Multivariate Unobserved Components Siem Jan Koopman Vrije - - PowerPoint PPT Presentation

Multivariate Unobserved Components

Siem Jan Koopman Vrije Universiteit Amsterdam Tinbergen Institute

EMM lecture — 2nd 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

1985 1990 1995 2000 13.90 13.95 14.00 14.05 14.10 14.15 14.20 14.25

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

1985 1990 1995 2000 14.0 14.2

HP trend

1985 1990 1995 2000 −0.01 0.00 0.01 0.02

HP cycle

1985 1990 1995 2000 13.9 14.0 14.1 14.2 14.3

STAMP trend

1985 1990 1995 2000 −0.01 0.00 0.01

STAMP cycle

1985 1990 1995 2000 14.0 14.2

AKR trend

1985 1990 1995 2000 −0.01 0.00 0.01

AKR cycle

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 yt = µt + ψt + εt, with

• trend µt: ∆dµt = ηt where d = 1 (RW) or d = 2 (IRW);
• cycle ψt:

AR(2) with complex roots as in Clark (87) or with (time-varying) stochastic trigonometric functions as in Harvey (85,89)

• irregular εt: white noise

State space framework Trend-cycle components are unobservables The dynamic properties of components can be characterised in Markovian form State space formulation yt = Zαt + εt, αt+1 = Tαt + Rζt, where αt is state vector and includes trend and cycle, εt ∼ N ID(0, G), ζt ∼ N ID(0, Q)

Example: IRW trend plus AR(2) cycle, that is, µt+1 = µt + βt, βt+1 = βt + ηt, and ψt+1 = φ1ψt + φ2ψt−1 + ξt. In state space form, αt+1 =

    

1 1 1 φ1 1 φ2

     αt +     

ηt ξt

     ,

with state vector αt = (µt βt ψt ψ∗

t )′ and observation vector

yt = (1 0 1 0)αt + εt.

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 αt (at) and its mean square error (Pt): vt = yt − Zat ft = ZPtZ′ + G kt = TPtZ′/ft at+1 = Tat + ktvt Pt+1 = TPtT ′ − ktk′

t/ft + RQR′

for t = 1, . . . , n and for some initialisation a1 and P1. We assume that all yt’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 yt is not available: vt = yt − Zat = ??? ft = ∞ (big !) kt = TPtZ′/ft = 0 at+1 = Tat + ktvt = Tat Pt+1 = TPtT ′ − ktk′

t/ft + RQR′ = TPtT ′ + RQR′

Kalman step reduced to a one-step prediction step. When consecutive yt’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

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

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

1985 1990 1995 2000 13.9 14.0 14.1 14.2 14.3 1985 1990 1995 2000 −0.02 0.00 0.02 −20 −10 10 20 0.0 0.1 0.2 −20 −10 10 20 0.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.5 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.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.5 1.0 TREND 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.5 1.0 CYCLE 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.5 1.0 IRREGULAR

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 µt = µ(m)

t

where ∆mµ(m)

t+1 = ζt,

ζt ∼ N ID(0, σ2

ζ ),

• r

µ(j)

t+1 = µ(j) t

+ µ(j−1)

t

, j = m, m − 1, . . . , 1, with µ(0)

t

= ηt as before. For m = 2 we have IRW with βt = 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 ψt is given by

ψt+1

ψ+

t+1

• = φ
• cos λ

sin λ − sin λ cos λ ψt ψ+

t

• +
• κt

κ+

t

• ,

with initial and disturbance distributions κt

i.i.d.

∼ N(0, σ2

κ),

ψ1

i.i.d.

∼ N(0, σ2

ψ),

κ+

t i.i.d.

∼ N(0, σ2

κ),

ψ+

1 i.i.d.

∼ N(0, σ2

ψ),

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 ψt = ψ(k)

t

where

  ψ(j)

t+1

ψ+(j)

t+1

  = φ

• cos λ

sin λ − sin λ cos λ

  ψ(j)

t

ψ+(j)

t

  +   ψ(j−1)

t

ψ+(j−1)

t

  ,

for j = 1, . . . , k, with ψ(0)

t

= κt and ψ+(0)

t

= κ+

t .

Further details see Harvey and Trimbur(2003) and Trimbur(2002).

Weights and gain functions of components

1985 1990 1995 2000 13.9 14.0 14.1 14.2 14.3 1985 1990 1995 2000 −0.02 0.00 0.02 −20 −10 10 20 0.0 0.1 0.2 −20 −10 10 20 0.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.5 1.0

Coincident cycle from a multiple time series

• Our analysis will be based on a multivariate model
• Includes series that are leading, lagging GDP
• We prefer not to choose lag-lengths for series a-priori
• Common cycle will be allowed to shift for individual time

series

Shifted cycles

1980 1985 1990 1995 −0.4 −0.2 0.0 0.2

estimated cycles gdp (red) versus cons confidence (blue)

1980 1985 1990 1995 −0.4 −0.2 0.0 0.2

estimated cycles gdp (red) versus shifted cons confidence (blue)

Shifted cycle In standard case, cycle ψt is

ψt+1

ψ+

t+1

• = φ
• cos λ

sin λ − sin λ cos λ ψt ψ+

t

• +
• κt

κ+

t

• The cycle

cos(ξλ)ψt + sin(ξλ)ψ+

t ,

is shifted ξ time periods to the right (when ξ > 0) or to the left (when ξ < 0). Here, −1

2π < ξ0λ < 1 2π (shift is wrt ψt)

More details in Runstler (2002) for idea of shifting cycles in multivariate unobserved components time series model of Harvey and Koopman (1997).

