Business cycle analysis and forecasting using advanced ...

Business cycle analysis and forecasting using advancedspectral methods and data-based low-order models

Andreas Groth1

Michael Ghil1,2 · Dmitri Kondrashov1 · Mickael Chekroun1

1University of California, Los Angeles

2Ecole Normale Superieure, Paris

35th International Symposium on ForecastingRiverside, California, June 2015

Motivation

Novel tools for both data analysis and modeling of business cycles1 Advanced spectral analysis of time series analysis2 Modeling tools for nonlinear and random dynamics

Combined study of modeling and data analysis

Andreas Groth (UCLA) SSA and EMR 35th ISF 1 / 18

Outline

1 Non-equilibrium Dynamic Model (NEDyM)Endogenous business cyclesCatastrophes and the state of the economy

2 Statistical data analysisUS business cyclesMultichannel singular spectrum analysis (M-SSA)Cyclical behavior vs. stochastic fluctuationsState-dependent fluctuations

3 Modeling and forecastingTheory and practiceEMR model fit to US trend residualsPrediction exercise I: The 2001 recessionPrediction exercise II: The 2008 recession


Non-equilibrium Dynamic Model (NEDyM)Endogenous business cycles

Endogenous dynamics: an alternativeexplanation for business cycles

Represents an economy with oneproducer, one consumer, one goods thatis used both to consume and invest.

Based on Solow (1956) model, in whichequilibrium constraints are replaced bydynamic relationships that involveadjustment delays.

NEDyM possesses endogenous businesscycles!

Production

Employment rate

Real wage

Price

2 4 6 8 10 12Time in years

a.u.

Investment

Hallegatte, Ghil, Dumas & Hourcade (2008), J. Econ. Behavior & Org.


Non-equilibrium Dynamic Model (NEDyM)Catastrophes and the state of the economy

Vulnerability paradox

A disaster that affects aneconomy during its recessionphase causes fewer long-termdamages than if it occurs duringan expansion!

Business cycle

-2 -1 0 1 2 3 4Time lag (years)

96

98

100

102

104

Prod

uctio

n (a

rbitr

ary

units

)

Economic losses due to a disaster

-2 -1 0 1 2 3 4Time lag (years)

0

10

20

Prod

uctio

n lo

sses

(%

GD

P)

Hallegatte & Ghil (2008), Ecological Economics


Outline





Statistical data analysisUS business cycles

Vertical lines are NBER-defined recessions

NBER Recession dating

“significant decline in economic activity spread acrossthe economy, lasting more than a few months,normally visible in real GDP, real income,employment, industrial production, andwholesale-retail sales.”

I translates into comovements and lead–lag structure


Statistical data analysisMultichannel singular spectrum analysis (M-SSA)

PCA: Spatial EOFs M-SSA: Spatio-temporal EOFs

Covariance matrixCi ,j =

⟨xi (t), xj(t)

⟩t

Ci ,j(τ) =⟨xi (t), xj(t + τ)

⟩t

EigendecompositionCi ,j ek = λk ek Ci ,j(τ) ek = λk ek

Empirical orthogonal functions (EOFs)ek(i) i – space ek(i , τ) i – space

τ – time

M-SSA relies on classical Karhunen-Loeve spectral decomposition of astochastic process as a linear combination of orthogonal functionsBroomhead & King (1986a,b) introduced SSA and M-SSA intodynamical systems analysis, following the Mane-Takens idea to reconstructdynamics via time-delayed embedding

Ghil et al. (2002), Advanced spectral methods for climatic time series, Reviews of Geophysics

Golyandina & Zhigljavsky (2013), Singular Spectrum Analysis for Time Series, Springer


Statistical data analysisCyclical behavior vs. stochastic fluctuations

Reconstruction

(a) US trend residuals

(b) M-SSA with 10spatio-temporal EOFs

(c) PCA with 2 spatial EOFs

Both reconstructions capture75% of the total variance.

