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DEWEY
HB31
I'
.M415
working paper
department
of
economics
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
EFFICIENT IV ESTIMATION FOR
AUTOREGRESSIVE MODELS
WITH CONDITIONAL HETEROGENEITY
Guido Kuersteiner
No.
99-08
March, 1999
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
EFFICIENT IV ESTIMATION FOR AUTOREGRESSIVE
MODELS WITH CONDITIONAL HETEROGENEITY
Guido M. Kuersteiner*
First Draft: December 1996
Current Version: December 1998
Massachusetts Institute of Techonolgy
Department of Economics
50 Memorial Drive, E52-251B
Cambridge, MA 02215
Tel (617) 253 2118
'This paper
my
Ph.D. dissertation at Yale University. I wish to thank Peter C.B. Philhps for
I have also benefited from comments by Donald Andrews, Oliver
Linton, Chris Sims and participants of the econometrics workshop at Yale and seminars at Barcelona,
Brussels, Chicago, CREST, Harvard, LSE, Minnesota, MIT, Northwestern, Oxford, Pennsylvania, Princeton and Stanford .All remaining errors are my own. Financial support from an Alfred P. Sloan Doctoral
Dissertation Fellowship is gratefully acknowledged.
is
part of
continued encouragement and support.
Abstract
This paper analyzes autoregressive time series models where the errors are assumed
weak additional restrictions on their distribuUnder these conditions Quasi Maximum Likelihood estimators of the autoregressive
parameters are no longer efficient in the GMM sense. The main result of the paper is
to be martingale difference sequences with
tion.
the construction of efficient semiparametric instrumental variables estimators for the autoregressive parameters.
The optimal instruments
are a linear function of the innovation
sequence.
shown that a frequency domain approximation of the optimal instruments leads to
an estimator which only depends on the data periodogram and an unknown linear filter.
Semiparametric methods to estimate the optimal ffiter are proposed.
The procedure is equi\/alent to GMM estimators where lagged observations are used
as instruments. The number of instruments is allowed to grow at the same rate as the
sample. No lag truncation parameters are needed to implement the estimator which makes
It is
it
particularly appealing from an applied point of view.
1.
Introduction
This paper develops new instrumental variables (IV) estimators for autoregressive time
series models when the errors are uncorrelated but not independent. The specification
includes error processes which are conditionally heteroskedastic of
unknown
functional
form. Efficiency gains are obtained without having to specify a model for the dependence in
the errors.
The setup
is
general enough to account for stylized facts in
series displaying features
many economic
time
such as thick tailed distributions and time dependent conditional
variances.
Classical efficiency results for the quasi
autoregressive parameters such as
Hannan
maximum
likelihood estimator
(QMLE)
of the
(1973) depend on independence of the errors.
In the more general case of conditional heteroskedasticity considered here the
QMLE does
GMM
lowerbound for the asymptotic covariance matrix which now
no longer attain a
depends on fourth moments of the innovation process. In Kuersteiner (1997) it is shown
how a decomposition of the higher moment terms leads to a lower bound for the covariance
matrix. An instrumental variables estimator based on this decomposition is shown to
achieve the lower bound for the covariance matrix in the class of IV estimators with
instruments that are linear in the innovations.
The
feasible
of Hayashi
GMM
estimators developed in this paper are similar to the estimators
and Sims (1983), Stoica, Soderstrom and Friedlander (1985) and Hansen and
Singleton (1991, 1996). In this literature lagged observations are used as instruments to
account for unmodelled
MA(q)
innovations which lead to inconsistent
OLS
estimators.
Here lagged observations are used as instruments to account for unmodelled conditional
heteroskedasticity of the error terms which renders
OLS
inefficient.
Apart form the different motivation for the use of instruments, inefficiency versus inconsistency, this paper extends the previous literature as it explicitly treats feasible estimation.
The number of instruments in our case is allowed to grow at the same rate as the sample
size. This is made possible at the cost of an additional restriction on the fourth order cumulants compared to the treatment in Kuersteiner (1997). The advantage of making this
assumption is that the estimator can be implemented without the need for a truncation
or bandwidth parameter for the number of instruments used.
Since the optimal instruments are unobservable they need to be estimated nonparametrically. Assumptions about the generating mechanism of the volatility process or more
generally the dependence in higher moments are replaced by smoothness assumptions for
higher order cumulant spectra of the errors. This setup allows for the treatment of dependence in higher moments as a nuisance parameter. Nonparametric estimators of this
nuisance parameter are used to construct the optimal instruments.
Other semiparametric procedures proposed to handle conditional heteroskedasticity
include Robinson (1987) and Newey (1991). No parametric assumptions about the form
of conditional heteroskedasticity are made in these treatments. However, in order to estimate the conditional variance these authors have to assume independent errors. This
assumption has precluded direct application of their techniques to the stochastic conditional variance case. Hidalgo (1992) relaxes the iid assumption for the errors but has
to assume instead that the conditional variance is a smooth function of an independent
stationary process.
GLS
metric
Hansen (1995)
He assumes
framework.
Brownian motion
in a semipara-
that the conditional variance process converges to a
Sample path continuity of the
in the limit.
model
treats the stochastic volatiHty
limit process then allows for
consistent kernel estimation of the conditional variance.
Hansen (1985) and Hansen, Heaton and Ogaki (1988) prove existance
lowerbound in the presence of condimensional
and
nonlinear
heteroskedasticity. The high
character of these instru-
More
generally
of instrumental variable estimators achieving a
ditional
GMM
ments has so far precluded implementation of such an estimator.
Here we limit ourselves to the implementation of optimal procedures in the much
smaller class of IV estimators with instruments that are linear functions of the observations. Ordinary least squares is a particular member of this class. In the general case of
conditional heteroskedasticity it is inefficient. This paper achieves the construction of a
feasible version of the most efficient IV estimator with hnear instruments.
The remainder of the paper is organized as follows. Section 2 describes the model assumptions. Section 3 develops the
domain approximation
for the
IV estimator
efficient
IV estimator
is
for the
AR{p) model.
derived in Section
4.
A frequency
Section 5 shows that a
semiparametric estimator with the same optimality properties can be constructed. Some
Monte Carlo simulations are reported in Secition 6 and concluding remarks are made
Section 7. Some important lemmas are quoted in Appendix A while the main results
the paper are proved in Appendix B.
in
of
Model
2.
We
by defining the stochastic environment of the model. Let {Q ,J^ ,P) he a. general
probability space and define a filtration J^t to be an increasing sequence of cr-fields such
that J^t ^ -^t+i C ^ V i. There is a doubly infinite sequence of random variables generating
the filtration Tf We assume that we observe a sample of size n of a univariate time series
yt where i = {1, ...,n}. More specifically, we assume that yt is generated by the following
autoregressive model
start
<l^{L)yt
where
(p'
is
=
et is
(01,
=
St
(1)
a martingale difference sequence generating Tt- Here
is
..., (f)p)
assumed that
the vector of parameters describing the
has
(l){L)
the parameter vector
(p.
all
roots outside the unit circle.
The martingale
correlation between the errors. However,
Rather we allow
for
dependence
(f){L)
= l — (^iL—... — (f)pU'.
mean equation
We
of the model.
It
are interested in estimating
difference assumption for et implies absence of
it is
in higher
not assumed that the errors are independent.
than second moments to account
for thick tails
and conditional heteroskedasticity.
Assumption A-1.
(T^,
(J
E {et) = cr^ <
(s)
<
DO for
E [e'te't^s) >^
Remark
s
>
(i)
oo,
et is strictly
(ii) (f>{L)
0, (iv)
some
a>
has
stationary and ergodic,
all
roots outside the unit
E [etet-s^t-r) =
for all
E {et
for s
^
r,
s,r
>
\
J^t-i)
= 0,E [e^
circle, (Hi) E{e'te't_s
0, (v)
^
\s\
\a {s)\
|
"
J-'t-i)
^'^)
=B<
=
oo,
s.
Assumption (iv) is added to the assumptions in Kuersteiner (1997) in order
to simplify the form of the optimal instruments. It is somewhat restrictive as it rules out
1.
-
some nonsymmetric parametric examples such as EGARCH. The IV estimators proposed
in Section 3 are still consistent and asymptotically normal if (iv) fails. However, in this
case they lose their optimality properties.
Remark
2.
It
should he emphasized that no assumptions about third moments are made.
In particular this allows for skewness in the error process.
Remark
GARCH
Asssumption (A-1) under certain conditions.
Nelson (1990) obtains sufficient conditions for stationarity and ergodicity of the GARCH(1,1)
model. The martingale difference property follows immediately from the definition of
a GARCH process. Assumption (A-liv) is shown to hold for the ARCH{p) case in
Milhoj (1985) for symmetric innovation densities. The same argument extends to the
GARCH (p.q) case. If the innovation distribution is normal then fourth moments are
known
3.
to exist for the
(p,q) processes satisfy
GARCH (1, 1)
+ 2'^if3i + 0\ <
case if 87!
1.
This condition
is
valid
and thus covers the ARCH case. In Milhoj (1 985) and BoUerslev (1 986) the autocorrelation structure a (s) is shown to be identical to the AR{p) and ARMA{maoc(p, q),q)
case for ARCH{p) and GARCH {p,q) respectively. This implies that the summability
condition holds if fourth moments exist. Similar arguments can be made to show that
stochastic volatility models satisfy the assumptions.
for
P=
,
Based on the results in Kuersteiner (1997), we will now introduce the optimal instrumental variables estimator for the AR{p) model. The estimator is constructed by reweighting the innovation sequence by the unconditional fourth moments a (k) + o"^ of the error
process.
these
3.
Without parametric assumptions about the form of conditional heterogeneity
moments
typically have to be estimated.
