Recursive Computation of the Conditional Probability Function of the Quadratic Exponential Model for Binary Panel Data

Abstract

We propose a general recursive algorithm for the computation of the conditional probability function of the quadratic exponential model for binary panel data given the total of the responses, which is a sufficient statistic for the individual intercept parameter. This recursion permits to implement conditional and pseudo-conditional maximum likelihood estimators of the parameters of this model, and related models such as the dynamic logit model, even when one or more statistical units are observed at many occasions. In this way we solve a typical problem in dealing with distributions with a complex normalizing constant. The advantage in terms of computational load with respect to standard techniques is assessed by simulation and illustrated by an application based on a popular dataset about brand loyalty.

Appendices

Appendix: Derivatives

In the following, we report the first and second derivatives for the relevant functions of the proposed recursive algorithm for the static logit, simplified, modified, and approximating QE models. These derivatives are then used to compute the score vector and Information matrix involved in the steps of the Newton-Raphson algorithm, in order to maximize the conditional log-likelihoods.

Static Logit Model

Concerning computation of the derivatives presented in Sect. 3.1 we can exploit the same recursive structure presented in Eq. (17), so that:

• 1. for $t=1$ the first derivatives are

  $$\begin{aligned} f^{(h)}_{1,0}({\varvec{\phi }})&= 0,\quad h=1,\ldots ,T_i,\\ f^{(h)}_{1,1}({\varvec{\phi }})&= \left\{ \begin{array}{ll} \exp (\phi _1), & h=1, \\ 0, & \text{otherwise}, \end{array}\right. \\ \end{aligned}$$

  and the second derivatives are

  $$\begin{aligned} f^{(h,j)}_{1,0}({\varvec{\phi }})&= 0,\quad h,j=1,\ldots ,T_i,\\ f^{(j,j)}_{1,1}({\varvec{\phi }})&= \left\{ \begin{array}{ll}\exp (\phi _1),& h=j=1,\\ 0, & \text{otherwise}; \end{array}\right. \end{aligned}$$

• 2. for $t=2,\ldots ,T_i$ and $s=1,\ldots ,t$, are the first derivatives are

  $$\begin{aligned} f^{(h)}_{t,0}({\varvec{\phi }})&= {} 0,\quad h=1,\ldots ,T_i,\\ f^{(h)}_{t,s}({\varvec{\phi }})&= {} \left\{ \begin{array}{ll} f_{t-1,s}^{(h)}({\varvec{\phi }})+f_{t-1,s-1}^{(h)}({\varvec{\phi }})\exp (\phi _t), &{} h=1,\ldots ,t-1,\\ f_{t-1,s-1}({\varvec{\phi }})\exp (\phi _t), &{} h=t, \\ 0, &{} \text{otherwise}, \end{array}\right. \end{aligned}$$

  and second derivatives are

  $$\begin{aligned}&f^{(h,j)}_{t,0}({\varvec{\phi }}) = 0, \quad h,j=1,\ldots ,T_i, \\&f^{(h,j)}_{t,s}({\varvec{\phi }}) = \left\{ \begin{array}{ll} f_{t-1,s}^{(h,j)}({\varvec{\phi }})+f_{t-1,s-1}^{(h,j)}({\varvec{\phi }})\exp (\phi _t), &{} h,j=1,\ldots ,t-1,\\ f_{t-1,s-1}^{(h)}({\varvec{\phi }})\exp (\phi _t), &{} h=1,\ldots ,t-1,\, j=t\\ f_{t-1,s-1}^{(j)}({\varvec{\phi }})\exp (\phi _t), &{} h=t,\,j=1,\ldots ,t-1, \\ f_{t-1,s-1}({\varvec{\phi }})\exp (\phi _t), &{} h=j=t, \\ 0, &{} \text{otherwise}. \end{array}\right. \end{aligned}$$

Simplified QE Model

As for the static model, define now the first and the second derivatives

$$\begin{aligned} g^{(h)}_{t,s}({\varvec{\phi }})= & {} \frac{\partial g_{t,s}({\varvec{\phi }})}{\partial \phi _h},\\ g^{(h,j)}_{t,s}({\varvec{\phi }})= & {} \frac{\partial ^2 g_{t,s}({\varvec{\phi }})}{\partial \phi _h\partial \phi _j}, \end{aligned}$$

respectively, where these quantities are computed for $h,j=1,\ldots ,T_i$, $s=0,\ldots ,t$, and $t=1,\ldots ,T_i$ and where we further define $\phi _{T_i+1}=\psi$ in order to include the derivative with respect to the state dependence parameter which is an additional argument of our function further to the $T_i$ elements of ${\varvec{\phi }}$.

