AR(1) model with skew-normal innovations

Abstract

In this paper, we consider an autoregressive model of order one with skew-normal innovations. We propose several methods for estimating the parameters of the model and derive the limiting distributions of the estimators. Then, we study some statistical properties and the regression behavior of the proposed model. Finally, we provide a Monte Carlo simulation study for comparing performance of estimators and consider a real time series to illustrate the applicability of the proposed model.

Appendix

Proof of Theorem 2

Consider the log-likelihood $L_{n}(\varvec{\theta })$ in (14), where for all $\varvec{x}\in \mathbb {R}^{2}$,

$$\begin{aligned} l(\varvec{x};\varvec{\theta })=- \frac{1}{2}\ln (\sigma ^{2})- \frac{y^{2}}{2\sigma ^{2}}+\ln \varPhi \left( \frac{\lambda y}{\sigma }\right) +c, \end{aligned}$$

(26)

with $y=x_{0}-\varphi x_{1}-\mu$ and c is a constant. Similar to Bondon’s approach, the basic technique of this proof is to control the behaviour of the first and second order in a Taylor expansion of $L_{n}(\varvec{\theta })$ about $\varvec{\theta }_{0}$. In some neighborhood $S_{0}$ of $\varvec{\theta }_{0}$ for almost all $\varvec{x}\in \mathbb {R}^{2}$, the function $l(\varvec{x};\varvec{\theta } )$ is twice continuously differentiable with respect to $\varvec{\theta }$. Then, for $\varepsilon >0, \left\| \varvec{\theta } -\varvec{\theta }_{0}\right\| <\varepsilon$, the Taylor expansion of $L_{n}(\varvec{\theta })$ about ${\varvec{\theta }}_{0}$ is

$$\begin{aligned} L_{n}(\varvec{\theta })= & {} L_{n}(\varvec{\theta }_{0})+\left( \varvec{\theta } -\varvec{\theta }_{0}\right) ^{\top }\frac{\partial }{\partial \varvec{\theta }}L_{n}(\varvec{\theta }_{0})+ \frac{1}{2}(\varvec{\theta } -\varvec{\theta }_{0})^{\top }V_{n} (\varvec{\theta } -\varvec{\theta }_{0})\nonumber \\&+ \frac{1}{2}(\varvec{\theta } -\varvec{\theta }_{0})^{\top }T_{n}(\varvec{\theta }^{*})(\varvec{\theta } -\varvec{\theta }_{0}), \end{aligned}$$

where

$$\begin{aligned} V_{n}=\frac{\partial ^{2}}{\partial \varvec{\theta } \partial {\varvec{\theta }}^\top } L_{n}(\varvec{\theta }_{0}), \, T_{n}(\varvec{\theta }^{*})=\frac{\partial ^{2}}{\partial \varvec{\theta } \partial {\varvec{\theta }}^\top }L_{n}(\varvec{\theta }^{*}) - \frac{\partial ^{2}}{\partial \varvec{\theta } \partial {\varvec{\theta } }^\top }L_{n}(\varvec{\theta }_{0}), \end{aligned}$$

and $\varvec{\theta }^{*}=\varvec{\theta }^{*}(X_{1}, \ldots , X_{n};\varvec{\theta })$ is an intermediate point between $\varvec{\theta }$ and $\varvec{\theta }_{0}$. According to Bondon’s approach, for proving the parts (i) and (ii), we must check conditions (A1)–(A4). To check condition (A1), by (26), we obtain

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial \varphi }l(\varvec{x};\varvec{\theta })&=\frac{x_{1}y}{\sigma ^{2}} - \frac{\lambda x_{1}}{\sigma }W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial }{\partial \mu }l(\varvec{x};\varvec{\theta })&=\frac{y}{\sigma ^{2}} - \frac{\lambda }{\sigma }W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial }{\partial \sigma ^{2}}l(\varvec{x};\varvec{\theta })&=- \frac{1}{2\sigma ^{2}} + \frac{y^{2}}{2\sigma ^{4}}- \frac{\lambda y}{2\sigma ^{3}}W \left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial }{\partial \lambda }l(\varvec{x};\varvec{\theta })&=\frac{y}{\sigma }W\left( \frac{\lambda y}{\sigma }\right) . \end{aligned} \end{aligned}$$

(27)

Finiteness of $\mathbb {E}X_{t}^{2}$ and $\mathbb {E}[Y^{k}W(\frac{\lambda _{0}Y}{\sigma _{0}})]$ for $k=0, 1$ where $Y\sim SN(0, \sigma , \lambda )$ imply that for $1\le i\le 4$

$$\begin{aligned} \mathbb {E}\left| \frac{\partial }{\partial \theta _{i}} l(X_{t}, X_{t-1};\varvec{\theta }_{0})\right| <\infty . \end{aligned}$$