Instrumental Variables Estimator
The parameter
in this
paper
vector
can be consistently estimated by OLS. Under the assumptions
OLS amounts
to an inefficient
IV
estimator in the class of
IV
estimators
with linear instruments. Kuersteiner (1997) derives the form of the optimal instrument as a
function of the fourth moments and the impulse response function of the underlying process
yt in
a
more general
context. In this section these
more general
results are specialized to
the autoregressive model.
Let zt G M.P be J^t-i measurable and square integrable, strictly stationary and ergodic.
Then, the instrument satisfies the moment condition E [(</> (L) yt) zt] = where (L) yt =
An instrumental variables estimator then
Et- Let y't = [yt,yt-i, ...,yt-p] and </> = [l, —<p']
(f>
.
is
defined as
=arg min
f=P+i
with an explicit solution in the case of an autoregressive model as
<t>=[ZY.i]'^ Z'Y
(1)
=
where Y'
Y=
that
[yi+p,
.
.
.
,yn]
+ e. By
Y-icf)
and
y!_-^
=
[yp,-.,yn-i] where
the ergodic theorem
As argued before instruments
follows
it
which are linear
zt
y'j
is 4> is
=
[yt,yt-i, ...,yt-p+i]
Note
consistent.
in past observations are considered.
Such instruments are equivalent to instruments that are linear functions of past innovations. This set up is formalized here. Using (f)~ (z) = X^jIq^^J-^''' *^^ time series yt can
be expressed as a linear filter of past shocks yt = Yl%.o''P(i>,j^t-j- Also, let
=E
ak
and
b'f^
=
(V'</.,fe-i,
with
,'4'4,,k-p)
In Kuersteiner (1997)
the z-th element of
it is
ip^^^
= ^ (f^) + (^^
(e^tetk)
=
afc^ of a^ Yl'kLi Wk,i\
k
for
shown that
for Zf
<
<
=
Then
0.
(2)
define
Yll^=i '^k^t-k
P^ =
[6i, ...,6^].
with a^ G
W such
ooVi the asymptotic distribution of
is
that for
normal
with mean zero and covariance matrix
—KX)
771
where
Am =
[o-i,
...,0^].
The
lower
bound
S=
where
It
O'k
=
is
\im{^' Pln^;n'Pm)-'
(3)
should be noted that under Assumption (A-1), Qrn = Dio-gioLi, ...,0^)then follows immediately that the optimal instrument achieving (3) has weights
it
A
^k/oi-k-
different
almost surely. This
is
way
of writing the optimal instrument
a special case of the
The optimal instrument matrix
Z
more general formulation
now
=
is zt
defined by setting z[
lim^^oo Pm^m^T'
in Kuersteiner (1997).
=
[zt,i, ...zt-p+i^p] with
t
It should be emphasized that the
Zt+i,k = Y^%zO'sr^^t-i and letting Z' = [zp, ...,Zn-i]
k -th instrument, i.e. the instrument for parameter (pj^ is lagged by k periods and has a
convolution filter which is also shifted by k elements.
is
.
The estimator introduced
be a function of the seqeunce
Define
(f)Q.
zt
=
not feasible and needs to be approximated. Let ho
{fcfc/ctfc}fc^i of optimal weights and the true parameter value
so far
is
Yfk'Ji'^ h/otk^t-ki^o) with
approximate version of
4> is
et{(f)o)
=
(p'oYt
and
let
Z=
[zp,...,Zn-i]
The
now
^(/io)=(zV_i)"'zV.
(4)
A feasible estimator is then obtained by replacing Hq with a consistent estimate h. The proof
of feasibility proceeds in two steps. In Section 4
and
in Section 5
we
establish that ^yn{^{h)
—
it is
4>{^o))
In the next two sections the Approximation (4)
domain
shown that
y/n{(f){ho)
—
</>)
=
Op(l)
=
Op{l).
is
represented in terms of frequency
This imposes no practical limitations since the freqeuncy domain version
by the exclusion of only a few observations at the beginning of the sample.
The difference is asymptotically of order Op{n~^) and therefore does not affect the first
differs
integrals.
from
(4)
order asymptotic properties of the estimator.
Formulating the estimator in the frequency domain however
offers the potential for im-
provements in terms of computational efficiency since the frequency domain formulation
6
can be used to implement FFT-algorithms. Using FFT algorithms reduces the computational complexity of ^{h) from 0{'n?) to 0(n log n). This improvement is substantial
in applications where the dataset is extremly large and the estimator is computed many
times.
Leading examples are model selection procedures
in forecasting applications
and
simulation studies.
4.
IV Estimation
in the
In this section a frequency
Frequency Domciin
domain approximation
to the optimal
IV estimator
Consider the inverse of the spectral density for the AR{p) model f~y (A)
and
let
[(/((e*"^)]
useful later
and
=
9yy{^>
'^)-
The
definition of the
=
is
^
derived.
|(?!)(e*^)|
spectrum of the squared errors
will
be
is
oo
/,2,2 (A)
=
(27r)-i
^{k)e--i\k
J2
(1)
fe=— oo
Define the lag operator a(A)
=
[e'^, ....,6*^^]
by a(A)*. Also introduce the matrix
A (A) =
and denote the complex conjugate transpose
a(A)*a (A)
.
Then
gyy{(t>, A)
can be represented
as
'
9yy(.(t>A)
1
a(A)
a{Xy
A{X)
Define
=
77(0, A)
d\ngyy{^,\)
(2)
and introduce the discrete Fourier transform of the data
the periodogram as In,yy (A)
With
these definitions
=
1^^;^
we turn
(A)
The main
Re
[In,zy (A)
-y= Y17=i Vt^
a
-itX
and
.
|
domain implementation of the
the beginning. It is easy to show that
1
=
=
to the frequency
mental variables estimator introduced at
4>
as cOy (A)
(A)]
d\
instru-
-1
Re[In,zy{\)\dX
+ Op{n-^)
(3)
contribution of the paper consists in deriving an approximation to this frequency
domain formulation that does only depend on observable quantities. In a first step we
decompose the instrument into observable data and an unobservable optimal filter. In the
next section it is shown how the unknown optimal filter can be consistently estimated.
For the purpose of this and the next section, we introduce the spaces L'^ [— 7r,7r] of
functions / [— 7r,7r] —> C^ such that J" |/| d\ < oo. Also, define the spaces C^ [— 7r,7r] of
functions / [— 7r,7r] —>
such that f is k times continuously differentiable. Throughout,
the function Rn (A) will denote a generic remainder term whose definition can change. We
start by approximating the discrete Fourier transform of the instrument variables ztj
:
:
W
^., (A)
=
l^,k (A)
e^^V(e-^^)a;, (A)
+ i?^,^ (A)
(4)
_-i
= ^-„ ^e-^>0+^) and^i?^^^) ^ „-i/2 ^oo^ _^^-,A;^^^^.
^^^
and ^{k) = {i/jQ/akjipi/ai+k,
Then let the
define the sequences ^ = {ipQj'ipi,
where
'A,
/^., (A)
_
}
}
j^^^
inner
product
= -^ Yl'^o ^J^
Rl^^s (A)
Unj (A) = Ylt=i-j ^te"^^
a compact form
—
^^^^n,j (A)
Y17=i ^te^^^-
We
be defined
for a
=
••'
Kmp)^^^^''
{sq, si,
...}
R^
f.
where
(A) in
=
[i?^
^
^^ ^^ analogous way u^^ (-A)
(A)
is
,
and
...i?^p (A)]
the complex conju-
(A) as
l.^
.
=
+ RriMk)^>')-
denote the vectors of stacked error terms by -R^ (A)
[^n,^(i)^^)'
K,i,^^^
gate of uJzi^ (A) Define
s
can now write the remainder term
-iX
<fc(A) = -W(A)0(e- )K,i,W
We
sequence
^^,1 (A)
=
l^ (A)
(5)
't/i.p
The
properties of
/,/,
determine the asymptotic distribution of the instrumental
(A)
The next lemma
ables estimator.
(A)
gives a representation of
operators. This shows that the smoothness of l^ (A)
vari-
terms of convolution
inherited from the smoothness of
AR(p) spectrum.
the
Lemma
4.1. Let l^^k (A)
sumption (A-1) and
assume that
Irj
is
(A) in
/,/,
=
(A)
-t-
1^
1
^^'^^ otk
—^
a{z)* has
= ZT=-oo ^j^''^'
(A)
all
{fa * v) (A)
and
-I-
-^r)
{(f),
A)
.
8.10, that
Ir,
(A)
G
.
Also
Then
^
= (^ - ^)
^^^h a,
The properties of
For fa & L} [— 7r,7r] and
fj.
satisfying As-
et
characteristic roots outside the unit circle.
Using the convolution operator
4.
,
.
B
Proof. See Appendix
Remark
(e?e?_fc)
(—A) can be represented as
J —IT
where fa
= E
being the coefficients of the power series expansion of 4>{z)~
ipj
=
(f){z)
l^ (A)
= E^o ^1^7^"'^^^'^''^
C'' [-7r,7r]
rj
E C^
Irj
*
a compact notation for
(A)
[—-n,-n]
can
it
l^i
(A) is
now be determined from
FoUand
follows from
Ij^
(A)
=
those of fa
(1984),
Theorem
oo
implying that J] j=0
\j\ < OO. While fa e L^ [-7r,7r] is
«fc+j
it is not necessary. Alternatively if a^ > a >
for some
sufRcient to obtain this result
a and
all
k then sup
a^.
|
|
< a
^
<
oo and Yl'^o
These arguments show that
central limit theorem in Appendix A.
if
f]
E
C"^ [— 7r,7r]
The
DFT
.
