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

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

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

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

Model-based trend-cycle decompositions with time-varying parameters – p. 5

Univariate UC Trend-Cycle Decomposition

yt = µt + ψt + εt

• Trend µt: ∆dµt = ηt;
• Irregular εt: White Noise;
• Cycle ψt: 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):

• ψt+1

ψ+

t+1

• = φ
• cos λ

sin λ − sin λ cos λ ψt ψ+

t

• +
• κt

κ+

t

• ,

κt, κ+

t ∼ NID(0, σ2 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

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

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.

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 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 µt = µ(m)

t

where ∆mµ(m)

t+1 = ζt,

ζt ∼ NID(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 and ηt is an IID sequence.

• For m = 2: IRW or smooth local linear trend or I(2) trend;
• For m = 2 and σζ = 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 kth 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

• ,

j = 1, . . . , k, with

• ψ(0)

t

ψ+(0)

t

• =
• κt

κ+

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

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

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

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)

Model-based trend-cycle decompositions with time-varying parameters – p. 13

Shifted cycles

In standard case, cycle ψt is generated by

• 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 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, yit, yit = µ(k)

it + δi

• cos(ξiλ)ψ(m)

t

+ sin(ξiλ)ψ+(m)

t

• + εit,

where

• 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

and ψ+(m)

t

;

• irregular εit.

Model-based trend-cycle decompositions with time-varying parameters – p. 15

Business cycle

Stock and Watson (1999) states that fluctuations in aggregate

• utput 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

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

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 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

Model-based trend-cycle decompositions with time-varying parameters – p. 18

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

Model-based trend-cycle decompositions with time-varying parameters – p. 19

The business cycle coincident indicator

Selected estimation results 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

Model-based trend-cycle decompositions with time-varying parameters – p. 20

Coincident indicator for Euro area business cycle

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

Model-based trend-cycle decompositions with time-varying parameters – p. 21

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

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

Model-based trend-cycle decompositions with time-varying parameters – p. 23

Role of time-varying parameters

Consider typical example of univariate trend-cycle decomposition: yt = µt + ψt + εt, t = 1, . . . , n, with

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

trigonometric functions (next slide);

• irregular εt: white noise.

In state space framework, the dynamic properties of components can be characterised in Markovian form: yt = Zαt + εt, αt+1 = Tαt + Rξt, where αt is state vector and includes trend and cycle.

Model-based trend-cycle decompositions with time-varying parameters – p. 24

Trend plus cycle decomposition

A decomposition model with an explicit cyclical term is given by yt = µt + ψt + εt, t = 1, . . . , n, ∆dµt+1 = ηt ⇒ µt+1 = µt + βt, βt+1 = βt + ηt+1, for d = 2

• ψt+1

˙ ψt+1

• = φ
• cos λ

sin λ − sin λ cos λ ψt ˙ ψt

• +
• κt

˙ κt

• εt ∼ NID(0, σ2

ε), ηt ∼ NID(0, σ2 η), κt, ˙

κt ∼ NID(0, σ2

κ).

The state vector for d = 2 is given by αt = ( µt βt ψt ˙ ψt ).

Model-based trend-cycle decompositions with time-varying parameters – p. 25

Time-varying parameters: deterministic functions

Time-varying cycle parameters: variance σ2

κ,t = fσ(t),

period ωt = fω(t), damping factor φt = fφ(t). Logit specification, with logit(x) = ex / (1 + ex): fσ(t) = exp

• cσ + γσ logit
• sσ(t − τσ)
• ,

fω(t) = 2 + exp

• cω + γω logit
• sω(t − τω)
• .

Spline specification: fσ(t) = exp

• cσ + w′

tδσ

• ,

fω(t) = 2 + exp

• cω + w′

tδω

• ,

c∗, γ∗, s∗, τ∗ are constants, wt is weight, δ∗ is coefficient vector.

Model-based trend-cycle decompositions with time-varying parameters – p. 26

State space representation

The measurement equation is time-invariant and is given by yt = Zαt + εt, Z =

• 1

O 1

• .

The state equation is time-varying: αt+1 = Ttαt + Rξt, ξt ∼ NID(0, Qt). The state αt is the d + 2 dimensional vector αt =

• µt

∆µt . . . ∆d−1µt ψt ˙ ψt

• .

