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Sensitivity of theportmanteau statistic in timeseries modelingAndy H. Lee , John S. Yick & Yer Van HuiPublished online: 02 Aug 2010.

To cite this article: Andy H. Lee , John S. Yick & Yer Van Hui (2001) Sensitivity ofthe portmanteau statistic in time series modeling, Journal of Applied Statistics,28:6, 691-702, DOI: 10.1080/02664760120059228

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Journal of Applied Statistics, Vol. 28, No. 6, 2001, 691-702

Sensitivity of the portmanteau statistic intime series modeling

ANDY H. LEE,1 JOHN S. YICK 2 & YER VAN HUI 3, 1Department ofEpidemiology and Biostatistics, School of Public Health, Curtin University ofTechnology, Perth, Australia, 2Statistics and Demography Unit, Northern TerritoryDepartment of Education, Darwin, Australia and 3Department of ManagementSciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

abstract The portmanteau statistic is commonly used for testing goodness-of-® t of timeseries models. However, this lack of ® t test may depend on one or several atypicalobservations in the series. We investigate the sensitivity of the portmanteau statistic in thepresence of additive outliers. Diagnostics are developed to assess both local and globalin¯ uence. Three practical examples demonstrate the usefulness of the proposed diagnostics.

1 Introduction

In many applications of time series, sample data are used to estimate the parametersof the assumed model, and structural relationships are tested statistically. It iscommon that estimation and the overall goodness-of-® t of an autoregressive movingaverage (ARMA) model may be aþ ected by one or several atypical observations.The unrecognized abnormality will lead to poor forecasts based on the estimatedmodel (Ledolter, 1989; Chen & Liu, 1993). The outliers also aþ ect the sensitivityanalysis where the eþ ects of minor changes to the data are monitored. It is thereforeimportant to determine whether the conformance of the hypothesized model isachieved throughout the series or distorted by a few particular observations. Theaim of this paper is to present eþ ective diagnostics for assessing the eþ ects ofaberrant observations on the portmanteau statistic.

The investigation of residuals has been well established in regression diagnostics.In time series analysis it is the residual autocorrelations that should be examined.

Correspondence: Andy H. Lee, Department of Epidemiology and Biostatistics, School of Public Health,Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia. E-mail: leea@health.curtin.edu.au.

ISSN 0266-4763 print; 1360-0532 online/01/060691-12 © 2001 Taylor & Francis Ltd

DOI: 10.1080/02664760120059228

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692 A. H. Lee et al.

A widely used diagnostic for checking overall model adequacy is the portmanteaustatistic Q (Box & Pierce, 1970) which accumulates the lag K squared residualautocorrelations. However, the portmanteau statistic and its variant such as theLjung- Box- Pierce statistic (Ljung & Box, 1978) are expected to be prone tooutliers. It has been documented in the literature that the existence of additiveoutliers can seriously bias the model coeý cients, whereas innovational outliers ingeneral have a much smaller eþ ect; see Ljung (1993). Therefore, we limit ourscope to additive outliers and assess the eþ ect of perturbations on the portmanteaustatistic in ARMA models. Two separate approaches based on global deletion andlocal in¯ uence analysis are proposed in the evaluations. The resulting diagnosticsare then demonstrated using several examples.

2 Assessing goodness-of-® t

We are concerned with diagnostic methods for assessing the in¯ uence of multipleand/or consecutive outliers on the goodness-of-® t through the portmanteau statis-tic. Consider the stationary and invertible ARMA model

u (B)yt 5 h (B)at

where B denotes the backward shift operator,

u (B) 5 1 2 u 1B 2 . . . 2 u pBp and h (B) 5 1 2 h 1B 2 . . . 2 h qB

q

u (B) and h (B) have all their roots outside the unit circle, and at is Gaussian whitenoise with zero mean and innovation variance r 2 . The well-known portmanteaustatistic (Box & Pierce, 1970) for testing goodness or lack of ® t is

Q 5 n +K

k 5 1

r2k[aà ]

where rk[aà ] is the lag k autocorrelation of the residuals aà s. If the orders ( p,q) arecorrectly speci® ed and n > K, the Q is distributed asymptotically as v 2 with degreesof freedom K 2 p 2 q. To improve the v 2 approximation of its null distribution,several variants of the portmanteau statistic have been proposed in the literature,including the Ljung-Box-Pierce statistic (Ljung & Box, 1978)

Q* 5 n(n + 2) +K

k 5 1

(n 2 k) 2 1r2k[aà ]

The additional terms involving n and k, however, may be regarded as nuisanceparameters in the assessment of local in¯ uence in Section 2.2. Without loss ofgenerality, we shall focus on the standard form Q in subsequent investigations.The eþ ect of a change in innovation variance on Q has been studied by Incla n(1992). Meanwhile, robust modi® cations of the portmanteau statistic for testingmodel adequacy have also been proposed (Li, 1988; Chan, 1994; Jiang et al.,1999).

