Bayesian model averaging for dynamic panels with an application to a trade gravity model

ABSTRACT

We extend the Bayesian Model Averaging (BMA) framework to dynamic panel data models with endogenous regressors using a Limited Information Bayesian Model Averaging (LIBMA) methodology. Monte Carlo simulations confirm the asymptotic performance of our methodology both in BMA and selection, with high posterior inclusion probabilities for all relevant regressors, and parameter estimates very close to their true values. In addition, we illustrate the use of LIBMA by estimating a dynamic gravity model for bilateral trade. Once model uncertainty, dynamics, and endogeneity are accounted for, we find several factors that are robustly correlated with bilateral trade. We also find that applying methodologies that do not account for either dynamics or endogeneity (or both) results in different sets of robust determinants.

KEYWORDS:

• Bayesian model averaging
• dynamic panels
• generalized method of moments
• gravity model
• model uncertainty
• robustness

JEL CLASSIFICATION:

• C11
• C15
• C23
• C52

Acknowledgments

We are grateful to the Editor, Esfandiar Maasoumi, the Associate Editor, and the anonymous referee for their constructive comments. We also thank Michael Binder, Steven Durlauf, Ivan Jeliazkov, Eduardo Ley, Roberto Samaniego, and participants at the 2013 Meetings of the Royal Economic Society and the 2013 North American Summer Meetings of the Econometric Society for helpful comments and suggestions. We also benefited from comments and suggestions on earlier drafts of the paper presented at the 2011 North American Summer Meetings of the Econometric Society, the 2011 NBER/NSF Seminar on Bayesian Inference in Econometrics and Statistics, and the 2009 International Panel Data Conference at the University of Bonn. The views expressed in this study are those of the authors and should not be attributed to the International Monetary Fund, its Executive Board, or its management.

Notes

¹The BMA framework has been applied in various areas of social sciences, including economics. In the growth empirics literature, work includes Brock and Durlauf (2001), Fernández et al. (2001a), and Sala-i-Martin et al. (2004). Other applications include biology Yeung et al. (2005), ecology (Wintle et al., 2003), public health (Morales et al., 2006), and toxicology (Koop and Tole, 2004).

²Several articles in the literature use different justifications to estimate nonparametric model likelihood functions. The likelihood function in Kim (2002) is based on a given set of moment conditions using the I-projection theory while Tsangarides (2004) proposes a quasi likelihood function justified only through large sample properties. Schennach (2005) and Ragusa (2007) use a generalized empirical likelihood method to propose a class of semiparametric likelihoods.

³Moral-Benito (2012) assumes that the country-specific effects are linearly dependent on the means of the time-varying regressors and independent of the time-invariant covariates. The proposed likelihood function eliminates the bias associated with the fixed effects estimator in dynamic panels by adopting a correlated random effects approach.

⁴Alternatively, model priors may reflect the researcher's view about the number of regressors that should be included, with a penalty that increases proportionally with the number of regressors included in the model (see Mitchell and Beauchamp, 1988, and Sala-i-Martin et al., 2004). In that context, the prior probability for model M_j is , where K is the total number of regressors, is the researcher's prior about the size of the model, and k_j is the number of included variables in model M_j.

⁵For example, as discussed in Schennach (2005), conditional on the existence of a set of parameters for which all the moment conditions used in the efficient GMM objective function hold, the approach in Kim (2002) is asymptotically equivalent to that of Schennach (2005).

⁶Raftery (1999) provides a detailed discussion on why the choice of unit information prior is reasonable in the framework of Bayesian model averaging or selection. This choice of parameter prior has the additional benefit that, when combined with the uniform model prior, it generally outperforms various combination of priors in terms of cross-validated predictive performance (see Eicher et al., 2011).

⁷A consistent estimate of the weighting matrix is used to replace

⁸Under the assumption that the candidate model M_j is the true DGP, the moment conditions for any possible explanatory variable should hold regardless of whether the variable belongs to the candidate model M_j or not. This approach is in line with the model selection procedure proposed by Andrews and Lu (2001).

⁹The theoretical R² of the generated model varies between 0.50 and 0.70.

¹⁰Posterior density plots show that the distribution of the posterior probabilities of the true model shifts toward 1 (see Chen et al. (2011)).

¹¹Note that for values of α of 0.95 (0.50) and of 0.05 in Table 1, the mean posterior probability of the true model is 0.218 (0.140). At the same time, as shown in Table 2, the correct model is preferred over the next best with ratios of 1.591 and 1.039. This suggests that our methodology selects the correct model even in the case where the model may not have a lot of posterior mass.

¹²In addition, for N = 2, 000, the variance over the 1, 000 replications is very small across the board with values less than 9 × 10⁻⁴ in many cases. Box plots describing the distribution of the parameter estimates of Table 5 suggest that as the sample size increases, the variance of the distribution decreases and the median converges to the true value (see Chen et al., 2011).

¹³Online Appendix A and B and the associated tables are available on the following link https://sites.google.com/site/cgtsangarides

¹⁴We present results for α = 0.95, but results from additional simulations for α = 0.50 (available upon request) show broadly similar to those in Tables 6 and 7. The LIBMA columns in Tables 6 and 7 correspond to the inclusion probabilities in Table 4 and the parameter estimates in Table 5, respectively.

¹⁵We thank the anonymous referee for suggesting this approach.

¹⁶Anderson (1979) and Bergstrand (1985) provide theoretical foundations for the use of gravity models in bilateral trade. Empirical applications include, inter alia, Frankel and Rose (2002), Klein and Shambaugh (2006), and Santos Silva and Tenreyro (2006).

¹⁷Also, omitted variable bias may originate from the correlation of any pro-trade omitted variables such as the multilateral resistance terms in Anderson and van Wincoop (2003) with the regressors in the gravity equation. The multilateral resistance terms—the price indices P_i and P_j in Eq. (23)—capture the notion that trade between two countries does not only depend on the characteristics of the countries, but also on the barriers between them and the rest of the world.

¹⁸Recent work on theoretical foundations for a dynamic gravity model specification includes Cuñat and Maffezzoli (2007) and Olivero and Yotov (2012).

¹⁹Functional form uncertainty is a separate issue related to model specification uncertainty, which does not belong to the class of BMA methodologies. Functional form uncertainty arises when the relevant explanatory variables are known, but there is uncertainty about mathematical representation of the relation between the dependent variable and the explanatory variables. In that context, Ranjan and Tobias (2007) use Bayesian inference in an attempt to address uncertainty about the functional form representing the relation between contract enforcement and trade. They extend the gravity model by generalizing the basic threshold tobit model by including country specific effects and a covariate for contract enforcement (without specifically providing the functional form for the latter).

²⁰The direct and indirect pegs are defined using an algorithm to associate bilateral exchange rate relations with anchor currencies. See Qureshi and Tsangarides (2010) for further details.

²¹Most articles in the literature investigate gravity models using classical estimation methods thus ignoring model uncertainty. They are typically based on a different set of assumptions and structures. For example, Rose (2000) uses ordinary least squares (OLS) and fixed effects; De Nardis and Vicarelli (2003), Bun and Klaassens (2002), and Micco et al. (2003) use dynamic panels estimation and GMM. It is nevertheless instructive to compare our results with the established literature. For a more direct comparison, we focus on estimation methods meant to address model uncertainty based on model averaging methodologies.

²²The EHP estimation is performed using the BMA for linear regression models routine (bireg) by Raftery (1995) available in the open source R software http://www.r-project.org. The MB estimation is performed using the replication code provided in www.moralbenito.com.