaptable to the present situation. Hannan's (1973) original paper provides a
ARM A
consistency proof for the estimators of an
model without assuming compactness.
Unfortunately, his technique for the Gaussian estimators does not readily generalize to
General consistency results are obtained by
the current context.
Zaman
Pollard (1989) and
(1989).
The
Wu
and
(1981), Pakes
stochastic equicontinuity arguments underlying
these proofs are not applicable in our context due to the discontinuities of the criterion
function on the boundary of the parameter space.
One
of the problems
compactification 9.
is
that the criterion function does not necessarily converge on the
The consistency proof used
here therefore proceeds by establishing
almost sure bounds for the criterion function along convergent sequences in 0.
It is
then
by analyzing convergent subsequences on an outcome
by outcome basis. This method was used by Brockwell and Davis (1987, p. 384) to prove
consistency for estimators for the ARMA model based on quadratic criterion functions.
The details of their proof rely heavily on nonegativity properties of quadratic forms. For
the IV estimators considered here such arguments are not available and a new proof is
presented. We start by making the following assumptions. Unless otherwise stated all
possible to circumvent uncertainty
conditions are for a e A, a fixed.
Assumption Cl. The
sequence of estimators
Assumption C2. Let the sets Bk((3
P The sets Bic (P can be taken as
.
)
=
Y^h=i a k£t-k o..s. where e t -k
sequences {afc}^L 1 such that
Let z t
all
A'
=
I
a €
A
)
for
k
G
= 1,2,
M.
d
is
defined by (3.3).
form a countable local base3 around
...
the set of balls with rational radius centered at
satisfies
liminf
inf
(3 n
Assumption (Al). Let A'
\\G(p n ,a)\\
>
where Bk(Po) c are the complements of Bk(Po). Assume that
Remark
6.
Assumption (Cl)
=
tency proof that \\G n (P n ,a)\\
(C2)
is
is
for k
is
A
However
.
1,2,
1
We show
in the consis-
implied by the assumptions on
a familiar identification condition which makes sure that the expectation of the
terion function is
(3
be the set of
A ^ 0.
the definition of the estimator.
almost surely
=
C
Gn
.
cri-
bounded away from zero outside a neighborhood of the true parameter.
this condition
does not hold for
all
a £ A.
instruments that satisfy the identification condition.
We therefore define the subset A' of
We require that this set be nonempty.
Condition (C2) strengthens Assumption (B2) by requiring that
liminf
||G(/?„,a)||
>
Condition (C2) requires in addition that the identification condition holds on the
entire parameter space. This imposes restrictions on z or a. A complete description of the
set A' is possible for a given parametric class C(/3, z). A characterization will be given for
holds.
t
the
3
ARMA
A
there
case.
collection of
is
a set
B
€
open subsets
B
B
X is called a base if for each open set O C X and each x £ O
B C O. A collection Bx of open sets containing a point x is called a
O containing x there is a B G Bx such that x e B C O. Every metric
of a space
such that x 6
x if for each open set
space has a countable base at each point (see Royden (1988),
local base at
p.
175).
Lemma
Assume
3.1.
m {a.fc}^=1
= argmin ||G n (/3n
[a\, ...,a
/3 n
]
,
(Al), (B1-B3), (C1-C2). Let z t
=
£ A' and e™
is
)||
[et-i,
consistent, (3 n —*
.
= lim™-^ A m ef
,£t-m]
.
Am =
with
a.s.
Then the estimator defined by
almost surely.
(3
Consistency of the IV estimator depends both on restrictions on the parameter space
and the instruments z
t
Assumption (C2)
.
The
restricts the class of allowable instruments.
conditions given are necessarily high level without further restrictions on the function
C(P,L). For practical purposes it is however important to characterize the set of instruments A' leading to consistent estimators. In the case of an ARMA(p,q) model it is
possible to give conditions on the sequences a £ A! This is done in the next proposition.
.
Proposition 3.2. Assume C(@,L) = 9o(L)/4> (L) is an ARMA(p,q) lag operator and
l
the parameter space G satisfies Assumption (B4). Let S = sp {x £
be
4> Q (L)x = 0}
l
the span of linearly independent solutions to the difference equation
A'x = 0} for A = [ax,...] and a £ A. If a £ A
A L = [x £
d = p + q then the following conditions are sufficient for a
1
I
nonsingular and YLkLi a k
row rank,
full
Remark
7.
¥"
AL n S =
Lemma
.
and *£%Ll Q*
shows that
(3.2)
<
then a £ A! IfO
+
° for
ARMA
q
fl
<
<f>
Ad =
with
:
£ A'
p then we need
.
If q
^'
:
(L)x
=
Define
0.
where
and A d
[a 1; ...,a d }'
> p >
A=
[a 1;
to be of
....]
models can be consistently estimated by
instrumental variables techniques provided that the instruments satisfy the specified
re-
The condition Y1T=\ a k ¥" is only needed to avoid problems at the boundary
of the parameter space and can be ignored if Q is restricted to a compact subset ofW*.
strictions.
We now
state additional assumptions that are sufficient to establish a result for the
= d\nC(p,e~ tX )/d/3
= (2n)~ J f](p ,X)e d\. It follows immediately that 6_fc = and bo = 0. Let
= Y^k=i a k e ~ lXk an d define the matrices Pm = [b\, ...,bm A'm = [ai,...,a m and
limiting distribution of
^((3 n —
and
(3
).
Introduce the notation
f](0, A)
lkX
1
6fc
^a(A)
],
a(l,l)+a A
«
<r(l,m)
—
£m.
(3.4)
••
<r(m, 1)
It is
]
easy to check that lim m P'm A m
=
x
(2ir)
a(m, m)
j rj(P
,
+a
4
X)l a (—X)'dX.
The
following conditions
are needed to prove the existence of a limiting distribution of /?„.
Assumption Dl. ^/nG n (0n ,a) =
Assumption D2. Define A" C
that A' n A" / 0.
o p (1)
A
as
.
A" = {a £
A |det J f](P
,
X)l a (-X)'dX
+
0}
.
As-
sume
The
limiting distribution of the instrumental variables estimator is stated in the next
-1
-4
_:l
=
For notational efficiency define lim m IX) <7
(Pm m )
m f2 m .Pm (.Prn m )
_
theorem.
o~
A
(P'
A)~ A€lA(A P)~
l
l
.
This notation
operators on infinite dimensional spaces.
will
P
P
P
'
be justified in the next section in terms of
Theorem 3.3. Assume (Al), (B1-B3), (CI, C2) and (Dl, D2). Let z = lim m _ 00 A'm e™
with A'm = [ai,...,a m
Then the estimator
,e{_ m
{afc}^ G -4'n.4" and e™ = [e -\,
has a limiting distribution given by
defined by j3 n = argmin ||G„(/3 n
t
]
t
,
.
.
.
.
