Inflated beta autoregressive moving average models

Abstract

In this paper, we introduce the inflated beta autoregressive moving average (I$\beta$ARMA) models for modeling and forecasting time series data that assume values in the intervals (0,1], [0,1) or [0,1]. The proposed model considers a set of regressors, an autoregressive moving average structure and a link function to model the conditional mean of inflated beta conditionally distributed variable observed over the time. We develop partial likelihood estimation and derive closed-form expressions for the score vector and the cumulative partial information matrix. Hypotheses testing, confidence interval, some diagnostic tools and forecasting are also proposed. We evaluate the finite sample performances of partial maximum likelihood estimators and confidence interval using Monte Carlo simulations. Two empirical applications related to forecasting hydro-environmental data are presented and discussed.

Appendix A Score vector and cumulative partial information matrix

In this appendix, we shall derive the partial score vector and the cumulative partial information matrix from (5). These are useful for the asymptotic theory and inference as well as numerical considerations.

1.1 A.1 Score vector and optimization algorithm

To obtain the partial score vector, we shall need to obtain the derivative of the log-likelihood $\ell (\varvec{\gamma })$ given in (5) with respect to each coordinate $\gamma _j$, with $j \in 1,\dots ,\kappa$, of the parameter $\varvec{\gamma }$. To obtain the derivative of $\ell (\varvec{\gamma })$ with respect to $\alpha _i$, $i=0,1$, observe that, in view of (5),

$$\begin{aligned}\frac{\partial c_t}{\partial \alpha _i}=(\mu _t-1)I_0(i)-\mu _tI_1(i)\quad \text{ and }\quad \frac{\partial \nu _t}{\partial \alpha _i}=\frac{(-1)^i(\alpha _{1-i}-1)(\mu _t-1)\mu _t}{c_t^2}.\end{aligned}$$

Now, for $i = 0,1$, it is straightforward to show that

$$\begin{aligned} \frac{\partial \ell (\varvec{\gamma })}{\partial \alpha _i} =\sum _{t=m+1}^{n} \frac{\partial \ell _t(\varvec{\gamma })}{\partial \alpha _i}&=\frac{1}{\alpha _i}I_i(y_t)+\bigg (\frac{1}{c_t}\left[ \frac{\partial c_t}{\partial \alpha _i}\right] + \phi (y_t^*-\mu _t^*) \left[ \frac{\partial \nu _t}{\partial \alpha _i}\right] \bigg )I_{(0,1)}(y_t), \end{aligned}$$

where $y_t^ {*} := \log \Big (\frac{y_t}{1-y_t}\Big ), \quad \mu _t^ {*} :=\psi (\nu _t\phi )-\psi \big ((1-\nu _t)\phi \big )$ and $\psi :(0,\infty )\rightarrow \mathbb {R}$ is the digamma function defined as $\psi (z)=\frac{d}{dz}\log \big (\Gamma (z)\big )$. The derivative with respect to $\phi$ is easy to obtain:

$$\begin{aligned} \frac{\partial \ell (\varvec{\gamma })}{\partial \phi }&=\sum _{t=m+1}^{n} \Big [\nu _t(y_t^*-\mu _t^*)+\log (1-y_t)-\psi \big ((1-\nu _t)\phi \big )+\psi (\phi )\Big ]I_{(0,1)}(y_t). \end{aligned}$$

For the remaining parameters, i.e., for $j \in 4,\dots ,\kappa$, by the chain rule, and since $\eta _t=g(\mu _t)$, $\displaystyle {\frac{d \mu _t}{d \eta _t} = \frac{1}{g'(\mu _t)}}$, so that

$$\begin{aligned} \frac{\partial \ell (\varvec{\gamma })}{\partial \gamma _j} = \sum _{t=m+1}^{n} \frac{1}{g^\prime (\mu _t)}\frac{\partial \ell _t(\varvec{\gamma })}{\partial \mu _t}\frac{\partial \eta _t}{\partial \gamma _j}\,. \end{aligned}$$

(A1)

Observe that $\frac{\partial \nu _t}{\partial \mu _t}=(\alpha _0-1)(\alpha _1-1)c_t^{-2}$ and

$$\begin{aligned} \frac{\partial \ell _t(\varvec{\gamma })}{\partial \mu _t}&=\bigg (\frac{\alpha _0-\alpha _1}{c_t}+\phi (y_t^*-\mu _t^*)\bigg [\frac{(\alpha _0-1)(\alpha _1-1)}{c_t^2}\bigg ] \bigg )I_{(0,1)}(y_t) \nonumber \\&\quad + \frac{1}{\mu _t}I_1(y_t)-\frac{1}{1-\mu _t}I_0(y_t). \end{aligned}$$

