Frequentist model averaging for threshold models

Abstract

This paper develops a frequentist model averaging approach for threshold model specifications. The resulting estimator is proved to be asymptotically optimal in the sense of achieving the lowest possible squared errors. In particular, when combining estimators from threshold autoregressive models, this approach is also proved to be asymptotically optimal. Simulation results show that for the situation where the existing model averaging approach is not applicable, our proposed model averaging approach has a good performance; for the other situations, our proposed model averaging approach performs marginally better than other commonly used model selection and model averaging methods. An empirical application of our approach on the US unemployment data is given.

Appendix

Lemma 1

Let ${\mathcal {W}}$ be a weight vector set which can be related to the sample size n. Define

$$\begin{aligned} w^{*}=\underset{w\in {{\mathcal {W}}}}{\text {argmin}}\left( L_n(w)+a_n(w) \right) . \end{aligned}$$

(17)

If

$$\begin{aligned}&\underset{w\in {{\mathcal {W}}}}{\sup }\frac{|a_n(w)|}{R_n(w)}\overset{p}{ \longrightarrow }0, \end{aligned}$$

(18)

$$\begin{aligned}&\underset{w\in {{\mathcal {W}}}}{\sup }\left| \frac{L_n(w)}{R_n(w)}-1\right| \overset{p}{\longrightarrow }0, \end{aligned}$$

(19)

and there exists a constant $\kappa _3$ such that

$$\begin{aligned} \inf \limits _{w\in {{\mathcal {W}}}}R_n(w) \ge \kappa _3 >0, \end{aligned}$$

(20)

then

$$\begin{aligned} \frac{L_n(w^*)}{\mathrm {inf}_{w\in {{\mathcal {W}}}}L_n(w)}\overset{p}{ \longrightarrow }1. \end{aligned}$$

(21)

Proof

From the definition of the infimum, there exist a non-negative series $\vartheta _{n}$ and a vector $w(n)\in {{\mathcal {W}}}$ such that $\vartheta _{n}\rightarrow 0$ and

$$\begin{aligned} \inf \limits _{w\in {{\mathcal {W}}}}L_{n}(w)=L_{n}(w(n))-\vartheta _{n}. \end{aligned}$$

(22)

In addition, it follows from (19) that

$$\begin{aligned} \inf \limits _{w\in {{\mathcal {W}}}}\frac{L_{n}(w)}{R_{n}(w)}= & {} \inf \limits _{w \in {{\mathcal {W}}}}\left( \frac{L_{n}(w)}{R_{n}(w)}-1\right) +1 \nonumber \\\ge & {} -\sup \limits _{w\in {{\mathcal {W}}}}\left| \frac{L_{n}(w)}{R_{n}(w)} -1\right| +1\overset{p}{\longrightarrow }1. \end{aligned}$$

(23)

From (20), (23) and $\vartheta _{n}\rightarrow 0$, we have

$$\begin{aligned} \inf \limits _{w\in {{\mathcal {W}}}}\frac{\left| L_{n}(w)-\vartheta _{n}\right| }{R_{n}(w)}\ge & {} \inf \limits _{w\in {{\mathcal {W}}}}\frac{ L_{n}(w)-\vartheta _{n}}{R_{n}(w)}\ge \inf \limits _{w\in {{\mathcal {W}}}}\frac{ L_{n}(w)}{R_{n}(w)}-\frac{\vartheta _{n}}{\mathrm {inf}_{w\in {{\mathcal {W}}} }R_{n}(w)} \nonumber \\\ge & {} -\sup \limits _{w\in {{\mathcal {W}}}}\left| \frac{L_{n}(w)}{R_{n}(w)} -1\right| +1-\frac{\vartheta _{n}}{\mathrm {inf}_{w\in {{\mathcal {W}}} }R_{n}(w)} \nonumber \\&\overset{p}{\longrightarrow }1. \end{aligned}$$

(24)

Now, by the definition of $w^{*}$, (18), (20), (22)–(24), and $\vartheta _{n}\rightarrow 0$, we have, for any $\delta >0$,

