Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued autoregressive processes

Abstract

To accurately and flexibly capture the dispersion features of time series of counts, we introduce the generalized Poisson thinning operation and further define some new integer-valued autoregressive processes. Basic probabilistic and statistical properties of the models are discussed. Conditional least squares and maximum quasi likelihood estimators are investigated via the moment targeting estimation methods for the innovation free case. Also, the asymptotic properties of the estimators are obtained. Conditional maximum likelihood estimation for the parametric cases are also discussed. Finally, some numerical results of the estimates and two real data examples are presented.

Appendix

The proofs of Proposition 1

Since (i)–(v) are easy to verify, we only prove (vi). By the law of total covariance, we have

$$\begin{aligned} \mathrm{Cov}(X_{t},X_{t-h})&=\mathrm{Cov}(E(X_t|X_{t-1},\ldots ),E(X_{t-h}|X_{t-1},\ldots ))+0\\&=\mathrm{Cov}[\phi (\lambda ,\kappa ) X_{t-1}+\mu _{\varepsilon },X_{t-h}]\\&=\phi (\lambda ,\kappa )\cdot \mathrm{Cov}(X_{t-1},X_{t-h})\\&=\cdots \\&=\phi (\lambda ,\kappa )^h\cdot \mathrm{Var}(X_{t-h}), \end{aligned}$$

Thus, the autocorrelation function $\rho (h)=\mathrm{Corr}(X_{t},X_{t-h})=\phi (\lambda ,\kappa )^h.$$\square$

The proofs of Theorems 1 and 2

These two theorems can be easily proved by verifying the regularity conditions of Theorems 3.1 and 3.2 in Klimko and Nelson (1978), we omit the details here. $\square$

The proof of Theorem 3

This is an application of the $\delta$-method; For completeness and reader’s convenience, we refer to Theorem A on p.122 of Serfling (1980) for a proof. $\square$

The proof of Theorem 4

First, we suppose $\varvec{\theta }_2$ is known. Let $\mathcal {F}_t=\sigma (X_0,X_1,\ldots ,X_t)$ be the $\sigma$-field generated by $\{X_0,X_1,\ldots ,X_t\}$. Denote

$$\begin{aligned} S_n^{(1)}(\varvec{\theta }_{2},\varvec{\theta }_1)=\sum _{t=1}^n V_{\varvec{\theta }_2}^{-1}(X_t|X_{t-1})(X_t-\phi X_{t-1}-\mu _{\varepsilon }) X_{t-1}. \end{aligned}$$

By direct calculation, we can see that

$$\begin{aligned}&E[V_{\varvec{\theta }_2}^{-1}(X_t|X_{t-1})(X_t-\phi X_{t-1}-\mu _{\varepsilon }) X_{t-1}|\mathcal {F}_{t-1}]\\&\quad =V_{\varvec{\theta }_2}^{-1}(X_t|X_{t-1})X_{t-1}E[(X_t-\phi X_{t-1}-\mu _{\varepsilon })|\mathcal {F}_{t-1}]\\&\quad =0, \end{aligned}$$

and

$$\begin{aligned}&E[S_t^{(1)}(\varvec{\theta }_2,\varvec{\theta }_1)|\mathcal {F}_{t-1}]\\&\quad =E[(V_{\varvec{\theta }_2}^{-1}(X_t|X_{t-1})(X_t-\phi X_{t-1}-\mu _{\varepsilon })X_{t-1} +S_{t-1}^{(1)}(\varvec{\theta }_2,\varvec{\theta }_1))|\mathcal {F}_{t-1}]\\&\quad =S_{t-1}^{(1)}(\varvec{\theta }_2,\varvec{\theta }_1). \end{aligned}$$

Thus, $\{S_t^{(1)}(\varvec{\theta }_2,\varvec{\theta }_1),\mathcal {F}_{t},~t\ge 0\}$ is a martingale. By (C1) and Theorem 1.1 in Billingsley (1961),

$$\begin{aligned}&\frac{1}{n}\sum _{t=1}^n V_{\varvec{\theta }_{2,0}}^{-2}(X_t|X_{t-1})(X_t-\phi X_{t-1}-\mu _{\varepsilon })^2X_{t-1}^2\\&\quad \overset{a.s.}{\longrightarrow }E\left( V_{\varvec{\theta }_{2,0}}^{-2}(X_1|X_{0})(X_1-\phi X_{0}-\mu _{\varepsilon })^2X_0^2\right) \\&\quad =E\left( E[V_{\varvec{\theta }_{2,0}}^{-2}(X_1|X_{0})(X_1-\phi X_{0}-\mu _{\varepsilon })^2X_0^2|X_0]\right) \\&\quad =E[V_{\varvec{\theta }_{2,0}}^{-1}(X_1|X_0)X_0^2]. \end{aligned}$$

