Periodic Seasonal Time Series Models with applications to U.S. macroeconomic data

Transcription

1 Periodic Seasonal Time Series Models with applications to U.S. macroeconomic data

2 ISBN Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul This book is no. 503 of the Tinbergen Institute Research Series, established through cooperation between Thela Thesis and the Tinbergen Institute. A list of books which already appeared in the series can be found in the back.

3 VRIJE UNIVERSITEIT Periodic Seasonal Time Series Models with applications to U.S. macroeconomic data ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. L.M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Economische Wetenschappen en Bedrijfskunde op woensdag 7 september 2011 om uur in de aula van de universiteit, De Boelelaan 1105 door Anastasia Irma Widyanti Hindrayanto geboren te Surabaya, Indonesië

4 promotor: copromotor: prof.dr. S.J. Koopman dr. M. Ooms

5 To my parents

6

7 Contents 1 Introduction 1 2 Periodic unobserved components (PUC) models Introduction Unobserved components models Periodic unobserved components models Periodic basic structural time series model Convenient state space representation of periodic BSM Periodic stochastic cycle model Simulation study on the periodic cycle component Periodic stochastic cycles and seasonal adjustment Parameter estimation and signal extraction Application to U.S. unemployment data Periodic UC model for U.S. unemployment series Cycle variance moderation in U.S. unemployment series Forecasting weights and forecasting performance Summary and conclusion Appendices Multivariate PUC models Introduction Multivariate periodic unobserved components models Periodic specification Estimation, testing and signal extraction Data description: U.S. employment series Empirical results for U.S. employment sectors Model specification within the class of MPUC models UC decomposition in the final model Residual diagnostics Comparison with the Krane and Wascher study

8 3.6 Summary and conclusion Appendix Periodic SARIMA models Introduction Periodic SARIMA models in state space representation Initialisation of periodic SARIMA models Empirical illustration: U.S. unemployment Data analysis Periodic SARIMA model for U.S. unemployment series Periodic UC model for U.S. unemployment series Estimation and forecast results Summary and conclusion Appendix Frequency-specific trigonometric seasonality Introduction Frequency specific basic structural model Properties of FS-BSM and measuring distance to FS-AM Stationary form of FS-BSM Dynamic properties of a FS-BSM Distance between FS models using MA coefficients Monte-Carlo studies for frequency specific models Empirical results Two examples comparing BSM and FS-BSM BSM vs FS-BSM in 75 time series How close are these (FS-) models to each other? Model fit comparison Summary and conclusions Appendices Conclusions 129 Bibliography 131 Samenvatting (Summary in Dutch) 139 Acknowledgements 143

9 Chapter 1 Introduction Many time series encountered in econometrics and in natural sciences display regular seasonal fluctuations. In the analysis of such series, the seasonal variation can be directly incorporated in the model, or removed from the series by seasonal adjustment methods which can implicitly or explicitly rely on seasonal models. This dissertation describes and analyses several types of periodic seasonal time series models and develops a new formulation which facilitates parameter estimation in the state space representation of the model. In periodic time series analysis, a set of (usually annual) time series is simultaneously analysed, and each individual series is exclusively related to a particular season. This approach is in contrast with the more widely adopted approach of treating a series as a single stochastic process with seasonal fluctuations (for example seasonal dummy variable). Concentrating on a time series that is associated with a specific season, such that it does not possess seasonal dynamics, may circumvent some of the intricate aspects of modelling seasonal variations. Periodic approaches have been investigated in the context of autoregressive moving average (ARMA) models and dynamic econometric models, see for example the book of Ghysels and Osborn (2001) or Franses and Paap (2004). Other researchers have investigated the periodic approach with various state space time series model formulations, e.g., Krane and Wascher (1999), Koopman and Ooms (2002, 2006), Proietti (2004), Bell (2004) and Penzer and Tripodis (2007). In this thesis we further explore the periodic analysis of both autoregressive moving average (ARMA) models and unobserved components (UC) time series models. The ARMA model is well-known and is covered in many textbooks on time series. The UC model decomposes a time series into components of interest including trend, seasonal, cycle and irregular, see Harvey (1989), Durbin and Koopman (2001) and Commandeur and Koopman (2007). We mainly focus on the analysis of seasonal macroeconomic time series where we extend the UC models by having periodic coefficients that are associated

10 2 CHAPTER 1. INTRODUCTION with the different components. Our contribution for periodic ARMA models lies in a convenient state-space formulation of these models that do not require (seasonal) differencing of the time series beforehand. Throughout the thesis, we pay special attention to identification of the parameters before we start the estimation procedure. Further we show that exact maximum likelihood estimation is feasible despite the large number of parameters that are typically encountered in this class of periodic models. We also link periodic and non-periodic model formulations by means of likelihood-ratio tests as these models are nested. Empirical illustrations are given by using monthly U.S. unemployment time series for univariate periodic models (Chapters 2 and 4) and quarterly U.S. employment time series for the multivariate periodic models (Chapter 3). A distinctive characteristic of periodic analysis is that we need to repeat a univariate analysis for each season or to take a multivariate approach by modelling the S time series simultaneously where S is the seasonal length. In this thesis, we advocate the use of univariate representation with time-varying parameters for periodic models. We demonstrate in Chapters 2 and 3 that this approach is identical to the multivariate representation of periodic models with fixed parameters. The advantage of using the univariate representation lies in the possibility of expanding the model for several time series together, so that we have multivariate periodic models as explained in Chapter 4. Chapter 5 of this thesis describes a special case of periodic UC model in which the basic structural model (BSM) is generalised by allowing the time-varying trigonometric terms associated with different seasonal frequencies to have different variances. We term the resulting class of model the frequency specific basic structural model (FS-BSM). The extended set of parameters in FS-BSM models can be estimated by standard maximum likelihood procedures based on the Kalman filter. We explore the dynamic properties of the FS-BSM and relate them to those of the frequency specific Airline model (FS-AM) as described in Aston et al. (2007). The periodicity in FS-BSM is imposed solely on the seasonal component so that each seasonal frequency has its own variance. This particular model has been discussed in detail by Harvey (1989) for the case of quarterly series. We extend the analysis of this model for monthly series and compare the estimation performance of the FS-BSMs with their FS-AMs counterpart. The relationships between coefficients in FS-AMs and FS-BSMs are highly non-linear and we rely on numerical techniques to investigate the connection between these models. We find empirically that both generalized models have properties that are very close to each other and that FS-BSMs lead to similar time series decompositions for a specific range of models in the FS-AMs. For this investigation, we employ a U.S. Census Bureau database of seasonal time series that has been analysed previously with the FS-AMs.

