Exact maximum likelihood estimation for

non-stationary periodic time series models

Irma Hindrayanto, Siem Jan Koopman∗ and Marius Ooms,

VU University Amsterdam

Tinbergen Institute Amsterdam

January 14, 2010

Abstract

Time series models with parameter values that depend on the seasonal index are commonlyreferred to as periodic models. Periodic formulations for two classes of time series models areconsidered: seasonal autoregressive integrated moving average and unobserved componentsmodels. Convenient state space representations of the periodic models are proposed tofacilitate model identification, specification and exact maximum likelihood estimation of theperiodic parameters. These formulations do not require a-priori (seasonal) differencing ofthe time series. The time-varying state space representation is an attractive alternative tothe time-invariant vector representation of periodic models which typically leads to a highdimensional state vector in monthly periodic time series models. A key development is ourmethod for computing the variance-covariance matrix of the initial set of observations whichis required for exact maximum likelihood estimation. The two classes of periodic models areillustrated for a monthly post-war U.S. unemployment time series.

Key words: State space methods; time-varying parameters; SARIMA models; UnobservedComponent models.

∗Corresponding author: prof. dr. S.J. Koopman, Faculty of Economics and Business Administration,Department of Econometrics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, TheNetherlands. Phone: +31 20 598 60 19; e-mail: s.j.koopman@feweb.vu.nl

1 Introduction

Seasonal time series with sample autocorrelation functions that change with the season arereferred to as periodic time series. To enable the identification of such dynamic characteristicsin a time series, Gladysev (1961) and Tiao & Grupe (1980) have formally defined periodicautocorrelations using a stationary vector representation of periodic univariate time series. Onceperiodic properties of a time series are detected, the time series analyst can consider time seriesmodels that allow for these periodic correlations. A model-based periodic time series analysisbecomes effective when appropriate methods and algorithms are developed for estimation anddiagnostic checking. This is the primary aim of our paper.

The dynamic properties of a particular seasonal time series model are governed by parametersthat are usually assumed fixed throughout a given time period. In the context of autoregressivemoving average (ARMA) models, the parameters associated with the autoregressive (AR) andmoving average (MA) lag polynomials are usually assumed fixed; see, for example, Brockwell &Davis (1993). In the context of unobserved components (UC) time series models, the parametersdriving the stochastic processes for the components are usually fixed as well; see, for example,Harvey (1989) and Kitagawa & Gersch (1996). In case these parameters are allowed to bedeterministic functions of the season index, the model becomes part of the class of periodiclinear time series models.

Various developments on periodic time series are given in the statistics and econometricsliterature. Maximum likelihood estimation methods for periodic ARMA models have beendiscussed by Vecchia (1985), Li & Hui (1988), Jimenez, McLeod & Hipel (1989) and Lund& Basawa (2000). Furthermore, many environmental and economic studies have given empiricalevidence that time series models require periodically changing parameters; see, for example,Osborn (1988), Osborn & Smith (1989), Bloomfield, Hurd & Lund (1994), Ghysels & Osborn(2001) and Franses & Paap (2004).

In this paper we present a convenient time-varying (univariate) state space representation ofperiodic time series models to enable exact maximum likelihood estimation. The time-invariant(multivariate) representation is only used for identification analysis. We consider periodicity inseasonal autoregressive integrated moving average (SARIMA) as well as unobserved componenttime series models. For both periodic model classes, we analyse the non-stationary time serieswithout a-priori differencing. The initial conditions for the non-stationary parts of the modelsare treated by an exact initial Kalman filter. In case of the periodic SARIMA model, we adopt amodified Kalman filter to compute the initial variance-covariance matrix of the stationary partof the time series. Once the initialisations are treated appropriately, exact maximum likelihoodestimation can be carried out using the standard Kalman filter. Standard software tools areavailable for our solution. The development of new software is not required.

The remainder of this paper is organised as follows. In Section 2 we discuss periodic timeseries models and provide details on the state space formulations for both periodic SARIMAand periodic UC models. In Section 3 we apply both periodic models to monthly postwar U.S.unemployment series and compare the results with their non-periodic counterpart. Section 4concludes and gives ideas for further research. The appendices contain further details.

2

2 Periodic time series models and state space formulations

The state space form provides a unified representation of a wide range of linear Gaussian timeseries models including ARMA and UC time series models; see, for example, Harvey (1989),Kitagawa & Gersch (1996) and Shumway & Stoffer (2000). The Gaussian state space formconsists of a transition equation for the m× 1 state vector αt and a measurement equation forthe N × 1 observation vector yt for t = 1, . . . , n. We formulate the model as in de Jong (1991),that is

αt+1 = Ttαt +Htεt, α1 ∼ N (a, P ) , t = 1, . . . , n, (1)

yt = Ztαt +Gtεt, εt ∼ NID(0, I), (2)

where εt is an independent sequence of standard normally distributed random vectors. Thematrices Tt, Ht, Zt and Gt are referred to as the state space system matrices. The initial statevector is α1 with mean vector a and variance matrix P . Model (1)-(2) is linear and driven byGaussian disturbances. Therefore, the state space model can be treated by standard time seriesmethods based on the Kalman filter; see, for example, Anderson & Moore (1979) and Durbin &Koopman (2001).

The log-likelihood function for the state space model (1)-(2) is given by

l = log p(y1, . . . , yn;ϕ) =n∑t=1

log p(yt|y1, . . . , yt−1;ϕ)

= −nN2

log(2π)− 12

n∑t=1

(log|Ft|+ v′tF

−1t vt

), (3)

where ϕ is the parameter vector of the statistical model under consideration. The innovationsvt and their variances Ft are computed via the Kalman filter for a given vector ϕ; see Durbin& Koopman (2001, Chapter 4). The computational details of the Kalman filter are discussed inKoopman, Shephard & Doornik (1999).

The variance matrix P of the initial state vector α1 may contain diffuse elements whennon-stationary components are included in αt. In this case, diffuse initialisation methods forthe Kalman filter exist to evaluate the exact or diffuse likelihood function, see Ansley & Kohn(1985), de Jong (1991), and Koopman (1997). The diffuse likelihood function is optimized toobtain exact maximum likelihood estimates of the parameters. The suite of state space routinesSsfPack 3.0 of Koopman, Shephard & Doornik (2008) offers a numerically stable implementationof the necessary diffuse likelihood computations. Furthermore, we employ the Broyden-Fletcher-Goldfarb-Shanno (BFGS) numerical optimisation algorithm of Fletcher (1987) to maximise thelikelihood function with respect to unknown parameters.

Both periodic SARIMA and periodic UC models can be represented in the state space for-mulation (1)-(2) as we show in § 2.1 and § 2.3, respectively. All computations in this paper areobtained using SsfPack 3.0 of Koopman et al. (2008) as implemented for the matrix program-ming language Ox of Doornik (2006).