(A2)

Substituting (A2) into expression (A1), we obtain a simple formula that allows the computation of $\partial \ell (\varvec{\gamma })/\partial \gamma _j$ for each remaining coordinate $\gamma _j$, by determining the derivatives $\partial \eta _t/\partial \gamma _j$, a much simpler task. We have

$$\begin{aligned} \frac{\partial \eta _t}{\partial \alpha } =1-\sum _{k=1}^q\theta _k \frac{1}{g^\prime (\mu _{t-k})} \frac{\partial \eta _{t-k}}{\partial \alpha } \quad \text{ and }\quad \frac{\partial \eta _t}{\partial \beta _l} = x_{tl}- \sum _{k=1}^q \theta _k \frac{1}{g^\prime (\mu _{t-k})} \frac{\partial \eta _{t-k}}{\partial \beta _l}, \end{aligned}$$

where $x_{tl}$ denotes the lth element of $\varvec{x}_t$, for $l=1,\dots ,r$. We also have, for $l=1,\dots ,p$, and $j=1,\dots ,q$,

$$\begin{aligned} \frac{\partial \eta _t}{\partial \varphi _l}=y_{t-l}- \sum _{k=1}^q \theta _k \frac{1}{g^\prime (\mu _{t-k})} \frac{\partial \eta _{t-k}}{\partial \varphi _l}\quad \text{ and }\quad \frac{\partial \eta _t}{\partial \theta _j}=r_{t-j} - \sum _{k=1}^q \theta _k \frac{1}{g^\prime (\mu _{t-k})} \frac{\partial \eta _{t-k}}{\partial \theta _j}. \end{aligned}$$

Let $T=\textrm{diag}\big \lbrace 1/g'\left( {\mu }_{m+1}\right) ,\dots , 1/g'\left( {\mu }_{n}\right) \big \rbrace$, ${{\textbf {a}}}=\left( \frac{\partial \eta _{m+1}}{\partial \alpha },\dots ,\frac{\partial \eta _n}{\partial \alpha }\right) ^\top$ and $\varvec{v}=\left( \frac{\partial \ell _{m+1}(\varvec{\gamma })}{\partial \mu _{m+1}},\dots ,\frac{\partial \ell _{n}(\varvec{\gamma })}{\partial \mu _{n}}\right) ^\top$. Finally, let R, P, Q be the matrices with dimension $(n-m)\times r$, $(n-m)\times p$ and $(n-m)\times q$, respectively, for which the (i, j)th elements are given by

$$\begin{aligned} R_{i,j}=\frac{\partial \eta _{i+m}}{\partial \beta _j}, \quad P_{i,j}=\frac{\partial \eta _{i+m}}{\partial \varphi _j}, \quad \text{ and } \quad Q_{i,j}=\frac{\partial \eta _{i+m}}{\partial \theta _j} \end{aligned}$$

and set $U_{\alpha }(\varvec{\gamma }):= {{\textbf {a}}}^\top T \varvec{v}$, $U_{\varvec{\beta }}(\varvec{\gamma }):=R^\top T \varvec{v}$, $U_{\varvec{\varphi }}(\varvec{\gamma }):=P^\top T \varvec{v}$ and $U_{\varvec{\theta }}(\varvec{\gamma }):=Q^\top T \varvec{v}$.

For $U_{\alpha _j}(\varvec{\gamma }):= \frac{\partial \ell (\varvec{\gamma })}{\partial \alpha _j}$, and $U_{\phi }(\varvec{\gamma }):= \frac{\partial \ell (\varvec{\gamma })}{\partial \phi }$, then the partial score vector is given by

$$\begin{aligned}U(\varvec{\gamma })= \big ( U_{\alpha _0}(\varvec{\gamma }),U_{\alpha _1}(\varvec{\gamma }), U_{\phi }(\varvec{\gamma }),U_{\alpha }(\varvec{\gamma }),U_{\varvec{\beta }}(\varvec{\gamma })^\top \, U_{\varvec{\varphi }}(\varvec{\gamma })^\top , U_{\varvec{\theta }}(\varvec{\gamma })^\top \big )^\top .\end{aligned}$$

