Mixed Poisson INAR(1) processes

Abstract

Overdispersion is a phenomenon commonly observed in count time series. Since Poisson distribution is equidispersed, the INteger-valued AutoRegressive (INAR) process with Poisson marginals is not adequate for modelling overdispersed counts. To overcome this problem, in this paper we propose a general class of first-order INAR processes for modelling overdispersed count time series. The proposed INAR(1) processes have marginals belonging to a class of mixed Poisson distributions, which are overdispersed. With this, our class of overdispersed count models have the known negative binomial INAR(1) process as particular case and open the possibility of introducing new INAR(1) processes, such as the Poisson-inverse Gaussian INAR(1) model, which is discussed here with some details. We establish a condition to our class of overdispersed INAR processes is well-defined and study some statistical properties. We propose estimators for the parameters and establish their consistency and asymptotic normality. A small Monte Carlo simulation to evaluate the finite-sample performance of the proposed estimators is presented and one application to a real data set illustrates the usefulness of our proposed overdispersed count processes.

Appendix

We here derive the asymptotic covariance matrix of the weak convergence given in Proposition 4. For this, we will use joint moments, which are defined by

$$\begin{aligned} \mu (s_1,\ldots ,s_r)\equiv E(X_tX_{t+s_1}\ldots X_{t+s_r}), \end{aligned}$$

where $0\le s_1\le \ldots \le s_r$ and r belongs to $\mathbb N$. Expressions of joint moments up to fourth order for INAR(1) processes were obtained by Schweer and Weiß (2014) (Theorem 3.3.1). We will use these expressions for computing the desired asymptotic covariance matrix.

Let $Y_n$, $Z_n$ and $W_n$ be as defined in Sect. 4 and define $g_n(i,j)=n{-}i{-}\frac{\alpha ^j}{1-\alpha ^j}\big (1-\alpha ^{j(n-i)}\big )$, for $i,j\in \mathbb N$. Using joint moments and their expressions given in Theorem 3.3.1 from Schweer and Weiß (2014), after some manipulations we obtain that

$$\begin{aligned} \text{ Var }(Y_n)= & {} n\text{ Var }(X_t)+2\sum _{t=1}^n\sum _{k=t+1}^n\text{ cov }(X_k,X_t)= n\kappa _2{+}2\sum _{t=1}^n\sum _{k=t+1}^n\big \{\mu (k-t){-}\mu ^2\big \}\\= & {} n\kappa _2\left\{ 1+2\dfrac{\alpha }{1-\alpha }-2\alpha ^2\dfrac{1-\alpha ^n}{n(1-\alpha )^2}\right\} , \end{aligned}$$

$$\begin{aligned} \text{ Var }(Z_n)= & {} n\text{ Var }(X_t^2)+2\sum _{t=1}^n\sum _{k=t+1}^n\text{ cov }\left( X_k^2,X_t^2\right) \\= & {} n(\mu _4-\mu _2^2)+2\sum _{t=1}^n\sum _{k=t+1}^n\left\{ \mu (0,k-t,k-t)-\left( \kappa _2+\mu ^2\right) ^2\right\} \\= & {} n\left( \mu _4-\mu _2^2\right) +2\dfrac{\alpha ^2}{1-\alpha ^2}g_n(1,2)\left\{ \kappa _4+\kappa _3(2\mu -1)-\kappa _2^2-2\mu \kappa _2\right\} \\&+\,2\dfrac{\alpha }{1-\alpha }g_n(1,1)(1+2\mu )(\kappa _3+2\mu \kappa _2), \end{aligned}$$

