Model-based trend-cycle decompositions with time-varying parameters - PowerPoint PPT Presentation

Model-based trend-cycle decompositions with time-varying parameters Siem Jan Koopman Kai Ming Lee Soon Yip Wong s.j.koopman@ klee@ s.wong@ feweb.vu.nl Department of Econometrics Vrije Universiteit Amsterdam and Tinbergen Institute Presentation for the Bank of Japan, Tokyo, 26 June 2006 Model-based trend-cycle decompositions with time-varying parameters – p. 1

Outline of talk • Trend-cycle decompositions: ◦ Nonparametric filters: Hodrick-Prescott, Baxter-King, Christiano-Fitzgerald; ◦ Model-based approaches using ARIMA models and unobserved components time series models. • Deterministic time-varying parameters: ◦ Smooth functions of time: logit, splines, etc. ◦ Time-varying linear state space model: Kalman filtering ◦ Some empirical results for the US economy. • Stochastically time-varying parameters: ◦ Asymmetry of business cycles ◦ Nonlinear state space model: Extended Kalman filtering ◦ Some empirical results for the US economy Model-based trend-cycle decompositions with time-varying parameters – p. 2

Some real data: Eurozone GDP Data 14.25 14.20 14.15 14.10 14.05 14.00 13.95 13.90 1985 1990 1995 2000 Model-based trend-cycle decompositions with time-varying parameters – p. 3

Some topics in Business Cycle Analysis Some issues related to business cycles: • dating of business cycles: peaks and throughs • coincident and leading indicators • prinicipal components and dynamic factor analysis • asymmetry and nonlinearities. Methods for tracking business cycle and growth: • Detrending methods (Hodrick-Prescott); • Bandpass filtering methods (Baxter-King, Christiano-Fitzgerald); • Model-based, univariate (Beveridge-Nelson, Clark, Harvey-Jaeger); • Model-based, multivariate, common cycles (VAR model, common features, UC model). Model-based trend-cycle decompositions with time-varying parameters – p. 4

Different trend-cycle decompositions 0.02 0.01 14.2 0.00 14.0 −0.01 HP trend HP cycle 1985 1990 1995 2000 1985 1990 1995 2000 14.3 0.01 14.2 14.1 0.00 14.0 STAMP trend 13.9 STAMP cycle −0.01 1985 1990 1995 2000 1985 1990 1995 2000 0.01 14.2 0.00 14.0 −0.01 AKR trend AKR cycle 1985 1990 1995 2000 1985 1990 1995 2000 Model-based trend-cycle decompositions with time-varying parameters – p. 5

Univariate UC Trend-Cycle Decomposition y t = µ t + ψ t + ε t • Trend µ t : ∆ d µ t = η t ; • Irregular ε t : White Noise; • Cycle ψ t : AR(2) with complex roots as in Clark (87) or with stochastic trigonometric functions as in Harvey (85,89); Trigonometric specification,enforces complex roots in AR(2): � � � � � � � � ψ t +1 cos λ sin λ ψ t κ t = φ + , ψ + ψ + κ + − sin λ cos λ t +1 t t κ t , κ + t ∼ NID(0 , σ 2 κ ) . Signal extraction is about (locally) weighting observations. Kalman filter gives the optimal weights for the given models. Model-based trend-cycle decompositions with time-varying parameters – p. 6

Weights and Gain Functions of Components 14.3 0.02 14.2 14.1 0.00 14.0 13.9 −0.02 1985 1990 1995 2000 1985 1990 1995 2000 0.2 0.5 0.1 0.0 0.0 −20 −10 0 10 20 −20 −10 0 10 20 1.0 1.0 0.5 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Model-based trend-cycle decompositions with time-varying parameters – p. 7

Band-pass Properties "Band-pass" refers to frequency domain properties of polynomial lag functions of time series (filters). In business cycle analysis, one is interested in filters for trend and cycles such that trend only captures the low-frequencies, cycle the mid-frequencies and irregular the high frequencies. 1.0 0.5 TREND 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 0.5 CYCLE 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 0.5 IRREGULAR 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Model-based trend-cycle decompositions with time-varying parameters – p. 8

Generalised trends: Butterworth filters Butterworth trend filters can be considered; they have a model-based representation and can be put in state space framework; see Gomez (2001). The m -th order stochastic trend is µ t = µ ( m ) where t ∆ m µ ( m ) ζ t ∼ NID(0 , σ 2 t +1 = ζ t , ζ ) , or µ ( j ) t +1 = µ ( j ) + µ ( j − 1) , “ j = m, m − 1 , . . . , 1 , t t with µ (0) = η t and η t is an IID sequence. t • For m = 2 : IRW or smooth local linear trend or I(2) trend; • For m = 2 and σ ζ = 1600 − 1 : the Hodrick-Prescott filter; • Higher value for m gives low-pass gain function with sharper cut-off downwards at a certain low frequency point. Model-based trend-cycle decompositions with time-varying parameters – p. 9