11 The remainder of this thesis contains four chapters. The highlights of each chapter are as follows. 3 Chapter 2: Periodic unobserved component (PUC) models This chapter introduces a general class of periodic unobserved components (PUC) time series models with stochastic trend and seasonal components and with a novel periodic stochastic cycle component. The general state space formulation of the periodic model allows for exact maximum likelihood estimation, signal extraction and forecasting. Also the consequences for model-based seasonal adjustment are discussed. The new periodic model is applied to post-war monthly U.S. unemployment series, from which we identify a significant periodic stochastic cycle. A detailed empirical periodic analysis is presented including a comparison between the performances of periodic unobserved components models and the non-periodic ones. Chapter 3: Multivariate PUC models This chapter analyses quarterly post-war U.S. non-farm employment for seven industrial sectors based on a new multivariate periodic unobserved components time series model. It includes sectoral trend, seasonal and irregular components with periodic features and parameter adjustments for the period of the Great Moderation. The main feature of our multivariate model is the common cycle component that enables the detection of possibly periodic cyclical behaviour in the U.S. employment rate. The investigation of periodicity in the cycle component of the time series is relevant since seasonal adjustment procedures need to take account of periodic features when they are present. We find that periodicity does exist in the sectoral trend and seasonal components but we do not detect it in the common cycle component. We compare our findings with an earlier empirical study on this subject. We investigate the robustness of our findings in detail by considering univariate and multivariate variations of our model and by presenting estimation results for sub-samples of our dataset. Chapter 4: Periodic SARIMA models In this chapter, state space formulations for periodic seasonal autoregressive integrated moving average (periodic SARIMA) are considered. Convenient state space representations of the periodic SARIMA models are proposed to facilitate model identification, specification and exact maximum likelihood estimation of the periodic parameters. These formulations do not require a-priori (seasonal) differencing of the time series. The time-varying state space representation is an attractive alternative to the

12 4 CHAPTER 1. INTRODUCTION time-invariant vector representation of periodic models which typically leads to a high dimensional state vector in monthly periodic time series models. A key development is our method of computing the variance-covariance matrix of the initial set of observations which is required for exact maximum likelihood estimation. We illustrate the use of periodic SARIMA models to fit and forecast a monthly post-war U.S. unemployment time series. Chapter 5: Frequency-specific trigonometric seasonality The basic structural time series model has been designed for the modelling and forecasting of seasonal time series. In this chapter we explore a generalisation of this model in which the time-varying trigonometric terms associated with different seasonal frequencies have different variances in their disturbances. The extended set of parameters is estimated by maximum likelihood procedures based on the Kalman filter. The contribution of this chapter is two-fold. The first is an elaborate description of the dynamic properties of this frequency specific basic structural model. The second is the relationship of this model to a comparable generalised version of the Airline model developed at the U.S. Census Bureau. By adopting a quadratic distance metric based on the restricted reduced form moving-average representation of the models, we conclude that the generalised models have properties that are very close to each other compared to their default counterparts. In some settings, the distance between the models is virtually zero so that the models can be regarded as observationally equivalent. An extensive empirical study on disaggregated monthly shipment and foreign trade series illustrates both the relations between the two classes of models, and the improvements obtained by adopting the frequency-specific extensions.

13 Chapter 2 Periodic unobserved components (PUC) models This chapter is based on Koopman, Ooms, and Hindrayanto (2009). 2.1 Introduction In this chapter we focus on periodic extensions of the univariate unobserved components (UC) time series models that are reviewed in Harvey (1989). This set of linear models describes a time series process as a sum of components that can be interpreted as trend, season and cycle. Each component is specified as an independent linear dynamic process. For example, the trend component can be associated with a random walk process while the cycle component can be modelled as a stationary ARMA process. Additional to lag polynomial coefficients, parameters consist of variances associated with disturbances that drive the random walk and ARMA processes. When these parameters are allowed to be deterministic functions of the season index, the resulting UC models are referred to as periodic unobserved components (PUC) models. Estimation procedures for the PUC models are based on exact maximum likelihood using computationally efficient state space methods. Given the general concepts of both UC and periodic models, there are many ways to specify a PUC model. If the PUC model is represented as a vector of independent time series where each element represents a particular season, the seasonal component is effectively eliminated. Therefore, the seasonal process can not be identified from an observed time series when remaining components are periodically independent for all seasons. Since we are particularly concerned with the decomposition of a time series into trend, seasonal and cycle, we propose a convenient periodic formulation of the UC model that preserves the ability to extract a seasonal component from a time series. These PUC models can still be represented in the vector form of Gladysev (1961) and

