Improved testing inferences for beta regressions with parametric mean link function

Abstract

Beta regressions are widely used for modeling random variables that assume values in the standard unit interval, (0, 1), such as rates, proportions, and income concentration indices. Parameter estimation is typically performed via maximum likelihood, and hypothesis testing inferences on the model parameters are commonly performed using the likelihood ratio test. Such a test, however, may deliver inaccurate inferences when the sample size is small. It is thus important to develop alternative testing procedures that are more accurate when the sample contains only few observations. In this paper, we consider the beta regression model with parametric mean link function and derive two modified likelihood ratio test statistics for that class of models. We provide simulation evidence that shows that the new tests usually outperform the standard likelihood ratio test in samples of small to moderate sizes. We also present and discuss two empirical applications.

Appendices

Appendix A

We provide below the observed and expected information matrices along with Fisher’s information matrix inverse. The observed information matrix is given by

$$\begin{aligned} J \equiv J(\varvec{\theta }) = \left( \begin{array}{ccc} J_{(\varvec{\beta },\varvec{\beta })} &{}\quad J_{(\varvec{\beta },\varvec{\gamma })} &{}\quad J_{(\varvec{\beta },\lambda )} \\ J_{(\varvec{\gamma },\varvec{\beta })} &{}\quad J_{(\varvec{\gamma },\varvec{\gamma })} &{}\quad J_{(\varvec{\gamma },\lambda )} \\ J_{(\lambda ,\varvec{\beta })} &{}\quad J_{(\lambda ,\varvec{\gamma })} &{}\quad J_{(\lambda ,\lambda )} \\ \end{array} \right) , \end{aligned}$$

where $J_{(\varvec{\beta },\varvec{\beta })} = X^{\top } [\varPhi T V^{*} + S T^2 (Y^{*} - M^{*})] T \varPhi X$, $J_{(\varvec{\beta },\varvec{\gamma })} = J_{(\varvec{\gamma },\varvec{\beta })}^{\top } = -X^{\top } [(Y^{*} - M^{*}) - \varPhi (M V^{*} + C)] T H Z$, $J_{(\varvec{\beta },\lambda )} = J_{(\lambda ,\varvec{\beta })}^{\top } = X^{\top } [\varPhi ^2 V^{*} T \varvec{\rho } - \varPhi (Y^{*} - M^{*}) \varvec{w}]$, $J_{(\varvec{\gamma },\varvec{\gamma })} = Z^{\top } \{H (M^2 V^{*} + 2M C + V^{\dagger }) + [ M (Y^{*} - M^{*}) + (Y^{\dagger } - M^{\dagger }) ] H^2 Q\} H Z$, $J_{(\varvec{\gamma },\lambda )} = J_{(\lambda ,\varvec{\gamma })}^{\top } = -Z^{\top } [(Y^{*} -M^{*}) - \varPhi (M V^{*} + C)] H \varvec{\rho }$, and $J_{(\lambda ,\lambda )} = [\varPhi ^2 V^{*} \varvec{\rho }^2 - \varPhi (Y^{*} - M^{*}) \varvec{\zeta } ]^{\top } \varvec{\imath }$. Here, $Y^{*} = {{\,{\mathrm{diag}}\,}}(y^{*}_1, \ldots , y_n^{*})$, $Y^{\dagger } = {{\,{\mathrm{diag}}\,}}(y_1^{\dagger }, \ldots , y_n^{\dagger })$, $M^{*} = {{\,{\mathrm{diag}}\,}}(\mu ^{*}_1, \ldots , \mu _n^{*})$, $M^{\dagger } = {{\,{\mathrm{diag}}\,}}(\mu _1^{\dagger }, \ldots , \mu _n^{\dagger })$, $V^{*} = {{\,\mathrm{diag}\,}}(\upsilon _1^{*}, \ldots , \upsilon _n^{*})$, $V^{\dagger } = {{\,\mathrm{diag}\,}}(\upsilon _1^{\dagger }, \ldots , \upsilon _n^{\dagger })$, $C = {{\,\mathrm{diag}\,}}(c^{*\dagger }_1,\ldots , c^{*\dagger }_n)$, $S = {{\,\mathrm{diag}\,}}(g''(\mu _1,\lambda ), \ldots , g''(\mu _n,\lambda ))$, $Q = {{\,\mathrm{diag}\,}}(h''(\phi _1), \ldots , h''(\phi _n))$, $\varvec{w} = (w_1, \ldots ,w_n)^{\top }$, $\varvec{\zeta } = (\zeta _1, \ldots ,\zeta _n)^{\top }$, where $w_t = \partial (\partial \mu _t/\partial \eta _{1t})/\partial \lambda$, $\zeta _t = \partial ^2\mu _t/\partial \lambda ^2$ and double primes denote second derivatives; see the end of this appendix for details.

