Asymptotics for the conditional self-weighted M-estimator of GRCA(1) models with possibly heavy-tailed errors

Abstract

Consider a generalized random coefficient AR(1) model, $y_t=\Phi _t y_{t-1}+u_t$, where $\{(\Phi _t, u_t)^\prime , t\ge 1\}$ is a sequences of i.i.d. random vectors, and a conditional self-weighted M-estimator of $\textsf {E}\Phi _t$ is proposed. The asymptotically normality of this new estimator is established with $\textsf {E}u_t^2$ being possibly infinite. Simulation experiments are carried out to assess the performance of the theory and method in finite samples and a real data example is given.

Appendix

In this Appendix, the proof of Theorem 2.1 is exhibited. We first give two lemmas, which will be used frequently in the proof. The first lemma is directly taken from Davis et al. (1992).

Lemma A.1

Let $\{V_n(\cdot )\}$ and $V(\cdot )$ be stochastic process on$R^p$ and suppose that $V_n(\cdot ){\mathop {\rightarrow }\limits ^{d}}V(\cdot )$ on $C(R^p)$. Let $\xi _n$ minimize $V_n(\cdot )$ and $\xi$ minimize $V(\cdot )$. If $V_n(\cdot )$ is convex for each n and $\xi$ is unique with probability one, then $\xi _n{\mathop {\longrightarrow }\limits ^{d}}\xi$ on $R^p$.

Lemma A.2

Under the conditions of Theorem 2.1, we have, as $n\rightarrow \infty ,$

• (i) $\frac{1}{n}\sum _{t=1}^n(w_t y_{t-1}^2){\mathop {\rightarrow }\limits ^{p}} \kappa ,\frac{1}{n}\sum _{t=1}^n(w_t^2 y_{t-1}^2){\mathop {\rightarrow }\limits ^{p}} \upsilon$;

• (ii) $\max _{1\le t\le n}\frac{|w_t y_{t-1}|}{\sqrt{n}}{\mathop {\rightarrow }\limits ^{p}} 0$;

• (iii) $\frac{1}{\sqrt{n}}\sum _{t=1}^n(w_t y_{t-1} \psi (u_t)){\mathop {\rightarrow }\limits ^{d}} N(0,\tau \upsilon )$.

Proof

By applying Assumption 2.4 with $y_t$ being stationary and ergodic, one can get (i) and (ii) hold true. Hence, we omit the proofs of (i) and (ii), and only give the proof of (iii).

Set $\zeta _{nt}=\frac{1}{\sqrt{n}}w_ty_{t-1}\psi (u_t)$. Note that $\{\zeta _{nt},1\le t\le n\}$ is a martingale difference sequence with respect to ${\mathcal {F}}_{t-1}$, and thus from (i), it follows

$$\begin{aligned} \sum _{t=1}^{n}\textsf {E}(\zeta ^2_{nt}|{\mathcal {F}}_{t-1})&=\frac{1}{n} \sum _{t=1}^n(w_t^2 y_{t-1}^2)\textsf {E}\psi ^2(u_t)\nonumber \\&=\tau \frac{1}{n} \sum _{t=1}^n(w_t^2 y_{t-1}^2){\mathop {\rightarrow }\limits ^{p}} \tau \upsilon ~.~ \end{aligned}$$

(A.1)

Letting $\xi _t=w_ty_{t-1}$, then for any $\eta >0$, we have

$$\begin{aligned}&\sum _{t=1}^n\textsf {E}(\zeta _{nt}^2 I(|\zeta _{nt}|>\eta )|{\mathcal {F}}_{t-1})\nonumber \\&\quad =\frac{1}{n} \sum _{t=1}^n\xi _t^2\textsf {E}(\psi ^2(u_t)I(|\xi _t\psi (u_t)|>\eta \sqrt{n}|{\mathcal {F}}_{t-1}))\nonumber \\&\quad \le \max _{1\le t \le n}\textsf {E}(\psi ^2(u_t)I(|\xi _t\psi (u_t)|>\eta \sqrt{n})|{\mathcal {F}}_{t-1})\frac{1}{n} \sum _{t=1}^n\xi _t^2. \end{aligned}$$

(A.2)

Notice that

$$\begin{aligned} I(|\xi _t\psi (u_t)|>\eta \sqrt{n})\le I(|\psi (u_t)|>\eta M)+I\left( \frac{|\xi _t|}{\sqrt{n}}>\frac{1}{M}\right) , \end{aligned}$$

and it implies that, for any $M>0$ and $1\le t \le n$,

$$\begin{aligned}&\textsf {E}\left( \psi ^2(u_t)I\left( \left| \frac{\xi _t\psi (u_t)}{\sqrt{n}}\right|>\eta \right) \bigg |{\mathcal {F}}_{t-1}\right) \nonumber \\&\quad \le \textsf {E}(\psi ^2(u_t)I(|\psi (u_t)|>\eta M))+\textsf {E}\left. \left( \psi ^2(u_t)I\left( \frac{|\xi _t|}{\sqrt{n}}>\frac{1}{M}\right) \right| {\mathcal {F}}_{t-1}\right) \nonumber \\&\quad \le \textsf {E}(\psi ^2(u_t)I(|\psi (u_t)|>\eta M))+\tau \max _{1\le t \le n}I\left( \frac{|\xi _t|}{\sqrt{n}}>\frac{1}{M}\right) \nonumber \\&\quad =\textsf {E}\left( \psi ^2(u_t)I\left( |\psi (u_t)|>\eta M\right) \right) +\tau I\left( \max _{1\le t \le n}\frac{|\xi _t|}{\sqrt{n}}>\frac{1}{M}\right) , \end{aligned}$$