Irj
oik+j
\j\
< a-'^T=okj\\j\ <oo
(A) is sufficiently
smooth
to apply the
representation of the discrete Fourier transforms of the instruments in terms of the
for the
data allows to obtain a frequency domain version of
4>
without the need to go
through an explicit calculation of the instruments in the time domain. The approximation
relies on the fact that convolutions in the time domain are transformed into multiplications
8
=
domain and the
by (f){e'^^).
in the frequency
multiphcation of
The
fact that the residuals
W
lOz,,
by k periods leading to
transforms in the following
u,
^z, (A)
cl^{e-'^)uy (A) l^ (A)
(A)
,-tAp
where Rn(X)
=
e~*^'^w^^ (A)
way
-iX
a (A)* ©i?^ (A) and
©
is
the element by element product.
=
=
e'^PcOz,
where the symmetry of
Rn{X),
(6)
The correspond-
is
(-A)
e'^cvz,
u>z (A)*
+
'^z, (A)
ing expression for the conjugate transpose of Uz (A)
+ R^{-X)
4,{e'^)u;y{X)*l^{-X)
(7)
(-A)
Z^ (A) is used.
discrete Fourier transform approximation for Zt introduced at the beginning of
produces an asymptotically equivalent estimator based on the periodogram of
this section
yt
is then obtained from (4)
and stacking the resulting
discrete Fourier transform of the instrument matrix
by lagging each
The
can be computed by a simple
ujy (A)
and an unknown
It is
filter.
convenient to define
/i^((^,A)
= Re
l^{-X)(t>{e'^)a{-\)
and
h{(l),X)
By
=Re
l^{-X)<P{e'^)
.
substituting for equations (6) and (7), (3) can be approximated by
-1
kho) =
^n,yy (A) /l^ (^, A)
dX
In,yy (A)
/
h{(f),
X)dX.
(8)
J— IT
It is
shown
in the proof of Proposition (4.2) that 0(/io)
1
(t){ho)
-
<^o
=
—
4^
=
Op[n
-^Z^)
and
-1
•n,yy (A)/i^((^,A)dA
J
In,ee (A)
Ir,
(A)
dX
+
Op {n'^'^
such that consistency follows from ergodicity and the fact that
E
f
In,ee (A)
Ir^
(A)
dX
=
a^ f
J —TT
transparent from equation
/^
(A)
dX
=
0.
J —IT
depends
on knowledge of the true parameter values and the correlation structure of the squared
errors. Feasible versions of 4>{ho) will be discussed in Section 5 below.
Under the assumption that the weight matrix Re [Z^ (—A) (e*'^)] is known, the asIt is
(8)
that ^(/iq)
is
infeasible as
it
stands, since
it
</>
ymptotic distribution of
(A. 4).
The next
^(/lo) is
now a
straight forward consequence of
proposition summarizes this result.
Lemmas
(A. 2)
and
Proposition 4.2. Let (f>{L)yt = et where all roots of (f)(L) are outside the unit circle. If
et satisfies Assumption (A-1) then for ^ defined in (1) and 4>{ho) defined in (8) we have
4){ho) — 4> = Op (n"-^/^) and
^(4>iho)-<Po)^N{0,a-^E)
where E
is
defined in
(3).
Proof. See Appendix B
The remainder of the paper
will
now be concerned with
parametric estimator with the same distribution as
5.
the construction of a semi-
(f).
Adaptive Estimation
To develop a feasible efficient IV procedure,
Re [l^ (A) (p (e~*^)] a^^d h^ {(f>, A) = Re \l^ (A)
cf)
has to be established that h
it
{&~''^)
is
called adaptive.
No
apart from efficiency issues,
is
A
semiparametric
confusion should arise between this
use of the terminology and the literature on feasible local
Bickel (1982), Kreiss (1987), Linton (1993)
=
a (A)] can be replaced by consistent
estimates without affecting the limiting properties of the estimator.
estimator having this property
A)
(cf),
minimax estimators such
and Steigerwald (1994). The main
as
difference,
the fact that here a nonparametric correction to the crite-
is made while the local minimax
Newton Raphson improvement to a consistent
rion function
literature
first
makes a nonparametric one step
stage estimator.
Different approaches to prove adaptiveness are used in the semiparametric literature.
is used in Robinson (1987,1988) in the context oiiid models and partially
models and by Hidalgo (1992) in the context of time series regression models. Newey
(1991) applies similar techniques as Robinson (1987) to the instrumental variables case for
iid data. Andrews (1994) develops a general methodology based on stochastic equicontinuity arguments and applies it to the partially linear framework. Andrews' approach will
Direct calculation
linear
be used here to break the proof into two parts. First, it is shown that a nonparametric
estimate h{^,X) converges to h{(f)Q,X) uniformly with probability one. The second step is
to established that uniformly in a shrinking neighborhood of the true filter h{(pQ,X) the
distribution of an estimator is arbitrarily close to the distribution of the estimator based
on the true filter.
This argument will now be formalized. Let
where C is the complex plane. Then introduce a
n = |/i:
[-7r,7r]
/i
= Re
L (-A)
(^(e'^)
l^
:
[— 7r,7r]
—
>
C^ and
set of functions
4>
'
H defined
;Re[Z^ (-A)] ,Re ^ (e'^)
["TTj^] ~^
C
as
e C^ [-n
,
Tr]\
.
(1)
where
C'^ [— tt,
tt]
the L°° Sobolev
denotes the space of k times continuously differentiable functions. Define
norm
of order one as
1=
sup
11/
(A) 11+
sup
A£[-7r,7r]
AG[-7r,7r]
10
I^W
where ||.||
on 7^ as
is
the Euclidean matrix
norm
=
p{hi,h2)
{HjP)
is
a complete metric space.
Lemma
We proceed
(4.1) that
l^
A)
((/>,
G C"
defined by
-
||i,/,,i
If
.
+
li,,2\\l
—
1101
(trAA*)
-
'
.
Introduce the metric
?f'2lll
given by (1) and
is
(f)
[-tt,it]
\\A\\
/,/,
by
(5)
then
it
follows from
Therefore h{(t>,X) G H.
by defining the estimator for h (</>, A) We have established that we can
= {Y_iY-i)~^Y-iY or from its frequency
obtain a consistent estimate 4> for example from
function of some fixed parameter value
before.
Residuals
as
a
domain analog introduced
(p are obtained as in Kreiss (1987) from
.
cf)
e«
{<f^)
=
such that the estimated error
measurable part (0
—
et
£t
((/'o)
{(f))
n
(</>- 0o)
-^yt-p)
{yt-i,
We
form the following
t=p+k+l
else
dr,
>
where the sequence d„
for all
n with dn =
O {n
^1'^+^) for
is
<
some
z^
<
1/2.
The
dn are used to avoid "too large" values for a^^(^). Truncation was
More
introduced by Bickel (1982) in the context of score estimation.
our context
J-'t-i
statistics.
=
oikQ))
truncation numbers
(2)
can be decomposed into the true error and the
,yt-p)-
{yt-i,
(po)
+
closely related to
Hidalgo's (1992) semiparametric frequency domain estimator.
Simulation
experiments indicate that the truncation playes no role in practice and can therefore be
ignored in applications.
Next, an estimate for b^
=
(27r)~^
J^^
A) e^^^dX
77 {4>,
is
The
needed.
vector hk contains
we
the impulse response function of the AR{p) model evaluated at different points. Here
want to express
definition of
6fc
77
b^ directly as a function of the underlying yli?-parameters.
(0, A) in (2)
4)~'^
and the expansion
{z)
=
with ip^j
Yl''p4>,j'^^
=
From
for j
the
<
0,
can be written as
bk
'^4>,k-p
where the
all s
>
coefficients ip^^j satisfy the recursion ip^^
and
tp^^Q
coefficients of the
=
1
Let
(see Kreiss, 1987).
polynomial expansion of
(f)~^
ifJ^
(z)
.
—
(pi^p^^s-i
This vector
"
0<^.o
<^1
1
<?^2
-01
—
(t>p4'4,,s-p
is
=
first
^o^
p
+
1
the solution to
"
"
"
1
—
denote the vector of the
1
^<t>A
(3)
1
-
11
.
^*,P
.
_
_
which
denoted by
is
defined in
pxp+1
(3).
t/)?
Then,
=
^~^ei where
ei is the first unit vector
bp+i denote the vector of coefficients
let
selector matrix ^i picking the last
p elements from a p
and
^o
^p^^2
+1
x
1
$
is
the matrix
'^4>,p+\-
vector
Using a
we have
1
hp+i
=
Si^l
=
r^5i$-^ei
1
<t>V
where we define T^ = (/>! if p = 1. In a similar way we obtain bp+s = T^Si^ ^ei. The
vectors h\
.b^ can now be expressed as functions of the underlying parameters by
.
.
bk
where the convention T9
=
with the indicator function
coefficient b^
is
Ip
=
is
{.}
=
5p,fcr("^"'t°''^-^1^5i$-iei
assumed and the
1 if
selector matrix Sp^k
the expression inside the bracket
continuous in the underlying parameters for
all finite
is
is
defined by
true.
The
Fourier
k and can therefore
be consistently estimated from a consistent estimate ^.
A nonparametric estimate of h (cj), A) is now defined as
n—p—l
and
hr (^,.A
No
=Re
(A)0(e
-i\
(4)
is needed. The reason is, that h (</>, A) is already a conbounded sequence and a twice continuously differentiable function. In
implicitly contain a bandwidth since for every
inside the stationary region
additional kernel smoothing
volution between a
fact,
the bk
(f)
they will decay to zero quickly.
We
need the following matrix h^{4',X), whose elements are continuous func^) and which is defined by
will also
tions of
hn{4'-,
h^^
The
(^, a)
= Re
[l^ (A)
^ (e-^^) a (A)
success of a semiparametric estimator depends on the ability to uniformly estimate
the weights aj^. Additional assumptions about the
moments
of the driving error process
on fourth moments such conditions necessarily
involve higher than fourth moments. Here we prove uniform convergence by a mean square
argument which necessitates summability assumptions on eighth moments. The following
assumption is sufficient to prove the main result.
are needed to assure this. Since aj depends
12
Assumption B-1. Let
Cs.,.e (ii,
.
