Model-based trend-cycle decompositions. with time-varying parameters

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1 Model-based trend-cycle decompositions with time-varying parameters Siem Jan Koopman Kai Ming Lee Soon Yip 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

2 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

3 Some real data: Eurozone GDP Data Model-based trend-cycle decompositions with time-varying parameters p. 3

4 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

5 Different trend-cycle decompositions HP trend STAMP trend HP cycle STAMP cycle AKR trend 0.01 AKR cycle Model-based trend-cycle decompositions with time-varying parameters p. 5

6 Univariate UC Trend-Cycle Decomposition Trend µ t : d µ t = η t ; Irregular ε t : White Noise; y t = µ t + ψ t + ε t 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 ψ + t+1 ) κ t,κ + t NID(0,σ 2 κ). = φ [ cos λ ] ( sin λ sin λ cosλ Signal extraction is about (locally) weighting observations. Kalman filter gives the optimal weights for the given models. ψ t ψ + t ) + ( κ t κ + t ) Model-based trend-cycle decompositions with time-varying parameters p. 6,

7 Weights and Gain Functions of Components Model-based trend-cycle decompositions with time-varying parameters p. 7

8 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 TREND CYCLE IRREGULAR Model-based trend-cycle decompositions with time-varying parameters p. 8

9 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) t where m µ (m) t+1 = ζ t, ζ t NID(0,σ 2 ζ ), or with µ (0) t µ (j) t+1 = µ(j) t + µ (j 1) t, j = m,m 1,...,1, = η t and η t is an IID sequence. For m = 2: IRW or smooth local linear trend or I(2) trend; For m = 2 and σ ζ = : 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

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

11 Weights and Gain Functions of Components Model-based trend-cycle decompositions with time-varying parameters p. 11

12 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

13 Shifted cycles estimated cycles gdp (red) versus cons confidence (blue) estimated cycles gdp (red) versus shifted cons confidence (blue) Model-based trend-cycle decompositions with time-varying parameters p. 13

14 Shifted cycles In standard case, cycle ψ t is generated by ( ψ t+1 ψ t+1 + ) [ = φ cos λ sin λ sin λ cos λ ] ( ψ t ψ t + ) + ( κ t κ + 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 0λ < 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

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

16 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 : 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

17 Eurozone Economic Indicators GDP IPI Interest rate spread Construction confidence indicator Consumer confidence indicator Retail sales unemployment Industrial confidence indicator Retail trade confidence indicator Model-based trend-cycle decompositions with time-varying parameters p. 17

18 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 = (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 σ2 κ) total number is 4N = 4 9 = 36 Model-based trend-cycle decompositions with time-varying parameters p. 18

19 Decomposition of real GDP GDP Euro Area Trend slope Cycle irregular Model-based trend-cycle decompositions with time-varying parameters p. 19

20 The business cycle coincident indicator Selected estimation results series load shift R 2 d gdp 0.31 indutrial prod Unemployment industriual c construction c retail sales c consumer c retail sales int rate spr Model-based trend-cycle decompositions with time-varying parameters p. 20

21 Coincident indicator for Euro area business cycle Model-based trend-cycle decompositions with time-varying parameters p. 21

22 Coincident indicator for growth tracking economic activity growth is done by growth indicator we compare it with EuroCOIN indicator EuroCOIN is based on generalised dynamic factor model of Forni, Hallin, Lippi and Reichlin (2000, 2004) it resorts to a dataset of almost thousand series referring to six major Euro area countries we were able to get a quite similar outcome with a less involved approach by any standard Model-based trend-cycle decompositions with time-varying parameters p. 22

23 EuroCOIN and our growth indicator Coincident Eurocoin Model-based trend-cycle decompositions with time-varying parameters p. 23

24 Role of time-varying parameters Consider typical example of univariate trend-cycle decomposition: y t = µ t + ψ t + ε t, t = 1,...,n, with trend µ t : d µ t = η t where d = 1 (RW) or d = 2 (IRW); cycle ψ t : AR(2) with complex roots or with stochastic trigonometric functions (next slide); irregular ε t : white noise. In state space framework, the dynamic properties of components can be characterised in Markovian form: y t = Zα t + ε t, α t+1 = Tα t + Rξ t, where α t is state vector and includes trend and cycle. Model-based trend-cycle decompositions with time-varying parameters p. 24

