Identification of Periodic Moving-Average Models

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Identification of Periodic Moving-Average ModelsTaylan A. Ula Ph.D. a & Abdullah A. Smadi Ph.D. ba Department of Systems Engineering , Yeditepe University , Kayisdagi, Istanbul, Turkeyb Department of Mathematics , Qatar University , Doha, QatarPublished online: 02 Sep 2006.

To cite this article: Taylan A. Ula Ph.D. & Abdullah A. Smadi Ph.D. (2003) Identification of Periodic Moving-Average Models,Communications in Statistics - Theory and Methods, 32:12, 2465-2475, DOI: 10.1081/STA-120025388

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©2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.

MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016

COMMUNICATIONS IN STATISTICS

Theory and Methods

Vol. 32, No. 12, pp. 2465–2475, 2003

Identification of Periodic

Moving-Average Models

Taylan A. Ula, Ph.D.1,* and Abdullah A. Smadi, Ph.D.2

1Department of Systems Engineering, Yeditepe

University, Kayisdagi, Istanbul, Turkey2Department of Mathematics, Qatar University,

Doha, Qatar

ABSTRACT

ARMA models with seasonally-varying parameters and orders,

known as periodic ARMA (PARMA) models, have found wide

applications in modeling of seasonal processes. This article considers

the identification of orders of periodic MA (PMA) models. The

identification is based on the cut-off property of the periodic auto-

correlation function (PeACF). We derive an explicit expression for

the asymptotic variance of the sample PeACF to be used in establish-

ing its bands. A simulated example is also provided which agrees well

with the theoretical results.

Key Words: Periodic models; PARMA model; PMA model;

Identification; Autocorrelation function.

*Correspondence: Taylan A. Ula, Ph.D., Department of Systems Engineering,

Yeditepe University, 81120 Kayisdagi, Istanbul, Turkey; Fax: 90-216-5780400;

E-mail: [email protected].

2465

DOI: 10.1081/STA-120025388 0361-0926 (Print); 1532-415X (Online)

Copyright & 2003 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

Mixed autoregressive (AR) moving-average (MA), ARMA, modelswith seasonally-varying parameters and orders are known as periodicARMA (PARMA) models, and have found wide applications inmodeling of seasonal processes. This study considers the identificationof orders in PARMA models. The identification of periodic AR (PAR)models has already been resolved (Sakai, 1982) and reconsidered by theauthors in an article submitted for a possible journal publication. Thisarticle considers the identification of periodic MA (PMA) models. Themore complicated problem of mixed PARMA identification is currentlyunder investigation. For the identification of PMA and PAR models,Box–Jenkins (Box and Jenkins, 1976) identification techniques forordinary MA and AR models, employing the cut-off property of theautocorrelation function (ACF) and the partial autocorrelation function(PACF), respectively, are extended to the periodic case, employingthe analogous cut-off property of the periodic ACF (PeACF) and theperiodic PACF (PePACF). We derive an explicit expression for theasymptotic variance of the sample PeACF to be used in establishing itsbands over the cut-off region. We could not obtain such an explicitexpression from the vector results in Anderson and Vecchia (1993).Instead we took a more simplified univariate approach and verified forsome special cases that the two approaches give the same variance overthe cut-off region.

The univariate, !-period, !¼ 2, 3, . . . , PARMA model of varyingorders [ p(1), q(1); p(2), q(2); . . . ; p(!), q(!)], denoted by PARMA!

[ p(�), q(�)], �¼ 1, . . . ,!, is given for all integers k as

Xk!þ� ¼ �ð�Þ1 Xk!þ��1 þ � � � þ �ð�Þ

pð�ÞXk!þ��pð�Þ

þ ak!þ� � �ð�Þ1 ak!þ��1 � � � � � �ð�Þqð�Þak!þ��qð�Þ, ð1:1Þ

where fak!þ�g is a periodic white noise process with independently andnormally distributed terms with mean zero and periodic variances �2

að�Þ.That is, we assume a Gaussian PARMA process {Xk!þ�}. Here, for amonthly series, for example, !¼ 12, � is the month index and k is theyear index. For !¼ �¼ 1, we have the ARMA ( p, q) model. It is assumedwithout loss of generality here that EðXk!þ�Þ ¼ 0: For EðXk!þ�Þ ¼ ��,Xk!þ� is replaced by Xk!þ� � ��. The parameters ��,�

ð�Þ1 , . . . ,�ð�Þ

pð�Þ,�ð�Þ1 , . . . , �ð�Þqð�Þ, and �2

að�Þ are all assumed to be periodic in � with period !.It is further assumed that the process is periodic stationary, i.e., the firstand second-order moments of the process are finite and periodic with

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period !, for which AR parameters are to satisfy certain conditions (Ula,1990; Ula and Smadi, 1997; Vecchia, 1985).

