Two-Stage Bayesian Model Averaging in Endogenous Variable Models

Abstract

Economic modeling in the presence of endogeneity is subject to model uncertainty at both the instrument and covariate level. We propose a Two-Stage Bayesian Model Averaging (2SBMA) methodology that extends the Two-Stage Least Squares (2SLS) estimator. By constructing a Two-Stage Unit Information Prior in the endogenous variable model, we are able to efficiently combine established methods for addressing model uncertainty in regression models with the classic technique of 2SLS. To assess the validity of instruments in the 2SBMA context, we develop Bayesian tests of the identification restriction that are based on model averaged posterior predictive p-values. A simulation study showed that 2SBMA has the ability to recover structure in both the instrument and covariate set, and substantially improves the sharpness of resulting coefficient estimates in comparison to 2SLS using the full specification in an automatic fashion. Due to the increased parsimony of the 2SBMA estimate, the Bayesian Sargan test had a power of 50% in detecting a violation of the exogeneity assumption, while the method based on 2SLS using the full specification had negligible power. We apply our approach to the problem of development accounting, and find support not only for institutions, but also for geography and integration as development determinants, once both model uncertainty and endogeneity have been jointly addressed.

Keywords:

• Bayesian Model Averaging
• Endogeneity
• Instrumental variables
• Posterior Predictive P-Values

JEL Classification:

• C52

ACKNOWLEDGMENT

We thank David Albouy and Francesco Trebbi for kindly sharing their data as well as Chris Papageorgiou for helpful comments. Lenkoski's research is supported by DFG research grant 1653. Raftery's research was supported by NIH Grants R01 HD-54511 and R01 GM-084163, NSF grant no. llS-0534094, and NSF grant no. ATM-0724721.

Notes

See, e.g., Fernández et al., 2001; Sala-I-Martin et al., 2004; Ciccone and Jarocinski, 2007, and Eicher et al., 2010;.

A similar heuristic panel approach is introduced by Hineline (2007) to examine the growth/inflation relationship. Morales-Benito (2009) provides statistical foundations for a panel BMA approach.

We use the term development accounting in the broad sense, referring to studies that seek to examine the determinants of differences in levels of per capita income. Previous development accounting approaches differ in their emphases, such as physical capital (King and Levine, 1994), human capital (Klenow and Rodrigues-Clare, 1997), as well as total factor productivity (TFP; Caselli, 2005). In our application we focus only on studies that sought to explain to differences in per capita incomes based on integration, institutions, and geography since the above approaches do not feature instrument uncertainty.

From a decision theory perspective, one can imagine cases where a policy-maker might be interested in a variable even when its posterior inclusion probability is below 50%; see Brock and Durlauf (2001) and Brock et al. (2003).

Our computations use the bicreg function from the BMA R package (Raftery et al., 2005), since the Bayesian Information Criteria (BIC) closely approximates the posterior model probability under the UIP Kass and Wasserman (1995)Raftery (1995). This allows for BIC approximations of the integrated likelihood in both the first and second stages. For a given value of θ⁽ⁱ⁾, the quantity pr(L _j |θ⁽ⁱ⁾, M _i , D) is given by a regression model with a particular UIP and is therefore well approximated by the BIC. In addition, we can approximate pr(L _j |M _i , D) with and calculate BIC relative to this . The conditional independence assumptions of (M _i , L _j ) enable us to set . Thus in our BIC approximation the likelihood is maximized at defined in Eq. (7). Note that BIC can be calculated from the associated R ² of the regression. By calculating the model probability ν _{i, j} conditional on the fitted value of W, we note that the resulting R ² is equivalent to using the “Generalized R ²” suggested by Pesaran and Smith (1994) for scoring the second stage model L _j .

RST report neither first stages nor tests of instrument restrictions beyond the core specification.