3

2.1 Periodic SARIMA model in state space representation

Maximum likelihood estimation of yearly stationary periodic ARMA models has been discussedby Vecchia (1985) (using the conditional likelihood), Li & Hui (1988) (extending the exactlikelihood of Ansley (1979)), Jimenez et al. (1989) (using a state space method for the exactlikelihood), Lund & Basawa (2000) and others. In this subsection we discuss the periodicSARIMA model and its state space representation in order to construct the exact likelihoodfunction.

There are different state-space representations of periodic SARIMA models. This paper fo-cuses on the univariate time-varying state space form for periodic SARIMA(p, d, q)(P,D,Q)Smodels. The multivariate time-invariant state space representation following the idea of Glady-sev (1961) is not attractive from a computational point of view since the dimension of thestate-space matrices can be large for modest orders of p, q and S.

We show our state space analysis for a general periodic version of SARIMA model. In thismodel, the observations yt follow a periodic SARIMA(p, d, q)(P,D,Q)S process given by

φp,s(L)ΦP,s(LS)(1− L)d(1− LS)Dyt = θq,s(L)ΘQ,s(LS)εt, εt ∼ NID(0, σ2ε,s), (4)

for t = SD+d+1, . . . , n and s = 1, . . . , S. We define L as the lag operator such that Liyt = yt−i.The periodic lag polynomials are defined as follows,

φp,s(L) = 1− φ1,sL− · · · − φp,sLp, (5)

Φp,s(LS) = 1− Φ1,sLS − · · · − ΦP,sL

SP (6)

θq,s(L) = 1 + θ1,sL+ · · ·+ θq,sLq, (7)

ΘQ,s(LS) = 1 + Θ1,sLS + · · ·+ ΘQ,sL

SQ, (8)

where φi,s,Φi,s, θi,s, Θi,s and σ2ε,s are coefficients that vary deterministically across the S different

periods of the year. The seasonal length is S with S = 4 for quarterly data and S = 12 formonthly data. In this paper, the differencing operator for the ‘levels’, d, takes values 0, 1, or2, while the differencing operator for the ‘seasonals’, D, takes values 0 or 1. The non-periodicSARIMA model is retrieved by restricting the parameters to be equal for all seasons. We denotethe differenced time series

y∗t = (1− L)d(1− LS)Dyt, (9)

where y∗t is a periodic ARMA(p∗, q∗) process with p∗ = p+ SP and q∗ = q + SQ.Next we represent the periodic SARIMA model (4) directly in state space form for yt.

Depending on the order of d and D, we can describe yt in terms of yt−i, (1 − L)dyt−i and y∗t .For example, if d = D = 1, we would have y∗t = (1−L)(1−LS)yt = (1−L−LS +LS+1)yt andafter some minor re-arrangements, we get

yt = yt−1 + (1− L)yt−S + y∗t . (10)

The stationary part associated with y∗t can be treated similarly as in the periodic ARMA state-space approach of Jimenez et al. (1989). The expression for yt as in (10) is incorporated directly

4

in the state space framework. As a result, our analysis for yt starts at t = 1 instead of t =SD + d+ 1. The non-stationary variables such as yt−i and (1− L)dyt−i are placed in the statevector and treated by diffuse initialisations following Ansley & Kohn (1985), de Jong (1991),Koopman (1997), Aston & Koopman (2006) and Koopman et al. (2008).

The elements of the state space form equations (1)-(2) are described as follows. Again,depending of the order of d and D, the state vector αt differs for each case. We define α∗t as thestate vector for d = D = 0 so that y∗t = yt. In this case, we have

α∗t = (y∗t , φ2,sy∗t−1 + · · ·+ φp,sΦP,sy

∗t−p∗+1 + θ1,sεt + · · ·+ θq,sΘQ,sεt−q∗+1, (11)

φ3,sy∗t−1 + · · ·+ φp,sΦP,sy

∗t−p∗+2 + θ2,sεt + . . . θq,sΘQ,sεt−q∗+2,

. . . , φp,sΦP,sy∗t−1 + θq,sΘQ,sεt)′,

with the dimension of α∗t equal to m = max(p∗, q∗ + 1). For other combinations of d and D, wehave the following state vectors:

D = 0; d = 1; αt = (yt−1, α∗t )′, (12)

D = 0; d = 2; αt = (yt−1, (1− L)yt−1, α∗t )′, (13)

D = 1; d = 0; αt = (yt−1, . . . , yt−S , α∗t )′, (14)

D = 1; d = 1; αt = (yt−1, (1− L)yt−1, . . . , (1− L)yt−S , α∗t )′, (15)

D = 1; d = 2; αt = (yt−1, (1− L)yt−1, (1− L)2yt−1, . . . , (1− L)2yt−S , α∗t )′, (16)

where the term y∗t in the state vector α∗t changes according to the orders of d and D, but thestructure of α∗t stays the same. The MA parameters are included in the disturbance vector,which is given by

Htεt =(

01×(SD+d), εt+1, θ1,sεt+1, . . . , θm−1,sεt+1

)′. (17)

The transition matrix Tt has dimension (SD+ d+m)× (SD+ d+m) and Zt is a row vector ofdimension 1 × (SD + d + m). For seasonal models with D = 1, the Tt and Zt matrices can bedefined via the matrix

(Tt

Zt

)=

1 1 01×(S−1) 1 1 0 0 . . . 00 1 01×(S−1) 1 1 0 0 . . . 00 0 01×(S−1) 1 1 0 0 . . . 00 0 IS−1 0 0 0 0 . . . 00 0 01×(S−1) 0 φ1,s 1 0 . . . 00 0 01×(S−1) 0 φ2,s 0 1 . . . 0...

......

......

.... . .

0 0 01×(S−1) 0 φm−1,s 0 0 . . . 10 0 01×(S−1) 0 φm,s 0 0 . . . 01 1 01×(S−1) 1 1 0 0 . . . 0

,

d = 2d = 1d = 0

(18)

where Zt is the last row of the matrix in (18) and Tt consists of the remaining rows. The r × ridentity matrix is denoted by Ir and an r × c matrix of zeros is denoted by 0r×c. For d = 2,

5

Tt and Zt are given by the full matrix (18), while for for d = 1 and d = 0 they are given bythe block matrices starting from the second row and column and the third row and column,respectively. The variance matrix of the state disturbances, HtH

′t = Var(Htεt), is given by

HtH′t =

(0(SD+d)×(SD+d) 0(SD+d)×m

0m×(SD+d) H∗tH∗ ′t

), (19)

where the m×m stationary part of this variance matrix is given by H∗tH∗ ′t . Since the measure-

ment equation has no error term, GtG′t = 0. Finally, the mean of the initial state vector is givenby a = E(α1) = 0(SD+d+m)×1 and the corresponding variance matrix is given by

P =

(κISD+d 0(SD+d)×m

0m×(SD+d) P †m×m

), with κ→∞. (20)

The matrix P † is the unconditional variance matrix for the stationary part of the state vector.The variable κ represents the diffuse initialisation for the non-stationary part of the state; seethe discussion in Koopman (1997).