Generalised cycle: band-pass filters Same principles can be applied to the cycle. The generalised k th order cycle is given by ψ t = ψ ( k ) , where t � ψ ( j ) � � � � ψ ( j ) � � ψ ( j − 1) � cos λ sin λ t +1 t t = φ + , ψ +( j ) ψ +( j ) ψ +( j − 1) − sin λ cos λ t +1 t t j = 1 , . . . , k, with � ψ (0) � � � κ t t = . ψ +(0) κ + t t Higher orders ensure smoother transitions, more details are reported in Trimbur (2002), Harvey & Trimbur (2004). Model-based trend-cycle decompositions with time-varying parameters – p. 10

Weights and Gain Functions of Components 14.3 0.02 14.2 14.1 0.00 14.0 13.9 −0.02 1985 1990 1995 2000 1985 1990 1995 2000 0.2 0.5 0.1 0.0 0.0 −20 −10 0 10 20 −20 −10 0 10 20 1.0 1.0 0.5 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Model-based trend-cycle decompositions with time-varying parameters – p. 11

Measuring business cycle from multiple time series Azevedo, Koopman and Rua (JBES, 2006) consider tracing the business cycle based on • a multivariate model with generalised components (band-pass filter properties) • data-set includes nine time series (quarterly, monthly) that may be leading/lagging GDP • a model where all equations have individual trends but share one common “business cycle” component. • a common cycle that is allowed to shift for individual time series using techniques developed by Rünstler (2002). Model-based trend-cycle decompositions with time-varying parameters – p. 12

Shifted cycles 0.2 0.0 −0.2 estimated cycles gdp (red) versus cons confidence (blue) −0.4 1980 1985 1990 1995 0.2 0.0 −0.2 estimated cycles gdp (red) versus shifted cons confidence (blue) −0.4 1980 1985 1990 1995 Model-based trend-cycle decompositions with time-varying parameters – p. 13

Shifted cycles In standard case, cycle ψ t is generated by � � � � � � � � ψ t +1 cos λ sin λ ψ t κ t = φ + ψ + ψ + κ + − sin λ cos λ t +1 t t The cycle cos( ξλ ) ψ t + sin( ξλ ) ψ + t , is shifted ξ time periods to the right (when ξ > 0 ) or to the left (when ξ < 0 ). Here, − 1 2 π < ξ 0 λ < 1 2 π (shift is wrt ψ t ) More details in Rünstler (2002) for idea of shifting cycles in multivariate unobserved components time series model of Harvey and Koopman (1997). Model-based trend-cycle decompositions with time-varying parameters – p. 14

The basic multivariate model Panel of N economic time series, y it , � � y it = µ ( k ) cos( ξ i λ ) ψ ( m ) + sin( ξ i λ ) ψ +( m ) it + δ i + ε it , t t where • time series have mixed frequencies: quarterly and monthly; • generalised individual trend µ ( k ) for each equation; it • generalised common cycle based on ψ ( m ) and ψ +( m ) ; t t • irregular ε it . Model-based trend-cycle decompositions with time-varying parameters – p. 15

Business cycle Stock and Watson (1999) states that fluctuations in aggregate output are at the core of the business cycle so the cyclical component of real GDP is a useful proxy for the overall business cycle and therefore we impose a unit common cycle loading and zero phase shift for Euro area real GDP . Time series 1986 – 2002 : quarterly GDP industrial production unemployment (countercyclical, lagging) industrial confidence construction confidence retail trade confidence consumer confidence retail sales interest rate spread (leading) Model-based trend-cycle decompositions with time-varying parameters – p. 16

Eurozone Economic Indicators 14.30 GDP Retail sales IPI unemployment Interest rate spread Industrial confidence indicator Construction confidence indicator Retail trade confidence indicator 14.25 Consumer confidence indicator 14.20 14.15 14.10 14.05 14.00 13.95 13.90 1990 1995 2000 Model-based trend-cycle decompositions with time-varying parameters – p. 17

Details of model, estimation • we have set m = 2 and k = 6 for generalised components • leads to estimated trend/cycle estimates with band-pass properties, Baxter and King (1999). • frequency cycle is fixed at λ = 0 . 06545 ( 96 months, 8 years), see Stock and Watson (1999) for the U.S. and ECB (2001) for the Euro area • shifts ξ i are estimated • number of parameters for each equation is four ( σ 2 i,ζ , δ i , ξ i , σ 2 i,ε ) and for the common cycle is two ( φ and σ 2 κ ) • total number is 4 N = 4 × 9 = 36 Model-based trend-cycle decompositions with time-varying parameters – p. 18