]
)||
M0n - Po) ± N{0,a-\P'A)Proof. See Appendix
Remark
8. If (5 n is
1
X{IA{A'P)-
1
)
B
PML criterion function then the
obtained from minimizing a Gaussian
asymptotic covariance matrix is a~ (P P)~ P QP(P P)~ l Such an estimator therefore
corresponds to an IV estimator where A = P. This shows that Gaussian estimators have the
4
l
.
interpretation of inefficient
TV
GMM estimators when the innovations are conditionally
or
heteroskedastic.
The main result of the paper will now be developed
bound for the covariance matrix
two
in
steps.
We
obtain a
first
lower
a- 4 (PA)- A'flA(A'P)l
This lower bound
in the next section.
is
1
(3.5)
then used to construct an optimal instrumental
variables estimator.
4.
Covariance Matrix Lowerbound
Finding a lower bound
infinite
for (3.5)
poses certain technical difficulties having to do with the
fourth order cumulant matrix
fi
m
,
first
by holding
lated infinite dimensional problem. In particular
operator Q, associated with
We
first
Lemma
Qm
way
in a
discuss the properties of
Qm
4.1. Let
Proof. See Appendix
Qm
Invertibility of
briefly
We
dimensional nature of the instrument space.
f2
be defined as in
m
we
m
investigate the properties of the
and then by looking
fixed
to be defined, has a well
for all fixed
(3.4).
m. This
behaved
done
is
Then, fi" 1 exists for
all
inverse.
in the next
Lemma.
m.
*'
B
for all
at a re-
establish that the infinite dimensional
m
however
is
not enough to show that
Q
is
We
invertible.
review the theory of invertible operators (see Gohberg and Goldberg (1980),
p. 65.
For two Banach spaces B\ and B2 denote the set of bounded linear operators mapping
B\
into £?2 by
L(B\,B2). Then
A
€ L(B\,B2) is invertible if there exists an operator
x € B x and AA~ l y = y for all y G B 2 Let
x G B{\ Then A is invertible if KerA =
A~ G L(B 2 ,Bi) such that A~ Ax = x for all
KerA — {x G B\ Ax = 0} and Imi = {Ax
{0} and \mA = B 2
l
l
:
.
.
:
.
now choose B\,B 2
Following Hanani, Netanyahu and Reichaw (1968) we
spaces whose points are sequences of real numbers denoted by x
{j/ii 2/2, ---}
-
Define the
norm
||x||
2
=
Q^il^il
a^2
)
-
Then
B
norm and is denoted by
bounded under the
2
defined by the infinite dimensional matrix A = (dij),i,j =
that are
||.||
10
is
2
I
.
=
the space of
An
1,2,
....
as linear
{xi,x 2 ,---} and y
operator
all
A
:
such that y
=
sequences
l
2 1—
>
= Ax
2
I
G
is
I
for all
operator
It
I
A
Note that
in
2
x G
.
invertible
is
2
This can be written element by element as
the only solution to
if
Gohberg and Goldberg (1980) it
is thus enough to show KerA =
Consider
Ax =
is
now
KerA L
follows
—
2
A
for
:
I
=Imi
2
/
>
x
,
A
= Y1T a xj f° r an The
= {0,0, ....} and Im^l = 2
*•
i,j
I
= ^°° Xjyj. From Theorem
a Hilbert space with inner product (x,y)
is
I
yi
for a self adjoint
.
11.4
operator A.
selfadjoint.
Qm
the following infinite dimensional operator associated with
.
Define
image for all x G I 2 by b{ = lim m _oo Y1T a i,j xj
where otij is defined in (2.3). In other words Q, is the infinite dimensional matrix such that
x
has the same elements as Q m We
any left upper corner sub matrix of dimension
Q
the operator
component- wise by
its
m m
use arguments similar to the ones in the proof of
Lemma
4.2. Let
fi
m
be defined as in
Then Q G
(3.4).
.
Lemma
(4.1) to establish invertibility.
L(l
,l
2
_1
and
)
f7
exists.
Proof. See Appendix B
Remark
The
9.
fact that the
G I 2 depends on
The interpretation of the summability condition
image of
the summability properties of a(k,l).
is
that the instruments et
Q
square summable,
is
become unrelated
fix
i.e.
moments
in their fourth
,
as the time spread
between them increases.
By
the Closed
-1
O
is
]ij
=uj
follows that
-1
where
[fi
Graph Theorem (Gohberg and Goldberg
_1
bounded,
i.e., ||fi
||
1
=
(1980),
su P||x|| <i ||^~ 1;r ||2
"^
°°
Lemma
we want
4.3. Let
tlm^m =
to establish that the inverse fi^
fl
where I stands for an
m—
>
•
m
be as defined
in (3.4).
1
m
approximates
Define Cl^
tends to
fi
_1
such that
as
infinity.
m—
>
In
oo.
Q^Clm = Im and
Im Vm. Let
wm =
Oas
Theorem X.4.2) it also
Thus sup, \uJij\ < oo
j.
Next, we need to establish properties of the matrix fi" 1 as
particular
-
infinite
nm
(T
4
I
(4.1)
j
dimensional identity matrix. Then
Q^
J
exists
and £7^
:
II
oo.
Proof. See Appendix B
Remark
inverse
fi
10.
_1
Lemma
(4.3)
provides an algorithm to approximate the infinite dimensional
.
d dimensional product of sequence spaces l\ = 2 x ... x I 2 Define the
infinite dimensional matrix P = [b\, ...] by stacking elements of the sequence {&fc}^=i £ ^5Introduce notation for the reverse operation of extracting a sequence form the rows of a
-1
_1
matrix by defining b(P) := {b k }^=1 Define the matrix 2 = (P'n P)
Using this notation we can state our next theorem which establishes a lower bound for
We
define the
I
.
.
.
the covariance matrix.
11
Theorem
a{P'A) e A"
A
any a €
4.4. For
A'
let
=
P
and
[ai,...]
1
l
then the matrix (P'A)~ A'VlA{A P)~
'
1
{P'A)- l A'£lA{A'P)- - {P'n^P)-
where
>
Remark
trix of
for
as previously defined.
If
>
1
stands for positive semi-definite.
Proof. See Appendix
is
Q
and
satisfies
B
11. If a G A'
DA"
an estimator based on
IV estimators
then {P' A)~ l AQ.A{A' P)~ l
a.
However,
ma-
the asymptotic covariance
important to point out that the lowerbound
in the class of all instruments which are linear functions of the innova-
and have an innovation
tion process
it is
is
filter in
A" The
.
construction of the lower
bound does
not involve consistency restrictions for the instruments. In order to construct an efficient
estimator in practice it has to be established that the optimal instrument does in fact
satisfy consistency restrictions.
5.
Optimal Instrumental Variables Estimators
Theorem
(4.4)
immediately leads to the construction of an
optimal instrument
the optimal
is
filter also results in
examples such as the
Theorem
determined by the linear
5.1.
sumption (B4).
satisfies a € A'
Theorem
ARM A class this
Assume C{0,L) —
timator for the
is
.
It is
IV estimator. The
not a priori true that
indeed the case.
and the parameter space
9{L)/(f){L)
A = P'fi -1 then the sequence a — a^P'Q*
A". We will write a(A) € A' n A".
