Comparing seasonal components for structural time series models

International Journal of Forecasting 16 (2000) 247–260www.elsevier.com/ locate / ijforecast

Comparing seasonal components for structural time series models

*Tommaso ProiettiDepartment of Statistics, University of Udine, Via Treppo 18, 33100 Udine, Italy

Abstract

This paper discusses several encompassing representations for linear seasonal models in the structural framework. Theirtime and frequency domain properties are ascertained in a unifying framework, casting particular attention on the notion of‘smoothness’ of the seasonal component. The shape of the forecast function is compared with that arising from a number ofexponential smoothing algorithms. Finally, we investigate whether the specification of the seasonal model is likely to affectthe out-of-sample predictive performance of the basic structural model. We conclude that the latter depends upon the featuresof the time series under investigation, and in particular on the degree of smoothness of the seasonal pattern. 2000International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved.

Keywords: Exponential smoothing; Forecasting; Kalman filter; Seasonal adjustment; Smoothness

21. Introduction effect at time t, and e | NID(0, s ). Severalt e

dynamic linear specifications for g have beentIt is a fact of life that seasonal fluctuations proposed in the structural approach, all comply-

account for a major part of the variation of a ing with a definition of seasonality in terms ofwide spectrum of economic, social, and en- predictions, as the ‘‘part of the series which,vironmental phenomena. This explains the at- when extrapolated, repeats itself over any one-tention that they have received in the time series year time period and averages out to zero overliterature. such a time period’’ (Harvey, 1989, p. 301).

This paper focuses on seasonal models for a The feature shared by these models is that thetime series, y , t 5 1,2, . . . ,T, observed witht seasonal sums, U 5 S(L)g , where L is the lagt t

s21periodicity s, admitting the following repre- operator and S(L) 5 1 1 L 1 ? ? ? 1 L is thesentation: summation operator, has an invertible MA

representation of order s 2 2 at most.y 5 m 1 g 1 e (1)t t tWhen m is replaced by a stochastic trend, m ,t

where m is an intercept, g denotes the seasonalt model (1) is referred to (Harvey, 1989) as thebasic structural model (BSM). In the sequel wewill adopt the local level specification for this*Tel.: 139-432-249-581; fax: 139-432-249-595.

E-mail address: [email protected] (T. Proietti) component:

0169-2070/00/$ – see front matter 2000 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved.PI I : S0169-2070( 00 )00037-6

248 T. Proietti / International Journal of Forecasting 16 (2000) 247 –260

Algorithms are then derived for the au-m 5 m 1 b 1ht t21 t21 t (2) tocovariance function of the seasonal sumsb 5 b 1 zt t21 t S(L)g (Section 5). The actual order of the MAt

where h and z are mutually and serially representation for the seasonal sums and thet t

independent disturbances with zero mean and role of rank deficiencies in the covariance2 2variance s and s , respectively. matrix of the disturbances are dealt with.h z

In this paper we review several equivalent In Section 6 we address the issue of charac-representations for g that encompass the linear terising the smoothness of a seasonal model; ont

models of stochastic seasonality proposed in the adoption of the Froeb and Koyak (1994) coeffi-structural time series literature; particular em- cient, it is possible to show that the HS model isphasis will be placed on the specifications most smoother than the trigonometric one, whichcommonly employed in empirical work such as suffers from excess power at the Nyquist fre-the Harrison and Stevens’ (1976, HS hence- quency. Now, it is usually thought that seasonalforth) model, Harvey’s (1989) trigonometric fluctuations should be slowly varying, as theseasonal model, Harvey and Todd’s (1983, HT underlying institutional factors and preferenceshenceforth) dummy seasonal model, and a change in the long run, but do not fluctuate‘crude seasonal’ specification. wildly. By a formal argument we argue, how-

We set off with the class of random walk ever, that smoothness alone cannot provide aseasonal models (Section 2), by which the criterion upon which to judge the suitability of aeffects associated with each season are allowed particular model.to vary over time according to a multivariate The last section is devoted to forecasting withrandom walk with restrictions on the distur- the BSM under four different specifications ofbance covariance matrix; the class of tri- the model for seasonality. We discuss how thegonometric seasonal models is dealt with in forecasts and their updating are related toSection 3, where we argue that the trigonomet- alternative ad hoc and model-based forecastingric and the HS specifications are particular cases procedures, and whether the out-of-sample pre-of the same underlying model and are obtained dictive performance of the BSM is sensitive toby imposing suitable restrictions on the co- the choice of the seasonal model.variance matrix of the innovations of the multi-variate random walk. We also derive the restric-tions under which the two are equivalent. 2. The random walk representation of aSection 4 illustrates an encompassing repre- seasonal modelsentation for the quarterly case. In Section 5 weconsider the West and Harrison (1997, WH In the time domain, a fixed (deterministic)

