A covariate-driven beta-binomial integer-valued GARCH model for bounded counts with an application

Abstract

This paper considers the modeling problem of the weekly number of districts with new cases of cryptosporidiosis infection, and proposes a covariate-driven beta-binomial integer-valued GARCH model with a logit transformation to illustrate such bounded integer-valued time series data with extra-binomial variation and high volatility. We establish the existence of the stationary and ergodic solution by imposing a contraction condition on its conditional mean process and a Markov structure on the incorporated covariate process, consider the conditional maximum likelihood (CML) estimator for the parameter vector and discuss its asymptotic properties, conduct a simulation study to examine the finite sample performance of the CML estimator for the proposed model with three data generating mechanisms of the covariate process. Finally, an application to the weekly number of districts with new cases of cryptosporidiosis infection is considered to illustrate the superior performance of the proposed model.

Appendix: Proof

Theorem 1 Denote ${\varvec{Z_t}}=(Y_t,{\varvec{X}}_t)^{\top }$ and $\nu _t=\mathrm{{logit}}(p_{t})$. Then $\nu _t=w+\alpha \nu _{t-1}+\beta Y_{t-1}+f(\varvec{X}_{t-1}, {\varvec{\gamma }})$ If $\varvec{{\widetilde{Z}}_t}=({\widetilde{Y}}_t,\widetilde{p}_t,\widetilde{\varvec{X}}_t)^{\top }$ exists, then $E\Vert \varvec{{\widetilde{Z}}_t} \Vert ^{s}<\infty$ because $Y_t$ is finite.

Denote $\mathscr {B}(\mathcal {B})=1-\alpha \mathcal {B}$ with $\mathcal {B}$ being lag operator, then $\nu _t$ can be rewritten as $\mathscr {B}(\mathcal {B}) \nu _t=w+\beta Y_{t-1}+f({\varvec{X}}_{t}, {\varvec{\gamma }}).$ Denote $\Psi (\mathcal {B})=\mathscr {B}^{-1}(z)=1/(1-\alpha \mathcal {B})=\sum _{i=0}^{+\infty }\psi _{i}\mathcal {B}^i$, where $\psi _0=1$ and $\psi _i=\alpha ^i,\forall i\ge 1.$ Hence, $\nu _t={w}/(1-\alpha )+\beta \Psi (\mathcal {B}) Y_{t-1}+\Psi (\mathcal {B})f({\varvec{X}}_{t},{\varvec{\gamma }})$ by $\mathscr {B}(1)=1-\alpha$. Thus, ${\varvec{Z_t}}$ can be rewritten as a Bernoulli-shift structure, which is denoted as ${\varvec{Z_t}}:= G({\varvec{Z_{t-1}}},{\varvec{Z_{t-2}}},\cdots ; {\varvec{\xi _{t}}}) =\big (F_{p_t}^{-1}(u_t),h({\varvec{X}}_{t-1},\epsilon _t)\big )^{\top },$ where ${\varvec{\xi _{t}}} =\big (F_{p_t}^{-1}(u_t),\epsilon _t\big )^{\top }$ is an i.i.d. sequence by (4) and (5). It is easy to see condition (3.3) of Theorem 3.1 in Doukhan and Wintenberger (2008) holds.

In the following, we show that ${\varvec{Z_t}}$ satisfies the conditions (3.1) and (3.2) of Theorem 3.1 in Doukhan and Wintenberger (2008). Define $\Vert \varvec{Z}_t \Vert _\tau =|Y_t|+\tau \Vert {\varvec{X}}_t\Vert$, if $\varvec{Z}_t=(Y_t,{\varvec{X}}_t)^{\top }$, $\tau >0$ and ${\varvec{X}}_t$ is a d-dimension vector. Then, for any two deterministic sequences $\{{{\varvec{z}}_{{\varvec{t-i}}}}\}$ and $\{{{\widetilde{{\varvec{z}}}}_{t-i}}\}$, $i=1,2,\cdots$, we obtain