Since ${\mathbb {E}}\left( \partial \ell _t(\mu _t, \phi _t)/\partial \mu _t\right) = {\mathbb {E}}\left( \partial \ell _t(\mu _t, \phi _t)/\partial \phi _t \right) = 0$, Fisher’s information matrix is given by

$$\begin{aligned} K \equiv K(\varvec{\theta }) = \left( \begin{array}{ccc} K_{(\varvec{\beta },\varvec{\beta })} &{}\quad K_{(\varvec{\beta },\varvec{\gamma })} &{}\quad K_{(\varvec{\beta },\lambda )} \\ K_{(\varvec{\gamma },\varvec{\beta })} &{}\quad K_{(\varvec{\gamma },\varvec{\gamma })} &{}\quad K_{(\varvec{\gamma },\lambda )} \\ K_{(\lambda ,\varvec{\beta })} &{}\quad K_{(\lambda , \varvec{\gamma })} &{}\quad K_{(\lambda ,\lambda )} \\ \end{array} \right) , \end{aligned}$$

where $K_{(\varvec{\beta },\varvec{\beta })} = X^{\top } \varPhi ^2 V^{*} T^2 X$, $K_{(\varvec{\beta },\varvec{\gamma })} = K_{(\varvec{\gamma },\varvec{\beta })}^{\top } = X^{\top } \varPhi (M V^{*} + C) T H Z$, $K_{(\varvec{\beta }, \lambda )} = K_{(\lambda ,\varvec{\beta })}^{\top } = X^{\top } \varPhi ^2 V^{*} T \varvec{\rho }$, $K_{(\varvec{\gamma },\varvec{\gamma })} = Z^{\top } H (M^2 V^{*} + 2M C + V^{\dagger }) H Z$, $K_{(\varvec{\gamma }, \lambda )} = K_{(\lambda , \varvec{\gamma })}^{\top } = Z^{\top } \varPhi (M V^{*} + C) H \varvec{\rho }$, and $K_{(\lambda , \lambda )} = \varvec{\rho }^{\top } \varPhi ^2 V^{*} \varvec{\rho }$.

In large samples and under the usual regularity conditions for maximum likelihood estimation, we have

$$\begin{aligned} \left( \begin{array}{l} \hat{\varvec{\beta }}\\ \hat{\varvec{\gamma }}\\ {\hat{\lambda }} \end{array} \right) \sim N_{k} \left( \begin{array}{l} \left( \begin{array}{l} \varvec{\beta } \\ \varvec{\gamma } \\ \lambda \end{array} \right) , K^{-1} \end{array} \right) , \end{aligned}$$

approximately, where $\hat{\varvec{\beta }}$, $\hat{\varvec{\gamma }}$, and ${\hat{\lambda }}$ are the MLEs of $\varvec{\beta }$, $\varvec{\gamma }$, and $\lambda$, respectively. In what follows, we shall use a result on inverses of partitioned matrices given by (Rao 1973, p. 33) to obtain a closed-form expression for $K^{-1}$.

Consider the symmetric matrix given by

$$\begin{aligned} I \equiv I(\varvec{\beta }, \varvec{\gamma }) = \left( \begin{array}{cc} I_{(\varvec{\beta },\varvec{\beta })} &{}\quad I_{(\varvec{\beta },\varvec{\gamma })}\\ I_{(\varvec{\gamma },\varvec{\beta })} &{}\quad I_{(\varvec{\gamma },\varvec{\gamma })} \\ \end{array} \right) , \end{aligned}$$

where $I_{(\varvec{\beta },\varvec{\beta })}= K_{(\varvec{\beta },\varvec{\beta })}$, $I_{(\varvec{\beta },\varvec{\gamma })}=K_{(\varvec{\beta },\varvec{\gamma })}$, $I_{(\varvec{\gamma },\varvec{\beta })}=K_{(\varvec{\gamma },\varvec{\beta })}$, and $I_{(\varvec{\gamma },\varvec{\gamma })}=K_{(\varvec{\gamma },\varvec{\gamma })}$. We denote its inverse as

$$\begin{aligned} I^{-1} \equiv I^{-1}(\varvec{\beta }, \varvec{\gamma }) = \left( \begin{array}{cc} I^{(\varvec{\beta },\varvec{\beta })} &{}\quad I^{(\varvec{\beta },\varvec{\gamma })}\\ I^{(\varvec{\gamma },\varvec{\beta })} &{}\quad I^{(\varvec{\gamma },\varvec{\gamma })} \\ \end{array} \right) . \end{aligned}$$

Let $\varDelta = X^{\top } \varPhi (M V^{*} + C) T H Z Z^{\top } H^{\top } T^{\top } (M V^{*} + C)^{\top } \varPhi ^{\top } X (X^{\top } \varPhi ^2 V^{*} T^2 X)^{-1}$. It can be shown that

$$\begin{aligned} I^{(\varvec{\beta },\varvec{\beta })} = (X^{\top } \varPhi ^2 V^{*} T^2 X)^{-1} \left( I_{p} + \frac{\varDelta }{\omega } \right) , \end{aligned}$$

with

$$\begin{aligned} \omega&= Z^{\top } H (M^2 V^{*} + 2M C + V^{\dagger }) H Z - Z^{\top } H^{\top } T^{\top } (M V^{*}\\&\quad +\,\, C)^{\top } \varPhi ^{\top } X (X^{\top } \varPhi ^2 V^{*} T^2 X)^{-1} X^{\top } \varPhi (M V^{*} + C) T H Z, \end{aligned}$$