Various results and our findings about PeACF for the PMA processare given in Sec. 2. Section 3 presents some simulation results.

2. PERIODIC AUTOCORRELATION FUNCTION

For the PARMA process fXk!þ�g in Eq. (1.1), the periodic autocovar-iance function (PeACVF) for season � and backward lag l� 0 is defined as

�lð�Þ ¼ CovðXk!þ�,Xk!þ��lÞ ¼ E½ðXk!þ� � ��ÞðXk!þ��l � ��lÞ�:

Then the PeACF is

�lð�Þ ¼ EXk!þ� � ��ffiffiffiffiffiffiffiffiffiffiffi

�0ð�Þp

!Xk!þ��l � ��lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�0ð�� l Þp

!" #¼ EðZk!þ�Zk!þ��lÞ

¼�lð�Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�0ð�Þ�0ð�� l Þp , l � 1, ð2:1Þ

which is also both the PeACVF and the PeACF of the periodicallystandardized series fZk!þ�g. It can be seen that �lð�Þ is not symmetricwith respect to the time lag l as �lð�Þ is not necessarily equal to ��lð�Þunless l is a multiple of the period !: However, it can be shown that

�lð�Þ ¼ EðZk!þ�Zk!þ��lÞ ¼ EðZk!þ��lZk!þ��l�ð�l ÞÞ ¼ ��lð�� l Þ:

Furthermore, as for the ACF of a stationary ARMA model, it can beshown for a periodic stationary PARMA model that liml!1 �lð�Þ ¼ 0(Smadi, 1994; p. 29). The cut-off property of �lð�Þ for an arbitrary season� following the MA[q(�)] process

Xk!þ� ¼ ak!þ� � �ð�Þ1 ak!þ��1 � � � � � �ð�Þqð�Þak!þ��qð�Þ

is that �lð�Þ ¼ 0 for all l>q(�). This follows easily from Eq. (1.1) byshowing that �lð�Þ ¼ 0 for all l>q(�). This is analogous to the cut-offproperty of the ACF of an ordinary MA process.

Let fX1,X2, . . . ,XN!g be a realization of size N! from fXk!þ�g (say,N years of monthly data, with !¼ 12). Then the sample PeACF isdefined, by analogy with the sample ACF, as

rlð�Þ ¼�̂�lð�Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�̂�0ð�Þ�̂�0ð�� l Þp , l � 1, ð2:2Þ

Identification of Periodic MA Models 2467

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where �̂�lð�Þ is the sample PeACVF defined as

�̂�lð�Þ ¼1

N

XN�1

k¼0

ðXk!þ��X�ÞðXk!þ��l � X��lÞ,

in which X� ¼ 1=NPN�1

k¼0 Xk!þ� is the sample mean for season �. Due tothe end effect, the terms in �̂�lð�Þ are set to zero whenever k!þ �� l < 1.The number of such terms is ½ðl � �Þ=!� þ 1, where [c] stands for theintegral part of c. In Pagano (1978) it is proved that, for all � and l,�̂�lð�Þ are consistent, converge almost surely and in mean square to �lð�Þ,and are asymptotically independent, joint normal, unbiased, and efficient.X� is an unbiased estimator of ��, and it can also be shown that it isconsistent under periodic stationarity assumption.

For stationary processes, the asymptotic joint normality and unbia-sedness of the sample ACF, rl, as defined by Eq. (2.2) with !¼ �¼ 1,have been shown by Bartlett (1946), and asymptotic variance–covariancematrix has been specified, in which

VarðrlÞ ffi1

N

X1m¼�1

f�2m þ �mþl�m�l � 4�l�l�m�l þ 2�2m�2l g ð2:3Þ

(Box and Jenkins, 1976; p. 34). These results were then generalized toperiodic stationary processes by Anderson and Vecchia (1993), consider-ing the stationary vector representation of a periodic stationary process.Letting Rl ¼ ðrlð!Þ, rlð1Þ, . . . , rlð!� 1ÞÞT , they have shown that Rl isasymptotically (large N ) joint normal and unbiased, and the asymptoticvariance–covariance matrix is

Wl ¼1

N

X1m¼�1

hFm�

lFm�m�l

þ Fmþl�lFm�l�

m�l

� FlðI þ�lÞFmFmþl�

m�Fm�

lFm�l�m�l

ðI þ��lÞFl

þ1

2FlðI þ�l

ÞF2m�

mðI þ��l

ÞFl

i, ð2:4Þ

where Fm ¼ diag½ �mð!Þ, �mð1Þ, . . . , �mð!� 1Þ�, and � is the orthogonal!�! cyclic permutation matrix

� ¼

0 1 0 0 � � � 00 0 1 0 � � � 0... ..