$$\begin{aligned}&\Pr \left\{ \left| \frac{\inf _{w\in {{{\mathcal {W}}}}}L_{n}(w)}{ L_{n}(w^{*})}-1\right|>\delta \right\} =\Pr \left\{ \frac{ L_{n}(w^{*})-\inf _{w\in {{{\mathcal {W}}}}}L_{n}(w)}{L_{n}(w^{*})}>\delta \right\} \nonumber \\&\quad =\Pr \left\{ \frac{\inf _{w\in {{\mathcal {W}}}}\left( L_{n}(w)+a_{n}(w)\right) -a_{n}(w^{*})-\inf _{w\in {{{\mathcal {W}}}} }L_{n}(w)}{L_{n}(w^{*})}>\delta \right\} \nonumber \\&\quad \le \Pr \left\{ \frac{L_{n}(w(n))+a_{n}(w(n))-a_{n}(w^{*})-L_{n}(w(n))+\vartheta _{n}}{L_{n}(w^{*})}>\delta \right\} \nonumber \\&\quad \le \Pr \left\{ \frac{|a_{n}(w(n))|}{L_{n}(w^{*})}+\frac{ |a_{n}(w^{*})|}{L_{n}(w^{*})}+\frac{\vartheta _{n}}{L_{n}(w^{*})}>\delta \right\} \nonumber \\&\quad \le \Pr \left\{ \frac{|a_{n}(w(n))|}{\inf _{w\in {{\mathcal {W}}}}L_{n}(w)}+ \frac{|a_{n}(w^{*})|}{L_{n}(w^{*})}+\frac{\vartheta _{n}}{ L_{n}(w^{*})}>\delta \right\} \nonumber \\&\quad =\Pr \left\{ \frac{|a_{n}(w(n))|}{L_{n}(w(n))-\vartheta _{n}}+\frac{ |a_{n}(w^{*})|}{L_{n}(w^{*})}+\frac{\vartheta _{n}}{L_{n}(w^{*})}>\delta \right\} \nonumber \\&\quad \le \Pr \left\{ \sup _{w\in {{\mathcal {W}}}}\frac{|a_{n}(w)|}{ L_{n}(w)-\vartheta _{n}}+\sup _{w\in {{\mathcal {W}}}}\frac{|a_{n}(w)|}{L_{n}(w)} +\sup _{w\in {{\mathcal {W}}}}\frac{\vartheta _{n}}{L_{n}(w)}>\delta \right\} \nonumber \\&\quad \le \Pr \left\{ \sup _{w\in {{\mathcal {W}}}}\frac{|a_{n}(w)|}{R_{n}(w)} \sup _{w\in {{\mathcal {W}}}}\frac{R_{n}(w)}{\left| L_{n}(w)-\vartheta _{n}\right| }+\sup _{w\in {{\mathcal {W}}}}\frac{|a_{n}(w)|}{R_{n}(w)} \sup _{w\in {{\mathcal {W}}}}\frac{R_{n}(w)}{L_{n}(w)}\right. \nonumber \\&\qquad \left. +\sup _{w\in {{\mathcal {W}}}}\frac{\vartheta _{n}}{R_{n}(w)} \sup _{w\in {{\mathcal {W}}}}\frac{R_{n}(w)}{L_{n}(w)}>\delta \right\} \nonumber \\&\quad =\Pr \left\{ \sup _{w\in {{\mathcal {W}}}}\frac{|a_{n}(w)|}{R_{n}(w)}\left[ \inf _{w\in {{\mathcal {W}}}}\frac{\left| L_{n}(w)-\vartheta _{n}\right| }{R_{n}(w)}\right] ^{-1}+\sup _{w\in {{\mathcal {W}}}}\frac{|a_{n}(w)|}{R_{n}(w)} \left[ \inf _{w\in {{\mathcal {W}}}}\frac{L_{n}(w)}{R_{n}(w)}\right] ^{-1}\right. \nonumber \\&\qquad \left. +\frac{\vartheta _{n}}{\inf _{w\in {{\mathcal {W}}}}R_{n}(w)} \left[ \inf _{w\in {{\mathcal {W}}}}\frac{L_{n}(w)}{R_{n}(w)}\right] ^{-1}>\delta \right\} \nonumber \\&\quad \rightarrow 0. \end{aligned}$$

(25)

Therefore, $\mathrm {inf}_{w\in {{\mathcal {W}}}}L_{n}(w)/L_{n}(w^{*}) \overset{p}{\longrightarrow }1$, which implies (21). $\square$

Proof of Theorem 1

First, from the fact that $X_{(m)}(\gamma )$ is of full column rank, we have $tr\hat{P}(w)=trP^{*}(w)\le 2\sum _{m=1}^{M}w_{m}k_{m}$. Let $\hat{A} (w)=I_{n}-\hat{P}(w)$, so that

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_{n}(w)=&\, \Vert Y-\hat{\mu }(w)\Vert ^{2}\Big (1+2\frac{tr\hat{P}(w) }{n}\Big ) \\ =&\, L_{n}(w)+\Vert e\Vert ^{2}+2\mu ^{\prime }(\hat{A}(w)-A^{*}(w))e+2\mu ^{\prime }A^{*}(w)e \\&+\,2\big (\sigma ^{2}trP^{*}(w)-e^{\prime }P^{*}(w)e\big )+2e^{\prime } \big (P^{*}(w)-\hat{P}(w)\big )e \\&+\,2trP^{*}(w)\big (\Vert A^{*}(w)Y\Vert ^{2}/n-\sigma ^{2}\big ) \\&+\,2trP^{*}(w)\big (\Vert \hat{A}(w)Y\Vert ^{2}-\Vert A^{*}(w)Y\Vert ^{2}\big )/n. \end{aligned} \end{aligned}$$