14 6 CHAPTER 2. PERIODIC UNOBSERVED COMPONENTS (PUC) MODELS Tiao and Grupe (1980) but the components do not consist of periodically independent processes since each component remains linearly dependent on a common underlying stochastic process for all periods. We argue that the seasonal component can still be identified in a PUC model and we show how to estimate it. The specification of the periodic model remains linear and no modifications of the Kalman filter equations are needed. Hence our approach facilitates the de-trending, seasonal adjustment and trend-cycle decomposition of a time series based on a periodic model. In the empirical illustration, we apply the periodic model to a long monthly time series of postwar U.S. unemployment (January 1948 until December 2006). The unemployment series is chosen as it is a key variable in economics. More importantly, it is also subject to seasonal variation since labour supply and labour demand in important sectors of the economy depend on seasonal factors such as school calendars, summer work, Easter/Christmas shopping, weather, winter breaks, etc. In economic literatures, unemployment dynamics are often found to be periodic, both theoretically and empirically, see for example Osborn and Smith (1989), Ooms and Franses (1997), Krane and Wascher (1999), Matas-Mir and Osborn (2004), and Van Dijk et al. (2003). Furthermore, it is well-known that unemployment data is highly dependent on business cycle dynamics and therefore cycle components also need to be considered in the analysis. Most important for this chapter, it is argued that unemployment is also subject to significant periodic serial correlation, in particular in the cyclical component. We present a comprehensive treatment of a general class of PUC models that include trend, seasonal and stationary cyclical components, and apply this to model the U.S. unemployment dataset. We discuss and implement many aspects of time series analysis for this class of PUC models, including seasonal adjustment, trend-cycle decomposition, diagnostic checking of prediction errors and forecasting. Periodic time series models have been introduced as early as 1955 in the article of Hannan (1955) and found widespread interest in geophysics and environmental empirical studies. Periodic dynamic regression models for economic time series have been applied since the 1930 s. Mendershausen (1937) gave an early overview. Osborn and Smith (1989) introduced the periodic time series framework in dynamic macroeconomic models while Ghysels and Osborn (2001) and Franses and Paap (2004) discussed a wider spectrum of periodic models and applications in econometrics. In the context of autoregressive models, Boswijk and Franses (1996) derived tests for periodic stochastic non-stationarity and Burridge and Taylor (2004) developed simulation based seasonal unit root tests in the presence of periodic heteroskedasticity. Exact maximum likelihood estimation methods for periodic ARMA models have been discussed by Jimenez et al. (1989) and Lund and Basawa (2000). Anderson and Meerschaert (2005) provided asymptotic theory for efficient moment based estimation. Most of these studies have

15 2.2. UNOBSERVED COMPONENTS MODELS 7 explored periodic versions of ARMA models. Earlier periodic extensions of the UC models have been explored by Krane and Wascher (1999), Koopman and Ooms (2002, 2006), Proietti (2004), Bell (2004) and Penzer and Tripodis (2007). These authors have considered straightforward applications of the vector representation of PUC models or they considered only specific parameters to be periodic. More specifically, Proietti (2004) considered a UC model with the trend component modelled as a weighted average of separate independent random walks for each season, Penzer and Tripodis (2007) considered a seasonal component with a periodic variance, while Koopman and Ooms (2002, 2006) explored different periodic specifications of the UC model using standard available software. In the context of ARMA based UC models, Krane and Wascher (1999) developed a multivariate periodic UC model involving a common cyclical component with a seasonally varying effect on quarterly U.S. employment growth. In the context of RegComponent UC models Bell (2004) considered the effect of seasonal heteroskedasticity in the irregular component on seasonal adjustment, see also Findley (2005). Different types of periodicity in a UC model imply different optimal seasonal adjustment filters. It is therefore interesting to investigate how we can identify different types of periodicity both in theory and in practice, also for seasonal adjustment. The remaining part of the chapter is organised as follows. The next section reviews the UC model. The third section introduces PUC models and in particular the novel periodic version of the stochastic cycle model. We also discuss possible implications for seasonal adjustment. The fourth section presents an analysis of monthly U.S. unemployment series using both periodic and non-periodic UC models where we reveal significant periodicity in the cycle component. We also take account of the cycle variance moderation in postwar U.S. unemployment in the different models. The fifth section concludes. 2.2 Unobserved components models In this section we introduce UC time series models. The established notation following Harvey (1989) and Durbin and Koopman (2001) will be used throughout the chapter. The univariate UC time series model that is particularly suited for many economic data sets is given by y t = µ t + γ t + ψ t + ε t, ε t NID(0, σ 2 ε), t = 1,..., n, (2.1) where µ t, γ t, ψ t and ε t represent trend, seasonal, cyclical and irregular components, respectively. The trend and seasonal components are modelled by linear dynamic stochastic processes driven by random disturbances. The cycle is based on a stochastic trigonometric function that relies on a damping factor, frequency and random disturbances.