(28)

The process $U_{t}=\frac{\partial }{\partial \varvec{\theta } }l(X_{t}, X_{t-1};\varvec{\theta }_{0})$ is strictly stationary and ergodic, because $(X_{t})$ is strictly stationary and ergodic. Therefore, from the pointwise ergodic Theorem for stationary sequences and (28), we have

$$\begin{aligned} n^{-1}\frac{\partial }{\partial \varvec{\theta }_{0}}L_{n}(\varvec{\theta } )=n^{-1}\sum \limits _{t=2}^{n}\frac{\partial }{\partial \varvec{\theta }} l(X_{t}, X_{t-1};\varvec{\theta }_{0})\overset{a.s}{\longrightarrow }\mathbb {E}U_{t}, \end{aligned}$$

hence, for checking condition (A1), we must prove $\mathbb {E}U_{t}=0$. Since $Y_{t}=Z_{t}-\mu _{0}\sim SN(0, \sigma , \lambda )$ and $X_{t-i}$ and $Z_{t}$ are independent when $i>0$ ($X_{t}$ is causal), we have

$$\begin{aligned} \mathbb {E}(Z_{t}-\mu _{0})= & {} b_{0}\sigma _{0}\delta _{0}, \nonumber \\ \mathbb {E}(Z_{t}-\mu _{0})^{2}= & {} \sigma _{0}^{2}, \nonumber \\ \mathbb {E}\left[ W\left( \frac{\lambda _{0}(Z_{t}-\mu _{0)}}{\sigma _{0}}\right) \right]= & {} \frac{b}{\sqrt{1+\lambda _{0}^{2}}}, \nonumber \\ \mathbb {E}\left[ (Z_{t}-\mu _{0})^{k} W\left( \frac{\lambda _{0}(Z_{t}-\mu _{0)}}{\sigma _{0}}\right) \right]= & {} 0, \quad (k \hbox { is odd}) \\ \mathbb {E}\left[ \left( Z_{t}-\mu _{0}\right) ^{2} W \left( \frac{\lambda _{0}\left( Z_{t}-\mu _{0}\right) }{\sigma _{0}}\right) \right]= & {} \frac{b\sigma _{0}^{2}}{\left( 1+\lambda _{0}^{2}\right) \sqrt{1+\lambda _{0}^{2}}}, \nonumber \\ \mathbb {E}[X_{t-1} (Z_{t}-\mu _{0})]= & {} 0, \nonumber \\ \mathbb {E}\left[ X_{t-1} W\left( \frac{\lambda _{0}(Z_{t}-\mu _{0)}}{\sigma _{0}} \right) \right]= & {} m\frac{b}{\sqrt{1+\lambda _{0}^{2}}},\nonumber \end{aligned}$$

(29)

By (27) and (29), we get $\mathbb {E}U_{t}=0$ and condition (A1) is checked.

For checking condition (A2), according to (27), we obtain all the second order partial derivatives of $l(\varvec{x};\varvec{\theta })$ with respect to $\varvec{\theta }$ as follows:

$$\begin{aligned} \begin{aligned} \frac{\partial ^{2}}{\partial \varphi ^{2}}l(\varvec{x};\varvec{\theta })&=- \frac{x_{1}^{2}}{\sigma ^{2}}- \frac{\lambda x_{1}}{\sigma }\frac{\partial }{\partial \varphi } W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \varphi \partial \mu }l(\varvec{x};\varvec{\theta })&=- \frac{x_{1}}{\sigma ^{2}}- \frac{\lambda x_{1}}{\sigma }\frac{\partial }{\partial \mu }W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \varphi \partial \sigma ^{2}}l(\varvec{x};\varvec{\theta })&=- \frac{x_{1}y}{\sigma 4}+ \frac{\lambda x_{1}}{2\sigma ^{3}}W\left( \frac{\lambda y}{\sigma }\right) - \frac{\lambda x_{1}}{\sigma }\frac{\partial }{\partial \sigma ^{2}}W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \varphi \partial \lambda }l(\varvec{x};\varvec{\theta })&=- \frac{x_{1}}{\sigma }W\left( \frac{\lambda y}{\sigma }\right) - \frac{\lambda x_{1}}{\sigma } \frac{\partial }{\partial \lambda }W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \mu ^{2}}l(\varvec{x};\varvec{\theta })&=- \frac{1}{\sigma ^{2}}- \frac{\lambda }{\sigma }\frac{\partial }{\partial \mu } W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \mu \partial \sigma ^{2}}l(\varvec{x};\varvec{\theta })&=- \frac{y }{\sigma ^{4}}+ \frac{\lambda }{2\sigma ^{3}}W\left( \frac{\lambda y}{\sigma }\right) - \frac{\lambda }{\sigma }\frac{\partial }{\partial \sigma ^{2}}W \left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \mu \partial \lambda }l(\varvec{x};\varvec{\theta })&=- \frac{W( \frac{\lambda y}{\sigma })}{\sigma }- \frac{\lambda }{\sigma }\frac{\partial }{\partial \lambda }W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \sigma ^{4}}l(\varvec{x};\varvec{\theta })&=\frac{1}{2\sigma ^{4}}- \frac{y^{2}}{\sigma ^{6}}+ \frac{3}{4}\frac{\lambda y}{\sigma ^{5}}W\left( \frac{\lambda y}{\sigma }\right) - \frac{\lambda y}{2\sigma ^{3}}\frac{\partial }{\partial \sigma ^{2}}W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \sigma ^{2}\partial \lambda }l(\varvec{x};\varvec{\theta })&=- \frac{y}{\sigma ^{3}}W\left( \frac{\lambda y}{\sigma }\right) - \frac{\lambda y}{2\sigma ^{3}}\frac{\partial }{\partial \lambda }W\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial ^{2}}{\partial \lambda ^{2}}l(\varvec{x};\varvec{\theta })&=\frac{y}{\sigma } \frac{\partial }{\partial \lambda }W\left( \frac{\lambda y}{\sigma }\right) , \end{aligned} \end{aligned}$$

(30)

where $y=x_{0}-\varphi x_{1}-\mu , W(.)=\frac{\phi (.)}{\varPhi (.)}$ and

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial \varphi }W\left( \frac{\lambda y}{\sigma }\right)&=\frac{\lambda ^{2}}{\sigma ^{2}}x_{1}y W\left( \frac{\lambda y}{\sigma }\right) + \frac{\lambda }{\sigma }x_{1} W^{2}\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial }{\partial \mu }W\left( \frac{\lambda y}{\sigma }\right)&=\frac{\lambda ^{2}}{\sigma ^{2}}yW\left( \frac{\lambda y}{\sigma }\right) + \frac{\lambda }{\sigma }W^{2}\left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial }{\partial \sigma ^{2}}W\left( \frac{\lambda y}{\sigma }\right)&=\frac{\lambda ^{2}}{2\sigma ^{4}}y^{2}W\left( \frac{\lambda y}{\sigma }\right) + \frac{\lambda y}{2\sigma ^{3}}W^{2} \left( \frac{\lambda y}{\sigma }\right) , \\ \frac{\partial }{\partial \lambda }W\left( \frac{\lambda y}{\sigma }\right)&=- \frac{\lambda }{\sigma ^{2}}y^{2}W\left( \frac{\lambda y}{\sigma }\right) - \frac{y}{\sigma }W^{2}\left( \frac{\lambda y}{\sigma }\right) . \end{aligned} \end{aligned}$$

(31)

Since for $\varvec{\theta }_{0}\in \varOmega \cap A, \mathbb {E}X_{t}^{2}$ and $a_{k}= \mathbb {E}[Y^{k}W^2(\frac{\lambda _{0}Y}{\sigma _{0}})]$ for $k=0, 1, 2$, are finite, we deduce

$$\begin{aligned} \mathbb {E}\left| \frac{\partial ^{2}}{\partial \theta _{i}\partial \theta _{j}}l(X_{t}, X_{t-1}; \varvec{\theta }_{0})\right| <\infty , \quad 1\le i\le j\le 4. \end{aligned}$$

(32)

The process $V_{t}=\frac{\partial ^{2}}{\partial \varvec{\theta } \partial {\varvec{\theta }}^\top }l(X_{t}, X_{t-1}; \varvec{\theta }_{0})$ is strictly stationary and ergodic. By the pointwise ergodic Theorem, we get

$$\begin{aligned} n^{-1}V_{n}\overset{a.s}{\longrightarrow }V, \end{aligned}$$

where $V=-\mathbb {E}(\frac{\partial ^{2}}{\partial \varvec{\theta } \partial {\varvec{\theta }}^\top }l(X_{t}, X_{t-1};\varvec{\theta }_{0}))$. By (29)–(32), the matrix V is given by

$$\begin{aligned} V=\left( \begin{array}{cccc} \left( \text {Cum}_{{\widetilde{X}}, {\widetilde{X}}}(0) +m^{2}\right) c_{1} &{}\quad mc_{1} &{}\quad mc_{2} &{}\quad mc_{3}\\ mc_{1} &{}\quad c_{1} &{}\quad c_{2} &{}\quad c_{3}\\ mc_{2} &{}\quad c_{2} &{}\quad \frac{1}{2\sigma _{0}^{4}}+ \frac{\lambda _{0}^{2}}{4\sigma _{0}^{4}}c_{4} &{}\quad - \frac{\lambda _{0}}{2\sigma _{0}^{2}}c_{4}\\ mc_{3} &{}\quad c_{3} &{}\quad - \frac{\lambda _{0}}{2\sigma _{0}^{2}}c_{4} &{}\quad c_{4} \end{array} \right) . \end{aligned}$$