$$\begin{aligned}&E\big (\Vert G({{{\varvec{z}}_{{\varvec{t-1}}}}}, {{{\varvec{z}}_{{\varvec{t-2}}}}},\cdots ; { {{\varvec{\eta }}_{{\varvec{t}}}}}) - G({{{\widetilde{{\varvec{z}}}}_{{\varvec{t-1}}}}},{ {\widetilde{\varvec{z}}_{{\varvec{t-2}}}}},\cdots ; {{{\varvec{\eta }}_{{\varvec{t}}}}}) \Vert _\tau \big )\\&\quad =E\big ( \vert F_{p_t}^{-1}(u_t)-F_{{\widetilde{p}}_t}^{-1}(u_t) \vert + \tau \Vert h({\varvec{x}}_{t-1},\epsilon _t)-h(\widetilde{\varvec{x}}_{t-1},\epsilon _t) \Vert \big )\\&\quad \le \int _{0}^{1}|F_{p_t}^{-1}(u)-F_{{\widetilde{p}}_t}^{-1}(u)|du + \tau \rho \Vert {\varvec{x}}_{t-1}-\widetilde{{\varvec{x}}}_{t-1}\Vert \\&\quad = |p_t-{\widetilde{p}}_t|+\tau \rho \Vert {{\varvec{x}}}_{t-1}- \widetilde{{\varvec{x}}}_{t-1}\Vert \text {by Lemma 4 in Chen et al. (2022)}\\&\quad \le \dfrac{1}{4}|\nu _t-{{\widetilde{\nu }}}_t|+\tau \rho \Vert {\varvec{x}}_{t-1}-\widetilde{\varvec{x}}_{t-1}\Vert \text {by Lemma 4 in Chen et al. (2020)}\\&\quad \le \dfrac{|\beta |}{4}\left| \Psi (\mathcal {B})(y_{t-1}-\widetilde{y}_{t-1})\right| \\&\qquad + \dfrac{1}{4}\left| \Psi (\mathcal {B}) \left( f({\varvec{x}}_{t-1},{\varvec{\gamma }})-f(\widetilde{\varvec{x}}_{t-1},{\varvec{\gamma }})\right) \right| +\tau \rho \Vert {\varvec{x}}_{t-1}-\widetilde{\varvec{x}}_{t-1}\Vert \\&\quad \le \sum _{i=0}^{\infty }\dfrac{|\beta | |\psi _i|}{4}|y_{t-i-1}-{\widetilde{y}}_{t-i-1}| \\ {}&\qquad +\tau \left( \dfrac{L}{4\tau }\sum _{i=0}^{\infty }|\psi _i| \Vert {\varvec{x}}_{t-i-1}-\widetilde{\varvec{x}}_{t-i-1}\Vert +\rho \Vert {\varvec{x}}_{t-1}-\widetilde{\varvec{x}}_{t-1}\Vert \right) \\&\quad \le \sum _{i=1}^{+\infty }\varsigma _i \Vert \varvec{z}_{t-i}-\widetilde{\varvec{z}}_{t-i} \Vert , \end{aligned}$$

where $\varsigma _1=\max \big (4^{-1}|\beta |, (4\tau )^{-1}L+\rho \big )$, $\varsigma _i =\max \big (4^{-1}|\beta | , (4\tau )^{-1} L\big )|\psi _{i-1}| =\max \big (4^{-1}|\beta | , (4\tau )^{-1} L\big )|\alpha |^{i-1}$, $\forall i\ge 2.$ Hence, (3.1) in Doukhan and Wintenberger (2008) holds.

Note that $|\beta |<4(1-|\alpha |)$ and $|\alpha |<1$, hence, $4^{-1}|\beta |\sum _{i=1}^{+\infty } |\alpha |^{i-1}=|\beta |/(4(1-|\alpha |))<1$. If $L = 0$, $\sum _{i=1}^{+\infty } \varsigma _i = \max \big (4^{-1}|\beta |,\rho \big )+ \sum _{i=2}^{+\infty } 4^{-1}|\beta | |\alpha |^{i-1} =\max \left( |\beta |/(4(1-|\alpha |)), \rho +|\alpha \beta |/(4(1-|\alpha |))\right) <1$ by $\rho <1-{|\alpha \beta |}/\big (4(1-|\alpha |)\big )$, $|\beta |<4(1-|\alpha |)$ and $|\alpha |<1$. If $L>0$, choosing $\tau$ arbitrarily large such that $\rho + (4\tau )^{-1} L \sum _{i=1}^{+\infty } |\alpha |^{i-1} =\rho + {L}/\big ({4\tau }(1-|\alpha |)\big )<1$. To sum up, $\sum _{i=1}^{+\infty }\varsigma _i<1$, i.e., (3.2) in Doukhan and Wintenberger (2008) hold. Therefore, $\{\varvec{Z}_t\}$ is weakly dependent. This concludes the proof.

