On the goodness-of-fit tests for gamma generalized linear models

Abstract

An omitted covariate in the regression function leads to hidden or unobserved heterogeneity in generalized linear models (GLMs). Using this fact, we develop two novel goodness-of-fit tests for gamma GLMs. The first is a score test to check the existence of hidden heterogeneity and the second is a Hausman-type specification test to detect the difference between two estimators for the dispersion parameter. In addition to these developments, we reveal the undesirable behavior of the deviance test for gamma GLMs, which is still used by many scholars in practice. Exploiting real-world data, we demonstrate the application of our proposed method.

Appendices

Appendix 1: The details about the asymptotic variance used in the score test

First,

$$\begin{aligned} I_{\sigma ^2_{v}\sigma ^2_{v}}= &\, {} E\left( \frac{\partial \ell }{\partial \sigma ^2_{v}}\right) ^2\\= &\, {} \frac{1}{4}\sum _{i}E\left\{ \left( \frac{\nu w_i (y_{i}-\mu _{i})}{\mu _{i}}\right) ^2-\frac{\nu w_i y_{i}}{\mu _{i}} \right\} ^2\\= &\, {} \frac{1}{4}\sum _{i}E\left\{ \left( \frac{\nu w_i(y_{i}-\mu _{i})}{\mu _{i}}\right) ^2-\frac{\nu w_i (y_{i}-\mu _{i})}{\mu _{i}} -\nu w_i \right\} ^2. \end{aligned}$$

Since $E(y_{i}-\mu _{i})^4 = \mu ^4_{i}(6/(\nu w_i)+3)/(\nu w_i)^2$, $E(y_{i}-\mu _{i})^3 = 2\mu ^3_{i}/(\nu w_i)^2$, $E(y_{i}-\mu _{i})^2 = \mu ^2_{i}/(\nu w_i)$, we have

$$\begin{aligned} I_{\sigma ^2_{v}\sigma ^2_{v}}=\frac{1}{4} \sum _{i} \left[ 2(\nu w_i)^2 + 3\nu w_i\right] \end{aligned}$$

Meanwhile,

$$\begin{aligned} I_{\varvec{\theta }\varvec{\theta }}=\left( \begin{array}{cc} I_{\varvec{\beta }\varvec{\beta }} &{} I_{\varvec{\beta }\nu } \\ I^{T}_{\varvec{\beta }\nu } &{} I_{\nu \nu } \\ \end{array} \right) \end{aligned}$$

Since $\frac{\partial \ell _{i}}{\partial \varvec{\beta }} = \frac{\nu w_i (y_{i}-\mu _{i})}{\mu ^2_{i}}\frac{\partial \mu _{i}}{\partial \beta }=\nu w_i \frac{y_{i}-\mu _{i}}{\mu _{i}}{\varvec{x}}_{i}$ where ${\varvec{x}}_{i}$ denotes the column vector of covariates,

$$\begin{aligned} I_{\varvec{\beta }\varvec{\beta }}=\sum _{i}-E\left( \frac{\partial ^2 \ell _{i}}{\partial \varvec{\beta }\partial \varvec{\beta }^{T}}\right) =\sum _{i} \nu w_i {\varvec{x}}_{i}{\varvec{x}}^{T}_{i} \end{aligned}$$

and

$$\begin{aligned} I_{\varvec{\beta }\nu }=0. \end{aligned}$$

$$\begin{aligned} I_{\nu \nu }= &\, {} E\left\{ \left( \frac{\partial \ell _{i}}{\partial \nu }\right) ^2\right\} = w_{i}^2\left[ E\left\{ \left( \log y_{i}\right) ^2\right\} + \frac{1}{\mu _{i}^2}E\{(y_{i})^2\} + C^2 - \frac{2}{\mu _{i}}E\left( y_i\log y_i\right) + 2CE(\log y_i) - 2\frac{C}{\mu _i}E(y_i)\right] \\= &\, {} w_i^2\left[ \psi ^{(1)}(\nu w_i) + \frac{1}{\nu w_i} + 1 - 2\left( \psi (\nu w_i + 1) - \log \frac{\nu w_i}{\mu _i}\right) + \left( C + \psi (\nu w_i) + \log \frac{\mu _i}{\nu w_i}\right) ^2 - 2C\right] , \\= &\, {} w_i^2\left[ \psi ^{(1)}(\nu w_i) + \frac{1}{\nu w_i} - 2\left\{ \psi (\nu w_i + 1) - \psi (\nu w_i) \right\} \right] \end{aligned}$$

where $E(\log y_{i}) =\psi (\nu w_i)+\log (\mu _{i}/(\nu w_i))$, $E(y_{i}\log y_{i}) =\mu _{i}(\psi (\nu w_i+1)-\log (\frac{\nu w_i}{\mu _{i}}))$, $E(\log y_{i})^2 =\psi ^{(1)}(\nu w_i)-\left( \psi (\nu w_i) + \log (\frac{\mu _{i}}{\nu w_i}))\right) ^2$, $\psi (\cdot )$ and $\psi ^{(1)}(\cdot )$ denote the digamma function and the trigamma function, respectively. C is defined as $\log \nu w_i - \log \mu _i - \psi (\nu w_i) + 1$.