16 8 CHAPTER 2. PERIODIC UNOBSERVED COMPONENTS (PUC) MODELS The simplest form of a UC model, the so-called local level model, is obtained from equation (2.1) where γ t and ψ t are zero for all t. The trend can be specified as the random walk process µ t+1 = µ t + η t, η t NID(0, σ 2 η), (2.2) for t = 1,..., n. By adding a slope term β t, that is also generated by a random walk to equation (2.2), we obtain the so-called local linear trend model, µ t+1 = µ t + β t + η t, η t NID(0, σ 2 η), β t+1 = β t + ζ t, ζ t NID(0, σ 2 ζ ), (2.3) for t = 1,..., n, where the trend and slope disturbances, η t and ζ t, respectively, are mutually uncorrelated sequences from a Gaussian density with zero mean and variance ση 2 for η t and σζ 2 for ζ t. If σζ 2 is zero, we have ζ t = 0 and β t+1 = β t = β for all t. This implies a random walk plus drift process for the trend µ t. When ση 2 is zero as well, a deterministic linear trend is obtained for µ t. If only ση 2 is zero, then we have a so-called smooth trend model or an integrated random walk process. This implies that µ t follows a random walk process where = 1 L is the difference operator and L is the lag operator with L p y t = y t p. To take account of the seasonal variation in the time series y t, the seasonal component γ t is included. A deterministic seasonal component should have the property that it sums to zero over the previous year to ensure that it is not confounded with the trend. Flexibility of the seasonal component is achieved when it is allowed to change over time. This can be established by adding a disturbance term (with mean zero) to the sum of the S seasonal effects over the past year. In this way we obtain the stochastic dummy variable form of the seasonal component as given by S S (L)γ t+1 = ω t, ω t NID(0, σ 2 ω), (2.4) where S S (L) is the summation operator defined as S S (L) = 1 + L + + L S 1. Other (trigonometric) models for γ t can be considered in this set-up, but they all share the assumption that S S (L)γ t has a zero conditional expectation; see Section 5.2. Since economic time series are often subject to cyclical dynamics, a stochastic cycle may be included in the model and be specified as ( ) ( ) ( ) ( ) ψ t+1 cos λ sin λ ψ t κ t = ρ +, 0 < ρ < 1, 0 < λ < 2π/S, (2.5) sin λ cos λ ψ t+1 ψ t κ t where the period of the cycle is given by 2π/λ, with λ < 2π/S to avoid confounding with the seasonal component. This dynamic process can be written as an ARMA(2,1) process with complex roots in the autoregressive polynomial. It therefore generates

17 2.3. PERIODIC UNOBSERVED COMPONENTS MODELS 9 a cyclical pattern in the theoretical autocorrelation function of process (2.5); Harvey (1985) provides more details. Although the average length of the cycle is fixed, individual realisations of the cycle can show considerable variation in length and amplitude. 2.3 Periodic unobserved components models Univariate seasonal time series y t are considered with seasonal length S (S = 12 for monthly data). Seasonal time series are often analysed by seasonal autoregressive moving average (SARMA) models and by the UC models of the previous section. The standard formulations of these models assume that all parameters are constant through time. In case the models are periodic, the parameters are allowed to vary with the season. As a result, the number of parameters increases by a multiple of S. This section develops a statistical periodic time series approach for univariate UC models with trend, season, cycle and irregular. We start our analysis with the simplest periodic basic structural model (BSM) which contains only three components: trend, season and irregular. Considering the moments of this model, we show that the univariate time-varying-parameter BSM does not correspond to a multivariate constant-parameter BSM. We also introduce a novel periodic stochastic cycle component. Further we derive the moments and show that exact ML estimation can be implemented without additional identifying restrictions Periodic basic structural time series model Consider a univariate basic structural time series model (BSM) with periodic variances for the disturbances associated with trend, seasonal and irregular components. This model can be expressed by y t = µ t + γ t + ε t, ε t NID(0, σ 2 ε,s), µ t+1 = µ t + η t, η t NID(0, σ 2 η,s), γ t+1 = S 2 j=0 γ t j + ω t, ω t NID(0, σ 2 ω,s), (2.6) for t = 1,..., n, n = n S, where σε,s, 2 ση,s 2 and σω,s 2 are the variances for ε t, η t and ω t respectively, and for season s = 1,..., S. To simplify notation for the multivariate representations of the model, we assume we have n complete years of data, but this assumption is not essential for the subsequent statistical analysis. Once model (2.6) is written as a multivariate process (each equation is for a particular season), it is shown below that the periodic BSM (2.6) does not reduce to a standard multivariate local level model for yearly observations. Denote y s,t as the observation for period s and year t such that y t y s,t, where t = (S 1)t + s for t = 1,..., n S,

18 10 CHAPTER 2. PERIODIC UNOBSERVED COMPONENTS (PUC) MODELS t = 1,..., n and s = 1, 2,..., S. For the case of S = 2 we have where y 1,t = µ 1,t + ε 1,t, y 2,t = µ 2,t + ε 2,t (2.7) µ 1,t = µ t + γ t, µ 2,t = µ t+1 + γ t+1, ε 1,t = ε t, ε 2,t = ε t+1, (2.8) so that the trend becomes µ 1,t +1 = µ 1,t + η 1,t, µ 2,t +1 = µ 2,t + η 2,t. (2.9) The trend disturbances in (2.9) include the seasonal disturbances by construction, since it follows that η 1,t = η t + η t+1 ω t + ω t+1, η 2,t = η t+1 + η t+2 ω t+1 + ω t+2. (2.10) In matrix form, the above model can be written as yt = µ t + ε t, (2.11) µ t +1 = µ t + η t, (2.12) where we denote the vectors as yt = (y 1,t, y 2,t ), µ t = (µ 1,t, µ 2,t ), ε t = (ε 1,t, ε 2,t ) and ηt = (η 1,t, η 2,t ). The variance matrix of the total disturbance vector (ε 1,t, ε 2,t, η 1,t, η 2,t ) is given by σε, σε, ση,1 2 + ση,2 2 + σω,1 2 + σω,2 2 ση,2 2 σω, ση,2 2 σω,2 2 ση,1 2 + ση,2 2 + σω,1 2 + σω,2 2. (2.13) Further, [( ) ( ) ] ( ) E η 1,t +1 η 2,t +1 η 1,t η 2,t = 0 σ 2 η,1 σ 2 ω,1 0 0 (2.14) and E [( η 1,t +j η 2,t +j ) ( η 1,t η 2,t ) ] = for j > 1. (2.15) Since (2.14) is not a zero matrix, the level disturbance vector ηt follows a moving average process. Therefore, we do not obtain a standard multivariate version of the local level model with a serially independent sequence of ηt. Unfortunately, it means we cannot estimate this simple model using standard software for multivariate basic structural models. This is a disadvantage compared to other PUC models, see the discussion in Koopman and Ooms (2006).