Theorem 2(1). Denote ${l}_t(\varvec{\eta })=\log {P_{n}(X_t|\mathcal {F}_{t-1})}:=\log {P_{n}}$ and ${\widetilde{l}}_t({\varvec{\eta }}) =\log {P_n(Y_t|{\widetilde{p}}_t({\varvec{\eta }}))} =\log {\widetilde{P}_n},$ where $\mathrm{{logit}}\big (\widetilde{p}_t(\varvec{\eta })\big )=w+\alpha \mathrm{{logit}}\big (\widetilde{p}_{t-1}(\varvec{\eta })\big )+\beta {\widetilde{Y}}_{t-1} +f(\widetilde{\varvec{X}}_{t-1},{\varvec{\gamma }}), ~t \in \mathbb {Z}$ and $\forall \eta \in \Omega$, $\mathrm{{logit}}(p_t)$ satisfies iterative relationship

$$\begin{aligned} \mathrm{{logit}}({\widetilde{p}}_t)&=\sum \limits _{j=0}^{t-1}\alpha ^j \big (w+\beta {\widetilde{Y}}_{t-1-j}+f(\varvec{{\widetilde{X}}}_{t-1-j},{\varvec{\gamma }})\big ) +\alpha ^{t-1} \mathrm{{logit}}({\widetilde{p}}_{0})\\&\quad \rightarrow \dfrac{w}{1-\alpha }, n\rightarrow \infty . \end{aligned}$$

Hence, there exists a positive constant $\delta$ such that $\dfrac{w}{1-\alpha }-\delta \le \mathrm{{logit}}({\widetilde{p}}_t) \le \dfrac{w}{1-\alpha }+\delta$. Thus,

$$\begin{aligned} \delta _1=:\dfrac{\exp \left( \dfrac{w}{1-\alpha }-\delta \right) }{1+\exp \left( \dfrac{w}{1-\alpha }-\delta \right) }\le {\widetilde{p}}_t \le \dfrac{\exp \left( \dfrac{w}{1-\alpha }+\delta \right) }{1+\exp \left( \dfrac{w}{1-\alpha }+\delta \right) }:=\delta _2. \end{aligned}$$

Note that $E({\widetilde{Y}}_t)=nE({\widetilde{p}}_t)$, thus, $n\delta _1\le E({\widetilde{Y}}_t)\le n\delta _2$, i.e., there exists a positive constant $\delta _3$ such that $c_1:=n\delta _1+\delta _3\le {\widetilde{Y}}_t\le \delta _3+ n\delta _2=:c_2.$

Note that $\widetilde{l}_t(\varvec{\eta })$ is a measurable function of ${X}_t$ for all $\varvec{\eta } \in \Omega$, and is continuous in an open and convex neighbourhood $N(\varvec{\eta }_0)$ of $\varvec{\eta }_0$, then there at least exists a point $\overline{\varvec{\eta }} \in N(\varvec{\eta }_0)$ such that ${l}_t(\varvec{\eta })$ attains the maximum value at $\overline{\varvec{\eta }}$. Thus,

$$\begin{aligned}&E\left( \sup \limits _{\varvec{\eta } \in N(\varvec{\eta }_0)} {\widetilde{l}}_t(\varvec{\eta })\right) =E_{\overline{\varvec{\eta }}}\left( \log {P_n(\widetilde{Y}_t ={\widetilde{y}}_t|{\widetilde{p}}_t)}\right) \\&\quad \propto E_{\overline{\varvec{\eta }}}\left[ \log \big (\Gamma (\widetilde{y}_t+\tau {\widetilde{p}}_t)\big )\right. \\ {}&\qquad \left. +\log \big (\Gamma (n-\widetilde{y}_t+\tau (1-{\widetilde{p}}_t)\big ) +\log \dfrac{1}{\widetilde{p}_t(1-{\widetilde{p}}_t)} +\log \dfrac{1}{{\widetilde{y}}_t!(n-\widetilde{y}_t)!} \right] \\&\quad \le E_{\overline{\varvec{\eta }}}\left[ \log \big (\Gamma (n+\tau \delta _2)\big ) +\log \big (\Gamma (n+\tau (1-\delta _1)\big ) \right. \\&\qquad \left. +\log \dfrac{1}{\delta _1(1-\delta _2)} +\log \dfrac{1}{c_1!(n-c_2)!} \right] <\infty . \end{aligned}$$

