badp: Bayesian model averaging for dynamic panels with weakly exogenous regressors

Model setup

Moral-Benito (2016) considers the following model specification:

\[y_{it}=\alpha y_{it-1}+x'_{it}\beta+\eta_{i}+\zeta_{t}+v_{it} \tag{1}\]

where \(y_{it}\) is the dependent variable, \(i\) \((=1,...,N)\) indexes entity (ex. country), \(t\) \((=1,...,T)\) indexes time, \(x_{it}\) is a \(k \times 1\) vector of growth determinants, \(\beta\) is the corresponding parameter vector, \(\eta_{i}\) is an entity-specific fixed effect, \(\zeta_{t}\) is a period-specific shock and \(v_{it}\) is a shock to the dependent variable. To address the issue of reverse causality the model is built on the assumption of weak exogeneity, that can be formalized as

\[\mathbb{E}(v_{i,t}|y^{t-1}_{t},x^{t}_{i},\eta_{i})=0 \tag{2}\]

where \(y^{t-1}_{t}=(y_{i,0},...,y_{i,t-1})'\) and \(x^t_{i}=(x_{i,0},...,x_{i,t})'\). Accordingly, weak exogeneity implies that the current values of the regressors, lagged dependent variable, and fixed effects are uncorrelated with the current shocks, while they are all allowed to be correlated with each other at the same time. On the assumption of weakly exogenous regressors, Moral-Benito (2013) augmented equation (1) with additional reduced-form equations capturing the unrestricted feedback process:

\[x_{it}=\gamma_{t0}y_{i0}+...+\gamma_{tt-1}y_{it-1}+\Lambda_{t1}x_{i1}+...+\Lambda_{tt-1}x_{it-1}+c_{t}\eta_{i}+\vartheta_{it} \tag{3}\]

where \(t=2,\dots ,T;\) \(c_{t}\) is the \(k\times 1\) vector of parameters. For \(h<t\), \(\gamma_{th}\) is a \(k\times 1\) vector \((y_{th}^{1},\dots,y_{th}^{k})'\) \(h=0,\dots,T-1\); \(\Lambda_{th}\) is a \(k\times k\) matrix of parameters, and \(\vartheta_{it}\) is a \(k\times 1\) vector of prediction errors. The initial observations are defined with

\[y_{i0}=c_{0}\eta_{i}+\upsilon_{it} \tag{4}\]

\[x_{i1}=\gamma_{10}y_{i0}+c_{1}\eta_{i}+\vartheta_{it} \tag{5}\]

where \(c_{0}\) is a scalar, \(c_{1}\) and \(\gamma_{10}\) are \(k\times 1\) vectors and \(\eta_{i}\) are the individual effects. The mean vector and the covariance matrix of the joint distribution of the initial observations and the individual effects are unrestricted. (The method outperforms the Arellano–Bond estimator, see Moral-Benito et al. 2019.)

For the model setup given in equations (1) and (3)-(5), Moral-Benito (2013) derived the log-likelihood function:

\[\log f(data|\theta) \propto -\frac{N}{2}\log\det(B^{-1}D\Sigma D'B'^{-1})-\frac{1}{2}\sum_{i=1}^{N}\{R'_{i}(B^{-1}D\Sigma D'B'^{-1})^{-1}R_{i}\} \tag{6}\]

where \(\theta\) denotes parameters to be estimated, \(R_{i}=(y_{io},x'_{i1},y_{i1},\dots,x'_{iT},y_{iT})'\) are vectors of observed variables, and \(\Sigma=diag[\sigma^{2}_{\eta},\sigma^{2}_{\upsilon_{0}},\Sigma_{\vartheta_{1}},\sigma^{2}_{\upsilon_{1}},...,\Sigma_{\vartheta_{T}},\sigma^{2}_{\upsilon_{T}}]\) is the block-diagonal variance-covariance matrix. Matrix B is given by:

\[B=\begin{bmatrix} 1&0&0&0&0&\dotsc& 0&0&0\\ -\gamma_{10}&I_{k}&0&0&0&\dotsc&0&0&0\\ -\alpha&-\beta'&1&0&0&\dotsc&0&0&0\\ -\gamma_{20}&-\Lambda_{21}&-\gamma_{21}&I_{k}&0&\dotsc&0&0&0\\ 0&0&-\alpha&-\beta'&1&\dotsc&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&0&0&0\\ -\gamma_{T0}&-\Lambda_{T1}&-\gamma_{T1}&-\Lambda_{T2}&-\gamma_{T2}&\dotsc&-\gamma_{TT-1}&I_{k}&0\\ 0&0&0&0&0&\dotsc&-\alpha&-\beta'&1\\ \end{bmatrix} \tag{7}\]

and matrix D is given by:

\[D=\begin{bmatrix} (c_{0}&c'_{1}&1&c'_{2}&1&\dotsc&c'_{T}&1)' & I_{T(k+1) + 1} \end{bmatrix}. \tag{8}\]

The model setup in equations (1) and (3)-(5) requires that in addition to the parameters of interest \(\alpha\) and \(\beta\), the parameters \(\gamma_{ij}\) and \(\Lambda_{km}\) need to be estimated. To make the optimization of likelihood computationally feasible, Moral-Benito (2013) developed Simultaneous Equations Model (SEM) setup where the nuisance parameters are incorporated in the variance-covariance matrix. In the SEM setup, the model is defined by \(1 + (T - 1)k + T\) equations:

\[\begin{cases} \eta_i = \phi_i y_{i0} + x'_{i1} \phi_1 + \epsilon_i & \\ x_{it} = \pi_{t0} y_{i0} + \pi_{t1} x_{i1} + \pi^w_t x_{i1} + \xi_{it}, & t = 2, ..., T \\ y_{it} = \alpha y_{it-1} + x'_{it} \beta + \phi_0 y_{i0} + x'_{i1} \phi_1 + w'_i \delta + \epsilon_i + v_{it}, & t = 1, ..., T \end{cases} \tag{9}\]

This setup can be rewritten in a matrix form:

\[B R_i = C z_i + U_i, \tag{10}\]

where:

\[z_i = [y_{i0}, x'_{i1}, w'_i]' \tag{11}\]

is the vector of strictly exogenous variables,

\[R_i = [y_{i1}, y_{i2}, ..., y_{iT}, x'_{i2}, x'_{i3}, ..., x'_{iT}]', \tag{12}\]

\[U_i = [\epsilon_i + v_{i1}, \epsilon_i + v_{i2}, ..., \epsilon_i + v_{iT}, \xi'_{i2}, \xi'_{i3}, ..., \xi'_{iT}]' \tag{13}\]

and matrices \(B\) and \(C\) contain coefficients \(\alpha\), \(\beta\), \(\phi_0\), \(\phi_1\). Since these matrices are not connected to the error, we simply note that they are defined in such a way that the equation (10) is equivalent to the SEM setup. The main difference of the SEM setup is that equations for \(x_{it}\) now depend only on \(y_{i0}\) and \(x_{i1}\) and not on \(y_{is}\) and \(x_{is}\) for other periods \(s\).

Following Moral-Benito (2013), and writing \(\ell(\theta) = \log L(\theta)\) for the log-likelihood throughout, we can then define the log-likelihood function as:

\[\ell(\theta) \propto - \frac{N}{2} \log \det \Omega(\theta) - \frac{1}{2} tr \{ \Omega(\theta)^{-1} (R - Z \Pi(\theta))' (R - Z \Pi(\theta)) \} \tag{14}\]

where \(R\) and \(Z\) are matrices containing vectors \(R_i\) and \(z_i\) respectively and:

\[\Pi(\theta) = B^{-1} C \tag{15}\]

\[U^*_i(\theta) = B^{-1} U_i \tag{16}\]

\[\Omega(\theta) = Var(U^*_i) = B^{-1} \cdot Var(U_i) \cdot B'^{-1} = B^{-1} \Sigma B'^{-1} \tag{17}\]

It is possible to obtain analytical solutions for the MLEs of some parameters. Then the formula for the log-likelihood function can be simplified to:

\[\ell(\theta) \propto - \frac{N}{2} \log \det \Sigma_{11} - \frac{1}{2} tr \{ \Sigma_{11}^{-1} U_1' U_1 \} - \frac{N}{2} \log \det (\frac{H}{N}) \tag{18}\]

where \(U_1\) is a matrix of errors connected only to dependent variables, \(\Sigma_{11}\) is a part of the \(\Sigma\) matrix:

\[\Sigma=var(U_{i})=var\left(\frac{U_{i1}}{U_{i2}}\right)=\begin{bmatrix} \Sigma_{11}&\Sigma_{12}\\ \Sigma_{21}&\Sigma_{22}\\ \end{bmatrix}. \tag{19}\]

and \(H = (R_2 + U_1 F_{12})' Q (R_2 + U_1 F_{12})\) with \(R_2\) being a matrix of regressor vectors \([x'_{i2}, x'_{i3}, ..., x'_{iT}]\) and \(F_{12} = - \Sigma_{11}^{-1} \Sigma_{12}\).