For models with D = 0, the Tt and Zt matrices can be defined by the following matrix:

(Tt

Zt

)=

1 1 1 0 0 . . . 00 1 1 0 0 . . . 00 0 φ1,s 1 0 . . . 00 0 φ2,s 0 1 . . . 0...

......

.... . .

0 0 φm−1,s 0 0 . . . 10 0 φm,s 0 0 . . . 01 1 1 0 0 . . . 0

,

d = 2d = 1d = 0

(21)

where Zt is the last row of the above matrix and Tt consists of the remaining rows. The variancematrix of the state disturbances, HtH

′t, is then given by (19) with D = 0. Similar to the previous

case, the measurement equation has no error term, GtG′t = 0. The mean of the initial statevector is given by a = E(α1) = 0(d+m)×1 and the variance matrix of the initial state is given by(20) with D = 0.

2.2 Initialisation of periodic SARIMA models

Next we need to compute the m × m (unconditional) variance matrix P † for the stationaryelements in the initial state vector. We notice that matrix P † is the lower sub-matrix of Pwhich is defined in (20). The standard but inefficient method of solving for P † is by using thetime-invariant first-order vector autoregressive representation following Gladysev (1961) andTiao & Grupe (1980). Computational details of solving P † for non-periodic stationary ARMAmodels in state space form are given by, for example, Harvey (1989). To show that solvingfor P † using the time-invariant vector method is not computer memory efficient, we give anexample of a periodic MA(1) model for y∗t . In this case the multivariate time-invariant statespace representation has a state vector of length 2S and is given by

α∗t∗ =(y∗1,t∗ . . . y∗S,t∗ θ1,2ε1,t∗ . . . θ1,SεS−1,t∗ θ1,1εS,t∗

)′,

6

where y∗i,t∗ indicates the observation at season i of year t∗. The initial variance matrix P † of α∗1is the solution of

(I − T ⊗ T )vec(P †) = vec(HH ′), (22)

where I is the identity matrix, T is the transition matrix and HH ′ is the variance-covariancematrix of the state disturbances of the time-invariant state space representation. In general,solving equation (22) requires excessive memory as the length of the state vector for a periodicMA(q) model is (q + 1)S. Therefore, solving P † requires the inversion of matrix (I − T ⊗ T )which has dimension (q+1)2S2×(q+1)2S2. The inverse computations become computationallyinefficient for high values of q or S.

We suggest an alternative and more general method for the computation of the necessarysubmatrix P † of P using the Kalman filter without additional analytical work and without anyadditional programming effort. First we construct the time-varying state space matrices Tt,Zt, Ht and Gt as in (18)-(19) and we take arbitrary values for the mean vector and variancematrix of the initial state vector α1; for example, we can take a = 0 and P = I. We define Ptas the (unconditional) variance matrix of the state vector αt with the same partitioning as in(20) so that P †t is also implicitly defined. Then we apply the Kalman filter that is used for thecomputation of (3) but here we apply it to a long series of missing observations. The Kalmanfilter for this series of missing observations effectively carries out the computations for solvingthe set of S matrix equations

Pj+1 = TjPjT′j +HjH

′j , j = 1, 2, . . . , (23)

simultaneously with respect to the matrices P †k+1, . . . , P†k+S for j = k, . . . , k + S − 1 with P †k ≡

P †k+S and for any k > S. The solution is obtained recursively as part of the Kalman filterfor missing values. These recursive computations continue until convergence, that is, whenP †k+1 ≈ P †k−S+1 for a large value of k. We use the Frobenius norm to determine numericalconvergence. More specifically, let D∗ be the difference matrix between two matrices and let d∗ijbe the entry of matrix D∗ with i, j = 1, . . . ,m. Numerical convergence is then reached when

||D∗|| = ||P †k+1 − P†k−S+1|| =

√√√√ m∑i=1

m∑j=1

|d∗ij |2 < δ, (24)

with δ very small, say 10−9. Since the variance matrices Pj in (23) are associated with missingvalues, the actual index j is not relevant but the seasonal index associated with j is obviouslyof key importance. Therefore, after convergence, we make sure to take P † equal to P †k for anindex k that satisfies (24) and that corresponds to the period of the first observation in thedata set. This solution for P † is equivalent to the solution method implied by (22) and which isonly feasible for parsimonious periodic models without seasonal lags. The recursive method isrelated to solving Riccati equations, see Anderson & Moore (1979, p. 39, (1.10)-(1.12)). Varga(2008) recently surveyed and developed extensions to numerically stable solution methods formore general discrete-time periodic Riccati equations.

To adopt the method for exact maximum likelihood estimation of non-stationary models, thenon-stationary part of P has to be initialised using the diffuse variable κ. We have implemented

7

this approach using the exact initialisation methods of Koopman (1997) which are part of theprocedures in SsfPack 3.0, see Koopman et al. (2008). These initialisation methods are alsoimplemented in the time series package RATS, see Doan (2004).

When we return to our example for the periodic MA(1) model, the 2× 1 state vector of theunivariate time-varying state-space framework is given by

α∗t =(y∗t θ1,sεt

)′,

for any seasonal length s. The 2 × 2 initial variance matrix P † can be calculated analytically,but using the Kalman filter, we only need 5 iterations to solve P † in less than 1 second. Morecomplex periodic models would need larger number of iterations, but given the modern computertechnology and speed, our method is highly feasible.

2.3 Periodic unobserved components time series models

We base our discussion of the periodic UC model on the basic structural time series model(BSM) of Harvey (1989). The BSM consists of the trend, season and irregular components andits periodic version can be expressed by

yt = µt + γt + εt, εt ∼ NID(0, σ2ε,s),

µt+1 = µt + ηt, ηt ∼ NID(0, σ2η,s),

γt+1 = −∑S−2

j=0 γt−j + ωt, ωt ∼ NID(0, σ2ω,s),

(25)

for t = 1, . . . , n with n = n∗S, where σ2ε,s, σ

2η,s and σ2

ω,s are the variances for εt, ηt and ωt

respectively, and which vary by the season s = 1, . . . , S. It is assumed for convenience ofnotation that we have n∗ complete years of data.

By focussing on the moments of yt implied by the periodic BSM model, it is shown byKoopman, Ooms & Hindrayanto (2009) that univariate and multivariate representations of theperiodic model are not necessarily equivalent. Here we develop two convenient ways of placing aperiodic UC model into state space: a univariate time-varying and a multivariate time-invariantrepresentation. Both formulations enable exact maximum likelihood estimation and the estima-tion of the state vector by filtering and smoothing. In Appendix A we derive the second ordermoments of the periodic BSM and we argue that not all parameters of periodic UC models areautomatically identified.

Periodic UC models can be formulated via a univariate measurement equation (N = 1)and time varying system matrices Tt, Ht, Zt and Gt for t = 1, . . . , n. Alternatively they canbe represented by a multivariate measurement equation (N = S) for y∗t∗ with constant systemmatrices T , H, Z and G. Next we discuss these two convenient state space representations forthe periodic BSM (25) and based on (1)-(2) .