Since $\Vert e\Vert ^{2}$ is unrelated to w and Condition (20) with ${\mathcal {W}}={\mathcal {H}}_{n}$ is implied by Condition (7), according to Lemma 1, Theorem 1 is valid if

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }|\mu ^{\prime }A^{*}(w)e|/R_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(26)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }|e^{\prime }P^{*}(w)e-\sigma ^{2}trP^{*}(w)|/R_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(27)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }|L_{n}^{*}(w)/R_{n}^{*}(w)-1| \overset{p}{\longrightarrow }0, \end{aligned}$$

(28)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }|trP^{*}(w)(\Vert A^{*}(w)Y\Vert ^{2}/n-\sigma ^{2})|/R_{n}^{*}(w)\overset{p}{\longrightarrow } 0, \end{aligned}$$

(29)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |\mu ^{\prime }\big (P^{*}(w)- \hat{P}(w)\big )e\big |/R_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(30)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |e^{\prime }\big (P^{*}(w)-\hat{ P}(w)\big )e\big |/R_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(31)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }|L_{n}(w)-L_{n}^{*}(w)|/R_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(32)

and

$$\begin{aligned} \underset{w\in {\mathcal {H}}_{n}}{\sup }\big |trP^{*}(w)\big (\Vert A^{*}(w)Y\Vert ^{2}-\Vert \hat{A}(w)Y\Vert ^{2}\big )\big |/nR_{n}^{*}(w) \overset{p}{\longrightarrow }0. \end{aligned}$$

(33)

(26)–(28) can been shown by following the proof of Theorem $\text {1}^{\prime }$ of Wan et al. (2010). Therefore, we only need to verify (29)–(33). First, we prove (29). Note that

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }|trP^{*}(w)\big (\Vert A^{*}(w)Y\Vert ^{2}/n-\sigma ^{2}\big )|/R_{n}^{*}(w) \\&=\underset{w\in {\mathcal {H}}_{n}}{\sup }\left\{ \frac{trP^{*}(w)}{ nR_{n}^{*}(w)}\big |\Vert \mu -P^{*}(w)Y\Vert ^{2}+\Vert e\Vert ^{2}+2\mu ^{\prime }A^{*}(w)e-2e^{\prime }P^{*}(w)e-n\sigma ^{2}\big | \right\} \\&\le \underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{L_{n}^{*}(w)}{ R_{n}^{*}(w)}\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{trP^{*}(w)}{ n}+\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{2|\mu ^{\prime }A^{*}(w)e| }{R_{n}^{*}(w)}\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{trP^{*}(w) }{n} \\&\qquad +\frac{|\Vert e\Vert ^{2}-n\sigma ^{2}|}{\sqrt{n}}\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{1}{R_{n}^{*}(w)}\underset{w\in {\mathcal {H}} _{n}}{\sup }\frac{trP^{*}(w)}{\sqrt{n}} \\&\qquad +\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{2|e^{\prime }P^{*}(w)e-\sigma ^{2}trP^{*}(w)|}{R_{n}^{*}(w)}\underset{w\in {\mathcal {H}} _{n}}{\sup }\frac{trP^{*}(w)}{n} \\&\qquad +2\sigma ^{2}\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{1}{ R_{n}^{*}(w)}\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{tr^{2}P^{*}(w)}{n}. \end{aligned}$$

By the central limit theorem, we have ${|\Vert e\Vert ^{2}-n\sigma ^{2}|}/ \sqrt{n}=O_{p}(1)$. In addition, it follows from (7) and ( 9) that

$$\begin{aligned} \underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{1}{R_{n}^{*}(w)}=o_{p}(1),\ \ \ \underset{w\in {\mathcal {H}}_{n}}{\sup }tr^{2}P^{*}(w)/n=O(1)\ \ \text { and}\ \ \ \underset{w\in {\mathcal {H}}_{n}}{\sup }trP^{*}(w)/n=o(1). \end{aligned}$$

Together with (26)–(28), (29) is obtained.