The PMLE of $\varvec{\gamma }$, $\widehat{\varvec{\gamma }}$, is obtained as a solution of the non-linear system $U(\varvec{\gamma })=\varvec{0}$, where $\varvec{0}$ is the null vector in $\mathbb {R}^{\kappa }$. There is no closed form solution for such a system and, hence, PMLE must be obtained numerically (Nocedal and Wright 1999). In this work, we use the so-called Broyden–Fletcher–Goldfarb–Shanno (BFGS) method (Press et al. 1992). In practice, to calculate $\widehat{\varvec{\gamma }}$ from a sample, we initialize $r_{t}=0$ and $\mu _t=0$ for $t\le \max \{p,q\}$ and calculate $\mu _t$ and $r_t$ for $t>\max \{p,q\}$ recursively from the data using (4). The BFGS algorithm also requires initialization of the parameters. The starting values of $\alpha$, $\varvec{\beta }$ and $\varvec{\varphi }$ were set as the OLS estimate of

$$\begin{aligned} g(y_t) = \alpha +\varvec{x}_t^{\top } \varvec{\beta }+ \sum _{i=1}^p \varphi _i y_{t-i} + \text{ error } \text{ term }, \end{aligned}$$

restricted to the observations where $y\in (0,1)$. The vector parameter $\varvec{\theta }$ is initialized as a null vector, as in Bayer et al. (2017), while inflation parameters $\alpha _0$ and $\alpha _1$ were initialized as the sample proportion of zeroes and ones, respectively.

1.2 A.2 Cumulative partial information matrix

In this appendix we derive the cumulative partial information matrix, given by

$$\begin{aligned} K_n(\varvec{\gamma }) = -\sum ^n_{t=m+1}\mathbb {E}\left( \frac{\partial ^2 \ell _t(\mu _t,\phi )}{\partial \varvec{\gamma }\partial \varvec{\gamma }^\top } \Bigm \vert {{\mathscr {F}}}_{t-1} \right) . \end{aligned}$$

Since direct knowledge of the unconditional distribution of the proposed model is not obtainable, $K_n$ will be the first step toward finding the asymptotic variance-covariance matrix related to the PMLE. In this case, under suitable assumptions (Fokianos and Kedem 2004), there exists a non-random information matrix, denoted by $K(\varvec{\gamma })$, such that the weak convergence

$$\begin{aligned} \frac{K_n(\varvec{\gamma })}{n}\underset{n\rightarrow \infty }{\longrightarrow } K(\varvec{\gamma }), \end{aligned}$$

holds, where $K(\varvec{\gamma })$ is a positive definite and invertible matrix. The matrix $K(\varvec{\gamma })^{-1}$ is the asymptotic variance-covariance matrix related to the PMLE, presented in (6).

For $i,j\in \{4,\dots ,\kappa \}$ (that is, $\gamma _j\notin \{\alpha _0,\alpha _1,\phi \}$), it can be shown that

$$\begin{aligned} \frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \gamma _i \partial \gamma _j}&= \frac{\partial }{\partial \mu _t} \left( \frac{\partial \ell _t(\varvec{\gamma })}{\partial \mu _t}\frac{\partial \mu _t}{\partial \eta _t} \frac{\partial \eta _t}{\partial \gamma _j}\right) \frac{\partial \mu _t}{\partial \eta _t} \frac{\partial \eta _t}{\partial \gamma _i} \\&= \left[ \frac{\partial ^2 \ell _t(\varvec{\gamma })}{\partial \mu _t^2}\frac{\partial \mu _t}{\partial \eta _t} \frac{\partial \eta _t}{\partial \gamma _j} + \frac{\partial \ell _t(\varvec{\gamma })}{\partial \mu _t}\frac{\partial }{\partial \mu _t}\left( \frac{\partial \mu _t}{\partial \eta _t} \frac{\partial \eta _t}{\partial \gamma _j} \right) \right] \frac{\partial \mu _t}{\partial \eta _t} \frac{\partial \eta _t}{\partial \gamma _i}\,. \end{aligned}$$

Since by Lemma A.1, $\mathbb {E}\big (\partial \ell _t(\varvec{\gamma })/\partial \mu _t \mid {{\mathscr {F}}}_{t-1}\big )=0$, we arrive at

$$\begin{aligned} \mathbb {E}\left( \frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \gamma _i \partial \gamma _j} \Bigm \vert {{\mathscr {F}}}_{t-1} \right)&= \frac{1}{g'(\mu _t)^2} \mathbb {E}\left( \left. \frac{\partial ^2 \ell _t(\varvec{\gamma })}{\partial \mu _t^2} \right. \Bigm \vert {{\mathscr {F}}}_{t-1} \right) \frac{\partial \eta _t}{\partial \gamma _j} \frac{\partial \eta _t}{\partial \gamma _i}. \end{aligned}$$