$$\begin{aligned} \text{ Var }(W_n)= & {} n\text{ Var }(X_{t-1}X_t)+2\sum _{t=1}^{n-2}\sum _{k=t+2}^n\text{ cov }(X_{k-1}X_k,X_{t-1}X_t)\\&+\,2\sum _{t=1}^{n-1}\text{ cov }(X_{t-1}X_t,X_tX_{t+1})\\= & {} \,n\left\{ \mu (0,1,1)-\mu (1)^2\right\} +2\sum _{t=1}^{n-2}\sum _{k=t+2}^n\left\{ \mu (1,k-t,k-t+1)-\mu (1)^2\right\} \\&+\,2\sum _{t=1}^{n-1}\left\{ \mu (1,1,2)-\mu (1)^2\right\} \\= & {} 2g_n(2,2)\dfrac{\alpha ^3}{1-\alpha ^2}\left\{ \alpha ^3(\kappa _4-3\kappa _3+\kappa _2(2-3\kappa _2))\right. \\&\left. +\,\,\alpha ^2(\kappa _3-\kappa _2)(2+\mu )+2\alpha \kappa _2^2+\mu (\kappa _3-\kappa _2)\right\} \\&+ 2g_n(2,1)\dfrac{\alpha }{1-\alpha }\left\{ \alpha ^3(\kappa _3-\kappa _2)(1+\mu )+\alpha ^2\mu (\kappa _3-\kappa _2)\right. \\&\left. +\,\,\alpha ^2\kappa _2(1+\mu )^2+2\alpha \mu \kappa _2(1+\mu )+\mu ^2\kappa _2\right\} \\&+ \,\alpha ^2\{\kappa _4-3\kappa _3+\kappa _2(2-3\kappa _2)\}(n+2\alpha ^2(n-1))\\&+\,\alpha (\kappa _3-\kappa _2)\{n(\alpha +(1+\alpha )(1+2\mu ))\\&+ \,2(n-1)(\mu (1+\alpha )+\alpha ^2(3+2\mu ))\}+\alpha \kappa _2\{n(1+\mu )(1+3\mu )\\&+\,2(n-1)(1+\mu )(2\mu +\alpha (1+\mu ))\}\\&+\, \mu ^2\kappa _2(n(4+\alpha )-2)+\alpha ^2\kappa _2^2(5n-2)+n\kappa _2(\kappa _2-2\alpha \mu ^2), \end{aligned}$$

$$\begin{aligned} \text{ cov }(Y_n,Z_n)= & {} nE(X_t^3)+\sum _{t=1}^{n-1}\sum _{k=t+1}^nE(X_tX_k^2)\\&+\,\sum _{k=1}^{n-1}\sum _{t=k+1}^nE(X_tX_k^2)-n^2E(X_t)E(X_t^2)\\= & {} n\mu _3+\sum _{t=1}^{n-1}\sum _{k=t+1}^n\mu (k-t,k-t)+\sum _{k=1}^{n-1}\sum _{t=k+1}^n\mu (0,t-k)-n^2\mu \mu _2\\= & {} n(\kappa _3+2\kappa _2\mu )+(\kappa _3-\kappa _2)\dfrac{\alpha ^2}{1-\alpha ^2}g_n(1,2)\\&+\dfrac{\alpha }{1-\alpha }g_n(1,1)(\kappa _2+4\mu \kappa _2+\kappa _3), \end{aligned}$$

$$\begin{aligned} \text{ cov }(Y_n,W_n)= & {} nE(X_{t-1}X_t^2)+(n-1)E(X^2_{t-1}X_t)+\sum _{t=1}^{n-2}\sum _{k=t+2}^nE(X_tX_{k-1}X_k)\\&+\,\sum _{k=1}^{n-1}\sum _{t=k+1}^nE(X_tX_{k-1}X_k)-n^2E(X_t)E(X_{t-1}X_t)\\= & {} n\mu (1,1)+(n-1)\mu (0,1)+\sum _{t=1}^{n-2}\sum _{k=t+2}^n\mu (k-t-1,k-t)\\&+\,\sum _{k=1}^{n-1}\sum _{t=k+1}^n\mu (1,t-k+1)-n^2\mu \mu (1)\\= & {} \,\alpha (n\alpha +n-1)(\kappa _3-\kappa _2)+(2n-1)\kappa _2(\alpha +\mu (1+\alpha ))\\&+\,(\kappa _3-\kappa _2)\dfrac{\alpha ^3}{1-\alpha ^2}g_n(2,2)\\&+\,\kappa _2\dfrac{\alpha }{1-\alpha }g_n(2,1)(\mu +\alpha (1+\mu ))\\&+\,\dfrac{\alpha }{1-\alpha }g_n(1,1)\{\alpha ^2(\kappa _3-\kappa _2) +\,\alpha \kappa _2(1+\mu )+\mu \kappa _2\}, \end{aligned}$$