(A.3)

which leads to

$$\begin{aligned}&\max _{1\le t \le n}\textsf {E}\left. \left( \psi ^2(u_t)I\left( \left| \frac{\xi _t\psi (u_t)}{\sqrt{n}}\right|>\eta \right) \right| {\mathcal {F}}_{t-1}\right) \\&\quad \le \textsf {E}(\psi ^2(u_t)I(|\psi (u_t)|>\eta M))+\tau I\left( \max _{1\le t \le n}\frac{|\xi _t|}{\sqrt{n}}>\frac{1}{M}\right) . \end{aligned}$$

Hence, we have

$$\begin{aligned}&\textsf {E}\left\{ \max _{1\le t \le n}\textsf {E}(\psi ^2(u_t)I\left. \left( \left| \frac{\xi _t\psi (u_t)}{\sqrt{n}}\right|>\eta \right) \right| {\mathcal {F}}_{t-1})\right\} \\&\quad \le \textsf {E}(\psi ^2(u_t)I(|\psi (u_t)|>\eta M))+\tau \textsf {P}\left( \max _{1\le t \le n}\frac{|\xi _t|}{\sqrt{n}}>\frac{1}{M}\right) .~ \end{aligned}$$

By virtue of $\textsf {E}\psi ^2(u_1)<\infty$, for any $\varepsilon >0$, there exists some $M=M(\varepsilon )>0$ large enough such that

$$\begin{aligned} \textsf {E}(\psi ^2(u_t)I(|\psi (u_t)|>\eta M))<\varepsilon , \end{aligned}$$

which implies

$$\begin{aligned}&\textsf {E}\left\{ \max _{1\le t \le n}\textsf {E}\left. \left( \psi ^2(u_t)I\left( \left| \frac{\xi _t\psi (u_t)}{\sqrt{n}}\right|>\eta \right) \right| {\mathcal {F}}_{t-1}\right) \right\} \nonumber \\&\quad \le \tau \textsf {P}\left( \max _{1\le t \le n}\frac{|\xi _t|}{\sqrt{n}}>\frac{1}{M(\varepsilon )}\right) +\varepsilon .~ \end{aligned}$$

(A.4)

Thus, it follows from (A.4) and (i),

$$\begin{aligned} \limsup _{n\rightarrow \infty }\textsf {E}\left( \max _{1\le t \le n}\textsf {E}\left. \left( \psi ^2(u_t)I\left( \left| \frac{\xi _t\psi (u_t)}{\sqrt{n}}\right| >\eta \right) \right| {\mathcal {F}}_{t-1}\right) \right) \le \varepsilon , \end{aligned}$$

and an application of the arbitrariness of $\varepsilon$ leads to

$$\begin{aligned} \limsup _{n\rightarrow \infty } \textsf {E}\left( \max _{1\le t \le n}\textsf {E}\left. \left( \psi ^2(u_t)I\left( \left| \frac{\xi _t\psi (u_t)}{\sqrt{n}}\right| >\eta \right) \right| {\mathcal {F}}_{t-1}\right) \right) =0. \end{aligned}$$

Hence, we have

$$\begin{aligned} \max _{1\le t \le n}\textsf {E}\left. \left( \psi ^2(u_t)I\left( \left| \frac{\xi _t\psi (u_t)}{\sqrt{n}}\right| >\eta \right) \right| {\mathcal {F}}_{t-1}\right) =o_p(1),~ \end{aligned}$$

(A.5)

which, coupled with (A.2) and (i), yields

$$\begin{aligned} \sum _{t=1}^n\textsf {E}(\zeta _{nt}^2 I(|\zeta _{nt}|>\eta )|{\mathcal {F}}_{t-1})=o_p(1).~ \end{aligned}$$

(A.6)

Then by applying the martingale central limit theorem, together with (A.1) and (A.6), we have

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^n(w_t y_{t-1} \psi (u_t)){\mathop {\rightarrow }\limits ^{d}} N(0,\tau \upsilon ), \end{aligned}$$

and thus the proof of Lemma A.2 is finished.   $\square$

Now we begin to prove Theorem 2.1.

Proof of Theorem 2.1

Denote ${{\hat{\beta }}}_n=\sqrt{n}({{\hat{\phi }}}_{SM}-\phi _0)$ and

$$\begin{aligned} L_n(\mu )=\sum _{t=1}^n w_t\left( \rho \left( u_t-\frac{\mu y_{t-1}}{\sqrt{n}}\right) -\rho (u_t)\right) , \end{aligned}$$

(A.7)

where $\mu \in {\mathbb {R}}$. It is clear that ${{\hat{\beta }}}_n$ is the minimizer of $L_n(\mu )$.

