Journal of Statistical and Econometric Methods, vol.1, no.3, 2012, 63-78
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2012
Autoregressive Process Parameters Estimation
under Non-Classical Error Model
S. Ramzani1 , M. Babanezhad∗2 and M.A. Mohseni3
Abstract
Error in measuring time varying data setting is one important source
of bias in estimating of time series modeling parameters. When the
measurement error model is non-classic, this raises the question whether
the different measurement error model strategy might differently affect
the estimation of the time series modeling parameters. In this article,
we investigate this in Autoregressive (AR) model parameters estimation
under the non-classical measurement error model. We compare the
parameters estimation of the AR model under the classical and nonclassical error models. We perform analytically this on the AR model of
order p. Further, we confirm this through simulation study specifically
on the AR model of order 1.
Mathematics Subject Classification: 62M10
Keywords: Non-classical measurement error model, Autoregressive process,
Parameter estimation
1
2
3
Department of Statistics, Faculty of Sciences, Golestan University, Gorgan, Golestan,
Iran, e-mail: sakineh.ramzani@gmail.com
Department of Statistics, Faculty of Sciences, Golestan University, Gorgan, Golestan,
Iran, ∗ Corresponding author, e-mail: m.babanezhad@gu.ac.ir
Department of Statistics, Faculty of Sciences, Golestan University, Gorgan, Golestan,
Iran, e-mail: azim mohseni@yahoo.com
Article Info: Received : July 18, 2012. Revised : August 27, 2012
Published online : November 30, 2012
64
Autoregressive process parameters estimation under non-classical error model
1
Introduction
Similar to other types of data, time series data can be prone to measurement error (e.g., measuring of population abundances or density, air pollution
or temperatures levels, disease rates, medical indices) [1, 2]. That is a variable of interest is observed with some measurement error and modeled as an
unobserved component. Since various error models are already known to be
classical measurement error model [3, 4], (a measurement error is classical if it
is independent of unobserved variable), while in many situation measurement
error might follows the model as follows,
y t = γ 0 + γ 1 x t + ut
t = 1, ..., T
(1)
where the observed error prone variable yt is assumed to be related to the true
unobserved xt through the above model [2, 3]. This error model is referred
to as the non-classical measurement error model or error calibration model [3,
4]. Model (1) is the classical measurement error model when γ0 = 0, γ1 = 1.
This implies that E(Yt |Xt = xt ) = γ0 + γ1 xt , unlike the classical error model,
suggesting that Yt is a possibly biased measure of Xt . Here, it is assumed that
Xt and Ut are independent in each time t [3, 4]. For example [5] examined
the time series studies of air pollution (effect of P M10 ) and mortality, if Xt
and Yt are respectively the average personal exposure to P M10 and measured
ambient P M10 concentration on day t, then the slope γ1 measures the change
in personal exposure per unit change in a measured ambient concentration and
the intercept γ0 represents personal exposure to particles that does not drive
from external sources, but arises from particle clouds generated by personal
activities or unmeasured micro-environments.
Since Autoregressive process has a clear structure in presence measurement
errors, therefore it is a popular choice for modeling of time series in many fields.
This is especially true in population dynamics where AR(1), AR(2) models are
often employed (see e.g., [6, 7]). [4] develops tests to identification of measurement error in long and short memory series. [8] studies the parameters
estimation of Autoregressive models using naive, ARMA and corrected naive
methods under the classical error model and heteroskedastic measurement errors. [9] studies the asymptotic properties of estimators based on Yule-Walker
equations in autoregressive models with measurement error.
65
S. Ramzani, M. Babanezhad and M.A. Mohseni
The goal of this paper is to estimate the parameters of AR(p) process under
the non-classical error model (1), and compare the bias and corresponding
standard errors of the estimators with the classical error model.
In model (1), {xt } the true time series agent which is unobserved, instead
we observed {yt }, and {ut } is measurement error with heteroskedastic variance
i.e. σu2t .
