Influence diagnostics in log-linear integer-valued GARCH models

Abstract

Integer-valued generalized autoregressive conditional heteroscedasticity (GARCH) models have played an important role in time series analysis of count data. To model negatively autocorrelated time series and to accommodate covariates without restrictions, the log-linear integer-valued GARCH model has recently been proposed as an alternative to the existing models. In this paper, we study a local influence diagnostic analysis in the log-linear integer-valued GARCH models. The slope-based diagnostic and stepwise curvature-based diagnostics in a framework of the modified likelihood displacement are proposed. Under five perturbation schemes the corresponding local influence measures are derived. Two simulated data sets and a real-world example are analyzed to illustrate our method. In addition, the fitted model for this example has a negative coefficient for one of the two covariates, which is particularly illustrative of the extra flexibility of the considered model.

Appendices

Appendix 1. The first and second derivatives of $\nu _t$

$$\begin{aligned} \frac{\partial \nu _t}{\partial \alpha _0}&=1+\sum _{j=1}^q\beta _j\frac{\partial \nu _{t-j}}{\partial \alpha _0},\\ \frac{\partial \nu _t}{\partial \alpha _i}&=\log (X_{t-i}+1)+\sum _{j=1}^q\beta _j\frac{\partial \nu _{t-j}}{\partial \alpha _i},~~~i=1,\ldots ,p,\\ \frac{\partial \nu _t}{\partial \beta _j}&=\nu _{t-j}+\sum _{m=1}^q\beta _m\frac{\partial \nu _{t-m}}{\partial \beta _j},~~~j=1,\ldots ,q,\\ \frac{\partial \nu _t}{\partial \gamma _s}&=C_{t,s}+\sum _{j=1}^q\beta _j\frac{\partial \nu _{t-j}}{\partial \gamma _s},~~~s=1,\ldots ,r,\\ \frac{\partial ^2\nu _t}{\partial \alpha _i\partial \alpha _{i^*}}&=0,~~~i, i^*=0,1,\ldots ,p,\\ \frac{\partial ^2\nu _t}{\partial \alpha _i\partial \beta _j}&=\frac{\partial \nu _{t-j}}{\partial \alpha _i}+\sum _{m=1}^q\beta _m\frac{\partial ^2\nu _{t-m}}{\partial \alpha _i\partial \beta _j},~~~i=0,1,\ldots ,p,j=1,\ldots ,q,\\ \frac{\partial ^2\nu _t}{\partial \beta _j\partial \beta _{j^*}}&=\frac{\partial \nu _{t-j}}{\partial \beta _{j^*}}+\frac{\partial \nu _{t-{j^*}}}{\partial \beta _j}+\sum _{m=1}^q\beta _m\frac{\partial ^2\nu _{t-m}}{\partial \beta _j\partial \beta _{j^*}},~~~j,j^*=1,\ldots ,q.\\ \frac{\partial ^2\nu _t}{\partial \alpha _i\partial \gamma _s}&=0,~~~i=0,1,\ldots ,p,s=1,\ldots ,r,\\ \frac{\partial ^2\nu _t}{\partial \beta _j\partial \gamma _s}&=\frac{\partial \nu _{t-j}}{\partial \gamma _s},~~~j=1,\ldots ,q,s=1,\ldots ,r,\\ \frac{\partial ^2\nu _t}{\partial \gamma _s\partial \gamma _{s^*}}&=0,~~~s,s^*=0,1,\ldots ,r. \end{aligned}$$

We obtain $\partial \nu _t/\partial \theta$ and $\partial ^2\nu _t/\partial \theta \partial \theta ^\top$ from above equations, and then obtain $\partial ^2l_t(\theta )/\partial \theta \partial \theta ^\top$ with $\partial ^2l_t(\theta )/\partial \theta _k\partial \theta _l$ given in (4) as the $(k,l)$-th element.