As shown in Moral-Benito (2013), the calculation of the likelihood function requires only the knowledge of \(\Sigma_{11}\) and \(\Sigma_{12}\). \(\Sigma_{11}=\sigma^{2}_{\epsilon}\iota\iota'+diag\{\sigma^{2}_{\upsilon 1},\dots,\sigma^{2}_{\upsilon T}\}\) is a classical error-component, where \(\iota\) denotes \(T\times1\) vector of ones. Finally, the \(T\times (T-1)k\) matrix capturing the feedback process is given by:

\[\Sigma_{12}=\begin{bmatrix} \phi'_{2}+\psi'_{21}&\phi'_{3}+\psi'_{31}&\dots&\phi'_{T}+\psi'_{T1}\\ \phi'_{2}&\phi'_{3}+\psi'_{32}&\dots&\phi'_{T}+\psi'_{T2}\\ \phi'_{2}&\phi'_{3}&\dots&\phi'_{T}+\psi'_{T3}\\ \vdots&\vdots&\ddots&\vdots\\ \phi'_{2}&\phi'_{3}&\dots&\phi'_{T}+\psi'_{T,T-1}\\ \phi'_{2}&\phi'_{3}&\dots&\phi'_{T}\\ \end{bmatrix} \tag{20}\]

where:

\[cov(\epsilon_{t},\xi_{it})=\phi_{t} \tag{21}\]

\[cov(v_{ih},\xi_{it})=\begin{cases} \psi_{ht} \text{ if } h<t\\ \mathbf{0} \text{ otherwise}\\ \end{cases} \tag{22}\]

and \(\phi_{t}\), \(\psi_{th}\), and \(\mathbf{0}\) are \(k\times1\) vectors.

Within the BMA approach developed by Moral-Benito (2016), the matrix \(\Sigma_{12}\) is computed for each model using all considered regressors. This framework treats each estimated model as nested within the model that includes all covariates. Consequently, the matrix \(R\) contains regressors that are excluded from the model being estimated.

However, the package also allows the user to employ a non-nested approach, i.e., an approach that utilizes the \(\Sigma_{12}\) matrix and the matrix \(R\) constructed only from the regressors included in the model being estimated. This approach has two advantages. First, the optimization problem is simpler and therefore less error-prone, as the number of parameters to be estimated is smaller. In addition, it substantially improves the speed of maximum likelihood estimation. Second, this approach may be more appropriate in cases where some regressors are unrelated to others, as using the complete set of regressors may lead to overspecification. The implementation of the non-nested approach is described in Section Estimation of the model space.

Bayesian model averaging

Given the likelihood function in (18), henceforth denoted as \(L(\text{data}|\theta_{i}, M_{i})\) for a specific model \(i\), it is possible to utilize Bayesian model averaging (for an introduction to BMA see Raftery 1995; Raftery et al. 1997; Kass and Raftery 1995; Doppelhofer and Weeks 2009; Amini and Parmeter 2011; Beck 2017; Fragoso et al. 2018) (BMA). To achieve that, we first estimate all possible variants of equation:

\[Y=f(X,\theta,v) \tag{23}\]

where \(Y\) is a vector of dependent variable, \(X\) is a matrix of potential determinants, \(\theta\) is a parameter vector, and \(v\) is a stochastic term. All the variants include a lagged dependent variable; therefore, with \(K\) regressors, there are \(2^{K}\) possible models that can be estimated. Each of these models can be assigned a posterior model probability; however, the marginal (integrated) likelihood, \(L(\text{data}|M_{i})\), must first be computed. Moral-Benito (2012) utilizes approach developed by Raftery (1995) and Sala-I-Martin et al. (2004) based on the Bayesian information criterion (BIC) approximation.

The Bayes factor for models \(M_{i}\) and \(M_{j}\), \(B_{ij}=\frac{L(\text{data}|M_{i})}{L(\text{data}|M_{j})}\), can be approximated using Schwartz criterion:

\[S=\log L(\text{data} \mid\hat{\theta_{i}},M_{i})-\log L(\text{data}|\hat{\theta_{j}},M_{j})-\frac{k_{i}-k_{j}}{2}\log (N) \tag{24}\]

where \(L(\text{data}|\hat{\theta}_{i}, M_{i})\) and \(L(\text{data}|\hat{\theta}_{j}, M_{j})\) are the maximum likelihood values for models \(i\) and \(j\), respectively. The terms \(k_{i}\) and \(k_{j}\) denote the number of regressors in models \(i\) and \(j\). Bayesian information criterion is given by:

\[BIC=-2S=-2\log B_{ij}. \tag{25}\]

Given null model \(M_{0}\)

\[B_{ij}=\frac{L(y|M_{i})}{L(y|M_{j})}=\frac{\frac{L(y|M_{i})}{L(y|M_{0})}}{\frac{L(y|M_{j})}{L(y|M_{0})}}=\frac{B_{i0}}{B_{j0}}=\frac{B_{0j}}{B_{0i}} \tag{26}\]

and

\[2\log B_{ij}=2[\log B_{0j} - \log B_{0i}]=BIC_{j}-BIC_{i}. \tag{27}\]

The posterior model probability (PMP) of model \(j\) given the data is

\[\mathbb{P}(M_{j}|y)=\frac{L(\text{data}|M_{j})\mathbb{P}(M_{j})}{\sum_{i=1}^{2^K}L(\text{data}|M_{i})\mathbb{P}(M_{i})} \tag{28}\]

where \(\mathbb{P}(M_{j})\) denotes prior model probability. In other words, the PMP represents the share of model \(j\) in the total posterior probability mass. Combining equations (25)-(28) we get:

\[\mathbb{P}(M_{j}|y)=\frac{L(\text{data}|M_{j})\mathbb{P}(M_{j})}{\sum_{i=1}^{2^K}L(\text{data}|M_{i})\mathbb{P}(M_{i})} =\frac{B_{j0}\mathbb{P}(M_{j})}{\sum_{i=1}^{2^K}B_{i0}\mathbb{P}(M_{i})} \tag{29}\]

Finally, using the result that

\[B_{j0}=\exp{(-\frac{1}{2}BIC_{j})} \tag{30}\]

we can calculate posterior model probability as

\[\mathbb{P}(M_{j}|\text{data})=\frac{\exp{(-\frac{1}{2}BIC_{j})}\mathbb{P}(M_{j})}{\sum_{i=1}^{2^K}\exp{(-\frac{1}{2}BIC_{i})} \mathbb{P}(M_{i})}. \tag{31}\]

BMA statistics

With PMPs, we can calculate useful BMA statistics. Let’s denote by \(\pi_k\) the random variable which is equal to one if the \(k^{th}\) regressor should be considered as the determinant of the dependent variable. The posterior inclusion probability (PIP) for the regressor is given by:

\[\mathbb{P}(\pi_k = 1 |\text{data}) = \sum_{j=1}^{2^K} \mathbf{1} (k^{th} \text{ regressor is in model } M_{j}) \cdot \mathbb{P}(M_{j}|\text{data}) \tag{32}\]

where the indicator function \(\mathbf{1}\) is equal to one if the regressor is part of the model \(M_j\) and zero otherwise. In other words, the PIP tells us how likely it is that the given regressor has impact on the variable of interest.

Another interesting statistic is the posterior mean (PM) of a given parameter \(\beta_k\). Let’s denote by \(\pi_{\beta}\) the random variable which is equal to one if the given parameter is present in the model, and zero otherwise. The posterior mean of \(\beta\) is given by:

\[\mathbb{E}(\beta_k|\text{data})=\sum_{j=1}^{2^K}\widehat{\beta_k}_{j} \cdot \mathbb{P}(M_{j}, \pi_{\beta_k} = 1 |\text{data}) \tag{33}\]

where \(\widehat{\beta_k}_{j}\) is the value of the coefficient \(\beta_k\) in model \(j\). It tells us what is the mean (or expected) value for the parameter taking into account all considered models. Note that in equation (33) the value assigned to \(\widehat{\beta_k}_{j}\) for models in which \(\beta_k\) is not present is immaterial, because the probability \(\mathbb{P}(M_{j}, \pi_{\beta_k} = 1 |\text{data})\) is zero for those models anyway. In the posterior variance below this is no longer the case, and the coefficient of an excluded regressor must be set to zero.