To prove (30), we observe that

$$\begin{aligned} \begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |\mu ^{\prime }\big (P^{*}(w)- \hat{P}(w)\big )e\big |/R_{n}^{*}(w) \\&\quad \le \frac{1}{\xi _{n}^{*}}\underset{w\in {\mathcal {H}}_{n}}{\sup }\big [ \Vert \mu \Vert ^{2}e^{\prime }\big (P^{*}(w)-\hat{P}(w)\big )^{2}e\big ] ^{1/2} \\&\quad \le \frac{1}{\xi _{n}^{*}}\frac{\Vert \mu \Vert }{\sqrt{n}}\frac{ \Vert e\Vert }{\sqrt{n}}\ n\underset{1\le m\le M}{\max }\lambda _{\max }(P_{(m)}^{*}-\hat{P}_{(m)}). \end{aligned} \end{aligned}$$

By Conditions (8) and (10), (30) is verified.

Note that

$$\begin{aligned} L_{n}(w)=\Vert e\Vert ^{2}+\Vert \hat{A}(w)\mu \Vert ^{2}+\Vert \hat{A} (w)e\Vert ^{2}-2e^{\prime }\hat{A}(w)\mu -2e^{\prime }\hat{A}(w)e+2\mu ^{\prime }\hat{A}^{2}(w)e, \end{aligned}$$

so

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup } |L_{n}(w)-L_{n}^{*}(w)|/R_{n}^{*}(w)\overset{p}{\longrightarrow }0\Leftrightarrow \\&\quad \underset{w\in {\mathcal {H}}_{n}}{\sup } \big |2\mu ^{\prime }\big (P^{*}(w)-\hat{P}(w)\big )\mu +2\mu ^{\prime }\big (P^{*}(w)-\hat{P}(w)\big )e \\&\qquad -\mu ^{\prime }\big (P^{*}(w)+\hat{P}(w)\big )\big (P^{*}(w)-\hat{P} (w)\big )\mu \\&\qquad -e^{\prime }\big (P^{*}(w)+\hat{P}(w)\big )\big (P^{*}(w)-\hat{P}(w)\big )e \\&\qquad -2\mu ^{\prime }P^{*}(w)\big (P^{{*}}(w)-\hat{P}(w)\big )e \\&\qquad -2\mu ^{\prime }\big (P^{{*}}(w)-\hat{P}(w)\big )\hat{P}(w)e\big |/R_{n}^{*}(w)\overset{p}{\longrightarrow }0. \end{aligned}$$

Thus, if

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |\mu ^{\prime }\big (P^{*}(w)+ \hat{P}(w)\big )\big (P^{*}(w)-\hat{P}(w)\big )\mu \big |/R_{n}^{*}(w) \overset{p}{\longrightarrow }0, \end{aligned}$$

(34)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |e^{\prime }\big (P^{*}(w)+\hat{ P}(w)\big )\big (P^{*}(w)-\hat{P}(w)\big )e\big |/R_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(35)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |\mu ^{\prime }P^{*}(w)\big ( P^{*}(w)-\hat{P}(w)\big )e\big |/R_{n}^{*}(w)\overset{p}{ \longrightarrow }0, \end{aligned}$$

(36)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |\mu ^{\prime }\big (P^{*}(w)- \hat{P}(w)\big )\hat{P}(w)e\big |/R_{n}^{*}(w)\overset{p}{\longrightarrow } 0, \end{aligned}$$

(37)

and

$$\begin{aligned} \underset{w\in {\mathcal {H}}_{n}}{\sup }\big |\mu ^{\prime }\big (P^{*}(w)- \hat{P}(w)\big )\mu \big |/R_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(38)

then (32) is valid. From Condition (8) and the following result

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |e^{\prime }\big (P^{*}(w)+ \hat{P}(w)\big )\big (P^{*}(w)-\hat{P}(w)\big )e\big |/R_{n}^{*}(w) \\&\quad \le \frac{1}{2\xi _{n}^{*}}\underset{w\in {\mathcal {H}}_{n}}{\sup }\big | e^{\prime }\big [\big (P^{*}(w)+\hat{P}(w)\big )\big (P^{*}(w)-\hat{P}(w)\big )\\&\qquad +\,\big (P^{*}(w)-\hat{P}(w)\big )\big (P^{*}(w)+\hat{P}(w)\big )\big ]e\big | \\&\quad \le \frac{\Vert e\Vert ^{2}}{2\xi _{n}^{*}}\underset{w\in {\mathcal {H}} _{n}}{\sup }\lambda _{\max }\big [\big (P^{*}(w)+\hat{P}(w)\big )\big (P^{*}(w)-\hat{P}(w)\big )\\&\qquad +\,\big (P^{*}(w)-\hat{P}(w)\big )\big (P^{*}(w)+\hat{P}(w)\big )\big ] \\&\quad \le \frac{\Vert e\Vert ^{2}}{\xi _{n}^{*}}\underset{w\in {\mathcal {H}} _{n}}{\sup }\big [\lambda _{\max }\big (P^{*}(w)+\hat{P}(w)\big )\lambda _{\max }\big (P^{*}(w)-\hat{P}(w)\big )\big ] \\&\quad \le \frac{\Vert e\Vert ^{2}}{\xi _{n}^{*}}\underset{w\in {\mathcal {H}} _{n}}{\sup }\big [\lambda _{\max }\big (P^{*}(w)\big )+\lambda _{\max }\big ( \hat{P}(w)\big )\big ]\sum _{m=1}^{M}w_{m}\lambda _{\max }(P_{(m)}^{*}-\hat{ P}_{(m)}) \\&\quad \le \frac{2}{\xi _{n}^{*}}\frac{\Vert e\Vert ^{2}}{n}n\underset{1\le m\le M}{\max }\lambda _{\max }(P_{(m)}^{*}-\hat{P}_{(m)}), \end{aligned}$$