Appendix 2. The required derivatives in Sect. 4.3

$$\begin{aligned} \frac{\partial l(\theta |\omega )}{\partial \omega }&=\sum _{t=1}^n(X_t-\exp (\nu _t))\frac{\partial \nu _t}{\partial \omega },\\ \frac{\partial ^2l(\theta |\omega )}{\partial \theta \partial \omega ^\top }&=\sum _{t=1}^n\left[ (X_t-\exp (\nu _t))\frac{\partial ^2\nu _t}{\partial \theta \partial \omega ^\top }-\exp (\nu _t)\frac{\partial \nu _t}{\partial \theta }\frac{\partial \nu _t}{\partial \omega ^\top }\right] ,\\ \frac{\partial ^2l(\theta |\omega )}{\partial \omega \partial \omega ^\top }&=-\sum _{t=1}^n\exp (\nu _t)\frac{\partial \nu _t}{\partial \omega }\frac{\partial \nu _t}{\partial \omega ^\top },\\ \frac{\partial ^2l(\theta |\omega )}{\partial \theta \partial \theta ^\top }&=\sum _{t=1}^n\left[ (X_t-\exp (\nu _t))\frac{\partial ^2\nu _t}{\partial \theta \partial \theta ^\top }-\exp (\nu _t)\frac{\partial \nu _t}{\partial \theta }\frac{\partial \nu _t}{\partial \theta ^\top }\right] , \end{aligned}$$

where

$$\begin{aligned} \frac{\partial \nu _t}{\partial \omega }&=\frac{\partial \omega _t}{\partial \omega }+\sum _{j=1}^q\beta _j\frac{\partial \nu _{t-j}}{\partial \omega },\\ \frac{\partial ^2\nu _t}{\partial \alpha _i\partial \omega ^\top }&=0,~~~i=0,1,\ldots ,p,\\ \frac{\partial ^2\nu _t}{\partial \beta _j\partial \omega ^\top }&=\frac{\partial \nu _{t-j}}{\partial \omega ^\top }+\sum _{m=1}^q\beta _m\frac{\partial ^2\nu _{t-m}}{\partial \beta _j\partial \omega ^\top },~~~j=1,\ldots ,q,\\ \frac{\partial ^2\nu _t}{\partial \gamma _s\partial \omega ^\top }&=0,~~~s=1,\ldots ,r, \end{aligned}$$

$\partial \nu _t/\partial \theta$ and $\partial ^2\nu _t/\partial \theta \partial \theta ^\top$ are given in Appendix 1, and

$$\begin{aligned} \frac{\partial \omega _t}{\partial \omega }=(0,\ldots ,0,1,0,\ldots ,0)^\top \hbox {with 1 in the }t\hbox {-th position}. \end{aligned}$$

These derivatives will be evaluated at $\theta =\hat{\theta }$ and $\omega =\omega _0$.

Appendix 3. The required derivatives in Sect. 4.5

$$\begin{aligned} \frac{\partial l(\theta |\omega )}{\partial \omega }=&\sum _{t=1}^n\left[ (\nu _t-\psi (X_t+1+\omega _t))\frac{\partial \omega _t}{\partial \omega }+(X_t+\omega _t-\exp (\nu _t))\frac{\partial \nu _t}{\partial \omega }\right] ,\\ \frac{\partial ^2l(\theta |\omega )}{\partial \theta \partial \omega ^\top }=&\sum _{t=1}^n\left[ \frac{\partial \nu _t}{\partial \theta }\frac{\partial \omega _t}{\partial \omega ^\top }-\exp (\nu _t)\frac{\partial \nu _t}{\partial \theta }\frac{\partial \nu _t}{\partial \omega ^\top }+(X_t+\omega _t-\exp (\nu _t))\frac{\partial ^2\nu _t}{\partial \theta \partial \omega ^\top }\right] ,\\ \frac{\partial ^2l(\theta |\omega )}{\partial \omega \partial \omega ^\top }=&\sum _{t=1}^n\left[ \left( \frac{\partial \omega _t}{\partial \omega }\frac{\partial \nu _t}{\partial \omega ^\top }+\frac{\partial \nu _t}{\partial \omega }\frac{\partial \omega _t}{\partial \omega ^\top }\right) -\exp (\nu _t)\frac{\partial \nu _t}{\partial \omega }\frac{\partial \nu _t}{\partial \omega ^\top }\right. \\&~~~+\left. (X_t+\omega _t-\exp (\nu _t))\frac{\partial ^2\nu _t}{\partial \omega \partial \omega ^\top }-\psi '(X_t+1+\omega _t)\frac{\partial \omega _t}{\partial \omega }\frac{\partial \omega _t}{\partial \omega ^\top }\right] ,\\ \frac{\partial ^2l(\theta |\omega )}{\partial \theta \partial \theta ^\top }=&\sum _{t=1}^n\left[ (X_t+\omega _t-\exp (\nu _t))\frac{\partial ^2\nu _t}{\partial \theta \partial \theta ^\top }-\exp (\nu _t)\frac{\partial \nu _t}{\partial \theta }\frac{\partial \nu _t}{\partial \theta ^\top }\right] , \end{aligned}$$