$I_p$ denoting the $p \times p$ identity matrix. Additionally,

$$\begin{aligned} I^{(\varvec{\beta },\varvec{\gamma })}=(I^{(\varvec{\gamma },\varvec{\beta })})^{\top } = - \frac{(X^{\top } \varPhi ^2 V^{*} T^2 X)^{-1} X^{\top } \varPhi (M V^{*} + C) T H Z}{\omega } \end{aligned}$$

and $I^{(\varvec{\gamma },\varvec{\gamma })} = \omega ^{-1}$. We obtain

$$\begin{aligned} K^{-1} \equiv K^{-1}(\varvec{\theta }) = \left( \begin{array}{ccc} K^{(\varvec{\beta },\varvec{\beta })} &{}\quad K^{(\varvec{\beta },\varvec{\gamma })} &{}\quad K^{(\varvec{\beta },\lambda )} \\ K^{(\varvec{\gamma },\varvec{\beta })} &{}\quad K^{(\varvec{\gamma },\varvec{\gamma })} &{}\quad K^{(\varvec{\gamma },\lambda )} \\ K^{(\lambda ,\varvec{\beta })} &{}\quad K^{(\lambda , \varvec{\gamma })} &{}\quad K^{(\lambda ,\lambda )} \\ \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} K^{(\varvec{\beta },\varvec{\beta })}&= I^{(\varvec{\beta },\varvec{\beta })} + \varOmega [I^{(\varvec{\beta },\varvec{\beta })} X^{\top } \varPhi ^2 V^{*} T \varvec{\rho } + I^{(\varvec{\beta },\varvec{\gamma })}Z^{\top } \varPhi (M V^{*} + C) H \varvec{\rho }] \\&\quad \times {[}\varvec{\rho }^{\top } T^{\top } (V^{*})^{\top } (\varPhi ^2)^{\top } X I^{(\varvec{\beta },\varvec{\beta })} + \varvec{\rho }^{\top } H^{\top }(M V^{*} + C)^{\top } \varPhi ^{\top } Z I^{(\varvec{\gamma },\varvec{\beta })}],\\ K^{(\varvec{\gamma },\varvec{\gamma })}&= I^{(\varvec{\gamma },\varvec{\gamma })} + \varOmega [I^{(\varvec{\gamma },\varvec{\beta })} X^{\top } \varPhi ^2 V^{*} T \varvec{\rho } + I^{(\varvec{\gamma },\varvec{\gamma })}Z^{\top } \varPhi (M V^{*} + C) H \varvec{\rho }] \\&\quad \times {[}\varvec{\rho }^{\top } T^{\top } (V^{*})^{\top } (\varPhi ^2)^{\top } X I^{(\varvec{\beta },\varvec{\gamma })} + \varvec{\rho }^{\top } H^{\top }(M V^{*} + C)^{\top } \varPhi ^{\top } Z I^{(\varvec{\gamma },\varvec{\gamma })}],\\ K^{(\varvec{\beta },\varvec{\gamma })}&= (K^{(\varvec{\gamma },\varvec{\beta })})^{\top } = I^{(\varvec{\beta },\varvec{\gamma })} + \varOmega [I^{(\varvec{\beta },\varvec{\beta })} X^{\top } \varPhi ^2 V^{*} T \varvec{\rho } + I^{(\varvec{\beta },\varvec{\gamma })} Z^{\top } \varPhi (M V^{*}\\&\quad +\,\, C) H \varvec{\rho } ] \times [ \varvec{\rho }^{\top } T^{\top } (V^{*})^{\top } (\varPhi ^2)^{\top } X I^{(\varvec{\beta },\varvec{\gamma })} + \varvec{\rho }^{\top } H^{\top }(M V^{*}\\&\quad +\,\, C)^{\top } \varPhi ^{\top } Z I^{(\varvec{\gamma },\varvec{\gamma })}],\\ K^{(\varvec{\beta },\lambda )}&= (K^{(\lambda ,\varvec{\beta })})^{\top } = - \varOmega [ I^{(\varvec{\beta },\varvec{\beta })} X^{\top } \varPhi ^2 V^{*} T \varvec{\rho } + I^{(\varvec{\beta },\varvec{\gamma })} Z^{\top } \varPhi (M V^{*} + C) H \varvec{\rho } ],\\ K^{(\varvec{\gamma },\lambda )}&= (K^{(\lambda ,\varvec{\gamma })})^{\top } = - \varOmega [ I^{(\varvec{\gamma },\varvec{\beta })} X^{\top } \varPhi ^2 V^{*} T \varvec{\rho } + I^{(\varvec{\gamma },\varvec{\gamma })} Z^{\top } \varPhi (M V^{*} + C) H \varvec{\rho }],\\ K^{(\lambda ,\lambda )}&= \varOmega , \end{aligned}$$

with

$$\begin{aligned} \varOmega&= \{ \varvec{\rho }^{\top } \varPhi ^2 V^{*} \varvec{\rho } - {[}\varvec{\rho }^{\top } T^{\top } (V^{*})^{\top } (\varPhi ^2)^{\top } X I^{(\varvec{\beta },\varvec{\beta })} X^{\top } \varPhi ^2 V^{*} T \varvec{\rho } + \varvec{\rho }^{\top } H^{\top } (M V^{*}\\&\quad +\,\, C)^{\top } \varPhi ^{\top } Z I^{(\varvec{\gamma },\varvec{\beta })} X^{\top } \varPhi ^2 V^{*} T \varvec{\rho } + \varvec{\rho }^{\top } T^{\top } (V^{*})^{\top } (\varPhi ^2)^{\top } X I^{(\varvec{\beta },\varvec{\gamma })} Z^{\top } \varPhi (M V^{*} \\&\quad +\,\,C) H \varvec{\rho } + \varvec{\rho }^{\top } H^{\top } (M V^{*} + C)^{\top } \varPhi ^{\top } Z I^{(\varvec{\gamma },\varvec{\gamma })} Z^{\top } \varPhi (M V^{*} +C) H \varvec{\rho } ] \}^{-1}. \end{aligned}$$