The state space matrices of the univariate time-varying parameter form of (25) forN = S = 2are given by

T =

(1 00 −1

), Ht =

(0 ση,t 00 0 σω,t

), (26)

Z =(

1 1), Gt =

(σε,t 0 0

), (27)

8

where ση,t, σω,t and ση,t vary deterministically according to the season. Note that the matricesHt and Gt are time-varying, while T and Z are constant over time. The state vector is given byαt = (µt, γt)′ and the disturbance vector is given by εt = (εt, ηt, ωt)′. The initialisation of αtis diffuse with a is a vector of 0’s and P is κI2 with κ→∞.

The multivariate time-invariant state space form of model (25) can be written as:

α∗t∗+1 = T ∗α∗t∗ +H∗ε∗t∗ , α∗1 ∼ N (a∗, P ∗) , t∗ = 1, . . . , n∗, (28)

y∗t∗ = Z∗α∗t∗ +G∗ε∗t∗ , ε∗t∗ ∼ NID(0, I). (29)

To simplify notation we consider model (25) for S = 2, where y∗t∗ = (yt, yt+1)′, t = 1, 1 + S, 1 +2S, . . . , 1 + (n∗ − 1)S. We derive convenient expressions for α∗t∗ , ε

∗t∗ , T

∗, H∗, Z∗ and G∗ asfollows. The measurement equations are given by

yt = µt + γt + εt, εt ∼ NID(0, σ2ε,1),

yt+1 = µt+1 + γt+1 + εt+1 = µt + ηt − γt + ωt + εt+1, εt+1 ∼ NID(0, σ2ε,2).

Further, we take α∗t∗ = (µt, ηt, γt, ωt)′ as the state vector and it follows from the transitionequations that

µt+2 = µt+1 + ηt+1 = µt + ηt + ηt+1, ηt+1 ∼ NID(0, σ2η,2),

γt+2 = −γt+1 + ωt+1 = γt − ωt + ωt+1, ωt+1 ∼ NID(0, σ2ω,2).

The state space matrices are then given by

T ∗ =

1 1 0 00 0 0 00 0 1 −10 0 0 0

, H∗ =

0 0 ση,2 0 0 00 0 0 ση,1 0 00 0 0 0 σω,2 00 0 0 0 0 σω,1

, (30)

Z∗ =

(1 0 1 01 1 −1 1

), G∗ =

(σε,1 0 0 0 0 00 σε,2 0 0 0 0

), (31)

with vector ε∗t∗ = (εt, εt+1, ηt+1, ηt, ωt+1, ωt)′ for t∗ = 1, 2, . . . , n∗ and t = 1, 1 +S, 1 + 2S, . . . , 1 +(n∗ − 1)S so that α∗t∗+1 = (µt+2, ηt+2, γt+2, ωt+2)′. The initialisation of the state vector isdiffuse for µ1 and γ1, while η1 and ω1 should be initialised by their variances. a∗ is a vector of0’s and P ∗ in this case is given by the matrix

P ∗ =

κ 0 0 00 σ2

η,1 0 00 0 κ 00 0 0 σ2

ω,1

, with κ→∞.

Note that the multivariate time-invariant system is observationally equivalent to the univariatetime varying system. In particular, the Gaussian likelihood of model (30)-(31) is exactly equalto the likelihood of model (26)-(27). For both specifications, it is clear that there is only onetrend for the whole observed series. Although the multivariate representation may suggest that

9

we have a separate trend for each season, the state vector α∗t∗ only has a single trend, µt, and asingle seasonal component, γt.

The periodic BSM can be extended by the inclusion of additional components (such as acycle) or by increasing the seasonal length in the univariate version of periodic UC models.The state space model with time-varying system matrices provides a general framework for thispurpose. However, when including a periodic stochastic cyclical component as in § 3.4 below,special attention must be given to its initialisation, see Koopman et al. (2009) for details. Forthe multivariate representation of the periodic UC model with a cycle, the initialisation issueis somewhat tedious but manageable. Only some elementary calculations in linear algebra arerequired. The merit of the multivariate specification is its dependence on time-invariant systemmatrices. It allows the examination of the dynamic properties of the time series in a straight-forward manner. The merit of the univariate specification is its straightforward initialisationtreatment. The drawback is its dependence on time-varying system matrices.

Finally, we focus on an identification problem related to periodic models with S = 2; seeAppendix A for technical details. In case of bi-annual periodic models, the number of unknownparameters is equivalent to the number of moment equations. However, the rank condition is notsatisfied. Fortunately, the reduced rank problem does not occur when S > 2. For the periodicBSM in equation (25), we have S(S + 1) linear equations to estimate 3S parameters. It is clearthat for S ≥ 3 we have more non-zero moment equations than parameters. The order-conditionfor identification is therefore satisfied. In practice, only 3S unique equations exist in the systemand all parameters can be identified exactly from these. The derivations are not given here tosave space. We conclude that the parameters are exactly identified in the two standard periodicUC models for all S > 2. Extension of the periodic BSM with a periodic stochastic cyclecomponent as defined in the empirical section does not lead to further identification problems.

3 Empirical illustration: U.S. unemployment

3.1 Data analysis

In this section we analyse a long monthly time series of U.S. unemployment data using bothperiodic SARIMA and periodic UC time series models. The data consists of seasonally un-adjusted monthly U.S. unemployment levels (in thousands of persons) for age 16 years orover. The estimation sample, from January 1948 until December 2007, has 720 observations.This series is published by the U.S. Bureau of Labor Statistics (BLS) and is obtained viahttp://www.bls.gov/webapps/legacy/cpsatab11.htm.

Figure 1 presents time series plots of log monthly unemployment. The top left panel showsthe data month by month, while the top right panel presents the data in multivariate form,year by year for each month. The monthly unemployment series is slowly trending upwards andcontains a pronounced cyclical pattern. The yearly series are smoother than the monthly seriesand clearly have common dynamics.

The middle and bottom rows of Figure 1 show a selection of the periodic sample autocor-relations of yearly changes in U.S. unemployment growth rates, ∆∆12yt, where yt is the log ofU.S. unemployment series at time t. An accurate definition of periodic autocorrelations is given

10

1960 1980 2000

8

9

log(U.S. unemployment)

1960 1980 2000

8

9

10Jan July

Feb Aug

Mar Sept

Apr Oct

May Nov

June Dec

0 25 50 75 100 125

−0.5

0.0

0.53PCor of dd12 log(unemp)

0 25 50 75 100 125

−0.5

0.0

0.56PCor of dd12 log(unemp)

0 25 50 75 100 125

−0.5

0.0

0.59PCor of dd12 log(unemp)

0 25 50 75 100 125

−0.5

0.0

0.512PCor of dd12 log(unemp)

Figure 1: U.S. unemployment (in logs, seasonally unadjusted), 1948.1 − 2007.12. Top row: Monthlytime series (left) and yearly time series for each month of the year (right) for s = 1, . . . , 12. Middle andbottom rows: Periodic autocorrelations of ∆∆12yt (monthly changes in the yearly growth rates of U.S.unemployment) for lags of 1 to 120 months, for s = 3, 6, 9, 12.

