The Growth-Volatility Relationship: New Evidence From ... · The Growth-Volatility Relationship: New Evidence From Stochastic Volatility in Mean Models Matthieu LEMOINE 1 Christophe

IntroductionModels

Empirical resultsConclusion

Estimation Methodology

The Growth-Volatility Relationship: New Evidence FromStochastic Volatility in Mean Models

Matthieu LEMOINE 1 Christophe MOUGIN 2

1DGEI-DEMS-SEPS, Banque de France, Paris

2Institut d’Etudes Politiques, Paris

September 27th, 2010

Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship

IntroductionModels



General overview

The great moderation which began in the mid-1980’s had 3 explanations:

I good monetary policy (good policies)

I improved inventory management (good practices)

I a decline in the volatility of exogenous shocks (good luck)

Recent shocks changed the diagnosis (Canarella et al. 2008):

I good luck seems to be the most likely explanation

I US and UK economies have entered a new regime of high volatility

The 2007-2008 financial crisis has raised the following questions:

I how will the end of the great moderation impact the growth?

I can short-term policies mitigate this impact?

These questions have put foreground the debate about the relationshipbetween growth and volatility.


IntroductionModels



The theoretical debate

In the literature on the investment demand

I negative link: a high uncertainty should decrease investment, because suchexpenditures are irreversible and can be delayed (Pindyck 1991)

I positive link: when shocks are positively serially correlated and adjustmentcosts are fixed, replacement investment is procyclical (Cooper et al., 1999)

In endogenous growth models

I negative link: in the AK model with convex adjustment costs of Barlevy(2004)

I positive link: in the stochastic endogenous growth model of Jones et al.(2005) (in the likely case of a risk aversion larger than 1)

Following the Schumpeterian view, other endogenous growth modelsunderline the role of productivity improving activities (PIA)

I positive link: lower opportunity costs of PIA than short-term investmentsduring recessions (Aghion and Saint Paul, 1998)

I negative link: credit market imperfections hamper innovation andreorganization during recessions (Aghion et al., 2005)


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The previous empirical evidence

Studies based on panel datasets

I negative link: macro data for 92 countries in the period 1960-1985 withvarious controls in Ramey and Ramey (1995)

I positive link: sectoral data for 47 countries in the period 1970-1992 inImbs (2007)

Studies based on time series

I positive link: GARCH-M model applied to industrial production of the UKin the period 1948-1991 in Caporale and McKiernan (1996)

I insignificant link: GARCH-M model applied to US GDP in the period1947-2006 in Fang and Miller (2008)


IntroductionModels



Content of the paper

Here, we propose a new empirical approach to address this issue:

I a stochastic volatility in mean (SV-M) model

I an estimation methodology based on sequential Monte-Carlo (SMC)methods

I an application of SV-M and GARCH-M (for the sake of comparison)models to G7 output series in the time period 1960-2009

We get 3 results:

I a significantly positive relationship in Germany and Italy and insignificantin other countries

I results are preferable relative to those of GARCH-M models (fit,assumptions about distributions)

I a positive impact of unexpected volatility on output growth


IntroductionModels



Outline

Introduction

Models

Empirical results

Conclusion



IntroductionModels



SV-M model

The SV-M model is the following{yt = c +

∑pi=1 αiyt−i + δht + σ∗ exp

(ht2

)εt

ht = φht−1 + ηt

I output growth (yt) is explained by its lags, log-volatility (ht), andinnovations,

I output innovations have instantaneous volatility (σ∗)2 exp(ht),

I ht is a stationary AR(1) process with persistence φ,

I δ expresses the relation between growth and its log-volatility.

Very close to Koopman & Uspensky (2002): growth is explained bylog-volatility instead of volatility itself


IntroductionModels



Expected and unexpected volatilities

Output growth is related to both expected and unexpected volatility:

yt = yt|t−1 + δ(ht − ht|t−1

)+ σ∗ exp

(ht

2

)εt

with the expected log-volatility (conditional to past states and observations):

ht|t−1 = φht−1

Advantage of SV-M models relative to GARCH-M ones, which do not takeinto account the unexpected volatility...