2.1 Global in¯ uence

A practical approach to sensitivity analysis involves the removal of individual cases;see, for example, Cook & Weisberg (1982) for a review. However, ordinary casedeletion is inappropriate in time series. The problem can be eþ ectively handled by

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The portmanteau statistic 693

treating the observation as missing data prior to parameter estimation. Jones (1980)considered the maximum likelihood estimation of ARMA models with missingobservations based on the Kalman ® lter. This approach has been extended byothers, including Harvey (1989), Go mez & Maravall (1994) and Cipra & Romera(1997). To predict the missing values, we follow the state- space representation ofKohn & Ansley (1986), which provides consistent and asymptotically-eý cientestimates. Alternatively, the E-M algorithm or other recursive methods (Nieto &Martõ nez, 1996) can be used to impute the missing values. A summary ofprocedures can be found in Basu & Reinsel (1996). Recently Go mez et al. (1999)suggested two alternative strategies for ARIMA models. Based on the missing dataapproach, global in¯ uence for the portmanteau statistic Q(t ) is evaluated from theincomplete series á y1, . . . , yt 2 1 , yt +1, . . . , yn ñ . It can be quanti® ed with respect tothe asymptotic v 2 reference distribution.

2.2 Local in¯ uence

A separate approach to in¯ uence is due to Cook (1986), who introduced the localin¯ uence methodology as a general tool for assessing the eþ ect of minor departuresfrom model assumptions. This latter approach relies on geometric diþ erentiationrather than complete case deletion. Ledolter (1990) applied the local in¯ uencemethod to outlier detection in time series via the normal curvature of the likelihooddisplacement surface. It was found that the resulting diagnostic for which thecurvature is maximized is given by the vector of diþ erences between observed andinterpolated values. Instead of relying on the likelihood displacement criterion, westudy the eþ ect of additive outlier peturbations on the portmanteau statistic.Consider the additive outlier perturbation ARMA model

yt 5 zt + d x t u (B)zt 5 h (B)at

where yt is the observed value, x 5 á x 1 , . . . , x n ñ T Î X are the perturbations withscale d, and X denotes the open set of relevant perturbations. In the manner of Wu& Luo (1993), the geometric surface of interest is formed by Q( x ), the portmanteaustatistic under perturbation x . Here the null point x 0 5 á 0, . . . , 0 ñ T Î X representsno perturbation so that Q( x 0) gives the observed Q statistic. Unlike the likelihooddisplacement surface, such a perturbation-formed portmanteau statistic surfaceQ( x ) does not have zero ® rst derivative at x 0 , so that its slope can be used toexamine local in¯ uence. To locate the direction of largest local change, weapproximate Q( x ) by its tangent plane at x 0 . The desired direction is then givenby the direction of maximum slope on this tangent plane over X . Such a directionvector, Q ¢ 5 ­ Q( x )/ ­ x T evaluated at x 0 , will serve as our diagnostic tool. It can beshown that

á Q ¢t ñ 5 +

K

k 5 1( S[k]t +

n

i

aà 2i 2 2aà t +

n

i

aà iaà t 2 k ) / ( +n

i

aà 2i )2

where

S[k]t 5­ R n

i aà iaà t 2 k

­ x T½ x 0

are model dependent; formulae for three examples are given in the Appendix.Sampling properties of these direction vectors remain to be developed, but do not

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694 A. H. Lee et al.

seem crucial at the diagnostic stage. Warning limits of 6 Z a /Ï n have also beensuggested for informal calibration (Lawrance, 1988).

2.3 Global eþ ect of local perturbations

Since Q ¢t signi® es how to perturb the postulated model to obtain the greatest local

change in Q, the sensitivity of Q to the induced perturbations in direction Q ¢t can

be further assessed by pro® ling Q( x ) against the perturbation scale d. Thecharacteristics of this curve should be informative for further investigation on therelationship between local and global in¯ uences.