9henceforth) representation, which also encom- seasonal pattern is modelled as: g 5 x d, wheret t

passes the HT (1983) dummy seasonal model d is an s 3 1 vector containing the seasonals 9and the ‘crude seasonal’ specification. effects hd j , and x 5 [D , . . . ,D ], withj j51 t 1t st

The novel contribution of this paper is to D 5 1 in season j and 0 otherwise. The d ’sjt j

establish a common ground for comparing the measure the effect associated with the cor-time and frequency domain properties of responding season and are restricted to sumseasonal models. The key result is the stationary up to zero, in order to ensure that modelrepresentation for S(L)g in terms of a moving (1) is identifiable. Denoting e 5t j,s

average of order at most s 2 2 of linear combi- [0, . . . , 0, 1, 0, . . . , 0]9, a vector of length s withnations of a multivariate white noise process. 1 in the jth position and 0 elsewhere, and

T. Proietti / International Journal of Forecasting 16 (2000) 247 –260 249

i 5 [1, 1, . . . , 1]9, an s 3 1 vector of ones, wes 12 F ] 9Ghave that x 5 e , t 5 j mod s; moreover, the V 5 s I 2 i i . (6)t j,s v s s ss9zero-sum constraint is expressed as: i d 5 0.s

A class of models of stochastic seasonality is This model is such that U | MA(s 2 2), sincet

derived by letting the coefficients d change over the theoretical autocovariance of U at lag s 2 2t

time according to a multivariate random walk: is nonzero and, precisely, defining c 5k2 21E(U U ), c 5 s s . Representation (3) ist t2k s22 v9g 5 x d , d 5 d 1 v , v | NID(0, V). (3)t t t t t21 t t amenable since the disturbances in v aret

The stochastic counterpart of the zero sum season-specific. In Proietti (1998) it is argued9constraint, i d 5 0, is enforced by that it can be generalised so as to account fors

seasonal heteroscedasticity in a simple fashion.9 9i V 5 0 . (4)s s The constraint (4) can be made explicit

˜ ˜9 9 9This implies i d 5 i d , which for i d 5 0 expressing V 5 QVQ9, for V unconstraineds t s t21 s 0

implies in turn that S(L)g is a stationary zero and wheret

mean process. Thus, the seasonal model arisesQ 5 [I , 2 i ]9. (7)s21 s21as a systematic sample of a multivariate random

walk, whose components measure the effectThis allows us to reparameterise the seasonalassociated with a particular season. The con-model in terms of s 2 1 effects: indeed, settingstraint (4) induces a singularity in the covar- 21d̃ 5 (Q9Q) Q9d , it is possible to write:t tiance matrix V that enforces one cointegration

relation between the d ’s.jt ˜˜ ˜ ˜ ˜ ˜˜ 9g 5 x d , d 5 d 1 v , v | NID(0, V ) (8)t t t t t21 t tBy repeated substitution from (3):

s21 ˜where d is an (s 2 1) 3 1 vector containing thet9S(L)g 5Ox dt t2j t2j seasonal effects associated with the first s 2 1j50

˜ ˜˜ 9 9seasons, and x 5 x Q 5 [D , . . . ,D ], witht t 1t s21,ts21s22 s21˜ ˜D 5 D 2 D , i.e. D 5 1, for t 5 j mod s,9 9 jt jt st jt5OOx v 1 Ox dS Dt2j t2j2k t2j t2s11

j50 k5j j50 ˜ ˜D 5 0, for t ± j mod s, D 5 2 1, for t 5jt jt

9 ˜5 x v 1 (x 1 x )9v 1 ? ? ? s mod s. Hence, x 5 e , t 5 j ± s mod s, andt t t t21 t21 t j,s21

x̃ 5 2 i , t 5 s mod s. In particular, the HSt s211 (x 1 ? ? ? 1 x )9v . (5)t t2s12 t2s12 2 21 2˜model has V 5 s (Q9Q) 5 s [I 2v v s2121The last row of (5) states that S(L)g | MA(q), 9s i i ].t s21 s21

with q # s 2 2. As a matter of fact, there maybe additional rank deficiencies implying forinstance (x 1 ? ? ? 1 x )9V 5 09. In gener- 3. The trigonometric seasonal modelt t2s12

al, rank(V) 5 q 1 1 is necessary but not suffi-cient for S(L)g | MA(q); an illustration is pro- In the frequency domain, a fixed seasonalt

vided in Section 5. pattern is formulated in terms of s 2 1 effectsModel (3) is time-invariant if V is a Toeplitz associated with the amplitude of deterministic

matrix; in such circumstance the autocovariance sine and cosine waves defined at the seasonalfunction of U 5 S(L)g does not depend on the frequencies, l 5 2pj /s, j 5 1,2, . . . ,[s /2],t t j

season. Moreover, U is invertible if V is where [s /2] 5 s /2 for s even and (s 2 1) /2 if st

positive semidefinite. The HS model is a special is odd. If these effects are collected in the vector9case of (3), which is obtained by choosing: t, we write: g 5 z t, witht t


