Modelling the Business Cycle Siem Jan Koopman - - PowerPoint PPT Presentation

Modelling the Business Cycle

Siem Jan Koopman

s.j.koopman@feweb.vu.nl

Vrije Universiteit Amsterdam Tinbergen Institute

Modelling the Business Cycle – p. 1

Eurozone GDP Data

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

Modelling the Business Cycle – p. 2

Topics in Business Cycle Analysis

• 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 to detect 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).

Modelling the Business Cycle – p. 3

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

Modelling the Business Cycle – p. 4

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:

• 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.

Modelling the Business Cycle – p. 5

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

Modelling the Business Cycle – p. 6

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 Modelling the Business Cycle – p. 7

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 ∼ 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 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.

Modelling the Business Cycle – p. 8

Generalised Cycle

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 ‘m

• .

Higher orders ensure smoother transitions. Further details: Trimbur (2002), Harvey & Trimbur (2003).

Modelling the Business Cycle – p. 9

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

Modelling the Business Cycle – p. 10

Measuring a common cycle from a multiple time series

• Analysis is based on a multivariate model
• Data-set includes series that are leading, lagging GDP
• We prefer not to choose leads & lags a-priori
• Common cycle will be allowed to shift for individual time series

using techniques developed by Rünstler (2002).

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)

Modelling the Business Cycle – p. 11

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).

Modelling the Business Cycle – p. 12

The basic multivariate model

Panel of N economic time series, yit, yit = µit + δiψt + εit, εit ∼ NID(0, σ2

i,ε)

The time series have mixed frequencies (quarterly and monthly) 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

)

• irregular εit.

Modelling the Business Cycle – p. 13

Business cycle

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)

Modelling the Business Cycle – p. 14

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

Modelling the Business Cycle – p. 15

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

Modelling the Business Cycle – p. 16

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

Modelling the Business Cycle – p. 17

The business cycle coincident indicator

Noteworthy features:

• GDP is quarterly, estimated components are monthly
• Euro area potential growth has declined after major recession of

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)

• 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

Modelling the Business Cycle – p. 18

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

Modelling the Business Cycle – p. 19

Coincident indicator for Euro area business cycle

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

Modelling the Business Cycle – p. 20

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

Modelling the Business Cycle – p. 21

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

Modelling the Business Cycle – 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

Modelling the Business Cycle – p. 23

Concluding Remarks

• 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 our

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

incorporate information from GDP without ad-hoc construction of artificial monthly GDP series

• cycle is based on nine key economic time series
• cycle appears to be in line with common wisdom

Modelling the Business Cycle – p. 24