3 Applications

3.1 AR(1) with consecutive outliers: simulated series

An arti® cial AR(1) series of n 5 100 observations is generated using S-Plus functionarima.sim with u 5 0.5 and innovation variance r 2 5 1. Fitting the AR(1) modelgives u à 5 0.497 (0.087), r à 2 5 0.851. The portmanteau statistic Q 5 16.24 at k 5 20is readily available from arima.diag. To ensure stability of Q, henceforth K is takento be 20 lags. Two consecutive additive outliers are created by adding three to the49th and 50th observations. The contaminated series, plotted in Fig. 1, hasu à 5 0.507 (0.087), r à 2 5 1.074 and Q( x 0) 5 17.09.

To assess global in¯ uence, missing values are introduced one at a time. Theresulting likelihood is maximized based on Kalman ® ltering applied to its state-space representation (Kohn & Ansley, 1986). The method of initializing theKalman ® lter recursions is that given by Bell & Hillmer (1987). Figure 2 showsQ(t) with a horizontal reference line drawn at the null state value Q( x 0). Asexpected, Q(49) is relatively large in the time series plot but case 50 does not appearprominent, an artifact of the masking eþ ect due to the adjacent outliers.

Figure 3 plots the absolute value of the normalized diagnostic Q ¢t . Both spurious

Fig. 1. Contaminated AR(1) series.

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The portmanteau statistic 695

Fig. 2. Q(t) for contaminated AR(1) series.

Fig. 3. Q ¢t for contaminated AR(1) series.

observations located at time points 49 and 50 have exceeded the warning limit of1.96/Ï n 5 0.196, suggesting they are locally in¯ uential on the test statistic. Wenext plot Q( x ) against the perturbation scale d for the direction Q ¢ in Fig. 4, wherethe range of d is chosen to be 6 3r à . It can be seen that a small perturbation aboutthe size of 1

2 r à in direction Q ¢t (which is dominated by its 49th and 50th components)

would help reduce Q(x 0) towards the value of 16.24 prior to contamination.

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696 A. H. Lee et al.

Fig. 4. Q( x ) in direction Q ¢ for contaminated AR(1) series.

3.2 IMA(1,1) with reallocation outliers: retail sales of automobile dealers

The term reallocation outliers is due to Wu et al. (1993) and can be considered asadditive outliers whose magnitudes sum to zero. They used 65 monthly observations( January 1985 to May 1990) from the estimated retail sales of automotive dealers(seasonally adjusted) published in the Survey of Current Business Statistics, USDepartment of Commerce. For this series, shown in Fig. 5, Wu et al. (1993)identi® ed the observation for September 1986 (t 5 21) as a single additive outlier,

Fig. 5. Retail sales of automobile dealers.

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The portmanteau statistic 697

Fig. 6. Q(t) for retail sales series.

Fig. 7. Q ¢t for retail sales series.

whereas observations for December 1986 (t 5 24) and January 1987 (t 5 25) consti-tuted a reallocation outlier pair: sales in September 1986 appeared to be unusuallyhigh, sales in the following December and January were merely a reallocation withno overall change in sales volume. The model ® tted is IMA(1,1), with estimatedmean 0.118 (0.028), h à 5 0.868 (0.079), r à 2 5 2.405 and Q( x 0) 5 15.01.

The plot of Q(t) in Fig. 6 suggests that the observed portmanteau statistic can besubstantially distorted by the additive outliers. Indeed, if all three observations aretreated as missing, Q increases to 19.51. However, only Q ¢

21 in Fig. 7 is appreciably

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698 A. H. Lee et al.

Fig. 8. Q( x ) in direction Q ¢ for retail sales series.

greater than the warning limit of 0.243, which supports the classi® cation ofobservation 21 as a single additive outlier. The anomaly in the process occurringat the successive time points 24 and 25 exerts minor impact on Q locally, which isconsistent with the reallocation property that their combined net disturbances tothe series are zero. To quantify further the global eþ ects of the local perturbations,Q( x ) is plotted against d in Fig. 8. It is worth noting that an optimal Q can beattained by perturbing the series simultaneously in direction Q ¢ with a magnitudeof about r à 2 .

3.3 ARMA(1,1) with isolated outliers: chemical process concentration readings

We next consider the Series A taken from Box et al. (1994), which contains 197readings of concentration in a chemical process observed every 2 h (Fig. 9). Boxet al. (1994, p. 214) ® tted an ARMA(1,1) model to this series, with u à 5 0.921(0.042), h à 5 0.581 (0.083), r à 2 5 0.098, and Q 5 25.37 at k 5 20 indicates theARMA(1,1) model has provided a reasonable ® t to the data. However, using aniterative robust ® tting procedure, LucenÄ o (1998) found two potential isolated out-liers at times t 5 43 and t 5 64. It is therefore of interest to scrutinize the contributionof such isolated outliers to the overall ® t in terms of the portmanteau statistic.