Hence, by Corollary 3.2 in Hall and Heyde (1980) and the central limit theorem of martingale, we have,

$$\begin{aligned} \frac{1}{\sqrt{n}}S_n^{(1)}(\varvec{\theta }_{2,0},\varvec{\theta }_{1}) \overset{L}{\longrightarrow }N(0,E[V_{\varvec{\theta }_{2,0}}^{-1}(X_1|X_0)X_0^2]). \end{aligned}$$

Similarly, denote $S_n^{(2)}(\varvec{\theta }_{2},\varvec{\theta }_1)=\sum _{t=1}^n V_{\varvec{\theta }_2}^{-1}(X_t|X_{t-1})(X_t-\phi X_{t-1}-\mu _{\varepsilon })$. Then, we can verify that $\{S_t^{(2)}(\varvec{\theta }_2,\varvec{\theta }_1),\mathcal {F}_{t},~t\ge 0\}$ is also martingale. Similar to the previous discussion, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}S_n^{(2)}(\varvec{\theta }_{2,0},\varvec{\theta }_1) \overset{L}{\longrightarrow }N(0,E(V_{\varvec{\theta }_{2,0}}^{-1}(X_1|X_{0}))). \end{aligned}$$

By Cramer-Wold device, for any $\varvec{c}^\textsf {T}=(c_1,c_2) \in \mathbb {R}^3$ and $(c_1,c_2)\ne (0,0)$, we have

$$\begin{aligned} \frac{\varvec{c}^\textsf {T}}{\sqrt{n}}\left( \begin{array}{c} S_n^{(1)}(\varvec{\theta }_{2,0},\varvec{\theta }_1)\\ S_n^{(2)}(\varvec{\theta }_{2,0},\varvec{\theta }_1) \end{array}\right)&=\frac{1}{\sqrt{n}}\sum _{i=1}^2 c_iS_n^{(i)}(\varvec{\theta }_{2,0},\varvec{\theta }_1)\\&=\frac{1}{\sqrt{n}}\sum _{t=1}^n V_{\varvec{\theta }_{2,0}}^{-1}(X_t|X_{t-1})(X_t-\phi X_{t-1}-\mu _{\varepsilon })(c_1 X_{t-1}+c_2)\\&\overset{L}{\longrightarrow } N(0,E[V_{\varvec{\theta }_{2,0}}^{-1}(X_1|X_0)(c_1 X_{0}+c_2)^2]), \end{aligned}$$

implying

$$\begin{aligned} \frac{1}{\sqrt{n}}\left( \begin{array}{c} S_n^{(1)}(\varvec{\theta }_{2,0},\varvec{\theta }_1)\\ S_n^{(2)}(\varvec{\theta }_{2,0},\varvec{\theta }_1) \end{array}\right)&\overset{L}{\longrightarrow } N\left( \varvec{0},\varvec{T}\right) . \end{aligned}$$

Now, we replace $V_{\varvec{\theta }_2}(X_t|X_{t-1})$ with $V_{\hat{\varvec{\theta }}_2}(X_t|X_{t-1})$, where $\hat{\varvec{\theta }}_2$ is a consistent estimator of ${\varvec{\theta }}_2$. Then we want to prove that

$$\begin{aligned} \frac{1}{\sqrt{n}}\left( \begin{array}{c} S_n^{(1)}(\hat{\varvec{\theta }}_2,\varvec{\theta }_1)\\ S_n^{(2)}(\hat{\varvec{\theta }}_2,\varvec{\theta }_1) \end{array}\right)&\overset{L}{\longrightarrow } N\left( \varvec{0},\varvec{T}\right) . \end{aligned}$$

(A.2)

To prove (A.2), we need to check that

$$\begin{aligned} \frac{1}{\sqrt{n}}S_n^{(i)}(\hat{\varvec{\theta }_2}, \varvec{\theta }_1)-\frac{1}{\sqrt{n}}S_n^{(i)}({\varvec{\theta }_{2,0}}, \varvec{\theta }_1)\overset{P}{\longrightarrow }0,~i=1,2. \end{aligned}$$

Let $R_n(\varvec{\theta }_2)=(1/\sqrt{n})S_n^{(1)}(\varvec{\theta }_2,\varvec{\theta }_1)$. Then for any $\varepsilon >0$ and $\delta >0$, we have

$$\begin{aligned} P(|R_n(\hat{\varvec{\theta }}_2)-R_n(\varvec{\theta }_{2,0})|>\varepsilon )&\le P(|\hat{\psi }-\psi _0|>\delta )+P(|\hat{\sigma }_{\varepsilon }^2-\sigma _{0}^2|>\delta )\\&\quad +P(\sup _D|R_n(\widetilde{\varvec{\theta }}_2)-R_n(\varvec{\theta }_{2,0})|>\varepsilon ), \end{aligned}$$

where $\widetilde{\varvec{\theta }}_2=(\widetilde{\psi },\widetilde{\sigma }_{\varepsilon }^2)^\textsf {T}, D:=\{|\widetilde{\psi }-\psi _0|<\delta ,|\widetilde{\sigma }_{\varepsilon }^{2}-\sigma _0^2|<\delta \}.$ If $\hat{\varvec{\theta }}_2$ is a consistent estimator of ${\varvec{\theta }}_2$, then we just need to prove that

$$\begin{aligned} P\left( \sup _D|R_n(\widetilde{\varvec{\theta }}_2)-R_n(\varvec{\theta }_{2,0} )|>\varepsilon \right) \overset{P}{\longrightarrow }0. \end{aligned}$$