. ... ..

. ... ..

.

0 0 0 0 � � � 11 0 0 0 � � � 0

266664

377775

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with ��l� ð�T

Þl, �0 being the !�! identity matrix, and

�ldiag½c!, c1, . . . , c!�1��l

¼ diag½cl, clþ1, . . . , clþ!�1�,

where fctg is any sequence of constants satisfying ck!þt ¼ ct for all k.The variance–covariance matrix Wl for the sample PeACF vector Rl

has been derived in Anderson and Vecchia (1993) for the purpose ofobtaining the asymptotic distribution of the discrete Fourier transformof the sample PeACF. For our purpose, we just need an expression forthe variance of the sample PeACF. We could not obtain from Eq. (2.4)an explicit expression for the variance. Instead we took a more simplifiedunivariate approach.

Deriving the exact moments of rlð�Þ in Eq. (2.2), which involves thesample means, variances, and autocovariances, is a formidable task. InSmadi (1994) an approximate solution for the first- and second-ordermoments of rlð�Þ were obtained by pretending that the sample meansand variances, X� and �̂�0ð�Þ, in rlð�Þ are equal to their populationcounterparts, �� and �0ð�Þ, respectively. This assumption willobviously be well justified only for large samples due to the consistencyof these estimators. In that case, rlð�Þ can be taken as

1

N

XN�1

k¼0

Zk!þ�Zk!þ��l ,

where Zk!þ� is as defined in Eq. (2.1). It then follows that rlð�Þ isasymptotically unbiased with asymptotic variance

Varðrlð�ÞÞ ffi1

N

X1m¼�1

½ �m!ð�Þ�m!ð�� l Þ þ �m!þlð�Þ�m!�lð�� l Þ�

¼1

N

1þ ð �lð�ÞÞ

2þ 2

X1m¼1

f �m!ð�Þ�m!ð�� l Þ

þ �m!þlð�Þ�m!�lð�� l Þg

!: ð2:5Þ

The derivations are available in Smadi (1994; pp. 33–35), andsummarized in Smadi and Ula (1998). For asymptotic normality, werely on the result in Anderson and Vecchia (1993).

We verified for some special cases that Eqs. (2.4) and (2.5) give thesame variance over the cut-off region. For example, for a two-period


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case, it follows from Eq. (2.4) that for odd lags l,

Varðrlð1ÞÞ ffi1

N

X1m¼�1

½ �2mð1Þ�2mð2Þ þ �2mþlð1Þ�2m�lð2Þ

� �lð1Þ½�mð2Þ�mþlð1Þ þ �mð2þmÞ�m�lðmÞ�

þ �2l ð1Þ½ �22mð1Þ þ �22mð2Þ��

and for even lags l,

Varðrlð1ÞÞ ffi1

N

X1m¼�1

½ �22mð1Þ þ �2mþlð1Þ�2m�lð1Þ

� 2�lð1Þ�2mð1Þ½ �2mþlð1Þ þ �2m�lð2Þ� þ 2�lð1Þ�lð2Þ�22mð1Þ�:

On the other hand, if Eq. (2.5) is used, only the first two terms areretained in these equations. The remaining last two terms containing thethird- and higher-order autocorrelations disappear for l > qð1Þ if the firstseason follows a MA (q (1)) process, for example, for which �lð1Þ ¼ 0 forl > qð1Þ. As an other example, setting !¼ �¼1 in Eq. (2.4) gives theBartlett formula (2.3), and in Eq. (2.5) gives

VarðrlÞ ffi1

N

X1m¼�1

f�2m þ �mþl�m�lg,

which differs from Eq. (2.3) by the term �4�l�l�m�l þ 2�2m�2l , which for

a MA (q) process is equal to zero for l > q for which �l ¼ 0.As another verification, assuming that the time lag l is large and

hence �lð�Þ ffi 0, a (large-lag) formula for Varðrlð�ÞÞ is obtained fromEq. (2.5), by setting �m!þlð�Þ to zero, as


N

X1m¼�1

�m!ð�Þ�m!ð�� l Þ,

which, by setting ! ¼ � ¼ 1, reduces to the corresponding formula forstationary processes, VarðrlÞ ffi

P1

m¼�1 �2m=N (Box and Jenkins, 1976,p. 35).