The posterior variance of the parameter \(\beta_k\) is equal to:

\[Var(\beta_k|\text{data}) = \sum_{j=1}^{2^K}Var(\beta_{k,j}|\text{data},M_{j}) \cdot \mathbb{P}(M_{j}, \pi_{\beta_k} = 1 |\text{data}) + \sum_{j=1}^{2^K}\left[\widehat{\beta}_{k,j}-\mathbb{E}(\beta_k|\text{data})\right]^{2} \cdot \mathbb{P}(M_{j}|\text{data}) \tag{34}\]

where \(Var(\beta_{k,j}|\text{data},M_{j})\) denotes the conditional variance of the coefficient \(\beta_k\) in model \(M_{j}\) (in other words assuming that the model \(M_j\) is the true model), and where \(\widehat{\beta}_{k,j} \equiv 0\) whenever \(\beta_k\) is not included in model \(M_{j}\). The first sum is the expected within-model variance and the second is the between-model variance of the coefficient estimates. Note the difference between the weights in the two sums. Models that exclude \(\beta_k\) contribute nothing to the first sum, because the coefficient is known to be zero in those models and its conditional variance vanishes. They do, however, contribute \([0-\mathbb{E}(\beta_k|\text{data})]^{2}\,\mathbb{P}(M_{j}|\text{data})\) to the second sum: exclusion is treated as the point mass \(\beta_k = 0\), and the distance between that point mass and the posterior mean is a genuine source of posterior uncertainty. Weighting the second sum by \(\mathbb{P}(M_{j}, \pi_{\beta_k} = 1 |\text{data})\) instead would omit these terms and understate the posterior variance, the more so the lower the posterior inclusion probability of the regressor. Posterior standard deviation (PSD) of \(\beta_k\) is then defined as the square root of the variance:

\[SD(\beta_k | \text{data}) = \sqrt{ Var(\beta_k | \text{data}) } \tag{35}\]

Alternatively, one might be interested in the values of the mean and variance on the condition of inclusion of a given parameter, i.e. assuming that it is definitely a part of the model. Note that this is usually determined by the presence of a related regressor. The conditional posterior mean (PMcon) for a parameter \(\beta_k\) is given by:

\[\mathbb{E}(\beta_k | \pi_{\beta_k}=1,\text{data})=\frac{\mathbb{E}(\beta_k|\text{data})}{\mathbb{P}(\pi_{\beta_k} = 1|\text{data})}. \tag{36}\]

Similarly, the conditional variance is:

\[Var(\beta_k|\pi_{\beta_k}=1,\text{data})=\frac{Var(\beta_k|\text{data})+\mathbb{E}(\beta_k|\text{data})^2}{\mathbb{P}(\pi_{\beta_k}=1|\text{data})}-\mathbb{E}(\beta_k|\pi_{\beta_k}=1,\text{data})^2 \tag{37}\]

and so the conditional standard deviation (PSDcon) is:

\[SD(\beta_k | \pi_{k}=1,\text{data}) = \sqrt{ Var(\beta_k | \pi_{k}=1,\text{data}) } \tag{38}\]

The BMA statistics allow the assessment of the robustness of the examined regressors. Raftery (1995) classifies a variable as weak, positive, strong, and very strong when the posterior inclusion probability (PIP) is between 0.5 and 0.75, between 0.75 and 0.95, between 0.95 and 0.99, and above 0.99, respectively. Raftery (1995) also refers to the variable as robust when the absolute value of the ratio of posterior mean (PM) to posterior standard deviation (PSD) is above 1, indicating that the regressor improves the power of the regression. Masanjala and Papageorgiou (2008) propose a more stringent criterion, where they require the statistic to be higher than 1.3, while Sala-I-Martin et al. (2004) argue for 2, corresponding to \(90\%\) and \(95\%\), respectively.

Model priors and jointness

To perform BMA one needs to specify prior model probability (for a thorough discussion of model priors see Sala-I-Martin et al. 2004; Ley and Steel 2009; George 2010; Eicher et al. 2011). The package offers two main options. The first is binomial model prior (Sala-I-Martin et al. 2004):

\[\mathbb{P}(M_{j})=\left(\frac{EMS}{K}\right)^{k_{j}}\left(1-\frac{EMS}{K}\right)^{K-k_{j}} \tag{39}\]

where \(EMS\) is the expected model size and \(k_{j}\) is a number of regressors in model \(j\). If \(EMS = \frac{K}{2}\), the binomial model prior simplifies to a uniform model prior with \(\mathbb{P}(M_{j}) = \frac{1}{2^K}\) for every \(j\), meaning that all models are assumed to have equal probabilities. The second is binomial-beta model prior (Ley and Steel 2009) given by:

\[\mathbb{P}(M_{j}) \propto \Gamma(1+k_{j}) \cdot \Gamma\left(\frac{K-EMS}{EMS}+K-k_{j}\right). \tag{40}\]

where \(\Gamma\) is the gamma function. In the context of the binomial-beta prior \(EMS = \frac{K}{2}\) corresponds to equal probabilities on model sizes.

In order to account for potential multicollinearity between regressors one can use dilution prior introduced by George (2010). The dilution prior involves augmenting the model prior (binomial or binomial-beta) with a function that accounts for multicollinearity:

\[\mathbb{P}_{D}(M_{j}) \propto \mathbb{P}(M_{j})|COR_{j}|^{\omega} \tag{41}\]

where \(\mathbb{P}_{D}(M_{j})\) is the diluted model prior, \(|COR_{j}|\) is the determinant of the correlation matrix of regressors in model \(j\), and \(\omega\) is the dilution parameter. The lower the correlation between regressors, the closer \(|COR_{j}|\) is to one, resulting in a smaller degree of dilution.

To determine whether regressors are substitutes or complements, various authors have developed jointness measures (to learn more about jointness measures, we recommend reading Doppelhofer and Weeks 2009; Ley and Steel 2007; Hofmarcher et al. 2018 in that order). Assuming two different covariates \(a\) and \(b\), let \(\mathbb{P}(a\cap b)\) be the posterior probability of the inclusion of both variables, \(\mathbb{P}(\overline{a}\cap \overline{b})\) the posterior probability of the exclusion of both variables, \(\mathbb{P}(\overline{a}\cap b)\) and \(\mathbb{P}(a\cap \overline{b})\) denote the posterior probability of including each variable separately. The first measure of jointness is simply \(\mathbb{P}(a\cap b)\). However, this measure ignores much of the information about the relationships between the regressors. Doppelhofer and Weeks (2009) measure is defined as:

\[J_{DW}=\text{log}\left[\frac{\mathbb{P}(a\cap b) \cdot \mathbb{P}(\overline{a}\cap \overline{b})}{\mathbb{P}(\overline{a}\cap b) \cdot \mathbb{P}(a\cap \overline{b})}\right]. \tag{42}\]

If \(J_{DW} < -2\), \(-2 < J_{DW} < -1\), \(-1 < J_{DW} < 1\), \(1 < J_{DW} < 2\), and \(J_{DW} > 2\), the authors classify the regressors as strong substitutes, significant substitutes, not significantly related, significant complements, and strong complements, respectively. Jointness measure proposed by Ley and Steel (2007) is given by:

\[J_{LS}=\frac{\mathbb{P}(a\cap b)}{\mathbb{P}(\overline{a}\cap b)+\mathbb{P}(a\cap \overline{b})}. \tag{43}\]

The measure takes values in the range \([0, \infty)\), with higher values indicating a stronger complementary relationship. Finally, Hofmarcher et al. (2018) measure of jointness is:

\[J_{HCGHM}=\frac{(\mathbb{P}(a\cap b)+\rho) \cdot \mathbb{P}(\overline{a}\cap \overline{b})+\rho)-(\mathbb{P}(\overline{a}\cap b)+\rho) \cdot \mathbb{P}(a\cap \overline{b})+\rho)}{(\mathbb{P}(a\cap b)+\rho) \cdot \mathbb{P}(\overline{a}\cap \overline{b})+\rho)+(\mathbb{P}(\overline{a}\cap b)+\rho) \cdot \mathbb{P}(a\cap \overline{b})+\rho)+\rho}. \tag{44}\]

Hofmarcher et al. (2018) advocate the use of the Jeffreys (1946) prior, which results in \(\rho=\frac{1}{2}\). The measure takes values from -1 to 1, where values close to -1 indicate substitutes, and those close to 1 complements.