(5.1) together
l
a consistent estimator. However for important parametric
If
(~l
efficient
A = P'Q~
1
filter
with Theorem
(3.3)
and Theorem
1
)
satisfies
(4.4) establish that the
ARMA model constructed with instruments satisfying A'
same type
Q
As-
defined by the rows of
=
P'Sl
-1
IV
A
es-
achieves
Hansen and Singleton (1991) but under the weaker
martingale difference sequence assumptions on et detailed in Assumption (Al).
Feasible versions of the optimal IV procedure have to be based on approximations
Such approximations replace unobserved et by observed
of the optimal instrument z
residuals i t = yt — ]Cj=i c (0O'J)yt-j for t = 1, ...,n where e\ = y\. Feasible versions of e
a lowerbound of the
t
as in
.
t
by substituting f3 for a first stage consistent estimator 0. Gaussian PMLE
procedures which are consistent but inefficient in our context can be used to generate first
are obtained
stage estimators.
Instruments are then given by z t
restriction
= X^=i &j£t-j- The
empirical analog of the
moment
now becomes
t-i
G n (/3,a) =
-^i
11
t=l
An
algebraically equivalent formulation of (5.1)
Gn (0,a) =
(27T)-
1
t
^cfy _ ,
t
is
r C-\0,eJ — IT
12
(5.1)
J
j=0
given by
lX
nm {X)d\
)h{(3 Q ,\)I
where In ,yyW is the data periodogram and the
iX
l
Mi^o.A) = li,{-X)C- {0o^ ) with
h(X)
filter
:
— C
[— n,n]
>
is
defined as
oo
Va^.
WA) =
3=1
The
instrument are given by
coefficients of the optimal
oo
dj
= 22 b k<^kj
fc=l
the Fourier coefficient of the derivative of the log spectral density of y t and u)kj
l
The b^ coefficients have simple interpretations in
the kj-th entry of the inverse Q~
where
is
6fc is
.
an AR(p) model for example they are equivalent
and can therefore be computed easily. It can also be
noted that the Gaussian estimators are obtained by setting a,j = bj
It is shown in Kuersteiner (1997) that a sufficient condition for the validity of the
special parametric models. In the case of
to the impulse response function
approximation
that the coefficients of the instruments satisfy
is
oo
X^'lNfcl <°ofor
fc
= l,...,d.
(5.2)
3=1
The
following
lemma shows
that under strengthened summability restrictions on the
fourth order cumulants Condition (5.2)
estimator of the
ARMA(p,q)
Theorem
Assume C(/3,L)
5.2.
sumption (B4).
symmetry
is
optimal instrumental variables
satisfied for the
model.
=
6(L)/4>(L) and the parameter space
Strengthen Assumption (Alv) to ^2r=i ^27=1
this implies
£~
1
]T~ 1
r \a (s, r)\
= B < oo.
If
A
=
a
s
\
satisfies
As-
= B < oo. By
then a = a^'fT
s r )l
(
>
P'9.~ l
1
)
satisfies (5.2).
G
G
Feasible versions of the optimal estimator are then obtained by replacing
n (fi,a) by
_1
lX
n ((3,a) where in
(/9, e
) and f3 is a consistent
n (0,a) we replace h(P ,X) by ^(A)C
G
first
stage estimate.
The
challenging part
moments
additional restrictions on the
In that particular case
it
is
is
to estimate l^(X) consistently. For a case with
of et this has been done in Kuersteiner (1997).
possible to estimate l^(\) consistently without the need to
introduce bandwidth or truncation parameters.
that in that particular case
In the
fi
_1
is
The
diagonal such that
simplification
a,j
=
comes from the
fact
bj/ctjj.
more general case the elements u^j can be estimated from a sample analog
of
the approximation matrix $1^ defined in (4.1). Using similar arguments as in Kuersteiner
(1997)
it
can be shown that the elements of
estimated as long as m/y/n
—
the truncated estimate of the
The development
»
0.
a,j
this
matrix can be uniformly consistently
From Cl^ we can obtain
coefficients
by setting
a,j
estimates of
=
is
beyond the scope
future research.
13
We
then form
5Zfc=i ^k^kj-
bandwidth selection
paper and will be left for
of a fully feasible estimator requires an optimal
procedure for the parameter m. This
u>kj.
of this
A. Appendix
The
following
Lemma
Lemmas
-
Lemmas
are used to derive the asymptotic distribution of the
A.l. Under Assumption (Al)
for each
mG
{1, 2...}
m fixed,
,
IV
estimators.
the vector
n
1
E
JV(0,fi)
£ t £t-l, ...,£*££-
i=l
with
We
Proof.
a(m,
cr(m,m)
1)
note that individually
Now
ferences.
a(l,rri)
—
'm
&
ct(1,1)+ct 4
=
define Y{
all
show that
y =Q
_1 / 2
t
y
s
and
Rm
G
for all I
t
'm
—
]
=
-
1
Then
t
are martingale dif-
\
we have -j^YL^ Yt =* ^(0,
!)
where now
be
easily evaluated to
is
E{Y
also
efet-iej-
a(l,l)+a 4
cr(l,m)
£*£t-i£t-
2_2
els
-t -m
aim,
cr(m,m)
-E'
Next we note that
t
for
any
£
G
Mm
such that
linearity of the conditional expectation
therefore apply a martingale
the
m
2
ere
c t-i
t
-.2
'
This
1
so that Yt is
Tt-i) =
To show that -4= Yl Yt => N(0,Q) it is enough
such that tl
Q = EY Y(.
>
the terms etSt-k with k
[etet-i, ...,etet-
a vector martingale difference sequence.
to
+ a4
sum ^2
t
Ynt =
-4=
Yl t
Y
t
.
CLT
I i
=
1
1,
^ fixed, £
and the fact that
and Heyde
(see Hall
]
m
,
is
t
fixed
is
CLT
is
<r
4
a martingale by
and
We
finite.
Theorem
1980,
Checking the conditions of the
F
+
3.2,
can
p. 52) to
straightforward and
therefore omitted.
Lemma
Assume
A. 2. Let In yz (A) = u} n ^(\)u> niZ (-X). In<eE (A) is the periodogram of{e\,
,e n }
Assumption (A\) and that y = Y2T=o' l; 3 £ t-3 w^h ^(A) = Sjio V j e_lAj
.
.
.
^
e t satisfy
;
x
t
^
such that YITLq \J\ l^jl < °°so ^ et z t = Y^T=o ai e t-j with a € A. Let q (.) be a function
on [— 7r,7r] —» C with absolutely summable Fourier coefficients {c/j,— oo < k < oo} such
that q (A)
P
asn-t
= Ejt-oo Cje~ iXj Then
v^(27r)
^
In,yz (A) c (A)
n
n,e>0
any
r /n
dA -
,
£2 (A)
C(/?
,
AX (A) dX >V
<e
oo.