Finally, we note the following results:

$$\begin{aligned} g'(\mu _t,\lambda )&= \dfrac{\lambda (1 - \mu _t)^{-(\lambda + 1)}}{(1 - \mu _t)^{-\lambda } - 1},\\ g''(\mu _t,\lambda )&= \frac{\lambda - (1 -\mu _t)^{\lambda } \lambda (1 + \lambda )}{(\mu _t - 1)^2 [(1 - \mu _t)^{\lambda } - 1]^2},\\ \dfrac{\partial \mu _t}{\partial \eta _{1t}}&= \exp (\eta _{1t})(1 + \lambda \exp (\eta _{1t}))^{\frac{-(1 + \lambda )}{\lambda }},\\ \dfrac{\partial }{\partial \lambda }\left( \dfrac{\partial \mu _t}{\partial \eta _{1t}}\right)&= \frac{\exp (\eta _{1t})(1 + \lambda \exp (\eta _{1t}))^{-2 - \frac{1}{\lambda }}}{\lambda ^2} \big [ \big (-\lambda \exp (\eta _{1t})(1 + \lambda ) \\&\quad +\,\, (1 + \lambda \exp (\eta _{1t}))\log (1 + \lambda \exp (\eta _{1t})\big )\big ],\\ \rho _t&= \frac{\partial \mu _t}{\partial \lambda }= \frac{1}{\lambda }\left[ \frac{1}{e^{-\eta _{1t}} + \lambda } - \frac{\log (1 + \lambda e^{\eta _{1t}})}{\lambda }\right] (1 + \lambda e^{\eta _{1t}})^{-1/\lambda },\\ \zeta _t&= \dfrac{\partial ^2 \mu _t }{\partial \lambda ^2} = \frac{\big (1 + \lambda e^{\eta _{1t}}\big )^{-\frac{1}{\lambda } - 2}}{\lambda ^4} \big \{ \big (1 + \lambda e^{\eta _{1t}}\big ) \log \big (1 + \lambda e^{\eta _{1t}}\big )\\&\quad \times \big [2 \lambda \big (1 + e^{\eta _{1t}}(1 + \lambda )\big )- \big (1 + \lambda e^{\eta _{1t}}\big ) \log \big (1 + \lambda e^{\eta _{1t}}\big )\big ] \\&\quad - \lambda ^2 e^{\eta _{1t}}\big ((3 \lambda + 1) e^{\eta _{1t}} + 2 \big )\big \}. \end{aligned}$$

Appendix B

In this appendix, we provide details on the derivation of the corrected likelihood test statistics for the varying precision beta regression model with parametric mean link function. Note that $\bar{\varvec{q}}$ is obtained from

$$\begin{aligned} \varvec{q} = {\mathbb {E}}_{\varvec{\theta _1}} [\varvec{U}(\varvec{\theta _1})(\ell (\varvec{\theta _1}) - \ell (\varvec{\theta }))] = \left[ \begin{array}{cccc} {\mathbb {E}}_{\varvec{\theta _1}} [\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1})\ell (\varvec{\theta _1})] - {\mathbb {E}}_{\varvec{\theta _1}} [\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1})\ell (\varvec{\theta })]\\ {\mathbb {E}}_{\varvec{\theta _1}} [\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1})\ell (\varvec{\theta _1})] - {\mathbb {E}}_{\varvec{\theta _1}} [\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1})\ell (\varvec{\theta })]\\ {\mathbb {E}}_{\varvec{\theta _1}} [\varvec{U}_{\lambda } (\varvec{\theta _1})\ell (\varvec{\theta _1})] -{\mathbb {E}}_{\varvec{\theta _1}} [\varvec{U}_{\lambda } (\varvec{\theta _1})\ell (\varvec{\theta })]\\ \end{array} \right] . \end{aligned}$$