Without loss of generality, suppose that unobserved time series {xt } is
mean zero Autoregressive process model of order p, AR(p),
Xt = φ1 Xt−1 + ... + φp Xt−p + et
t = p + 1, ..., T
(2)
where et is white noise (Random variables e1 , e2 , ..., en are independent and
identically distributed with a mean of zero. This implies that the variables all
have the same variance σe2 and Cor(ei , ej ) = 0 for all i 6= j. If, in addition,
the variables also follow a normal distribution (i.e., et ∼ N (0, σe2 )) the series
is called Gaussian white noise). This paper is summarized as follows. In the
next Section, the homoskedastic and heteroskedastic measurement errors are
described. In Section 3, we obtain the AR estimators and theirs variances
using different method under the non-classical error model. We compare the
extracted estimators with that [8] examined under the classical error model.
Section 4 provides the result of simulation studies. It should be noted that
method based on the ARMA result among the existing (naive and corrected
naive methods) is free from the estimation of σ̂u2t .
2
Hemosckedastic and Heteroskedastic Measurement Errors
Hemosckedastic measurement error refers to case that the variance of measurement error is constant. While historically constant measurement error
variances have been used, there are many situations that changing variances
be allowed, i.e., with respect to definition of the Heteroskedastic measurement error, we are allowed for the fact that measurement error variances could
change across observation. The variances of the Heteroskedastic measurement
errors introduce as a function of sampling effort and unobserved variable, i.e.
66
Autoregressive process parameters estimation under non-classical error model
σu2t (xt ) which may depend on xt through a function h(xt ) and may depend on
the sampling effort at time t. For asymptotic purpose it is assumed when the
measurement error is Heteroskedastic, σu2 is defined as the limit average of σu2t
as follows [8],
PT
2
t=1 σut
= σu2
(3)
limT →∞
T
3
Estimators under the Non-Classical Error
Model
In this section, we obtain the parameter estimation of AR process using
different methods. To make clear our example, we focus on the AR(p) and
provide our simulation on AR (1). We first assume Hemosckedastic measurement errors, we obtain four estimators so called the propose ARMA, naive
estimators, and corrected estimators.
3.1
Gold Estimator (GE)
Suppose {Xt }, is an AR(p) process as model (2) with the unknown parameter φu = [φ1 , , φp ]0 and σe2 . Parameter φbGold , without presence the measurement
error and using Yule- walker estimation method (YW), derived from the following equations,
b Gold = 0,
[b
γ(1) , , γ
b(p) ]0 − Γφ
−1 φ0u Ĝφu
=0
+
σe
σe3
b p × p sample covariance
whereb
γ(k) , is lag k of sample Autocovariance from Γ,
b are defined as follows,
matrix and G




γ
b(0) −b
γ(1) −b
γ(2) · · · −b
γ(p)
γ
b(0) · · · γ
b(p−1)




−b
γ
γ
b
γ
b
·
·
·
γ
b
(1)
(0)
(1)
(p−1)


.
.
...
b =  ..
b= .
.
..  , G
Γ
.
.
.
 .

.
.
.
.
.
.
.


γ
b(p−1) · · · γ
b(0)
−b
γ(p) γ
b(p−1) γ
b(p−2) · · · γ
b(0)
67
S. Ramzani, M. Babanezhad and M.A. Mohseni
3.2
Naive Estimator (NE)
Using some regularity conditions, the convergence of sample Autocovariances can be determined as follows [10]
p
γ
by,|t−s| → γ12 γ|t−s|
and
p
γ
by,(0) → γ12 γ(0) + σu2
p
∀t 6= s
∀t = s
where → stands for convergence in probability. As T → ∞, the parameters
estimating equation for φ is following;
  2
γ1 γ
b(0) + σu2 · · ·
γ12 γ
b(1)
 ..  
..
...
 . −
.

γ12 γ
b(p−1)
γ12 γ
b(p)
···
γ12 γ
b(p−1)
..
.
γ12 γ
b(0) + σu2

φ1
  .. 
  .  = 0.

φp
2
Hence, the naive estimators (φbnaive , σ
be,naive
) under the error model (1) and
using YW methods will converge in probability to the following equations;
and
p
φbnaive → (γ12 Γ + σu2 I)−1 γ12 γ T = (γ12 Γ + σu2 I)−1 γ12 ΓφGold ,
p
2
σe,naive
→ γ12 γ(0) + σu2 − γ12 γ T (γ12 Γ + σu2 I)−1 γ12 γ.