9z 5 [cos l t,sin l t, . . . ,cos l t,sin l t, . . . , sequel model (9) with this specification for Kt 1 1 j j

will be referred to as the TS model.cos l t,sin l t].[s / 2] [s / 2] Let us turn now to the relationship between

(3) and the trigonometric seasonal model. In theWhen s is even, the last element of the vector islight of the relationship z 5 H9x , the tri-t tzero and is dropped, so that z always containstgonometric model of stochastic seasonality cans 2 1 elements. Due to the periodic nature of the

9 9be parameterised as (3), since g 5 z t 5 x Ht ,t t t t tsine and cosine functions, there exists a linearand (9) implies that d 5 Ht is generated as at tmapping of the z ’s into the x ’s: z 5 H9x , fort t t tmultivariate random walk with innovation co-the (s 2 1) 3 s matrix H9 5 [z , z , . . . ,z ]1 2 svariance matrix:(Canova & Hansen, 1995). By trigonometric

s21identities o z 5 0 . V 5 HKH9. (12)j50 t2j s21

The stochastic trigonometric seasonal model9Note that by trigonometric identities i H 5sis obtained by letting t evolve as a multivariatet

90 , so that (4) is automatically satisfied.s21RW:An interesting issue is what specification of K

t 5 t 1 k , k | NID(0 , K). (9) ensures that the trigonometric model is co-t t21 t t s21

incident with the HS model in (3) and (6): inview of (12) we must haveThe seasonal model proposed by Hannan,

Terrell and Tuckwell (1970) arises for K diag- 12 F ] 9Gs I 2 i i 5 HKH9.onal, such that the diagonal elements vary with v s s ssthe frequency. It has been employed by Canova

Premultiplying and postmultiplying both sidesand Hansen (1995) and by Hylleberg and Paganrespectively by H9 and H, and solving for K,(1997) to explore issues relating to testing for

2 21 2˜gives K 5 s (H9H ) . For s odd, K 5 s I ,v v s21unit roots at the seasonal frequencies. Harvey2 2˜with s 5 2s /(s 2 1), whereas for s even,v v(1989) uses a different representation of this

2 2 2˜ ˜K 5 s diag[1, 1, . . . , 1, 1 /2], with s 5 2s /v v vmodel, according to which the seasonal effect ats. Hence, the last element of t has disturbancettime t results from the combination of [s /2]variance that is one-half of the other elements.stochastic cycles:In the Harvey’s representation of the tri-

[s / 2] g gonometric seasonal model (10), this amountsjt2g 5O g , F Gt jt to setting Var(v ) 5 s /2 in (11).*g s / 2,t vjti51

cos l sin l g vj j j,t21 j,t5 1F GF G F G* *2 sin l cos l g v 4. Illustration: quarterly seasonalityj j j,t21 jt

(10)In the quarterly case (s 5 4) the assumption

that V 5 hV j is a Toeplitz matrix implies thej 5 1, . . . ,[s /2]; for s even, the last component, ij

following representation:defined at l 5 p, reduces tos / 2

V(1 1 L)g 5 v (11)s / 2,t s / 2,t1 s 2 (1 1 2s) s

s 1 s 2 (1 1 2s)*v and v are a set of serially and mutually 2jt jt 5 ßv 2 (1 1 2s) s 1 sindependent sequences and if it is assumed that 3 42 s 2 (1 1 2s) s 1they have common variance s , this representa-v

2tion is equivalent to (9) with K 5 s I . The (13)v s21


2which depends on just two parameters, ß and longer interpreted as the disturbance pertainingv

s, the former acting as a scale parameter; the to the jth season.latter is constrained to lie in the interval (21, 0) The time series properties of the parame-in order to ensure that V is positive semidefi- terisation (14) can be ascertained from the

2nite. The restrictions V 5 2 ß (1 1 2s) and following result:31 v2

V 5 ß s arise as a consequence of the zero s2141 v

sum constraint imposed on the columns of V. In 9S(L)g 5 e OFt 1,s t2jj50general, V depends on [s /2] parameters, e.g. on

js21 s21six parameters for monthly processes.k † k95 e OOP v 1OP FS DNote that (13) encompasses three commonly 1,s t2j t2s11

j50k50 k50employed representations: the HS model, which

† † †2 2 5 v 1 (v 1 v ) 1 ? ? ?arises for s 5 2 1/3, ß 5 3s /4, the TS 1t 1,t21 2,t21v v

model, arising for s 5 2 1/2, and the tri- † †1 (v 1 ? ? ? 1 v )1,t2s12 s21,t2s12gonometric seasonal model with disturbance