M-SSA reconstruction issmoother and shows a clearlead-lag structure; i.e. itidentifies temporal dynamics.

PCA identifies onlyinstantaneous comovements.

(a) US trend residuals

−0.2

−0.1

0

0.1

0.2

(b) RCs 1−10 of M−SSA

−0.2

−0.1

0

0.1

0.2

(c) RCs 1−2 of PCA

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

−0.2

−0.1

0

0.1

0.2


Statistical data analysisCyclical behavior vs. stochastic fluctuations

Statistical significance test

H0 : coupled red noiseprocesses, AR(1); detrended

Multiple US aggregates

M-SSA identifies regularbehavior that cannot begenerated by random shocksalone

Regular behavior is associatedwith comovements across theentire economy

Groth, Ghil, Hallegatte & Dumas(2012), FEEM working paper 26.2012;to appear in JBCMA

Analsis of US GDP alone

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

(a) Spectrum of eigenvalues

Frequency (cycles/year)

λ

0 0.5 1 1.5 2

0.05

0.1

0.15

0.2

0.25

0.3

0.35


(b) Power spectral density

PS

D

−20 0 20−0.02

0

0.02

0.04Covariance function

Time lag in quarters

Analysis of multiple US aggregates

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2


λ


Statistical data analysisState-dependent fluctuations

Phase-dependent volatility

I Vulnerability paradox,NEDyM model

A disaster that affects an economyduring its recession phase causes

fewer long-term damages than if itoccurs during an expansion!

Hallegatte & Ghil (2008), EcologicalEconomics

Groth, Ghil, Hallegatte & Dumas (2012),FEEM working paper 26.2012; to appearin JBCMA

Groth, Dumas, Ghil & Hallegatte (2015),AGU monograph

Reconstruction with leading EOF pair(a) GDP and its reconstruction with M−SSA RCs 1−2

−0.2

−0.1

0

0.1

0.2

(b) Local variance fraction of M−SSA PCs 1−2

0

0.2

0.4

0.6

0.8

1

(c) Local variance fraction of M−SSA PCs 3−150

0

0.2

0.4

0.6

0.8

1

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

The “signal” fraction is largest during the recessions

The “noise” fraction is largest during the expansions


Outline





Modeling and forecastingTheory and practice

Question: System of closed equations which governs the evolution ofpartial observations x(t)?

Theoretical solution

Mori-Zwanzig formalism from statistical mechanics – generalized Langevinequation:

dx

dt= F (x)︸︷︷︸

(nonlinear) self-interactions

+

∫ t

0G (t, s, x(s))ds︸︷︷︸

←→ cross-interactions ←→

+ η(t)︸︷︷︸stochastic forcing

Mori (1965), Zwanzig (1964), Chorin et al. (1999)

x – observed slow variables

η – unobserved fast variables, uncorrelated with x

F – Markovian part

G – non-Markovian part (memory effects) F ,G in practice difficulty to obtain


Modeling and forecastingTheory and practice

Practical solution – Multi-level regression

Empirical model reduction (EMR) can approximate theoretical solution

xn+1 − xn =

Regression︷︸︸︷(F + A xn + B[xn, xn]

)δt +

Residuals︷︸︸︷r

(0)n δt

r(0)n+1 − r

(0)n = L(1)[xn, r

(0)n ]δt + r

(1)n δt

...

r(m−1)n+1 − r

(m−1)n = L(m−1)[xn, r

(0)n , . . . , r

(m−1)n ]δt + r

(m)n δt

Kravtsov, Kondrashov & Ghil (2005,2009); Kondrashov, Chekroun & Ghil (2015)

x – observed variables

r(m) – regression residuals account for unobserved variables(each additional level represents faster scales)

A,B,F , L(m) – regression coefficients; found through top-to-bottom procedure

Stopping criterion: r(m) becomes i.i.d.; e.g. uncorrelatedAndreas Groth (UCLA) SSA and EMR 35th ISF 11 / 18

