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Introduction Application Conclusion
Bayesian Model Averaging in Meta-AnalysisA Simple Application
Tomas Havranek
Czech National BankCharles University in Prague
MAER-Net Colloquium, 10 September 2015, Prague
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 1 / 17
Introduction Application Conclusion
What Is BMA?
• A method used to deal with model uncertainty• Many explanatory variables→ problems with model
selection• Sequential t-tests problematic (each conditional on the
previous one)• BMA runs many regressions with different subsets of the
explanatory variables• Weighted by goodness of fit and model complexity
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 2 / 17
Introduction Application Conclusion
How Does It Work in Practice?
• We can estimate many OLS regressions with differentsubsets and weigh them by adjusted R2
• Problem: usually billions of subsets• Bayesian methods make computation easier• Monte Carlo Markov Chain Algorithm typically used• Estimated in R or Matlab using the BMS package
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 3 / 17
Introduction Application Conclusion
Application: Elasticity of Intertemporal Substitution
The EIS reflects households’ willingness to substituteconsumption between time periods in response to changes inthe expected real interest rate.
u(c) =c1− 1
EIS − 11− 1
EIS
; if EIS = 1⇒ u(c) = log c.
Crucial in models involving intertemporal choice:
• monetary policy,• fiscal policy,• portfolio choice,• computing the social cost of carbon emissions, and more.
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 4 / 17
Introduction Application Conclusion
The Elasticity Varies Across Countries.
EIS ∈ [0.3, 0.5]
EIS ∈ (0.5, 0.7]
EIS ∈ [0.1, 0.3)
EIS < 0.1
no data
EIS > 0.7
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 5 / 17
Introduction Application Conclusion
What Explains the Differences?
Country-level variablesStock market participation: Euler equation valid for assetholders.GDP per capita: necessities hard to substitute across timeperiods.Credit availability: financial constraints hamper intertemporalsubstitution.Real interest: the elasticity does not have to be constant.Rule of law: institutions can affect financial decisions.
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 6 / 17
Introduction Application Conclusion
The Elasticity Varies Across Methods.
−5 0 5 10estimate of the EIS
Yogo (2004)Sarantis and Stewart (2003)
Sakuragawa and Hosono (2010)Rodriguez et al. (2002)
Pagano (2004)Osano and Inoue (1991)
Okubo (2011)Ogaki et al. (1996)
Noda and Sugiyama (2010)Nieh and Ho (2006)
Koedijk and Smant (1994)Kim and Ryou (2012)
Jimenez−Martin and deFrutos (2009)Ito and Noda (2012)
Ho (2004)Hamori (1996)
Fuse (2004)Chyi and Huang (1997)
Campbell and Mankiw (1991)Campbell (2003)Campbell (1999)
Bosca et al. (2006)
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 7 / 17
Introduction Application Conclusion
Variables Coded (1)
UtilityEpstein-Zin =1 if the estimation differentiates between the EIS and the
coefficient of relative risk aversion.Habits =1 if habits in consumption are assumed.Nonsep.durables
=1 if the model allows for nonseparability between durablesand nondurables.
Nonsep. public =1 if the model allows for nonseparability between private andpublic consumption.
Nonsep. trad-ables
=1 if the model allows for nonseparability between tradablesand nontradables.
DataNo. of house-holds
The logarithm of the number of cross-sectional units used inthe estimation (households, cohorts, countries).
No. of years The logarithm of the number of years of the data period usedin the estimation.
Average year The logarithm of the average year of the data period.Micro data =1 if the coefficient comes from a micro-level estimation.Annual data =1 if the data frequency is annual.Monthly data =1 if the data frequency is monthly.
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 8 / 17
Introduction Application Conclusion
Variables Coded (2)
DesignQuasipanel =1 if quasipanel (synthetic cohort) data are used.Inverse estima-tion
=1 if the rate of return is the dependent variable in the esti-mation.
Asset holders =1 if the estimate is related to the rich or asset holders.First lag instru-ment
=1 if the first lags of variables are included among instru-ments.
No year dum-mies
=1 if year dummies are omitted in micro studies using thePanel Study of Income Dynamics.
Income =1 if income is included in the specification.Taste shifters The logarithm of the number of controls for taste shifters.
Variable definitionTotal consump-tion
=1 if total consumption is used in the estimation.