Hence,

$$\begin{aligned} \dfrac{1}{T}\sum \limits _{t=1}^{T} {\widetilde{l}}_t(\varvec{\eta })\rightarrow E\widetilde{l}_t(\varvec{\eta }) \end{aligned}$$

(A.1)

in probability as $T\rightarrow \infty$ by Proposition 2 in Kristensen and Rahbek (2005). By Jensen’s inequality and Assumption 2,

$$\begin{aligned} E(\widetilde{l}_t(\varvec{\eta }))-E(\widetilde{l}_t(\varvec{\eta }_0)) =E\left. \log {\frac{P_{n}(\widetilde{Y}_t|\widetilde{p}_{t})_{\varvec{\eta }}}{P_{n}(\widetilde{Y}_t|\widetilde{p}_t)_{\varvec{\eta }_0}}}\right. \le \log {E\frac{P_{n}(\widetilde{Y}_t|\widetilde{p}_{t})_{\varvec{\eta }}}{P_{n}(\widetilde{Y}_t|\widetilde{p}_{t})_{\varvec{\eta }_0}}}=0 \end{aligned}$$

with equality if and only if ${P_{n}(\widetilde{Y}_t|\widetilde{p}_{t})_{\varvec{\eta }}}= {P_{n}(\widetilde{Y}_t|\widetilde{p}_{t})_{\varvec{\eta }_0}}.$ Thus, $E{{\widetilde{l}}}_t(\varvec{\eta })$ attains a strict local maximum at $\varvec{\eta }_0$. Then the consistence of $\hat{{\varvec{\eta }}}_T^{cml}$ can be obtained by $\text {(I)}:=\Vert {\ell }_T({\varvec{\eta }}) - \overline{{\widetilde{\ell }}~}({\varvec{\eta }})\Vert _{\Omega } ~{{\mathop {\longrightarrow }\limits ^{a.s.}}} ~0,~T\rightarrow \infty .$ Note that

$$\begin{aligned} \text {(I)}&\le \Vert {\ell }_T({\varvec{\eta }}) - {\widetilde{\ell }}_T({\varvec{\eta }})\Vert _{\Omega } + \Vert {\widetilde{\ell }}_T({\varvec{\eta }}) -\overline{{\widetilde{\ell }}~}({\varvec{\eta }})\Vert _{\Omega }\\&{}\le \underbrace{\frac{1}{T}\sum \limits _{t=2}^{T} \Vert l_t({\varvec{\eta }}) - {\widetilde{l}}_t({\varvec{\eta }}) \Vert _{\Omega } }_{\text {(II)}} + \underbrace{\Vert \frac{1}{T}\sum \limits _{t=2}^{T}\widetilde{l}_t({\varvec{\eta }}) -E\widetilde{l}_t({\varvec{\eta }})\Vert _{\Omega } }_{\text {(III)}}. \end{aligned}$$

Using (A.1), ${\text {(III)}}~{{\mathop {\longrightarrow }\limits ^{a.s.}}} ~0, ~T\rightarrow \infty .$ By Lemma 2.1 in Straumann and Mikosch (2006) and Assumption 3, $\forall t\ge N, N \in \mathbb {N}$, $\sum _{t=N}^{\infty }\zeta _t \Vert p_t({\varvec{\eta }}) - {\widetilde{p}}_t({\varvec{\eta }}) \Vert _{\Omega }$ converges a.s., hence $\sum _{t=N}^{\infty }\Vert l_t({\varvec{\eta }}) - {\widetilde{l}}_t({\varvec{\eta }}) \Vert _{\Omega }<\infty .$ Thus, $\text {(II)}~{{\mathop {\longrightarrow }\limits ^{a.s.}}}~0, ~T\rightarrow \infty$. Therefore, we have $\text {(I)}~{{\mathop {\longrightarrow }\limits ^{a.s.}}} ~0, ~T\rightarrow \infty$. Then, the CML estimator $\hat{{\varvec{\eta }}}_T^{cml}$ is strongly consistent with respect to ${\varvec{\eta }}_0.$ Hence, part (1) holds.