11

by McLeod (1994) and detailed in Appendix B. These sample autocorrelation coefficients areclearly periodic. They differ significantly from month to month. For example, for March (middleleft panel), there is a pronounced short cyclical movement of approximately 24 months startingwith 7 positive autocorrelations, while we see a much longer cyclical movement for June (middleright panel), starting with 14 negative autocorrelations. There is also a significant differencebetween March and June for the seasonal autocorrelation at lag 12. The periodicity in theautocorrelation structure is our main motivation to analyse log U.S. unemployment using peri-odic time series models. Other recent studies that analysed the periodic interaction of cyclicalchanges and seasonal patterns in other macro-economic time series are for example Van Dijk,Strikholm & Terasvirta (2003) and Matas-Mir & Osborn (2004).

3.2 Cycle variance moderation in U.S. unemployment

The structural volatility change in the 1980s is a well documented stylised fact occurring inseveral U.S. macroeconomic time series, see Kim & Nelson (1999), McConnell & Perez-Quiros(2000), Stock & Watson (2003), Sensier & van Dijk (2004) and Kim, Nelson & Piger (2004). Theexplanation of this decrease in volatility is still a matter of discussion. In monetary economics,the decrease in U.S. unemployment volatility can be associated with a change in monetary policy.Warne & Vredin (2006) find support for this explanation in a bivariate Structural Vector Au-toregressive (SVAR) model for inflation and unemployment, where volatility breaks are capturedby endogenous two-regime Markov-Switching.

We take the approach of Sensier & van Dijk (2004) by allowing for a breakpoint in thestochastic process governing U.S. unemployment. In the periodic SARIMA model, as we do notdistinguish between different variance components, we allow for different values of the σ2

ε,s’s inthe two subsamples, namely σ2

ε,1,s and σ2ε,2,s.

In the periodic UC model, we found that only by varying the shocks of the cycle component,discussed below in § 3.4 equation (36), large increases in the likelihood are obtained. Formally,we extend the model for the variances of the cycle component as σ2

κ,τ,s where τ is given by

τ =

{1, for t in the period from 1948.1 to h.122, for t in the period from (h+ 1).1 to 2007.12

where h is the breakpoint year. This specification effectively adds an additional periodic variancefactor to our model. The largest increase in the likelihood was found by having the break inthe early 1980s, which is in line with results in the literature. The differences in the likelihoodsfor different breakpoints in the early 1980s are small. We decided to fix the breakpoint in 1982.Warne & Vredin (2006) found additional variance switches before 1982, but for the purpose ofthis paper we found one variance break to be satisfactory.

3.3 Periodic SARIMA model for U.S. unemployment

In order to select a suitable SARIMA model form, we initially applied a seasonal difference filterto the observations. The resulting series does not show a clearly discernible trend, see Figure 2.However, a cycle could be readily perceived, which motivated the inclusion of an AR(2) term. To

12

take account of some remaining serial correlation, we included low order MA terms in both theseasonal and non-seasonal polynomials. This results in a SARIMA(2, 0, 1)(0, 1, 1)S with time-varying mean, which appears to be satisfactory in fitting the log of monthly U.S. unemploymentseries.

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

−0.5

0.0

0.5

1.0D12(log_unemployment)

0 10 20 30 40 50 60 70 80 90 100 110 120

−0.5

0.0

0.5

1.0ACF−D12(log_unemployment)

Figure 2: Seasonally differenced log unemployment series (∆12yt) and its ACF.

Let the seasonally differenced time series y∗t = ∆Syt be a periodic SARIMA(2, 0, 1)(0, 0, 1)Sprocess, which we write as

y∗t = βs + φ1,s(y∗t−1 − βs) + φ2,s(y∗t−2 − βs) + εt + θ1,sεt−1 + Θ1,sεt−S , (32)

for t = S + 1, . . . , n, s = 1, . . . , S, with εt ∼ NID(0, σ2ε,s), where βs, φ1,s, φ2,s, θ1,s, Θ1,s and

σ2ε,s are coefficients that vary deterministically across the S different periods of the year. Note

that in model (32), we omit the term θ1,sΘ1,sεt−(S+1) from the formal SARIMA(2, 0, 1)(0, 0, 1)Smodel. Further we rewrite model (32) in terms of the levels, yt, as follows

yt = βs + yt−S + y†t , (33)

where y†t = y∗t − βs = ∆Syt − βs = yt − yt−S − βs and we cast the periodic SARIMA model(32) directly in a state space form for yt with t = 1, . . . , n. The state space matrices are givenin Appendix C. With S = 12 we have 84 unknown parameters, βs, φ1,s, φ2,s, θ1,s,Θ1,s, σε,1,s andσε,2,s for s = 1, 2, . . . , 12 including the variance moderation term analogous to the specificationdescribed in § 3.2.

Given the state space form for yt in Appendix C, maximum likelihood estimation of theparameters φ1,s, φ2,s, θ1,s,Θ1,s, σε,1,s and σε,2,s is based on the same procedures as discussed in§ 2.1 and § 2.2. Note however that our periodic SARIMA model does not provide estimates ofseparate components. Since the periodic growth rates, βs, are included in the state vector, they

13

are effectively marginalised out of the likelihood function. Although the remaining number of 72parameters is large, it should be emphasised that we have 60 years of monthly data. Empirically,it turns out to be quite feasible to estimate this model.

0 2 4 6 8 10 12

1

2 φ1,s

0 2 4 6 8 10 12

0

1φ

2,s

0 2 4 6 8 10 12

−1

0

1θ

1,s

0 2 4 6 8 10 12

−1.0

−0.5

0.0Θ

1,s

0 2 4 6 8 10 12

0.025

0.050

0.075σ

ε ,s 1948−1981

0 2 4 6 8 10 12

0.025

0.050

0.075σ

ε ,s 1982−2007

Figure 3: Estimated parameters for periodic SARIMA model as in equations (32)-(33) for log U.S. unem-ployment with s = 1, . . . , 12. Top: φ1,s, φ2,s. Middle: θ1,s, Θ1,s. Bottom: σε,1948−1981,s, σε,1982−2007,s.All estimated parameters are plotted with ± 2 standard errors.

Figure 3 gives a graphical presentation of the parameter estimates of the periodic SARIMAmodel for U.S. unemployment with two standard errors bands. We see significant fluctuations inparameter estimates across the different months of the year which suggests that a non-periodicSARIMA specification is implausibly restrictive.

