Multivariate Unobserved Components Siem Jan Koopman Vrije - PowerPoint PPT Presentation

Multivariate Unobserved Components Siem Jan Koopman Vrije Universiteit Amsterdam Tinbergen Institute EMM lecture — 2 nd meeting ESF-EMM Network, Rome, November 20, 2003

• Univariate unobserved components • Multivariate extension • Case I: Time-varying rank-reductions and “convergence” • Case II: Coincident indicator for business cycle • Non-Gaussian extension • Case III: Round-the-Clock price discovery in cross-listed stocks • Case IV: Modelling defaults

Let’s start with Eurozone GDP data 14.25 14.20 14.15 14.10 14.05 14.00 13.95 13.90 1985 1990 1995 2000

Much literature on business cycles and growth • dating of business cycles (Markov-switching models) • prinicipal components analysis (Stock and Watson, Forni, Hallin, Lippi and Reichlin) • convergence and synchronisation (economic theory, empirical studies) • asymmetry and nonlinearities (econometrics) • coincident and leading indicators (economics)

Aim is the detection of 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, UC model)

Motivation Undertaking fiscal and monetary policies requires information about the state of the economy. Given the mixed signals in economic data, the assessment of the economic situation is a challenging task. Our aim is to extract relevant information through statistical rigorous methods in order to provide a clear signal regarding current and future economic developments.

Different univariate 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

Multivariate model-based approach We will adopt a multivariate model with a common cycle for different economic time series Economic time series are often not available at the same and/or desired frequency. We aim to reconcile a high frequency business cycle indicator without disregarding data recorded at lower frequencies. For example, GDP is an important variable for business cycle assessments. Sometimes GDP discarded because it is a quarterly variable: Stock and Watson (1989) and Eurocoin (2001).

Approach is based on • unobserved components time series model w/common cycle • multiple time series observed at different frequencies (M/Q) and possibly observed at different time-intervals • maximum likelihood estimation via the Kalman filter • estimated cycle with possibly band-pass filter properties • individual cycles can be shifted * phase shifts are estimated * no a priori classification of lead-lag relationships

Further contributions of paper the estimated common cycle factor is the business cycle indicator (proxy to monthly output gap) growth rate indicator can also be obtained novel approach is applied to Euro area using nine key economic variables contrasts with other Euro area coincident indicators ...

A univariate trend-cycle decomposition y t = µ t + ψ t + ε t , 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 as in Clark (87) or with (time-varying) stochastic trigonometric functions as in Harvey (85,89) • irregular ε t : white noise

State space framework Trend-cycle components are unobservables The dynamic properties of components can be characterised in Markovian form State space formulation y t = Zα t + ε t , α t +1 = Tα t + Rζ t , where α t is state vector and includes trend and cycle, ε t ∼ N ID (0 , G ) , ζ t ∼ N ID (0 , Q )

Example: IRW trend plus AR(2) cycle, that is, µ t +1 = µ t + β t , β t +1 = β t + η t , and ψ t +1 = φ 1 ψ t + φ 2 ψ t − 1 + ξ t . In state space form,     1 1 0 0 0 0 1 0 0 η t     α t +1 =  α t +  ,     0 0 1 φ 1 ξ t       0 0 0 0 φ 2 t ) ′ and observation vector with state vector α t = ( µ t β t ψ t ψ ∗ y t = (1 0 1 0) α t + ε t .

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 next. For more details, Durbin and Koopman (2001)

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 . We assume that all y t ’s are observed.

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. For example, in case of missing observations, Kalman filter can handle them. For state space, forecasting is a missing observations problem (future observations are missing)

Kalman filter When observation y t is not available: v t = y t − Za t = ??? f t = ∞ (big !) k t = TP t Z ′ /f t = 0 a t +1 = Ta t + k t v t = Ta t P t +1 = TP t T ′ − k t k ′ t /f t + RQR ′ = TP t T ′ + RQR ′ Kalman step reduced to a one-step prediction step. When consecutive y t ’s are missing: multi-step forecasting !

Treatment of missing values Kalman filter can incorporate missing values in a time series Related algorithms such as smoothing and simulation can be adapted accordingly Forecasting is a special case of treatment of missing values

Decomposition with missing observations 14.3 0.01 14.2 14.1 0.00 14.0 filtered −0.01 with missing 13.9 1985 1990 1995 2000 1985 1990 1995 2000 14.3 0.01 14.2 14.1 0.00 14.0 smoothed −0.01 with missing 13.9 1985 1990 1995 2000 1985 1990 1995 2000 14.3 0.01 14.2 14.1 0.00 14.0 smoothed −0.01 no missing 13.9 1985 1990 1995 2000 1985 1990 1995 2000

Trend and cycle estimation Signal extraction is all about weighting observations In fact, it is about locally weighting observations State space methods carry out “optimal” weighting Algorithms available to get weights (Koopman and Harvey, 2003)

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

Problem: model-based ok, but no band-pass properties ”Band-pass” refers to frequency domain properties of polyno- mial 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

Incorporate Butterworth filters for trend 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 ∼ N ID (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 as before. t For m = 2 we have IRW with β t = µ (1) . t Higher value for m gives low-pass gain function with sharper cut-off downwards at certain low frequency point.

Generalised cycle for model-based band-pass Same principle can be applied to cycle component. Standard cycle component ψ t is given by � ψ t +1 � � � � � � � ψ t κ t cos λ sin λ = φ + , ψ + ψ + κ + − sin λ cos λ t +1 t t with initial and disturbance distributions i.i.d. i.i.d. N (0 , σ 2 N (0 , σ 2 ∼ κ ) , ∼ ψ ) , κ t ψ 1 i.i.d. κ + ψ + i.i.d. N (0 , σ 2 N (0 , σ 2 ∼ ∼ κ ) , ψ ) , t 1 Dynamic properties of cycle can be characteristed via the auto- covariance function.

Generalised cycle for model-based band-pass 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  = φ  +  , t t ψ +( j ) ψ +( j ) ψ +( j − 1) − sin λ cos λ t t t +1 for j = 1 , . . . , k , with ψ (0) = κ t and ψ +(0) = κ + t . t t Further details see Harvey and Trimbur(2003) and Trimbur(2002).

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