Efficient MCMC estimation of inflated beta regression models

Abstract

This paper introduces a new and computationally efficient Markov chain Monte Carlo (MCMC) estimation algorithm for the Bayesian analysis of zero, one, and zero and one inflated beta regression models. The algorithm is computationally efficient in the sense that it has low MCMC autocorrelations and computational time. A simulation study shows that the proposed algorithm outperforms the slice sampling and random walk Metropolis–Hastings algorithms in both small and large sample settings. An empirical illustration on a loss given default banking model demonstrates the usefulness of the proposed algorithm.

Appendix

1.1 Parameterization of the inverse-gamma distribution

If X has an inverse-gamma distribution with shape a and scale b, then the density function is parameterized as

$$\begin{aligned} \mathcal {IG}(x \vert a, b) = \frac{x^{-(a+1)}}{{\varGamma }(a) b^a} e^{-\frac{1}{xb}}. \end{aligned}$$

The mean is $\mathbb {E}(X) = (b(a-1))^{-1}$ for $a>1$, and the variance is $\mathbb {V}(X)=(b^2(a-1)^2(a-2))^{-1}$ for $a>2$.

1.2 Quasi-likelihood quantities

The following describes the method to obtain consistent estimates of $\beta$ and $\phi$ (or equivalently, $\psi$). Following Papke and Wooldridge (1996), the first step is to maximize the following log Bernoulli quasi-likelihood function

$$\begin{aligned} L_{QL}(b) = \sum _{t=1}^{n_{01} } \left[ y_t \log \left( \frac{ \exp (x_t^\top b) }{ 1 + \exp (x_t^\top b)} \right) + (1-y_t) \log \left( 1 - \frac{ \exp (x_t^\top b) }{ 1 + \exp (x_t^\top b)} \right) \right] \end{aligned}$$

(23)

with respect to the $K_2 \times 1$ vector b. In (23), $t=1,\ldots ,n_{01}$ indexes the $N_{01}$ subsample, and $n_{01}$ is the cardinality of $N_{01}$. Note that the referenced paper expresses the log QL function in terms of an arbitrary function $G(\cdot )$; in this paper, $G( x_t^\top b) = \text {logit}^{-1}(x_t^\top b) = \frac{ \exp (x_t^\top b) }{ 1 + \exp (x_t^\top b)}$, which leads to (23). Papke and Wooldridge (1996) show that the maximizing vector, denoted as $\widehat{\beta }$, is a consistent estimator of $\beta$, as long as $\mathbb {E}(y_t) = \frac{\exp (x_t^\top \beta )}{1+\exp (x_t^\top \beta )}$ for $t=1,\ldots ,n_{01}$ (which holds due to (3) and (8) holding for $i \in N_{01}$). Next, under the assumption that $\mathbb {V}(y_t) = \delta ^2 \mu _t (1-\mu _t)$ for $t=1,\ldots ,n_{01}$ with $\delta ^2 = \frac{1}{1+\phi }$ (which is true given (9) for $i \in N_{01}$), the authors also show that the estimator

$$\begin{aligned} \widehat{\delta }^2 = \frac{ \sum _{t=1}^{n_{01}} \tilde{u}_t^2}{n_{01} - K_2}, \end{aligned}$$

is consistent for $\delta ^2$, where

$$\begin{aligned} \tilde{u}_t = \frac{y_t - \widehat{\mu }_t}{ \sqrt{ \widehat{\mu }_t(1-\widehat{\mu }_t) } } \end{aligned}$$

and

$$\begin{aligned} \widehat{\mu }_t = \frac{\exp (x_t^\top \widehat{\beta } )}{1+\exp (x_t^\top \widehat{\beta })}. \end{aligned}$$

Then, by the continuous mapping theorem, the estimator $\widehat{\phi }= \frac{1-\widehat{\delta }^2}{\widehat{\delta }^2}$ is consistent for $\phi$. The vector $\widehat{\psi }$ in Sect. 3.1 is defined as $\widehat{\psi } = (\widehat{\beta }^\top , \widehat{\phi })^\top$.

1.3 Information matrix quantities

Partition $\widehat{K}$ as

$$\begin{aligned} \widehat{K}&= \left( \begin{array}{cc} \underset{(K_2 \times K_2)}{\widehat{K}_{\beta \beta }} &{} \underset{(K_2 \times 1)}{\widehat{K}_{\beta \phi } }\\ \underset{(1 \times K_2)}{\widehat{K}_{\phi \beta } }&{} \underset{(1 \times 1)}{ \widehat{K}_{\phi \phi } } \end{array} \right) . \end{aligned}$$

(24)

Following Ferrari and Cribari-Neto (2004), the quantities in (24) are defined as

$$\begin{aligned} \widehat{K}_{\beta \beta }&= \widehat{\phi }X_{01}^\top W_{01} X_{01} \nonumber \\ W_{01}&= \text {diag}(w_1, \ldots , w_{n_{01}})\nonumber \\ w_{t}&= \widehat{\phi } \left[ \psi '( \widehat{\mu }_t \widehat{\phi }) + \psi '( (1-\widehat{\mu }_t) \widehat{\phi }) \right] \left[ \widehat{\mu }_t (1 - \widehat{\mu }_t) \right] ^2 \end{aligned}$$

(25)

$$\begin{aligned} \widehat{K}_{\beta \phi }&= \widehat{K}^\top _{ \phi \beta } = X_{01}^\top T_{01}C_{01} \nonumber \\ T_{01}&= \text {diag}(\widehat{\mu }_1(1-\widehat{\mu }_1), \ldots , \widehat{\mu }_{n_{01}}(1-\widehat{\mu }_{n_{01}} ) ) \nonumber \\ C_{01}&= (c_1, \ldots , c_{n_{01}})^\top \nonumber \\ c_t&= \widehat{\phi } \left[ \psi '( \widehat{\mu }_t \widehat{\phi })\widehat{\mu }_t - \psi '( (1-\widehat{\mu }_t) \widehat{\phi }) (1 - \widehat{\mu }_t) \right] \nonumber \\ \widehat{K}_{\phi \phi }&= \text {trace}(D_{01})\nonumber \\ D_{01}&= \text {diag}(d_1, \ldots , d_{n_{01}}) \nonumber \\ d_{t}&= \psi '( \widehat{\mu }_t \widehat{\phi }) \widehat{\mu }_t^2 + \psi '( (1-\widehat{\mu }_t) \widehat{\phi }) (1 - \widehat{\mu }_t)^2 - \psi '(\widehat{\phi }), \end{aligned}$$

(26)

where $t=1,\ldots , n_{01}$, $X_{01}=(x_1, \ldots , x_{n_{01}})^\top$ is an $n_{01} \times K_2$ matrix of covariates for the observations in $N_{01}$, and $\psi '(\cdot )$ denotes the trigamma function. Note that the quantities above are defined over $N_{01}$, while the quantities in Ferrari and Cribari-Neto (2004) are defined over the full sample. Also, the analogous quantities for (25) and (26) in Ferrari and Cribari-Neto (2004) are expressed for an arbitrary link function $g( \mu _t)$. In this paper, $g(\mu _t) = \text {logit}(\mu _t) = \log (\frac{\mu _t}{1-\mu _t})$, and $g'(\mu _t) = \frac{1}{\mu _t(1-\mu _t)}$ (the first derivative), so plugging these quantities into the formulas in the referenced paper results in the expressions for (25) and (26).