The second-order derivatives of $\ell _t(\varvec{\gamma })$ with respect to $\mu _t$ is given by

$$\begin{aligned} \frac{\partial ^2 \ell _t(\varvec{\gamma })}{\partial \mu _t^2}&=\bigg (\frac{-(\alpha _0-\alpha _1)^2}{c_t^2}+\phi (\alpha _0-1)(\alpha _1-1)\frac{\partial }{\partial \mu _t} \bigg [\frac{(y_t^*-\mu _t^*)}{c_t^2}\bigg ]\bigg )I_{(0,1)}(y_t) \\&\quad -\frac{1}{\mu _t^2}I_1(y_t)-\frac{1}{(1-\mu _t)^2}I_0(y_t). \end{aligned}$$

Observe that

$$\begin{aligned} \frac{\partial \mu _t^*}{\partial \mu _t}=\frac{\partial \mu _t^*}{\partial \nu _t}\frac{\partial \nu _t}{\partial \mu _t}=\frac{\phi (\alpha _0-1)(\alpha _1-1)}{c_t^2} \Big [\psi '(\nu _t\phi )+\psi '\big ([1-\nu _t]\phi \big )\Big ]. \end{aligned}$$

We have, for $y_t\in (0,1)$,

$$\begin{aligned} \frac{\partial }{\partial \mu _t}\left[ \frac{y_t^*-\mu _t^*}{c_t^2}\right]&= -\frac{2(\alpha _0-\alpha _1)}{c_t^4}(y_t^*-\mu _t^*)\\&\quad -\frac{\phi (\alpha _0-1)(\alpha _1-1)}{c_t^4} \Big [ \psi '(\nu _t\phi )+\psi '\big ([1-\nu _t]\phi \big ) \Big ], \end{aligned}$$

hence

$$\begin{aligned} \mathbb {E}\bigg (\frac{\partial ^2 \ell _t(\varvec{\gamma })}{\partial \mu _t^2} \Bigm \vert {{\mathscr {F}}}_{t-1} \bigg )&= \frac{\alpha _0\mu _t+\alpha _1(1-\mu _t)}{\mu _t(\mu _t-1)}-\frac{(\alpha _0-\alpha _1)^2}{c_t}\\&\quad -\frac{\big [\phi (\alpha _0-1)(\alpha _1-1)\big ]^2}{c_t^3}\Big [\psi '(\nu _t\phi )+\psi '\big ([1-\nu _t]\phi \big )\Big ]. \end{aligned}$$

Second mixed derivatives related to $\alpha _0$ and $\alpha _1$ are obtained through direct differentiation of the log-likelihood. We have, for $i\in \{0,1\}$ and $j\in \{4,\dots ,\kappa \}$

$$\begin{aligned} \frac{\partial ^2 \ell _t(\varvec{\gamma })}{\partial \gamma _j\partial \alpha _i}= \bigg [\frac{\partial }{\partial \gamma _j}\bigg (\frac{1}{c_t}\left[ \frac{\partial c_t}{\partial \alpha _i}\right] \bigg )+ \frac{\partial }{\partial \gamma _j}\bigg (\phi (y_t^*-\mu _t^*) \left[ \frac{\partial \nu _t}{\partial \alpha _i}\right] \bigg )\bigg ] I_{(0,1)}(y_t), \end{aligned}$$

which, by Lemma (A.1), yields

$$\begin{aligned} \mathbb {E}\bigg (\frac{\partial ^2 \ell _t(\varvec{\gamma })}{\partial \gamma _j\partial \alpha _i} \Bigm \vert {{\mathscr {F}}}_{t-1} \bigg )= \bigg [ \frac{\partial }{\partial \gamma _j}\bigg (\frac{1}{c_t}\left[ \frac{\partial c_t}{\partial \alpha _i}\right] \bigg ) -\phi \frac{\partial \nu _t}{\partial \alpha _i}\frac{\partial \mu _t^*}{\partial \mu _t}\frac{\partial \mu _t}{\partial \eta _t}\frac{\partial \eta _t}{\partial \gamma _j}\bigg ]c_t. \end{aligned}$$