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log-GARCH-M model

The log-GARCH(1,1) in mean model is the following{yt = c +

∑pi=1 αiyt−i + δ log(σ2

t ) + ut

log(σ2t ) = ω + ξ log(σ2

t−1) + ψ log(u2t−1)

I σ2t is the expected volatility, its logarithm depends on lags and past innovations.

I GDP growth is explained by its lags, expected log-volatility, and innovations,

I δ expresses the relation between growth and its expected log-volatility.

Very close to GARCH in mean models used by Caporale & Mc Kiernan (1996):focus on log-volatility instead of volatility itself.

We can rewrite it in a form similar to the SV-M one:{yt = c +

∑pi=1 αiyt−i + δht + σ∗ exp

(ht2

)εt

ht = φht−1 + ψηt−1

with φ = ξ + ψ and ηt = log(ε2t )− E

[log(ε2

t )].


IntroductionModels



Outline

Introduction

Models

Empirical results

Conclusion



IntroductionModels



Data and specification tests

Dataset

I GDP growth rates of G7 countries (GE, FR, IT, UK, US, JP and CA)

I time period 1960q2-2009q2, except for Germany (1968q2-2009q2) andCanada (1961q2-2009q2)

I sources: Eurostat and OECD

Specification tests

I number of lags in the mean equation determined with the SIC criterionand the Ljung-Box test

I Jarque-Bera test: residuals normality not rejected for SV-M models,rejected for FR and UK log-GARCH-M models

I better fit (higher log-likelihood, lower SIC) of SV-M models relative tolog-GARCH-M ones


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Parameters estimates for the SV-M model

Legend: standard errors are written in italic and p-values at the 5% level are underlined.


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Parameters estimates for the SV-M model

Volatility parameters

I high persistence (φ) in all countries except Japan

I largest variations of volatility (ση) for the UK

Growth-volatility relationship

I positive relationship (δ) in Germany and Italy

I insignificant in other countries


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Parameters estimates for the log-GARCH-M model

Legend: standard errors are written in italic and p-values at the 5% level are underlined.


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Comparison with the log-GARCH-M modelParameters of the log-GARCH-M model

I high persistence (φ) in all countries and the French log-volatility is nearlyintegrated

I largest variations in volatility (ψ) for the UK

I the growth-volatility relationship is significant for GE, IT, FR, the UK andthe US

I Student tests might deliver false results, because the normality of residualsis not warranted

The comparison suggests a positive impact of unexpected volatility

I the high persistence implies a high weight of expected volatility relative tothe unexpected one

var(φht−1) =φ2

1− φ2(σ∗)2 � (σ∗)2 = var(ηt)

I estimates of δ are generally lower for log-GARCH-M models than SV-Mones


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Estimates of the log-volatility in G7 economies

Legend: for the SV-M model, smoothed estimates of the log-volatility (black line) with their 95%confidence intervals (dotted lines); for the log-GARCH-M model, log-volatilities (grey line).


IntroductionModels



Estimates of the log-volatility in G7 economies

The end of the great moderation

I we observe the great moderation with a prompt and significant decrease ofthe volatility around 1983 in the US

I more gradual decrease in other countries and no significant decrease in JP

I clear increase of the volatility after the financial crisis of 2007-2008

Comparison with the results of Stock and Watson (2005)

I they estimate an AR model of the G7 output growth rates withnon-stationary SV innovations

I similar results for US, UK and IT; slight differences for FR, GE and JP

I their sample covers the period 1960-2002 and does not include the end ofthe great moderation


IntroductionModels



Outline

Introduction

Models

Empirical results

Conclusion



IntroductionModels



ConclusionMain results

I first application of SV-M models to the growth-volatility issue

I a better fit of SV-M relative to log-GARCH-M

I significant and positive relationship for GE and IT, insignificant for otherG7 countries