Regarding the first derivatives of the function defined in (19), we exploit the same recursion:

• 1. when $t=1$ compute

  $$\begin{aligned} g_{1,a,s,v}^{(h)}({\varvec{\phi }},\psi )=\left\{ \begin{array}{ll} \exp (\phi _1+a\psi ),&{} h=1,\\ a\exp (\phi _1+a\psi ), &{} h=T_i+1, \end{array}\right. \end{aligned}$$

  for $a=0,1$ and $s=v=1$ and $g_{1,a,s,v}^{(h)}=0$ in all other cases;

• 2. when $t=2,\ldots ,T_i$ consider the following cases:

  • for $s=1,\ldots ,t-1$ and $v=0$:

    $$\begin{aligned}&g^{(h)}_{t,a,s,v}({\varvec{\phi }},\psi )= g_{t-1,a,s,0}^{(h)}({\varvec{\phi }},\psi )+g_{t-1,a,s,1}^{(h)}({\varvec{\phi }},\psi ), \quad h=1,\ldots ,t-1,T_i +1; \end{aligned}$$

  • for $s=1,\ldots ,t$, $v=1$,

    $$\begin{aligned}&g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi ) \\&\quad =\left\{ \begin{array}{ll} g_{t-1,a,s-1,0}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t)+ g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=1,\ldots ,t-1,\\ \\ g_{t,a,s,v}({\varvec{\phi }},\psi ),&{}h=t,\\ \\ g_{t-1,a,s-1,0}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad +g_{t-1,a,s-1,1}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=T_i+1;\end{array}\right. \end{aligned}$$

  • $g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi )=0$ in all other cases.

Regarding the second derivatives, we have:

• 1. when $t=1$ compute

  $$\begin{aligned} g_{1,a,s,v}^{(h,j)}({\varvec{\phi }},\psi )=\left\{ \begin{array}{ll} \exp (\phi _1+a\psi ),&{} h=j=1,\\ a\exp (\phi _1+a\psi ),&{} h=1,\,j=T_i+1,\\ a\exp (\phi _1+a\psi ), &{} h=T_i+1,\,j=1,\\ a\exp (\phi _1+a\psi ), &{} h,j=T_i+1, \end{array}\right. \end{aligned}$$

  for $a=0,1$ and $s=v=1$ and $g_{1,a,s,v}^{(h)}=0$ in all other cases.

• 2. when $t=2,\ldots ,T_i$ consider the following cases:

  • for $s=1,\ldots ,t-1$ and $v=0$,

    $$\begin{aligned}&g^{(h,j)}_{t,a,s,v}({\varvec{\phi }},\psi )= g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\psi )+g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\psi ), \quad h,j=1,\ldots ,t-1,T_i+1; \end{aligned}$$

  • for $s=1,\ldots ,t$, $v=1$,

    $$\begin{aligned}&g_{t,a,s,v}^{(h,j)}({\varvec{\phi }},\psi )\\&\quad =\left\{ \begin{array}{ll} g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h,j=1,\ldots ,t-1,\\ \\ g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi ), &{} h=1,\ldots ,t-1,\,j=t,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), \\ \quad +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=1,\ldots ,t-1,\,j=T_i+1,\\ \\ g^{(j)}_{t,a,s,v}({\varvec{\phi }},\psi ), \quad \quad h=t,\,j=1,\ldots ,t,T_i+1,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t)+ g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad +g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=T_i+1,\,j=1,\ldots ,t-1\\ \\ g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi ), &{} h=T_i+1,\,j=t,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t) + g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ) +g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad \quad \quad \quad +g_{t-1,a,s-1,1}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=T_i+1,\,j=T_i+1; \end{array}\right. \end{aligned}$$

  • $g_{t,a,s,v}^{(h,j)}({\varvec{\phi }},\psi )=0$ in all other cases.

