We propose to use the analytical generalised least squares (GLS) transformation matrix of Galbraith and Zinde-Walsh (1992) to correct finite sample estimation error of MA(q) processes parameters estimates. Our method may be considered as an iteration of the analytical indirect inference estimator (AIIE) of Galbraith and Zinde-Walsh (1994). Its potential is explored through a series of Monte Carlo experiments.MA models, Analytical indirect inference, GLS.

Patrick Richard

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 4 
  
Improving the accuracy of the analytical indirect inference estimator for MA 
models. 
  
 
 
Patrick Richard  
GREDI, Universite de Sherbrooke. 
Abstract 
We propose to use the analytical generalised least squares (GLS) transformation matrix of Galbraith and Zinde-Walsh 
(1992) to correct finite sample estimation error of MA(q) processes parameters estimates. Our method may be 
considered as an iteration of the analytical indirect inference estimator (AIIE) of Galbraith and Zinde-Walsh (1994). Its 
potential is explored through a series of Monte Carlo experiments.
This research was funded, in part, by a grant from the Fonds Québécois de Recherche sur la Société et la Culture. I thank Russell Davidson, 
John Galbraith, James MacKinnon, David Stephens and Victoria Zinde-Walsh as well as the participants at the 2006 CIREQ Ph.D. student 
conference for their comments on various versions of this paper. All errors remain mine. 
Citation: Patrick Richard, (2009) ''Improving the accuracy of the analytical indirect inference estimator for MA models.'', Economics 
Bulletin, Vol. 29 no.4 pp. 2795-2802. 
Submitted: Jun 19 2009.   Published: November 09, 2009. 
 
     1. Introduction
It is well known that estimators for MA(q) models parameters are biased in ﬁnite
samples. For example, Cordeiro and Klein (1994) derive analytical expressions for
the bias of the maximum likelihood estimator (MLE). Bias corrected estimators
can be derived from these equations, but they become complicated as the order of
the MA process increases.
The bias of an estimator typically is inversely related to sample size. Thus,
biased estimators often have quite large ﬁnite sample estimation errors, that is,
the value of the parameter estimated in a given small sample may be very far from
the true value. In this paper, we introduce a simple way to estimate this error and
obtain a corrected estimate.
Our method is especially well suited for the analytical indirect inference esti-
mator (AIIE) of Galbraith and Zinde-Walsh (1994). Although not as popular as
MLE, this estimator, which exploits the analytical binding functions that exist be-
tween the parameters of MA and AR models, is nevertheless often used in applied
work, see for example, Tkacz (2007). Among its advantages is the fact that it is
more robust to misspeciﬁcation than MLE. It may also be very useful when one
uses a simulation-based estimatior which requires estimation of an MA process at
each step, see for example, Hryshko (2006) for such an application.
The theoretical foundation of the AIIE is that any invertible MA(q) process has
an AR(∞) form which can be approximated arbitrarily well in ﬁnite samples by
an autoregression of order p. Among other things, this means that if the ultimate
goal is to forecast future values of the process, then one can simply use an AR(p)
model. However, the AR(p) approximation, and consequently the forecasts, may
be quite imprecise if p is small, so that it may still be preferable to estimate the
MA(q) process in small samples. If, on the other hand, the MA(q) process arises
from the formulation of a theoretical model, as it does in expectation models such
as the one used by Tkacz (2007), then estimation of its parameters is necessary
to compute the GLS estimator. Tests of the rational expectation hypothesis in
currency exchange markets such as proposed by Hansen and Hodrick (1980) also
give rise to regression models with MA error terms.
12. GLS correction for MA(q) models
Consider n observations of an invertible MA(q) process:
ut = θ1εt−1 + θ2εt−2 + ... + θqεt−q + εt
where the εts are iid innovations with mean 0 and ﬁnite variance σ2
ε. Let ˆ θ be a
consistent estimator of the q-vector of true parameters θ0. Let θ0
j denote the true
value of the jth parameter and assume that ˆ θj = θ0
j +bj where bj is the estimation
error of the jth element of ˆ θ. Also, let Ψ(θ0) be the GLS transformation matrix of
Galbraith and Zinde-Walsh (1992) evaluated at θ0 such that Ψ(θ0)Ψ>(θ0) = Σ−1,
where Σ = E(uu>) and u is a n-vector with typical element ut. Let ν = Ψ>(ˆ θ)u
denote the n × 1 vector of residuals obtained when Ψ is evaluated at ˆ θ.
Observation 1.
Assuming that ε0 = ε−1 = ... = ε−q+1 = 0, which is harmless asymptotically,
νt =
∞ X
i=1
γiνt−i + εt (1)
where
γi =
min{i,q} X
k=1
−γi−kθ
0
k − bi, γ0 = 0. (2)
Proof
The element in position i,j of the matrix Ψ>(ˆ θ) is determined by:
hi,j =