we obtain (35). Similarly, (31), (34) and (38) can be verified. On the other hand, analogous to the proof of (30), one can obtain (36) and (37).

Further, it can be shown that

$$\begin{aligned} \begin{aligned}&\underset{w\in {\mathcal {H}}_{n}}{\sup }\big |trP^{*}(w)\big (\Vert A^{*}(w)Y\Vert ^{2}-\Vert \hat{A}(w)Y\Vert ^{2}\big )\big |/nR_{n}^{*}(w) \\&\quad \le \underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{trP^{*}(w)}{n} \underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{|\Vert A^{*}(w)Y\Vert ^{2}-\Vert \hat{A}(w)Y\Vert ^{2}|}{R_{n}^{*}(w)} \\&\quad \le a_{1}\underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{|\Vert A^{*}(w)Y\Vert ^{2}-\Vert \hat{A}(w)Y\Vert ^{2}|}{R_{n}^{*}(w)}, \end{aligned} \end{aligned}$$

where the last step is from Condition (9). Observe that

$$\begin{aligned}&|\Vert A^{*}(w)Y\Vert ^{2}-\Vert \hat{A}(w)Y\Vert ^{2}| \\&\quad =\, |2\mu ^{\prime }(\hat{P}(w)-P^{*}(w))\mu +\mu ^{\prime }(P^{*}(w)+ \hat{P}(w))(P^{*}(w)-\hat{P}(w))\mu \\&\qquad +2e^{\prime }(\hat{P}(w)-P^{*}(w))e+e^{\prime }(P^{*}(w)+\hat{P} (w))(P^{*}(w)-\hat{P}(w))e \\&\qquad +4\mu ^{\prime }(\hat{P}(w)-P^{*}(w))e+2\mu ^{\prime }P^{*}(w)(P^{*}(w)-\hat{P}(w))e \\&\qquad +2\mu ^{\prime }(P^{*}(w)-\hat{P}(w))\hat{P }(w)e|, \end{aligned}$$

so from (30), (31) and (34)–(38 ), we have

$$\begin{aligned} \underset{w\in {\mathcal {H}}_{n}}{\sup }\frac{|\Vert A^{*}(w)Y\Vert ^{2}-\Vert \hat{A}(w)Y\Vert ^{2}|}{R_{n}^{*}(w)}\overset{p}{ \longrightarrow }0. \end{aligned}$$

Thus, we obtain (33). This completes the proof of Theorem . $\square$

The following lemma is used in the proof of Theorem 2.

Lemma 2

For any $\hat{\gamma }_{(m)}$ and $\gamma _{(m)}^{*} \in \Gamma$ and any random variable Y, if Assumptions (a.3) and (a.4) are satisfied, and

$$\begin{aligned} |E(Y|z_i=\gamma ,\hat{\gamma }_{(m)})|\le \bar{E}, \end{aligned}$$

(39)

where $\bar{E}$ is a finite constant, then

$$\begin{aligned} E\big (Y|\text {I}(z_{i}\le \gamma _{(m)}^{*} )-\text {I}(z_{i}\le \hat{\gamma }_{(m)})|\big )=O(n^{-\rho }). \end{aligned}$$

(40)

Proof

The proof is similar to that of Lemma A.1 in Hansen (2000).

$$\begin{aligned} \begin{aligned} \frac{\partial E(Y\text {I}(z_{i}\le \gamma )|\hat{\gamma }_{(m)})}{\partial \gamma }&=\int _{-\infty }^{+\infty }y\frac{\partial \int _{-\infty }^{\gamma }f(y,z|\hat{\gamma }_{(m)})\,dz}{\partial \gamma }dy \\&=\int _{-\infty }^{+\infty }yf(y,\gamma |\hat{\gamma }_{(m)})dy \\&=\int _{-\infty }^{+\infty }yf_{1}(y|\gamma ,\hat{\gamma } _{(m)})f_{2}(\gamma |\hat{\gamma }_{(m)})dy \\&=f_{2}(\gamma |\hat{\gamma }_{(m)})E(Y\big |z_{i}=\gamma ,\hat{\gamma } _{(m)}), \end{aligned} \end{aligned}$$