(15)variances varying across the frequencies, K 52 2 2 2 2

s21 kdiag(s ,s ,s ), for which s 5 2 s /ßv v v v v 9 9which follows from e o P 5 i . Hence,1 1 2 2 1,s k50 s2 2 2and ß 5 s 1 s . S(L)g | MA(q), q # s 2 2.v v v t1 2† 2 † † †9 9Setting V 5 s r r , with r 5 [2i , (s 2v s21

1)]9, the specification considered by WH (1997,5. The West–Harrison representation p. 243) is obtained; with a terminology bor-

rowed from Roberts (1982), see also Section 7,Some seasonal models proposed in the litera- this will be referred to as the ‘crude seasonal’

ture cannot be represented as (3) with a time specification (CS henceforth). Its interesting†invariant V; nevertheless, they admit the so- feature is that, although rank(V ) 5 1, U |t

called WH (1997) representation: MA(s 2 2). In fact, by the last row of (15), the† theoretical autocovariance of U at lag s 2 2 ist9g 5 e F , F 5 PF 1 v , 2 †t 1,s t t t21 t

s (s 2 1). This illustrates that rank(V ) 5 q 1 1v† †v | NID(0, V ) (14) is not sufficient to ensure U | MA(q).t t

†9 9The constraint i V 5 0 implies that thes s†9 9where i V 5 0 , and P is the permutation model (14) can be reparameterised in terms of as s

matrix: first order vector autoregression with dimensions 2 1. This amounts to writing F 5 AC , for ant t0 Is21 s21 s 3 (s 2 1) transformation matrix A with theP 5F G91 0s21 9property i A 5 09. Hence,s

sn1jcharacterised by the periodic property: P 5 9 *g 5 (e A)C , C 5 TC 1 v ,t 1,s t t t21 tjP , for integer n and j 5 1, . . . ,s.* *v | NID(0, V ) (16)The connection with representation (3) is t

t212 29 9easily established writing e 5 e P , andj,s 1,s where T 5 A PA, A denoting the M-P inverse

t21 † 2 † 2defining the process F 5 P d , so that v 5 * 9of A, and V 5 A V A .t t tt21 † t21 t21 9 †P v , and V 5 P VP , provided V is If we choose A 5 Q, as given in (7), thent

t219 9time invariant. Obviously, as i P 5 i , (4) 9 9e Q 5 e , so the measurement equations s 1,s 1,s21†9 9 9implies i V 5 0 and i F 5 0. For the HS 9reduces to g 5 e C . Moreover, the transi-s s s t t 1,s21 t

† † tion matrix takes the form:model V is given by (6), although v is nojt


of the initial state C would be 0 and00 Is22 s22 k j j 9*T 5 . o T V T , respectively. For the HS and theF G j5092 i s21 *TS models the latter is proportional to kV ,This is equivalent to dropping the last element, whereas for the DS and CS models it is aF , from (14) and replacing F 5 2 periodic function of k, with period equal to s;st s,t21

s21 however, when 2 k is a multiple of s, it iso F in the last but one recursion,j51 j,t2121† 9proportional to k(I 2 s i i ) for the CSF 5 F 1 v ; hence, the transition s21 s21 s21s21,t s,t21 s21,t

model and to a tridiagonal matrix with k on themodel for the first s 2 2 elements is unchangedmain diagonal and 2 0.5k on the subdiagonals.whereas that for the element in position (s 2 1)

† † Assuming that the process has started in the*is amended. Finally, v 5 [v , . . . ,v ]9.t 1t s21,tremote past amounts to letting k go to infinityThe dummy seasonal (DS) model proposedin the expressions above. An analytic algorithmby HT (1983) is formulated so as to require that(termed the exact initial Kalman filter) forthe sum of the seasonal effects over a one yearinferences in the presence of such (often termedspan is a zero mean white noise innovation:diffuse) initial conditions is presented in Koop-

g 5 2 g 2 ? ? ? 2 g 1 v (17)t t21 t2s11 t man (1997).or equivalently S(L)g 5 v , where v | NID(0,t t t

2s ). The DS model is actually a particular casev

† 2of model (14) which has V 5 s rr9, withv 6. The smoothness of a seasonal modelr 5 [0, . . . , 0, 2 1, 1]9. As a matter of fact, it

