The COUNTREG Procedure

Panel Data Analysis

Subsections:

Panel Data Poisson Regression with Fixed Effects
Panel Data Poisson Regression with Random Effects
Panel Data Negative Binomial Regression with Fixed Effects
Panel Data Negative Binomial Regression with Random Effects

Panel Data Poisson Regression with Fixed Effects

The count regression model for panel data can be derived from the Poisson regression model. Consider the multiplicative one-way panel data model,

$y_{it} \sim \mbox{Poisson}(\mu _{it})$

where

$\mu _{it} = \alpha _{i} \lambda _{it} = \alpha _{i} \exp (\mathbf{x}_{it}’\bbeta ),\; \; i=1,\ldots ,N, \; \; t=1,\ldots ,T$

Here, $\alpha _{i}$ are the individual effects.

In the fixed-effects model, the $\alpha _{i}$ are unknown parameters. The fixed-effects model can be estimated by eliminating $\alpha _{i}$ by conditioning on $\sum _{t} y_{it}$ .

In the random-effects model, the $\alpha _{i}$ are independent and identically distributed (iid) random variables, in contrast to the fixed effects model. The random-effects model can then be estimated by assuming a distribution for $\alpha _{i}$ .

In the Poisson fixed-effects model, conditional on $\lambda _{it}$ and parameter $\alpha _{i}$ , $y_{it}$ is iid Poisson-distributed with parameter $\mu _{it}=\alpha _{i}\lambda _{it}=\alpha _{i} \exp (\mathbf{x}_{it}’\bbeta )$ , and $x_{it}$ does not include an intercept. Then, the conditional joint density for the outcomes within the $i$ th panel is

$\begin{eqnarray*} P[y_{i1},\ldots ,y_{iT_{i}}|\sum _{t=1}^{T_{i}}y_{it}] & = & P[y_{i1},\ldots ,y_{iT_{i}},\sum _{t=1}^{T_{i}}y_{it}] / P[\sum _{t=1}^{T_{i}}y_{it}] \\ & = & P[y_{i1},\ldots ,y_{iT_{i}}]/P[\sum _{t=1}^{T_{i}}y_{it}] \end{eqnarray*}$

Because $y_{it}$ is iid Poisson( $\mu _{it}$ ), $P[y_{i1},\ldots ,y_{iT_{i}}]$ is the product of $T_{i}$ Poisson densities. Also, $(\sum _{t=1}^{T_{i}} y_{it})$ is Poisson( $\sum _{t=1}^{T_{i}} \mu _{it}$ ). Then,

$\begin{eqnarray*} P[y_{i1},\ldots ,y_{iT_{i}}|\sum _{t=1}^{T_{i}}y_{it}] & = & \frac{\sum _{t=1}^{T_{i}} (\exp (-\mu _{it}) \mu _{it}^{y_{it}} / y_{it}! )}{\exp (-\sum _{t=1}^{T_{i}} \mu _{it}) \left( \sum _{t=1}^{T_{i}} \mu _{it} \right)^{\sum _{t=1}^{T_{i}} y_{it} } / \left( \sum _{t=1}^{T_{i}} y_{it} \right)!} \\ & = & \frac{\exp (-\sum _{t=1}^{T_{i}} \mu _{it}) \left( \prod _{t=1}^{T_{i}} \mu _{it}^{y_{it}} \right) \left( \prod _{t=1}^{T_{i}} y_{it}! \right) }{\exp ( -\sum _{t=1}^{T_{i}} \mu _{it}) \prod _{t=1}^{T_{i}} \left( \sum _{s=1}^{T_{i}} \mu _{is} \right)^{y_{it}} / \left( \sum _{t=1}^{T_{i}} y_{it} \right)!} \\ & = & \frac{(\sum _{t=1}^{T_{i}} y_{it})!}{(\prod _{t=1}^{T_{i}} y_{it}!)} \prod _{t=1}^{T_{i}} \left(\frac{\mu _{it}}{\sum _{s=1}^{T_{i}} \mu _{is}}\right)^{y_{it}} \\ & = & \frac{(\sum _{t=1}^{T_{i}} y_{it})!}{(\prod _{t=1}^{T_{i}} y_{it}!)} \prod _{t=1}^{T_{i}} \left(\frac{\lambda _{it}}{\sum _{s=1}^{T_{i}} \lambda _{is}}\right)^{y_{it}} \\ \end{eqnarray*}$