3.4 Periodic unobserved component model for U.S. unemployment

We have shown that U.S. unemployment is subject to periodic dynamics and therefore wecontinue our analysis by considering a periodic UC model, consisting of trend, season and cyclecomponents. The disturbance variances of the trend, season, cycle and irregular componentsare all periodic. The periodic UC model for the empirical section is given by

yt = µt + γt + ψt + εt, t = 1, . . . , n, (34)

with the trend (µt) and season (γt) are defined as follows,

µt+1 = µt + βt + ηt, ηt ∼ NID(0, σ2η,s),

βt+1 = βt + ζt, ζt ∼ NID(0, σ2ζ,s),

SS(L)γt+1 = ωt, ωt ∼ NID(0, σ2ω,s),

(35)

14

where SS(L) is the summation operator defined as SS(L) = 1 + L + · · · + LS−1. The cyclecomponent (ψt) is defined as,(

ψt+1

ψ+t+1

)= ρs

(cosλ sinλ− sinλ cosλ

)(ψt

ψ+t

)+

(κt

κ+t

), κt, κ

+t ∼ NID(0, σ2

κ,τ,s), (36)

with variances σ2κ,1,s in January 1948 to December 1981 and σ2

κ,2,s from January 1982 onwardsfollowing in Sensier & van Dijk (2004). A restriction on the damping terms 0 <

∏Ss=1 ρs < 1

is imposed to ensure that the stochastic process ψt is stationary. Further we assume thatthe irregular term (εt) has variances σ2

ε,s and that all disturbances are mutually uncorrelatedsequences from a Gaussian density for s = 1, . . . , S.

The above model is cast into a state space form containing 7S + 1 unknown parameters,ση,s, σζ,s, σω,s, σκ,1,s, σκ,2,s, σε,s, ρs and λ for s = 1, . . . , S. The parameters are estimated bynumerically maximising the exact log-likelihood. In our case with S = 12, we originally estimate85 parameters. Since the parameters ση,s, σζ,s, and σε,s are mostly estimated as nearly zeroes,we fixed them at zero for all seasons except for σζ,9 and the number of free parameters thenbecomes 50. The estimated components, associated parameters, and their interpretation aresimilar as presented in Koopman et al. (2009).

3.5 Estimation and forecast results

In our analysis of monthly postwar U.S. unemployment series, we discover periodicity in mostparameters also in the seasonal ones. Figure 1 clearly indicates periodicity of the autocorrelationfunction. To fit the U.S. unemployment monthly series, we use the periodic SARIMA model asin (32)-(33) and the (perioidc) UC model as in (34)-(36). For the non-periodic SARIMA modelwe fit the time series with several SARIMA models starting with AR order one until 13. Wechoose the model with smallest value for Akaike information criterion (AIC) and AIC with finitesample correction (AICc) which corresponds to an unrestricted SARIMA(3,0,1)(0,1,1)12 as ourfinal non-periodic SARIMA model.

Table 1 reports the log-likelihood value at the parameter estimate, AIC, AICc, and somediagnostic tests for the one-step-ahead prediction errors. These estimation results are for modelswith variance moderation. Following Brockwell & Davis (1993), the AIC and AICc statisticsare defined as AIC = −2 logL + 2p and AICc = −2 logL + (2np)/(n − p − 1). The Akaikecriteria indicate to choose the periodic models in favour of their non-periodic counterparts, butthe corrected Akaike criteria choses the non-periodic SARIMA above the periodic SARIMAmodel. By introducing the variance moderation in 1982, we eliminate heteroskedasticity andnon normality problems in the prediction errors which as can be concluded from the N and H

statistics: the normality N statistic is based on the sample skewness and kurtosis as describedin Bowman & Shenton (1975); the heteroskedasticity H statistic is the classical test of Goldfeld& Quandt (1965). The reported nominal p-values are indicative as they do not account for thevariance moderation.

Comparing the results of SARIMA and UC models in general is a non-standard exercise.Since the models are not nested, we cannot directly compare the log-likelihood values for theestimated parameters. Taking the AIC(c) as an indicative goodness-of-fit criterion, it appears

15

Table 1: Estimation results for models with variance moderation in 1982

SARIMA PSARIMA UC PUC(3,0,1)(0,1,1)S (2,0,1)(0,1,1)†S

logL 1157.59 1226.52 1182.88 1238.23AIC -2299.18 -2309.04 -2353.76 -2376.46AICc -2298.98 -2292.79 -2353.64 -2368.83N 2.6769 5.3150 2.5656 4.6877(p-value) (0.262) (0.070) (0.277) (0.096)H 0.9117 0.9088 0.9585 0.9923(p-value) (0.381) (0.364) (0.687) (0.941)p 8 72 6 50‡

n∗S 720 720 720 720

Notes: Data sample used for estimation purpose is from January 1948 until December 2007. PSARIMA = periodic

SARIMA; PUC = periodic UC. † : the cross term θ1,sΘ1,sεt−(S+1) of PSARIMA model is omitted. logL = log-

likelihood. AIC = Akaike Information Criterion. AICc = AIC with finite sample correction. N is a normality test

on the prediction errors; Na∼ χ2(2); H is a heteroskedasticity test on the prediction errors; H

a∼ F (n∗S/2, n∗S/2);

p is the number of parameters. ‡ : the number of parameters in periodic UC model is calculated as 49 + 12 (cycle

variance moderation parameters σκ,s,1982) − 11 (zeros in σζ,s). n∗S is the number data points with n∗ = 60 and

S = 12. The lowest AIC and AICc between periodic and non-perioidc version for both type of models are printed

in boldface.

that the periodic UC model outperforms the non-periodic UC model since it has the lowestAIC(c) values, but the periodic SARIMA model is not superior to its non-periodic version, seeTable 1. However, the differences between SARIMA model and non-parsimonious PSARIMAmodel are not statistically significant at normal significance levels, both measured using in-sample fit as in Table 1 (which could favor extensive models) and using out-of-sample forecastingaccuracy as in Table 2 (which could favor parsimonious models). Note that all four modelspresented in the paper have satisfactory residual diagnostics.

Based on the parameter estimates up to December 2000, the periodic UC model has thebest one-step-ahead out-of-sample forecasting performance from January 2001 until December2008 according the overall RMSE. Using the forecast accuracy test of non-nested models byDiebold & Mariano (1995), we find insignificant differences between the forecasts generated bythe (periodic) UC and (periodic) SARIMA models, see Table 2. S1,3 is the Diebold-Mariano(DM) test applied to UC and SARIMA model, S1,4 is DM test applied to UC and PSARIMAmodel, S2,3 is DM test applied to PUC and SARIMA model, S2,4 is DM test applied to PUCand PSARIMA model, and S3,4 is DM test applied to SARIMA and PSARIMA model. Noneof the overall test for DM procedure is significant. As the size of the Diebold-Mariano testmight be difficult to control, see e.g. Harvey, Leybourne & Newbold (1998), and the forecasterror variance estimates are only based on 8 years, the results have to be interpreted with care.Overall, the differences seem insignificant.

The optimal model is likely to be a more parsimonious PSARIMA specification which could

16

also be estimated using the algorithm of Section 2. More parsimonious periodic SARIMAmodels can be formulated in different ways, for example different models with equal parameterscan be specified for subsets of months, see Penzer & Tripodis (2007) with their applicationfor economic time series. Alternatively, SARIMA models with periodic parameters as smoothfunctions of the month-of-the-year can be considered, see Anderson & Vecchia (1993) withtheir application for geophysical time series. These examples of parsimonious periodic SARIMAmodel specifications fall outside the scope of this paper. The empirical example in this paperillustrates the applicability of our technique, which enables comparisons between parsimoniousspecifications against the unrestricted periodic model using exact maximum likelihood, evenwhen the number of periods is comparatively large.