2
Also the asymptotic bias of σe,naive
will converge in probability to;
p
2
2
σ
be,naive
−σ
be,Gold
→ ((γ12 − 1)γ(0) ) + σu2 − γ12 γ T (γ12 Γ + σu2 I)−1 γ12 γ + γ T (Γ)−1 γ.
For example when p = 1, i.e. AR(1), it is implied that,
γ2γ
p
φbnaive →
γ12 γ(1)
= λφGOld ,
γ12 γ(0) + σu2
where λ = γ 2 γ1 (0)
2.
1 (0) +σu
Using the results of [8], the naive estimator of φ under the classical error model
(γ0 = 0, γ1 = 1),
γ(1)
p
= λφGOld ,
φbnaive →
γ(0) + σu2
where λ =
γ(0)
2.
γ(0) +σu
68
Autoregressive process parameters estimation under non-classical error model
3.3
ARMA Estimator
ARMA method is based on the fact that if Xt is an AR(p) process with
parameters φ and ut is i.i.d with mean zero and constant variance σu2 , then
([11, 12]),
φ(B)(yt − γ0 ) = γ1 et + φ(B)ut = θ(B)bt ,
follows ARM A(p, p), where parameters in the ARMA model are the same as
in the original Autoregressive model. Also bt is white noise with mean zero
and variance σb2 , φ(z) = 1−φ1 z −...−φp z p , θ(z) = 1−θ1 z − −θp z p and B is the
backward shift operator with B j Wt = W(t−j) . A simple approach to estimate
φARM A , is to fit an ARM A(p, p) model to the observed {Yt − γ 0 } series. The
asymptotic properties of φbARM A , can be computed. For example, when p=1
([10], chapter 8),

!−1 
!
!
−1
2 −1
b
(1 − φ )
(1 + φθ)
φARM A
φ
∼
.
, T −1
=N
−1
−1
b
(1 + φθ)
(1 − θ2 )
θARM A
θ
It can be conclude φbARM A ∼
= AN
φ,
(1+φθ)2 (1−φ2 )
T (φ+θ)2
, where AN is denoted ap-
proximately normal. There are two solutions for θ1 , but only one leads to a
stationary process. Defining,
γ12 σe2 + σu2 (1 + φ21 )
,
k=
φ1 σu2
then,
θ1 =
−k +
and
θ1 =
−k −
√
√
k2 − 4
,
2
if 0 < φ1 < 1
k2 − 4
, when − 1 < φ1 < 0.
2
Therefore for classical error model, one can obtain the similar result as
non-classical one just by substituting σe2 instead of γ12 σe2 , because γ1 et is white
noise with mean zero and variance γ12 σe2 .
3.4
Corrected Naive Estimator (CNE)
The problem with the naive estimator arises from the bias in the estimator
69
S. Ramzani, M. Babanezhad and M.A. Mohseni
of γy,(0) . For the eliminate bias, we obtain the following corrected estimator;
PT
−1
T
bY − σ
φbCN E = Γ
bu2 Ip
γ
bY,(1) , ..., γ
bY,(p)
2
σ
but
. Like moment estimator in linear regression, φbCN E can
where
= t=1
T
have problems for small samples. One reason is the distribution of the corrected estimator of γy,(0) , γ
by,(0) − σ
bu2 can have positive probability around zero,
so the estimate of γy,(0) may be negative (Bounaccorsi, 2010). To gard this
bY − σ
problem,we propose a modified version of this estimator; if Γ
bu2 Ip is non
positive definite, then φbCN E := φbnaive . In line with the proposition of [8], the
estimator φbCN E is consistent if,
PT
σ
b2 p
2
(4)
limT →∞ σ
bu = t=1 ut → σu2
T
σ
bu2
Theorem 1 Consider model AR(1), assuming independence Ut and σU2 t of
Xt , and using the following equations,
PT
2
γ(0)
γ12 γ(0)
t=1 σut
2
σu = limT →∞
, λ́ =
,
,λ = 2
2
T
γ1 γ(0) + σu
γ(0) + σu2
and
σσ2u2
= limT →∞ var
σ
bu2
, σu4
= limT →∞
P
σu4t
, E(u4 ) = limT →∞
T
t
P
t
E(u4t )
,
T
then the asymptotic variance of φbCN E under non-classical error model is as
follows,
"
#
2
2
2
4
4)
2
−
λ
φ
φ
1
(1
−
φ
)
σ
E(u
u
γ14 1 − φ2
+
+ 2 σσ2u2 .