*can be written as (16) with V 5 In order to ascertain the time domain prop-2 9s e e . The DS model can also be erties of the various seasonal models, let usv s21,s21 s21,s21

represented as in (3) with a time varying derive the autocovariance function of the2 2(t21) 2(t21)covariance matrix: V 5 s P rr9P . seasonal sums, U 5 S(L)g . Without loss oft v t t

On the other hand, the HS model can be generality, let us assume t 5 ns, for a positive2*represented as (16) with V 5 s (I 2 integer n, in (5), so thatv s21

21 9s i i ).s21 s21

U 5 v 1 (v 1 v ) 1 ? ? ?Table 1 recapitulates the structure of the t st s,t21 s21,t21† *matrices V and V for the four models

1 (v 1 v 1 ? ? ?s,t2s12 s21,t2s12considered so far. The specification is com-1 v )pleted by the first two moments of the initial 2,t2s12

distributions, which can be stated in terms ofwhere v is the jth element of v . IntroducingC , as F 5 QC . If the process had started at jt t0 0 0

the s 3 (s 2 1) matrixtime 2 k, the mean and the covariance matrix

Table 1aDisturbance covariance matrices for four standard seasonal models

† *Model V V

2 21 2 219 9HS s (I 2 s i i ) s (I 2 s i i )v s s s v s21 s21 s212 2 2 29TS s HH9 s Q HH9Qv v2 † † 29 9CS s r r s i iv v s21 s212 2 9DS s rr9 s e ev v s21,s21 s21,s21

a † 2 219Note: i 5 [1, 1, . . . , 1]9, r 5 [2i , (s 2 1)]9, r 5 [0, . . . , 0, 2 1, 1]9, e 5 [0, . . . , 0, 1]9, Q 5 (Q9Q) Q9,s s21 s21,s21

Q 5 [I , 2 i ]9, and the matrix H is given in Section 3.s21 s21


¯0 0 ? ? ? ? ? ? 0 0 1 Thus, exp 6(l ) is approximately the ratio be-0 0 ? ? ? ? ? ? 0 1 1 tween the geometric mean of the spectral den-0 0 ? ? ? ? ? ? 1 1 1S 5 sity at the low frequencies and the geometric: : ? ? ? ? ? ? : : :

mean at higher frequencies, and can be inter-3 40 0 ? ? ? ? ? ? 1 1 1preted as the ratio between the average long-run0 1 ? ? ? ? ? ? 1 1 1variance to the average short-run variance. Notevt that when U is white noise, the smoothnesstvt21U 5 [vec(S9)]9 . coefficient is zero everywhere.t :3 4

The left panel of Fig. 1 shows the naturalvt2s12

logarithm of the standardised power spectra ofThe autocovariance at lag k, c , is computedk U for the HS, TS and CS models when s 5 12.tas follows: The second has higher density at the frequency

c 5 E(U U ) p, whereas CS displays local minima at thek t t2k

seasonal frequencies, l , j 5 1, . . . ,6. The rightj95 [vec(S9)]9(I ^ V)vec(S ) (18)s21 k ¯panel shows the behaviour of (19) for l rangingwhere S is obtained shifting the columns of S from p /24 to p /2: the message conveyed isk

k positions to the right and filling with columns that the HS model is smoother than the TSof zeros: model; as a matter of fact, this results from the

fact that the TS model is characterised byIs2k‘excess’ power at p ; furthermore, both are0 :S 5 S .s21,kk outperformed, in terms of smoothness, by CS.3 40s2k21,s2k For the DS model, the smoothness coefficient is

¯identically 0 for all l.When the seasonal model is parameterisedIt is usually thought that seasonality shouldaccording to (14), the autocovariance function

be a smoothly evolving component (see, e.g.,9 9 9for U is (18) with S 5 [a , a , . . . ,a ], wheret 1 2 s21

Den Butter & Fase, 1991), as the institutionala is an s 3 1 vector with j unit elements at thej

factors and preferences change slowly acrosstop and zeros at the bottom, and V replaced by† time. Hence, there is a strong a priori forV .

preferring smooth representations. This can beIn the frequency domain, the various modelsdisputed, of course, as smoothness alone cannotcan be compared in terms of the spectral densitybe advocated to state the superiority of aof the seasonal sums, U :tseasonal model as is argued below; usuallys221 several criteria are employed jointly, which]f(l) 5 c 1Oc cos lk , l [ [0, p].F G0 k2p k51 makes finding an optimal seasonal model a verycomplex task.One possibility is to compare them with respect

That smoothness itself does not provide ato the measure of smoothness introduced bycriterion for the selection of a particular season-Froeb and Koyak (1994): in particular, the