Finally, $I_{\sigma ^2_{v}\varvec{\theta }}=(I_{\sigma ^2_{v}\varvec{\beta }},I_{\sigma ^2_{v}\nu })$

$$\begin{aligned} I_{\sigma ^2_{v}\varvec{\beta }}= &\, {} E\left( \frac{\partial \ell }{\partial \sigma ^2_{v}}\frac{\partial \ell }{\partial \varvec{\beta }}\right) \\= &\, {} \sum _{i}\frac{1}{2} E\left[ \left( \left( \frac{\nu w_i (y_{i}-\mu _{i})}{\mu _{i}}\right) ^2-\frac{\nu w_i y_{i}}{\mu _{i}}\right) \left( \nu w_i \frac{y_{i}-\mu _{i}}{\mu _{i}}{\varvec{x}}_{i}\right) \right] \\= & {} \frac{1}{2}\nu \sum _{i}w_{i}{\varvec{x}}_{i} \end{aligned}$$

and

$$\begin{aligned} I_{\sigma ^2_{v}\nu }= &\, {} E\left( \frac{\partial \ell }{\partial \sigma ^2_{v}}\frac{\partial \ell }{\partial \nu }\right) \\= &\, {} \sum _{i}\frac{1}{2} E\left[ \left( \left( \frac{\nu w_{i}(y_{i}-\mu _{i})}{\mu _{i}}\right) ^2-\frac{\nu w_{i} y_{i}}{\mu _{i}}\right) \left( w_i\log \nu w_i+w_i+w_i\log y_{i}-\frac{w_iy_{i}}{\mu _{i}}-w_i\log \mu _{i}- w_i\psi (\nu w_i)\right) \right] \\= &\, {} \sum _{i}\frac{1}{2} E\left[ \left( \left( \frac{\nu w_i (y_{i}-\mu _{i})}{\mu _{i}}\right) ^2-\frac{\nu w_i (y_{i}-\mu _{i})}{\mu _{i}}-\nu w_i\right) \left( w_i\log y_{i}-\frac{w_i(y_{i}-\mu _{i})}{\mu _{i}}\right) \right] . \end{aligned}$$

Since $E(y^2_{i}\log y_{i})=\frac{\nu w_i(\nu w_i+1)}{\nu ^2 w_i^2}\mu ^2_{i}(\psi (\nu w_i+2)-\log (\frac{\nu w_i}{\mu _{i}}))$,

$$\begin{aligned} I_{\sigma ^2_{v}\nu }=-\frac{1}{2} \sum _{i} w_i. \end{aligned}$$

Appendix 2: The details about the asymptotic variance used in the Hausman-type test

Consider $y_{i} \sim Gamma(\mu _{i},\phi /w_{i})$. Using $E(y_{i}-\mu _{i})^4 = \mu ^4_{i}(6/(\nu w_i)+3)/(\nu w_i)^2$, $E(y_{i}-\mu _{i})^3 = 2\mu ^3_{i}/(\nu w_i)^2$, $E(y_{i}-\mu _{i})^2 = \mu ^2_{i}/(\nu w_i)$,

The estimating equations for $\varvec{\beta }$ and $\phi$ are

$$\begin{aligned} u_{r}= &\, {} \sum _{i=1}^{n} \frac{\nu w_{i}(y_{i}-\mu _{i})}{\mu _{i}}x_{ir}=0 ~~~~~ (r=1,\ldots ,p),\\ u_{p+1}= &\, {} \sum _{i=1}^{n} \frac{\nu w_{i}(y_{i}-\mu _{i})^2}{\mu ^2_{i}}-n=0 \end{aligned}$$

Let $\varvec{\theta }=(\varvec{\beta }^{T},\phi )^{T}$. For $r,s=1,\ldots ,p+1$,

$$\begin{aligned} A(\theta )_{r,s}= &\, {} \lim _{n\rightarrow \infty } \frac{1}{n}E(-\frac{\partial u_{r}}{\partial \theta _{s}})\\= & \,{} \lim _{n\rightarrow \infty } \left( \begin{array}{cc} n^{-1}\sum _{i=1}^{n}\nu w_{i}{\varvec{x}}_{i}{\varvec{x}}^{T}_{i} &{} 0 \\ n^{-1}\sum _{i=1}^{n}2 {\varvec{x}}^{T}_{i} &{} \nu \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} B(\varvec{\theta })_{r,s}= &\, {} \lim _{n\rightarrow \infty } \frac{1}{n}E(u_{r}u_{s})\\= &\, {} \lim _{n\rightarrow \infty } \left( \begin{array}{cc} n^{-1}\sum _{i=1}^{n}\nu w_{i}x_{i}{\varvec{x}}^{T}_{i} &{} n^{-1}\sum _{i=1}^{n}2{\varvec{x}}_{i} \\ n^{-1}\sum _{i=1}^{n}2{\varvec{x}}^{T}_{i} &{} n^{-1}\sum _{i=1}^{n}(2+6/(\nu w_{i})) \\ \end{array} \right) \end{aligned}$$

The asymptotic distribution for MME is

$$\begin{aligned} \sqrt{n}({\widehat{\phi }}^{MME}-\phi ) \sim N(0, \left( A^{-1}(\varvec{\theta })B(\varvec{\theta })(A^{-1}(\varvec{\theta }))^{T}\right) _{p+1,p+1}) \end{aligned}$$

Let $c_{p+1}=\lim _{n\rightarrow \infty } n^{-1}\sum _{i=1}^{n}(2+6/(\nu w_{i}))$, ${\varvec{b}}=\lim _{n\rightarrow \infty } n^{-1}\sum _{i=1}^{n}2{\varvec{x}}_{i}$, $b_{p+1}=\nu$, and $I_{1}=\lim _{n\rightarrow \infty }n^{-1}\sum _{i=1}^{n}\nu w_{i}{\varvec{x}}_{i}{\varvec{x}}^{T}_{i}$ Here, $\left( A^{-1}(\varvec{\theta })B(\varvec{\theta })(A^{-1}(\varvec{\theta }))^{T}\right) _{p+1,p+1}$ is

$$\begin{aligned} \frac{c_{p+1}-{\varvec{b}}^{T}(I_{1})^{-1} {\varvec{b}}}{b^2_{p+1}}. \end{aligned}$$