From (4) and (5), we have

$$\begin{aligned} {\mathbb {E}}_{\varvec{\theta }} [\varvec{U}_{\varvec{\beta }}(\varvec{\theta })\ell (\varvec{\theta })]&= {\mathbb {E}}_{\varvec{\theta }}\big \{ X^{\top } \varPhi T (\varvec{y}^{*} - \varvec{\mu }^{*}) [(\varvec{y}^{*} - \varvec{\mu }^{*})^{\top }(\varPhi M - I_n) \\&\quad +\,\, (\varvec{y}^{\dagger } -\varvec{\mu }^{\dagger })^{\top }(\varPhi - 2I_n) + \varvec{b}^{\top }] \varvec{\imath } \big \}\\&= X^{\top } \varPhi T \big \{\mathbb {E}_{\varvec{\theta }} [(\varvec{y}^{*} - \varvec{\mu }^{*})(\varvec{y}^{*} -\varvec{\mu }^{*})^{\top }](\varPhi M - I_n) \\&\quad +\,\, {\mathbb {E}}_{\varvec{\theta }}[(\varvec{y}^{*} - \varvec{\mu }^{*})(\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })^{\top }] (\varPhi - 2I_n) \\&\quad +\,\, {\mathbb {E}}_{\varvec{\theta }}[(\varvec{y}^{*} - \varvec{\mu }^{*})] \varvec{b}^{\top } \big \}\varvec{\imath } \\&= X^{\top } \varPhi T \left\{ V^{*}(\varPhi M - I_n) + C(\varPhi - 2I_n) \right\} \varvec{\imath }. \end{aligned}$$

For all $t \not = u$, $y_t$ and $y_u$ are independent, and ${\mathbb {E}}_{\theta _1}(y_t^{*} - \mu _t^{{*}(1)}) = 0$. Therefore, ${\mathbb {E}}_{\theta _1}[(y_t^{*} - \mu _t^{*(1)}) (y_u^{*} - \mu _u^{*})] = 0$ and ${\mathbb {E}}_{\theta _1}[(y_t^{*} - \mu _t^{*(1)}) (y_t^{*} - \mu _t^{*})] = {\mathbb {E}}_{\theta _1}[(y_t^{*} - \mu _t^{*(1)}) (y_t^{*} - \mu _t^{*(1)})] + {\mathbb {E}}_{\theta _1}[(y_t^{*} - \mu _t^{*(1)}) (\mu _t^{*(1)} - \mu _t^{*})] = {\mathbb {E}}_{\theta _1}[(y_t^{*} - \mu _t^{*(1)})^2] = \upsilon _t^{*(1)}$. Evaluation at $\theta _1$ is indicated by the superscript ‘(1)’.

After some algebra, we arrive at

$$\begin{aligned} {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1})\ell (\varvec{\theta })] = X^{\top } \varPhi ^{(1)} T^{(1)} \big \{ V^{*(1)}(\varPhi M - I_n) + C^{(1)}(\varPhi - 2I_n) \big \} \varvec{\imath }. \end{aligned}$$

Thus,

$$\begin{aligned} {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1})\ell (\varvec{\theta _1})] - {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1})\ell (\varvec{\theta })]&= X^{\top } \varPhi ^{(1)} T^{(1)} \big \{ V^{*(1)}(\varPhi ^{(1)} M^{(1)} - I_n) + C^{(1)}(\varPhi ^{(1)}\\&\quad - 2I_n) \big \}\varvec{\imath } - X^{\top } \varPhi ^{(1)} T^{(1)} \big \{ V^{*(1)}(\varPhi M - I_n)\\&\quad +\,\, C^{(1)}(\varPhi - 2I_n) \big \}\varvec{\imath }\\&= X^{\top } \varPhi ^{(1)} T^{(1)} \big \{ V^{*(1)}(\varPhi ^{(1)} M^{(1)} - \varPhi M)\\&\quad +\,\, C^{(1)}(\varPhi ^{(1)} - \varPhi ) \big \}\varvec{\imath }. \end{aligned}$$

Using Equations (4) and (6), we obtain

$$\begin{aligned}&{\mathbb {E}}_{\varvec{\theta }}[\varvec{U}_{\varvec{\gamma }}(\varvec{\theta })\ell (\varvec{\theta })] \\&\quad = {\mathbb {E}}_{\varvec{\theta }} \big \{ Z^{\top } H [M(\varvec{y}^{*} -\varvec{\mu }^{*}) + (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })][(\varvec{y}^{*} - \varvec{\mu }^{*})^{\top } (\varPhi M - I_n) \\&\qquad + (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })^{\top } (\varPhi - 2I_n) + \varvec{b}^{\top }]\varvec{\imath } \big \} \\&\quad = Z^{\top } H \big \{ M {\mathbb {E}}_{\varvec{\theta }}[(\varvec{y}^{*} -\varvec{\mu }^{*}) (\varvec{y}^{*} - \varvec{\mu }^{*})^{\top }](\varPhi M - I_n)\\&\qquad + M {\mathbb {E}}_{\varvec{\theta }}[(\varvec{y}^{*} - \varvec{\mu }^{*})(\varvec{y}^{\dagger } -\varvec{\mu }^{\dagger })^{\top }](\varPhi - 2I_n)\\&\qquad + {\mathbb {E}}_{\varvec{\theta }} [(\varvec{y}^{\dagger } -\varvec{\mu }^{\dagger })(\varvec{y}^{*} -\varvec{\mu }^{*})^{\top }](\varPhi M - I_n) \\&\qquad + {\mathbb {E}}_{\varvec{\theta }} [(\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })(\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })^{\top }](\varPhi - 2I_n) \big \} \varvec{\imath }\\&\quad = Z^{\top } H \left\{ M V^{*} (\varPhi M - I_n) + M C (\varPhi - 2I_n) + C(\varPhi M - I_n) + V^{\dagger }(\varPhi - 2I_n) \right\} \varvec{\imath }\\&\quad = Z^{\top } H \left\{ (M V^{*} + C) (\varPhi M - I_n) + (M C + V^{\dagger }) (\varPhi - 2I_n) \right\} \varvec{\imath }. \end{aligned}$$