Writing $\ \displaystyle {\frac{\partial c_t}{\partial \gamma _j}=\frac{\partial c_t}{\partial \mu _t}\frac{\partial \mu _t}{\partial \eta _t}\frac{\partial \eta _t}{\partial \gamma _j}=\frac{\alpha _0-\alpha _1}{g'(\mu _t)}\frac{\partial \eta _t}{\partial \gamma _j}}$, we have

$$\begin{aligned} \frac{\partial }{\partial \gamma _j}\bigg (\frac{1}{c_t}\left[ \frac{\partial c_t}{\partial \alpha _i}\right] \bigg )=\frac{1}{c_tg'(\mu _t)} \bigg [ (-1)^i+\big [(\mu _t-1)I_0(i)-\mu _tI_1(i)\big ]\frac{\alpha _0-\alpha _1}{c_t} \bigg ]\frac{\partial \eta _t}{\partial \gamma _j}, \end{aligned}$$

and thus $\displaystyle {\mathbb {E}\bigg (\frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \gamma _j \partial \alpha _i} \Bigm \vert {{\mathscr {F}}}_{t-1} \bigg ) =\frac{s_t^{(i)}}{g'(\mu _t)}\frac{\partial \eta _t}{\partial \gamma _j},}$ where

$$\begin{aligned} s_t^{(i)}:&= \frac{(-1)^i\phi ^2(\alpha _0-1)^{i+1}(\alpha _1-1)^{2-i}(1-\mu _t)\mu _t}{c_t^3} \Big [\psi '(\nu _t\phi )+\psi '\big ([1-\nu _t]\phi \big )\Big ]\nonumber \\&\quad +(-1)^i+ +\big [(\mu _t-1)I_0(i)-\mu _tI_1(i)\big ]\frac{\alpha _0-\alpha _1}{c_t}. \end{aligned}$$

(A3)

For $j\in \{4,\dots ,\kappa \}$, it is easy to show that

$$\begin{aligned} \frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \gamma _j \partial \phi }&=-\frac{\partial }{\partial \gamma _j}\Big [ \nu _t\mu _t^*+\psi \big ([1-\nu _t]\phi \big )\Big ]I_{(0,1)}(y_t)\nonumber \\&=-\bigg (\frac{(1-\alpha _0)(1-\alpha _1)}{c_t^3g'(\mu _t)}\Big [ c_t\mu _t^*+\phi (1-\alpha _1)\psi '(\nu _t\phi )-\phi \alpha _1\psi '\big ([1-\nu _t]\phi \big )\Big ]\frac{\partial \eta _t}{\partial \gamma _j}\bigg ) I_{(0,1)}(y_t). \end{aligned}$$

(A4)

Observe that, except for the indicator function, all terms in (A4) are ${{\mathscr {F}}}_{t-1}$-measurable, so that

$$\begin{aligned} \mathbb {E}\bigg (\frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \gamma _j \partial \phi }\Bigm \vert {{\mathscr {F}}}_{t-1}\bigg )=\frac{d_t}{g'(\mu _t)}\frac{\partial \eta _t}{\partial \gamma _j}, \end{aligned}$$

where

$$\begin{aligned} d_t:=\frac{(1-\alpha _0)(1-\alpha _1)\phi }{c_t^2} \Big [ (1-\nu _t)\psi '\big ([1-\nu _t]\phi \big )-\nu _t \psi '(\nu _t\phi )\Big ]. \end{aligned}$$

(A5)

For $i\in \{0,1\}$,

$$\begin{aligned} \frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \alpha _i \partial \phi }&=\bigg ((y_t^*-\mu _t^*)\frac{\partial \nu _t}{\partial \alpha _i} +\bigg [\phi \psi '\big ([1-\nu _t]\phi \big )-\nu _t\frac{\partial \mu _t^*}{\partial \nu _t} \bigg ]\frac{\partial \nu _t}{\partial \alpha _i}\bigg )I_{(0,1)}(y_t). \end{aligned}$$

The first term has conditional expectation 0 (Lemma A.1), so that

$$\begin{aligned} \mathbb {E}\bigg (\frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \alpha _i \partial \phi } \Bigm \vert {{\mathscr {F}}}_{t-1}\bigg )=&\frac{(-1)^i(\alpha _{1-i}-1)(\mu _t-1)\mu _t \phi }{c_t^2}\Big [ (1-\nu _t)\psi '\big ([1-\nu _t]\phi \big ) - \nu _t\psi '(\nu _t\phi ) \Big ]. \end{aligned}$$