avar(φbCN E ) =
+
2
2
T
λ
γ(0)
γ(0)
γ(0)
By assuming variance σu2 to be constant and E(u4 ) = δσu4 ,
"
#
2 2
2
2
−
λ
(1
−
λ́)
[φ
(δ
−
1)
+
1]
φ
1
γ14 1 − φ2
+ 2 σσ2u2 +
avar(φbCN E ) =
,
T
λ
γ(0)
λ́2
and under normality measurement error, i.e δ = 3;
1
avar(φbCN E ) =
T
"
γ14
#
2
2
2
φ
2
−
λ
(1
−
λ́)
[2φ
+
1]
.
+ 2 σσ2u2 +
1 − φ2
λ
γ(0)
λ́2
70
Autoregressive process parameters estimation under non-classical error model
( For more details of proof see Appendix).
Finally using the proposition (3) of [8], it can be concluded that the asymptotic variance of φbCEE has approximately normal distribution. Under the classical error model, similar results can be observed by substituting γ0 = 0, γ1 = 1.
4
Simulation study
In this section, simulation studies are carried out for AR(1) process. In
order to investigate the bias of the estimators and theirs standard errors, a
simulation study is done for the case where {xt } is AR(1) process. The simulation is done in three steps:
• At first step, T segment was simulated from AR(1) for {xt }
• At second step, by assuming normality of the measurement error ut , yt
was simulated from model (1) for all the different values of γ1 between
(0.5, 2).
• At third step, the biases and standard errors of estimators are evaluated,
with 500 replication for steps (1) and (2), given different values of β =
0.1, 0.3, 0.5, 0.7, 0.9 and T = 100, 200, 500, λ = 0.5, 0.75, 0.9 under the
non-classical error model.
Figure 1, represents the simulation result under the non-classical measurement error model. The left and right panels are respectively bias and standard
error of the estimators. It is clear that biases are decreased when the values
of T are increased, but bias behavior of the estimators is uniform by changing
values of the γ1 , that is it shows the low influence of the γ1 . Also it can be
observed that the bias of the naive estimator is more than the other estimators.
Similarly, We observe that the influence of γ1 and T respectively in the naive
and ARMA estimators standard error.
Figure 2 shows the second scenarios of the obtained estimator through the
considered values of φ. It can be seen that, for the small values of φ, the bias of
the ARMA estimator is more than the other estimators, because identification
from of the ARMA model be harder.
71
S. Ramzani, M. Babanezhad and M.A. Mohseni
0.5
1.0
1.5
Average S.E by T=200
1.0
1.5
S.E
Average bias by T=200
0.0 0.4 0.8
gamma1
−0.06
gamma1
2.0
0.5
1.0
1.5
gamma1
2.0
2.0
cee
ARMA
naive
Gold
0.05
−0.12
1.5
0.15
Average S.E by T=500
S.E
Average bias by T=500
−0.04
gamma1
1.0
2.0
cee
ARMA
naive
Gold
gamma1
cee
ARMA
naive
Gold
0.5
1.5
2.0
cee
ARMA
naive
Gold
0.5
S.E
3.0
1.5
cee
ARMA
naive
Gold
0.0
−0.06
1.0
−0.14
bias
0.5
bias
Average s.E by T=100
cee
ARMA
naive
Gold
−0.14
bias
Average bias by T=100
0.5
1.0
1.5
2.0
gamma1
Figure 1: bias of the estimators of the Autoregressive process of order one for
γ1 = (0.5, 2) and T = 100, 200, 500 and φ = 0.5, λ = 0.75.
Figure 3 represents the simulation result of the corresponding estimator’s
standard error. It clearly can be seen that by changing the considered values of
φ, the naive and ARMA estimators considerably vary. This might be because
of the impact of increasing of the standard error of γˆ1 .