Model space parameters

The first element of the list contains the estimated MLEs of the parameters for each considered model. Each column represents a single model, and rows correspond to the parameters. For example, to display the first 10 parameters for 5 models we can call:

full_model_space$params[1:10, 1:5]

##                  [,1]        [,2]        [,3]        [,4]        [,5]
## alpha      1.06606411  1.03313490  1.08838653  1.05826552  1.05283625
## phi_0     -0.05662058 -0.05538199 -0.10945004 -0.09877528 -0.06439057
## err_var    0.08915495  0.05254540  0.07600435  0.03944992  0.08031349
## dep_var_1  0.19712536  0.16637122  0.19715552  0.16940538  0.19274306
## dep_var_2  0.17880573  0.18696579  0.18038940  0.18922278  0.17934814
## dep_var_3  0.16883232  0.17030650  0.17068787  0.17104659  0.16957551
## dep_var_4  0.16472270  0.16983566  0.16771000  0.17241047  0.16632818
## beta_ish           NA  0.11973691          NA  0.12212375          NA
## beta_sed           NA          NA -0.02733797 -0.02145665          NA
## beta_pgrw          NA          NA          NA          NA  0.00956287

NA value means that the corresponding parameter is not present in the given model.

Model space statistics

The second element of the list provides:

• the values of the likelihood function at the estimated MLE (first row),
• the Bayesian information criterion (second row),
• and the standard errors for the parameters describing linear relations, i.e. the beta parameters from equation (9), for each model (remaining rows).

full_model_space$stats[, 1:5]

##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,] -4.051184e+02 -318.60696994 -3.331794e+02 -241.94845848 -3.365417e+02
##  [2,]  3.740825e-03    0.01176951  9.639695e-03    0.03235431  9.205777e-03
##  [3,]  8.522610e-02    0.08377425  1.009219e-01    0.10180894  8.072344e-02
##  [4,]  0.000000e+00    0.03077398  0.000000e+00    0.03193838  0.000000e+00
##  [5,]  0.000000e+00    0.00000000  6.839816e-02    0.07235671  0.000000e+00
##  [6,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  3.708785e-02
##  [7,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
##  [8,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
##  [9,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [10,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [11,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [12,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [13,]  7.242499e-02    0.14729930  1.446555e-01    0.23261588  7.954644e-02
## [14,]  0.000000e+00    0.08702753  0.000000e+00    0.08971462  0.000000e+00
## [15,]  0.000000e+00    0.00000000  1.252883e-01    0.14627880  0.000000e+00
## [16,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  6.476474e-02
## [17,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [18,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [19,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [20,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [21,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [22,]  0.000000e+00    0.00000000  0.000000e+00    0.00000000  0.000000e+00
## [23,]  8.271174e+00   21.06161495  1.215665e+01   25.23327780  1.227343e+01
## [24,]  8.800000e+01   90.00000000  9.000000e+01   92.00000000  9.000000e+01
## [25,]  7.000000e+00    9.00000000  9.000000e+00   11.00000000  9.000000e+00

Again, each column represents a single considered model.

Two types of standard errors are provided, both derived from the Hessian of the maximized log-likelihood function. The first type consists of the regular standard errors, calculated using the inverse of the observed information matrix:

\[I(\hat{\theta}) = -\frac{\partial^2 \ell(\hat{\theta})}{\partial \theta \partial \theta'} \tag{45}\]

where \(\hat{\theta}\) are the estimated MLE parameters, \(I(\hat{\theta})\) is the information matrix and \(\ell(\hat{\theta}) = \log L(\hat{\theta})\) is the log-likelihood. The variance covariance matrix is given by:

\[Var(\hat{\theta}) = I(\hat{\theta})^{-1}, \tag{46}\]

and the standard errors by

\[\text{SE}(\hat{\theta}) = \sqrt{\text{diag}(Var(\hat{\theta}))}. \tag{47}\]

where the square root is obviously applied separately to each coordinate of the vector with diagonal values. The second type are the robust standard errors or heteroscedasticity consistent standard errors. To understand how they work, we first have to rewrite the equation (18) in a form which will display the contribution of each entity on the likelihood value. First note that:

\[\ell(\theta) \propto - \frac{1}{2} tr \{ \Sigma_{11}^{-1} U_1' U_1 \} - \sum_{i=1}^{N} \frac{1}{2} \left( \log \det \Sigma_{11} + \log \det (\frac{H}{N}) \right) \tag{48}\]

Now, because of the cyclic property of the trace we can rewrite the first term as:

\[- \frac{1}{2} tr \{ \Sigma_{11}^{-1} U_1' U_1 \} = - \frac{1}{2} tr \{ U_1 \Sigma_{11}^{-1} U_1' \} = - \frac{1}{2} \sum_{i=1}^N u_i \Sigma_{11}^{-1} u_i' \tag{49}\]

where \(u_i\) is a row vector corresponding to the data relating to the single entity \(i\). Hence, the entire likelihood function can be rewritten as a sum of contributions from each entity:

\[\ell(\theta) \propto \sum_{i=1}^{N} -\frac{1}{2} (\log \det \Sigma_{11} + \log \det (\frac{H}{N}) + u_i \Sigma_{11}^{-1} u_i') \tag{50}\]

From there we can see that the contribution of a single entity \(i\) is:

\[\ell_i(\theta) \propto -\frac{1}{2} (\log \det \Sigma_{11} + \log \det (\frac{H}{N}) + u_i \Sigma_{11}^{-1} u_i'). \tag{51}\]

Now if we consider a multivariate function \(\boldsymbol{\ell}(\theta)\) of all such contributions (with single contributions as its coordinates), we can find its gradient at the MLE: \(G(\hat{\theta}) = \frac{\partial \boldsymbol{\ell}(\hat{\theta})}{\partial \theta}\). Then the robust variance is:

\[Var_{R}(\hat{\theta}) = I(\hat{\theta})^{-1} \cdot G'(\hat{\theta}) G(\hat{\theta}) \cdot I(\hat{\theta})^{-1} \tag{52}\]

and the robust standard errors are given by:

\[\text{SE}_{R}(\hat{\theta}) = \sqrt{\text{diag}(Var_{R}(\hat{\theta}))}. \tag{53}\]

where the square root is again applied to each coordinate separately.

The structure of equation (51) has a consequence worth noting. Splitting it as \(\ell_i(\theta) = c(\theta) + b_i(\theta)\), with \(c = -\frac{1}{2}(\log \det \Sigma_{11} + \log \det (H/N))\) the same for every entity and \(b_i = -\frac{1}{2} u_i \Sigma_{11}^{-1} u_i'\) specific to entity \(i\), the scores are \(s_i = c' + b_i'\). They sum to zero at the maximum, so \(\sum_i b_i' = -N c'\) and

\[G'(\hat{\theta}) G(\hat{\theta}) = \sum_{i=1}^{N} s_i s_i' = \sum_{i=1}^{N} (b_i' - \bar{b}')(b_i' - \bar{b}')' \tag{54}\]

The common component cancels: the matrix depends only on how the scores vary across entities. Any parameter entering the likelihood only through \(c(\theta)\) therefore contributes nothing, and \(G'(\hat{\theta}) G(\hat{\theta})\) is rank deficient however many entities are observed. In the model spaces bundled with the package the scores span between 8% and 29% of the parameter directions; summary() reports the proportion for a fitted model space.

Because the matrix vanishes outside the directions it spans, the sandwich restricted to that block equals the profile variance obtained by profiling the remaining parameters out, so the reported robust standard errors, those of \(\alpha\) and the \(\beta\) parameters, are well defined. What the construction discards is the score covariance involving the unspanned directions, which is set to zero rather than estimated. The standard errors based on \(I(\hat{\theta})\) alone do not involve \(G'(\hat{\theta}) G(\hat{\theta})\) and are unaffected; they rest instead on the likelihood being correctly specified.

Parallelization

The optim_model_space function is the most computationally intensive part of the package. Therefore, the function provides an option for parallel computing. If the user’s data contains only a few regressors, the sufficient option is

model_space <- optim_model_space(
  df            = data_prepared,
  dep_var_col   = gdp,
  timestamp_col = year,
  entity_col    = country,
  init_value    = function(n) rep(0.5, n)
)

However, for larger datasets, it is better to take advantage of parallel computing. Then the numerical optimization used to find MLEs can be carried out in separate parallel processes, one model at a time per worker. To do this, first load the parallel package and set up a cluster.

library(parallel)
# Leave one core free so the session remains responsive.
cores <- max(1, detectCores() - 1)
cl <- makeCluster(cores)
# Only needed when init_value draws at random; set.seed() does not reach
# the workers, where the per-model optimization runs.
clusterSetRNGStream(cl, 20240812)

Then the user just needs to provide this cluster to the function:

model_space <- optim_model_space(
  df            = data_prepared,
  dep_var_col   = gdp,
  timestamp_col = year,
  entity_col    = country,
  init_value    = function(n) rep(0.5, n),
  cl            = cl
)

stopCluster(cl)

Precomputed model spaces

Even with parallelization, optim_model_space call may be time-consuming. Hence, for users who wish to explore badp without recomputing the model space, we provide already computed model space objects included with the package:

• full_model_space is the model space built with the entire data used by Moral-Benito (2016),
• small_model_space is a smaller model space built with only a subset of regressors.