Model-based trend-cycle decompositions with time-varying parameters – p. 27

Details of the state space representation

The required state space form is yt = Zαt + εt, αt+1 = Ttαt + Rξt, ξt ∼ NID(0, Qt). where αt =

• µt

∆µt . . . ∆d−1µt ψt ˙ ψt

• and

Tt =

• M

O O Ct

• ,

M = Id +

• O

Id−1 O

• ,

Ct = φ

• cos λt

sin λt − sin λt cos λt

• ,

Qt =    σ2

η

σ2

κ,t

σ2

κ,t

   , R =

• O

I3

• .

Model-based trend-cycle decompositions with time-varying parameters – p. 28

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

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. 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

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: www.ssfpack.com.

Model-based trend-cycle decompositions with time-varying parameters – p. 31

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, http://research.stlouisfed.org

Model-based trend-cycle decompositions with time-varying parameters – p. 32

Results I: basic decomposition (time-invariant models)

Lik AICC BIC full 719.37

• 1428.46
• 1411.63

GDP I 331.55

• 652.52
• 639.50

II 390.32

• 770.09
• 756.96

full 517.10

• 1023.92
• 1007.09

Investment I 243.37

• 476.18
• 463.15

II 279.92

• 549.29
• 536.17

full 97.66

• 185.23
• 162.72

Unemployment I

• 1.15

12.48 31.39 II 130.65

• 251.12
• 232.12

full 2206.89

• 4403.69
• 4381.18

IPI I 1015.57

• 2020.96
• 2002.06

II 1258.02

• 2505.86
• 2486.87

Model-based trend-cycle decompositions with time-varying parameters – p. 33

Results II: time-varying cycle volatility and period

For GDP en unemployment we obtain:

Fixed parameters Time-varying: spline Time-varying: logit GDP Lik 719.37 Lik 738.35 Lik 741.20 AICC

• 1428.46

AICC

• 1446.72

AICC

• 1459.17

BIC

• 1411.63

BIC

• 1400.82

BIC

• 1422.78

Unemployment Lik 97.66 Lik 147.00 Lik 152.24 AICC

• 185.23

AICC

• 265.37

AICC

• 282.08

BIC

• 162.72

BIC

• 202.73

BIC

• 232.77

Model-based trend-cycle decompositions with time-varying parameters – p. 34

US Gross Domestic Product

1950 1960 1970 1980 1990 2000 −0.05 0.00 0.05

Spline function

1950 1960 1970 1980 1990 2000 −0.05 0.00 0.05

Logit function

1950 1960 1970 1980 1990 2000 0.005 0.010 0.015 1950 1960 1970 1980 1990 2000 0.0050 0.0075 0.0100 1950 1960 1970 1980 1990 2000 4 6 1950 1960 1970 1980 1990 2000 5.0 7.5 10.0

Model-based trend-cycle decompositions with time-varying parameters – p. 35

US unemployment

1950 1960 1970 1980 1990 2000 0.0 2.5

Spline function

1950 1960 1970 1980 1990 2000 2

Logit function

1950 1960 1970 1980 1990 2000 0.2 0.3 0.4 1950 1960 1970 1980 1990 2000 0.1 0.2 1950 1960 1970 1980 1990 2000 5.0 7.5 10.0 12.5 1950 1960 1970 1980 1990 2000 4 5

Model-based trend-cycle decompositions with time-varying parameters – p. 36

Results III: time-varying cycle volatility

Spline specification Logit specification GDP Lik 747.27 Lik 738.74 AICC

• 1475.70

AICC

• 1460.81

BIC

• 1445.75

BIC

• 1434.11

Investment Lik 542.95 Lik 535.79 AICC

• 1067.06

AICC

• 1054.92

BIC

• 1037.11

BIC

• 1028.22

Unemployment Lik 161.16 Lik 150.55 AICC

• 304.06

AICC

• 384.88

BIC

• 263.66

BIC

• 248.95

IPI Lik 2307.17 Lik 2307.29 AICC

• 4596.08

AICC

• 4598.37

BIC

• 4555.67

BIC

• 4562.43

Model-based trend-cycle decompositions with time-varying parameters – p. 37

Spline specification

1950 1960 1970 1980 1990 2000 −0.05 0.00 0.05

Cycle GDP

1950 1960 1970 1980 1990 2000 0.005 0.010 0.015

Volatility

1950 1960 1970 1980 1990 2000 −0.1 0.0 0.1

Investment

1950 1960 1970 1980 1990 2000 0.01 0.02 0.03 0.04 1950 1960 1970 1980 1990 2000 2 4