Thus, the conditional log-likelihood function of the fixed-effects Poisson model is given by

$\mathcal{L} = \sum _{i=1}^{N} \left[ \ln \left( (\sum _{t=1}^{T_{i}}y_{it})! \right) - \sum _{t=1}^{T_{i}}\ln (y_{it}!) + \sum _{t=1}^{T_{i}}y_{it}\ln \left(\frac{\lambda _{it}}{\sum _{s=1}^{T_{i}}\lambda _{is}}\right) \right]$

The gradient is

$\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial \bbeta } & = & \sum _{i=1}^{N} \sum _{t=1}^{T_{i}} y_{it}x_{it} - \sum _{i=1}^{N} \sum _{t=1}^{T_{i}} \left[ \frac{y_{it} \sum _{s=1}^{T_{i}} \left( \exp (\mathbf{x}_{is}\bbeta ) \mathbf{x}_{is} \right)}{\sum _{s=1}^{T_{i}} \exp (\mathbf{x}_{is}\bbeta )} \right] \\ & = & \sum _{i=1}^{N} \sum _{t=1}^{T_{i}} y_{it} (\mathbf{x}_{it}-\mathbf{\bar{x}}_{i}) \end{eqnarray*}$

where

$\mathbf{\bar{x}}_{i} = \sum _{s=1}^{T_{i}} \left( \frac{\exp (\mathbf{x}_{is}\bbeta )}{\sum _{k=1}^{T_{i}} \exp (\mathbf{x}_{ik}\bbeta )} \right) \mathbf{x}_{is}$

Panel Data Poisson Regression with Random Effects

In the Poisson random-effects model, conditional on $\lambda _{it}$ and parameter $\alpha _{i}$ , $y_{it}$ is iid Poisson-distributed with parameter $\mu _{it}=\alpha _{i}\lambda _{it}=\alpha _{i} \exp (\mathbf{x}_{it}’\bbeta )$ , and the individual effects, $\alpha _{i}$ , are assumed to be iid random variables. The joint density for observations in all time periods for the $i$ th individual, $P[y_{i1},\ldots ,y_{iT}|\lambda _{i1},\ldots ,\lambda _{iT_{i}}]$ , can be obtained after the density $g(\alpha )$ of $\alpha {_ i}$ is specified.

Let

$\alpha _{i} \sim \mbox{iid}\; \mathrm{gamma}(\theta ,\theta )$

so that $E(\alpha _{i})=1$ and $V(\alpha _{i}) = 1/\theta$ :

$g(\alpha _{i}) = \frac{\theta ^{\theta }}{\Gamma (\theta )} \alpha _{i}^{\theta -1}\exp (-\theta \alpha _{i})$

Let $\lambda _{i} = (\lambda _{i1},\ldots ,\lambda _{iT_{i}})$ . Because $y_{it}$ is conditional on $\lambda _{it}$ and parameter $\alpha _{i}$ is iid Poisson( $\mu _{it}=\alpha _{i} \lambda _{it}$ ), the conditional joint probability for observations in all time periods for the $i$ th individual, $P[y_{i1},\ldots ,y_{iT_{i}}|\lambda _{i},\alpha _{i}]$ , is the product of $T_{i}$ Poisson densities:

$\begin{eqnarray*} P[y_{i1},\ldots ,y_{iT_{i}}|\lambda _{i},\alpha _{i}] & = & \prod _{t=1}^{T_{i}} P[y_{it}| \lambda _{i}, \alpha _{i}]\\ & = & \prod _{t=1}^{T_{i}}\left[ \frac{\exp (-\mu _{it}) \mu _{it}^{y_{it}}}{y_{it}!} \right] \\ & = & \left[ \prod _{t=1}^{T_{i}} \frac{e^{-\alpha _{i}\lambda _{it}}(\alpha _{i}\lambda _{it})^{y_{it}}}{y_{it}!} \right] \\ & = & \left[ \prod _{t=1}^{T_{i}} \lambda _{it}^{y_{it}}/y_{it}! \right] \left( e^{-\alpha _{i} \sum _{t} \lambda _{it}} \alpha _{i}^{\sum _{t} y_{it}} \right) \end{eqnarray*}$