¯ al model can be seen from the quarterly case,coefficient of smoothness at cutoff frequency ls 5 4. In such a case, under the assumption thatis the quantity:V is a Toeplitz matrix, V is given as in (13)

l̄ p

and the smoothness coefficient depends upon1 1¯ ] ]]6(l ) 5 Eln f(l)dl 2 Eln f(l)dl. the structural parameter s alone. In Fig. 2 the¯ ¯l p 2 l¯0 l behaviour of 6(p /8) for s varying in the range

(21, 0), for which V is positive semidefinite, is(19)


Fig. 1. Log-standardised spectra of U , ln[ f(l) /c ], for three seasonal models (left panel) and Froeb and Koyak smoothnesst 0¯ ¯coefficient, 6(l ) versus l (right panel).

Fig. 2. 6(p /8) as a function of the structural parameter s.

presented. The shape of the function makes there are two maxima, one for s 5 2 1, whichclear that the solution to the problem of max- corresponds to a non-invertible process at fre-imising smoothness is trivial: as a matter of fact, quency p /2, and one for s 5 0, for which U ist


˜ ˜˜ ñon-invertible at frequency p. We can also see y 5 m 1 lb 1 Ft1l ut t t hl mod sj,t

that the TS model, which has s 5 2 1/2, is less˜where y 5 E( y uY ), Y 5 hy , . . . ,y j; thet1l ut t1l t t 1 tsmooth than the HS model (s 5 2 1/3).

updating equations for its components are re-spectively:

˜7. Forecasting with the BSM ˜ ˜m 5 m 1 b 1 a vt t21 t21 m t

˜ ˜b 5 b 1 a vt t21 b t

This section aims at discussing two fun- ˜ ˜F 5 F 1 · v , j 5 1,2, . . . ,s 2 1jt j11,t21 j tdamental issues arising with respect to the˜ ˜F 5 F 1 · vst 1,t21 s tforecasts generated by the BMS with gt

(20)specified alternatively as HS, TS, DS, CS; in thefirst place we would like to establish how they ˜ ˜˜where m 5 E(m uY ), b 5 E(b uY ), F 5t t t t t t jtrelate to those produced by alternative ad hoc ˜ ˜˜E(F uY ), v 5 y 2 (m 1 b 1 F ) is thejt t t t t21 t21 1,t21and model-based forecasting procedures; sec-

one-step-ahead forecast error, and a , a , · ,m b jondly, we would like to investigate whether thej 5 1, . . . ,s, are smoothing factors satisfying theBSM predictive performance is sensitive to thefollowing constraints:choice of the seasonal model.

sTo address the first issue we shall refer to the0 # a ,a , 1, O · 5 0, a 1 · # 1.m b j m 1WH representation (14). Usually, the hyper-

j512 2 2 2parameters, s , s , s , and s , will have to bee h z v

The latter two constraints can be proved ana-estimated by maximising the diffuse likelihoodlytically; the first can be shown numerically, seefunction under the assumption that the distur-Anderson and Moore (1979, p. 156), by explor-bances, e , h , z and v , are mutually andt t t ting the behaviour of a and a for all hy-m bserially uncorrelated Gaussian. Details are givenperparameter combinations. It should be noticedin Koopman (1997); in the illustrations pre-that, by virtue of the zero sum constraint on thesented later in this section a computer pro-· ’s, the last updating equation can be droppedjgramme implementing Koopman’s exact initial

ãnd the last but one rewritten as F 5 2s21,tKalman filter has been written in the object s21 õ F 1 · v , which amounts to usingj51 j,t21 s21 toriented matrix programming language Ox 2.1representation (16).of Doornik (1998) and is available from the

The shape of the forecast function is co-author. We note in passing that estimation of theincident with that of a number of exponentialDS and TS is available in Stamp 5.0 (Koop-smoothing and model-based algorithms, namely:man, Harvey, Doornik & Shephard, 1995),

whereas the package Ssfpack, linked to the Ox1. Holt–Winters type forecasting procedure —code (Koopman, Shephard & Doornik, 1999),

Crude seasonal model (Roberts, 1982): forfeatures also HS, although relies upon an ap-three given smoothing constants,proximate solution to the initialisation of the0 # a ,a ,r , 1,m bfilter.· 5 r, · 5 2 r /(s 2 1), j 5 2, . . . ,s.The Kalman filter is the basic algorithm for 1 j

inference and prediction; when it has reached asteady state (see, e.g., Harvey, 1989, pp. 118– 2. Corrected Additive Seasonal Holt–Winters125), the forecast function of the BSM takes on (Newbold, 1988, p. 115): for three giventhe following form: smoothing constants, 0 # A,B,C , 1,


a 5 A 1 C(1 2 A) /s, a 5 AB, The noticeable feature is that the · ’s de-m b j

crease linearly for j . 1.s 2 1]]· 5 C(1 2 A),1 s In the structural framework, the pattern of