Proof. First an expression
-i/2
for
^™
y t e~
lXt
for i?n (A)
=
In yz
^
(X)—In ,ez
(A) ip(X)
is
obtained. Let u^j, (A)
=
be the discrete Fourier transform of the data. Then
u n>y
(A)
= zKAK, £
(A)
+ n-^J^iPj^Unj
14
(A)
(a.i;
.
Unj
where
= ET=l-j £te~ iXt ~ ET=1 £
(A)
«
e_tAt such that
oo
Rn
(A) :=
-
In ,yz (A)
= u> 2
</>(A)In EZ (A)
,
(-A) n"
1
/2
^^e~
2Aj
^
(A)
j=0
Note that
(2tt)
-1
J'
Rn
Then using the Markov
E\fn
-1
Lemma
term
is
aj-et-j
l
it is
YlkLi
enough to consider
/
<
J2fc=i \°k\
<E
A
Then
any
for
0. It
(27T)-
also follows that etZt
oo
oo
fc=l
/=0
m=-oo
V ^^
°° and
£'Jn e2 (A)
is
eR d
A TV (
r*
/
?
dX
,
•'— T
Proof. First note that
oo
1/2
0,
j\ In>EZ (A) dA =
<
l^/l
e t satisfy
^izm\\l\^0
°°
"
Assumption (j4l)and
such that l'i=\,
°°
°°
J^afc/afcai
l_i
/=1
\
\a k
J2
H^o Kl
= w n ,e(A)u; n]Z (— A). Assume
(A)
with a
n 1/2 (27T)" 1
E£o Et=i Em=-oo aki>ic™.£t-k (er-l - £n-M-r)
<2supa^
dA
bounded from
A. 3. Let 7ni£z
= J]~
let 2t
inequality
i?„ (A) ^ (A)
(2ir)
since the last
= n~
dA
(A) ? (A)
;
fc
=l
1
^
1 /2
(2n)~ 1 [*n Jn ez (A)
=1 e t z t such that En
a martingale difference sequence. However z t = Y2kL\ a k&t-k
1
n" 1
,
Lemma (A.l) is not possible.
zj = YT=i a k£t-k such that lo™ z {\) = n"
such that a direct application of
For a fixed
and I™£z (\)
that for
all e
m we introduce
=
u> n ^(X)u>™
>
z
1
(— A). From Billingsley (1968, Theorem
4.2)
1/2
it is
J*=1 z
enough
m e - lAfc
t
to
show
0,
lim
limsup
P
» 1/2 /
nC
2
(A)-/n £2 (A))dA >
e
,
where
/7f
Tl
£'(i^
(A)
-
/„, e2 (A))dA
= n- 1 ' 2 J2
OO
Et
a#tet-k
t=l fc>m
Since Ea^etSt-k
—
it is
enough
to consider
n
n
~ ljE
E E^'
afce ' £t - fc )
t=l fc>m
Applying
The
Lemma
2
- n_1
oo
oo
EEE
i=l
takafaw -»Oasm-4
0o.
j>mk>m
(A.l) then gives the result.
Lemmas
are used in the consistency proof to show that the criterion funcnon zero almost surely when evaluated along any convergent sequence of parameter
estimates that do not converge to the true parameter value.
tion
following
is
Lemma
A. 4. Assumption (Bl) implies that c(@,j)
Zj\c(P,j)\ <coforall(3ee.
15
=
(27r)
-1
JC~
1
(P, A)e
tAj
dA
satisfies
d,
Proof. Since
C' ^^) = C~
1
l
\c((3,j)\=r
From dC~ l {t3, X)/d\ = C~ 2 ((3,
oo
it
follows that
Lemma
u
...,a
1
follows that
it
(27T)-
fdC- 1 (0,X)/dXeiX^dX
1
X)dC{(3, X)/dX and the fact that C{0, X) satisfies
£j
\c((3,j)\
has absolutely summable Fourier coefficients. Rearrangover j then gives the result.
A.5. Assume (Al), (B1-B3), (Cl, C2).
e? =
{a k }^=l € A' and
m },
(A.2)
dC~ (P,X)/dX
and summing
ing (A.2)
{a
1
-it)
{(3,
[e t -l,-
limm _*oo
Then
,£t-m]'
-
=
Let zt
any
for
Xmef
with A'm
=
G 0,
Gn (P,a)
-»
(3
almost surely.
G(/3, a)
Proof. Without loss of generality assume that zt € K. Let Ey z s = j yz (t — s), and
cum(y t ,zs ,y q ,z r ) = K yyzz (t — s,t — q,t — r). Then, form Assumption (Al) and the proof of
Theorem 2.8.1 in Brillinger (1981) it follows that ]T] |7 y2 (j)| < oo and J^sqr K yyzz( s cl> r )\ <
oo. Let Xn = G n ((3,a) — EG n {j3,a). Since EG n ((3,a) — G([3,a) as n — oo it is enough to
show that Xn —
almost surely. This follows from verifying the conditions of Lemma 3
in Gaposhkin (1980). Using the short hand notation Cj = c(/3,j) we have
t
•
>
\
>
>
>
n
n
n
n
"-'£££$:
EXl =
[l y2 {t-r) lyz { q
-s)
t=l s=t+l <j=l r=t+l
+"fyy(t
K
<
n~ 2
~ <lhzz( r ~ S )+ K yyzz {t -
Y—
*
'
2
=
d _t
K
'
V—
n
s,t
(1
*
- q,t -
S,t
- ^)cJ2 <
n
j=—n
r)] C s -tC - r
q
Km'
1
2
with
Kq =
=
and K\
(J2T=-oc
K
Ej=-oo
UyzOn) +
Next consider
°)-
n\
Xn
Xni =
-
nni
X ni
E(Xn -
2
)
for
n/2 < n a
_1
V^
Y^
z—
z—
'
y
c s - t {y t z s
- Ey
+
zs )
t
n
Ew
<
I
«!»«(*» 9
n. First
n
n
ni
V"
z— z—
+
lZT=-oo hyyU)\ET=~oo hzzU)\
c s _ t {y t z s
-
£:y t z s
Km-
{n
- m]
'
s=n\ t=s+\
5=1 t=s+\
with
E
n_1
n
n
Yl
2
5 «-*(Vt^
- ^2/^)
s=ni t=s+l
\
Kon~ J2 J2
2
<
I
2
° s-t
<
2
t=n\ s=t+l
/
Since
E
I
\
\
it
^^ V— V
h??,iL
Z
'
^—
c s _ t (y t z,
-
5=1 t=s+l
(ni
/<i
~
n)
n 2l n\
<
o(l) almost surely
Ki <
Xni
by
2
)
oo such that
<
Lemma
16
K
2
- m)n" 2
(n
3 of
2
A'!(7i- (n
/
E{Xn -
Xn =
<
)
/
follows that there exists a constant
such that
£j/ t z s )
Gaposhkin (1980)
- m))
r)
<
Lemma
A. 6. Assume (Al), (B1-B3), (C1-C2).
ande™ =
Let z
= lim™-^ A m ef
t
Then
Am =
with
any convergent sequence/3n 6
[et-i,
,£t-m]
m {ofc}^! G
9 with p n —> /3 € 9 there exists an event E with probability one such that i) if (3 £ 9 then
for all outcomes in E, G n (/3 n ,a) — G(P,a) and ii) if (3 € 9 then liming ||G„(/? n ,a)|| >
for all outcomes in E.