.
.
,
ifc-i)
he the k-th order cumulant of the error process
Then
et.
^(1 + \tj\) \c,,„, {ti,..., tk-i)\ <
•
J]]
tl
=
oo, for all j
1,
...,
k
-
1
and k
=
2, 3, ..,8
t-k-l
Assumption (B-1) implies that higher order cumulant spectra
assumption enables us to state the following
Proposition 5.1. Let hn{4>n^ -^) he as deRned
assume that ^„ -^ (pQ in probability. Then
sup
of order eight exist. This
result.
Assumptions (A-1, B-1) hold and
in (4), let
hn{(t>n^>^)-h{(t)Q,X)
=Op(l)
AG[-7r,7r]
^
n
—
Proof. See Appendix B
We proceed to define the semiparametric estimator ^„{hn) by replacing Hq
with a nonparametric estimate (4) We will establish that
=
n
as
P
Also
oo.
[hn{(l>n->
P
(p
^'H] -^
^)
1
f
/i„((/)„,
as
A),
/i
((/>o,
A)
>^) —
any
for
>
)
(5
>
as
>
oo and
^ oo.
n
/i((/)q,
A)
.
Vn
By
applying
Lemma
(A. 2)
In,yy (A)
h
it
((/),
(^^„(^n)
-
(ho))
4>n
=
can be shown that ior h
A)
=
In,yy (A)
Rc
Op (1)
(5)
.
eH
[l^ (A) (/.(e-'^)a (A)]
+In,ee (A) h^^ (0, \)
+
h
((/.,
(6)
A) i?„ (A)
where the remainder term Rn (A) is such that ^Jn f Rn{X)'^ (A) dX = Op
uous function (A) with absolutely summable Fourier coefficients. Let
(1) for
(7)
any contin-
<;
h^^
such that h^^
{(pQ,
A)
then follows
= Re
= Re [l^ (—A)] =
V
(5)
icf>,X)
(^, a)
Ir/
= Re
[l^
(-A)
(A)
and
[i^
(-A)
4>{e'^)<t>^\e'^)
4>{e'^)<Po'i<^'^)
if
£^ In,yy (A)
(A^(^„, A)
-
h^
(</>o,
A)) dX
=
Op{l)
(8)
and
j^In,e.{X){h^^Q>^,X)-lr,{X))dX
=
Op{l)
/l„(A)(/i„(^„,A)-/i(<^o,A))c!A
=
Op{l).
13
(9)
(10)
(8)
can be established easily with the help of Proposition
^"
-
/„,,, (A) (/i^(,^„, A)
<
sup
r In,yy{X)dX
/l^(0„,A)-^-((^O,A)
J —n
,^A^
sup
2
Ae[-7r,7r]
last
h
=
=
{a a){b b)
expression goes to zero by (5.1)
(9)
we work with
hn{4>n, A),
is a positive scalar and the second
where a and b are two conformable vectors. The
and the fact that sup;^£t_ ^.^i ||a(A)|| is bounded. To
inequality uses the fact, that In,yy (A)
first
inequality uses tr{ab ba)
prove
||a(A)||%j,(0)^0
sup
/V(-A)^(e^^)-Z^,o(-A).^o(e
Ae[—tt.tt]
where the
by the following argument
h^ (00, A)) d\
Ae(-7r,7r]
<
(5.1)
0^
Vn
=
cP{e'^),
=
(h)
the metric space {H,p) defined in
= Me'^)
ct>^
y/n
(1).
Also
let
Hq
-
Hq)
dX
=
h{(j)Q,X)
,
and
In,ee (A) {h^^
{(f),
A)
-
/,,,o)
+ Rn (A)
(h
J —TT
for
(5
h G H. Following Andrews (1994),
>
any given
??,
>
e
limsupPf
Vn (hn)
-
Vn (ho)
>
< limsupPf
Vn (hn)
-
Vn (ho)
> &,hn€H,p{hn,ho) <6
^
n—>(x>
+ lim sup P
<
there exists a
1?)
'
(hn ^ 7i oi p{hn, ho)
sup
\\vn {h)
>
6
— f„
(/io)||
>
>
=0.
i?
|
<
£-
\h€n,p{hr,M)<i
we have established
in Proposition (5.1) that
lim sup
n—too
Therefore
^
hmsup P
^-"^
Since
(9) follows if for
such that
P (Jin^H
V
or p (hn,hQ]
^
6]
^
if
limsupP
"-°°
sup
ll^n (/i)
—
'"n (^o)ll
>
<^
'i^
(11)
I
yft.e-H,p(/i„,/io)«5
then the following theorem can be established.
^
Theorem
5.2. Let hn{^ny^)
defined in (4). Let assumption (A-1) hold and let ^„
be a previous estimator for which ^„ —> (pQ in probability or almost surely. Then, the
semiparametric estimator 4>(hn) defined by
^ [hn
=
In,yy{X)hlQ)^,X)dX
)
•TT
/
J
14
./— TT
/„,yy (A) /i„(^„,
A)dA
.
has a limiting distribution characterized by
^
(4,{hn)
-
4>o)
^N{0,a-^E)
.
Proof. See Appendix B
This result establishes the
feasibility of
a semiparametric estimator that improves on
the efficiency of the conventional Gaussian estimator in the presence of higher order depen-
The frequency domain
dence.
representation allows to avoid estimating the instruments
each observation in the sample. Instead an optimal
for
filter
applied to the periodogram
of the data leads to an asymptotically equivalent procedure. Moreover, the fact that the
optimal
a convolution integral in the frequency
filter itself is
domain
solves the
problem
of truncating the approximation of the optimal instrument at a given lag in a natural
and elegant way and eliminates the need
for lag
truncation parameters for the number of
instruments used.
Monte Carlo Simulations
6.
Monte Carlo experiment is reported. To keep the exposition as simple
on an AR{1) model. We consider what the efficiency gains/losses of
In this section a small
as possible
the
IV
we
focus
estimator are relative to a correctly specified likelihood procedure and relative to
OLS.
The
following questions are of interest:
estimator achieve efficiency gains,
how much
how
Under what circumstances does the optimal IV
QMLE and
big are they relative to the Gaussian
by not specifying the true likelihood. These questions are analyzed for
mechanism is an ARCH{1) process.
generate samples of size n = 256, n = 512 and n = 1024 from the following model
is lost
the case where the true generating
We
yt
where
Ut
~
et is
generated by the
ARCH{1)
A''(0,1). Starting values are yo
=
=
+ et
(fyyt-i
process
and
ej
eq
(i)
= Ufh/ where ht = 'Yq + 7i£t_i with
= 0. Small sample properties of three
be defined below are evaluated for different values of 0, 7;^ G [0, 1)
from Milhoj (1985) that asymptotic normality established in previous chapters
different estimators to
It is clear
only obtains for values of 7^ € [0, -^1/3). Nevertheless, simulation results are reported for
parametrizations outside this interval in order to analyze the robustness of the proposed
IV
procedure to departures from the assumptions. The parameter 70
is
fixed at
.1
for all
experiments.
The parameter
is
(f)
^
mator
is
is
The
least squares esti-
=
Yl^=2ytyt-i/Y17=2yt-i- ^^^ optimal instrumental variables
obtained from the consistent first stage estimator 4>oLS ^^
denoted by 0„
estimator
estimated by three different estimators.
OLS
IT
In,yy (X) h"" {4)QLS, >^)dX
15
/
J — -K
In,yyWH^OLS^^)d>^
where h^
ated by
and
A)
{(f),
h{(p, A)
computed
are
the Ukelihood estimator
(1),
obtained from maximizing
is
(j)^
7i;n =
70,
/(</.,
as explained in Section 5. If the data are gener-
-^E (in
and /it = 7o + li^l-iEngle (1982) to maximize the likelihood.
with
St
=
yt
—
We
<t>yt-i
Figure
Figure
QMLE
1
shows the potential
+
^t
use the
(2)
^)
BHHH
algorithm described in
1
efficiency gains of the
as a function of the autoregressive parameter
IV estimator relative to the Gaussian
</>.
The
efficiency gains are
from the asymptotic covariance matrix when the generating mechanism
plicitly,
the asymptotic covariance matrix oi
c^Ils (</',7o,7i)
=
=
is (1).
computed
More ex-
can be expressed as
(p^
^^-^ E'^'^^.+i
(3)
7i)^ and a,+i = 27§7'i+V[(l - 7i)^(l - 37?)] + o"^. The asymptotic
covariance matrix for the optimal IV estimator can be obtained from (3). It is given by
where a^
(70/I
-
fv-i
^/y
(<?!',
7o,7i)
=
"I
-1
-^E^'V\
(4)
i=0
Figure
plots cr^y
1
(</>,
.1,7;^)
/cr^^^
(c;!),
.l,7i) for
cf)
6
[0,1)
and
different values of 71.
' OLS
These theoretical gains are contrasted to the empirical efficiency of the estimators 4>n
and 4>n based on 3000 replications for sample sizes 256, 512 and 1024. The results
(j)^
are summarized in Table 1.
Table 1
,
As expected,
parameter
sive
is
gains for the
above
.5.
IV
estimator are achieved for models where the autoregres-
This conforms with the theoretical analysis based on asymptotic
approximations. For the sample sizes considered here, the theoretical efficiency gains are
not achieved completely.
The
improves with the sample
to 512. It
is
The most
significant increase takes place
also interesting to note that the
for values of 7^
dependence
table shows that the relative efficiency of the
size.