· 5 2 C(1 2 A) /s, j 5 2, . . . ,s.j coefficients associated with v in the updatingt

formulae depends on the variance parametersThe pattern of the · ’s is the same as for the 2 2 2 2j s , s , s , s ; their analytic derivation ine h z vcrude seasonal model (constant across j forterms of these quantities involves the solution ofj . 1), but the updating coefficients arean algebraic Riccati equation for the one-step-functionally related to the three smoothingahead state covariance matrix, which will usual-constants.ly prove quite complicated. Nevertheless, we3. Holt–Winters type forecasting procedure —can conclude that the structural approach isEWR seasonal model (Roberts, 1982): formore flexible, in that the pattern of the updatingthree given smoothing constants,coefficient is allowed to adapt to the nature of0 # a ,a ,r , 1,m b the series under investigation. Imposing patterns

s21 j22 to the · ’s, as in Roberts’ crude and ERWj· 5 1 2 r , · 5 2 r (1 2 r),1 jschemes, may yield forecasts with higher mean

j 5 2, . . . ,s. square error. Furthermore, the representation ofthe components, and in particular their ortho-The seasonal model is based on the notion ofgonality, prevents the insurgence of anomalousExponentially Weighted Regression (EWR)forecasting schemes, such as those arising forand it is advocated on the grounds that thethe Airline model when u , 1 2 2s / [(s 2 1)(1 1information content of the previous monthsQ)], in which circumstance · is greater than1on the current seasonal effect dies awayone, and it would be difficult to consider (20) asgeometrically. The notion of EWR can be

2 an error learning procedure, since the innova-further employed to restrict a 5 1 2 a , andm2 tions v would enter the updating of the seasonalta 5 (1 2 a) , where a [ (0, 1).b

s effect amplified by a factor greater than 1.4. Airline model: DD y 5 (1 1uL)(1 1 QL )j ,s t t2

2 1 #u, Q , 0, j | NID(0, s ). The updat-t

ing coefficients are given in Box, Pierce and 7.1. The airline seriesNewbold (1987, p. 281) and can be rewritten

The series of natural logarithms of monthlyas follows:passenger totals in international air travel (Box,

s 1 1Jenkins & Reinsel, 1994), available fromF ]]Ga 5 (1 1u ) 1 2 (1 1 Q)m 2sJanuary 1949 to December 1960, is considered

1 21 here to illustrate the pattern of the smoothing]1 (1 1 Q), a 5 s (1 1u )(1 1 Q)bs factors in (20) for the BMS. The estimatedˆparameter values of the Airline model are u 5s 2 1

2]]· 5 (1 1 Q) ˆ1 ˆ2 0.402, Q 5 2 0.557, s 5 0.00135, and thespattern of coefficients · is given in Table 1 ofs 2 1 j

]]2 (1 1u )(1 1 Q), Box et al. (1987). In our notation, · 5 0.257,2s 1

· 5 0.097 2 0.020( j 2 1); moreover, a 5 0.50j ms 1 1 2 2( j 2 1)]]]]]· 5 (1 1u )(1 1 Q) and a 5 0.02.j b2s

The corresponding estimates for four spe-1] cifications of the BSM, differing only for the2 (1 1 Q).s


Table 2 similar pattern, but with a more jagged appear-5Parameter estimates (multiplied by 10 ) and prediction ance; finally, the DS specification is character-

error variance ised by a lower value for · and by · ¯ 0 for1 j2 2 2 2ˆ ˆ ˆ ˆModel s s s s pev j 5 3, . . . ,10. Moreover, for HS, TS and CSh z v e

a 5 0.460 and a 5 0.001, whereas for DSHS 2902 0 219 2482 0.00138 m b

TS 2983 0 36 2344 0.00139 a 5 0.679 and a 5 0.001.m b

DS 6995 0 641 1295 0.00152 Hence, the forecasts produced by DS areCS 2865 0 18 2595 0.00138 likely to differ from TS, HS and CS, which are

in turn almost indistinguishable. On the otherhand, they will differ from those of the Airlinemodel, which updates respectively by a greater

seasonal model, are reported in Table 2, along and a lesser amount the slope and the seasonals.with the prediction error variance (pev), which Fig. 4 compares the forecasts for lead times uprepresents the variance of the one-step-ahead to 24 months, generated by the Airline and theprediction errors, v , in the steady state and is BMS models with HS and DS specificationst

2ˆdirectly comparable to s . (estimated using the observations up to Dec.The steady state coefficients · are plotted in 1959). For this particular illustration the HSj

Fig. 3. For the HS and the CS models their model minimises the sum of the squares of thepattern is virtually identical; TS displays a forecast errors.

