Integer-valued time series model order shrinkage and selection via penalized quasi-likelihood approach

Abstract

This paper proposes a penalized maximum quasi-likelihood (PMQL) estimation that can solve the problem of order selection and parameter estimation regarding the pth-order integer-valued time series models. The PMQL estimation can effectively delete the insignificant orders in model. By contrast, the significant orders can be retained and their corresponding parameters are estimated, simultaneously. Moreover, the PMQL estimation possesses certain robustness hence its order shrinkage effectiveness is superior to the traditional penalized estimation method even if the data is contaminated. The theoretical properties of the PMQL estimator, including the consistency and oracle properties, are also investigated. Numerical simulation results show that our method is effective in a variety of situations. The Westgren’s data set is also analyzed to illustrate the practicability of the PMQL method.

Appendix

To prove the asymptotic properties of the PMQL estimator, we need the following regularity conditions.

• (C.1) $0<\sum ^p_{i=1}\alpha _i<1$;

• (C.2) $\mathrm {E}|X_t|^4<\infty$;

• (C.3) $a_n=O(n^{-1/2})$;

• (C.4) $b_n=o(1)$.

(C.1) ensures that the sequence $\{X_t\}$ is strictly stationary and ergodic; (C.2) guarantees that the PMQL estimator possesses asymptotic properties; According to (C.3) and (C.4), the PMQL criterion function is dominated by the quasi-likelihood criterion function and ensure that the PMQL estimator is $\sqrt{n}$ consistent and asymptotically normal. To prove Theorem 2.1, the following lemma is needed.

Lemma A1

Under conditions (C.1)–(C.2), as $n\rightarrow \infty$ we have

$$\begin{aligned} \frac{1}{\sqrt{n}}S_n(\varvec{\beta }_{0})\mathop {\longrightarrow } \limits ^\mathrm{D}\mathrm {N}(\mathbf{0 },\varvec{\Sigma }(\varvec{\beta }_{0})), \end{aligned}$$

and $S_{n}(\varvec{\beta }_{0})$ is defined in (2).

Proof of Lemma A1

Assuming that $\varvec{\theta }$ is known, we set ${\mathcal {F}}_{n-1}=\sigma \left\{ X_{0},X_{1},\ldots ,X_{p}\right\}$, for

$$\begin{aligned} S_{n}^{(1)}(\varvec{\beta })=\sum ^{n}_{t=1}V_{\varvec{\theta }}^{-1} \left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right) \left( X_{t}-\sum ^{p}_{j=1} \alpha _{j}X_{t-j}-\mu \right) X_{t-1}, \end{aligned}$$

calculating the conditional expectation we have

$$\begin{aligned}&\mathrm {E}\left( V_{\varvec{\theta }}^{-1}\left( X_{t}|X_{t-1},\ldots ,X_{t-p} \right) \left( X_{t}-\sum ^{p}_{j=1}\alpha _{j}X_{t-j}-\mu \right) X_{t-1} \big |{\mathcal {F}}_{n-1}\right) \\ =&V_{\varvec{\theta }}^{-1}\left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right) \mathrm {E} \left( \left( X_{t}-\sum ^{p}_{j=1}\alpha _{j}X_{t-j}-\mu \right) X_{t-1} \big |{\mathcal {F}}_{n-1}\right) \\ =&0, \end{aligned}$$

then

$$\begin{aligned} \mathrm {E}\left( S_{n}^{(1)}(\varvec{\beta })\big |{\mathcal {F}}_{n-1} \right) =S_{n-1}^{(1)}(\varvec{\beta }), \end{aligned}$$

that means $\left\{ S_{n}^{(1)}(\varvec{\beta }),{\mathcal {F}}_{n},n\ge 0\right\}$ is a martingale. By $\mathrm {E}|X_{t}|^{4}<\infty$, the strict stationarity of $\{X_{t}\}$ and the ergodic theorem, we can derive that

$$\begin{aligned} \frac{1}{n}\sum ^{n}_{t=1}&V_{\varvec{\theta }}^{-2}\left( X_{t}|X_{t-1}, \ldots ,X_{t-p}\right) \left( X_{t}-\sum ^{p}_{j=1}\alpha _{j}X_{t-j} -\mu \right) ^{2}X_{t-1}^{2}\mathop {\longrightarrow }\limits ^{a.s.}\\&\mathrm {E}\left( V_{\varvec{\theta }}^{-2}\left( X_{p}|X_{p-1}, \ldots ,X_{0}\right) \left( X_{t}-\sum ^{p}_{j=1}\alpha _{j}X_{p-j} -\mu \right) ^{2}X_{p-1}^{2}\right) \\ =&\mathrm {E}\left( V_{\varvec{\theta }}^{-1}\left( X_{p}|X_{p-1}, \ldots ,X_{0}\right) X_{p-1}^{2}\right) =\sigma _{11}. \end{aligned}$$