>
IV
procedure maintains
\/l/3. In fact the gains are strongest
when both
ARCH
estimator
size
256
properties even
autocorrelation and
in the conditional variance are strong.
Figure 2 shows the empirical densities of the three estimators
no
its
IV
from
4)^
,
4>n
^^d ^„
when
effects are present.
Figure 2
The graph confirms the information summarized
identical
and
(^„
in the tables:
The
three estimators are
under iid conditions. Figure 3 shows the empirical distributions of
= .9 and 7^ = .9.
for a sample size of 1024 when
</>
Figure 3
16
4>n
>
4'n
Here 0„ clearly dominates the two other estimators in terms of efficiency and mean
and median unbiasedness. The IV estimator has surprisingly good properties even though
the asymptotic theory used for
construction does not hold for this set of parameter
its
values.
Table 2 contains the means and medians
on 3000
is
bias tends to be largest for the
(f).
For a fixed
other hand,
7.
4'n
^'^'^
^n
when n
=
512 based
is little
largest
(p, it is
affected
2
IV estimator, but the
smaller than the difference of the former with
with
^^^
,
replications.
Table
The
^^^^
^„
for
when
=
7^
4>^
.5.
difference
between
(/>„
and ^„
The bias for 0„
and ^„ increases
The bias of the ML estimatoi, on the
.
by the parametrization of the model.
Conclusions
This paper develops
efficient
IV estimators
for autoregressive
models with martingale
novations. Popular parametric examples of such innovation processes are
A
and stochastic
volatility models.
these effects in
many macroeconomic and
The paper shows
vast empirical literature
that
GMM
OLS.
ARCH, GARCH
documents the presence
of
financial time series.
that estimation of the autoregressive parameters by standard Gaussian
ML techniques leads to inefficient estimators. An important result
inate
in-
in Kuersteiner (1997)
is
estimators based on lagged instruments improve efficiency and therefore domIt is
shown how
to construct the best
GMM estimator
based on instruments
that are linear in past observations.
A common
GMM
problem of
procedures based on a large number of instruments is
bad small sample performance. This is mainly due to estimates of a high dimensional
weight matrix. We introduce a novel decomposition of the weight matrix which leads to an
their
orthogonahzation of the instrument space. This decomposition has the advantage that
it
implemented by using FFT algorithms. Moreover, the
estimator does not require a bandwidth choice for the number of instruments used and is
can be computationally
efficiently
therefore straight forward to use in practice.
The
small sample properties of the procedures developed are very promising.
mators are equivalent to conventional
OLS
The
esti-
even when the innovations are iid and strictly
dominate OLS when the innovations are conditionally heteroskedastic. These results hold
for sample sizes as small as 250 observations and are therefore relevant for macroeconomic
time
series.
17
A. Appendix
Lemma
-
Lemmas
A.l. Under Assumption (A-1) the following statements hold:
a) for each
m 6 N+\ {1} m fixed,
the vector
,
-^
Yl^=i i^t^t-i, ,£t£t-m] =^
N{0,Q) with
a{l)+a^
n=
(T{m)
+ (7^
Leram.a A. 2. Let In,yy (A) be the periodogram of {yi,. ,yn} and /„_££
odogram of {ei ,...,£„} Assume et satisfy Assumption {A — 1) and that yt
.
.
(A) is the peri-
=
YI'jLq i^j^t-j
with spectral density ^gyyiPot^) such that Yl'jLo |'0j| \j\ < o°- Let (.) be any continuous
tt] —^M. with absolutely summable Fourier coefficients {zk,—oo <k < oo}
<;
function on [— IT,
then for any
77,
P(y^
as
n
e
,
>
J
In,yy (A)
<;
{\)
dX -
^J
/„,,, (A)
<e
dX >r/
gyy{pQ, X)q (A)
00.
Proof. The proof is the same
minor modifications.
Lemma
A. 3.
(Billingsley)
pose that for each
m
Xr,
p.
388, with
Let Xmn, Yn be random variables defined on {Q,,J^,P)
Xm
—^
1
and Davis (1987),
as the proof in Brockwell
as
—
n
00 and that
>
Xm
—^
X
as
m
.
Sup-
-^ 00. Suppose
further that
hmsup
lim
for each positive
Lemma
e.
A. 4. Let
Then Yn
^>-
X
as
P {\Xmn - i^n| >
n —f
00.
In,yy{X) be the periodogram of {yi,.
riodogram of {£!,...,£„}. Suppose the
Y^'jLoi'j^t-j with spectral density
any continuous even function on
=
e}
satisfy
et
.
.
,yn} and
Assumption {A —
[—n,Tr]
such that
fc=i
and f^^
q (A) QyyiPo,
X)dX
=
0,
then
In,yy (A) ? (A)
dA
T
with bk
A iV
0,
\
= Ejl-oo 7yy {k - j) Zj.
18
I)
is
the pe-
and that
yt
=
that Yl'jLo
°°- ^^^ "^(O ^^
l^il bl <
with Fourier coefficients {zk, —00 < k < 00}
^gyyiPoA) such
—> M
/n,ee (A)
4
^
t,_i
fc=i
Qfet^
,
B. Appendix
-
Proofs
Proof of Lemma 4.1
— —T
Y^°^_
J
<
First
oo-
we show that /„
Next note
(A)
for a typical
€ Li
which follows from J^
[—tt,7t]
\fa (A)|
element k
—n
oo
/jr
-•^
oo
oo
oo
j=0 l=-oo
j=0 /=-oo
oo
oo
I
/ oo
^
I
oo
gJA(j+fc)
such that the result follows.
Proof of Proposition 4.2 Using equation
can be written as Re[In,zy
W] =
+4>{e-'^)li.
(4)
In,yy (A) Re[h^{(f),
(-A)
u;,
i-X)
Rl^
and the
A)](?!)
(A)
definition for
h^{(f).,
A), /n,2y (A)
+ /^ (A) In,ee (X) + Re[Rn (A)]
+
l^
(-A)
where
|<^(e'^)|' |i?^_^ (A)|'
+i?i,^(-A)i?2(A)
and
a;,(A)
The two remainder terms R^ ^
quences
uct i?^
Un,j (A)
g
= rne"^>e(A)+<^(A).
and R^
(A)
(A)
are defined next.
First define the se-
= {ipQ,ip-y,...} and Tpik) = {^o/<^fc! V'l/^i+A:) •••}• Then let the inner prod= -^Y^'jLoSj^~^^''Un,j
be defined for a sequence s = {so,si,...} where
= Yl7=i-j £te*'^* — Y17=i ^te'^*. We can now write the remainder term R^ (A) in a
Tp
W
(A)
compact form
Rl
where R^
-
(A)
'
=
(A)
=
-l^
(A) 0(e-^^)Hi,^(A)
^(uW' ...,R^
[i?-^
+ a (X)* © i?;^_- (A)
such that i?^ (A)
r, ,(A)]
is
a
p x
vector. This leads
1
'
to
Re[In,zyiX)]dX
=
/
+
In,yyiX)Re[h%(j),X)]dX(f>
/"
Ir^
(A) /„,.. (A)
+
dA
J —ir
Alternatively one has also Re[In,yz (A) a (—A)]
with RP, (A)
=
(l)-\e-^^)uj, (A)
Rl i-X)
+
r Re[i?„ (A)]dA.
(1)
J —TV
=
Rj, (A)
19
In,yy (A) Re[/i^((?!>, A)]
R^ (-A) Now
.
if
-I-
Re[i?^ (A)
/Re [i?^
a(— A)]
(A)] <r(A)dA
=
dX
=
^/^),
Op (n
(f)
+
where
has absolutely summable Fourier coefficients,
<;(A)
follows that
it
(f)
=
Opin^^f^) and
-1
In,yy{X)Re[h''{cj,,X)]dX
Vn[j)-(j)j
x-v/n
r
(A) In,.e (A) c?A
Ir,
Since l^ (A) G C'^
[-7r,7r]
and
(?!>(e^'^)
G
J —TV
C'^ [-tTjTt]
/
/•IT
^^(A)J„,eaA)tiA^7V
V^J
where the matrix Ym^i
'^i
W
{^n
>
s'^')
follows
it is
Lemma
(A. 4) that
A
(A)
{^n
,
i\l
,
has typical element
e*'^')
=
J2'^J+kvl'l'4-,J%,J+\k-l\-
j=0
k,l
(4.1)
from
OO
1=1
Lemma
it
0,^a,(z,(A),e'^^)(z,(A),e'^^)
J^ai{l^iX),e'^')(lr,{X),e
Using
T Re[i?„ {X)]d\
+
J — TT
easy to show that
5^a,(/,(A),e^^A(z,(A),e'^^-
faiX-Ov{<l>,Ov{^,X) d^dX + -,
/
77
a'*
a
A)
77 (<^,
A)
dX
^/n X^tt -^ (A) dX = Op (1) We discuss the four different types
of remainder terms separately. Since Z^ (A) and Z^ (A) (j){e^^) have absolutely summable
The
result then follows
if
.
Fourier coefficients and are uniformly bounded on [— 7r,7r] the proof of
be applied to the remainder terms involving
R\ ^
Lemma
To show that -Jnj^^(f)~^{e^^)u)^{—X)R'^{X)dX
=
Op(l)
is
it
enough to show that
element by element
Ey/n [%-'ie^^)u,{-X)Rl,,{X)dX
0.
J —IT
Using the definition of
/?^ (A)
EV^
one has
r r'
(e^^)'^e
J — IT
<
(- A) Rlf,
(A)
dX
E^ Jr l^^k{X)ct>'\e'^)<p{e-'^)u,{-X)R\^^{X)dX
—IT
+Ey/n
£ r'
(f")
-. (-A) n- V2e-A'=iii
20
(A. 2) can
(A) alone.
