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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
Empirical resultsConclusion
Estimation Methodology
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.
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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)
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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)
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
Outline
Introduction
Models
Empirical results
Conclusion
Estimation Methodology
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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...
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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 )].
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
Outline
Introduction
Models
Empirical results
Conclusion
Estimation Methodology
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
Parameters estimates for the SV-M model
Legend: standard errors are written in italic and p-values at the 5% level are underlined.
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
Parameters estimates for the log-GARCH-M model
Legend: standard errors are written in italic and p-values at the 5% level are underlined.
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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).
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
Outline
Introduction
Models
Empirical results
Conclusion
Estimation Methodology
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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)
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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.
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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)
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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∗.
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship
IntroductionModels
Empirical resultsConclusion
Estimation Methodology
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
)
Matthieu LEMOINE , Christophe MOUGIN The Growth-Volatility Relationship