Hence,

$$\begin{aligned} {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1})\ell (\varvec{\theta _1})] - {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1})\ell (\varvec{\theta })]&= Z^{\top } H^{(1)} \big \{ (M^{(1)} V^{*(1)}+ C^{(1)}) (\varPhi ^{(1)} M^{(1)} - I_n) \\&\quad +\,\, (M^{(1)} C^{(1)} + V^{\dagger (1)} ) \times (\varPhi ^{(1)} - 2I_n) \big \} \varvec{\imath }\\&\quad - Z^{\top } H^{(1)} \big \{ (M^{(1)} V^{*(1)} + C^{(1)}) (\varPhi M - I_n) \\&\quad +\,\, (M^{(1)} C^{(1)} + V^{\dagger (1)}) (\varPhi - 2I_n) \big \} \varvec{\imath } \\&= Z^{\top } H^{(1)} \big \{ (M^{(1)} V^{*(1)} + C^{(1)}) (\varPhi ^{(1)} M^{(1)} - \varPhi M) \\&\quad +\,\, (M^{(1)} C^{(1)} + V^{\dagger (1)}) (\varPhi ^{(1)} - \varPhi ) \big \} \varvec{\imath }. \end{aligned}$$

Similarly, from (4) and (7) it follows that

$$\begin{aligned} {\mathbb {E}}_{\varvec{\theta }}[\varvec{U}_{\lambda }(\varvec{\theta })\ell (\varvec{\theta })]&= {\mathbb {E}}_{\varvec{\theta }} \big \{ \varvec{\rho }^{\top } \varPhi (\varvec{y}^{*} - \varvec{\mu }^{*})[(\varvec{y}^{*} - \varvec{\mu }^{*})^{\top } (\varPhi M - I_n) \\&\quad +\,\, (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })^{\top }(\varPhi - 2I_n) + \varvec{b}^{\top }] \varvec{\imath } \big \}\\&= \varvec{\rho }^{\top } \varPhi \big \{ {\mathbb {E}}_{\varvec{\theta }}[(\varvec{y}^{*} - \varvec{\mu }^{*})(\varvec{y}^{*} -\varvec{\mu }^{*})^{\top }] (\varPhi M - I_n)\\&\quad +\,\, {\mathbb {E}}_{\varvec{\theta }}[(\varvec{y}^{*} - \varvec{\mu }^{*})(\varvec{y}^{\dagger }-\varvec{\mu }^{\dagger })^{\top }](\varPhi - 2I_n) \big \} \varvec{\imath }\\&= \varvec{\rho }^{\top } \varPhi \big \{ V^{*}(\varPhi M - I_n) + C (\varPhi - 2I_n) \big \} \varvec{\imath }. \end{aligned}$$

Thus,

$$\begin{aligned} {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\lambda }(\varvec{\theta _1})\ell (\varvec{\theta _1})] - {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\lambda }(\varvec{\theta _1})\ell (\varvec{\theta })]&= \varvec{\rho }^{\top (1)} \varPhi ^{(1)} \big \{ V^{*(1)} (\varPhi ^{(1)} M^{(1)} - I_n) \\&\quad +\,\, C^{(1)} (\varPhi ^{(1)}- 2I_n) \big \} \varvec{\imath } - \varvec{\rho }^{\top (1)} \varPhi ^{(1)} \big \{ V^{*(1)} (\varPhi M - I_n) \\&\quad +\,\, C^{(1)} (\varPhi - 2I_n) \big \} \varvec{\imath }\\&= \varvec{\rho }^{\top (1)} \varPhi ^{(1)} \big \{ V^{*(1)} (\varPhi ^{(1)} M^{(1)} - \varPhi M ) \\&\quad +\,\, C^{(1)} (\varPhi ^{(1)}- \varPhi ) \big \} \varvec{\imath }. \end{aligned}$$

We shall now move to the derivation of ${\bar{\varUpsilon }}$, which is obtained from

$$\begin{aligned} \varUpsilon =\left[ \begin{array}{ccc} {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\beta }}^{\top }(\varvec{\theta })] &{}\quad {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\gamma }}^{\top }(\varvec{\theta })] &{}\quad {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1}) \varvec{U}_{\lambda }^{\top }(\varvec{\theta })]\\ {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\beta }}^{\top }(\varvec{\theta })] &{}\quad {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\gamma }}^{\top }(\varvec{\theta })] &{}\quad {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1}) \varvec{U}_{\lambda }^{\top }(\varvec{\theta })]\\ {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\lambda }(\varvec{\theta _1}) \varvec{U}_{\varvec{\beta }}^{\top }(\varvec{\theta })] &{}\quad {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\lambda }(\varvec{\theta _1}) \varvec{U}_{\varvec{\gamma }}^{\top }(\varvec{\theta })] &{}\quad {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\lambda }(\varvec{\theta _1}) \varvec{U}_{\lambda }^{\top }(\varvec{\theta })] \end{array} \right] . \end{aligned}$$