Then, the joint density for the $i$ th panel conditional on just the $\lambda$ can be obtained by integrating out $\alpha _{i}$ :

$\begin{eqnarray*} P[y_{i1},\ldots ,y_{iT_{i}}|\lambda _{i}] & = & \int _{0}^{\infty } P[y_{i1},\ldots ,y_{iT}|\lambda _{i},\alpha _{i}] g(\alpha _{i}) d\alpha _{i} \\ & = & \frac{\theta ^{\theta }}{\Gamma (\theta )} \left[ \prod _{t=1}^{T_{i}} \frac{\lambda _{it}^{y_{it}}}{y_{it}!} \right] \int _{0}^{\infty } \exp (-\alpha _{i} \sum _{t} \lambda _{it}) \alpha _{i}^{\sum _{t} y_{it}} \alpha _{i}^{\theta -1} \exp (-\theta \alpha _{i}) d\alpha _{i} \\ & = & \frac{\theta ^{\theta }}{\Gamma (\theta )} \left[ \prod _{t=1}^{T_{i}} \frac{\lambda _{it}^{y_{it}}}{y_{it}!} \right] \int _{0}^{\infty } \exp \left[ -\alpha _{i} \left( \theta + \sum _{t} \lambda _{it} \right) \right] \alpha _{i}^{\theta + \sum _{t} y_{it}-1} d\alpha _{i} \\ & = & \left[ \prod _{t=1}^{T_{i}} \frac{\lambda _{it}^{y_{it}}}{y_{it}!} \right] \frac{\Gamma (\theta + \sum _{t} y_{it})}{\Gamma (\theta )} \\ & & \times \left(\frac{\theta }{\theta +\sum _{t} \lambda _{it}} \right)^{\theta } \left(\theta + \sum _{t} \lambda _{it} \right)^{-\sum _{t} y_{it}} \\ & = & \left[ \prod _{t=1}^{T_{i}} \frac{\lambda _{it}^{y_{it}}}{y_{it}!} \right] \frac{\Gamma (\alpha ^{-1}+ \sum _{t} y_{it})}{\Gamma (\alpha ^{-1})} \\ & & \times \left(\frac{\alpha ^{-1}}{\alpha ^{-1}+\sum _{t} \lambda _{it}} \right)^{\alpha ^{-1}} \left(\alpha ^{-1} + \sum _{t} \lambda _{it} \right)^{-\sum _{t} y_{it}} \end{eqnarray*}$

where $\alpha (=1/\theta )$ is the overdispersion parameter. This is the density of the Poisson random-effects model with gamma-distributed random effects. For this distribution, $E(y_{it})=\lambda _{it}$ and $V(y_{it})=\lambda _{it} +\alpha \lambda _{it}^{2}$ ; that is, there is overdispersion.

Then the log-likelihood function is written as

$\begin{eqnarray*} \mathcal{L} & = & \sum _{i=1}^{N} \left\{ \sum _{t=1}^{T_{i}} \ln (\frac{\lambda _{it}^{y_{it}}}{y_{it}!}) + \alpha ^{-1} \ln (\alpha ^{-1}) -\alpha ^{-1} \ln (\alpha ^{-1}+\sum _{t=1}^{T_{i}}\lambda _{it}) \right\} \\ & & + \sum _{i=1}^{N} \left\{ - \left( \sum _{t=1}^{T_{i}}y_{it} \right) \ln \left(\alpha ^{-1}+\sum _{t=1}^{T_{i}}\lambda _{it}\right) \right. \\ & & \left. \hspace*{0.3in} + \ln \left[\Gamma \left(\alpha ^{-1}+ \sum _{t=1}^{T_{i}}y_{it} \right)\right] -\ln (\Gamma (\alpha ^{-1})) \right\} \end{eqnarray*}$