where f, $f_{1}$ and $f_{2}$ are density functions. Let $C=\bar{f}_{2}\bar{ E}$. By Lagrange’s mean value theorem, there exists a $\tilde{\gamma }_{(m)}$ between $\gamma _{(m)}^{*}$ and $\hat{\gamma }_{(m)}$ such that

$$\begin{aligned}&E(Y\text {I}(z_{i}\le \hat{\gamma }_{(m)})|\hat{\gamma }_{(m)})-E(Y\text {I} (z_{i}\le \gamma _{(m)}^{*})|\hat{\gamma }_{(m)}) \nonumber \\&\quad =f_{2}(\tilde{\gamma }_{(m)}|\hat{\gamma }_{(m)})E(Y\big |z_{i}=\tilde{\gamma }_{(m)},\hat{\gamma }_{(m)})(\hat{\gamma }_{(m)}-\gamma _{(m)}^{*}) \nonumber \\&\quad \le C|\hat{\gamma }_{(m)}-\gamma _{(m)}^{*}|. \end{aligned}$$

(41)

Define $f_3(\gamma )$ as the density of $\hat{\gamma }_{(m)}$. By (41) and Assumptions (a.3) and (a.4), we have

$$\begin{aligned}&E\big (Y|\text {I}(z_{i}\le \gamma _{(m)}^{*})-\text {I}(z_{i}\le \hat{ \gamma }_{(m)})|) \\&\quad = \int ^{\bar{\gamma }}_{\underline{\gamma }}E\big (Y|\text {I}(z_{i}\le \gamma _{(m)}^{*})-\text {I}(z_{i}\le \hat{\gamma }_{(m)})|\big |\hat{\gamma } _{(m)}\big ) f_3(\hat{\gamma }_{(m)}) d\hat{\gamma }_{(m)} \\&\quad = \int _{\underline{\gamma }}^{\gamma _{(m)}^{*}}E\big (Y\big (\text {I} (z_{i}\le \gamma _{(m)}^{*})-\text {I}(z_{i}\le \hat{\gamma }_{(m)})\big ) \big |\hat{\gamma }_{(m)}\big ) f_3(\hat{\gamma }_{(m)}) d\hat{\gamma }_{(m)} \\&\qquad +\int ^{\bar{\gamma }}_{\gamma _{(m)}^{*}}E\big (Y\big (\text {I}(z_{i}\le \hat{\gamma }_{(m)})- \text {I}(z_{i}\le \gamma _{(m)}^{*})\big )\big |\hat{ \gamma }_{(m)}\big ) f_3(\hat{\gamma }_{(m)}) d\hat{\gamma }_{(m)} \\&\quad \le \int ^{\bar{\gamma }}_{\underline{\gamma }}C|\hat{\gamma }_{(m)}-\gamma _{(m)}^{*}| f_3(\hat{\gamma }_{(m)}) d\hat{\gamma }_{(m)}=O(n^{-\rho }). \end{aligned}$$

The proof of Lemma 2 is completed. $\square$

Proof of Theorem 2

Note that $\mu ^{\prime } A^{*}(w)e=\mu ^{\prime }e-\mu ^{\prime } P^{*}(w)e$. From the proof of Theorem 1 and the fact that $\mu ^{\prime }e$ is unrelated to w, Theorem 2 is valid if

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_n}{\sup } |e^{\prime }P^{*}(w)e-\sigma ^2trP^{*}(w)|/Q_n^{*}(w)\overset{p}{ \longrightarrow }0, \end{aligned}$$

(42)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_n}{\sup }|\mu ^{\prime }P^{*}(w)e|/Q_{n}^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(43)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_n}{\sup } |L_n^{*}(w)/Q_n^{*}(w)-1|\overset{p }{\longrightarrow }0, \end{aligned}$$

(44)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_n}{\sup } |trP^{*}(w)(\Vert A^{*}(w)Y\Vert ^2/n-\sigma ^2)|/Q^{*}_n(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(45)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_n}{\sup } \big |\mu ^{\prime }\big (P^{*}(w)-\hat{P}(w)\big )e\big |/Q_n^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(46)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_n}{\sup } \big |e^{\prime }\big (P^{*}(w)-\hat{P}(w)\big )e\big |/Q_n^{*}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(47)

$$\begin{aligned}&\underset{w\in {\mathcal {H}}_n}{\sup } |L_n(w)-L_n^{*}(w)|/Q_n^{*}(w) \overset{p}{\longrightarrow }0, \end{aligned}$$

(48)

and

$$\begin{aligned} \underset{w\in {\mathcal {H}}_n}{\sup } \big |trP^{*}(w)\big (\Vert A^{*}(w)Y\Vert ^2 -\Vert \hat{A}(w)Y\Vert ^2\big )\big | /nQ_n^{*}(w)\overset{p}{\longrightarrow }0. \end{aligned}$$

(49)

Because $x_i$ contains the lag values of $y_i$, the proofs of (42)–(44) are different from those of (26)–(28).

