COMMON FEATURES IN TIME SERIES WITH BOTH DETERMINISTIC AND STOCHASTIC SEASONALITY
Gianluca Cubadda
Dipartimento di Scienze Economiche Gestionali e Sociali, Università del Molise, Via De Sanctis, 86100 Campobasso, Italy (
[email protected])
Keywords and Phrases: common features; seasonality; codependence.
JEL Classification: C32; C52.
ABSTRACT
This paper extends the notions of common cycles and common seasonal features to time series having deterministic and stochastic seasonality at different frequencies. The conditions under which quarterly time series with these characteristics have common features are investigated, various representations are presented and statistical inference is discussed. Finally, the analysis is applied to study comovements between different components of consumption and income using UK data.
1. INTRODUCTION
An important advance in the analysis of comovements among economic time series is the concept of common cycles (Vahid and Engle, 1993; 1997). This notion is complementary to cointegration since it focuses on common transitory components in the multivariate Beveridge-Nelson (henceforth BN, 1981) decomposition of difference-stationary variables. Examples of reasoning from economic theory which were elucidated by the common cycles analysis include the excess sensitivity of consumption to current income (Vahid and Engle, 1993), the comovements among sectoral outputs (Engle and Issler, 1995), and the existence of an international business cycle under alternative exchange rate regimes (Andreano and Savio, 1996). In a set of variables being I(1) solely at the zero frequency, the presence of common cycles requires that the first differences of these variables have the serial correlation common feature as defined by Engle and Kozicki (1993). However, when we observe economic variables in sub-annual intervals, first differences are often insufficient to achieve stationarity of the related time series. If the seasonal patterns of variables are well described by deterministic functions of time, common cycles analysis needs only minor modifications (Engle and Hylleberg, 1996). In contrast, if seasonal differencing is needed in order to make quarterly time series stationary, the existence of common cycles in the multivariate Hylleberg-Engle-Granger-Yoo (henceforth HEGY, 1990) decomposition of these variables requires that their seasonal differences are codependent of order, at most, three (Cubadda, 1999).1 Moreover, in this case the use of seasonally adjusted data cannot be suggested, since the presence of common cycles is in general not invariant to univariate linear filtering.
1
A set of stationary time-series is codependent if there exist some linear combinations of these series following a vector moving-average (VMA) of lower order than others. The codependence order is q if these linear combinations follow a VMA(q) process. See Gouriéroux and Paucelle (1989), and Vahid and Engle (1997) for discussions on codependence.
1
In order to distinguish between deterministic and non-stationary stochastic seasonal behaviors, extensions of unit root tests to seasonal frequencies are available (Hylleberg et al., 1990; Canova and Hansen, 1995). However, there is still no unanimous consensus on which type of seasonality is prevailing in economic variables (Beaulieu and Miron, 1993; Hylleberg et al., 1993). A situation that is often encountered in practice is finding evidence of unit roots at the zero and some, but not all, seasonal frequencies. The aim of this paper is twofold. First, the common cycles analysis is extended to quarterly time series which are not necessarily all I(1) or I(0) at the seasonal frequencies. Secondly, since these time series may have a deterministic periodic component at the seasonal frequency where the order of integration is zero, the test for common deterministic seasonal features by Engle and Hylleberg (1996) is adapted to this case. 2 The main conclusion is that the presence of common stochastic and deterministic cycles imposes a certain reduced-rank structure on the parameter matrices of the error-correction model. Hence, the suggested approach may be applied to identify a parsimonious statistical model of a multiple seasonal time series through parameter reduction. As shown by Reinsel and Ahn (1992), imposing reduced-rank structure in VAR models can yield considerable improvements in prediction performances. This paper is organized as follows. Section 2 outlines the motivation for this paper in details. Section 3 deals with common cycles and common seasonal features among time series with both stochastic and deterministic seasonality. Section 4 tackles inferential problems. In Section 5 the analysis is applied to UK data of income and components of consumption in order to test the MankiwVahid-Engle (Mankiw, 1982; Vahid and Engle, 1997) model of durables and nondurables consumption augmented with seasonal preferences. Section 6 presents conclusions. 2
This approach is relevant to the empirical analysis of theoretical ideas concerning the response of the economy to deterministic seasonal perturbations in preferences and technology. See Miron (1996) for a detailed survey of this class of models.
2
2. MOTIVATION Let x t (t = 1,...,T ) be an n-vector of quarterly time series being I(1) at frequency zero and having deterministic periodic components at the seasonal frequencies. Hence, these series follow the data generating process (DGP)
4
∆1 xt = ∑ λ x ,i d i ,t + C ( L)ε t i =1
where ∆1 = (1 − L), L is the lag operator, λ x,i (i = 1,...,4) is an n-vector, d1,t = 1, d 2,t = cos(tπ ), d 3,t = cos(tπ 2), d 4,t = sin( tπ 2), C (L) is a polynomial n×nmatrix such that C 0 = I n , ∞
∑ jC
j