Since $\displaystyle {\frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \phi ^2}=\Big [\psi '(\phi )-\nu _t^2\psi '(\nu _t\phi )- (1-\nu _t)^2\psi '\big ([1-\nu _t]\phi \big )\Big ]I_{(0,1)}(y_t),}$ we have

$$\begin{aligned} \mathbb {E}\bigg (\frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \phi ^2} \Bigm \vert {{\mathscr {F}}}_{t-1}\bigg )=c_t\Big [\psi '(\phi )-\nu _t^2\psi '(\nu _t\phi )-(1-\nu _t)^2\psi '\big ([1-\nu _t]\phi \big )\Big ]. \end{aligned}$$

Finally, for $i,j\in \{0,1\}$,

$$\begin{aligned} \frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \alpha _j \partial \alpha _i}=\bigg [ \phi (y_t^*-\mu _t^*)\frac{\partial ^2\nu _t}{\partial \alpha _j \partial \alpha _i}-\phi \frac{\partial \nu _t}{\partial \alpha _i}\frac{\partial \mu _t^*}{\partial \alpha _j} -\frac{1}{c_t^2}\frac{\partial c_t}{\partial \alpha _i}&\frac{\partial c_t}{\partial \alpha _j} \bigg ]I_{(0,1)}(y_t)-\frac{1}{\alpha _i^2}I_i(j)I_i(y_t). \end{aligned}$$

Upon observing that $P(y_t=i)=\alpha _i(1-i+(-1)^i\mu _t)$, it follows that

$$\begin{aligned} \mathbb {E}\bigg (\frac{\partial ^2\ell _t(\varvec{\gamma })}{\partial \alpha _j \partial \alpha _i}\Bigm \vert {{\mathscr {F}}}_{t-1}\bigg )&= \frac{I_i(j)(i-1+(-1)^{i}\mu _t)}{\alpha _i}-c_t\bigg [ \phi \frac{\partial \nu _t}{\partial \alpha _i}\frac{\partial \mu _t^*}{\partial \alpha _j}+\frac{1}{c_t^2}\frac{\partial c_t}{\partial \alpha _i}\frac{\partial c_t}{\partial \alpha _j} \bigg ]\\&=\frac{I_i(j)(i-1+(-1)^{i}\mu _t)}{\alpha _i}\\&\quad -c_t\phi ^2\Big [\psi '(\nu _t\phi )+\psi '\big ([1-\nu _t]\phi \big )\Big ] \frac{\partial \nu _t}{\partial \alpha _i}\frac{\partial \nu _t}{\partial \alpha _j}-\frac{1}{c_t}\frac{\partial c_t}{\partial \alpha _i}\frac{\partial c_t}{\partial \alpha _j}. \end{aligned}$$

For $i,j\in \{0,1\}$, let

$$\begin{aligned} A_{\{i,j\}}&:=\textrm{diag}\left\{ \mathbb {E}\bigg ( \frac{\partial \ell _{m+1} (\mu _{m+1},\varphi )}{\partial \alpha _i\partial \alpha _j} \Bigm \vert {{\mathscr {F}}}_{m}\bigg ), \dots ,\mathbb {E}\bigg ( \frac{\partial \ell _{n} (\mu _{n},\varphi )}{\partial \alpha _i\partial \alpha _j} \Bigm \vert {{\mathscr {F}}}_{n-1}\bigg ) \right\} ,\\ B_i&:=\textrm{diag}\left\{ \mathbb {E}\bigg (\frac{\partial ^2\ell _{m+1}(\varvec{\gamma })}{\partial \alpha _i \partial \phi } \Bigm \vert {{\mathscr {F}}}_{m}\bigg ),\dots , \mathbb {E}\bigg (\frac{\partial ^2\ell _{n}(\varvec{\gamma })}{\partial \alpha _i \partial \phi } \Bigm \vert {{\mathscr {F}}}_{n-1}\bigg )\right\} ,\\ C&:=\textrm{diag}\left\{ \mathbb {E}\bigg (\frac{\partial ^2\ell _{m+1}(\varvec{\gamma })}{\partial \phi ^2} \Bigm \vert {{\mathscr {F}}}_{m}\bigg ),\dots ,\mathbb {E}\bigg (\frac{\partial ^2\ell _n(\varvec{\gamma })}{\partial \phi ^2} \Bigm \vert {{\mathscr {F}}}_{n-1}\bigg )\right\} ,\\ V&:=\textrm{diag}\left\{ \mathbb {E}\bigg (\frac{\partial ^2\ell _{m+1}(\varvec{\gamma })}{\partial \mu _{m+1}^2} \Bigm \vert {{\mathscr {F}}}_{m}\bigg ),\dots ,\mathbb {E}\bigg (\frac{\partial ^2\ell _n(\varvec{\gamma })}{\partial \mu _n^2} \Bigm \vert {{\mathscr {F}}}_{n-1}\bigg )\right\} , \end{aligned}$$