4 Summary and conclusion

The primary aim of this paper is to develop estimation methods for time series models withperiodically varying parameters including those associated with the seasonal dynamics. Theestimates are obtained by the exact maximum likelihood method with diffuse initialisations forthe non-stationary components in the model. We further develop a feasible estimation methodfor periodic SARIMA models, which does not require a-priori differencing of the data.

The periodic models under investigation can be extended or restricted in several ways. Inshorter samples and for higher frequencies, placing other (smoothness) restrictions on the intra-yearly patterns of the periodic coefficients may be necessary to obtain sufficiently parsimoniousmodels. Extensions of the statistical analysis are possible within our framework. FollowingKurozumi (2002) and Busetti & Harvey (2003), stationarity tests for the periodic componentsin the model can be developed. Finally, the periodic approach can be extended to multivariatetime series models without essential modification of the algorithm. It is however an empiricalquestion whether such extensions will result in a better fit and in more effective interpretationsof the results.

Appendices

A Moments and identification of periodic BSM for S = 2

To investigate whether all parameters in the UC models are identified, we use systems of momentconditions. The moment conditions are based on the autocovariance function of the stationaryform of the UC models. Consider model (25) with S = 2 where we have 6 parameters toestimate, namely σε,1, σε,2, ση,1, ση,2, σω,1 and σω,2. The stationary form of model (25) is basedon yearly differences ∆Syt = (1− LS)yt of the observations yt. For S = 2, that is

∆2yt+i = ηt+i−2 + ηt+i−1 − ωt+i−2 + ωt+i−1 + εt+i − εt+i−2, (37)

for i = . . . ,−2,−1, 0, 1, 2, . . . and t = 1, S + 1, 2S + 1, . . . . It follows that E[∆2yt+i] = 0, ∀ i.

17

Tab

le2:

Acc

urac

yte

stfo

ron

e-st

ep-a

head

out-of

-sam

ple

fore

cast

ing

betw

een

peri

odic

and

non-

peri

odic

mod

els

for

Janu

ary

2001

untilD

ecem

ber

2008

RM

SED

iebo

ld-M

aria

note

st

UC

PU

CSA

RIM

AP

SAR

IMA

S1,3

S1,4

S2,3

S2,4

S3,4

(1)

(2)

(3)

(4)

Jan

0.04

800.

0470

0.04

300.

0353

1.17

911.

5167

0.73

770.

9767

1.08

31Fe

b0.

0263

0.03

350.

0250

0.04

210.

2927

-1.3

769

1.69

54-0

.763

1-1

.348

4M

ar0.

0338

0.02

810.

0351

0.03

64-0

.657

8-0

.356

8-1

.010

4-0

.712

7-0

.186

2A

pr0.

0298

0.02

910.

0317

0.05

83-0

.342

5-2

.005

1-0

.323

5-2

.317

1-2

.081

5M

ay0.

0449

0.04

810.

0419

0.05

250.

8274

-1.1

569

1.55

26-0

.680

0-1

.128

7Ju

n0.

0288

0.03

900.

0203

0.03

291.

0426

-0.5

327

1.63

050.

6638

-1.2

623

Jul

0.02

430.

0329

0.02

230.

0284

1.49

62-1

.069

01.

5464

0.51

31-1

.513

5A

ug0.

0371

0.03

150.

0349

0.03

890.

4956

-0.2

854

-0.4

668

-0.7

877

-1.4

211

Sep

0.02

540.

0278

0.02

270.

0254

1.39

41-0

.009

90.

7632

0.40

95-0

.686

9O

ct0.

0369

0.02

390.

0403

0.04

03-1

.839

6-0

.917

4-1

.692

2-1

.679

9-0

.005

8N

ov0.

0271

0.02

160.

0246

0.03

211.

5308

-0.7

691

-0.4

413

-1.6

388

-1.1

336

Dec

0.04

020.

0338

0.04

190.

0465

-0.3

039

-0.7

531

-1.7

605

-1.6

374

-1.3

017

Ove

rall

test

0.03

440.

0330

0.03

390.

0402

0.35

93-0

.630

40.

1309

-0.6

240

-0.8

367

Note

s:D

ata

sam

ple

use

dfo

rpara

met

ers

esti

mati

on

isfr

om

January

1948

unti

lD

ecem

ber

2000.Si,j

are

the

Die

bold

-Mari

ano

(DM

)te

sts

for

equal

one-

step

-ahea

dM

SE

fore

cast

acc

ura

cyof

two

non-n

este

dm

odel

sw

her

easy

mpto

tica

lly

each

Si,j∼N

(0,1

).T

he

crit

ical

valu

eat

5%

signifi

cance

level

for

the

DM

test

isgiv

enat±

1.9

6.

Sig

nifi

cant

neg

ati

ve

(posi

tive)

valu

efo

rSi

indic

ate

sth

at

fore

cast

sgen

erate

dby

the

firs

tm

odel

are

more

(les

s)acc

ura

teth

an

the

fore

cast

sfr

om

seco

nd

model

.S

1,3

is

DM

-tes

tapplied

toU

Cand

SA

RIM

Am

odel

,S

1,4

isD

Mte

stapplied

toU

Cand

PSA

RIM

Am

odel

,S

2,3

isD

M-t

est

applied

toP

UC

and

SA

RIM

Am

odel

,S

2,4

isD

M-t

est

applied

toP

UC

and

PSA

RIM

Am

odel

,andS

3,4

isD

M-t

est

applied

toSA

RIM

Aand

PSA

RIM

Am

odel

.N

one

of

the

over

all

test

for

DM

pro

cedure

issi

gnifi

cant.

RM

SE

=

Root

Mea

nSquare

Err

or.

Low

est

RM

SE

for

the

four

model

sand

signifi

cant

valu

efo

rD

M-t

est

are

giv

enin

bold

face

.

18

The yearly autocovariance function for t = 1, S + 1, 2S + 1, . . . is given by

Γ0 = E

[(∆2yt

∆2yt+1

)(∆2yt

∆2yt+1

)′]

=

(σ2η,1 + σ2

η,2 + σ2ω,1 + σ2

ω,2 + 2σ2ε,1 σ2

η,2 − σ2ω,2

σ2η,2 − σ2

ω,2 σ2η,1 + σ2

η,2 + σ2ω,1 + σ2

ω,2 + 2σ2ε,2

), (38)

Γ1 = E

[(∆2yt

∆2yt+1

)(∆2yt−2

∆2yt−1

)′]=

(−σ2

ε,1 σ2η,1 − σ2

ω,1

0 −σ2ε,2

), (39)

Γj = E

[(∆2yt

∆2yt+1

)(∆2yt−2j

∆2yt+1−2j

)′]= 0, for j = 2, 3, 4, . . . , (40)

which is equivalent to the autocovariance function of a vector moving average process with onelag. Note that the autocovariance matrix of lag 1, Γ1, is not symmetric, namely

Γ−1 = E

[(∆2yt

∆2yt+1

)(∆2yt+2

∆2yt+3

)′]=

(−σ2

ε,1 0σ2η,1 − σ2

ω,1 −σ2ε,2

)= Γ′1. (41)

These moment expressions reconfirm that we do not have a standard multivariate local levelmodel for y∗t∗ , see the discussion in Koopman et al. (2009).