where

$$\begin{aligned} \frac{\partial \nu _t}{\partial \omega }&=\sum _{i=1}^p\frac{\alpha _i}{X_{t-i}+1+\omega _{t-i}}\frac{\partial \omega _{t-i}}{\partial \omega }+\sum _{j=1}^q\beta _j\frac{\partial \nu _{t-j}}{\partial \omega },\\ \frac{\partial ^2\nu _t}{\partial \alpha _0\partial \omega ^\top }&=0,\\ \frac{\partial ^2\nu _t}{\partial \alpha _i\partial \omega ^\top }&=\frac{1}{X_{t-i}+1+\omega _{t-i}}\frac{\partial \omega _{t-i}}{\partial \omega ^\top }+\sum _{j=1}^q\beta _j\frac{\partial ^2\nu _{t-j}}{\partial \alpha _i\partial \omega ^\top },~~~i=1,\ldots ,p,\\ \frac{\partial ^2\nu _t}{\partial \beta _j\partial \omega ^\top }&=\frac{\partial \nu _{t-j}}{\partial \omega ^\top }+\sum _{m=1}^q\beta _m\frac{\partial ^2\nu _{t-m}}{\partial \beta _j\partial \omega ^\top },~~~j=1,\ldots ,q,\\ \frac{\partial ^2\nu _t}{\partial \gamma _s\partial \omega ^\top }&=0,~~~s=1,\ldots ,r,\\ \frac{\partial ^2\nu _t}{\partial \omega \partial \omega ^\top }&=-\sum _{i=1}^p\frac{\alpha _i}{(X_{t-i}+1+\omega _{t-i})^2}\frac{\partial \omega _{t-i}}{\partial \omega }\frac{\partial \omega _{t-i}}{\partial \omega ^\top }+\sum _{j=1}^q\beta _j\frac{\partial ^2\nu _{t-j}}{\partial \omega \partial \omega ^\top },\\ \frac{\partial \nu _t}{\partial \alpha _i}&=\log (X_{t-i}+1+\omega _{t-i})+\sum _{j=1}^q\beta _j\frac{\partial \nu _{t-j}}{\partial \alpha _i},~~~i=1,\ldots ,p, \end{aligned}$$

$\partial \nu _t/\partial \alpha _0,\partial \nu _t/\partial \beta _j(j=1,\ldots ,q)$ and $\partial ^2\nu _t/\partial \theta \partial \theta ^\top$ are given in Appendix 1, $\psi (x)=\mathrm{d}\log \Gamma (x)/\mathrm{d}x=(\mathrm{d}\Gamma (x)/\mathrm{d}x)/\Gamma (x)$ is the digamma function, $\psi '(x)$ is the first derivative of $\psi (x)$ and is known as the trigamma function. These derivatives will be evaluated at $\theta =\hat{\theta }$ and $\omega =\omega _0$.