$\varvec{s}_i:=(s_{m+1}^{(i)},\dots ,s_n^{(i)})^\top$ and $\varvec{d}:=(d_{m+1},\dots ,d_n)^\top$, where $s_t^{(i)}$ and $d_t$ are given in (A3) and (A5), respectively. Thus, the joint cumulative partial information matrix for $\varvec{\gamma }$ based on a sample of size n is

$$\begin{aligned} K_n(\varvec{\gamma }) := - \left( \begin{array}{cccccccc} K_{(\alpha _0,\alpha _0)} &{}K_{(\alpha _0,\alpha _1)}&{}K_{(\alpha _0,\phi )} &{}K_{(\alpha _0,\alpha )} &{}K_{(\alpha _0,\varvec{\beta })} &{}K_{(\alpha _0,\varvec{\varphi })} &{}K_{(\alpha _0,\varvec{\theta })}\\ K_{(\alpha _1,\alpha _0)} &{}K_{(\alpha _1,\alpha _1)}&{}K_{(\alpha _1,\phi )} &{}K_{(\alpha _1,\alpha )} &{}K_{(\alpha _1,\varvec{\beta })} &{}K_{(\alpha _1,\varvec{\varphi })} &{}K_{(\alpha _1,\varvec{\theta })}\\ K_{(\phi ,\alpha _0)} &{}K_{(\phi ,\alpha _1)}&{}K_{(\phi ,\phi )} &{}K_{(\phi ,\alpha )} &{}K_{(\phi ,\varvec{\beta })} &{}K_{(\phi ,\varvec{\varphi })} &{}K_{(\phi ,\varvec{\theta })}\\ K_{(\alpha ,\alpha _0)} &{}K_{(\alpha ,\alpha _1)}&{}K_{(\alpha ,\phi )} &{}K_{(\alpha ,\alpha )} &{}K_{(\alpha ,\varvec{\beta })} &{}K_{(\alpha ,\varvec{\varphi })} &{}K_{(\alpha ,\varvec{\theta })}\\ K_{(\varvec{\beta },\alpha _0)} &{}K_{(\varvec{\beta },\alpha _1)}&{}K_{(\varvec{\beta },\phi )} &{}K_{(\varvec{\beta },\alpha )} &{}K_{(\varvec{\beta },\varvec{\beta })} &{}K_{(\varvec{\beta },\varvec{\varphi })} &{}K_{(\varvec{\beta },\varvec{\theta })}\\ K_{(\varvec{\varphi },\alpha _0)} &{}K_{(\varvec{\varphi },\alpha _1)}&{}K_{(\varvec{\varphi },\phi )} &{}K_{(\varvec{\varphi },\alpha )} &{}K_{(\varvec{\varphi },\varvec{\beta })} &{}K_{(\varvec{\varphi },\varvec{\varphi })} &{}K_{(\varvec{\varphi },\varvec{\theta })}\\ K_{(\varvec{\theta },\alpha _0)} &{}K_{(\varvec{\theta },\alpha _1)}&{}K_{(\varvec{\theta },\phi )} &{}K_{(\varvec{\theta },\alpha )} &{}K_{(\varvec{\theta },\varvec{\beta })} &{}K_{(\varvec{\theta },\varvec{\varphi })} &{}K_{(\varvec{\theta },\varvec{\theta })} \end{array} \right) , \end{aligned}$$