-4'
[oi, ...,a
,
]
•
for
>
Remark
sequence
(5 n
G
The behavior
12.
It is
.
of n (f3 n ,a) as P n approaches the boundary depends on the
therefore not possible to describe the limit of
n (P n ,a) for all con-
G
vergent sequences. Possible behavior includes convergence to a constant, divergence as a
random variable which can have unit root
random behavior. All we need to show however
is that \\G n (P n ,a)\\ stays away from zero for large enough n with probability one.
The
idea is therefore to distinguish between random and nonrandom limits and to show that
nonrandom limits involve constants that are bounded away from zero.
nonrandom function ofn, convergence
to a limit
or near unit root asymptotics or explosive
Proof,
\\l3
n
For each
i)
>
e
-P\\<6
by continuity of
will
{j3'
,
z)
C -1
(/3,
A) at
f5
Xn (p) = G n (p,a)-EG n (P,a)
€ 9. For
such that
/3
c'
=
c(/3',j).
and define
||/3'
—
letting
Xn
<
=
sup,,
<
such that Yl'jLiJ
loss of generality
Xn = sup,,
\Xn {0)\
,,
,
.
|
c(/?', j
)
<
I
We
oo.
assume z £ K. Let
t
Since
EG n (p',a)
Xn —
show that
to
polynomial
6 the lag
c((3',j)
- G(P,a)\ < e f \fyz (X)\ d\ it is enough
Xn (j) = %%'{ y zt+j - Jyz (-j)
>
-»
almost
t
^
n" 1
sup
'
IgJI
\Xn (j)\ < Kon-
,_,,(£~
^j"
1
2
\X n (j)\
2
[
j=0
||/3'-/3|<«
where if
P\\
Without
G(P',a) and |G(/?',a)
Thus
uq
A
has an expansion with coefficients
use the short hand notation
surely.
>
such that for n
,
|/3'-/3|<5
C~
>
oo and 8
Bup|C~ 1 G9 ,A)-C- 1 09,A)|<€
sup
l
<
there exists an no
and
\j=0
j\.
From
lE'ATnl
< (EX%) 1/2 we
consider
EXl<K*n-*Y:r 2 (EXn (tf).
3=0
Since
EXn {j) 2
<nJ2\lyy(kh z M+lyAkh yz (k)\+nY,\K4(3,k,l)\
j,k,l
for all j there is
consider
Xnni
=
a Xisuch that
sup
\\P
-p\\<6
EX£ <
\Xn (p')
K2T1
where
- Xni (P')\
K-i
= ^K\Kq.
such that
1/2
ni
Xn ni
,
K
(n
-
ni ) (nn,)-
2
1
\
J^J' \Xni {j)\'
J=0
17
For n/2
<
n\
< n
n—j
\.t=max(n 1 -j,l)
ii=0
Now
^(n-ni) 2
(nni)
2
E^Tj
2
\Xni
2
(j)\
<
K
2
(n
-n
)n
x
2
3=0
and
2
n—
3=0
together with
now
ni). It
y=max(ni— j',1)
£ (K + Z) 2 < £F 2 +2 (EY EZ 2
Fatou's
Lemma
follows from
For part
e
was
Gaposhkin
Let c™
\\G n (p n ,a)
>
||
almost surely where lim := lim inf„
.
By
0).
> and lim V ar(G n (0 n
limVar(G„(0 n ,a)) > implies
,
l
((3 n X)e
=
i
the Toeplitz
lX]
,
>
E"=o(^i/V")
Xn
< hmP(||G„(/? n ,a)|| 2 =
0)
cases that need to be considered are lim V'ar(G n .(/? n a))
= JC-
2
that for
By
>
Lemma
The only two
that
+EZ 2 implies that £X 2 ni < K n~ 2 (nalmost surely. The
(1980) that X n —
2
)
arbitrary.
we show that lim
ii)
3 in
< P(lim||G n (/? n ,a)|| 2 =
0.
1/2
2
then follows since
result
/
d\. For the
first
b y the second inequality in (A. 3). Let to?
E, £S=i+j utffoi-j* Lemma it then follows
££;«? £E=i+;W-j*t = Op (l).
leads to F(||G n (/? n ,a)||
that lim \\EG n (/3 n ,a)\\
1
P(!|G n (/3 n ,a)||
2
case note that
< e) —
> 0, since
>
<
e)
7 y,C?'))
we
that Ylj=i
w]'^yzU)
This implies that
for
for
any
any
e
e
<
>
< P(\\Gn ((3n ,a) -
EXn
have
*
such
c™/J]"=0
and
£X = O^"
2
1
)
Z}j=i7yz(i) sucn that
G n (f3 n ,a) = Op (Z]=
oo. For the
EG n
=
~
=
.vhich
second case one needs to show
2
{(3 n ,a)\\
>
2
||£G*„(/? n ,a)||
-
e)
such that the result follows from the Markov inequality after taking lim on both sides.
To
see this
we
first
show that
= ]mVar(G n (l3 n ,a))
<£»
\\mY™=Q (cy/y/n) 2 =
0.
Now
<(=
follows immediately from
n— 1
Ti—l
I
.
n—1
< Var(G n (P n ,a)) = £(- £c?X„(j)) 2 < ^(^/V^) 2 -Ej^Xn (j) 2
II
j=0
-
0.
(A.3)
3=0
3=0
< iim^"=0 (c"/ v/n) 2 < oo. Then there exists a subsequence n k
with n k < n k+ and k —> oo such that
< YJj=o(^j k \/^k) 2 < °°> V ar (G nk ((3 nk a ))
only if 3a 6 2
V/c. Note that Var(G nk (0nk ,a)) =
exists and Var(G nic (p nk ,a)) >
such that y = Ej=i ajUt-j almost surely. But this is impossible since there is no
To show
assume
=4>
i
I'
,
I
t
x €
2
I
such that e t
=
Yji=i xj £ t-j almost surely (see the Proof of
18
Lemma
4.2).
This
,
a)
hmVar(G n ((3n ,a)) = 0.
lim inf \\EG n (0 n a) - G(/3n a) f =
Therefore liminf ^IjLo^j/v^) 2
contradicts
implying that
since
,
,
=
1/2
2
\EG n ((3 n ,a)-G(P n ,a)\\ < (^jr\%\
2
j>n
\
/
1
/
\
I
1/2
\
V~^
£j KzH
M
.\||2
/
\j>n
J
At the same time lim ||G(/3„,a)|| > by the identification assumptions such that lim
2
= 0) = 0.
0. This shows that hjn.P(||G n (/? n ,a)||
Appendix
B.