Bayesian model averaging: The bma function

The bma function enables users to perform Bayesian model averaging using the object obtained with the optim_model_space function. The round parameter specifies the decimal place to which the BMA statistics should be rounded in the results.

bma_results <- bma(full_model_space, round = 3)

The weighting parameter selects the approximation to the marginal likelihood \(L(\text{data}|M_j)\) used to weight the models. Writing

\[A_j = \hat{\ell}_j - \frac{k_j}{2}\log(NT),\]

where \(\hat{\ell}_j\) is the maximized log-likelihood of model \(M_j\), \(k_j\) its number of linear parameters, \(N\) the number of entities and \(T\) the number of periods, three of the four available options are the same construction with different learning rates \(\eta\), namely \(L(\text{data}|M_j) \propto \exp(\eta A_j)\):

• "mb2016" (the default), \(\eta = 1/N\). This is the approximation computed by the implementation accompanying Moral-Benito (2016), and the option that reproduces the results published there.
• "mb2012", \(\eta = 1\), i.e. \(\exp\{-\text{BIC}_j/2\}\) with the \(NT\) entity-period observations treated as the sample, as stated in equations (24)-(30) of Moral-Benito (2012).
• "nt", \(\eta = 1/(NT)\), averaging over entity-periods rather than entities. This is the rate that would be internally consistent with the \(\log(NT)\) penalty, and it tempers \(T\) times more strongly than "mb2016".

The fourth option, "uip", instead alters the penalty rather than the rate, setting \(L(\text{data}|M_j) \propto \exp\{\hat{\ell}_j - \frac{k_j}{2}\log(N)\}\) and so taking the entity rather than the entity-period as the unit of information. The unit information prior underlying the approximation (Kass and Raftery 1995) is defined through the Fisher information for a single observation, and the likelihood factorizes over entities, so the entity is the natural unit.

The choice matters. Any \(\eta \neq 1\) yields a tempered posterior over the model space, proportional to \(L(\text{data}|M_j)^{\eta}\), which lies outside the BIC approximation that motivates the criterion. A rate held fixed as the sample grows rescales all log weights by a common factor, so the posterior still concentrates on the best model in the space, only more slowly for \(\eta < 1\) and faster for \(\eta > 1\). A rate that shrinks with the sample is different in kind: under \(\eta = 1/N\) the log weight of a model at Kullback-Leibler distance \(c_j\) per entity satisfies \(\eta A_j = A_j/N \rightarrow -c_j\), so the weights converge to the fixed positive constants \(\exp(-c_j)\) and the posterior does not concentrate however large \(N\) becomes. Conversely \(\eta = 1\) concentrates the posterior sharply, and on likelihoods of this kind it can place nearly all of the posterior mass on a single model, in which case model averaging reduces to model selection. We retain "mb2016" as the default so that results remain comparable with the published literature, and recommend that users examine the sensitivity of their conclusions to this choice. Switching between the options requires no re-estimation, because all of them are recovered from the same fitted model space:

bma_results_mb2012 <- bma(full_model_space, round = 3, weighting = "mb2012")

Any other rate can be supplied through the eta argument, which overrides weighting. This is the intended way to check how much a conclusion depends on the rate:

bma_results_eta <- bma(full_model_space, round = 3, eta = 0.1)

The package deliberately offers no option that estimates \(\eta\) from the data. The magnitude adjustment developed for misspecified and composite likelihoods (Chandler and Bate 2007; Ribatet et al. 2012) would set \(\eta = \operatorname{rank}(J)/\operatorname{tr}(H^{-1}J)\), where \(J = \sum_{i=1}^{N} s_i s_i'\) is the outer product of the entity-level scores. For this likelihood \(J\) is substantially rank deficient, because parameters entering it only through terms common to every entity contribute nothing; in the bundled model spaces the scores span between 8% and 29% of the parameter directions, which summary() reports. The adjustment would then be calibrated on a small part of the parameter space with no way to assess what the rest contributes. The ingredients are stored with every fitted model space, so the rate can be computed and inspected directly by anyone who wants it.

The bma function returns a list containing 19 elements. However, most of these elements are only required for other functions. The main objects of interest are the two tables with the BMA statistics. The results obtained with binomial model prior are first on the list.

bma_results[[1]]

##           PIP     PM   PSD  PSDR  PMcon PSDcon PSDRcon    %(+)
## gdp_lag    NA  0.917 0.083 0.142  0.917  0.083   0.142 100.000
## ish     0.773  0.063 0.046 0.065  0.081  0.035   0.062 100.000
## sed     0.717  0.031 0.060 0.082  0.044  0.066   0.093  69.531
## pgrw    0.714  0.018 0.032 0.057  0.025  0.035   0.067  99.609
## pop     0.990  0.121 0.068 0.110  0.122  0.067   0.110 100.000
## ipr     0.657 -0.033 0.033 0.048 -0.050  0.028   0.052   0.000
## opem    0.766  0.034 0.030 0.034  0.044  0.027   0.033 100.000
## gsh     0.751 -0.013 0.041 0.098 -0.017  0.047   0.113  28.906
## lnlex   0.864  0.086 0.083 0.145  0.100  0.081   0.152 100.000
## polity  0.678 -0.056 0.046 0.054 -0.083  0.031   0.046   0.000

PIP denotes the posterior inclusion probability, PM denotes the posterior mean, PSD denotes the posterior standard deviation, and PSDR denotes the posterior standard deviation calculated using robust standard errors. These are the four main results of BMA with respect to the assessment of individual regressors. PMcon, PSDcon, and PSDRcon denote the posterior mean, posterior standard deviation, and posterior standard deviation based on robust standard errors, respectively, conditional on the inclusion of the variable. Users should base their interpretation of the results on conditional BMA statistics only when they believe that certain regressors must be included. Finally, for a given parameter we can consider all models that include this parameter, and check if it has a positive or negative value. \(\%(+)\) denotes the percentage of models with positive value for a given parameter across all models that include that parameter. A value of \(\%(+)\) equal to \(0\%\) or \(100\%\) indicates coefficient sign stability.

The PIP for all the regressors shows that none of them can be considered very strong according to the classification by Raftery (1995). This also applies to the population variable (pop), which has a PIP of 0.990 due solely to approximation. The ratios of PM to PSD and PSDR give a broadly similar ordering. In particular, for the absolute value of the PM to PSDR ratio, only the population variable exceeds 1.3, while investment (ish) and the democracy index (polity) are above 1. This finding led Moral-Benito (2016) to emphasize the fragility of economic growth determinants. The only variable that can be considered robust across all metrics is the lagged GDP (gdp_lag). However, the results change when using the binomial-beta model prior, which is included as the second object in the bma list.

bma_results[[2]]

##           PIP     PM   PSD  PSDR  PMcon PSDcon PSDRcon    %(+)
## gdp_lag    NA  0.917 0.084 0.156  0.917  0.084   0.156 100.000
## ish     0.954  0.074 0.035 0.063  0.078  0.032   0.062 100.000
## sed     0.939  0.049 0.061 0.083  0.052  0.061   0.084  69.531
## pgrw    0.939  0.023 0.033 0.064  0.025  0.034   0.066  99.609
## pop     0.998  0.101 0.066 0.119  0.102  0.066   0.119 100.000
## ipr     0.924 -0.043 0.029 0.048 -0.047  0.027   0.048   0.000
## opem    0.953  0.036 0.025 0.029  0.038  0.024   0.029 100.000
## gsh     0.948 -0.017 0.043 0.099 -0.018  0.044   0.101  28.906
## lnlex   0.974  0.114 0.080 0.156  0.117  0.079   0.157 100.000
## polity  0.929 -0.077 0.036 0.049 -0.083  0.030   0.046   0.000

In the case of the binomial-beta model prior, the PIPs for all the regressors increase. Population is classified as very strong, while all other regressors are classified as strong or positive according to posterior inclusion probabilities. There are also considerable changes in the PM to PSD and PSD_R ratios. The absolute value of the PM to PSD ratio exceeds two for investment and the democracy index, and is above 1.3 for population, investment price (ipr), trade openness (open), and life expectancy (lnlex). However, these results are less pronounced when using robust standard errors, with only population, trade openness, and the democracy index remaining above 1.3. Consequently, the results are not robust with respect to the choice of prior model specification. The reasons behind these differences will become clear once other functionalities of the package are explored.

The last object in the list is a table containing the prior and posterior expected model sizes for the binomial and binomial-beta model priors. Importantly, these numbers reflect only the number of regressors in a model and do not include the lagged dependent variable, which is present in every model by construction.

bma_results[[16]]

##               Prior model size Posterior model size
## Binomial                   4.5                6.910
## Binomial-beta              4.5                8.558

The results show that, after observing the data, the posterior expected number of included regressors is around seven under the binomial model prior and eight and a half under the binomial-beta model prior. These numbers may seem high; however, they are driven by relatively substantial PIPs. This illustrates the importance of focusing on both posterior inclusion probabilities and the ratios of posterior mean to posterior standard deviation when assessing the robustness of the regressors.