The gradient is

$\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial \bbeta } & = & \sum _{i=1}^{N} \left\{ \sum _{t=1}^{T_{i}} y_{it}\mathbf{x}_{it} -\frac{\alpha ^{-1}\sum _{t=1}^{T_{i}} \lambda _{it} \mathbf{x}_{it}}{\alpha ^{-1}+\sum _{t=1}^{T_{i}} \lambda _{it}}\right\} \\ & & -\sum _{i=1}^{N} \left\{ \left( \sum _{t=1}^{T_{i}} y_{it} \right) \frac{\sum _{t=1}^{T_{i}} \lambda _{it} \mathbf{x}_{it}}{\alpha ^{-1}+\sum _{t=1}^{T_{i}} \lambda _{it}} \right\} \\ \frac{\partial \mathcal{L}}{\partial \bbeta } & = & \sum _{i=1}^{N} \left\{ \sum _{t=1}^{T_{i}} y_{it}\mathbf{x}_{it} -\frac{(\alpha ^{-1}+\sum _{t=1}^{T_{i}} y_{it}) (\sum _{t=1}^{T_{i}}\lambda _{it}\mathbf{x}_{it})}{\alpha ^{-1}+\sum _{t=1}^{T_{i}} \lambda _{it}} \right\} \end{eqnarray*}$

and

$\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial \alpha } & = & \sum _{i=1}^{N} \left\{ -\alpha ^{-2} \left[ [1+ \ln (\alpha ^{-1})] - \frac{(\alpha ^{-1}+\sum _{t=1}^{T_{i}} y_{it})}{(\alpha ^{-1})+ \sum _{t=1}^{T_{i}}\lambda _{it}} - \ln \left(\alpha ^{-1} + \sum _{t=1}^{T_{i}} \lambda _{it} \right) \right] \right\} \\ & + & \sum _{i=1}^{N} \left\{ -\alpha ^{-2} \left[ \frac{\Gamma (\alpha ^{-1}+ \sum _{t=1}^{T_{i}} y_{it})}{\Gamma (\alpha ^{-1} +\sum _{t=1}^{T_{i}} y_{it})} -\frac{\Gamma (\alpha ^{-1})}{\Gamma (\alpha ^{-1})} \right] \right\} \end{eqnarray*}$

where $\lambda _{it} = \exp (\mathbf{x}_{it}’\bbeta )$ , $\Gamma ’(\cdot ) = d \Gamma (\cdot )/d (\cdot )$ and $\Gamma ’(\cdot )/\Gamma (\cdot )$ is the digamma function.

Panel Data Negative Binomial Regression with Fixed Effects

This section shows the derivation of a negative binomial model with fixed effects. Keep the assumptions of the Poisson-distributed dependent variable

$y_{it}\sim Poisson\left(\mu _{it}\right)$

But now let the Poisson parameter be random with gamma distribution and parameters $\left(\lambda _{it},\delta \right)$ ,

$\mu _{it}\sim \Gamma \left(\lambda _{it},\delta \right)$

where one of the parameters is the exponentially affine function of independent variables $\lambda _{it}=\exp \left(\mathbf{x}_{it}’\beta \right)$ . Use integration by parts to obtain the distribution of $y_{it}$ ,

$\begin{eqnarray*} P\left[y_{it}\right]& = & \int _{0}^{\infty }\frac{e^{-\mu _{it}}\mu _{it}^{y_{it}}}{y_{it}!}f\left(\mu _{it}\right)d\mu _{it}\\ & = & \frac{\Gamma \left(\lambda _{it}+y_{it}\right)}{\Gamma \left(\lambda _{it}\right)\Gamma \left(y_{it}+1\right)}\left(\frac{\delta }{1+\delta }\right)^{\lambda _{it}}\left(\frac{1}{1+\delta }\right)^{y_{it}} \end{eqnarray*}$

which is a negative binomial distribution with parameters $\left(\lambda _{it},\delta \right)$ . Conditional joint distribution is given as

$\begin{eqnarray*} P[y_{i1},\ldots ,y_{iT_{i}}|\sum _{t=1}^{T_{i}}y_{it}]& =& \left(\prod _{t=1}^{T_{i}}\frac{\Gamma \left(\lambda _{it}+y_{it}\right)}{\Gamma \left(\lambda _{it}\right)\Gamma \left(y_{it}+1\right)}\right)\\ & & \times \left(\frac{\Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}\right)\Gamma \left(\sum _{t=1}^{T_{i}}y_{it}+1\right)}{\Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}\right) \end{eqnarray*}$