Proof of
Proofs
-
Lemma
<liminf
n
From Lemma
From
3.1
2
(A. 5)
it
n
If
E
the definition of
<limsup
||G n (/? n ,a)||
n
=
||G n (/? n ,a)||
'
follows that
(3 n it
2
2
<limsup
||G n (/? n ,a)||
G n (Po,a)
follows that
limsup
Let
||£'G T ,.(/3 T1
\\G n (0 o ,a)\\
n
=
—> G(/3 ,a)
almost surely. Thus
=
lim ||G n (/? n ,a)||
n
(B.l)
.
almost surely.
(B.2)
be the probability one event in Lemma (A. 6). Now consider the sequence /3 n G 0.
there exists a subsequence /3 n
(3 Q then by compactness of
Pn does not converge to
such that
—
(3 nk
>
(B.2). Therefore
j3
n
/3
Lemma
G O. By
fyz (X)
=
nfc (/?nfc
>
,a)||
a.s. contradicting
->P .*
Proof of Proposition 3.2 We
that
G
(A. 6) liminffc
only prove that Assumption (C2) holds.
=
lX
C(f3 ,e- )l a (-\) where l a (X)
G(0,a)
= {2*y
r
l
J—
We
first
note
€
such
zXk
such that
T,kLi a k^
lX
i>((3,e-
(\)d\.
)l a
TV
Letting
V(/?o,eit is
clear that ip((3 ,e~
We
lX
)
need to show that
that G(P, a)
=
0.
Now
for
=
a = G-
C(0,e~
any
(/?,e-
a )G(/3
=
such that G((3 Q ,a)
1
for
1
)
/3
and denominator degrees equal
)
=
9(e'
iX
lA
)
0.
)/<p(e~
iX
)
there
lX
the polynomial tp(0, e~ )
€
to
iX
,e-
p
+ q. The
is
is
no other
j3
rational with nominator
orthogonality conditions can
now be
written
as
(27T)-
1
p
{i>{(3,
e-'
A
)
-
l)/ a (-A)dA
=
(B.3)
0.
J — 7r
We want to show that the only function tp(., e~ %x — 1
)
:
[—it,
it]
— C satisfying this condition
>
is
^(.,eIf
the assumptions of
is /?
Lemma
(3.2)
iA
)-l =
hold then the only value
-
19
(B.4)
0.
for
which
ip((3,
e~ lX )
—
1
=
,a)||''
Now showing that ip((3, e~ lX — l = is equivalent to showing (p(e~ lX )9o(e~ lX / (e~ tX )~
lX
is not zero for any A G [— 7r,7r] for (3 G 0. Here
9(e~ lX =
since this polynomial 6{e~
~ tA
~* A
-tA
polynomial
of an ARMA(p,p + q) process with parae
e
s
tne
a
</>(e
)#o(
S
)/^o(
)
)
)
^
)
meters
<f)
01
,
...^>o, P
Yl'jLo i)jZ~
j
%X:
*
)
^
$ii ••> ^p+9-
^
the coefficients
For
denote the impulse response function of this process by
lX
where dependence of
>p+
q
we denote the
9{e~ lX ) by 9
We
such that (p{e~
+
1
=
)9o(e~
lX /
_zA
(e
'</>
)
)
cp is
]'
^o,i^'j-i
_- _
[ip
[ai,...,a Q ],
=
implies
B=
be written as
[ipi,...,xp
and x
where
[a q+ i,...]
Rx =
=
q]
=
xp
9q
—
l
...]
,
[ip
by
ip
=
restriction
(B.5)
o.
and the vector of
coefficients in
(B.6)
0.
1
—
,...,%l> q ]
9
and
[V' 9 +i>
A
=
where
Define
Aq =
]
by the identification conditions.
such that Equations (B.5) and (B.6) can
[9',0,...]'
R
and R22 contains the
known
^o.pV'j-p
-,a„
[ai,
Establishing (B.4) amounts to showing that
•••]
suppressed for notational efficiency.
then Condition (B.3) has a matrix representation
a'xp
[V'g+i,
has a one sided Fourier expansion
satisfy the well
ipj
dimensional vector
infinite
[9i,...,9 q
-
on
xjij
the coefficients
i>i
If
'4>
)
B
q
R22
coefficients of (B.5).
The
result
now
follows
if
R
is
of full rank.
then R22 = I and R is of full rank if A q is of
p =
with x = tp. The
then the system reduces to Bx =
and #22^ =
full rank. If q =
=
condition ^22^
is satisfied for any of the p linearly independent solutions to (B.5). It
1 n N(R22) =
is therefore necessary that B is of full row rank and B
0, where ./V(i?22) is
the p-dimensional null space of #22 and B
is the orthogonal complement of the linear
and
span of the row vectors of B. Thus the only solution is x = 0. Finally if both q ^
>
>
between
and
consider
Define
we
need
to
distinguish
First
^then
<
q.
p.
q
p
p
q
p
We
can distinguish three cases.
If
y
1
D=
and
y
0,p-l
-<k0,1
0,p
C
J
1
0,P J
such that
R
where Ad
D
is
=
[ai,...,a d ],
B=
invertible such that \R\
R
[a d +i,.-.},
=
l>
Ad B
=
C
b
c
-
[(0,C)' ,0,
A d -C'D- B
of full rank. On
l
It
...]
and
D
conformingly. Note that
can be checked that
C'D~ B =
l
the other hand if q < p then we need
if Ad is
L
such
again that [A q ,B} D N{C,D) = 0. Then there is no element x G N(C,D),
=
=
=
required.
0as
implies
x
such that Rx
that [A q ,B]x
The previous arguments can break down on the boundary of 9 because it is possible
lX
iX
that xjj(P, e~ ) has an expansion with constant coefficients. In that case /^ %p(@, e~ )l a (-\)d\
such that
is
of full rank
i/O
2U
Kl a (0)
for
lim inf n
some constant K.
/?»
C-
l
(Pn,
therefore need to require / a (0)
>
e-^)fyz (X)dX
Proof of Theorem 3.3 A
=
op (I)
We
familiar
for
mean
G
e
/3 n
and
{§pG n {f3 n )]^iG n {(3n
Here
-MfG n ((3)
2
(27r)-
the multivariate
is
Gn(/£)
,
e
- lA
v^(/3 n -/3
)]-
<9A
)/d/3/ a (A)dA
and (/^-/?
the matrix with rows jSfGn(P)' for
mean value expansion can be made
at a different point
First,
/^ dlnC(/3
1
e 99.
)
)
M=a
/?
value expansion leads to
= (M + op (l))[VnG„(/3 +
where
-
n
in order to insure that
7^
=
i
)
=
op (l) for
1, ...,d. It is
well
i
=
l,...,d.
known
that
exact by evaluating each row -gjfG n (l3)
l
j3 n
.
convergence of
to
gf-Gn (/?«)'
M can be shown by the same arguments as con-
vergence of G((3, a) noting that both y t and z t are strictly stationary and d In C{(3, e~ lX )/d(3
is uniformly continuous on [— n, n] X U for U C 0, U compact, (5 Q € 0. A set C/ with these
properties exists by local compactness of the parameter space.