19 2.3. PERIODIC UNOBSERVED COMPONENTS MODELS Convenient state space representation of periodic BSM By focussing on the moments of y t implied by the periodic BSM model in equation (2.6), we have shown that univariate and multivariate representations of the periodic model are not necessarily equivalent. Here we develop two convenient ways of placing a PUC model into state space form: a univariate time-varying and a multivariate time-invariant representation. Both formulations enable exact maximum likelihood estimation and the estimation of the state vector by filtering and smoothing. The general state space representation is summarised in Appendix 2.A, while the Kalman filter, smoother, and the likelihood evaluation are summarised in Appendix 2.B, 2.C and 2.D, respectively. We derive the second order moments of the periodic BSM for S = 2 in Appendix 2.E and for S = 3 in Appendix 2.F and we argue that not all parameters of PUC models are automatically identified. PUC models can be formulated via a univariate measurement equation (N = 1) and time varying system matrices T t, H t, Z t and G t for t = 1,..., n. Alternatively they can be represented by a multivariate measurement equation (N = S) for yt with constant system matrices T, H, Z and G. Next we discuss these two convenient state space representations for the periodic BSM (2.6) and based on (2.35)-(2.36). The state space matrices of the univariate time-varying parameter form of (2.6) for N = S = 2 are given by T = ( ( Z = ) ( ) 0 σ η,t 0, H t =, (2.16) 0 0 σ ω,t ) ( ) 1 1, G t = σ ε,t 0 0, (2.17) where σ η,t, σ ω,t and σ η,t vary deterministically according to the season. Note that the matrices H t and G t are time-varying, while T and Z are constant over time. The state vector is given by α t = (µ t, γ t ) and the disturbance vector is given by ɛ t = (ε t, η t, ω t ). The initialisation of α t is diffuse with a is a vector of 0 s and P is κi 2 with κ. The multivariate time-invariant state space form of model (2.6) can be written as: α t +1 = T α t + H ɛ t, α 1 N (a, P ), t = 1,..., n, (2.18) yt = Z αt + G ɛ t, ɛ t NID(0, I). (2.19) To simplify notation we consider model (2.6) for S = 2, where y t = (y t, y t+1 ), t = 1, 1 + S, 1 + 2S,..., 1 + (n 1)S. We derive convenient expressions for α t, ɛ t, T, H, Z and G as follows. The measurement equations are given by y t = µ t + γ t + ε t, y t+1 = µ t+1 + γ t+1 + ε t+1 = µ t + η t γ t + ω t + ε t+1, ε t NID(0, σ 2 ε,1), ε t+1 NID(0, σ 2 ε,2).

20 12 CHAPTER 2. PERIODIC UNOBSERVED COMPONENTS (PUC) MODELS Further, we take αt = (µ t, η t, γ t, ω t ) as the state vector and it follows from the transition equations that µ t+2 = µ t+1 + η t+1 = µ t + η t + η t+1, η t+1 NID(0, σ 2 η,2), γ t+2 = γ t+1 + ω t+1 = γ t ω t + ω t+1, ω t+1 NID(0, σ 2 ω,2). The state space matrices are then given by σ η, T = , σ η,1 0 0 H = σ ω,2 0, (2.20) σ ω,1 ( ) ( ) Z =, G σ ε, =, (2.21) σ ε, with vector ɛ t = (ε t, ε t+1, η t+1, η t, ω t+1, ω t ) for t = 1, 2,..., n and t = 1, 1 + S, 1 + 2S,..., 1 + (n 1)S so that αt +1 = (µ t+2, η t+2, γ t+2, ω t+2 ). The initialisation of the state vector is diffuse for µ 1 and γ 1, while η 1 and ω 1 should be initialised by their variances. a is a vector of 0 s and P in this case is given by the matrix κ P 0 ση, =, with κ. 0 0 κ σω,1 2 Note that the multivariate time-invariant system is observationally equivalent to the univariate time varying system. In particular, the Gaussian likelihood of model (2.20)- (2.21) is exactly equal to the likelihood of model (2.16)-(2.17). For both specifications, it is clear that there is only one trend for the whole observed series. Although the multivariate representation may suggest that we have a separate trend for each season, the state vector α t only has a single trend, µ t, and a single seasonal component, γ t. The periodic BSM can be extended by the inclusion of additional components (such as a cycle) or by increasing the seasonal length in the univariate version of PUC models. The state space model with time-varying system matrices provides a general framework for this purpose. However, when including a periodic stochastic cyclical component as in subsection below, special attention must be given to its initialisation. For the multivariate representation of the PUC model with a cycle, the initialisation issue is somewhat tedious but manageable. Only some elementary calculations in linear algebra are required. The merit of the multivariate specification is its dependence on time-invariant system matrices. It allows the examination of the dynamic properties of the time series in a straightforward manner. The merit of the univariate specification