For a season following a MAðqð�ÞÞ process, we derive from Eq. (2.5)the following results for l > qð�Þ, (Smadi, 1994, pp. 36–37):


N, qð�Þ < ! ð2:6aÞ


N1þ 2

Xkm¼1

�m!ð�Þ�m!ð�� l Þ

!;

k! � qð�Þ < ðkþ 1Þ!, k ¼ 1, 2, . . . ð2:6bÞ

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Setting !¼ �¼ 1, it can be seen that, for l>q(�)¼q, Eq. (2.6a)reduces to that for the white noise process with VarðrlÞ ffi 1=N, l � 1,and Eq. (2.6b) reduces to that for a MA(k) process with

VarðrlÞ ffi1

N1þ 2

Xkm¼1

�2m

!, l > k

These are the well-known formulas for the identification of whitenoise and MA processes, respectively, for stationary processes (Boxand Jenkins, 1976; p. 35).

The identification bands for the MAðqð�ÞÞ process utilize rlð�Þ beingasymptotically normal with zero mean and variance Eq. (2.6) over thecut-off region l > qð�Þ. Following the same methodology applied to thesample ACF of a stationary process, for a season � which follows aMAðqð�ÞÞ process, we start checking values of qð�Þ successively, startingwith qð�Þ ¼ 1. Then for large N, as long as qð�Þ < !, Eq. (2.6a) impliesthat rlð�Þ, for l > qð�Þ, is normally distributed with mean zero andvariance 1/N, so that the 95% band ð�1:96=

ffiffiffiffiN

p, 1:96=

ffiffiffiffiN

pÞ is applied

to those autocorrelations. If qð�Þ � !, which is unlikely for moderateor large values of !, Eq. (2.6b) should be utilized, with �lð�Þ estimatedby rlð�Þ. Letting s2� to denote the sample estimate of Varðrlð�ÞÞ, the 95%band is ð�1:96s�, 1:96s�Þ which should be applied for rlð�Þ for l > qð�Þ. InSec. 3, it will be shown through simulation that the relative frequency ofthe autocorrelations falling outside the corresponding bands is indeedvery close to the theoretical value of 5%.

3. A SIMULATION STUDY

To illustrate the previous results, we consider the following PARMA4

(2, 2; 0, 1; 3, 0; 0, 4) model

X4kþ1 ¼ 0:8X4ðk�1Þþ4þ0:9X4ðk�1Þþ3þa4kþ1�0:4a4ðk�1Þþ4�0:9a4ðk�1Þþ3

X4kþ2 ¼ a4kþ2þ0:8a4kþ1

X4kþ3 ¼ 1:2X4kþ2�0:7X4kþ1þ0:5X4ðk�1Þþ4þa4kþ3

X4kþ4 ¼ a4kþ4�0:5a4kþ3�0:7a4kþ2þ0:3a4kþ1�1:1a4ðk�1Þþ4

which is chosen to be periodic stationary and invertible (for determiningperiodic stationarity and invertibility of such processes, see, for example,Ula (1990)). The first season follows an ARMA(2, 2) model, the second aMA(1), the third an AR(3), and the last season follows a MA(4) model.


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Table

1.

Averagesample

PeA

CF,r lð�Þ.

�2 að�Þ¼

1

N¼30

N¼100

N¼500

l\�

12

34

12

34

12

34

1.342

.329

.552

�.436

.361

.347

.573

�.451

.372

.353

.588

�.461

2.278

�.000

�.154

�.193

.315

�.001

�.179

�.193

.311

.001

�.184

�.205

3.358

.012

.225

�.041

.403

.000

.228

�.049

.399

�.002

.223

�.041

4�.220

�.033

�.385

�.334

�.217

�.010

�.421

�.370

�.212

�.002

�.423

�.355

5�.070

�.026

�.351

.002

�.079

�.006

�.365

.010

�.068

.003

�.359

.000

6�.348

.010

.095

.004

�.397

.000

.103

.004

�.405

.002

.121

�.001

7�.333

�.013

�.092

�.005

�.347

�.004

�.121

�.005

�.345

.002

�.118

�.004

8.082

�.021

.197

�.021

.106

�.009

.256

.005

.111

�.002

.265

�.004

9�.100

�.007

.192

.013

�.103

�.001

.222

�.007

�.121

�.003

.228

�.000

10

.209

�.008

�.050

.006

.244

.007

�.063

�.001

.258

.002

�.072

�.007

%Rel.freq.