(A)
-
^^^
dX
(e ''^) and applying
term can be analyzed by setting q {X) = l^ (A) (^ ^ (e*"^)
the proof of Lemma (A. 2). Next look at a typical element k of the second term
The
first
(f)
i^i
EV^
^
/
^ -^
Then
for j,
Z,
and r
oo
oo
J
,,
n
EE E E S^'.
(='- -
-"--) --"<'-'-'*"<iA
j=0 1=0 r=l t=l "'=+J
fixed such that
l<r — k + l<n
1/2
E
fc+/
Now summing
i,
over
'-"j-n-k+l
m gives
j,
U^) u>, i-X) n-V2 Y^ ^.e-'^Wnj
This holds true
1/2
'
for all
k
=
l,...,p.
(A) ? (A)
dX
<sup24/vv2j;j:
^,-^/
j=0 /=0
"fc+j
lJl-0.
Next consider
r Rli-X)Rl,{X)dX
J—TT
Vfc(A)0(e-'^)|i?i(A)|'dA
r R\ (-A) n-1/2 V Ae-^(^+'=)C/„,,
+Ey/n
^
V-TT
such that the
first
OLj
term goes to zero by the proof of
(A)
dA
+k
Lemma
The second term then
(A. 2).
is
OO
E^
/
y J^e-Mi+fc)c/^
Ri (_A) n-1/2
.
(A)
dX
/TT
< En
(fA
-TT
<
n-^l''
'^
/TT
1=0 j=0
EE
"j+'=
E\Un,l{->^)\\UnjW\dX
/=o j=0
OO
.
\
1/2
""
1/2
< n
(^it^n,,(A)r
0!j+k
/TI"
< n-V2
CO
OO
ElV'dmin(/,n)^/2E
'^
Z=0
j=0
21
mm{j,n)^^^dX^O.
dX
The remaining terms can be shown
to go to zero
by the same arguments and the proofs
are omitted. This completes the proof of the proposition
Lemma
=
^ Z7=i+p+i £ti<t>)^t-M, ""^ = Ea* {4>) with e^cp) defined
satisfying Assumptions (A-1) and (B-1) and any fixed S < oo it follows that
(2), £t
B.l. For a*
(</.)
sup
maxt;ar(n^/^
Proof. Let yt
=
,yt-p\ and
[yt,
m,^
and a"^
=
ai^^{n
=
— — p)/n.
et {(pf Et-i
4>q
=
=
fi'n,y...y{qi,
94=0
=
sup
\ai
sup
'^Y^\^g^---(J}g^\\il'^yy{qi,...,qi,l)\.
<l>^N6{<l>o)
n'^ Y17=i+p+l
{(f))
-
q^=Q
'
--yt-i-qi
-
Eyt-q^
--yt-i-q^.
Then
ai^^l
(2)
54=0
can be bounded by
s
fmax(|(?!)i_o|
^
it
yt-qi
p
/
so that
write
^'Yl^i^'" ^giyt-qiyt-q2yt-l-q3yt-l-q4-
p
(2)
we can
1
-,94,0
<
Expression
0(1).
p
gi=0
define
=
vec{ct)(f)'y{yty't0yt-iy't_i)vec{(t)(f)')
p
Now
af^\)
vec{(l)cj)')'E{ytyt®yt-iy't-i)vec{(j)(j)')
=
((P)'^
-
Then
=[l,(t)']'.
</>
Letting
I
|a,* {(())
in
+ ^)
p
p
XI
'
'
'
J
^91=0
'
remains to show var
n
|An,y...2/(9ij
XI
|An,2/...2/(9i>
•••>94,0|
94=0
•)94)0|
=
0{n~^)
for all
/.
Now
var
\jj!n^y„,y\
is
n
X X
"~^
4
COv{yt-q^
yt-l-q^,ys-qs
--ys-l-qs)-
t=l+l+p s=l+k+p
We
"~^
can substitute
n
n
X
Yl
for the definition of yj
oo
^
'4'j^t-j
which leads to
oo
X X
t=l+l+ps=l+k+pji=0
=
^Jl
^J8^°^(^*-9l-Jl
^t-l-qi-H,£s~qs.-h
'
'
'
^s-k-qs-js)
J8=0
(3)
The
covariances can be represented by eighth and lower order cumulants by considering
the following matrix
X{t,sJ,qi -ji,...,q8-J8)
=
^t-qi-jl
^t-q2-h
^t-l-q3-J3
^t-l~q4-J4
^s—qz—jb
^s-qe—je
^s-l—qr-jr
^s-l-qs—js
22
.
with typical element Xij where reference to the time indices
Brillinger (1981),
Theorem
Then from
suppressed.
is
2.3.2,
4
4
coz;(JJ
j=l
Xi
,
J,
n
JJX2j) = X]
ctim(Xij,z,j G Vs)
Vs€v
V
j=l
where cum{Xij,i,j € Vs) is the joint cumulant of all the Xij with indices
the sum is over all indecomposable partitions v of the table
(1,1)
(1,4)
(2,1)
(2,4)
i,j
S Vs and
•
A
given in Brillinger (1981), p.20. We note that
cumulants are zero. Second cumulants are nonzero only if the time indices of et
definition of indecomposable partitions
all first
By
coincide.
indecomposability of partitions there
from both rows
n
-1
is
in
X
in
each product Ylv
is
one cumulant with elements
at least
£v^'^''^i-^i,j^'''i3
^
Es Et co'u(n)=i ^i,j, 11^=1 ^2,j) = Es cou(nl=i ^l,j, nj=i
dices for
^Y
^s)-
strict stationarity
^2j) where the time
in-
have been normalized to zero. The summability assumption (B-1) implies that
t
Es<^^''^(nt=i ^ij,n7=i-^2,j) < oo uniformly in I, j\,...,js- This shows that the term (3)
is of order n~^ uniformly in I as had to be shown.
Proof of Proposition
for
any
ry,
5.1 For the
first
part of the proposition
we have
to
show that
A^^ ((^g) is
contained
>
e
lim
P{ sup
hr,
>
(0„,A)-/i(0o,A)
77)
<
e.
Ae[-7r,7r]
This holds
there
if
a ^ and a neighborhood
is
A^^
lim
P(
Consistency of
considered.
4>
hn{(l),X)
implies P{(p
€
-h{(po,X)
A^^ {4>o))
of
(pQ
such that
—
1
>
>
rj)
+
lim
P{(j)
so that only the
^ Ng
first
{(f)o))
<
e.
term needs to be
From
hni(f),X)
Re
<
sup
sup
{(J)q)
parameter space and
in the interior of the stationary region of the
-
h{(f)Q,X)
f^(A)</>(e-^)
l^ (A) </.(e-'^)
-
Re hfi{X)Ue-^^)
/^,o (A)
-iA>
M^~n
+
1
i^(-X)^{e'^)-l^^o{-X)Me'^)
and
l^{X)cp{e-^^j-l^,o{X)Me-'^)
<
kW-h,o{X)
<t>{e-^^)
+||Z^,o(A)||
23
0(e-^)-(/)o(e-^^)
,
it is
enough to show that
while
\(l){e'-^)
To estabhsh
element
aj^tj)
=
^ip,k
—
(e''^)|
(f>Q
sup
sup
A
4>€Ns{4>o)
<
(A)
—
ltp,kfi
Eel{4i)el_A(j)).
sup
on Ns
57/2
sup^^sup^^g^y^^^^^^
(A)
-
li,
Z,/,
(A)
Now
let ctj
/^,o (A)
=
=
Op (1)
Ee1e1_j,
Then, using the definition of
sup
'i/',fe
(A)
—
Z^,fc,o
=
l^fi (A)
Op (1)
by uniform continuity of ^q
{4>o)
—
-
(A)
bj^k
it is
{^^^) o^i
enough to look
denotes the
A;-th
[-t'",'?'']
at a typical
element of
bj
and
l^^k (A)
(A)
Ae[-7r,7r](;6eiVi(<^o)
sup
n—p—1
oo
j=i
j=i
sup
n—p—l
<
sup
sup
Yl
(^j(^) ^hk,4>-(^j,lhk,4>
+ (^jibj,k,ct>-aj^bj^kfi)e
'^^
(4)
j=i
+ sup
A
j>n—p
We note that sup;^ Er>n-p cq\ks>e-''^ <
sup,,
afn-'/' T.T>n-pj"'
\hkfi\
=
o{n-''').
Next
n—p—\
n—p—l
<
(5)
n—p—l
+
E
3
It is
therefore
enough to show that
(5)
and
(6)
{oc-\bi^k,^-b,,kfl))e-''^^
(6)
=1
go to zero uniformly on [— 7r,7r] as 6
—
>
0.
First consider (6).
n—p—l
sup
{(f)Q,n))dn
Ae[-7r,7r]
<
sup
Ae[-7r,7r]
n—p—l
1
+-4
(^
sup
(7)
A£[-7r,7r]
24
where, for
E Ng
cf)
{(po)
the finite Fourier approximation of
,
77 {(f>,
^) converges uniformly
such that
n—p—1
sup
yZ
Ae[-7r,7r]
J
=
sup
ihk,4>
-
^j,kfi) e'
i\j
=l
(77 {cp,
X)-fj
(</>o,
A))fc
-
Y,
AG[-7r,7r]
<
ibj^k,4>
-
bj,k,o)
e-'^'
]=n—p
sup
\7]{4>,X)-f]((f)Q,X)\f.