and

$$\begin{aligned} \text{ cov }(Z_n,W_n)= & {} nE(X_{t-1}X_t^3)+(n-1)E(X^3_{t-1}X_t)+\sum _{t=1}^{n-2}\sum _{k=t+2}^nE(X_t^2X_{k-1}X_k)\\&+\,\sum _{k=1}^{n-1}\sum _{t=k+1}^nE(X_{k-1}X_kX_t^2)-n^2E(X_t^2)E(X_{t-1}X_t)\\= & {} n\mu (1,1,1)+(n-1)\mu (0,0,1)+\sum _{t=1}^{n-2}\sum _{k=t+2}^n\mu (0,k-t-1,k-t)\\&+\,\sum _{k=1}^{n-1}\sum _{t=k+1}^n\mu (1,t-k+1,t-k+1)-n^2\mu _2\mu (1)\\= & {} \alpha (n\alpha ^2+n-1)\{\kappa _4-3\kappa _3+\kappa _2(2-3\kappa _2)\}\\&+\,\alpha (n\alpha +n-1)(\kappa _3-\kappa _2)(3+2\mu )\\&+\,\mu \{n(\alpha (1+\alpha )+2)-(1+\alpha )\}(\kappa _3-\kappa _2)\\&+\,(2n-1)\kappa _2\{\mu ^2+2\alpha \kappa _2+(1+\mu )(\alpha +\mu (1+2\alpha ))\}\\&+\,\dfrac{\alpha ^3}{1-\alpha ^2}g_n(2,2)\{\kappa _4-3\kappa _3+\kappa _2(2-3\kappa _2)\\&+\,2(\kappa _3-\kappa _2)(1+\mu )+2\kappa _2^2\}\\&+\,\dfrac{\alpha }{1-\alpha }g_n(2,1)\{(\kappa _3-\kappa _2)(\mu +\alpha (1+\mu ))\\&+\,\alpha \kappa _2(1+\mu )^2+\kappa _2\mu (1+\mu )(1+\alpha )+\mu ^2\kappa _2\}\\&+\,\dfrac{\alpha ^2}{1-\alpha ^2}g_n(1,2)\{\alpha ^3(\kappa _4-3\kappa _3+\kappa _2(2-3\kappa _2))\\&+\,\alpha ^2(\kappa _3-\kappa _2)(2+\mu )+2\alpha \kappa _2^2+\mu (\kappa _3-\kappa _2)\}\\&+\,\dfrac{\alpha }{1-\alpha }g_n(1,1)\{\alpha ^2(\kappa _3-\kappa _2)(1+2\mu )+\alpha \kappa _2(1+\mu )^2\\&+\,\mu \kappa _2(\mu +(1+\mu )(1+\alpha ))\}, \end{aligned}$$

where $\mu =E(X_n)$, $\kappa _j=E((X_n-\mu )^j)$ and $\mu _j=E(X_n^j)$, for $j=2,3,4$.

Let $\Sigma _n$ be the covariance matrix with the above terms. Hence, it follows that the covariance matrix $\Sigma$ of the Proposition 4 can be obtained by $\Sigma =\displaystyle \lim _{n\rightarrow \infty }\Sigma _n/n$.