From Eqs. (5), (6), and (7), we obtain

$$\begin{aligned} {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\beta }}^{\top }(\varvec{\theta })]&= {\mathbb {E}}_{\varvec{\theta _1}} \big \{ X^{\top } \varPhi ^{(1)} T^{(1)} (\varvec{y}^{*} - \varvec{\mu }^{*(1)}) [X^{\top } \varPhi T (\varvec{y}^{*} - \varvec{\mu }^{*})]^{\top } \big \} \\&= X^{\top } \varPhi ^{(1)} T^{(1)} V^{*(1)} T \varPhi X,\\ {\mathbb {E}}_{\varvec{\theta _1}}\big [\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\gamma }}^{\top }(\varvec{\theta })\big ]&= {\mathbb {E}}_{\varvec{\theta _1}}\big \{ X^{\top } \varPhi ^{(1)} T^{(1)} (\varvec{y}^{*} -\varvec{\mu }^{*(1)}) \big [Z^{\top } H \big [M(\varvec{y}^{*} - \varvec{\mu }^{*}) + (\varvec{y}^{\dagger } -\varvec{\mu }^{\dagger })\big ] \big ]^{\top } \big \}\\&= X^{\top } \varPhi ^{(1)} T^{(1)} \big \{ V^{*(1)}M + C^{(1)} \big \} H Z, \\ {\mathbb {E}}_{\varvec{\theta _1}}\big [\varvec{U}_{\varvec{\beta }}(\varvec{\theta _1}) \varvec{U}_{\lambda }^{\top }(\varvec{\theta })\big ]&= {\mathbb {E}}_{\varvec{\theta _1}} \big \{ X^{\top } \varPhi ^{(1)} T^{(1)} (\varvec{y}^{*} - \varvec{\mu }^{*(1)}) \big [\varvec{\rho }^{\top } \varPhi (\varvec{y}^{*} - \varvec{\mu }^{*})\big ]^{\top } \big \} \\&= X^{\top } \varPhi ^{(1)} T^{(1)} V^{*(1)} \varPhi \varvec{\rho }, \\ {\mathbb {E}}_{\varvec{\theta _1}}\big [\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\beta }}^{\top }(\varvec{\theta })\big ]&= {\mathbb {E}}_{\varvec{\theta _1}}\big \{ Z^{\top } H^{(1)} \big [M^{(1)} (\varvec{y}^{*} - \varvec{\mu }^{*(1)}) + (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger (1)})\big ] \\&\quad \times \big [X^{\top } \varPhi T (\varvec{y}^{*} - \varvec{\mu }^{*})\big ]^{\top } \big \} \\&= Z^{\top } H^{(1)} \big \{ M^{(1)} V^{*(1)} + C^{(1)} \big \} T \varPhi X, \\ {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1}) \varvec{U}_{\varvec{\gamma }}^{\top }(\varvec{\theta })]&= {\mathbb {E}}_{\varvec{\theta _1}} \big \{ Z^{\top } H^{(1)} [M^{(1)}(\varvec{y}^{*} - \varvec{\mu }^{*(1)}) + (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger (1)})] [Z^{\top } H [M(\varvec{y}^{*} - \varvec{\mu }^{*}) \\&\quad +\,\, (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })]]^{\top } \big \} \\&= Z^{\top } H^{(1)} \big \{ M^{(1)} V^{*(1)} M + (M^{(1)} + M) C^{(1)} + V^{\dagger (1)} \big \} H Z,\\ {\mathbb {E}}_{\varvec{\theta _1}}\big [\varvec{U}_{\varvec{\gamma }}(\varvec{\theta _1}) \varvec{U}_{\lambda }^{\top }(\varvec{\theta })\big ]&= {\mathbb {E}}_{\varvec{\theta _1}} \big \{ Z^{\top } H^{(1)} \big [M^{(1)} (\varvec{y}^{*} - \varvec{\mu }^{*(1)}) + (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger (1)})\big ] \big [\varvec{\rho }^{\top } \varPhi (\varvec{y}^{*} - \varvec{\mu }^{*})\big ]^{\top } \big \}\\&= Z^{\top } H^{(1)} \big \{ M^{(1)} V^{*(1)} + C^{(1)} \big \} \varPhi \varvec{\rho },\\ {\mathbb {E}}_{\varvec{\theta _1}}\big [\varvec{U}_{\lambda }(\varvec{\theta _1}) \varvec{U}_{\varvec{\beta }}^{\top }(\varvec{\theta })\big ]&= {\mathbb {E}}_{\varvec{\theta _1}} \big \{ \varvec{\rho }^{\top (1)} \varPhi ^{(1)} (\varvec{y}^{*} - \varvec{\mu }^{*(1)})\big [X^{\top } \varPhi T (\varvec{y}^{*} - \varvec{\mu }^{*})\big ]^{\top } \big \} \\&= \varvec{\rho }^{\top (1)} \varPhi ^{(1)} V^{*(1)} T \varPhi X, \\ {\mathbb {E}}_{\varvec{\theta _1}}\big [\varvec{U}_{\lambda }(\varvec{\theta _1}) \varvec{U}_{\varvec{\gamma }}^{\top }(\varvec{\theta })\big ]&= {\mathbb {E}}_{\varvec{\theta _1}} \big \{ \varvec{\rho }^{\top (1)} \varPhi ^{(1)} (\varvec{y}^{*} - \varvec{\mu }^{*(1)})\big [Z^{\top } H \big [M (\varvec{y}^{*} - \varvec{\mu }^{*}) + (\varvec{y}^{\dagger } - \varvec{\mu }^{\dagger })\big ]\big ]^{\top } \big \} \\&= \varvec{\rho }^{\top (1)} \varPhi ^{(1)} \big \{ V^{*(1)} M + C^{(1)} \big \} H Z,\\ {\mathbb {E}}_{\varvec{\theta _1}}[\varvec{U}_{\lambda }(\varvec{\theta _1}) \varvec{U}_{\lambda }^{\top }(\varvec{\theta })]&= {\mathbb {E}}_{\varvec{\theta _1}} \big \{ \varvec{\rho }^{\top (1)} \varPhi ^{(1)} (\varvec{y}^{*} - \varvec{\mu }^{*(1)})[\varvec{\rho }^{\top } \varPhi (\varvec{y}^{*} -\varvec{\mu }^{*})]^{\top } \big \} \\&= \varvec{\rho }^{\top (1)} \varPhi ^{(1)} V^{*(1)} \varPhi \varvec{\rho }.\\ \end{aligned}$$