Prior and posterior model probabilities

The model_pmp function allows the user to compare prior and posterior model probabilities over the entire model space in the form of a graph. The models are ranked from the one with the highest to the one with the lowest posterior model probability. The function returns a list with three objects:

1. a graph for the binomial model prior;
2. a graph for the binomial-beta model prior;
3. a combined graph for both binomial and binomial-beta model priors.

The user can retrieve each graph separately from the list; however, the function automatically displays a combined graph.

for_models <- model_pmp(bma_results)

The graphs demonstrate that most of the posterior probability mass is concentrated within just a couple of models. To view the results for only the best models, the user can use the top parameter.

for_models <- model_pmp(bma_results, top = 10)

The last graph for the binomial-beta prior is particularly illuminating in terms of explaining the very high values of posterior inclusion probabilities. Almost 70% of the posterior probability mass is concentrated in just one model; therefore, variables included in this model will have very high PIP values. The model in question will be identified after implementing model_sizes (and best_models, which is covered in Section Selecting the best models). Nevertheless, the results from the graph suggest that the best model is the one that includes all the regressors or none (because the prior value is around \(\frac{1}{9}\) on the plot).

The model_sizes function displays prior and posterior model probabilities on a graph for models of different sizes. The graphs exclude the lagged dependent variable; therefore, the model with zero regressors still includes the lagged dependent variable. Similarly to the model_pmp function it returns a list with three objects:

1. a graph for the binomial model prior;
2. a graph for the binomial-beta model prior;
3. a combined graph for both binomial and binomial-beta model priors.

Again, the user can retrieve each graph separately from the list; however, the function automatically displays a combined graph.

size_graphs <- model_sizes(bma_results)

The graph in panel b) again explains why PIPs are so high in the case of the binomial-beta model prior. The model with all the regressors accounts for almost 70% of the total posterior probability mass, while the remaining portion is concentrated on models with a high number of regressors. In contrast, the posterior probability mass for the binomial model prior is centered around models with seven regressors. This graph clearly illustrates the impact of changes in the model prior on posterior probabilities.

Selecting the best models

The best_models function allows the user to view a chosen number of the best models in terms of posterior model probability. The function returns a list containing nine objects:

1. An inclusion table stored as an array object;
2. A table with estimation results using regular standard errors, stored as an array object;
3. A table with estimation results using robust standard errors, stored as an array object;
4. An inclusion table stored as a knitr object;
5. A table with estimation results using regular standard errors, stored as a knitr object;
6. A table with estimation results using robust standard errors, stored as a knitr object;
7. An inclusion table stored as a gTree object;
8. A table with estimation results using regular standard errors, stored as a gTree object;
9. A table with estimation results using robust standard errors, stored as a gTree object;

The parameters estimate and robust pertain only to the results that will be automatically displayed after running the function. The parameter prior determines whether the models should be ranked according to posterior model probabilities calculated using the binomial (prior = "binomial") or binomial-beta (prior = "beta") model prior. To obtain the inclusion array for the 10 best models ranked with the binomial model prior, the user needs to run:

best_8_models <- best_models(bma_results, prior = "binomial", best = 8)
best_8_models[[1]]

##         'No. 1' 'No. 2' 'No. 3' 'No. 4' 'No. 5' 'No. 6' 'No. 7' 'No. 8'
## gdp_lag    1.00   1.000   1.000   1.000   1.000   1.000   1.000   1.000
## ish        1.00   1.000   1.000   1.000   1.000   1.000   1.000   0.000
## sed        1.00   1.000   1.000   0.000   1.000   1.000   1.000   1.000
## pgrw       1.00   1.000   1.000   1.000   0.000   1.000   1.000   1.000
## pop        1.00   1.000   1.000   1.000   1.000   1.000   1.000   1.000
## ipr        1.00   0.000   1.000   1.000   1.000   1.000   1.000   1.000
## opem       1.00   1.000   1.000   1.000   1.000   1.000   0.000   1.000
## gsh        1.00   1.000   1.000   1.000   1.000   0.000   1.000   1.000
## lnlex      1.00   1.000   1.000   1.000   1.000   1.000   1.000   1.000
## polity     1.00   1.000   0.000   1.000   1.000   1.000   1.000   1.000
## PMP        0.09   0.044   0.042   0.036   0.035   0.029   0.026   0.025

1 indicates the presence of a given regressor in a model, while the last row displays the posterior model probability of that model. To obtain a knitr table with estimation output with regular standard errors for best 3 models ranked with binomial-beta model prior, the user needs to run:

best_3_models <- best_models(bma_results, prior = "beta", best = 3)
best_3_models[[5]]

The comparison of the last two tables further highlights the importance of the model prior. The best model under the binomial model prior accounts for around 9% of the posterior probability mass, while the best model under the binomial-beta model prior accounts for over 69%. Finally, to obtain a gTree table with estimation output using robust standard errors for the top 3 models ranked by the binomial-beta model prior, the user needs to run:

best_3_models <- best_models(bma_results, prior = "beta", best = 3)
best_3_models[[9]]

The comparison of the last two tables and the estimation outputs with regular and robust standard errors demonstrates how the results change when switching between these two variance estimators.

Calculating jointness measures

Within the BMA framework, it is possible to establish the nature of the relationship between pairs of examined regressors using the jointness measures. This can be accomplished using the jointness function. The latest jointness measure, introduced by Hofmarcher et al. (2018), has been shown to address shortcomings of the earlier measures of Ley and Steel (2007) and Doppelhofer and Weeks (2009, see Section Model priors and jointness for the interpretations of jointness measures). Therefore, the Hofmarcher et al. (2018) measure is the default option in the jointness function.

jointness(bma_results)

##          ish   sed  pgrw   pop   ipr  opem   gsh lnlex polity
## ish       NA 0.217 0.208 0.531 0.151 0.262 0.244 0.366  0.182
## sed    0.806    NA 0.155 0.421 0.116 0.200 0.189 0.289  0.126
## pgrw   0.806 0.779    NA 0.416 0.124 0.199 0.187 0.284  0.132
## pop    0.906 0.874 0.874    NA 0.304 0.517 0.490 0.711  0.346
## ipr    0.782 0.758 0.759 0.846    NA 0.153 0.139 0.210  0.102
## opem   0.830 0.802 0.803 0.902 0.781    NA 0.241 0.373  0.170
## gsh    0.822 0.795 0.795 0.894 0.773 0.820    NA 0.341  0.154
## lnlex  0.865 0.836 0.836 0.944 0.811 0.863 0.854    NA  0.228
## polity 0.791 0.764 0.765 0.856 0.745 0.788 0.780 0.818     NA

Above the main diagonal the user can find the results for the binomial model prior, and below the results for the binomial-beta model prior. All the values in the table are positive, indicating complementary relationships between the regressors. Notably, the values for the binomial-beta prior are substantially higher than those for the binomial prior. This result is not surprising, as the model with all the regressors accounts for almost 70% of the total posterior probability mass.

To obtain the results for the Ley and Steel (2007) measure, the user should run:

jointness(bma_results, measure = "LS")

##           ish    sed   pgrw    pop   ipr   opem    gsh  lnlex polity
## ish        NA  1.470  1.448  3.286 1.230  1.678  1.603  2.213  1.324
## sed     9.550     NA  1.250  2.467 1.091  1.424  1.378  1.812  1.141
## pgrw    9.538  8.284     NA  2.437 1.100  1.416  1.369  1.792  1.146
## pop    20.257 14.940 14.952     NA 1.876  3.163  2.935  5.987  2.062
## ipr     8.387  7.441  7.491 12.021    NA  1.223  1.178  1.477  1.014
## opem   11.040  9.360  9.375 19.533 8.306     NA  1.583  2.212  1.292
## gsh    10.491  8.998  9.011 17.821 7.997 10.341     NA  2.061  1.242
## lnlex  14.093 11.422 11.424 35.097 9.744 13.896 12.958     NA  1.566
## polity  8.776  7.685  7.723 12.873 7.014  8.629  8.292 10.197     NA

The values are consistent with those obtained using the Hofmarcher et al. (2018) measure. All the regressors exhibit complementary relationships, which are visibly stronger under the binomial-beta model prior.