I contrary to log-GARCH-M, SV-M takes into account the impact of theunexpected volatility, which seems to be positive

I illustration of the end of the great moderation

Further research

I disentangling the impact of unexpected and expected volatilities

I distinguishing the long-run from the short-run fluctuations of outputgrowth

I incorporating these distinctions by merging the SV-M and unobservedcomponent models

I add control variables and test inverse causality


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Estimation of the SV-M model

Specificity of our model

I an unobserved variable ht

I non linear state space model→ no analytical form for the states estimates and the likelihood

The sequential Monte-Carlo (SMC) approach consists in simulatingsequentially the random variables ht :

I ht |y0:t−1 (predicted samples),

I ht |y0:t (filtered samples) and

I ht |y0:T (smoothed samples).

Then, following the Monte-Carlo principle, we can compute

I smoothed state estimates E[ht |y0:T ]

I the likelihood L(θ) = Eh[`0(h0, y0)|θ]∏T

t=1 Eh[`t(ht , y0:t)|y0:t−1; θ]that is maximized for parameter estimation


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Filtering: the SIR algorithm

The SIR algorithm proceeds sequentially in 2 steps: Consider for some t, afiltered sample {hi

t} distributed according to the law of ht |y0:t .

1. draw a sample {ηit+1} and use the transition equation

(ht+1 = φht + ηt+1) to get a predicted sample {hpr,it+1} distributed according

to the law of ht+1|y0:t .

2. re-sample this sample according the observation likelihood weights`t+1(hpr,i

t+1, y0:t+1), to get a filtered sample {hit+1} distributed according to

the law of ht+1|y0:t+1.

Our smoothing algorithm consists in re-weighting sequentially backward thefiltered samples, according to some transition density weights. (All details inDoucet et al. 2001)


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SIR in practice

Advantages:

I simple algorithm

I fast algorithm: O(N) for filtering, O(N2) for smoothing

I enables to compute state estimates + likelihood

Drawback: difficult to maximise the likelihood (likelihood estimate is noisy)

I no gradient computation available

I maximise with meta-heuristics (e.g. simulated annealing)

I unable to compute the Hessian matrix at the optimum (for standard errors)

We develop another likelihood computation: Hurzeler & Kunsch (1998), theapproximated likelihood.


IntroductionModels



Approximated likelihood

To avoid the noise issue,

I we use only one set of samples {hi0:T}...

I ... drawn for one unique set of parameters θ0...

I ... to compute all likelihoods for all θ.

This relies on importance sampling: the approximated likelihood L(θ, θ0) iscomputed using

L(θ) = L(θ0) Eh

[π0:T (h0:T , θ, θ0)

∣∣∣ y0:T ; θ0

]with the following importance weights

π0:T (h0:T , θ, θ0) =p(h0:T , y0:T ; θ)

p(h0:T , y0:T ; θ0)


IntroductionModels



Maximisation of the likelihood in practice

Advantages:

I fast computation of the maximised function: O(N)I the noise is frozen→ Newton methods can be applied to maximize

Drawbacks:

I require a smoothing step with a computational cost of O(N2)I local approximation for θ close to θ0

→ θ0 has to be close to θ∗

We iterate a 2-steps procedure

1. maximize the approximated likelihood

2. update θ0 with the previous maximum

This shall converge to the true likelihood maximum θ∗.


IntroductionModels



Implementation details for the SV-M model

The SIR algorithm requires:

I the following measurement densities

`t(ht , y0:t) , p(yt |ht , yt−p:t−1)

=1√

2πσ∗ exp(

ht2

) exp

(−

(yt − c −∑p

i=1 αiyt−i − δht)2

2(σ∗)2 exp(ht)

)I and the following transition densities

p(ht |ht−1) =1√

2πσηexp

(− (ht − φht−1)2

2σ2η

)


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The Growth-Volatility Relationship: New Evidence From ... · The Growth-Volatility Relationship: New Evidence From Stochastic Volatility in Mean Models Matthieu LEMOINE 1 Christophe