Figure 4 shows the presentation of the estimators behavior and corresponding standard errors by different values of the λ, where λ is function of γ1 and
γ2γ
σu2 , i.e., λ = γ 2 γ1 (0)
2 . As is shown, when λ values is increased, bias is de1 (0) +σu
creased. Because it might be the les impact of σu2 . The standard error of the
naive estimator is more than the ARMA estimator when the values of γ1 and
λ is simultaneously increased.
72
Autoregressive process parameters estimation under non-classical error model
1.5
−0.02
bias
2.0
0.5
1.0
1.5
Average bias by p=.5
Average bias by p=.7
−0.12
1.0
1.5
2.0
gamma
bias
gamma
−0.15 −0.05
gamma
cee
ARMA
naive
Gold
0.5
cee
ARMA
naive
Gold
−0.08
−0.04
1.0
−0.04
0.5
bias
Average bias by p=.3
cee
ARMA
naive
Gold
−0.12
bias
Average bias by p=.1
2.0
cee
ARMA
naive
Gold
0.5
1.0
1.5
2.0
gamma
−0.10
cee
ARMA
naive
Gold
−0.25
bias
Average bias by p=.9
0.5
1.0
1.5
2.0
gamma
Figure 2: bias of the estimators of the Autoregressive process of order one for
γ1 = (0.5, 2) and φ = 0.1, 0.3, 0.5, 0.7, 0.9 and T = 300, λ = 0.75.
73
S. Ramzani, M. Babanezhad and M.A. Mohseni
Average S.E by p=.3
1.0
1.5
S.E
0.5
1.0
1.5
1.0
1.5
2.0
gamma
0.20
Average S.E by p=.7
2.0
cee
ARMA
naive
Gold
0.05
S.E
Average S.E by p=.5
0.20
gamma
cee
ARMA
naive
Gold
0.5
1.0
2.0
gamma
0.05
S.E
0.5
cee
ARMA
naive
Gold
0.0
4
cee
ARMA
naive
Gold
0
S.E
8
2.0
Average S.E by p=.1
0.5
1.0
1.5
2.0
gamma
0.20
cee
ARMA
naive
Gold
0.05
S.E
Average S.E by p=.9
0.5
1.0
1.5
2.0
gamma
Figure 3: standard error of the estimators of the Autoregressive process of
order one for γ1 = (0.5, 2) and φ = 0.1, 0.3, 0.5, 0.7, 0.9 and T = 300, λ = 0.75.
74
Autoregressive process parameters estimation under non-classical error model
Average s.E by lamda=.5
1.0
1.5
S.E
0.4
1.0
1.5
2.0
1.5
cee
ARMA
naive
Gold
0.1
−0.06
1.0
0.3
Average S.E by lamda=.75
S.E
Average bias by lamda=.75
2.0
0.5
1.0
1.5
2.0
Average S.E by lamda=.9
−0.06
1.0
1.5
gamma1
2.0
S.E
Average bias by lamda=.9
0.05 0.15
gamma1
−0.02
gamma1
cee
ARMA
naive
Gold
0.5
0.5
gamma1
cee
ARMA
naive
Gold
0.5
bias
2.0
gamma1
−0.14
bias
0.5
cee
ARMA
naive
Gold
0.1
cee
ARMA
naive
Gold
−0.25 −0.10
bias
0.7
Average bias by lamda=.5
cee
ARMA
naive
Gold
0.5
1.0
1.5
2.0
gamma1
Figure 4: bias of the estimators of the Autoregressive process of order one for
γ1 = (0.5, 2) and λ = 0.5, 0.75, 0.9 and T = 300, φ = 0.5.
75
S. Ramzani, M. Babanezhad and M.A. Mohseni
5
Conclusion
This study concerns the estimation of parameter for AR(1) process in the
presence of non-classical error model. From theoretical point of view and also
simulation study, it is concluded that Corrected naive and Gold estimators
have similar behaviors from biases and the standard errors in the presence of
the non-classical measurement error model. Moreover, when the parameters
of the model are like that the model is markedly far from to be a white noise,
i.e., φ and γ1 are increased simultaneously, it can be seen that the bias and
standard error of the ARMA estimator closed to CNE and GE. Further the
naive estimator has not good performance, because with increasing γ1 parameter, the corresponding standard error is increased. A comparison between the
estimators, it can be obtained that Corrected naive estimator is more efficient
than the other estimators. These results show the trace of non-classical measurement error model and measurement error, and considering that tends to
be more valuable and informative.
