bayesian model averaging in meta-analysis · introduction application conclusion bayesian model...

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


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



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



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.



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



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.



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)



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.



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.



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.



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



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




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


(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




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




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



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


http://www.meta-analysis.cz


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.”


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