21 2.3. PERIODIC UNOBSERVED COMPONENTS MODELS 13 is its straightforward initialisation treatment. The drawback is its dependence on timevarying system matrices. Finally, we focus on an identification problem related to periodic models with S = 2; see Appendix 2.E for details. In case of bi-annual periodic models, the number of unknown parameters is equivalent to the number of moment equations. However, the rank condition is not satisfied. Fortunately, the reduced rank problem does not occur when S > 2. For the periodic BSM in equation (2.6), we have S(S + 1) linear equations to estimate 3S parameters. It is clear that for S 3 we have more non-zero moment equations than parameters. The order-condition for identification is therefore satisfied. In practice, only 3S unique equations exist in the system and all parameters can be identified exactly from these, see Appendix 2.F for technical details with S = 3. We conclude that the parameters are exactly identified in the two standard PUC models for all S > 2. Extension of the periodic BSM with a periodic stochastic cycle component as defined in the empirical section does not lead to further identification problems Periodic stochastic cycle model In this section we introduce a novel periodic version of the stochastic cycle component as part of the UC time series models. Macroeconomic time series often require a cyclical component in their specification in addition to the trend which is usually interpreted as a business cycle. In multivariate UC models the cycle component can be identified as a common stationary factor, where the corresponding factor loadings model the cyclical sensitivity of each constituent series. This approach was taken by Krane and Wascher (1999) and Azevedo et al. (2006). To identify the cycle in a univariate time series the variation of the component has to be restricted to frequencies within the business cycle range. Harvey (1985) has implemented this idea for UC models by the stochastic cycle component. Krane and Wascher (1999) make their model periodic by allowing the cyclical sensitivity to be seasonally dependent. Here we consider the cycle model of Harvey (1985) in equation (2.5) and extend it by having ρ and σ 2 κ periodic. To save space we only present the equations for a model with two periodic components, cycle and irregular, and with S = 2. We discuss the extension with a seasonal component below. We define for t = 1, S + 1, 2S + 1,..., (n 1)S + 1, y t = ψ t + ε t, ε t NID(0, σε,1), 2 y t+1 = ψ t+1 + ε t+1, ε t+1 NID(0, σε,2), 2 (2.22)

22 14 CHAPTER 2. PERIODIC UNOBSERVED COMPONENTS (PUC) MODELS where ( ) ( ) ( ) ( ) ψ t+1 cos λ sin λ ψ t κ t = ρ ψ t , κ sin λ cos λ ψ t + κ + t, κ + t NID(0, σκ,1), 2 ( ) ( ) ( ) ( t ) ψ t+2 cos λ sin λ ψ t+1 κ t+1 = ρ ψ t , κ sin λ cos λ ψ t+1 + κ + t+1, κ + t+1 NID(0, σκ,2), 2 t+1 (2.23) with mutually uncorrelated white noise disturbances (κ t, κ t+1 ) and (κ + t, κ + t+1). A restriction on the damping terms 0 < S s=1 ρ s < 1 ensures that the stochastic process ψ t is stationary. The periodic damping terms allow the cyclical sensitivity of y t to differ from season to season, as in Krane and Wascher (1999). The frequency of the stochastic cycle, λ, is common to all seasons s and ranges between 0 and 2π/S. This implies that the average period of the cycle is 2π/λ (as measured in semesters for S = 2, quarters for S = 4 and months for S = 12) while it is equal to 2π/(λS) in years. By making λ periodic, the model becomes nonlinear while we like to keep the focus on linear state space models. Also, the interpretation of the cycle would be more difficult when each season has its own frequency. The main idea of the PUC model is that we only have a common trend, seasonal, and cyclical component and therefore we only need a common cycle frequency. The dynamic properties of the stationary periodic cycle process are given by expressions for the variances and covariances of ψ t and ψ t+1. Following the approach of Gladysev (1961) and Tiao and Grupe (1980) these are derived from their vector autoregressive representation of order 1 and denoted by VAR(1). This yearly VAR(1) is obtained by substituting the expression for (ψ t+1, + ψ t+1) + in the second equation of (2.23). We get: Ψ t +1 = ΦΨ t + κ t, (2.24) ψ t+2 ρ 1 ρ 2 cos 2λ ρ 1 ρ 2 sin 2λ ρ 2 cos λ ρ 2 sin λ ψ t κ t+1 ψ t+2 + κ t+2 = ρ 1 ρ 2 sin 2λ ρ 1 ρ 2 cos 2λ ρ 2 sin λ ρ 2 cos λ ψ + t κ t + κ + t+1 κ t+2, κ + t+2 for t = 1, 2,..., n and for t = 1, S + 1, 2S + 1,..., (n 1)S + 1. First we derive the variance covariance matrix Λ 0 of Ψ t : E[Ψ t +1Ψ t +1] = Φ E[Ψ t Ψ t ]Φ + Σ κ Λ 0 = ΦΛ 0 Φ + Σ κ, κ + t κ + t+2 where ( ) Σ κ = diag σκ,2 2 σκ,2 2 σκ,1 2 σκ,1 2