Modified QE Model

Regarding the first derivatives of the function reported in Eq. (20), we exploit the same recursion:

• 1. with $t=1$

  • for $s=v=0$ compute

    $$\begin{aligned}&g_{1,a,s,v}^{(h)}({\varvec{\phi }},\psi )\\&\quad =\left\{ \begin{array}{ll} 0,&{} h=1,\\ (1-a)\exp[\phi _1+ (1-a)\psi ], &{} h=T_i+1; \end{array}\right. \end{aligned}$$

  • for $s=v=1$ compute

    $$\begin{aligned}&g_{1,a,s,v}^{(h)}({\varvec{\phi }},\psi )\\&\quad =\left\{ \begin{array}{ll} \exp (\phi _1+ a\psi ),&{} h=1,\\ a\exp (\phi _1+ a\psi ), &{} h=T_i+1; \end{array}\right. \end{aligned}$$

  • $g_{1,a,s,v}^{(h)}=0$ in all other cases.

• 2. with $t=2,\ldots ,T_i$ consider the following cases:

  • for $s=0,\ldots ,t-1$ and $v=0$,

    $$\begin{aligned}&g^{(h)}_{t,a,s,v}({\varvec{\phi }},\psi )\\&\quad =\left\{ \begin{array}{ll} 0, &{} h = 1,\ldots ,t-1,\\ \\ g_{t-1,a,s,0}^{(h)}({\varvec{\phi }},\psi )\exp (\psi ) + g_{t-1,a,s,0}\exp (\psi )\\ \quad +g_{t-1,a,s,1}^{(h)}({\varvec{\phi }},\psi ), &{} h=T_i+1;\\ \end{array}\right. \end{aligned}$$

  • for $s=1,\ldots ,t$, $v=1$,

    $$\begin{aligned}&g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi )\\&\quad =\left\{ \begin{array}{ll} g_{t-1,a,s-1,0}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ) \\ \quad +g_{t-1,a,s-1,0}({\varvec{\phi }},\psi )\exp (\phi _t)\\ \quad \quad +g_{t-1,a,s-1,1}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=1,\ldots ,t,\\ \\ g_{t-1,a,s-1,0}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad +g_{t-1,a,s-1,1}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=T_i+1;\end{array}\right. \end{aligned}$$

• 3. $g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi )=0$ in all other cases.

Regarding the second derivatives, we have:

• 1. for $t=1$ and $v=0$ compute

  $$\begin{aligned} g_{1,a,s,v}^{(h,j)}({\varvec{\phi }},\psi )=\begin{array}{ll} (1-a)\exp [(1-a)\psi ],&h,j=T_i+1; \end{array} \end{aligned}$$

• 2. for $t=1$ and $v=1$ compute

  $$\begin{aligned} g_{1,a,s,v}^{(h,j)}({\varvec{\phi }},\psi )=\left\{ \begin{array}{ll} \exp (\phi _1+ a\psi ),&{} h=j=1,\\ a\exp (\phi _1+ a\psi ),&{} h=1,\,j=T_i+1,\\ a\exp (\phi _1+ a\psi ), &{} h=T_i+1,\,j=1,\\ a\exp (\phi _1+ a\psi ), &{} h,j=T_i+1; \end{array}\right. \end{aligned}$$

  for $a=0,1$ and $s=v=1$ and $g_{1,a,s,v}^{(h)}=0$ in all other cases.

• 3. for $t=2,\ldots ,T_i$ consider the following cases:

  • for $s=1,\ldots ,t-1$ and $v=0$,

    $$\begin{aligned}&g^{(h,j)}_{t,a,s,v}({\varvec{\phi }},\psi )\\&\quad = \left\{ \begin{array}{ll} g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\psi ) +\exp (\psi )+g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\psi ) &{} h = j = 1,\ldots ,t-1;\\ \\ g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\psi ) +\exp (\psi )+g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\psi )\\ \quad + g_{t-1,a,s,0}^{(h)}({\varvec{\phi }},\psi )\exp (\psi ), &{} h=1,\ldots ,t-1,j = T_i+1;\\ \\ g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\psi ) +\exp (\psi )+g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\psi )\\ \quad + g_{t-1,a,s,0}^{(j)}({\varvec{\phi }},\psi )\exp (\psi ), &{} j=1,\ldots ,t-1,h = T_i+1;\\ \\ g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\psi ) +\exp (\psi )+g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\psi )\\ \quad + g_{t-1,a,s,0}^{(h)}({\varvec{\phi }},\psi )\exp (\psi ) + g_{t-1,a,s,0}^{(j)}({\varvec{\phi }},\psi )\exp (\psi )\\ \quad \quad + g_{t-1,a,s,0}({\varvec{\phi }},\psi )\exp (\psi ),&{} j = h = T_i+1. \end{array}\right. \end{aligned}$$