Modeling and forecastingEMR model fit to US trend residuals

Pre-processing

US trend residuals are first projected onto 3 leading spatial EOFs, which capture 93% ofthe total variance

EMR models are then fitted using the resulting 3 principal components (PCs)



Auto-covariance function estimates

Single-level model misses the auto-covariance structure

Multi-level models give a much better fit; i.e. they account for memory effects



Probability density function estimates

Quadratic EMR model captures skewness (S); i.e. non-Gaussian behavior



Quadratic multi-level EMR model

I US recessions (red lines) have larger magnitude than expansions (green lines)

Quadratic EMR model captures well this non-Gaussian feature


Modeling and forecastingPrediction exercise I: The 2001 recession

GDP retrospective forecast 1997–2007

EMR-model training interval 1954–1997

Quadratic EMR model captures best the recession in amplitude, shape, and timing

Mean prediction and 80% range from 500 EMR realizations; vertical lines are NBER-defined recessions


Modeling and forecastingPrediction exercise II: The 2008 recession

GDP retrospective forecast 2003–2014

EMR-model training interval 1954–2003

Quadratic EMR model captures best the recession in amplitude, shape, and timing

Mean prediction and 80% range from 500 EMR realizations; vertical lines are NBER-defined recessions


Summary

1 Theoretical model — NEDyMEndogenous dynamics: alternative explanation for business cyclesVulnerability paradox: economic loss depends on state, with higher

loss during expansion

2 Statistical data analysis — M-SSARegular behavior: Business cycles cannot be explained by random

shocks alone; not just spurious detrending resultsVolatility: “signal” fraction is largest during the recessions, where

the “noise” fraction is largest during expansionsI Confirms the vulnerability paradox

3 Prediction exercises — EMRMulti-level EMR model: captures better memory effectsQuadratic EMR model: captures better skewnessQuadratic multi-level EMR model: performs best in predicting the

amplitude, shape and timing of the recessions; inparticular at larger lead times


Literature

NEDyM Hallegatte, S., M. Ghil, P. Dumas, and J.-C. Hourcade, 2008: Business cycles, bifurcations andchaos in a neo-classical model with investment dynamics, Journal of Economic Behavior &Organization, 67, 57–77.Hallegatte, S. and M. Ghil, 2008: Natural disasters impacting a macroeconomic model withendogenous dynamics, Ecological Economics, 68, 582–592.Groth, A., P. Dumas, M. Ghil, and S. Hallegatte, 2015: Impacts of natural disasters on adynamic economy, in Extreme Events: Observations, Modeling, and Economics, AGU bookseries.

M-SSA Broomhead, D. S. and G. P. King, 1986a: Extracting qualitative dynamics from experimentaldata, Physica D, 20, 217–236.Broomhead, D. S. and G. P. King, 1986b: On the qualitative analysis of experimental dynamicalsystems, in Nonlinear Phenomena and Chaos, S. Sarkar (Ed.), 113–144.Ghil, M., et al. 2002: Advanced spectral methods for climatic time series, Reviews ofGeophysics, 40, 1–41.Golyandina, N. and A. Zhigljavsky, 2013: Singular Spectrum Analysis for Time Series, SpringerBerlin Heidelberg.Groth, A., M. Ghil, S. Hallegatte, and P. Dumas, 2012: The Role of Oscillatory Modes in U.S.Business Cycles, FEEM working paper 26.2012, to appear in OECD Journal of Business CycleMeasurement and Analysis.

EMR Kravtsov, S., D. Kondrashov, and M. Ghil, 2005: Multilevel regression modeling of nonlinearprocesses: Derivation and applications to climatic variability, Journal of Climate, 18, 4404–4424.Kravtsov, S., D. Kondrashov, and M. Ghil, 2009: Empirical model reduction and the modellinghierarchy in climate dynamics and the geosciences, in Stochastic physics and climate modelling.Cambridge University Press, 35–72.Kondrashov, D., M. D. Chekroun, and M. Ghil, 2015: Data-driven non-Markovian closuremodels, Physica D, 297, 33–55.


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