Hence, the conditional fixed-effects negative binomial log-likelihood is

$\begin{eqnarray*} \mathcal{L}& = & \sum _{i=1}^{N}\left[\log \Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}\right)+\log \Gamma \left(\sum _{t=1}^{T_{i}}y_{it}+1\right)-\log \Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)\right]\\ & & +\sum _{i=1}^{N}\sum _{t=1}^{T_{i}}\left[\log \Gamma \left(\lambda _{it}+y_{it}\right)-\log \Gamma \left(\lambda _{it}\right)-\log \Gamma \left(y_{it}+1\right)\right] \end{eqnarray*}$

The gradient is

$\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial \beta }& = & \sum _{i=1}^{N}\left[\left(\frac{\Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}\right)}{\Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}\right)}-\frac{\Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}{\Gamma \left(\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}\right)\sum _{t=1}^{T_{i}}\lambda _{it}\mathbf{x}_{it}\right]\\ & & +\sum _{i=1}^{N}\frac{\Gamma \left(\sum _{t=1}^{T_{i}}y_{it}+1\right)}{\Gamma \left(\sum _{t=1}^{T_{i}}y_{it}+1\right)}\\ & & +\sum _{i=1}^{N}\sum _{t=1}^{T_{i}}\left[\left(\frac{\Gamma \left(\lambda _{it}+y_{it}\right)}{\Gamma \left(\lambda _{it}+y_{it}\right)}-\frac{\Gamma \left(\lambda _{it}\right)}{\Gamma \left(\lambda _{it}\right)}\right)\lambda _{it}\mathbf{x}_{it}-\frac{\Gamma \left(y_{it}+1\right)}{\Gamma \left(y_{it}+1\right)}\right] \end{eqnarray*}$

Panel Data Negative Binomial Regression with Random Effects

This section describes the derivation of negative binomial model with random effects. Suppose

$y_{it}\sim Poisson\left(\mu _{it}\right)$

with the Poisson parameter distributed as gamma,

$\mu _{it}\sim \Gamma \left(\nu _{i}\lambda _{it},\delta \right)$

where its parameters are also random:

$\nu _{i}\lambda _{it}=\exp \left(\mathbf{x}_{it}’\beta +\eta _{it}\right)$

Assume that the distribution of a function of $\nu _{i}$ is beta with parameters $\left(a,b\right)$ :

$\frac{\nu _{i}}{1+\nu _{i}}\sim Beta\left(a,b\right)$

Explicitly, the beta density with $\left[0,1\right]$ domain is

$f\left(z\right)=\left[B\left(a,b\right)\right]^{-1}z^{a-1}\left(1-z\right)^{b-1}$

where $B\left(a,b\right)$ is the beta function. Then, conditional joint distribution of dependent variables is

$P[y_{i1},\ldots ,y_{iT_{i}}|\mathbf{x}_{i1},\ldots ,\mathbf{x}_{iT_{i}},\nu _{i}]=\prod _{t=1}^{T_{i}}\frac{\Gamma \left(\lambda _{it}+y_{it}\right)}{\Gamma \left(\lambda _{it}\right)\Gamma \left(y_{it}+1\right)}\left(\frac{1}{1+\nu _{i}}\right)^{\lambda _{it}}\left(\frac{\nu _{i}}{1+\nu _{i}}\right)^{y_{it}}$

Integrating out the variable $\nu _{i}$ yields the following conditional distribution function:

$\begin{eqnarray*} P[y_{i1},\ldots ,y_{iT_{i}}|\mathbf{x}_{i1},\ldots ,\mathbf{x}_{iT_{i}}]& = & \int _{0}^{1}\left[\prod _{t=1}^{T_{i}}\frac{\Gamma \left(\lambda _{it}+y_{it}\right)}{\Gamma \left(\lambda _{it}\right)\Gamma \left(y_{it}+1\right)}z_{i}^{\lambda _{it}}\left(1-z_{i}\right)^{y_{it}}\right]f\left(z_{i}\right)dz_{i}\\ & = & \frac{\Gamma \left(a+b\right)\Gamma \left(a+\sum _{t=1}^{T_{i}}\lambda _{it}\right)\Gamma \left(b+\sum _{t=1}^{T_{i}}y_{it}\right)}{\Gamma \left(a\right)\Gamma \left(b\right)\Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}\\ & & \times \prod _{t=1}^{T_{i}}\frac{\Gamma \left(\lambda _{it}+y_{it}\right)}{\Gamma \left(\lambda _{it}\right)\Gamma \left(y_{it}+1\right)} \end{eqnarray*}$

Consequently, the conditional log-likelihood function for a negative binomial model with random effects is

$\begin{eqnarray*} \mathcal{L}& = & \sum _{i=1}^{N}\left[\log \Gamma \left(a+b\right)+\log \Gamma \left(a+\sum _{t=1}^{T_{i}}\lambda _{it}\right)+\log \Gamma \left(b+\sum _{t=1}^{T_{i}}y_{it}\right)\right]\\ & & -\sum _{i=1}^{N}\left[\log \Gamma \left(a\right)+\log \Gamma \left(b\right)+\log \Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)\right]\\ & & +\sum _{i=1}^{N}\sum _{t=1}^{T_{i}}\left[\log \Gamma \left(\lambda _{it}+y_{it}\right)-\log \Gamma \left(\lambda _{it}\right)-\log \Gamma \left(y_{it}+1\right)\right] \end{eqnarray*}$

The gradient is

$\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial \beta }& = & \sum _{i=1}^{N}\left[\frac{\Gamma \left(a+\sum _{t=1}^{T_{i}}\lambda _{it}\right)}{\Gamma \left(a+\sum _{t=1}^{T_{i}}\lambda _{it}\right)}\sum _{t=1}^{T_{i}}\lambda _{it}\mathbf{x}_{it}+\frac{\Gamma \left(b+\sum _{t=1}^{T_{i}}y_{it}\right)}{\Gamma \left(b+\sum _{t=1}^{T_{i}}y_{it}\right)}\right]\\ & & -\sum _{i=1}^{N}\left[\frac{\Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}{\Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}\sum _{t=1}^{T_{i}}\lambda _{it}\mathbf{x}_{it}\right]\\ & & +\sum _{i=1}^{N}\sum _{t=1}^{T_{i}}\left[\left(\frac{\Gamma \left(\lambda _{it}+y_{it}\right)}{\Gamma \left(\lambda _{it}+y_{it}\right)}-\frac{\Gamma \left(\lambda _{it}\right)}{\Gamma \left(\lambda _{it}\right)}\right)\lambda _{it}\mathbf{x}_{it}-\frac{\Gamma \left(y_{it}+1\right)}{\Gamma \left(y_{it}+1\right)}\right] \end{eqnarray*}$

and

$\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial a}& = & \sum _{i=1}^{N}\left[\frac{\Gamma \left(a+b\right)}{\Gamma \left(a+b\right)}+\frac{\Gamma \left(a+\sum _{t=1}^{T_{i}}\lambda _{it}\right)}{\Gamma \left(a+\sum _{t=1}^{T_{i}}\lambda _{it}\right)}\right]\\ & & -\sum _{i=1}^{N}\left[\frac{\Gamma \left(a\right)}{\Gamma \left(a\right)}+\frac{\Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}{\Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}\right] \end{eqnarray*}$

and

$\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial b}& = & \sum _{i=1}^{N}\left[\frac{\Gamma \left(a+b\right)}{\Gamma \left(a+b\right)}+\frac{\Gamma \left(b+\sum _{t=1}^{T_{i}}y_{it}\right)}{\Gamma \left(b+\sum _{t=1}^{T_{i}}y_{it}\right)}\right]\\ & & -\sum _{i=1}^{N}\left[\frac{\Gamma \left(b\right)}{\Gamma \left(b\right)}+\frac{\Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}{\Gamma \left(a+b+\sum _{t=1}^{T_{i}}\lambda _{it}+\sum _{t=1}^{T_{i}}y_{it}\right)}\right] \end{eqnarray*}$