However, the Doppelhofer and Weeks (2009) measure yields a slightly different outcome:

jointness(bma_results, measure = "DW")

##          ish   sed   pgrw    pop    ipr   opem   gsh  lnlex polity
## ish       NA 0.051  0.020  0.005  0.018  0.030 0.008 -0.004  0.067
## sed    0.988    NA -0.023 -0.029  0.004 -0.008 0.000  0.004 -0.024
## pgrw   0.974 0.905     NA -0.001  0.047 -0.001 0.001 -0.007  0.010
## pop    1.019 0.957  0.987     NA -0.018 -0.023 0.049  0.154  0.012
## ipr    0.985 0.931  0.979  0.974     NA  0.048 0.019  0.025  0.036
## opem   1.012 0.938  0.949  0.991  1.006     NA 0.032  0.139  0.032
## gsh    0.972 0.931  0.941  1.042  0.961  0.991    NA  0.034  0.001
## lnlex  0.983 0.957  0.953  1.173  0.985  1.104 1.000     NA -0.056
## polity 1.013 0.903  0.939  0.994  0.967  0.979 0.934  0.905     NA

In this case, some pairs of regressors have negative values of the jointness measure under the binomial model prior; however, these values are very close to zero, indicating unrelated variables. Once again, the values for the binomial-beta model prior are higher, demonstrating how the results are influenced by the choice of model prior.

Visualizing model coefficients and posterior distributions

The coef_hist function allows the user to plot the distribution of estimated coefficients. It returns a list containing a number of objects equal to the number of regressors plus one. The first object in the list is a graph of the coefficients for the lagged dependent variable, while the remaining objects are graphs of the coefficients for the other regressors. The graph for the lagged dependent variable collects coefficients from the entire model space, whereas the graphs for the other regressors only collect coefficients from the models that include the given regressor (half of the model space).

There are two main options for visualizing the coefficient distributions. The first option uses a histogram. The coef_hist function provides the user with options for controlling the bin widths of the histogram (bin_method, bin_widths, use_bin_count, and bin_counts). The default is bin_method = "FD", which selects bin widths using the Freedman-Diaconis method.

coef_plots <- coef_hist(bma_results)
coef_plots[[1]]

The second option allows the user to plot kernel densities.

coef_plots2 <- coef_hist(bma_results, use_kernel = 1)
coef_plots2[[1]]

The choice of appropriate plotting options is left to the user’s preferences regarding the style of presentation and the size of the model space.

library(gridExtra)

## Warning: package 'gridExtra' was built under R version 4.5.2

grid.arrange(coef_plots[[1]], coef_plots[[2]], coef_plots2[[1]],
             coef_plots2[[2]], nrow = 2, ncol = 2)

One additional option available to the user with the coef_hist function is weighting coefficients by posterior model probabilities. This can be accomplished using the weight parameter, whose default value is set to NULL. Setting weight to "binomial" or "beta" results in plotting histograms based on the binomial or binomial-beta model prior, respectively. For example, in the case of the binomial-beta model prior, it can be observed that the value of the posterior mean is mainly driven by the model characterized by the highest posterior model probability.

coef_plots3 <- coef_hist(bma_results, weight = "beta")
coef_plots3[[1]]

The posterior_dens allows the user to plot posterior distributions of coefficients. Similarly to the coef_hist function, posterior_dens returns a list containing a number of objects equal to the number of regressors plus one. The first object in the list is a graph of the posterior distribution for the lagged dependent variable, while the remaining objects are graphs of the posterior distribution for the other regressors. The user needs to specify whether to plot posterior distributions based on the binomial (prior = "binomial") or binomial-beta (prior = "beta") model prior, as well as whether to use regular (SE = "standard") or robust (SE = "robust") standard errors used in the calculation of posterior objects.

distPlots <- posterior_dens(bma_results, prior = "binomial", SE = "standard")
grid.arrange(distPlots[[2]], distPlots[[3]], nrow = 2, ncol = 1)

Changing expected model size

The bma function calculates BMA statistics using both the binomial and binomial-beta model priors. By default, the bma function sets the expected model size (EMS) to \(K/2\), where \(K\) denotes the total number of regressors. The binomial model prior with \(EMS = K/2\) leads to a uniform model prior, assigning equal probabilities to all models. In contrast, the binomial-beta model prior with \(EMS = K/2\) assumes equal probabilities across all model sizes. However, the user can modify the prior model specification by changing the EMS parameter.

First, consider the consequence of concentrating prior probability mass on small models by setting \(EMS = 2\).

bma_results2 <- bma(full_model_space, round = 3, EMS = 2)

Before turning to the main BMA results, let us focus on the changes in the posterior probability mass with respect to model sizes.

bma_results2[[16]]

##               Prior model size Posterior model size
## Binomial                     2                4.560
## Binomial-beta                2                7.502

The results show that decreasing the prior expected model size led to a considerable decline in the posterior expected model size. The consequences of this change in the prior expected model size are best illustrated using the prior and posterior probability mass over model sizes.

size_graphs2 <- model_sizes(bma_results2)

For both the binomial and binomial-beta model priors, the prior probability mass is more concentrated on small model sizes. However, for the binomial model prior, the center of the posterior probability mass shifted to medium-sized models, while it remained on large models for the binomial-beta model prior. Nevertheless, the posterior model probability for the model with all regressors decreased from nearly 0.7 for \(EMS = 4.5\) to less than 0.3. There are also substantial changes in the distribution of the posterior probability mass over the model space.

model_graphs2 <- model_pmp(bma_results2)

Both panels of the graph show that the prior and posterior model probabilities have substantially decoupled from each other. This strongly indicates that the prior and the data are suggesting vastly different model choices. The tall blue spike represents the model with no regressors. The main BMA posterior statistic for the binomial model prior also experienced a significant change.

bma_results2[[1]]

##           PIP     PM   PSD  PSDR  PMcon PSDcon PSDRcon    %(+)
## gdp_lag    NA  0.923 0.087 0.128  0.923  0.087   0.128 100.000
## ish     0.483  0.043 0.050 0.061  0.088  0.035   0.061 100.000
## sed     0.420  0.015 0.048 0.066  0.035  0.068   0.098  69.531
## pgrw    0.414  0.009 0.025 0.044  0.023  0.036   0.066  99.609
## pop     0.964  0.142 0.068 0.099  0.147  0.064   0.097 100.000
## ipr     0.344 -0.019 0.031 0.040 -0.055  0.029   0.052   0.000
## opem    0.468  0.024 0.032 0.034  0.052  0.027   0.032 100.000
## gsh     0.459 -0.003 0.033 0.080 -0.007  0.049   0.118  28.906
## lnlex   0.637  0.053 0.074 0.120  0.082  0.078   0.141 100.000
## polity  0.372 -0.029 0.043 0.047 -0.079  0.032   0.045   0.000

Posterior inclusion probabilities drop considerably for all the regressors, except for population, which remains almost unchanged. Interestingly, the ratios for all variables declined, with population being the exception. The ratio for population remains above two for regular standard errors and 1.7 for robust standard errors. This outcome indicates that population performs relatively better in smaller models. The results for binomial-beta model prior are given below.

bma_results2[[2]]

##           PIP     PM   PSD  PSDR  PMcon PSDcon PSDRcon    %(+)
## gdp_lag    NA  0.917 0.083 0.147  0.917  0.083   0.147 100.000
## ish     0.838  0.067 0.043 0.064  0.080  0.034   0.062 100.000
## sed     0.796  0.038 0.061 0.082  0.047  0.064   0.090  69.531
## pgrw    0.795  0.020 0.032 0.060  0.025  0.034   0.066  99.609
## pop     0.992  0.114 0.068 0.114  0.115  0.068   0.114 100.000
## ipr     0.754 -0.037 0.032 0.049 -0.049  0.028   0.051   0.000
## opem    0.833  0.034 0.028 0.033  0.041  0.026   0.032 100.000
## gsh     0.822 -0.014 0.042 0.098 -0.017  0.045   0.108  28.906
## lnlex   0.902  0.096 0.083 0.150  0.107  0.081   0.155 100.000
## polity  0.769 -0.064 0.044 0.053 -0.083  0.030   0.046   0.000

The change in PIPs is again significant, though not as pronounced as in the case of the binomial model prior. Changes in the ratios are relatively small and irregular for both regular and robust standard errors. The most pronounced change is the drop in the value of the ratios for the democracy index (polity), indicating that this regressor performs better in larger models.

