Robust open Bayesian analysis: Overfitting, model uncertainty, and endogeneity issues in multiple regression models

Abstract

The paper develops a computational method to deal with some open issues related to Bayesian model averaging for multiple linear models: overfitting, model uncertainty, endogeneity issues, and misspecified dynamics. The methodology takes the name of Robust Open Bayesian procedure. It is robust because the Bayesian inference is performed with a set of priors rather than a single prior and open because the model class is not fully known in advance, but rather is defined iteratively by MCMC algorithm. Conjugate informative priors are used to compute exact posterior probabilities. Empirical and simulated examples describe the functioning and performance of the procedure. Discussions with related works are also accounted for.

Keywords:

• Bayesian model averaging
• endogeneity issues
• MCMC integrations
• misspecified dynamics
• multiple linear regression

JEL CLASSIFICATION:

• A1
• C01
• E02
• H3
• N01
• O4

Acknowledgments

I gratefully thank the two anonymous referees for their useful suggestions.

Notes

1 The Occam’s window usually reports the null model or the simplest model solution including the null model as the only one to be selected. See, for instance, Raftery et al. (1997).

2 See Section 2.4 for more details.

3 See Section 4.

4 In Bayesian analysis, this term is usually taken to mean that the posterior probability concentrates on the true model.

5 See Section 4.

6 See, for instance, Kass and Raftery (1995).

7 It would be a generalization of the standard objective prior of Zellner (1986).

8 Similar frameworks, with appropriate Bayesian empirical specifications, have been used to make inference and obtain posterior distributions among time-varying macroeconomic-financial variables in multicountry panel data (see, e.g., Canova and Ciccarelli (2004, 2009), Canova et al. (2007, 2012), and Pacifico (2019a, 2019b)).

9 Here, a consistent estimator or asymptotically consistent estimator is an estimator having the property that, as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to $\mathcal{C}.$

10 See, e.g., Ehrlich (1973).

11 The result follows the studies of Becker (1968) and Stigler (1970), who argue that the decision to engage in criminal activity is a rational choice determined by its costs and benefits relative to some legitimate opportunities.

12 See, e.g., Vandaele (1987).

13 I use $\tau =1\%$ for an enough posterior consistency with predictors $\le 15.$

14 I use $\dot{\tau }=0.5\%$ for a sufficient prediction accuracy in explaining the data with predictors $\le 15.$

15 Own computations.

16 When accounting for BMA procedure in linear regression models with many predictors, penalized regression techniques are often used to deal with overfitting and select promising variables for predicting a variable of interest.

17 Spike-and-slab regression is a Bayesian variable selection technique particularly useful in high dimensional datasets when the number of possible predictors is larger than the number of observations. See, e.g., Mitchell and Beauchamp (1988), Raftery and Madigan (1994), and George and McCulloch (1997) for further implementations.

18 Here, statistical significance refers to the concept of the standard statistical hypothesis testing with a significance level setting to 5%.

19 In spike-and-slab regression, the top best covariates in explaining the data are determined by minimizing the test-set Mean Squared Error (MSE) for its entire solution-path. The average number of times a predictor is selected in this way defines a value recorded from $0\%-100\%$ denoting its stability.

20 Own computations.

21 See, e.g., Becker (1968), Stigler (1970), and Brier and Fienberg (1980).

22 Own computations.

23 See, e.g., Weisberg (2005).

24 They are: Italy (IT); Spain (ES); France (FR); Germany (DE); Greece (GR); and Portugal (PT).

25 The vec operator transforms a matrix into a vector by stacking the columns of the matrix, one underneath the other.

26 Here, ${\mathfrak{X}}_{\ddot{f}}$ would correspond to the auxiliary indicator $\chi =({\chi }_k)$ in matrix form.

27 See, for instance, Pacifico (2019b) for an illustration of the conformation of the time-varying multicountry SPBVAR model and the exact form of the ${\beta }_{\ddot{f}t}$‘s, ${\ddot{\chi }}_{\ddot{f}t}$‘s, and the ${\ddot{G}}_{\ddot{f}}$‘s.

28 The Wishart distribution is a multivariate extension of ${\chi }^2$ distribution and, in Bayesian statistics, corresponds to the conjugate prior of the inverse covariance-matrix of a multivariate normal random vector.

29 See, e.g., Chib and Greenberg (1995).

30 Own computations.

31 The convergence has been found by amounting to about 1.5 draws per regression parameter.

32 The result function would be $L(Y_T|M_{\xi })=\int \mathcal{E}(Y_T|{\ddot{\varphi }}_1,M_{\xi })p({\ddot{\varphi }}_1|M_{\xi })d{\ddot{\varphi }}_1,$ where $p({\ddot{\varphi }}_1|M_{\xi })$ is the mixture density priors for submodels $M_{\xi }.$

33 The conditional projection for output growth is the one that the model would have obtained over the same period conditionally on the actual path of unexpected dynamics for that period.

34 The unconditional projection is the one that the model would obtain for output growth for that period only on the basis of historical information, and it is consistent with a model-based forecast path for the other variables.

35 See, for instance, Pacifico (2019a, 2019b), Groen et al. (2013), Koop and Korobilis (2012), and Raftery et al. (2010) for interesting implementations when studying macroeconomic-financial linkages in multicountry setups.