Food =1 if food is used as a proxy for nondurables.Stock return =1 if the rate of return is measured as stock return.Capital return =1 if the rate of return is measured as the return on capital.
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 9 / 17
Introduction Application Conclusion
Variables Coded (3)
EstimationExact Euler =1 if the exact Euler equation is estimated.ML =1 if maximum likelihood methods are used for estimation.TSLS =1 if two-stage least squares are used for estimation.OLS =1 if ordinary least squares are used for estimation.
PublicationSE The reported standard error of the estimate of the EIS.Publication year The logarithm of the year of publication of the study.Citations The logarithm of the number of per-year citations of the study
in Google Scholar.Top journal =1 if the study was published in one of the top five journals in
economics.Impact The recursive RePEc impact factor of the outlet.
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 10 / 17
Introduction Application Conclusion
Bayesian Model AveragingModel Inclusion Based on Best 5000 Models
Cumulative Model Probabilities
0 0.05 0.1 0.15 0.19 0.24 0.29 0.34 0.38 0.43 0.48 0.53 0.57 0.62 0.66 0.7 0.75 0.79 0.84 0.88 0.93
Stock market partic. GDP per capita
Credit availability Real interest
Rule of law Inverse estimation
Top journalNo. of years
Total consumption Stock return
OLSCapital return
Citations Asset holders
Nonsep. durables Monthly data
Exact Euler Quasipanel Epstein-Zin
MLFoodTSLS
First lag instrument No. of households No year dummies
HabitsImpact
Nonsep. tradables Micro data
IncomeAnnual data
Publication year Taste shifters Average year
Nonsep. public
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 11 / 17
Introduction Application Conclusion
Posterior Coefficient Distributions (1)
(a) GDP per capita
−0.2 −0.1 0.0 0.1 0.2 0.3 0.4
01
23
45
Marginal Density: GDP_per_capita (PIP 100 %)
Coefficient
Den
sity
Cond. EV2x Cond. SDMedian
(b) Credit availability
−0.3 −0.2 −0.1 0.0 0.1
01
23
45
67
Marginal Density: Credit_availability (PIP 100 %)
Coefficient
Den
sity
Cond. EV2x Cond. SDMedian
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 12 / 17
Introduction Application Conclusion
Posterior Coefficient Distributions (2)
(c) Real interest
−0.03 −0.02 −0.01 0.00 0.01 0.02
010
2030
4050
Marginal Density: Real_interest (PIP 100 %)
Coefficient
Den
sity
Cond. EV2x Cond. SDMedian
(d) Rule of law
−0.4 −0.2 0.0 0.2
01
23
4
Marginal Density: Rule_of_law (PIP 100 %)
Coefficient
Den
sity
Cond. EV2x Cond. SDMedian
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 13 / 17
Introduction Application Conclusion
Stock Market Participation
0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Marginal Density: Stock_market_partic. (PIP 100 %)
Coefficient
Den
sity
Cond. EV2x Cond. SDMedian
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 14 / 17
Introduction Application Conclusion
Economic Significance
Stock market participation and GDP per capita affect theelasticity a lot:
Variable Maximum effect Std. dev. effect
Stock market partic. 0.931 0.141GDP per capita 0.683 0.088Credit availability -0.119 -0.020Real interest -0.265 -0.019Rule of law -0.087 -0.012
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 15 / 17
Introduction Application Conclusion
Summary
In a Nutshell. . .
1 Bayesian model averaging is useful in regressions withmany explanatory variables.
2 It can be thought of as a generalization of the typicalpractice of conducting robustness checks with differentsets of explanatory variables.
3 BMA is easy to use!
Project Websitewww.meta-analysis.cz
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 16 / 17
Introduction Application Conclusion
For Further Reading
Koop, G. (2003): Bayesian Econometrics.Wiley, 1st. edition.
Havranek, T., M. Rusnak (2013): Transmission Lags ofMonetary Policy: A Meta-Analysis.International Journal of Central Banking: 9(4): pp. 39–76.
Havranek, T., R. Horvath, Z. Irsova, & M. Rusnak (2015):Cross-Country Heterogeneity in Intertemporal Substitution.Journal of International Economics: 96(1): pp. 100–118.
Reading list on RePEc: Google “meta-analysis in economics.”
T. Havranek (CNB, CUNI) Bayesian Model Averaging MAER-Net, 10 Sep 2015 17 / 17
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