ACKNOWLEDGEMENTS. This study was granted by Golestan University, Gorgan, Golestan, Iran.
Appendix
In this Section we shad light on some formulas that we have pointed out
in the text.
!
√
0
γ
by − γy
.
is
obtained,
where
γ
=
γ
,
...,
γ
The asymptotic behavior of T
y
y,(0)
y,(p)
σ
bu2 − σu2
The variance of φbCEE is determined by using the delta method.
P
Z
If Zt0 = [Yt2 , Yt Y!t+1 , ..., Yt Yt+p ] and Z = Tt t , then the asymptotic behavior
√
√
γ
by − γy
T
T Z. By taking the limit exists, we can
is
equivalent
to
σ
bu2 − σu2
√
evaluate limT →∞ Cov( T Z) = Q, where
Q=
"
Qγ Qγ,σu2
Qσu2 ,γ Qσ2 2
σu
#
.
76
Autoregressive process parameters estimation under non-classical error model
For computation of the matrix Q, we should obtain elements of the matrix Q,
i.e., limT →∞ Cov(b
γy,(p) , γ
by,(r) ) and limT →∞ Cov(b
γy(r) , σ
bu2 ). To do so, we obtain
limT →∞ Cov(b
γy,(p) , γ
by,(r) )
X
1 X
(Ys − E(Ys ))(Ys+r − E(Ys+r )))
= limT →∞ E( (Yt − E(Yt ))(Yt+p − E(Yt+p ))
T
t
s
− limT →∞ T E(b
γy,(p) )E(b
γy,(r) )
1 XX
= limT →∞ (
E ((γ1 Xt + Ut )(γ1 Xt+p + Ut+p )(γ1 Xs + Us )(γ1 Xs+r + Us+r )))
T t s
− γy,(p) γy,(r) .
Now it follows from the latter that,
E(γ1 Xt + Ut )(γ1 Xt+p + Ut+p )(γ1 Xs + Us )(γ1 Xs+r + Us+r ).
Assuming independence of Xt and Ut , we have,
= I(r = 0)[γ12 E(Xt Xt+p )E(Us2 )] + I(p = 0)[γ12 E(Xs Xs+r )E(Ut2 )]
2
+ I(p = r = 0)[E(Ut2 Us2 )] + I(p = r 6= 0)[E(Ut2 Ut+p
)]
+ I(s = t + p − r)[γ12 E(Xt Xs )E(Ut+p Us+r )]
+ I(s = t + p)[γ12 E(Xt Xs+r )E(Ut+p Us )]
+ I(s = t − r)[γ12 E(Xt+p Xs )E(Ut Us+r )]
+ I(s = t)[γ12 E(Xt+p Xs+r )E(Ut Us )] + [γ14 E(Xt Xt+p Xs Xs+r )].
Replacing in the previous equation, it can be concluded,
limT →∞ Cov(b
γy,(p) , γ
by,(r) )
X
X
1
I(r = 0)[γ12 E(Xt Xt+p )E(Us2 )])
= limT →∞ (
T t s
1 XX
+ limT →∞ (
I(p = 0)[γ12 E(Xs Xs+r )E(Ut2 )])
T t s
1 XX
I(p = r = 0)[E(Ut2 Us2 )])
+ limT →∞ (
T t s
1 XX
2
+ limT →∞ (
I(p = r 6= 0)[E(Ut2 Ut+p
)])
T t s
1 XX
+ limT →∞ (
I(s = t + p − r)[γ12 E(Xt Xs )E(Ut+p Us+r )])
T t s
77
S. Ramzani, M. Babanezhad and M.A. Mohseni
1 XX
+ limT →∞ (
I(s = t + p)[γ12 E(Xt Xs+r )E(Ut+p Us )])
T t s
1 XX
I(s = t − r)[γ12 E(Xt+p Xs )E(Ut Us+r )])
+ limT →∞ (
T t s
1 XX
+ limT →∞ (
I(s = t)[γ12 E(Xt+p Xs+r )E(Ut Us )])
T t s
1 XX 4
[γ E(Xt Xt+p Xs Xs+r )]) − γy,(p) γy,(r) .