Next, turn to -i/nG n (/3
it
)
=
1
v n(2^)" f*K
/
C-
l
{(3
,e-
lX
The
details are omitted.
)In yz {\)d\.
From Lemma
(A.2)
,
follows that
v^
Using
Lemma
lX
C-\f3 ,e- )In yz {\)d\ - v^
/
,
(A. 3) then
shows that
should be noted that
00
/
/nG n (/3
/„,«(A)dA
=
op (l).
tf(0,E£i££= 1 «*.za*ai) whe re
N
it
00
;=i fc=i
The
now
result
follows from
x
dlnC(f3 ,e-* )/dP
= ^Tb k e —i\k
fc=l
such that (27T)" 1
/^ (?lnC(/?
,
e- tA )/a/?/ a (A)'dA
= E^Li
&fc<-
symmetric since a(k,l) = E(e 2 £ -k£t-i) =
a(l,k) for k,l > 0, k ^ I. Then, by the Shur decomposition (see Magnus and Neudecker,
there exists an orthogonal matrix Sm such that Sm QrnSm = A m where
1988) for all
A m is diagonal with elements A"1 j = 1, ...,m. Now, for any x m G R m x m ^ 0, we have
Proof of
Lemma
4.1 First, note that
Qm
is
t
m
,
,
,
x m Q Tn x Tn
Qm
is
positive definite
Proof of
It
= E(^2 x i,m£tEt-i) 2 >
Lemma
where the inequality is strict by Assumption (Al). So
such that A™ >
Vj, m. This shows that fi m has full rank.
4.2
remains to show that
From Assumption (Al) it is clear that Qx 6
2
such that
kerfi = 0. Assume there is x €
Z
21
2
I
rr.
for all
/
{0,0,
2
x €
...}
I
.
and
Qx =
=
2
which can be written as E(Y^Zi x i £ t £ t
= 0. But this is
with probability one. Now
x e t £t-i
only possible if Y^ x l e t e t -i =
a.s. iie t £t-i =
a.s. or the functions et-i are linearly dependent a.s.
2
a /
such that ^aiSt-% =
If st-i are linearly dependent then 3a G I
a.s.
for all i = 2,3,... then Yl a £ t-i =
Without loss of generality a\ =^ 0. If ojj =
a.s.
Now assume a t ^
for at least one i = 2,3,.... such that
is trivially contradicted.
1
1
£t -i = -Qf YH^=2 a ^ £ t-i a s B ut then £J(£ f _i|JFt _ 2 ) = -a^ X^^2 a i £ t-i as. so that
Then
0.
also x'flx
^
%
,
i
-
£7(ej_i
]-7-t_2)
-
with positive probability.
7^
This contradicts the martingale difference
assumption.
On
the other hand
e t e t -i
if
=
a.s. for all
then e 2 e 2_
z
z
I
Y2k a (^)OI <
as
an y ' Therefore
the roles of k and
if
A;
By Assumption
4.3
-^ f° r
\
also
—
(Al) we
any fixed
for
therefore
/
Also E^
are reversed.
/
^
know that E^
a (k,l) —
as
I,
>
_1
Q
if
exists
)
=
=
0.
and
is
x
|c (/c,/)|
/c
—
>
oo.
< ^
thus
This holds
< B such that a(k,k) —
5^, 5^ and 5^ according to the
|cr (/c,
fc
oo. Define the infinite dimensional matrices
>
2
can only hold
.
Lemma
Proof of
=
But then E(e 2 e 2_ l
a.s.
Qx =
which contradicts Assumption (Al). Therefore
Thus kerfi = 0. Symmetry of f} now implies that Imfi
bounded on 2
for all
=
l
k)\
>
following partition
Cm,
=
fi
Then ir(5^5f2
<j
4
J)
—
as
>
Eoo
!
m
00.
m+ i£ZLik(M)l
Then
j '771
°12
°21
°22
cm
ir(52"}52-
s
0,
and tr(5g-
ct
4
/)
(5^-
define the infinite dimensional approximation matrix
Q. r
Clearly
l
m
Vt*
exists
Mm
by
Lemma
(fi-
1
and the partitioned inverse formula.
(4.1)
-0^- 1 ) =
fi^-
1
We now
have
(fi-^)Sl- 1
such that
-1
llfi
- Q* _1
ll
777
_1
<
-—
llfT
771
ll
where ||.|| is the matrix norm defined by \\A\\ =
is bounded. By the partitioned inverse formula
\\Q
- Q*
777
supi| x
|i
llQ
II
_1
ll
||j4x||
<x
2
.
'
First
show that Hf^" 1
!]
a -i
a~ 4 I
such that
eigenvalue
1/A™ < 00
l|Q
_1
||
<
*-H
Wl
<
+ <t~ 4 We have shown in Lemma (4.1) that the smallest
\™ of Q m is nonzero. Then by a familiar inequality for all x G R m x Q^x/x x <
\\Q
Vm
00 by
such that
Lemma
.
H^m
(4.2)
1
!]
<
00 since
for finite
m
all
norms
are equivalent.
and
1/2
00
lift
-
ft;
=
sup
y^
llxIKl
1
,n
<j(k,i)xi
+
00
E E
fc=m+l z=i
1
22
a
(k, I) xi
Also
oo
<
sup
Y^
y~"
INI<1
fc= i
J=m+ i
\^ \^ \a (k, l)\ —
2
(=m+l
Thus H^- 1 - Q*^ 1
ll*ll<i
->
I
\
>
as 7n
—
>
oo.
fc=l
m -> oo
as
1|
/_. a (k,l)\ 1 /
2_\
fc=m +i (=1
oo
oo
<
sup
\cr{k, l)\ \xi\ 4-
Proof of Theorem 4.4 For
all
m
fixed
it
follows from standard results that
m
for all
it follows that lim inf m x m >
since for any sequence {x m } such that x m >
holds
in
the
limit.
Since
both
p(P)
above
inequality
also
the
G I 1 and a(A) G i 1 it follows
from a bounded convergence argument that lim m P^Am exists and is finite. If a G A" then
But
The same arguments can be used
the inverse exists as well.
exists
and
to
show that lim m j4'm f2 mJ4 m
is finite.
_1
= (^m — S^S^ S'Ji) -1 where Uij are the elements
Let w™ be the elements of the approximating finite
of the infinite dimensional inverse Q~
dimensional inverse f^if i,j < m and otherwise. Since uj™ — Wjj for all i,jasm-»oo
Finally define
[il
]
m=
[wi,j]i,j<m
l
.