where, for $i,j\in \{0,1\}$, $K_{(\alpha _i,\alpha _j)} = \textrm{tr}(A_{\{i,j\}})$, $K_{(\alpha _i,\phi )}=K_{(\phi ,\alpha _i)}^\top =\textrm{tr}(B_i)$, $K_{(\alpha _i,\alpha )} =K_{(\alpha ,\alpha _i)} = \varvec{s}_i^\top T \varvec{a}$, $K_{(\alpha _i,\varvec{\beta })} =K_{(\varvec{\beta },\alpha _j)}^\top = \varvec{s}_i^\top T R$, $K_{(\alpha _i,\varvec{\varphi })} =K_{(\varvec{\varphi },\alpha _i)}^\top = \varvec{s}_i^\top T P$, $K_{(\alpha _i,\varvec{\theta })} =K_{(\varvec{\theta },\alpha _i)}^\top = \varvec{s}_i^\top T Q$, $K_{(\phi ,\phi )}=\textrm{tr}(C)$, $K_{(\phi ,\alpha )}=K_{(\alpha ,\phi )} = \varvec{d}^\top T \varvec{a}$, $K_{(\phi ,\varvec{\beta })} =K_{(\varvec{\beta },\phi )}^\top = \varvec{d}^\top T R$, $K_{(\phi ,\varvec{\varphi })} =K_{(\varvec{\varphi },\phi )}^\top = \varvec{d}^\top T P$, $K_{(\phi ,\varvec{\theta })} =K_{(\varvec{\theta },\phi )}^\top = \varvec{d}^\top T Q$, $K_{(\alpha ,\alpha )} = \varvec{a}^\top T^2 V \varvec{a}$, $K_{(\alpha ,\beta )} = K_{(\beta ,\alpha )}^\top = \varvec{a}^\top T^2 V R$, $K_{(\alpha ,\varphi )} = K_{(\varphi ,\alpha )}^\top = \varvec{a}^\top T^2 V P$, $K_{(\alpha ,\theta )} = K_{(\theta ,\alpha )}^\top = \varvec{a}^\top T^2 V Q$, $K_{(\varvec{\beta },\varvec{\beta })} = R^\top T^2 V R$, $K_{(\varvec{\beta },\varvec{\varphi })}=K_{(\varvec{\varphi },\varvec{\beta })}^\top = R^\top T^2 V P$, $K_{(\varvec{\beta },\varvec{\theta })}=K_{(\varvec{\theta },\varvec{\beta })}^\top = R^\top T^2 V Q$, $K_{(\varvec{\varphi },\varvec{\varphi })} = P^\top T^2 V P$, $K_{(\varvec{\varphi },\varvec{\theta })}=K_{(\varvec{\theta },\varvec{\varphi })}^\top = P^\top T^2 V Q$, $K_{(\varvec{\theta },\varvec{\theta })} = Q^\top T^2 V Q$.

Lemma A.1

With the notation in A.1,

$$\begin{aligned} \mathbb {E}\big ((y_t^*-\mu _t^*)I_{(0,1)}(y_t)\mid {{\mathscr {F}}}_{t-1}\big )=0\quad \text{ and }\quad \mathbb {E}\bigg (\frac{\partial \ell _t(\mu _t,\nu )}{\partial \mu _t} \Bigm \vert {{\mathscr {F}}}_{t-1}\bigg )=0. \end{aligned}$$

Proof

Observe that

$$\begin{aligned}\mathbb {E}\big ((y_t^*-\mu _t^*)I_{(0,1)}(y_t)\mid {{\mathscr {F}}}_{t-1}\big )=c_t\int _0^1\left( \frac{\log (x)}{\log (1-x)}-\mu _t^*\right) \texttt{b}(x;\nu _t,\phi )dx=0,\end{aligned}$$

by standard results on the beta distribution. Hence

$$\begin{aligned} \mathbb {E}\bigg (\frac{\partial \ell _t(\mu _t,\nu )}{\partial \mu _t} \Bigm \vert {{\mathscr {F}}}_{t-1}\bigg )=&\left( \frac{1}{\mu _t}\right) P(y_t=1 \mid {{\mathscr {F}}}_{t-1})-\left( \frac{1}{1-\mu _t}\right) P(y_t=0 \mid {{\mathscr {F}}}_{t-1})\\&+\left( \frac{\alpha _0-\alpha _1}{c_t}\right) P\big (y_t\in (0,1) \mid {{\mathscr {F}}}_{t-1}\big )\\ =&\left( \frac{\alpha _0-\alpha _1}{c_t}\right) c_t-\left( \frac{1}{1-\mu _t}\right) \alpha _0(1-\mu _t)+ \left( \frac{1}{\mu _t}\right) \alpha _1\mu _t=0, \end{aligned}$$

as asserted. $\square$