It is also very instructive to examine the jointness measures calculated under the new prior specification.

jointness(bma_results2, measure = "HCGHM", rho = 0.5, round = 3)

##          ish   sed  pgrw    pop    ipr   opem    gsh  lnlex polity
## ish       NA 0.021 0.008 -0.030  0.003  0.002  0.001 -0.012  0.021
## sed    0.441    NA 0.016 -0.146  0.043  0.007  0.011 -0.036  0.026
## pgrw   0.437 0.390    NA -0.155  0.053  0.012  0.011 -0.041  0.037
## pop    0.667 0.586 0.583     NA -0.281 -0.057 -0.072  0.253 -0.231
## ipr    0.391 0.355 0.361  0.503     NA  0.021  0.023 -0.065  0.072
## opem   0.483 0.430 0.430  0.657  0.391     NA  0.010  0.022  0.020
## gsh    0.467 0.419 0.418  0.636  0.378  0.464     NA -0.012  0.019
## lnlex  0.559 0.497 0.495  0.793  0.439  0.562  0.538     NA -0.072
## polity 0.413 0.364 0.369  0.532  0.340  0.405  0.390  0.453     NA

On the one hand, the results obtained with the binomial-beta model prior did not change in any significant manner. On the other hand, the results obtained with the binomial model prior changed substantially. The measure indicates that population is a substitute for both the investment price (ipr) and the democracy index, as well as, to a lesser extent, secondary education (sed) and population growth (pgrw).

Next, to consider the consequences of concentrating prior probability mass on large models, EMS was set to eight.

bma_results8 <- bma(full_model_space, round = 3, EMS = 8)
bma_results8[[16]]

##               Prior model size Posterior model size
## Binomial                     8                8.666
## Binomial-beta                8                8.944

The posterior model size increased for the binomial prior; however, it remained almost unchanged for the binomial-beta model prior. The most interesting aspect is the new graphs of prior and posterior probability mass over the model sizes.

size_graphs8 <- model_sizes(bma_results8)

In both cases, the posterior probability mass has concentrated near the models with all the regressors. However, in the case of the binomial-beta model prior, the model with all the regressors captures most of the posterior probability mass (almost 96%). This conclusion is further supported by the graphs of posterior model probability across the entire model space.

model_graphs8 <- model_pmp(bma_results8)

Panel (a) demonstrates that the change in the expected model size led to a substantial increase in the posterior model probability for the model with all regressors under the binomial model prior. It now accounts for over 70% of the total posterior probability mass. The increase in the expected model size also influenced the main BMA statistics.

bma_results8[[1]]

##           PIP     PM   PSD  PSDR  PMcon PSDcon PSDRcon    %(+)
## gdp_lag    NA  0.917 0.084 0.157  0.917  0.084   0.157 100.000
## ish     0.967  0.075 0.034 0.063  0.077  0.032   0.062 100.000
## sed     0.953  0.050 0.061 0.083  0.052  0.061   0.084  69.531
## pgrw    0.953  0.023 0.033 0.065  0.025  0.034   0.066  99.609
## pop     0.999  0.100 0.066 0.119  0.100  0.066   0.119 100.000
## ipr     0.942 -0.044 0.028 0.048 -0.047  0.027   0.048   0.000
## opem    0.965  0.036 0.025 0.029  0.037  0.024   0.029 100.000
## gsh     0.961 -0.017 0.043 0.099 -0.018  0.044   0.101  28.906
## lnlex   0.981  0.116 0.080 0.157  0.118  0.079   0.158 100.000
## polity  0.945 -0.079 0.035 0.049 -0.083  0.030   0.046   0.000

The PIPs increased considerably. Population is classified as very strong, while the other regressors are classified as strong or positive. Interestingly, all the ratios have improved as well, except for population. The change in the results for the binomial-beta model prior is less pronounced.

bma_results8[[2]]

##           PIP     PM   PSD  PSDR  PMcon PSDcon PSDRcon    %(+)
## gdp_lag    NA  0.917 0.085 0.160  0.917  0.085   0.160 100.000
## ish     0.994  0.077 0.031 0.062  0.077  0.031   0.062 100.000
## sed     0.992  0.054 0.060 0.082  0.054  0.060   0.082  69.531
## pgrw    0.992  0.024 0.033 0.066  0.024  0.034   0.066  99.609
## pop     1.000  0.096 0.065 0.121  0.096  0.065   0.121 100.000
## ipr     0.990 -0.046 0.027 0.047 -0.046  0.027   0.047   0.000
## opem    0.994  0.036 0.024 0.027  0.036  0.024   0.027 100.000
## gsh     0.994 -0.017 0.043 0.097 -0.017  0.043   0.098  28.906
## lnlex   0.997  0.121 0.078 0.158  0.121  0.078   0.158 100.000
## polity  0.991 -0.082 0.031 0.046 -0.083  0.030   0.046   0.000

With the increase in expected model size, population is classified as very strong, and all the other regressors are classified as strong in terms of the posterior inclusion probability criterion. Similarly to the case of the binomial prior, all the ratios increased except for population.

Again, it is instructive to examine the jointness measures.

jointness(bma_results8, measure = "HCGHM", rho = 0.5, round = 3)

##          ish   sed  pgrw   pop   ipr  opem   gsh lnlex polity
## ish       NA 0.840 0.841 0.931 0.819 0.865 0.857 0.896  0.826
## sed    0.975    NA 0.814 0.903 0.792 0.837 0.830 0.869  0.799
## pgrw   0.975 0.971    NA 0.904 0.793 0.838 0.830 0.869  0.800
## pop    0.988 0.984 0.984    NA 0.881 0.928 0.920 0.960  0.888
## ipr    0.972 0.968 0.968 0.980    NA 0.816 0.808 0.847  0.778
## opem   0.978 0.974 0.974 0.988 0.971    NA 0.854 0.894  0.823
## gsh    0.977 0.973 0.973 0.987 0.970 0.977    NA 0.886  0.815
## lnlex  0.983 0.979 0.979 0.993 0.976 0.983 0.982    NA  0.854
## polity 0.973 0.969 0.969 0.982 0.966 0.972 0.971 0.977     NA

The values of the measures show that all the regressors exhibit a very strong complementary relationship. This outcome, once again, underscores the importance of carefully considering the prior when interpreting jointness measures.

Dilution prior

One of the main issues associated with identifying robust regressors is multicollinearity. Some regressors may approximate the same underlying factor influencing the dependent variable. Multicollinearity may result from the absence of observable variables associated with a specific theory or from a theory failing to provide a unique candidate for a regressor. Moreover, some regressors may share a common determinant. Although Moral-Benito (2013) and Moral-Benito (2016) addressed this issue to some extent, researchers have another option to mitigate multicollinearity: the dilution prior proposed by George (2010) which was described in detail in Section Model priors and jointness.

To apply the dilution prior, the user must set dilution = 1 in the bma function. The user can also manipulate the dilution parameter \(\omega\). The default option is omega = 0.5, as recommended by George (2010).

bma_results_dil <- bma(
  model_space = full_model_space,
  round       = 3,
  dilution    = 1
  )

The effect of implementing the dilution prior is well depicted by the distribution of prior probability mass over the model sizes.

size_graphs_dil <- model_sizes(bma_results_dil)

The change in the prior distribution is more visible for the binomial-beta model prior. In panel b, the prior probability mass has decreased for larger models and increased for smaller models. However, this change is not uniform, as models characterized by the highest degree of multicollinearity are subject to the greatest penalty in terms of prior probability mass.

Before moving to the BMA statistics, it is instructive to examine the change in the omega parameter.

bma_results_dil01 <- bma(
  model_space = full_model_space,
  round       = 3,
  dilution    = 1,
  omega     = 0.1
)
size_graphs_dil01 <- model_sizes(bma_results_dil01)

As we can see, decreasing the value of \(\omega\) diminishes the impact of dilution on the model prior. Conversely, raising the omega parameter increases the degree of dilution.

bma_results_dil2 <- bma(
  model_space = full_model_space,
  round       = 3,
  dilution    = 1,
  omega     = 2
)
size_graphs_dil2 <- model_sizes(bma_results_dil2)

An especially strong impact can be seen for the binomial-beta prior.

However, even after giving such priority to the penalty for multicollinearity, the main BMA statistics remain stable.

bma_results_dil2[[2]]

##           PIP     PM   PSD  PSDR  PMcon PSDcon PSDRcon    %(+)
## gdp_lag    NA  0.922 0.085 0.147  0.922  0.085   0.147 100.000
## ish     0.735  0.056 0.045 0.063  0.076  0.034   0.062 100.000
## sed     0.641  0.030 0.057 0.076  0.046  0.065   0.090  69.531
## pgrw    0.687  0.019 0.032 0.057  0.028  0.035   0.067  99.609
## pop     0.993  0.122 0.070 0.114  0.122  0.069   0.114 100.000
## ipr     0.773 -0.039 0.032 0.051 -0.050  0.028   0.053   0.000
## opem    0.824  0.037 0.030 0.034  0.045  0.027   0.033 100.000
## gsh     0.840 -0.014 0.044 0.107 -0.016  0.047   0.117  28.906
## lnlex   0.768  0.086 0.085 0.152  0.112  0.081   0.165 100.000
## polity  0.613 -0.050 0.046 0.054 -0.081  0.031   0.046   0.000

Hence, we see that Moral-Benito (2016)’s claim about the fragility of growth regressors withstands the test of various manipulations in the model prior.