 
 
0 if i < j,
1 if i = j,
−
Pmin{i−1,q}
k=1 (θ0
k + bk)hi−k,j otherwise.
Thus, we have the following equations, where the 0 superscript is omitted for ease
of notation:
ν1 = ε1 (3)
ν2 = −b1ε1 + ε2 (4)
ν3 = (b
2
1 + θ1b1 − b2)ε1 − b1ε2 + ε3 (5)
2ν4 = (θ1b2+θ2b1+2b1b2−b
3
1−2θ1b
2
1−θ
2
1b1−b3)ε1+(b
2
1+θ1b1−b2)ε2−b1ε3+ε4 (6)
ν5 = (θ
3
1b1+3θ1b
3
1+3θ
2
1b
2
1+b
4
1−θ
2
1b2−2θ1θ2b1−4θ1b1b2−2θ2b
2
1−3b
2
1b2+θ1b3+θ3b1 (7)
+2b1b3 + θ2b2 + b
2
2 − b4)ε1 +
￿
−θ
2
1b1 − 2θ1b
2
1 − b
3
1 + θ1b2 + θ2b1 + 2b1b2 − b3
￿
ε2
+
￿
b
2
1 + θ1b1 − b2
￿
ε3 − b1ε4 + ε5.
Solving (3), (4), (5) and (6) for ε1, ε2, ε3 and ε4 and substituting in equation (7),
ν5 = −b1ν4 + (θ1b1 − b2)ν3 +
￿
−θ
2
1b1 + θ1b2 + θ2b1 − b3
￿
ν2 (8)
+
￿
θ
3
1b1 − θ
2
1b2 − 2θ1θ2b1 + θ1b3 + θ3b1 + θ2b2
￿
ν1 + ε5
which has the expected form. Further substitutions yield the stated result. •
The equations (2) are generalisations of the equations used by Galbraith and
Zinde-Walsh (1994) to develop their AIIE of MA parameters. This can be seen by
placing an original estimator such that ˆ θi = 0 for all i in the GLS transformation
matrix so that bi = −θ0
i for all i. If ˆ θ is the AIIE, then estimating bi through
equations (2) could be considered as an iteration of this estimator.
Based on observation 1, we propose to estimate bi for each parameter one at a
time and to deﬁne the corrected estimator in the following recursive way:
1. Use the initial estimate to obtain a vector of ﬁltered data: ˆ ν = Ψ>(ˆ θ)u.
2. Fit a long autoregression to ˆ νt. Then, estimate the error of ˆ θ1 as ˆ b1 = −ˆ γ1 and
compute the corrected estimator, ˜ θ1 ≡ ˆ θ1 −ˆ b1.
3. Use steps similar to 2 to get corrected estimates of any other parameter. That
is, compute the corrected estimators ˜ θj ≡ ˆ θj −ˆ bj where ˆ bj = −ˆ γj −
Pmin{i,q}
j=1 ˆ γi−j˜ θj
for j = 2,...,q.
This correction scheme can be iterated by using ˜ θ in the GLS transformation
matrix so as to obtain a new vector of ﬁltered data ˜ ν = Ψ>(˜ θ)u and going through
steps 2 and 3 again. Similar results exist for AR(p) and ARMA(p,q) models (see
Richard, 2007). Note that the proposed method is only valid if the MA process
described by ˆ θ is invertible since the analytical transformation matrix requires this
assumption. Also, notice that if one replaces ˆ θ by E(ˆ θ) in observation 1, then
equations (2) become bias equations and ˜ θ becomes a bias corrected estimator.
This avenue, which is a bit less intuitive than what we do here (indeed, it is not
clear how to link Ψ(E(ˆ θ)) and Ψ(ˆ θ)), is explored in Richard (2007).
33. Simulations
We now use Monte Carlo simulations to assess the quality of the GLS correction.
Throughout this section, error terms are drawn from a N(0,1) distribution and
the order of the autoregressions necessary to compute the AIIE and the GLS bias
correction are chosen with the Akaike information criterion. Because of its close
relation with the GLS correction, we focus on the AIIE of Galbraith and Zinde-
Walsh (1994). GLS correction of the MLE is considered in Richard (2007) and is
not in general recommendable.
Figures 1 and 2 consider the estimation of a MA(1) model with a sole parameter