The basic multivariate model Panel of N economic time series, yit, yit = µit + δiψt + εit εit ∼ N ID(0, σ2

i,ε)

The time series have mixed frequencies (quarterly and monthly)

quarterly GDP and monthly consumer confidence

1990 1995 2000 13.9 14.0 14.1 14.2 1999 2000 2001 14.20 14.22 14.24

Basic components: trend and cycle Separate trend components µi,t+1 = µit + βit + ηit, ηit ∼ N ID(0, σ2

i,η),

βi,t+1 = βit + ζit, ζit ∼ N ID(0, σ2

i,ζ),

The cycle component is common to all time series in the panel:

ψt+1

ψ+

t+1

• = φ
• cos λ

sin λ − sin λ cos λ ψt ψ+

t

• +
• κt

κ+

t

• .

Model is extended to get band-pass filter properties and allows for shifts in cycles. This model can be put in state space and is linear Estimation of parameters and of unobservables such as trends and cycles is based on Kalman filter and associated smoothing methods see Harvey (1989) and Durbin and Koopman (2001).

Final model shifted cycles in model via the formulation yit = µ(k)

it

+ δi

• cos(ξiλ)ψ(m)

t

+ sin(ξiλ)ψ+(m)

t

• + εit,

with generalised individual trend µ(k)

it ,

generalised common cycle ψ(m)

t

(with associated variable ψ+(m)

t

) and irregular εit.

Business cycle we follow Stock and Watson (1999): “... 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 ...” we therefore 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)

Time series 1986 – 2002

1990 1995 2000 13.90 13.95 14.00 14.05 14.10 14.15 14.20 14.25 14.30 GDP IPI Interest rate spread Construction confidence indicator Consumer confidence indicator Retail sales unemployment Industrial confidence indicator Retail trade confidence indicator

Details of model, estimation we have set m = 2 and k = 6 for generalised components leads to estimated trend/cycle estimates with band-pass prop- erties, 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 ξi are estimated number of parameters for each equation is four (σ2

i,ζ, δi, ξi, σ2 i,ε)

and for the common cycle is two (φ and σ2

κ)

total number is 4N = 4 × 9 = 36

Decomposition of real GDP

1990 1995 2000 13.9 14.0 14.1 14.2 GDP Euro Area Trend 1990 1995 2000 0.001 0.002 0.003

slope

1990 1995 2000 −0.01 0.00 0.01

Cycle

1990 1995 2000 −0.0050 −0.0025 0.0000 0.0025 0.0050

irregular

The business cycle coincident indicator several interesting features: GDP is quarterly, estimated components are monthly Euro area potential growth has declined after major recession

• f 1993 (before, growth was around 3.7% in annualised terms,

after it was 2.4%, falls within the 2.0 − 2.5 underlying the ECB monetary policy) GDP cycle in line with common wisdom regarding Euro area business cycle, ECB (2002)

series load shift R2

d

gdp −− −− 0.31 indutrial prod 1.18 6.85 0.67 Unemployment −0.42 −15.9 0.78 industriual c 2.46 7.84 0.47 construction c 0.77 1.86 0.51 retail sales c 0.26 −0.22 0.67 consumer c 1.12 3.76 0.33 retail sales 0.11 −4.70 0.86 int rate spr 0.57 16.8 0.22 business cycle tracks the turning points well historical minimum value is observed in Aug 1993, falls in most severe recession period of Euro area maximum value is in Jan 2001

Coincident indicator for Euro area business cycle

1990 1995 2000 −0.015 −0.010 −0.005 0.000 0.005 0.010

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) 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

Quarterly GDP growth and our growth indicator

1990 1995 2000 −0.0075 −0.0050 −0.0025 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150

EuroCOIN and our growth indicator

1990 1995 2000 −0.0075 −0.0050 −0.0025 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 Coincident Eurocoin

Revisions Real-time reliability of business cycle and growth indicators: in practice, indicators are subject to revisions over time due to data revisions and to their re-computation not possible to evaluate consequences of first potential source

• f revisions

we assess the second one by comparing smoothed and filtered versions of indicator

Smoothed and filtered estimates, revisions

1990 1995 2000 −0.02 −0.01 0.00 0.01 0.02

Smoothed cycle Filtered cycle

1990 1995 2000 −0.02 −0.01 0.00 0.01 0.02

revisions

Revisions contd Some revision statistics for cycle period sd ratio corr sign 1989 – 2002 0.84 0.55 0.72 1993 – 2002 0.66 0.75 0.84 take into account it is hard to estimate output gap in real-time

• nly with the increase of time, one can be more accurate about

cyclical position Orphanides and van Norden (2002) say whatever method is used, reliability is quite low

Smoothed and filtered estimates of growth, revisions

1990 1995 2000 −0.0075 −0.0050 −0.0025 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 filtered growth Coincident smoothed growth Coincident

Revisions contd Some revision statistics for growth period sd ratio corr 1989 - 2002 0.64 0.84 1993 - 2002 0.59 0.84 real-time reliability of growth is substantially higher than cycle we can not compare our statistics with EuroCOIN, they did not perform this evaluation

Finally, we have considered the cycle component with band-pass filter properties by having it as a common cycle component that allows for phase shifts this base cycle is associated with real GDP and it resumes to

• ur business cycle indicator

by discarding the irregular component of GDP, a smooth growth indicator can also be constructed fact that GDP is only available quarterly, we are able to incor- porate information from GDP without ad-hoc construction of artificial monthly GDP series

Finally contd, cycle is based on nine key economic time series cycle appears to be in line with common wisdom despite the inherent difficulty in assessing cyclical positions in real-time, our growth indicator reliability is shown to be good