+
V
rap
2
\bj^k,4
AGf-TT.TT
<
+2
e
^
sup
\bj,k,<p\
4>eNs{4>o)j=n-p
Then, letting
ctj
=
a~^ — a~^ the
term
first
in (7)
dominated by
is
n—p—1
<
sup
\Vk{4>,iA -'?fc(0o>M)M/^
Ag[— 7r,7r] J —TT
j=i
n—p—1
<
J—n
„_,
j=—n+p+l
,
<
The constant
continuity of
from which
1
Ce{5).
n—p—1
C is bounded by X^,=i
7)
it
< 00. From uniform
< Ejli Qj (-^o)
Wo)
— f] {(f>Q, ^) < e for some 5 >
(cj),
fj) on [— vr, tt] x A^^ ((/)q) we have \ri {(p, fi)
follows that J_^ ri{(j),fx) — f] ((f)Q,fj,)\i^dfi < 2'ne. The constant C can be
[.
bounded by (27r^°l^
goes to zero as
aij
n —
>
dj
00.
((/>o)~
+ 0-4)
independent of n and sup^g^^(^^) Ejln-p
Next we consider
(5). First
l^j.fc.^l
look at
Pj>-^jl
From
aj_0
a.s.
and
(j),
=
E[4>{L)yt)^ {(f){L)yt-j)'^
But, since
(f){L)yt
4>{L)yt has
=
(f){L)(f)Q
it
> a^ > since otherwise (j){L)yt =
an ARMA{p,p) process with parameters 0q
follows that a^^^
{L)et
is
nonzero variance contradicting
(«j,0)~^
-
(^J^
< Ci
(f>{L)yt
\aj,4,
-
=
a.s.
aj\
some constant Ci. Since Eej < 00 we can uniformly bound
C2 is a finite constant depending on (f>Q and Eef. Then
for
Then
\aj^^
n—p—1
n—p—1
j=l
25
—
aj\
by 6C2 where
^
1
where Y^=i
<
\^oM,<t'\
oo on
Ne
{(po)-
Now
turn to the
two terms of
first
(4)
n—p—l
n—p—l
<
Now
"j
(</•)
- ",i <
^
=
.
"
^
^
1
- ajj <
{4>)
\aj
Y^
a^-,,^)
("j (0)
l^i
where a^
sup
{(f))''^
|aj
(</))
\aj
{(f))
-
aj^^-^
aj, <^|
^j,k,4>\
and
——
- a]J + J'+P
n
the expected value of a*
^^.(^ is
-
((^)
|aj-
Then
.
n-p—l
n—p—l
w— p—
n
First note, that since Qj^^
replace
{ctj
((f))
is
bounded away from zero and aj
{(f>)~
<
cn^/^~'",
we can
aj^^)"^ by n^^'^~'" times a constant that does not affect the argument.
Then
the second term
n—p—l
^-i/2-i;
^
{j
+p)
\aj^^\ \bj^k,4>\
-^
n
as
^ CO
3=1
uniformly on
Ng
((^q)
The
•
term now
first
is
shown
to go to zero in probability
by looking
at
P{
The term
n}''^~^
X]?=f~
^n-p-l
The
last
term
is
ni/2-
sup
.
^i i^)
~
Y"-^~
^?<4
\^J i^)
l^j,fc,</>l
-
^U
\hk,4>\
>
n)
dominated by
^^
n—p—l
^
zero with probability tending to one
which follows form maxj P(n-^/^ sup j,g^(0
the first term is bounded by
n
)('-''*i d>
maxj P{sup^^j^^/^
if
~ '^1 d>^ >
s—^n-p-l
n"'
ry)
—
sup
>
0.
\
a^
By Markov's
|a*(<^)-a^^^|
4>&Ns{ct>o)
1/2
<
^^YT.
^
\hk,<l>\\^ar{n^^^i
26
sup
\a* {^)
-
-
al^\))
<
d„)
—
>
inequality
.
The
^
from
result then follows
nmaxfar(
This
shown
is
It
Lemma
l^j.fc.^l
sup
it is
|q;*
(0)
0{n-'')
—
if
= O (1)
a"^|)
.
(B.l).
remains to show that P(p(/i„(^„,
part
first
in
=
^
YTjZl
A), /i(<;6o)'^))
enough to show that sup;^g[_^
,r]
>
^^
^)
dhn{4>ni
Using the result from the
0.
^)/dX - dh
A)
(</)q,
/dX
=
Op (1)
Since
implies that
e C^[-7r,n]
f]{(l),X),^
V/i
from Folland (1984), Theorem
follows
~
Bernstein's Theorem, ^\j\
<
8.22e, that {ij^hj
\bj {(p)\
<
=
oo and d^r]{(j>,-T^)^ /dX^
{(j))
=J
d^i]{(i),T:)^
/dX^
(^T]((f),X)) e-'^^dX.
it
By
oo such that
^(^j)6,(</>)e'^^->^r,(</>,A)
uniformly on [— 7r,7r]
the
first
Since
that
.
Using these
part can be applied to the
{Ti.,p) is
PChn
and noting that
facts,
first
lr]{X)
€ C^ [— vr,7r]
,
the proof of
derivative.
P (p
a complete metric space
(
/i„(^„, A),
^((J^iq,
A)
]
>
6)
^
implies
eH) ^i.m
Proof of Theorem
the proof of
Lemma
5.2
(A. 2)
We
start
by obtaining an expression
for
Rn
(A) in (6).
we have
oo
^y
=
(A)
ct>o'
(e~'^) ^. (A)
+ n'^/^ ^V'^-e-^^^'C/^,-
(A)
j=0
Letting
R^
(A)
In,yy{X)
=
n.-^/^
Yl'^o ^je'^^^Unj (A) we can write
=
UJy{X)Uy{-X)
=
^y
=
In,yy (A)
(A)
^y (-A) a (A)
a (A)
4>o
(/.q
+ Uy (A) u, (-A) + cjy (A) 0o
+ In,ee (A) <Po'
+u,{X)-p^Ri{-X) +
00 (e
In the
same way we
In,yy (A)
also
=
[^"^)
+ ^e
(e'^)
(-A) hi
< (-A)
(A)
\Ri{-X)\'
^-^j
have
In,yy (A)
a (-A)
<^o
+ ^n,.. (A) <t>o^
27
(e^^)
+ o;, (A) R], (-A)
From
Let
Rn
Next write
= Re
(A)
(-A)
(A) i?i
a;,
=
A)
/iQ (</>0,
Re
In,yy (A)
+In,ee (A)
while a
filter
based on a different parameter
=
In,yy{\)h{(p,X)
(A)
Zr,,0
</>
(-A)
l^^Q
+
/i0|j
that ^/n
(<?!),
j Rn
coefficients.
= Re
A)
(A)
(-A)
(A)
dX
(11)
is
<;
Next
[l^p
In,yy{X)Re
=
Op(l) for
P
limsup
ra—oo
which
in turn follows
lim sup
[e'^j a (-A)
(pQ
/iQ (</'o,
(po
A) i?n (A)
{- X)
l^
(j)
+
(e'^^ a
h
{- X)
(t>Q
(^, A) Rr, (A)
<;
for all
sup
\h&n,pihM)<6
i?,
>
e
||f„ {h)
-
there exists a 5
Vn
>
(/io)||
<
'J?
>
such that
^
J
from
P
n->oo
as
(pQ
i^''^)]
all
if
(-A)I'
Then by the proof of Lemma (^.2) it follows
continuous (A) with absolutely summable Fourier
(j){e'^)(t)Q^
estabhshed
|i?i
expressed as
is
+In,ee (A) h^, (0, X)
with
+
(A)
based on the true parameter
In,yy (A) h^ {cpQ, A) for the filter
In,yy (A)
%^i?i
+ ^, (-A)
sup
^/n
<e
In,esW{h4,,{(f>,X)-lr^fiW)dX
[
(8)
J — IT
\h&H,p{h,ho)<6
and
P
lim sup
n-xx,
For
(9) define
sup
all
(A) [h
-
ho]
<£.
dX
(9)
the open neighborhood
ns = {h:
Now for
Rn
^/n
\hGn,p{h,ho)<6
h €
TCs
it is
[-7r,7r]
^RP \h
en,p{h,ho) < 6}
the case that h (A) € C^[— 7r,7r].
.
By Bernstein's theorem this implies
— ho is also continuous and
uniformly bounded with absolutely summable Fourier coefficients such that (9) follows
absolute summability of the Fourier coefficients of h. In turn h
from the proof of Lemma (A. 2).
Next turn to (8) The main idea of the proof
by parts is used to separate h from In^ee (A)
.
is
taken from Robinson where integration
-
<
r In,eeW{h4,,{<P,X)-lr^fiW)dX
sup y/n
h&Hs
J —TV
sup y/n
h&Hs
J —IT
+
sup
r
Vn
{In,,e (A)
-
EIn,ee (A)) {h^,
(</>,
A)
-
lr,,0
EIn,ee{X){h<t>,{(l),X)-lr,fiiX))dX
28
(A))
dX
Since EIn,ee (A)
note that J^^
=
Ir^fi
cr^ it
(A)
dA
sup y/n
heUs
r
and /^^
sup \/n
+
-
h,p^
{In,,e (A)
{(f),
EIn,ee (A))
{h^o
(</),
(V
^)
(<^'
{4>, Tt)
-
-
(A))
^'/.O
for
/i
/^,0 (tI"))
/
J —-K
(-^n.ee (A^)
Q
PIT
/
'dX
remains to show that y/n J^^ J_^
ity.
Let /^gg(^)
=
—
/„_££ (/i)
"
^^^"^
^'''°
—
{^n,e£ (/^)
ting Hg
=
=
^,
In,ee
we
(m
+
71")
The
/j,,o (7r)||
are uniformly
~
'^^
E^n,6e
(/^)) C^MC^A
dX.