23 2.3. PERIODIC UNOBSERVED COMPONENTS MODELS 15 and σψ,1 2 E(ψ t ψ t + ) 0 0 E(ψ t ψ t + ) σψ, Λ 0 = 0 0 σκ,1 2 0, σκ,1 2 with E(ψ t ψ t + ) = 0, σψ,1 2 = σ2 κ,2 + ρ 2 2σκ,1 2, (2.25) 1 ρ 2 1ρ 2 2 E(ψ t+1 ψ t+1) + = 0, σψ,2 2 = σ2 κ,1 + ρ 2 1σκ,2 2. (2.26) 1 ρ 2 1ρ 2 2 Subsequently, the periodic autocovariance function (ACVF) of y t for S = 2, t = 1, S + 1, 2S + 1,... and t = 1, 2,... is expressed in terms of σ 2 ψ,s, σ2 ε,s, ρ s and λ: Γ 0 = E Γ 1 = E =. Γ j = E = [( [( y t y t+1 y t y t+1 ) ( ) ( y t y t+1 y t 2 y t 1 ) ] ) ] = E = E [( [( y 1,t y 2,t y 1,t y 2,t ) ( ) ( ( ρ 1 ρ 2 cos(2λ)σψ,1 2 ρ 2 cos(λ)σψ,2 2 ρ 2 1ρ 2 cos(3λ)σψ,1 2 ρ 1 ρ 2 cos(2λ)σψ,2 2 ( [( y t y t+1 ) ( y t 2j y t+1 2j ) ] = E [( y 1,t y 2,t ) y 1,t y 2,t ) ] y 1,t 1 y 2,t 1 = ) ] ( σ 2 ψ,1 + σ2 ε,1 ρ 1 cos(λ)σ 2 ψ,1 ρ 1 cos(λ)σ 2 ψ,1 σ 2 ψ,2 + σ2 ε,2 ), (2.27), (2.28) ) ( y 1,t j y 2,t j ) ] ρ j 1ρ j 2 cos(2jλ)σψ,1 2 ρ j 1 1 ρ j 2 cos((2j 1)λ)σψ,2 2 ρ j+1 1 ρ j 2 cos((2j + 1)λ)σψ,1 2 ρ j 1ρ j 2 cos(2jλ)σψ,2 2 ), (2.29) for j = 1, 2,.... This means that the ACVF of y t in the second period of a year at a lag of 2j +k semesters, with k = 0, 1 and j = 0, 1, 2,..., is given by ρ j+1 1 ρ j 2 cos((2j +k)λ 2 ψ,1 ), see the left bottom element of the last right hand side matrix. The two complex roots of the AR part of the periodic stochastic cycle model are restricted to reflect business cycle behaviour, but the autocovariance function also shows an extra seasonal pattern that interacts with these cycles if ρ 1 and ρ 2 differ sufficiently. The autocovariance function of y t has a nonlinear structure in terms of σ ψ,s, ρ s and λ with s = 1, 2. Since σ ψ,s also depends on σ κ,s and ρ s, the structure of the ACVF becomes intricate. As a result, identification can not be analysed analytically and we

24 16 CHAPTER 2. PERIODIC UNOBSERVED COMPONENTS (PUC) MODELS therefore carry out some Monte Carlo experiments for Maximum Likelihood estimation to investigate whether the parameters can be estimated from simulated data. Model (2.22) is already in a standard time-varying state space form, which makes it very suitable for exact ML estimation using the prediction error decomposition, as in Durbin and Koopman (2001, Chapter 7). However, the construction of the exact likelihood function of models with a periodic cyclical component involves one non-standard step. The expression for σψ,1 2 is needed for the exact initialisation of the likelihood for the first observations, since (ψ 1, ψ 1 + ) is included in the corresponding state vector. The other terms in the likelihood for t = 2,... follow in a standard way from the timevarying Kalman filter equations as in Durbin and Koopman (2001, Chapter 7). The multivariate representation (2.24) is only used in the derivation of the variance of the initial state. The extension of the periodic cycle model to a general S = 2, 3,... is straightforward. The Φ matrix in the multivariate VAR(1) form (2.24) becomes S s=1 ρ S sc(sλ) s=2 ρ sc((s 1)λ)... ρ S C(λ), (2.30) where O 2(S 1) 2S ( cos sλ C(sλ) = sin sλ ) sin sλ, cos sλ for s = 1,..., S and the general solution for σψ,1 2 in (2.25) becomes { [ S 1 S ]} σψ,1 2 s=1 σκ,s 2 j=s+1 ρ2 j + σκ,s 2 = 1. (2.31) S s=1 ρ2 s The degree of non-stationarity or smoothness of the model based trends and cycles in (2.1) is sometimes viewed as too restrictive, see Gómez (2001) and Azevedo et al. (2006). A similar remark could be made regarding our PUC model. Higher order extensions for stochastic trends and cycles in the UC model were introduced by Harvey and Trimbur (2003) to allow for nearly ideal band pass filter properties in model-based component estimation. In the time domain, higher order trends and cycles lead to smoother estimates of cycle and trend. However, a periodic analysis of the higher order models falls outside the scope of this chapter. In larger models combining the periodic stochastic cycle component with periodic stochastic trends and seasonals, we have to make sure we exclude extreme cases where the parameters ρ s are nearly one and λ is very close to 2π/S, to avoid confusion with the seasonal component (2.4). We also need S larger than 2 to identify the periodic variances of the trend component (2.3) and seasonal component (2.4). We did not encounter such identification problems in our empirical application of Section 2.4.

25 2.3. PERIODIC UNOBSERVED COMPONENTS MODELS Simulation study on the periodic cycle component Monte-Carlo experiments to study the finite sample behaviour of the exact ML estimator are implemented for different values of the parameters σ κ,1, σ κ,2, ρ 1, ρ 2 and λ. We look at the distributions of the exact ML estimators of these parameters running the experiments times with 100 observations (equal to 50 years for S = 2). A smaller number of observations is not sufficient to produce reliable estimates in the simulation study. All computations and graphs in this chapter are made using recent versions of SsfPack, see Koopman et al. (1999) and programming language Ox, see Doornik (2006). Figure 2.1 presents the simulated densities for one representative set of parameter values which are based on empirical estimates for semi-annual postwar data of log U.S. unemployment, see Section 2.4 The true parameters are given by σ κ,1 = 0.03, σ κ,2 = 0.06, ρ 1 = 0.95, ρ 2 = 0.70 and λ = 0.3. The simulated means are close to the true values and the simulated densities are approximately normal. This confirms that the stochastic periodic cycle component does not lead to identification problems for parameter estimation. 50 σ κ,1 50 σ κ, ρ 1 4 ρ λ Figure 2.1: Simulated histograms and non-parametric density estimates of the exact ML parameter estimates in periodic cycle model (2.22) with true values: σ κ,1 = 0.03, σ κ,2 = 0.06, ρ 1 = 0.95, ρ 2 = 0.70 and λ = 0.3. σ ε,1 and σ ε,2 are fixed at.0005 and.0001 and not estimated. Based a on Monte-Carlo experiment of replications with 100 observations.