From Fig. 11, we found that the greatest local change in Q as measured by Q ¢depends to a large extent on observations 43 and 64, whose components are wellabove the warning limit of 0.14. But according to the Q(t) statistic presented in Fig.10, both points are not ¯ agged as globally in¯ uential. To con® rm the indicationsof the diagnostics, we examine the actual Q displacements due to the localperturbations. It is evident from Fig. 12 that no overall improvement in thegoodness-of-® t can be achieved by perturbing the series in the neighbourhood ofthe null point. Apparently the in¯ uence exerted by the isolated outliers has beencompensated by the rest of the series. This reinforces the implication of the abovein¯ uence diagnostics results.

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The portmanteau statistic 699

Fig. 9. Chemical process concentration readings.

Fig. 10. Q(t) for chemical process series.

4 Discussion

This paper is concerned with the sensitivity of the portmanteau statistic to additiveperturbations. It is well known that eþ ects of aberrant observations will generallydepend on the assumed model structure. Consequently, the diagnostics areexpected to be sensitive to model speci® cation, including the degree of diþ erencing.

We have addressed the local sensitivity of the portmanteau statistic through thetangent plane of the perturbation surface Q( x ) at the null point. An evaluation of

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Fig. 11. Q ¢t for chemical process series.

Fig. 12. Q( x ) in direction Q ¢ for chemical process series.

the normal curvature of Q( x ) in the manner of Wu & Luo (1993) appears logicallyto be the next step of analysis. However, as pointed out by Fung & Kwan (1997),when the slope is non-zero, the normal curvature of the test statistic is notscale invariant and thus ambiguous conclusions may be drawn regarding locallyin¯ uential observations. Therefore, application of the second-order curvatureapproach is not considered in our investigation.

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The portmanteau statistic 701

Acknowledgement

We are grateful to Prof. Ravishanker for providing the retail sales of automobiledealers series.

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Appendix: Expressions for S[k]t in Q ¢t

For the AR(1) model with n observations:

S[k]1 5 2 u à aà k +2

For 1< t < k + 1, S[k]t 5 aà t +k 2 u à (aà t +k +1)

S[k]k +1 5 aÃ2k +1 2 u à (aà 2k +2 + aà 2)

For k + 1< t < n 2 k, S[k]t 5 aà t 2 k + aà t +k 2 u à (aà t 2 k +1 + aà t +k +1)S[k]n 2 k 5 aà n 2 2k + aà n 2 u à (aà n 2 2k +1)For n 2 k< t < n, S[k]t 5 aà t 2 k 2 u à (aà t 2 k +1)S[k]n 5 aà n 2 k

For the MA(1) model with n observations:

S[k]1 5 2 2For 1< t < k + 1, S[k]t 5 h à S[k]t +1 + aà t +k 2 aà t +k +1

S[k]k +1 5 h S[k]k +2 + aà 2k +1 2 aà 2k +2 2 aà 2

For k + 1< t < n 2 k, S[k]t 5 h à S[k]t +1 + aà t 2 k + aà t +k 2 aà t 2 k +1 2 aà t +k +1

S[k]n 2 k 5 h à S[k]n 2 k +1 + aà n 2 2k + aà n 2 aà n 2 2k +1

For n 2 k< t < n, S[k]t 5 h à S[k]t +1 + aà t 2 k 2 aà t 2 k +1

S[k]n 5 aà n 2 k

For the ARMA(1,1) model with n observations:

S[k]1 5 2 2For 1< t < k + 1, S[k]t 5 h à S[k]t +1 + aà t +k 2 u à (aà t +k +1)S[k]k +1 5 h à S[k]k +2 + aà 2k +1 2 u à (aà 2k +2 + aà 2)For k + 1< t < n 2 k, S[k]t 5 h à S[k]t +1 + aà t 2 k + aà t +k 2 u à (aà t 2 k +1 + aà t +k +1)S[k]n 2 k 5 h à S[k]n 2 k +1 + aà n 2 2k + aà n 2 u à (aà n 2 2k +1)For n 2 k< t < n, S[k]t 5 h à S[k]t +1 + aà t 2 k 2 u à (aà t 2 k +1)S[k]n 5 aà n 2 k

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