According to Theorem 3.35 of White (1984), Assumption (a.1) implies that $x_{(m)i}x_{(m)i}^{\prime }\text {I}(z_{i}\le \gamma _{(m)}^{*})$ is stationary and ergodic. Further, Assumption (a.2) ensures $E|x_{(m)ij}x_{(m)ik}\text {I}(z_{i}\le \gamma _{(m)}^{*})|<\infty$. By Theorem 3.34 of White (1984), we have

$$\begin{aligned} \frac{X_{(m)}^{*\prime }X_{(m)}^{*}}{n}\overset{p}{\longrightarrow } \left( \begin{array}{cc} E(x_{(m)i}x_{(m)i}^{\prime }\text {I}(z_{i}\le \gamma _{(m)}^{*})) &{} 0 \\ 0 &{} E(x_{(m)i}x_{(m)i}^{\prime }\text {I}(z_{i}>\gamma _{(m)}^{*})) \end{array} \right) \equiv V_{(m)}, \end{aligned}$$

(50)

where $V_{(m)}$ is an invertible matrix. From Assumptions (a.1) and (a.2), $x_{i}\text {I}(z_{i}\le \gamma )e_{i}$ is a square integrable stationary martingale difference sequence. Therefore, by the central limit theorem for martingale difference sequence, we obtain $\frac{1}{\sqrt{n}}X_{(m)}^{*\prime }e\overset{d}{\longrightarrow }N(0,\sigma ^{2}V_{(m)})$. Thus, $\frac{ 1}{\sqrt{n}}X_{(m)}^{*\prime }e=O_{p}(1)$. Together with the fact that $k_{M^{*}}$ and M are bounded, it can be shown that

$$\begin{aligned} e^{\prime }P_{(m)}^{*}e=\frac{1}{\sqrt{n}}e^{\prime }X_{(m)}^{*}\left( \frac{X_{(m)}^{*\prime }X_{(m)}^{*}}{n}\right) ^{-1}\frac{1}{\sqrt{n}} X_{(m)}^{*\prime }e=O_{p}(1) \end{aligned}$$

(51)

and

$$\begin{aligned} trP^{*}(w)=\sum _{m=1}^{M}w_{m}trP_{(m)}^{*}\le 2\sum _{m=1}^{M}w_{m}k_{m}\le 2 k_{M^{*}}<\infty . \end{aligned}$$

(52)

From Condition (12), we have

$$\begin{aligned} \underset{w\in {\mathcal {H}}_{n}}{\sup }|e^{\prime }P^{*}(w)e-\sigma ^{2}trP^{*}(w)|/Q_{n}^{*}(w)\le \zeta _{n}^{*-1}\max _{1\le m\le M}|e^{\prime }P_{(m)}^{*}e|+2\zeta _{n}^{*-1}\sigma ^{2}k_{M^{*}}\overset{p}{\longrightarrow }0. \end{aligned}$$

(53)

Consequently, (42) is verified.

Under (51) and Condition (10), it can be shown that

$$\begin{aligned} |\mu ^{\prime }P^{*}(w)e|= & {} |e^{\prime }P^{*}(w)\mu \mu ^{\prime }P^{ *}(w)e|^{\frac{1}{2}} \le \Vert \mu \Vert |e^{\prime }P^{*2}(w)e|^{\frac{1}{2}} \nonumber \\\le & {} \Vert \mu \Vert \lambda _{\text {max}}^{1/2}\big (P^{*}(w)\big )|e^{\prime }P^{*}(w)e|^{1/2} =O_{p}(\sqrt{n}). \end{aligned}$$

(54)

Hence, (43) is valid by Condition (12).

For (44), similar to (54), it can be shown that

$$\begin{aligned} e^{\prime }P^{*2}(w)e=O_{p}(1) \end{aligned}$$

(55)

and

$$\begin{aligned} |\mu ^{\prime }P^{*2}(w)e|=O_{p}(\sqrt{n}). \end{aligned}$$

(56)

In addition,

$$\begin{aligned} trP^{*2}(w)\le \lambda _{\max }\big (P^{*}(w)\big )trP^{*}(w)\le 2k_{M^{*}}. \end{aligned}$$

(57)

Thus,

$$\begin{aligned} |L_{n}^{*}(w)-Q_{n}^{*}(w)|= & {} \big |\Vert P^{*}(w)e\Vert ^{2}-2\mu ^{\prime }A^{*}(w)P^{*}(w)e-\sigma ^{2}trP^{*2}(w)\big | \nonumber \\\le & {} \Vert P^{*}(w)e\Vert ^{2} + 2|\mu ^{\prime }P^{*}(w)e| + 2|\mu ^{\prime }P^{*2}(w)e|+ 2\sigma ^{2}k_{M^{*}} \\= & {} O_{p}(\sqrt{n}). \end{aligned}$$

Hence, (44) holds by Condition (12).