Unemployment

1950 1960 1970 1980 1990 2000 0.1 0.2 0.3 0.4 1950 1960 1970 1980 1990 2000 −0.1 0.0 0.1

IPI

1950 1960 1970 1980 1990 2000 0.005 0.010 0.015

Model-based trend-cycle decompositions with time-varying parameters – p. 38

Logit specification

1950 1960 1970 1980 1990 2000 −0.05 0.00 0.05

Cycle GDP

1950 1960 1970 1980 1990 2000 0.0050 0.0075 0.0100 0.0125

Volatility

1950 1960 1970 1980 1990 2000 −0.1 0.0 0.1

Investment

1950 1960 1970 1980 1990 2000 0.01 0.02 0.03 1950 1960 1970 1980 1990 2000 2 4

Unemployment

1950 1960 1970 1980 1990 2000 0.1 0.2 0.3 1950 1960 1970 1980 1990 2000 −0.1 0.0 0.1

IPI

1950 1960 1970 1980 1990 2000 0.005 0.010 0.015

Model-based trend-cycle decompositions with time-varying parameters – p. 39

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 fσ(t) = exp(χt), χt+1 = χt + 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

Asymmetric cycle process

Given the cycle process,

• ψt+1

˙ ψt+1

• = φ
• cos λ

sin λ − sin λ cos λ ψt ˙ ψt

• +
• κt

˙ κt

• ,

it follows that ˙ ψt =

∂ψt ∂(λt) (this is shown in the paper).

To have λ as function of ˙ ψt, cycle becomes asymmetric: ψt = a cos(λtt − b), λt =

• λa,

˙ ψt > 0 λd, ˙ ψt ≤ 0 ,

• r

ψt = a cos(λtt − b), λt = λ + γ ˙ ψt.

Model-based trend-cycle decompositions with time-varying parameters – p. 41

Deterministic examples

Stylized asymmetric business cycles.

1950 1955 1960 −1 1 1950 1955 1960 0.08 0.09 0.10 0.11

Model-based trend-cycle decompositions with time-varying parameters – p. 42

Nonlinear state space

When considering λt = λ + γ ˙ ψt, we need to consider the nonlinear state space model yt = Zαt + εt, αt+1 = T(αt) + ηt, with T(αt) =      

• 1

1 1

• O

O φ

• cos(λt)

sin(λt) − sin(λt) cos(λt)

• 

     αt, and λt = λ +

• γ
• αt.

Model-based trend-cycle decompositions with time-varying parameters – p. 43

Importance sampling

Estimation by ML is implemented using importance sampling techniques for nonlinear state space models. L(θ) =

• pθ(α, y)dα =

pθ(α, y) gθ(α|y) gθ(α|y)dα = gθ(y) pθ(α, y) gθ(α, y)gθ(α|y)dα. where gθ(y) is the likelihood of the approximating model. log ˆ L(θ) = log Lg(θ) + log ¯ w, ¯ w = 1 N

N

• i=1

pθ(α(i), y) gθ(α(i), y), where Lg(θ) is the likelihood from the approximating model, α(i) is drawn from gθ(α|y) using simulation smoothing.

Model-based trend-cycle decompositions with time-varying parameters – p. 44

Results IV: stochastic asymmetric cycles

Un Un (as) Inv Inv (as) GDP GDP (as) σ2

ε

7.70e−4 1.67e−3 – – – – σ2

ζ

1.13e−7 1.13e−7 1.23e−5 1.21e−6 8.29e−8 7.91e−8 σ2

κ

2.77e−2 2.48e−2 2.53e−4 2.44e−4 5.60e−5 5.45e−5 φ 0.988 0.989 0.963 0.968 0.950 0.953 ω 124.9 102.9 24.0 24.0 36.2 34.8 γ – 0.00738 – −0.36 – −0.91 Norm 146.1 164.6 12.0 8.0 7.8 7.9 Q(20) 73.5 69.4 28.3 23.8 24.5 23.4 LogL 153.8 163.6 428.5 432.3 584.5 586.9 W 24.9 7.6 5.0 LM 26.2 6.3 5.3 LR 19.6 8.6 4.8

Model-based trend-cycle decompositions with time-varying parameters – p. 45

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

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.278-290.

• 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