θ, meaning that the data is generated from the process yt = θεt−1 + εt with
εt ∼ NID(0,1), for all t = 1,2,...,N. It can be seen that the GLS correction
greatly improves the small sample properties of the AIIE estimator when θ is close
to the non-invertibility region and only slightly increases its mean squared error
(MSE) for moderate values. Iterating the GLS correction appears to be useful
in small samples when θ is extreme. On the other hand, ﬁgure 2 indicates that
there is no advantage (nor inconvenient) in using the GLS correction in very large
samples.
Figure 3 considers the estimation of an MA(2) process. The data is generated
from the process yt = θ1εt−1+θ2εt−2+εt with εt ∼ NID(0,1), for all t = 1,2,...,N.
The determinant of the MSE is plotted as a function of the modulus closest to 1
of the roots of the lag polynomial corresponding to the diﬀerent MA(2) processes
used. Evidently, the GLS correction is very useful, although iterating it does not
seem to produce any gain.
4. Conclusion
We propose a bias correction technique for the estimation of MA(q) models based
on the analytical GLS transformation matrix of Galbraith and Zinde-Walsh (1992).
Our simulations indicate that it may provide accuracy gains when used along with
the AIIE of Galbraith and Zinde-Walsh (1994). A similar technique can be devised
for AR(p) and ARMA(p,q) models but unreported simulations indicate that it does
not perform as well as the one proposed here. It is likely that one could use it
along with the AIIE for VMA models of Galbraith, Ullah and Zinde-Walsh (2002).
4References
Cordeiro, G. M. and R. Klein (1994). Bias correction in ARMA models, Statistics
and Probability Letters, 19, 169-76.
Galbraith, J. W. and V. Zinde-Walsh (1992). The GLS transformation matrix
and a semi-recursive estimator for the linear regression model with ARMA errors,
Econometric Theory, 8, p. 95-112.
Galbraith, J. W. and V. Zinde-Walsh (1994). A simple non-iterative estimator for
moving-average models, Biometrika, 81, p.143-55.
Galbraith, J. W., A. Ullah and V. Zinde-Walsh (2002). Estimation of the vector
moving-average model by vector autoregression, Econometric Reviews, 21, p. 205-
219.
Hansen, L. P. and R. J. Hodrick (1980). Forward exchange rates as optimal pre-
dictors of future spot rates: an econometric analysis, The Journal of Political
Economy, 88, 829-853.
Hryshko, D. (2006). Informational assumptions on income process and consump-
tion dynamics in the buﬀer stock model of savings, Working paper, University of
Alberta.
Richard, P. (2007). GLS bias correction for low order ARMA models, Working
paper, GREDI, Universite de Sherbrooke.
Tkacz, G. (2007). Gold prices and inﬂation, Bank of Canada working paper no.
2007-35.
5Figures
Figure 1. MSE, MA(1) model, N=50, 100 000 Monte Carlo samples.
Figure 2. MSE, MA(1) model, θ = −0.85, 100 000 Monte Carlo samples.
6Figure 3. Det(MSE), MA(2) model, N=50, 10 000 Monte Carlo samples.
7

Improving the accuracy of the analytical indirect inference estimator for MA models.

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Improving the accuracy of the analytical indirect inference estimator for MA models.

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