+ limT →∞ (
T t s 1
If Cov(b
γy,(p) , γ
by,(r) ) = qγrp , then using simple calculation for different form
of r, p = 0, 1, we will have;
qγ,00 = γ14 q00 + 4γ12 γ(0) σu2 + E(u4 ) − σu4
qγ,01 = γ14 q01 + 4γ12 γ(1) σu2
qγ,11 =
γ14 q11
+2
γ12 γ(0) σu2
+
γ12 γ(2) σu2
and also it implies Qγb,bσu2 = 0, because
+ limT →∞
PT
t=1
σu2t σu2t+1
T
1 XX
E(Yt Yt+p )b
σu2t ) − γy,(p) σu2 = 0.
limT →∞ Cov(b
γy,(p) , σ
bu2 ) = limT →∞ (
T t r
Now, is obtained the corrected naive estimator under AR(1) model, φbCN E =
using three variables of the Taylor expansion evaluated at (b
γy,(0) , γ
by,(1) , σ
bu2 ),
we have,
φ 2
1
(b
γy,(1) − φb
γy,(0) ) +
σ
b .
φbCN E = φGold +
γ(0)
γ(0) u
Then,
φ2 2
1 2
q
+
φ
q
−
2φq
+
var(φbCN E ) =
σσu2 .
γ11
γ00
γ01
2
2
T γ(0)
T γ(0)
γ
by,(0)
2,
γ
by,(0) −b
σu
Substituting var(φGold ) =
1
2
2 [q11 +φ q00 −2φq01 ]
T γ(0)
(where qrp is auto-covariance
between γ
bx,(p) and γ
bx,(r) ) in the above equation, it can be compute that,
"
P 2 2 #
2
1
−
φ
1
t σut σut+1
2 2
var(φbCN E ) = γ14
+
lim
+
2γ
σ
γ
+
γ
T
→∞
(0)
(2)
1 u
2
T
T γ(0)
T
h
i
1
1 2
2
2
2
2
4) − σ4 ) −
8φγ
γ
σ
φ
(4γ
γ
σ
+
+
E(u
(1)
(0)
1
u
1
u
u
2
2
T γ(0)
T γ(0)
+
φ2 2
σσu2 .
2
T γ(0)
Proof is proved with simplification.
78
Autoregressive process parameters estimation under non-classical error model
References
[1] A.J. Scott, T.M.F. Smit and R.G. Jones, The application of time series
methods to the analysis of repeated surveys, International Statistical Review, 45, (1977), 13-28.
[2] A.J. Scott and T.M.F. Smit, Analysis of repeated survays using time series
methods, Journal American Statistical Association, 69, (1974), 674-678.
[3] J.P. Bounaccorsi, Measurement error models, methods and applications,
New York, Chapman Hall/CRC, Taylor and Francis Group, 2010.
[4] K. Tanka, A unified approach to the measurement error problem in time
series Models, Econometric Theory, 18, (2002), 278-296.
[5] F. Dominici and S. Zeger, A measurement error model for time-series
studies of air pollution and mortality, Biostatistics, 1, (2000), 157-175.
[6] B. Dennis and M.L. Taper, Density dependence in time series observations
of natural populations: estimation and testing, Ecology, 64, (1994), 205224.
[7] D.W. Williams and A.M. Liebhold, Influence of weather on the synchrony
of gypsy moth (Lepidoptera: Lymantriidae) outbreaks in New England,
Envirnomental entomlogy, 24, (1995), 987-995.
[8] J. Staudenmayer and J.P. Bounaccorsi, Measurement error in linear
autoregressive models, Journal American Statistical Association, 471,
(2005), 841-852.
[9] K.C. Chanda, Asymptotic properties of estimators for autoregressive models with errors in variables, The Annals of Statistics, 24, (1996), 423-430.
[10] P.J. Brockwell and R.A. Davis , Time series: Theory and methods, Second
Edition, New York, Springer, 1991.
[11] G.E.P. Box, G.M. Jenkins and G. Reinsei Time Series Analysis: Forecasting and Control, 3rd ed., New York: Prentice-Hall, 1994.
[12] M. Pagano, Estimation of Models of Autoregressive Signal Plus White
Noise, The Annals of Statistics, 2, (1974), 99-108.