25 Trend plus cycle decomposition A decomposition model with an explicit cyclical term is given by y t = µ t + ψ t + ε t, t = 1,...,n, d µ t+1 = η t µ t+1 = µ t + β t, β t+1 = β t + η t+1, for d = 2 ( ) [ ]( ) ( ) ψ t+1 cos λ sin λ ψ t κ t = φ + ψ t+1 sin λ cos λ ψ t κ t ε t NID(0,σ 2 ε), η t NID(0,σ 2 η), κ t, κ t NID(0,σ 2 κ). The state vector for d = 2 is given by α t = ( µ t β t ψ t ψ t ). Model-based trend-cycle decompositions with time-varying parameters p. 25

26 Time-varying parameters: deterministic functions Time-varying cycle parameters: variance σ 2 κ,t = f σ (t), period ω t = f ω (t), damping factor φ t = f φ (t). Logit specification, with logit(x) = e x / (1 + e x ): Spline specification: ( f σ (t) = exp c σ + γ σ logit ( s σ (t τ σ ) )), ( f ω (t) = 2 + exp c ω + γ ω logit ( s ω (t τ ω ) )). f σ (t) = exp ( c σ + w tδ σ ), f ω (t) = 2 + exp ( c ω + w tδ ω ), c, γ, s, τ are constants, w t is weight, δ is coefficient vector. Model-based trend-cycle decompositions with time-varying parameters p. 26

27 State space representation The measurement equation is time-invariant and is given by y t = Zα t + ε t, [ ] Z = 1 O 1 0. The state equation is time-varying: α t+1 = T t α t + Rξ t, ξ t NID(0,Q t ). The state α t is the d + 2 dimensional vector ] α t = [µ t µ t... d 1 µ t ψ t ψ t. Model-based trend-cycle decompositions with time-varying parameters p. 27

28 Details of the state space representation The required state space form is where α t = y t = Zα t + ε t, α t+1 = T t α t + Rξ t, ξ t NID(0,Q t ). ] [µ t µ t... d 1 µ t ψ t ψ t and T t = [ ] M O, M = I d + O C t [ O I d 1 0 O ], C t = φ [ cos λ t sin λ t ] sin λ t, cos λ t Q t = ση σκ,t 2 0, R = 0 0 σκ,t 2 [ ] O. I 3 Model-based trend-cycle decompositions with time-varying parameters p. 28

29 Kalman filter Kalman filter is a key tool for state space time series analysis: prediction error decomposition likelihood evaluation diagnostic checking filtered estimates of trend and cycle source for smoothing algorithms (signal extraction) forecasting Kalman filter is given next. More details in Harvey (1989) and Durbin and Koopman (2001). Model-based trend-cycle decompositions with time-varying parameters p. 29

30 Kalman filter Recursion to evaluate predictor of state α t (a t ) and its mean square error (P t ): v t = y t Za t, f t = ZP t Z + G, k t = TP t Z /f t, a t+1 = Ta t + k t v t, P t+1 = TP t T k t k t/f t + RQR, for t = 1,...,n and for some initialisation a 1 and P 1. This is the basic algorithm and can be compared with OLS computation for standard regression model. Model-based trend-cycle decompositions with time-varying parameters p. 30

31 Kalman filter State space methods are useful; they offer a unified approach to standard time series analysis for dynamic regression, ARMA, UC models, etc. But there is more. When dealing with messy time series, state space methods provide appropriate tools for their treatment. In case of missing observations, Kalman filter can handle them. In state space, forecasting is a missing observations problem (future observations are missing). Univariate and multivariate treatments are the same. Implementations of algorithms are widely available: OxMetrics/SsfPack: Model-based trend-cycle decompositions with time-varying parameters p. 31

32 Some empirical results for the US economy Data description series frequency data range description GDP quarterly 1948:1-2004:2 log of the U.S. Real Gross Domestic Product series; seasonal adjusted. IN quarterly 1948:1-2004:2 log of the U.S. Fixed Private Investments series; seasonal adjusted. U monthly 1948:1-2004:6 U.S. Civilian Unemployment Rate; seasonal adjusted. IPI monthly 1948:1-2004:6 log of the U.S. Industrial Production Index; Index 1997=100; seasonal adjusted. Source: Federal Reserve Bank of St. Louis, Model-based trend-cycle decompositions with time-varying parameters p. 32

33 Results I: basic decomposition (time-invariant models) Lik AICC BIC full GDP I II full Investment I II full Unemployment I II full IPI I II Model-based trend-cycle decompositions with time-varying parameters p. 33

34 Results II: time-varying cycle volatility and period For GDP en unemployment we obtain: Fixed parameters Time-varying: spline Time-varying: logit GDP Lik Lik Lik AICC AICC AICC BIC BIC BIC Unemployment Lik Lik Lik AICC AICC AICC BIC BIC BIC Model-based trend-cycle decompositions with time-varying parameters p. 34