Fig. 3. Airline series, pattern of the updating coefficients · , j 5 1, . . . ,12, for four BSM specifications.j


Fig. 4. Airline series, plot of the observed values and forecasts for 1,2, . . . ,24 months ahead, from the origin December1959.

7.2. US Industrial production through a given forecast origin and l-steps-ahead forecasts, l 5 1, . . . ,12, are computed.

In order to address the second issue, namely The procedure is repeated shifting the forecastwhether the choice of the seasonal model affects origin by one month until the end of the samplethe BMS predictive performance, we consider a period is reached; this yields a total of 84set of monthly US industrial production series one-step-ahead forecast errors and 72 twelve-for two-digit manufacturing industries, along steps-ahead forecast errors for the four spe-with the Total, Durable and Nondurable Manu- cifications of the BSM.facturing series, for a number of 23 series The results can be summarised as follows: foraltogether. This set, which has been studied by the vast majority of series the predictive per-Miron (1996), is made available electronically formance of HS, TS and CS at all forecastby the Federal Reserve Board at the URL horizons is virtually indistinguishable and repre-www.bog.frb.fed.us and covers the sents a significant improvement over DS. Aperiod 1947.1–1996.12. typical case is the Nondurable Manufacturing

Comparison of out-of-sample multistep mean series plotted in Fig. 5 (logarithms) along withsquare forecast error (MSFE) has been based on the estimated spectral density of the seasonalthe following rolling forecast exercise: starting and regular differences. Especially at shorterfrom 1989.12, the four alternative specifications forecast horizons, the MSFE reduction fromof the BSM are estimated using the observations using HS rather than DS seems to be quite high,


Fig. 5. US Index of industrial production for Nondurable manufacturing and Tobacco, Jan. 1947–Dec. 1996: series in logs,estimated spectral density of DD ln y , and root mean square forecast errors at forecast horizons 1, . . . ,12.12 t

as is evident from the third panel. Notice that time series under investigation, and in particularthe root MSFE is practically identical for HS, the relative smoothness of the seasonal pattern;TS and CS. for a wide class of macroeconomic time series,

These results, however, should not be taken for which seasonality changes slowly, the DSas evidence that the DS specification should not displays the worst out-of-sample performancebe entertained. As a matter of fact, in the same and is not to be recommended; nevertheless,data set it is possible to present a case, the index none of the models considered in this com-of industrial production for Tobacco, in which parison can be advocated as the optimal season-the forecasting performance of the DS spe- al model: in fact, every specification captures acification is slightly superior (see the last panel different kind of seasonality and is not, strictlyof Fig. 5). The comparison of the spectral speaking, nested in other models.density of the DD ln y with that of the12 t

Nondurable series reveals that the smoothnessReferencesproperties of the two series are rather different;

for the latter the contribution of the lowerAnderson, B. D. O., & Moore, J. B. (1979). Optimalfrequencies is substantially greater than in the

filtering, Prentice-Hall, Englewood Cliffs, NJ.former case. Box, G. E. P., Jenkins, G. M., & Reinsel, G. C. (1994).

In conclusion, the performance of the four Time series analysis, forecasting and control, 3rd ed.,specifications depends upon the features of the Prentice-Hall, Englewood Cliffs, NJ.


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Doornik, J. A. (1998). Object-oriented matrix program- SsfPack 2.2. Econometrics Journal 2, 107–160.ming using Ox 2.0, Timberlake Consultants Press, Miron, J. A. (1996). The economics of seasonal cycles,London. MIT Press, Cambridge MA.

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seasonal dummies. The Statistician 37, 111–127.hypothesis. Journal of Econometrics 64, 97–122.Proietti, T. (1998). Seasonal heteroscedasticity and trends.Hannan, E. J., Terrell, R. D., & Tuckwell, N. E. (1970).

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type forecasting models. Management Science 28, 808–Harrison, P. J., & Stevens, C. F. (1976). Bayesian forecast-820.ing. Journal of the Royal Statistical Society, Series B

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economic time series with structural and Box–Jenkins Biography: Tommaso PROIETTI (BSc Economics, Uni-models: a case study. Journal of Business and Econ- versity of Perugia; MSc Statistics, London School ofomic Statistics 1, 229–307. Economics; PhD Statistics, University of London) is

Hylleberg, S., & Pagan, A. R. (1997). Seasonal integration Assistant Professor at the Statistics Department of theand the evolving seasonal model. International Journal University of Udine. His research interests are time seriesof Forecasting 13, 303–434. decomposition, modelling, and forecasting.

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Comparing seasonal components for structural time series models