(2).The proof of part 2 relies on the Taylor series expansion of the score vector around ${\varvec{\eta }}_0$. We have

$$\begin{aligned} \varvec{0}&=T^{-1/2} \dfrac{\partial {\widetilde{\ell }{({\hat{\varvec{\eta }}_T^{cml}})}}}{\partial {\varvec{\eta }}}\\&{}=T^{-1/2}\dfrac{\partial {\widetilde{\ell }{({{\varvec{\eta }}_0})}}}{\partial {\varvec{\eta }}} +\left( \dfrac{1}{T} \dfrac{\partial ^2{{{\widetilde{\ell }}}({\varvec{\eta }^{*})}}}{\partial {\varvec{\eta }}\partial {\varvec{\eta }^{\top }}} \right) \sqrt{T}(\hat{\varvec{\eta }}_T^{cml}-{\varvec{\eta }}_0), \end{aligned}$$

where $\varvec{\eta }^{\star }$ lies in between $\hat{\varvec{\eta }}_T^{cml}$ and ${\varvec{\eta }}_0$. According to Theorem 4.1.3 of Amemiya (1985), we need to show that

$$\begin{aligned}&T^{-1/2}\dfrac{\partial { \widetilde{\ell }{({{\varvec{\eta }}_0})}}}{\partial {\varvec{\eta }}} {{\mathop {\longrightarrow }\limits ^{d}}} N(\varvec{0},\varvec{I}(\varvec{\eta }_0)) ~\text {with}~\varvec{I}(\varvec{\eta }_0)=E\left[ \dfrac{\partial {\widetilde{l}_t({\varvec{\eta }_0})}}{\partial {\varvec{\eta }}} \dfrac{\partial { {\widetilde{l}}_t({\varvec{\eta }_0})}}{\partial {\varvec{\eta }^{\top }}} \right] ,\\&\dfrac{1}{T} \dfrac{\partial ^2{{{\widetilde{\ell }}}({\varvec{\eta }^{\star })}}}{\partial {\varvec{\eta }}\partial {\varvec{\eta }^{\top }}}\longrightarrow -{\varvec{I}}({\varvec{\eta }}_0)~ {\text {in probability}}, \end{aligned}$$

which can be obtained by the method in Theorem 4 (Chen et al. 2020), $\varvec{{{\widetilde{\eta }}}}_T^{cml}$ is asymptotic normality. Similar to Theorem 3 in Chen et al. (2022), we obtain

$\text {(a).}~~ T^{-1/2}\dfrac{\partial {\widetilde{\ell }{({{\varvec{\eta }}_0})}}}{\partial {\varvec{\eta }}} ~{{\mathop {\longrightarrow }\limits ^{d}}}~N(\varvec{0},\varvec{I}(\varvec{\eta }_0)) ~\text {with}~ \varvec{I}(\varvec{\eta }_0):=E\left[ \dfrac{\partial { {\widetilde{l}}_t({\varvec{\eta }_0})}}{\partial {\varvec{\eta }}} \dfrac{\partial { \widetilde{l}_t({\varvec{\eta }_0})}}{\partial {\varvec{\eta }^{\top }}} \right] .$

$\text {(b)}.~~ \dfrac{1}{T} \dfrac{\partial ^2{{{\widetilde{\ell }}}({\varvec{\eta }^{\star })}}}{\partial {\varvec{\eta }}\partial {\varvec{\eta }^{\top }}}~ {\mathop {\longrightarrow }\limits ^{p}}~ -{\varvec{H}}({\varvec{\eta }}_0):= E\dfrac{\partial ^2{{\widetilde{l}}_t({\varvec{\eta }_0)}}}{\partial {\varvec{\eta }}\partial {\varvec{\eta }^{\top }}},$  where $\varvec{\eta }^{\star }$ lies in between $\hat{\varvec{\eta }}_T^{cml}$ and ${\varvec{\eta }}_0$.

Thus, the result in part (2) holds.