35 US Gross Domestic Product 0.05 Spline function 0.05 Logit function Model-based trend-cycle decompositions with time-varying parameters p. 35

36 US unemployment Spline function Logit function Model-based trend-cycle decompositions with time-varying parameters p. 36

37 Results III: time-varying cycle volatility Spline specification Logit specification GDP Lik Lik AICC AICC BIC BIC Investment Lik Lik AICC AICC BIC BIC Unemployment Lik Lik AICC AICC BIC BIC IPI Lik Lik AICC AICC BIC BIC Model-based trend-cycle decompositions with time-varying parameters p. 37

38 Spline specification Cycle GDP Investment Unemployment IPI Volatility Model-based trend-cycle decompositions with time-varying parameters p. 38

39 Logit specification Cycle GDP Investment Volatility Unemployment IPI Model-based trend-cycle decompositions with time-varying parameters p. 39

40 Time-varying parameters: stochastic functions Instead of the deterministic spline and logit functions, we can also adopt stochastic functions of time. For example, a possible random walk specification is f σ (t) = exp(χ t ), χ t+1 = χ t + error. This seems to suggest that we want to model parameters such as variances, periods and autoregressive coefficients. We are not sure... However, there are other motivations to adopt stochastically time-varying parameters in a model. For example, asymmetric cycles... Model-based trend-cycle decompositions with time-varying parameters p. 40

41 Asymmetric cycle process Given the cycle process, ( ) [ ]( ) ψ t+1 cos λ sin λ ψ t = φ ψ t+1 sin λ cos λ ψ t + ( ) κ t, κ t it follows that ψ t = ψ t (λt) (this is shown in the paper). To have λ as function of ψ t, cycle becomes asymmetric: ψ t = acos(λ t t b), λ t = { λ a, ψ t > 0 λ d, ψ t 0, or ψ t = acos(λ t t b), λ t = λ + γ ψ t. Model-based trend-cycle decompositions with time-varying parameters p. 41

42 Deterministic examples Stylized asymmetric business cycles Model-based trend-cycle decompositions with time-varying parameters p. 42

43 Nonlinear state space When considering λ t = λ + γ ψ t, we need to consider the nonlinear state space model y t = Zα t + ε t, α t+1 = T(α t ) + η t, with T(α t ) = [ ] O O [ ] cos(λ t ) sin(λ t ) α t, φ sin(λ t ) cos(λ t ) and λ t = λ + [ ] γ α t. Model-based trend-cycle decompositions with time-varying parameters p. 43

44 Importance sampling Estimation by ML is implemented using importance sampling techniques for nonlinear state space models. pθ (α,y) L(θ) = p θ (α,y)dα = g θ (α y) g θ(α y)dα pθ (α,y) = g θ (y) g θ (α,y) g θ(α y)dα. where g θ (y) is the likelihood of the approximating model. log ˆL(θ) = log L g (θ) + log w, w = 1 N N i=1 p θ (α (i),y) g θ (α (i),y), where L g (θ) is the likelihood from the approximating model, α (i) is drawn from g θ (α y) using simulation smoothing. Model-based trend-cycle decompositions with time-varying parameters p. 44

45 Results IV: stochastic asymmetric cycles Un Un (as) Inv Inv (as) GDP GDP (as) σε e e 3 σζ e e e e e e 8 σκ e e e e e e 5 φ ω γ Norm Q(20) LogL W LM LR Model-based trend-cycle decompositions with time-varying parameters p. 45

46 Conclusions Some remarks: Tracking business cycle using a model-based approach is preferred and its feasibility in practical research is shown. Evidence for time-varying parameters is found. More theoretical and empirical work is needed to investigate the role of time-varying parameters in business cycle tracking. Model-based trend-cycle decompositions with time-varying parameters p. 46

47 Conclusions Some recent work: Tracking the business cycle of the Euro area: a multivariate model-based band-pass filter. 2006, by Joao Valle e Azevedo, Siem Jan Koopman and Antonio Rua, Journal of Business and Economic Statistics, Volume 24, No. 3, July 2006, pp Trend-cycle decomposition models with smooth-transition parameters: evidence from US economic time series. 2005, with K.M. Lee and S.Y. Wong, in D. van Dijk, C. Milas and P.A. Rothman (eds), Nonlinear Time Series Analysis of Business Cycles, Elsevier. Measuring asymmetric stochastic cycle components in U.S. macroeconomic time series. 2006, by S.J. Koopman and K. M. Lee, Tinbergen Institute Discussion paper. Model-based trend-cycle decompositions with time-varying parameters p. 47