The matrix V is positive definite, because $\det (V_{k})>0, \ 1\le k\le 4$, for $\varvec{\theta }_{0}\in \varOmega \cap A$ which reduce to (15) and (16), where $V_{k}$ is the $k\times k$ submatrix formed by deleting the last $(4-k)$ rows and last $(4-k)$ columns of V (Lemma 1, Searle 1971).

To prove condition (A3), similar to Bondon’s method, we shall prove there exist measurable functions $g_{i, j, k}:\mathbb {R}^{2}\rightarrow \mathbb {R} 1\le i, j, k\le 4$, such that

$$\begin{aligned} \left| \frac{\partial ^{3}}{\partial \theta _{i}\partial \theta _{j}\partial \theta _{k}}l(\varvec{x};\varvec{\theta })\right|<g_{i, j, k}(\varvec{x})\quad \hbox {and}\quad \mathbb {E} \left( g_{i, j, k}(X_{t}, X_{t-1})\right) <\infty , \end{aligned}$$

(33)

for almost all $\varvec{x}\in \mathbb {R}^{2}$. But all the third order partial derivatives of $l(\varvec{x};\varvec{\theta })$ with respect to $\varvec{\theta }$ take sum of the forms

$$\begin{aligned} h(\varvec{x};\varvec{\theta })=\frac{r\lambda ^{i}x_{1}^{j}y^{k}W^{l} \left( \frac{\lambda y}{\sigma }\right) }{\sigma ^{q}}, \end{aligned}$$

(34)

where r is an integer and i, j, k, l, q are nonnegative integers. Hence, to prove (33), it is sufficient to show that there exist a measurable function $g:\mathbb {R}^{2}\rightarrow \mathbb {R}$ such that

$$\begin{aligned} \left| h(\varvec{x};\varvec{\theta })\right| \le g(\varvec{x}), \quad \hbox {and}\quad \mathbb {E}\left( g(X_{t}, X_{t-1})\right) <\infty , \end{aligned}$$

for all $\varvec{\theta } \in N$ and for almost all $\varvec{x}\in \mathbb {R}^{2}$. Now, we choose N, which for all $\varvec{\theta } \in N$

$$\begin{aligned} \left| \varphi \right|<1, \left| \mu \right| \le 2\left| \mu _{0}\right| , \sigma >\frac{\sigma _{0}}{2}, \ \left| \lambda \right| <\left| \lambda _{0}\right| , W\left( \frac{\lambda y}{\sigma }\right) \le W\left( \frac{2}{\sigma _{0}}\left| \lambda _{0}y\right| \right) , \end{aligned}$$

(35)

then for all $(\varvec{\theta }, \varvec{x})\in N\times \mathbb {R}^{2}$, we have

$$\begin{aligned} \left| h(\varvec{x};\varvec{\theta })\right|\le & {} \frac{2^{q} \left| r\right| \left| \lambda _{0}\right| ^{i} \left| x_{1}\right| ^{j} \left| y\right| ^{k}}{\sigma _{0}^{q}}W^{l}\left( \left| \frac{2\lambda _{0}y}{\sigma _{0}}\right| \right) \\\le & {} \frac{2^{q} \left| r\right| \left| \lambda _{0}\right| ^{i} \left| x_{1}\right| ^{j}}{\sigma _{0}^{q} } (\left| x_{0}\right| +\left| x_{1}\right| +2\left| \mu _{0}\right| )^{k}(\sqrt{2/ \pi })^{l}. \quad \hbox {(a.e.)} \end{aligned}$$

Finiteness of $\mathbb {E}X_{t}^{k}, k\ge 1$, conclude that

$$\begin{aligned} \mathbb {E}(\left| X_{t}\right| +\left| X_{t-1}\right| +2\left| \mu _{0}\right| )^{k} \left| X_{t-1}\right| ^{i}<\infty , \end{aligned}$$

and then (A3) is proved. The method of checking condition (A4) is similar to Bondon’s approach. $\square$