(35)

3.1

2.7

4.2

3.0

5.4

4.4

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The white noise terms are assumed to be independently and normallydistributed with zero means (our basic assumptions) and unit variances(�2

að�Þ ¼ 1, �¼ 1, 2, 3, 4). One hundred realizations each of length N(years), i.e., 4�N values, for N¼ 30, 100, 500, are simulated from theabove 4-period PARMAmodel. For each realization, the sample PeACF,rlð�Þ, for �¼ 1, 2, 3, 4, and l¼ 1, . . . , 20, are computed. Their averagesover 100 realizations are also obtained. Table 1 shows these results (foronly up to l¼ 10 to save space). Only the second and fourth seasons herefollow a pure MA process, MA(1) and MA(4), respectively. For thesecond season, we expect a cut-off behavior for the sample PeACF forl>1, and for the fourth season for l>4. These can be observed from thevalues in Table 1. Only for these two seasons, we observe an apparentsudden drop in the values. For the second season, q¼ 1<!¼ 4, thereforeEq. (2.6a) follows, and the relative frequencies (rel. freq.) of rlð2Þ,l>q¼ 1, going outside the 95% band ð�1:96=

ffiffiffiffiN

p, 1:96=

ffiffiffiffiN

pÞ for

N¼ 30, 100, 500, are given in Table 1 as 3.1, 4.2, and 5.4%, respectively,which agree well with the theoretical asymptotic value of 5%. Therelative frequency is the percentage of autocorrelation values over allrealizations falling outside the corresponding bands. The asymptoticproperty also justifies the improvement in the values as N increases.For the fourth season, q¼ 4¼!<2!, therefore Eq. (2.6b) applies withk¼ 1, and the relative frequencies of rlð4Þ, l>q¼ 4, going outside thecorresponding 95% band for the three N values are obtained as 2.7, 3.0,

Table 2. Average sample PeACF values.

�2að�Þ ¼ 0:8, 0:5, 4:0, 0:2, N ¼ 100

rlð�Þ

l \� 1 2 3 4

1 .470 .404 .332 �.773

2 �.218 �.014 �.096 �.118

3 .362 .013 .042 .069

4 �.090 .008 �.103 �.108

5 .000 .000 �.193 �.006

6 �.158 �.008 .061 �.005

7 �.277 .014 �.023 .001

8 .065 �.008 .061 �.007

9 �.070 �.016 .112 .003

10 .114 �.011 �.033 .006

% Rel. freq. 4.8 4.1


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and 4.4%. The asymptotic formulas seem to be conservative for small N,in the sense of corresponding to a percentage greater than 95%. For thefirst season, and for N¼ 30, the relative frequency is computed by pre-tending that the model is MA(2), rather than ARMA(2,2), and obtainedfor l>2 as 35%. This is a very high value indicating that the model isnot MA(2).

To see the effect of variances on the results, the simulation is repeatedfor N¼ 100 and �2

að�Þ ¼ 0:8, 0:5, 4:0, 0:2 for �¼ 1, 2, 3, 4, respectively.The results are given in Table 2. The variances seem to have no significanteffect on the overall result, as theoretically expected. However, sincesmaller variances for seasons two and four (as compared to the previousvalue of 1) lead to more accurate simulation results, there is someimprovement in the relative frequencies.

4. CONCLUSIONS

An explicit expression for the asymptotic variance of the samplePeACF is derived to be used in establishing its bands for the PMAprocess over the cut-off region. Simulation results agree well with thetheoretical results.

ACKNOWLEDGMENT

The authors would like to thank the referee for helpful comments.

REFERENCES

Anderson, P. L., Vecchia, A. V. (1993). Asymptotic results for periodicautoregressive moving-average processes. J. Time Ser. Anal.14(1):1–18.

Bartlett, M. S. (1946). Theoretical specification of sampling properties ofautocorrelated time series. J. Royal Stat. Soc., Ser. B 8:27–41.

Box, G. E. P., Jenkins, G. M. (1976). Time Series Analysis: Forecastingand Control. San Francisco: Holden-Day.

Pagano, M. (1978). On periodic and multiple autoregressions. Ann.Statist. 6(6):1310–1317.

Sakai, H. (1982). Circular lattice filtering using Pagano’s method. IEEETrans. Acoust. Sp. Signal Process 30(2):279–287.

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Vecchia, A. V. (1985). Periodic autoregressive-moving average (PARMA)modeling with applications to water resources. Water Resour. Bull.21(5):721–730.


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Identification of Periodic Moving-Average Models

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