26 18 CHAPTER 2. PERIODIC UNOBSERVED COMPONENTS (PUC) MODELS The use of a PUC model generally entails a loss of estimation efficiency when the data generating process (DGP) is not periodic. Even when the DGP is correctly modelled by a PUC model, a misspecified non-periodic model can still yield estimates with lower mean squared error due to the higher sampling error in the PUC specification. This depends on both the sample size and the periodic variation in the true parameters. We investigate the loss of efficiency by simulating from a PUC model where the true values of the parameters ρ s and σ κ,s are close, comparing the root mean squared error (RMSE) of the estimators of the correctly specified PUC model and the misspecified restricted model. For completeness, we also compute the results of the estimators for non-periodic true parameter values. In case of a PUC model, we define the overall RMSE of â by RMSE(â) = 1 S S MSE(â s ), (2.32) where a represents a particular parameter (such as σ κ, ρ and λ), â is the maximum likelihood estimator of a and â s refers to the periodic estimator of a for period s. Panel A of Table 2.1 shows the RMSEs of the estimates for four pairs of true values for the parameters ρ 1 and ρ 2 while the other parameters, σ κ,1, σ κ,2 and λ, are fixed as in the Monte Carlo study of Figure 2.1. We repeat the Monte-Carlo experiment for different values of σ κ,s, s = 1, 2, with ρ 1, ρ 2, and λ fixed. Panel B of Table 2.1 shows the corresponding RMSEs of the restricted and unrestricted estimators. The restricted non-periodic parameter estimates of ρ s and σ κ,s are more accurate when the parameters of the model are close to non-periodic, ρ 1 = 0.95 and ρ 2 = 0.9 or σ κ,1 = and σ κ,2 =.06, and vice versa when ρ 1 = 0.95 and ρ 2 = 0.70 or σ κ,1 = 0.04 and σ κ,2 = We also report the precision of the model based estimators of the (periodic) first order autocorrelation τ(1) s, derived in (2.27), e.g. s=1 τ(1) 1 = ( ρ 1 cos(λ)σ 2 ψ,1) / ( (σ 2 ψ,1 + σ 2 ε,1)(σ 2 ψ,2 + σ 2 ε,2) ) 1/2. The periodic estimator of τ(1) generally outperforms in Panel A. In each replication, we also test for periodicity using an LR test with nominal 5% significance. The empirical rejection frequency is close to 5% for the nonperiodic DGP. In Panel A, when the RMSE of the unrestricted estimator of ρ s is just larger than the RMSE of the restricted estimator for ρ 1 = 0.95, ρ 2 = 0.9, the LR test rejects the constancy of ρ in 9% of the cases. For ρ 1 = 0.95, ρ 2 = 0.8, where both estimators yield nearly the same RMSE, the LR test rejects the non-periodicity in 26% of the replications. A similar pattern emerges in Panel B for σ κ,s. The unrestricted estimator outperforms the restricted estimator when σ κ,1 σ κ,2 is large enough, see the column

27 2.3. PERIODIC UNOBSERVED COMPONENTS MODELS 19 for (σ κ,1 = 0.04, σ κ,2 = 0.06). In this case the LR-test rejects the null of non-periodic σ κ,s s in 54% of the replications. This small Monte Carlo study shows that unrestricted periodic estimators are more precise for realistic periodic parameter values of the DGP and a relevant number of observations. Moreover, tests for nonperiodicity have substantial power in these cases Periodic stochastic cycles and seasonal adjustment Consider a model combining a seasonal component γ t as in (2.4) and a periodic stochastic cycle component ψ t as in (2.23). One might wonder what the seasonal component γ t represents when the ACVF of the cycle component ψ t shows seasonal patterns too. We agree with Bell and Hillmer (1984, p.292) that a seasonal adjustment be consistent with the information about seasonality present in the data being adjusted and with Canova and Ghysels (1994, p.1169) that cataloguing business cycle facts with seasonally adjusted data is improper unless the seasonal adjustment takes into account the particular form of interactions existing among the components of the series. On the other hand, we also want to address the issue that Cecchetti et al. (1997, p.891) described as the basic issue of whether the interaction term should be treated as seasonal or cyclical, and, at a more fundamental level, whether seasonal adjustment makes sense at all, when seasonals and cycles do not neatly decompose. The essential identifying restriction for the seasonal component γ t lies in its conditional expectation of zero over a year. The seasonality implied by the cycle ψ t moves up and down with the business cycle and is not restricted to sum to zero over a year and will not be picked up by estimates of γ t if the model is correctly specified. Estimates of the seasonal component γ t in an approximately correctly specified model do make sense and estimates can be used for seasonal adjustment, even if there is periodic interaction between seasonal and cyclical movements in the data. The seasonal adjustment method associated with our model is linear and straightforward to analyse. Standard theory of filtering and smoothing in state-space models provides algorithms to compute the minimum mean squared error estimator of the seasonal component. If the periodicity in the cycle is neglected, the model is misspecified and the annual changes in the estimated component ˆγ t reflect annual changes in the business cycle which make the seasonal estimates more difficult to interpret. Ooms and Franses (1997) showed the significance of this problematic interaction for several multiplicative non-periodic seasonal adjustment methods for U.S. unemployment. Application of the periodic stochastic cycle mitigates this problem considerably. The reduction in the variance of the seasonal component is in line with the maintained criterion of gradual year-on-year changes in seasonal adjustment factors implemented in automatic decomposition methods developed by the U.S. Census since Method II.