Following the martingale central limit theorem, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}S_{n}^{(1)}(\varvec{\beta })\mathop {\longrightarrow } \limits ^\mathrm {D}\mathrm {N}(0,\sigma _{11}). \end{aligned}$$

Similarly, we can prove $\left\{ S_{n}^{(i)}(\varvec{\beta }), {\mathcal {F}}_{n},n\ge 0\right\}$, $i=2,3,\ldots ,p+1$ is a martingale and

$$\begin{aligned} \frac{1}{\sqrt{n}}S_{n}^{(i)}(\varvec{\beta })\mathop {\longrightarrow } \limits ^\mathrm{D}\mathrm {N}(0,\sigma _{ii}). \end{aligned}$$

For any $\mathbf{c }=(c_{1},c_{2},\ldots ,c_{p+1})^{\mathrm {\textsf {T}}} \in {\mathbb {R}}^{p+1}\backslash (0,\ldots ,0)^{\mathrm {\textsf {T}}}$,

$$\begin{aligned} \frac{1}{\sqrt{n}}\mathbf{c }^{\mathrm {\textsf {T}}}\begin{pmatrix} S_{n}^{(1)}(\varvec{\beta })\\ \vdots \\ S_{n}^{(p+1)}(\varvec{\beta }) \end{pmatrix}=&\frac{1}{\sqrt{n}}\sum ^{n}_{t=1}V_{\varvec{\theta }}^{-1} \left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right) \left( X_{t}-\sum ^{p}_{j=1}\alpha _{j}X_{t-j} -\mu \right) \\&\cdot \left( c_{1}X_{t-1}+\cdots +c_{p}X_{t-p}+c_{p+1}\right) \\&\mathop {\longrightarrow }\limits ^\mathrm{D}\mathrm {N} \left( \mathbf{0 },\mathrm {E} \left[ V_{\varvec{\theta }}^{-1}\left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right) \left( c_{1}X_{t-1}+\cdots +c_{p}X_{t-p}+c_{p+1}\right) ^{2}\right] \right) . \end{aligned}$$

As $n\rightarrow \infty$, according to the Cramér–Wold device we have:

$$\begin{aligned} \frac{1}{\sqrt{n}}\begin{pmatrix} S_{n}^{(1)}(\varvec{\beta })\\ \vdots \\ S_{n}^{(p+1)}(\varvec{\beta }) \end{pmatrix}\mathop {\longrightarrow }\limits ^\mathrm{D}\mathrm {N}(\mathbf{0 }, \varvec{\Sigma }(\varvec{\beta })). \end{aligned}$$

Next, we replace $V_{\varvec{\theta }}\left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right)$ by $V_{\varvec{{\bar{\theta }}}}\left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right)$, where $\varvec{{\bar{\theta }}}$ is a consistent estimator of $\varvec{\theta }$, to prove

$$\begin{aligned} \frac{1}{\sqrt{n}}\begin{pmatrix} {\hat{S}}_{n}^{(1)}(\varvec{\beta })\\ \vdots \\ {\hat{S}}_{n}^{(p+1)}(\varvec{\beta }) \end{pmatrix}\mathop {\longrightarrow }\limits ^\mathrm{D}\mathrm {N}(\mathbf{0 }, \varvec{\Sigma }(\varvec{\beta })). \end{aligned}$$

(11)

To prove (11) holds, we need prove

$$\begin{aligned} \left( \frac{1}{\sqrt{n}}{\hat{S}}_{n}^{(i)}(\varvec{\beta }) -\frac{1}{\sqrt{n}}S_{n}^{(i)}(\varvec{\beta })\right) \mathop {\longrightarrow }\limits ^{\mathrm {P}}0,~i=1,2,\ldots ,p+1. \end{aligned}$$

(12)

We set $R_{n}(\varvec{\theta })=(1/\sqrt{n})S_{n}^{(1)}(\varvec{\beta })$, for any $\epsilon >0$ and $\delta >0$

$$\begin{aligned} \mathrm {P}\left( \big |R_{n}(\varvec{{\hat{\theta }}})-R_{n} (\varvec{\theta })\big |>\epsilon \right) \le&\sum ^{p}_{i=1}\mathrm {P} \left( |{\hat{\theta }}_{i}-\theta _{i}|>\delta \right) +\mathrm {P} \left( \big |{\hat{\delta }}_{z}^{2}-\delta ^{2}_{z}\big |>\delta \right) \\&+\mathrm {P}\left( \sup \limits _{D}\big |R_{n}(\varvec{\theta }_{1}) -R_{n}(\varvec{\theta })\big |>\epsilon \right) , \end{aligned}$$

where $\varvec{\theta }_{1}=(\sigma _{11}^{2}, \ldots ,\sigma _{p1}^{2},\sigma ^{2}_{1})^{\mathrm {\textsf {T}}}$, $D\triangleq \left\{ |\sigma ^{2}_{i1}-\sigma ^2_{i}|<\delta ,1\le i\le p,|\sigma ^{2}_{1} -\sigma ^{2}_{z}|<\delta \right\}$. If $\varvec{{\bar{\theta }}}$ is a consistent estimator of $\varvec{\theta }$, we only need to prove

$$\begin{aligned} \mathrm {P}\left( \sup \limits _{D}\big |R_{n}(\varvec{\theta }_{1})-R_{n} (\varvec{\theta })\big |>\epsilon \right) \mathop {\longrightarrow } \limits ^{\mathrm {P}}0. \end{aligned}$$