Setting $A_n=\frac{1}{\sqrt{n}}\sum _{t=1}^n w_t y_{t-1} \psi (u_t)$ and $B_t(\mu )=w_t \int _0^{-\frac{\mu y_{t-1}}{\sqrt{n}}}(\psi (u_t+s)-\psi (u_t))\mathrm{d}s,$ then (A.7) can be rewritten to be

$$\begin{aligned} L_n(\mu )=&-\mu A_n+\sum _{t=1}^nB_t(\mu )\nonumber \\ =&-\mu A_n+\sum _{t=1}^n\textsf {E}(B_t(\mu )|{\mathcal {F}}_{t-1})+\sum _{t=1}^n(B_t(\mu )-\textsf {E}(B_t(\mu )|{\mathcal {F}}_{t-1})). \end{aligned}$$

(A.8)

From Assumption 2.2, we have

$$\begin{aligned} \sum _{t=1}^n\textsf {E}(B_t(\mu )|{\mathcal {F}}_{t-1})&=\sum _{t=1}^n w_t \int _0^{-\frac{\mu y_{t-1}}{\sqrt{n}}}\textsf {E}(\psi (u_t+s))\mathrm{d}s\\&=\sum _{t=1}^n w_t \int _0^{-\frac{\mu y_{t-1}}{\sqrt{n}}}\lambda s(1+o(1))\mathrm{d}s\\&=\frac{\lambda \mu ^2}{2}\Big (\frac{1}{n} \sum _{t=1}^{n}w_ty_{t-1}^2\Big )(1+o(1)). \end{aligned}$$

Noting that $\{B_t(\mu )-\textsf {E}(B_t(\mu )|{\mathcal {F}}_{t-1}),1\le t \le n\}$ is a martingale difference sequence, then we have

$$\begin{aligned} \sum _{t=1}^n\textsf {E}(B_t^2(\mu )|{\mathcal {F}}_{t-1})&=\sum _{t=1}^n\textsf {E}\left( w_t^2\left( \int _0^{-\frac{\mu y_{t-1}}{\sqrt{n}}}\psi (u_t+s)-\psi (u_t)\mathrm{d}s\right) ^2\bigg |{\mathcal {F}}_{t-1}\right) \nonumber \\&\le \sum _{t=1}^n\textsf {E}\left( w_t^2 \int _0^{-\frac{\mu y_{t-1}}{\sqrt{n}}}\mathrm{d}s \int _0^{-\frac{\mu y_{t-1}}{\sqrt{n}}}(\psi (u_t+s)-\psi (u_t))^2\mathrm{d}s\bigg |{\mathcal {F}}_{t-1}\right) \nonumber \\&\le \sum _{t=1}^n w_t^2 \frac{|\mu y_{t-1}|}{\sqrt{n}}\left| \int _0^{-\frac{\mu y_{t-1}}{\sqrt{n}}}\textsf {E}(\psi (u_t+s)-\psi (u_t))^2\mathrm{d}s\right| \nonumber \\&=\frac{\mu ^2}{n} \sum _{t=1}^n w_t^2 y_{t-1}^2\cdot o(1). \end{aligned}$$

(A.9)

Combined with (A.9) and Lemma A.2, it leads to $\sum _{t=1}^n\textsf {E}B_t^2(\mu )\rightarrow 0,$ which implies

$$\begin{aligned}&\textsf {E}\Big (\sum _{t=1}^n(B_t(\mu )-\textsf {E}(B_t(\mu )|{\mathcal {F}}_{t-1}))\Big )^2\\&\quad =\sum _{t=1}^n\textsf {E}(B_t(\mu )-\textsf {E}(B_t(\mu )|{\mathcal {F}}_{t-1}))^2\\&\quad \le 2\sum _{t=1}^n\textsf {E}B^2_t(\mu )\rightarrow ~0.~ \end{aligned}$$

Hence, we have

$$\begin{aligned} L_n(\mu )=\frac{\lambda \mu ^2}{2}\left( \frac{1}{n} \sum _{t=1}^{n}w_ty_{t-1}^2\right) -\mu A_n+o_p(1), \end{aligned}$$

(A.10)

and this, together with Lemma A.2, gives

$$\begin{aligned} L_n(\mu ){\mathop {\rightarrow }\limits ^{d}} \frac{\lambda \mu ^2}{2}\kappa -\mu A, \end{aligned}$$

where $A\sim N(0,\tau \upsilon )$. Then, from Assumption 2.1 and Lemma A.1, we have

$$\begin{aligned} {{\hat{\beta }}}_n=\sqrt{n}({{\hat{\phi }}}_{SM}-\phi _0){\mathop {\rightarrow }\limits ^{d}} \frac{A}{\lambda \kappa }\sim N\left( 0,\frac{\tau \upsilon }{\lambda ^2\kappa ^2}\right) , \end{aligned}$$

which completes the proof.