By Markov inequality,

$$\begin{aligned}&P\left( \sup _D|R_n(\widetilde{\varvec{\theta }}_2)-R_n(\varvec{\theta }_{2,0}) |>\varepsilon \right) \\&\quad \le \frac{1}{\varepsilon ^2}E\left( \sup _D(R_n(\widetilde{\varvec{\theta }}_2) -R_n(\varvec{\theta }_{2,0}))^2\right) \\&\quad =\frac{1}{\varepsilon ^2}E\left( \sup _D\frac{1}{n}\sum _{t=1}^n (V_{\widetilde{\varvec{\theta }}_2}^{-1}(X_t|X_{t-1})-V_{\varvec{\theta }_{2,0}}^{-1} (X_t|X_{t-1}))^2(X_t-\phi X_{t-1}-\mu _{\varepsilon })^2X_{t-1}^2\right) \\&\quad =\frac{1}{\varepsilon ^2}E\left( \sup _D(V_{\widetilde{\varvec{\theta }}_2}^{-1} (X_1|X_{0})-V_{\varvec{\theta }_{2,0}}^{-1}(X_1|X_{0}))^2 (X_1-\phi X_{0}-\mu _{\varepsilon })^2X_{0}^2\right) \\&\quad =\frac{1}{\varepsilon ^2}E\left( \sup _D\frac{((\widetilde{\psi }- \psi _{0})X_0+(\widetilde{\sigma }_{\varepsilon }^2-\sigma _0^2))^2}{V_{\widetilde{\varvec{\theta }}_2}^{2}(X_1|X_{0})V_{\varvec{\theta }_{2,0}}^{2} (X_1|X_{0})}\left( X_1-\phi X_{0}-\mu _{\varepsilon }\right) ^2X_{0}^2\right) \\&\quad =\frac{1}{\varepsilon ^2}E\left( \sup _D\frac{((\widetilde{\psi }- \psi _0)X_0+(\widetilde{\sigma }_{\varepsilon }^2-\sigma _0^2))^2}{V_{\widetilde{\varvec{\theta }}_2}^{2}(X_1|X_{0})V_{\varvec{\theta }_{2,0}} (X_1|X_{0})}X_{0}^2\right) \\&\quad \le \frac{1}{\varepsilon ^2}\sup _D \left\{ [(\widetilde{\psi }- \psi _0)^2 m_1 +(\widetilde{\sigma }_{\varepsilon }^2-\sigma _0^2)^2 m_2 +2|\widetilde{\psi }-\psi _0||\widetilde{\sigma }_{\varepsilon }^2- \sigma _0^2| m_3]X_{0}^2\right\} \\&\quad \le \frac{C\delta ^2}{\varepsilon ^2}, \end{aligned}$$

where $m_i~(i=1,2,3)$ denote some finite moments of process $\{X_t\}$, C is a positive constant. Similar argument can be used for $1/\sqrt{n}S_n^{(2)}(\varvec{\theta }_2,\varvec{\theta }_1)$. For any fixed $\varepsilon >0$, letting $\delta \rightarrow 0$, we get our assertion which in turn establishes (A.2).

Finally, by the ergodic theorem, we have $\frac{1}{n}\varvec{P}\overset{P}{\longrightarrow }\varvec{T}$. After some algebra, we have,

$$\begin{aligned} (\hat{\varvec{\theta }}_{1,MQL}-\varvec{\theta }_{1,0})=\varvec{P}^{-1} \left( \begin{array}{c} S_n^{(1)}(\hat{\varvec{\theta }_2},\varvec{\theta }_{1,0})\\ S_n^{(2)}(\hat{\varvec{\theta }_2},\varvec{\theta }_{1,0}) \end{array}\right) . \end{aligned}$$

Therefore,

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\theta }}_{1,MQL}-\varvec{\theta }_{1,0})=\left( \frac{1}{n}\varvec{P}\right) ^{-1} \frac{1}{\sqrt{n}} \left( \begin{array}{c} S_n^{(1)}(\hat{\varvec{\theta }_2},\varvec{\theta }_{1,0})\\ S_n^{(2)}(\hat{\varvec{\theta }_2},\varvec{\theta }_{1,0}) \end{array}\right) \overset{L}{\longrightarrow }N(\varvec{0},\varvec{T}^{-1}). \end{aligned}$$

The proof is complete. $\square$

The proof of Theorem 5

Since this proof is similar to the proof of Theorem 4, we omit here. $\square$

The proof of Theorem 6

Using the same statements as in the proof of Theorem 3, we can get the conclusion. $\square$