To identify the parameters from the autocovariances, we need to solve a linear system ofmoment equations. Asymptotically, the expressions for the autocovariances determine the Gaus-sian likelihood that we use in estimation. If two instances of a time series model with differentparameters have the same autocovariance function and therefore the same spectrum and thesame moving average representation, the parameters cannot be identified by the Gaussian MLestimator, see Brockwell & Davis (1993, § 10.8) and Yao & Brockwell (2006).

Rewriting expressions (38)-(40) we get the system of moment equations for model (25)

1 1 1 1 2 00 1 0 −1 0 01 1 1 1 0 20 0 0 0 −1 01 0 −1 0 0 00 0 0 0 0 −1

σ2η,1

σ2η,2

σ2ω,1

σ2ω,2

σ2ε,1

σ2ε,2

=

Γ0(1, 1)Γ0(1, 2)Γ0(2, 2)Γ1(1, 1)Γ1(1, 2)Γ1(2, 2)

, (42)

where Γj(i, k) indicates the element on row i and column k of matrix Γj . The system of equations(42) clearly has multiple solutions as the the first matrix only has rank 5. The null space ofthis matrix is spanned by the vector (1 − 1 1 − 1 0 0)′. Hence, we have to impose arestriction on the first four parameters to obtain identification, for example

σ2η,1 − σ2

η,2 + σ2ω,1 − σ2

ω,2 = 0. (43)

Note that the restriction is not unique, we can also impose

σ2η,1 = σ2

η,2 or σ2ω,1 = σ2

ω,2. (44)

19

B Periodic correlations

Denote ys,t∗ as the observation for period s and year t∗ such that yt ≡ ys,t∗ , where t = (S−1)t∗+sfor t = 1, . . . , n∗S, t∗ = 1, . . . , n∗ and s = 1, 2, . . . , S. Sample periodic correlations have beendefined by, i.a., McLeod (1994). Consider the separate series in {yt∗} where

{yt∗} =(y1,t∗ y2,t∗ · · · yS,t∗

)′(45)

and standardise the series ys,t∗ separately by subtracting the periodic means and by dividing bythe periodic standard deviations to get {yt∗}, where

{yt∗} =(y1,t∗ y2,t∗ · · · yS,t∗

)′(46)

for s = 1, . . . , S, t∗ = 1, 2, . . . , where t∗ is the index in years. If we associate y1,t∗ with yt (seriesy at time t), y2,t∗ with yt+1 (series y at time t+ 1), . . . , and yS,t∗ with yt+S−1 (series y at timet+ S − 1), we can also view the standardised series as

{yt∗} =(yt yt+1 · · · yt+S−1

)′, for t = 1, S + 1, 2S + 1, . . . . (47)

Next, consider all the covariances between the standardised subseries, {yt∗}, and their lags{yt∗−j}, where

{yt∗−j} =(y1,t∗−j y2,t∗−j · · · yS,t∗−j

)′=(yt−jS yt+1−jS · · · yt+S−1−jS

)′, for t = 1, S + 1, 2S + 1, . . . .

For S = 2 the periodic correlations γi,s, i = 1, 2, . . ., s = 1, . . . , S, are selected from the multi-variate correlation matrices of yt∗ as follows:

E

[(y1,t∗

y2,t∗

)(y1,t∗

y2,t∗

)′]=

(1 γ1,−1

γ2,1 1

)

E

[(y1,t∗

y2,t∗

)(y1,t∗−j

y2,t∗−j

)′]=

(γ1,2j γ1,2j+1

γ2,2j+1 γ2,2j

), j = 1, 2, . . . .

We compute the sample periodic correlations by noting that the sample correlations ci,s, i =1, 2, . . . , (n−1)S, s = 1, . . . , S are associated with the population correlations γi,s defined abovefor S = 2 and without degrees of freedom corrections. The computation is therefore basic. ForS > 2 the computation is analogous.

C Periodic SARIMA(2,0,1)(0,1,1)S

For s = 1, . . . , S, the state vector αt of periodic SARIMA(2,0,1)(0,1,1)S model is defined asαt = (β1 . . . βS yt−1 . . . yt−S y

†t φ2,sy

†t−1 + θ1,sεt + Θ1,sεt−S+1 Θ1,sεt−S+2 . . .Θ1,sεt)′,

with corresponding disturbance vector given by

Htεt =(

01×2S εt+1 θ1,sεt+1 01×(S−2) Θ1,sεt+1

)′.

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The transition matrix Tt is therefore (3S + 1)× (3S + 1) and Tt and Zt are given by

Tt =

Ta 0S×(2S+1)

Tb Tc

0(S+1)×S Td

, Zt = first row of[Tb Tc

], (48)

Ta =

[0 IS−1

1 01×(S−1)

], Tc =

[01×(S−1) 1 1 01×S

IS−1 0(S−1)×1 0(S−1)×1 0(S−1)×S

],

Tb =

[1 01×(S−1)

0(S−1)×1 0(S−1)×(S−1)

], Td =

02×S

φ1,s

φ2,s

I2 02×S−2

0(S−2)×S 0(S−2)×1 0(S−2)×2 I(S−2)

01×S 0 01×2 01×(S−2)

.where 0r×c denotes a zero matrix with r rows and c columns and where Ir denotes an identitymatrix of dimension r.

The variance-covariance matrix of the state disturbances is given by

HtH′t =

0 01×2S 01×(S+1)

02S×1 02S×2S 02S×(S+1)

0(S+1)×1 0(S+1)×2S H∗tH∗ ′t

, (49)

where

H∗tH∗ ′t =

σ2ε,s θ1,sσ

2ε,s 01×(S−2) Θ1,sσ

2ε,s

θ1,sσ2ε,s θ2

1,sσ2ε,s 01×(S−2) θ1,sΘ1,sσ

2ε,s

0(S−2)×1 0(S−2)×1 0(S−2)×(S−2) 0(S−2)×1

Θ1,sσ2ε,s θ1,sΘ1,sσ

2ε,s 01×(S−2) Θ2

1,sσ2ε,s

. (50)

whereas Gt = 0. Finally, a = E(α1) = 0(3S+1)×1 and the variance matrix of the initial state isgiven by

P =

[κI2S 02S×(S+1)

0(S+1)×2S P †

], (51)

where the initial variance matrices of the growth rates βt and the levels yt are diffuse: κ→∞.The variance matrix of the stationary elements of the state, P †, can be constructed by a pre-runof the Kalman filter as explained in § 2.2.

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