Appendix C

In this appendix, we provide details on the derivation of the score test statistic for testing ${\mathcal {H}}_0: \lambda = 1$ (logit link).

The general form of the score test statistic is $S_R=U(\tilde{\varvec{\theta }})' K^{-1}(\tilde{\varvec{\theta }})U(\tilde{\varvec{\theta }}),$ where $U(\tilde{\varvec{\theta }})$ is the score vector and $K^{-1}(\tilde{\varvec{\theta }})$ is Fisher’s information inverse matrix, both evaluated at $\tilde{\varvec{\theta }}$. When the interest lies in testing ${\mathcal {H}}_0: \lambda =1$ in the varying precision beta regression model with parametric link function, the score test statistic can be expressed as

$$\begin{aligned} S_R=U_{\lambda }(\tilde{\varvec{\theta }})' {\tilde{K}}^{(\lambda \lambda )}U_{\lambda }(\tilde{\varvec{\theta }}), \end{aligned}$$

where

$$\begin{aligned} U_{\lambda }(\tilde{\varvec{\theta }}) = \sum _{t=1}^{n} \left\{ {\tilde{\phi }}_t (y^{*}_t-{\tilde{\mu }}^{*}_t)\left[ \dfrac{1}{2+e^{-{\tilde{\eta }}_{1t}} + e^{{\tilde{\eta }}_{1t}}} - \dfrac{\log (1 + e^{{\tilde{\eta }}_{1t}})}{1 + e^{{\tilde{\eta }}_{1t}}}\right] \right\} \end{aligned}$$

and

$$\begin{aligned} {\tilde{K}}^{(\lambda \lambda )}&= \{ \tilde{\varvec{\rho }}^{\top } {\tilde{\varPhi }}^2 {\tilde{V}}^{*} \tilde{\varvec{\rho }} - [\tilde{\varvec{\rho }}^{\top } {\tilde{T}}^{\top } ({\tilde{V}}^{*})^{\top } ({\tilde{\varPhi }}^2)^{\top } X {\tilde{I}}^{(\varvec{\beta },\varvec{\beta })} X^{\top } {\tilde{\varPhi }}^2 {\tilde{V}}^{*} {\tilde{T}} \tilde{\varvec{\rho }} \\&\quad +\,\, \tilde{\varvec{\rho }}^{\top } {\tilde{H}}^{\top } ({\tilde{M}} {\tilde{V}}^{*} + {\tilde{C}})^{\top } {\tilde{\varPhi }}^{\top } Z {\tilde{I}}^{(\varvec{\gamma },\varvec{\beta })} X^{\top } {\tilde{\varPhi }}^2 {\tilde{V}}^{*} {\tilde{T}} \tilde{\varvec{\rho }}\\&\quad +\,\, \tilde{\varvec{\rho }}^{\top } {\tilde{T}}^{\top } ({\tilde{V}}^{*})^{\top } ({\tilde{\varPhi }}^2)^{\top } X {\tilde{I}}^{(\varvec{\beta },\varvec{\gamma })} Z^{\top } {\tilde{\varPhi }} ({\tilde{M}} {\tilde{V}}^{*} + {\tilde{C}}) {\tilde{H}} \tilde{\varvec{\rho }} \\&\quad +\, \tilde{\varvec{\rho }}^{\top } {\tilde{H}}^{\top } ({\tilde{M}} {\tilde{V}}^{*} + {\tilde{C}})^{\top } {\tilde{\varPhi }}^{\top } Z {\tilde{I}}^{(\varvec{\gamma },\varvec{\gamma })} Z^{\top } {\tilde{\varPhi }} ({\tilde{M}} {\tilde{V}}^{*} + {\tilde{C}}) {\tilde{H}} \tilde{\varvec{\rho }} ] \}^{-1}. \end{aligned}$$

Under the null hypothesis and when n is large, $S_R$ is approximately $\chi ^2_1$ distributed.