By Markov inequation, we have

$$\begin{aligned}&\mathrm {P}\left( \sup \limits _{D}\big |R_{n}(\varvec{\theta }_{1}) -R_{n}(\varvec{\theta })\big |>\epsilon \right) \nonumber \\&\le \frac{1}{\epsilon ^{2}}\mathrm {E}\left( \sup \limits _{D}(R_{n} (\varvec{\theta }_{1})-R_{n}(\varvec{\theta }))^{2}\right) \nonumber \\&=\frac{1}{\epsilon ^{2}}\mathrm {E}\left( \sup _{D}\frac{1}{n}\sum ^{n}_{t=1} \left( V_{\varvec{\theta }_{1}}^{-1}\left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right) -V_{\varvec{\theta }}^{-1}\left( X_{t}|X_{t-1},\ldots ,X_{t-p}\right) \right) ^{2} \left( X_{t}-\sum ^{p}_{j=1}\alpha _{j}X_{t-j}-\mu \right) ^{2}X_{t-1}^{2}\right) \nonumber \\&=\frac{1}{\epsilon ^{2}}\mathrm {E}\left( \sup _{D} \left( V_{\varvec{\theta }_{1}}^{-1}\left( X_{p}|X_{p-1},\ldots ,X_{0}\right) -V_{\varvec{\theta }}^{-1}\left( X_{p}|X_{p-1},\ldots ,X_{0}\right) \right) ^{2} \left( X_{p}-\sum ^{p}_{j=1}\alpha _{j}X_{p-j}-\mu \right) ^{2}X_{p-1}^{2}\right) \nonumber \\&=\frac{1}{\epsilon ^{2}}\mathrm {E}\left( \sup _{D}\frac{\left( \sum ^{p}_{i=1}| \sigma ^{2}_{i1}-\sigma ^2_{i}||X_{p-i}|+(\sigma ^{2}_{1}-\sigma ^{2}_{z})\right) ^{2}}{V^{2}_{\varvec{\theta }_{1}}\left( X_{p}|X_{p-1},\ldots ,X_{0} \right) V^{2}_{\varvec{\theta }}\left( X_{p}|X_{p-1},\ldots ,X_{0}\right) } \left( X_{p}-\sum ^{p}_{j=1}\alpha _{j}X_{p-j}-\mu \right) ^{2}X_{p-1}^{2}\right) \nonumber \\&=\frac{1}{\epsilon ^{2}}\mathrm {E}\left( \sup _{D}\frac{\left( \sum ^{p}_{i=1}| \sigma ^{2}_{i1}-\sigma ^2_{i}||X_{p-i}|+(\sigma ^{2}_{1}-\sigma ^{2}_{z})\right) ^{2}}{V^{2}_{\varvec{\theta }_{1}}\left( X_{p}|X_{p-1},\ldots ,X_{0}\right) V_{\varvec{\theta }}\left( X_{p}|X_{p-1},\ldots ,X_{0}\right) }X_{p-1}^{2} \right) \nonumber \\&\le \frac{1}{\epsilon ^{2}}\mathrm {E}\left( \sup _{D}\frac{\left( \sum ^{p}_{i=1} \delta |X_{p-i}|+\delta \right) ^{2}}{(\sigma ^{2}_{z}-\delta )^{2}\sigma ^{2}_{z}} X_{p-1}^{2}\right) \nonumber \\&\le \frac{c\delta ^{2}}{\epsilon ^{2}.} \end{aligned}$$

(13)

where c is a finite positive constant. $\frac{1}{\sqrt{n}}S_{n}^{(i)}(\varvec{\beta }),~i=2,\ldots ,p+1$ can be discussed similarly. As $\delta \rightarrow 0$, the result holds up. $\square$

Proof of Theorem 2.1

By Lemma A1 and Theorem 1 of Wang et al. (2017), the result can be derived directly. $\square$

Proof of Lemma 2.1

The relevant proofs are similar to Lemma 1 of Wang et al. (2017). Hence, detailed proofs are omitted here. $\square$

Proof of Theorem 2.2

The proofs are similar to Theorem 2 of Wang et al. (2017). We omit here. $\square$