(fJ-))
d^ dX
is
bounded
in probabil-
fj.<X
/Lt> A
we work with y/n J^''
n-l
n
-
tt)
(10)
/q (/„,ee
(f^)
-
Eln^ee
(a^)) C^Ai
dX. Let-
show that
first
27r
£^/n,e£ {lA) df^^X
Eln^ee (yu)and define the function
1
(yu)
-
-E/„,ee (ju)) C?A
(</>,
^"^"'^^
/
EIn,ee
t{X,h)
Since /„,££
dA
/-A
^^'^° ^^' '^^
d^
It
"
||/i^g
(<?!',
sup y/n
hens
(A))
/^,o
{In,ee (m)
/
A) - lr,fl (A))|| and
ein Tis both ||^ (/i^^,
bounded by C6 for some constant C < oo such that
Now
-
A)
Now
r\
-Q^
sup y/n
h&He
0.
(V,
Q
J^^ (h^^ {(p, A) - Irjfl (A)) dX.
Next use integration by parts
is a'^
=
dX
A)
J — IT
TT
<
term
follows that the last
=
V
„27r
r(A, /xjl°,ee (Ms)
-
/
^(A, M)^n,e£
(A^)
^/^
dX
=
Op(l).
inner integral can be split into a part
"-1
"
f27r
-^0
5=1
"-1
/•27r(s+l)/n
= X]
[(7"(A,AXs)
/
-
T{X,ll))In,s6
ifJ-s)
+ r{X,IJ.)(In,,£
(/J-s)
-
In,ee
{^J'))]
dfM
r2iT/n
+
T(X,n)In,eei^)dlX
(11)
Jo
and o-2(^ ^^Zj^ r(A, /xj 27ri
< An
implying that
/(f"
t(A, n)dn). If M^ < A for all s
2TTt/n < 27r/n such that
"-1
/r27r
— y]^(A,/xJ-/
27r
n<T
Vni
<
i
then
<A—
29
T{X,fj,)diJ,
=
0(n-i/2)
27r(i
+
1)
> An and
1
uniformly in
A.
Using Markov's inequality we look at each term of (11) separately. First
"-1
u
yfnE
r27r(s+l)/n
{t^K Ms)
- t{X, lj))In,ee
{f^s) ^/^
l2TTs/n
"-1
IV —
<
/•27r(s+l)/n
p
Wn
sup
2ns/n
where sup^ \T{X,^g) —
=
t(A,/x)|
"-1
|r(A, Us)
-
t(A,
/x)
a'^dfj.
I
fJ-
A ^
if
+
[27rs/n, 27r(s
l)/n]
and
1
otherwise. Therefore
^
/•27r(s+l)/n
27r
727rs/n
V^l
A^
uniformly in A. Also
/•27r/n
^/nE
<a'
T{X,^)In,te{lj)d^
/
JO
where the bound
is
again uniform in
"-1
\fnE
"-1
2tt
^/n
A. Finally, since E{In,ee {^Js)
~
In,ee ifj))
=
0,
r2n(s+l)/n
T{X,jl){In,ee
~(
J2-n
s=l •'27rs/ra
(mJ "
-^n.ee (Ai))c^M
r27r{s+l)/n
g—l J2-Ks/n
where, from Brillinger (1981), p. 417, nvar(In,e£
2iTs/n < fi < 2n{s + l)/n. This shows that
n-l
uniformly in
A.
is
=
We
bounded
0{n~^) uniformly on
=
0{n-'/^)
can therefore consider
n
J —IT
is
In,ee{fJ'))
n
2n
which
—
j.2n
27r
y/nE
{/J^s)
in probability
n-l
^^(A,Ms)^°,£e(Ats) dX
s=l
by Markov's inequality if sup;^
bounded. From Brillinger (1981), Theorem
^ Yl^=i
5.10.1,
2
n-l
nE
nE
— ^T(A,/xJ/°_^^(AiJ
s=l
Jo
+
/
/
Jo
o/o
/e..e(/^l,Ai2>-j"l)c?/^l<^M2+C>(n
30
^)
'''(^^
f^s)^n,ee (Ms)
where the error is uniform in A. Then f^^{X)
bounded under assumption (A-1).
From Brillinger (1981), Theorem 5.10.1 and
=
cr'^
5.10.2,
and
it
/^..^ (/ii,/X2,
— /^i)
is
uniformly
follows immediately that
VnE
J —TI
is
bounded such that the second term
in (10)
is
the proof.
31
small in probability on Tis. This completes
References
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for
semiparametric econometric models via sto-
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Bickel, P.J. (1982)
On
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Brillinger,
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Modern Techniques and Their
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John
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&c
Heaton, J.C.
moment
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& Ogaki,
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&;
linear time
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Computing semiparametric
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&
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32
ARMA
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The Annals
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G.M.
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33
Table
Model
1:
Relative efRciency of
yt
:
=
(jnjt-i
+et
OLS
et
1'^
"-.ML
1'^
<Pn
9n
4>n
n
=
for
ARCH(l)
uth^'
'•iML
n =:512
== 256
innovations
ht
=
0.1
+ 7..eU
1'^
~,ML
(Pn
<Pn
n
=
1024
<!>
7i
0.0
0.0
0.9969
(0.0023)
(0.0058)
(0.0017)
(0.0041)
(0.0011)
(0.0029)
0.0
0.5
0.9918
0.5158
0.9947
0.4144
0.9984
0.3801
(0.0015)
(0.0158)
(0.0013)
(0.0134)
(0.0011)
(0.0126)
0.9619
0.4458
0.9631
0.3978
1.0294
0.9966
1.0126
0.9980
1.0107
0.0
0.9
0.9502
0.4432
(0.0031)
(0.0138)
(0.0043)
(0.0131)
(0.0046)
(0.0123)
0.5
0.0
0.9937
1.0272
0.9985
1.0087
0.9971
1.0106
(0.0030)
(0.0054)
(0.0021)
(0.0039)
(0.0016)
(0.0032)
0.5
0.5
0.9388
0.4761
0.9375
0.4515
0.9616
0.3658
(0.0056)
(0.0144)
(0.0054)
(0.0144)
(0.0082)
(0.0121)
0.5
0.9
0.9361
0.4083
0.9171
0.4478
0.9106
0.4759
(0.0078)
(0.0131)
(0.0096)
(0.0137)
(0.0108)
(0.0141)
1.0046
1.0190
1.0046
1.0096
1.0012
1.0025
(0.0040)
(0.0054)
(0.0029)
(0.0036)
(0.0022)
(0.0029)
0.5084
0.8958
0.4416
0.8590
0.3715
0.7
0.0
0.7
0.5
0.9071
(0.0070)
(0.0155)
(0.0081)
(0.0139)
(0.0076)
(0.0121)
0.7
0.9
5.5958
0.3998
0.8274
0.4470
0.8274
0.4470
(0.1854)
(0.0134)
(0.0120)
(0.0138)
(0.0120)
(0.0138)
1.0391
1.0097
1.0152
1.0101
1.0085
1.0022
(0.0069)
(0.0047)
(0.0053)
(0.0035)
(0.0037)
(0.0024)
0.5420
0.8522
0.4570
0.8308
0.4663
0.9
0.0
0.9
0.5
0.9209
(0.0101)
(0.0154)
(0.0094)
(0.0135)
(0.0094)
(0.0142)
0.9
0.9
0.7281
0.4352
0.5951
0.3486
0.4878
0.4247
(0.0123)
(0.0151)
(0.0113)
(0.0119)
(0.0107)
(0.0139)
is defined as Sfy/Spi^g or S]i^j^/SQj^g, respectively,
where S^ is the estimated variance of the estimator 4>- Numbers in parenthesis
are asymptotic standard deviations of the variance ratio. Results are
Relative Efficiency
based on 3000 replications.
34
Table
Model:
yt
=
(f)yt-i
Means and Medians
2:
+ et
et
= uthj
ht
1'^
"lOLS
= 0.1+7i£?_i
'-.ML
<Pn
4>n
<t>
7i
0.0
Mean
Median
Mean
Median
Mean
Median
-0.0007
-0.0006
-0.0006
-0.0005
-0.0007
-0.0007
0.0
0.5
-0.0025
-0.0014
-0.0024
-0.0016
-0.0009
-0.0009
0.0
0.9
0.0006
0.0000
0.0009
-0.0002
0.0011
0.0009
0.4978
0.4994
0.4969
0.4980
0.4977
0.4992
0.5
0.5
0.5
0.4927
0.4935
0.4908
0.4919
0.4987
0.4995
0.5
0.9
0.4770
0.4830
0.4694
0.4780
0.4945
0.4984
0.6962
0.6969
0.6948
0.6957
0.6962
0.6970
0.7
0.7
0.5
0.6933
0.6950
0.6913
0.6933
0.6979
0.7000
0.7
0.9
0.6779
0.6888
0.6733
0.6847
0.6945
0.6992
0.8965
0.8981
0.8947
0.8962
0.8965
0.8982
0.9
0.5
0.8948
0.8977
0.8933
0.8957
0.8982
0.8992
0.9
0.9
0.8850
0.8925
0.8860
0.8921
0.8984
0.8999
0.9
Samp
e size is 512.
Results are based on 3000 replications.
Relative Efficiency
0.95
0.9
0.85
0.8
0.75
AR
0.7-
0.2
0.4
0.6
0.8
Asymptotic efficiency of OLS relative to the IV estimator as a function of the
parameter
Generating mechanisms considered are from bottom to top: 7^ = .5, 7^ = .4,
Figure
1:
cj).
7i
=
.3
and 7^
=
.2.
35
0.65
0.70
0.75
0.80
0.85
OLS
Figure
2:
0.90
0.95
1.00
1.05
1.10
ARCH-ML
IV
Empirical density of parameter estimates for an AR(1) model with
the errors have
no
ARCH
effects
36
4>
=
.9
when
0.65
0.70
0.75
0.80
OLS
Figure
3:
0.85
0.90
0.95
1.00
1.05
1.10
ARCH-M
IV
Empirical densities of estimated
AR parameters
37
when
cj)
=
.9
and 7^
=
.9
7 U
/
b
Date Due
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MIT LIBRARIES
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