  • for $s=1,\ldots ,t$, $v=1$,

    $$\begin{aligned}&g_{t,a,s,v}^{(h,j)}({\varvec{\phi }},\psi )\\&\quad = \left\{ \begin{array}{ll} g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h,j=1,\ldots ,t-1,\\ \\ g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi ), &{} h=1,\ldots ,t-1,\,j=t,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), \\ \quad +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=1,\ldots ,t-1,\,j=T_i+1,\\ \\ g^{(j)}_{t,a,s,v}({\varvec{\phi }},\psi ), &{} h=t,\,j=1,\ldots ,t,T_i+1,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad +g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=T_i+1,\,j=1,\ldots ,t-1\\ \\ g_{t,a,s,v}^{(h)}({\varvec{\phi }},\psi ), &{} h=T_i+1,\,j=t,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ) +g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\psi )\exp (\phi _t+\psi )\\ \quad \quad +g_{t-1,a,s-1,1}({\varvec{\phi }},\psi )\exp (\phi _t+\psi ), &{} h=T_i+1,\,j=T_i+1; \end{array}\right. \end{aligned}$$

  • $g_{t,a,s,v}^{(h,j)}({\varvec{\phi }},\psi )=0$ in all other cases.

Approximating QE model

Regarding the first derivatives of the function in Eq. (21), compute:

• 1. with $t=1$,

  • for $s=v=0$ and $a=0,1$,

    $$\begin{aligned} g_{1,a,s,v}^{(h)}({\varvec{\phi }},\gamma )=\left\{ \begin{array}{ll} a\exp (a\nu _1)(- {\bar{r}}_{i1}), &{} h=T_i+1,\\ 0, &{} \text{otherwise}; \end{array}\right. \end{aligned}$$

  • for for $s=v=1$ and $a=0,1$,

    $$\begin{aligned} g_{1,a,s,v}^{(h)}({\varvec{\phi }},\gamma )=\left\{ \begin{array}{ll} \exp (\phi _1+a(\nu _1 + \gamma )),&{} h=1,\\ a\exp (\phi _1+a(\nu _1 + \gamma ))(1 - {\bar{r}}_{i1}), &{} h=T_i+1; \end{array}\right. \end{aligned}$$

  • $g_{1,a,s,v}^{(h)}=0$ in all other cases.

• 2. for $t=2,\ldots ,T_i$ consider the following cases:

  • for $s=1,\ldots ,t-1$ and $v=0$,

    $$\begin{aligned}&g_{t,a,s,v}^{(h)}({\varvec{\phi }},\gamma ) \\&\quad =\left\{ \begin{array}{ll} g_{t-1,a,s,0}^{(h)}({\varvec{\phi }},\gamma )+g_{t-1,a,s,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\nu _t), &{} h=1,\ldots ,t,\\ \\ g_{t-1,a,s,0}^{(h)}({\varvec{\phi }},\gamma )+g_{t-1,a,s,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\nu _t)\\ \quad +g_{t-1,a,s,1}({\varvec{\phi }},\gamma )\exp (\nu _t)(-{\bar{r}}_{it}), &{} h=T_i+1;\end{array}\right. \end{aligned}$$

  • for $s=1,\ldots ,t$, $v=1$,

    $$\begin{aligned}&g_{t,a,s,v}^{(h)}({\varvec{\phi }},\gamma ) \\&\quad =\left\{ \begin{array}{ll} g_{t-1,a,s-1,0}^{(h)}({\varvec{\phi }},\gamma )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad + g_{t-1,a,s-1,0}({\varvec{\phi }},\gamma )\exp (\phi _t) +g_{t-1,a,s-1,1}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma ), &{} h=1,\ldots ,t\\ \\ g_{t-1,a,s-1,0}^{(h)}({\varvec{\phi }},\gamma )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad +g_{t-1,a,s-1,1}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it}), &{} h=T_i+1;\end{array}\right. \end{aligned}$$

  • $g_{t,a,s,v}^{(h)}({\varvec{\phi }},\gamma )=0$ in all other cases.