>
and sup
<
|k>ij|
oo
Let Z?^
for ail i,j.
it
=
follows that for
df
sup m
3
-
.
e
>
uj™
BtJ
Then
>
is finite
oo
lim
The
function f
which
m (i,j) =
\u)ij\
+e
only for a
for all i,j.
finite
number
of
m
Now
oo
P'A Pm = lim Y" V btb'pQ
l
bib-ui^j
integrable for
is
all
m
and dominated by
6;6-
B
lt j
on the counting measure. Therefore by dominated convergence
YaL\ Z)j=i WjUiij as had to be shown.
also integrable
is
lim m P'm Q,^Prn
=
show that a(A) G A for A = P'Q.~ l From Assumption
(Al) it follows that Q maps l into l l To see this write Q — E + a 4 I where the matrix E
1
4
1
consists of elements a(k,l). For if! we have Qx = Ex + <7 :r with Ex €
because of the
summability restrictions on a(k,l). From Lemma 4.2 we know that for x G I 1 C / 2 we have
fr 1 ^ G 2 Assume n _1 x g 1 Then a; = ftfr 1 ^ = EQ _1 x + o- 4 f2 -1 a:. But Efr 1 ^ € i 1
Thus ||cr 4 Sl — a:|| 1 = ||x — Ef2 -1 :c|| <
+ IJSQ - rcj| 1 and \\x\1-l becomes unbounded
1
_1
because of ff4 fi x L But this contradicts the assumption that x G I 1 It follows that
Proof of Theorem
5.1
We
first
.
l
.
Z
;
Z
.
.
.
1
1
.-r
1
.
.
the image of
l
l
under fi"
1
also in
is
1
I
which
<
in turn implies that ]>Zfc=i \^ik\
°° f° r
_1
f2
of the Z-th unit vector. Since
P
This can be seen by considering the image under
now
l
an '
E
A
follows that P'fl~ G A.
Next we show that a(A) G A'. Recall that the optimal instrument is defined by
_1
2
The row rank of
A' = P'17
or A'VL = P' Interpret P as a set of d vectors in
=
A is therefore the same as the column rank of P. Remember that P
[61,62,...] with
it
I
.
23
.
=
bk
(27T)-
j^d\nC{(3 ,e-^)/d(3e^ k d\.
1
C(p ,e~^) =
For
^
- gg
dlnC(0o ,e-*)/dP=[^
we have
o (A)/</> o (A)
gg
.-.
'
Define the expansions of ^q (2) = Yl^Lo' P<p,j z an d #o H 2 ) = YlJLo^ej^ The coefficients
— ^o.i^Vj-i - •••- (t)o,p' P<f>,j-p = which
in the expansion satisfy the difference equation
1
''
t
A
has p linearly independent solutions.
j
<
,
similar expression for 4>gj. Set t/^
^
=
=
ip e
for
Then
0.
=
&fc
Any
^j
set of
=
d
p
+
•'•
Vv,fc-i
[
Vv,fc- P
g vectors &fcn&fc 2 >t>k d
]'•
4>e,k- q
independent because of the linear
linearly
is
>
••
V'e.jfc-i
=
and 6q{L)x = together with the requirement
independence of the solutions to 4> (L)x
and 6 q / 0. Thus P has full
that 4> (L) and # (L) have no common zeros and that 4> p ^
column rank. But this means that A has full column rank as well, thus establishing
Ad =
that
[ai,...,ad\
For the case where q
nonsingular.
is
=
we note
P
that
is
a
matrix of p linearly independent vectors. The space P 1 is spanned by vectors of the
form Vj = [0, ---,£jT — 1,0, ...]' where £ is a root of (p (L). This can be seen from x e
-
,
fc
= °A,--P ~ 1- Thus for a11 Jt has t0 nold that
Y^jLo^^j-^j = ° for a11
1
<Pq\L)L x = J]jfc(l - H L)~ L x = «*• (1 - ^ L)- L x = for at least one f" Now
_1
assume that 3x ^ with Q (L)x = and fi x € P It follows that x £
by stationarity
_1
-1
x
and Q x G 2 by invertibility of
Since Q~ x € P there must exist constants Cj such
that Q~ x = ^2jCjVj. Recursive solution for Cj shows that Cj — oo as j — oo. Thus
_1
2
which contradicts Q x = 53, 9j vj- Thus for any x ^ 0,
(L)x = it follows
53 CjUj ^
that A'x ^ 0.
P1
<=>
-
l
l
1
l
*
l
l
l
l
.
-1
1
I
.
cf>
x
fi-
I
.
l
>
>
/
•
<
</>
< p
then the matrix P' contains q rows which are determined by 6q (L). As
argued before P' has full row rank. By the previous argument it follows that for at least
If
one row
all
pi of
x^0,
We also
(f>
The sum
P'l
q
7^ 0.
P' and any x
{L)x
=
^
such that
l
we have P'9r x
^
cf>
=
(L)x
it
follows that pjfi
-1
2
^
Thus
0.
^
where i is an infinite dimensional vector of ones.
_1
l
such that
of the Fourier coefficients P'l is proportional to (f> (l)~ and 6>o(l)
-1
is not
Since A = P'fi
is contained in the linear span of the vectors of
and
need to establish A'l
P
orthogonal to
l it
follows that
A
also not orthogonal to
is
P
l.
l
Finally we show that a{A) 6 A". First, ff)(0, X)l a (-X)'dX = P'Q~ P and P
d
2
such that PE^ztz'A
*> 3^ G R
£ 7^
E(e z t z't ). Now, det P'Q^P =
,
eE{e1ztz' )l
t
a.s.
or
x
2
=
of A is
=
=
Then x 2
rank
for
0.
0.
t
< £(e x
,
1
a.s.
full
implies x t
so that £'z t
=
=
Awe can write
P
e
where e^
is
the
fc-th
=
£'z t
a.s.
Proof of Theorem 5.2 We need
Since
[e
p'
unit vector.
=
to
is
2
^E
[e
—id
IS
n~ P =
1
=
<=>
\Tt -\ =
a.s. is ruled out by Assumption (^41).
|^t-i] =
a.s. But we have shown before that the column
2
Then for x := £'zt
a.s. Now, clearly E
,
= Ex E
2
2
)
2
[e t
\F -\
t
)
=
2
}
impossible.
show that
p'Vl~ 1 VL
^Zfcli
1^1
\
a k\j
= P 'Sl -1 ^ 4 / + £).
'
f° r 3
—
1j
Define the vector
bounded.
4
=
fce fc
Then
p'4 = p'n-
1
24
4
(^ /
+ £)4-
(B.7)
Now, the sequence
a(P'f2
From
_1
6
)
(B.7)
A
<
P Ik
€
\
A
and
x €
M
is
d
-
a vector
Summing
Note that
Therefore, by the fact that
for all k.
l
of
Lemma
(5.2),
<
P n _1 E^fc
>
e
A
we have
p'
<
|.|
l
€
and the summability assumption
p'n-^kJ
where
T,£ k
P Q.
l
(.k
norm on
M.
over k gives
=
k \Y^iZi
d
.
<?
p.
Without
4
YlkLi
bitoik\
.
-
k
l
p' ar Y£k
p'n-^ik
loss of generality
P Qr
l
lu
<zr=i
we use
ph
This establishes the result
25
= sup^Xjl
iP'fT^J <
|x|
for
oo.
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05
Inequalities:
A
Generalization of Wu's
Date Due
MIT LIBRARIES
3 9080 01972 0934
'."