Following the same approach as above, the second derivatives are:

• 1. for $t=1$,

  • with $v=0$ compute

    $$\begin{aligned} g_{1,a,s,0}^{(h,j)}({\varvec{\phi }},\psi )= \begin{array}{ll} a\exp (a\nu _1)(-{\bar{r}}_{i1})^2, &{} h,j=T_i+1;\\ \end{array} \end{aligned}$$

  • while for $v=1$ we have

    $$\begin{aligned}&g_{1,a,s,1}^{(h,j)}({\varvec{\phi }},\psi )\\&\quad =\left\{ \begin{array}{ll} \exp (\phi _1+a(\nu _1 + \gamma )),&{} h=j=1,\\ a\exp (\phi _1+a(\nu _1 + \gamma ))(1-{\bar{r}}_{i1}),&{} h=1,\,j=T_i+1,\\ a\exp (\phi _1+a(\nu _1 + \gamma ))(1-{\bar{r}}_{i1}), &{} h=T_i+1,\,j=1,\\ a\exp (\phi _1+a(\nu _1 + \gamma ))(1-{\bar{r}}_{i1})^2, &{} h,j=T_i+1; \end{array}\right. \end{aligned}$$

  • $g_{1,a,s,v}^{(h,j)}({\varvec{\phi }},\psi ) = 0$ in other cases.

• 2. with $t = 2,\dots ,T_i$:

  • for $s=1\dots t-1$ and $v=0$, we have:

    $$\begin{aligned}&g_{t,a,s,v}^{(h,j)}({\varvec{\phi }},\gamma ) \\&\quad = \left\{ \begin{array}{ll} g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\gamma ) +g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\nu _t), &{} h,j=1,\ldots ,t\\ \\ g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\gamma )+g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\nu _t) \\ \quad +g_{t-1,a,s,1}^{(j)}({\varvec{\phi }},\gamma )\exp (\nu _t)(-{\bar{r}}_{it}), &{} h=T_i+1,\,j=1,\ldots ,t\\ \\ g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\gamma ) +g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\nu _t) \\ \quad +g_{t-1,a,s,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\nu _t)(-{\bar{r}}_{it}), &{} h=1,\ldots ,t,\,j=T_i+1,\\ \\ g_{t-1,a,s,0}^{(h,j)}({\varvec{\phi }},\gamma ) +g_{t-1,a,s,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\nu _t) \\ \quad +g_{t-1,a,s,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\nu _t)(-{\bar{r}}_{it}) + g_{t-1,a,s,1}^{(j)}({\varvec{\phi }},\gamma )\exp (\nu _t)(-{\bar{r}}_{it})\\ \quad \quad + g_{t-1,a,s,1}\exp (\nu _t)(-{\bar{r}}_{it})^2, &{} h=T_i+1,\,j=T_i+1; \end{array}\right. \end{aligned}$$

  • for $s=1\dots t-1$ and $v=1$, we have:

    $$\begin{aligned}&g_{t,a,s,v}^{(h,j)}({\varvec{\phi }},\gamma ) \\&\quad = \left\{ \begin{array}{ll} g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad + g^{(h)}_{t-1,a,s-1,0}({\varvec{\phi }},\gamma )\exp (\phi _t) + g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad \quad + g^{(j)}_{t-1,a,s-1,0}({\varvec{\phi }},\gamma )\exp (\phi _t) + g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad \quad \quad +g_{t-1,a,s,0}({\varvec{\phi }},\gamma )\exp (\phi _t) + g_{t-1,a,s,1}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma ), &{} h,j=1,\ldots ,t,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad + g_{t-1,a,s-1,1}^{(h)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it}) +g^{(j)}_{t-1,a,s-1,0}({\varvec{\phi }},\gamma )\exp (\phi _t) \\ \quad \quad + g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma ) \\ \quad \quad \quad + g_{t-1,a,s,1}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it}), &{} h=1,\ldots ,t,\,j=T_i+1,\\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t) +g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad + g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it}) \\ \quad \quad + g^{(h)}_{t-1,a,s-1,0}({\varvec{\phi }},\gamma )\exp (\phi _t) + g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma ) \\ \quad \quad \quad + g_{t-1,a,s,1}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it}), &{} h=1,\ldots ,t,\,j=T_i+1,\\ \\ \\ g_{t-1,a,s-1,0}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t) + g_{t-1,a,s-1,1}^{(h,j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )\\ \quad +g^{(h)}_{t-1,a,s-1,0}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it}) \\ \quad \quad + g_{t-1,a,s-1,1}^{(j)}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it})\\ \quad \quad \quad + g_{t-1,a,s-1,1}({\varvec{\phi }},\gamma )\exp (\phi _t+\nu _t+\gamma )(1-{\bar{r}}_{it})^2, &{}h=T_i+1,\,j=T_i+1; \end{array}\right. \end{aligned}$$

• 3. $g_{t,a,s,v}^{(h,j)}({\varvec{\phi }},\gamma )=0$ in all other cases.