The proof of (45) is similar to that of (29). From the proofs of (30)–(33), if

$$\begin{aligned} n\zeta _{n}^{*-1}\underset{1\le m\le M}{\max }\lambda _{\max }(P_{(m)}^{*}-\hat{P}_{(m)})\overset{p}{\longrightarrow }0, \end{aligned}$$

(58)

then (46)–(49) will hold. In the following, we will verify (58).

By Lemma 2, for the mth candidate model,

$$\begin{aligned} E|x_{(m)ij}x_{(m)ik}\big (\text {I}(z_{i}\le \gamma _{(m)}^{*})-\text {I} (z_{i}\le \hat{\gamma }_{(m)})\big )|=O(n^{-\rho }) \end{aligned}$$

uniformly in i. Hence,

$$\begin{aligned} \frac{X_{(m)}^{*\prime }X_{(m)}^{*}}{n}-\frac{\hat{X}_{(m)}^{\prime } \hat{X}_{(m)}}{n}=O_{p}(n^{-\rho }), \end{aligned}$$

(59)

and

$$\begin{aligned} \frac{(X_{(m)}^{*}-\hat{X}_{(m)})^{\prime }(X_{(m)}^{*}-\hat{X} _{(m)})}{n}=O_{p}(n^{-\rho }). \end{aligned}$$

(60)

From (50) and (59), it follows that

$$\begin{aligned} \frac{\hat{X}_{(m)}^{\prime }\hat{X}_{(m)}}{n}\overset{p}{\longrightarrow } V_{(m)}. \end{aligned}$$

(61)

Thus, by (50), (59) and (61), we obtain

$$\begin{aligned} \left( \frac{X_{(m)}^{*\prime }X_{(m)}^{*}}{n}\right) ^{-1}-\left( \frac{ \hat{X}_{(m)}^{\prime }\hat{X}_{(m)}}{n}\right) ^{-1}=O_{p}(n^{-\rho }). \end{aligned}$$

(62)

Note that

$$\begin{aligned} P_{(m)}^{*}-\hat{P}_{(m)}= & {} X_{(m)}^{*}[(X_{(m)}^{*\prime }X_{(m)}^{*})^{-1}-(\hat{X}_{(m)}^{\prime }\hat{X}_{(m)})^{-1}]X_{(m)}^{ *\prime } \nonumber \\&-(\hat{X}_{(m)}-X_{(m)}^{*})(\hat{X}_{(m)}^{\prime }\hat{X} _{(m)})^{-1}(\hat{X}_{(m)}-X_{(m)}^{*})^{\prime } \nonumber \\&-(\hat{X}_{(m)}-X_{(m)}^{*})(\hat{X}_{(m)}^{\prime }\hat{X} _{(m)})^{-1}X_{(m)}^{*\prime } \nonumber \\&-X_{(m)}^{*}(\hat{X}_{(m)}^{\prime } \hat{X}_{(m)})^{-1}(\hat{X}_{(m)}-X_{(m)}^{*})^{\prime } \nonumber \\\equiv & {} \Delta P_{(m)1}+\Delta P_{(m)2}+\Delta P_{(m)3}+\Delta P_{(m)4}. \end{aligned}$$

(63)

By using (60)–(62), we have

$$\begin{aligned} \lambda _{\max }(\Delta P_{(m)1})\le & {} \lambda _{\max }\left[ \left( \frac{ X_{(m)}^{*\prime }X_{(m)}^{*}}{n}\right) ^{-1}-\left( \frac{\hat{X} _{(m)}^{\prime }\hat{X}_{(m)}}{n}\right) ^{-1}\right] \lambda _{\max }\left( \frac{ X_{(m)}^{*\prime }X_{(m)}^{*}}{n}\right) \\= & {} O_{p}(n^{-\rho }), \\ \lambda _{\max }(\Delta P_{(m)2})\le & {} \lambda _{\max }\left[ \left( \frac{\hat{X} _{(m)}^{\prime }\hat{X}_{(m)}}{n}\right) ^{-1}\right] \lambda _{\max }\left( \frac{( \hat{X}_{(m)}-X_{(m)}^{*})^{\prime }(\hat{X}_{(m)}-X_{(m)}^{*})}{n} \right) \\= & {} O_{p}(n^{-\rho }), \end{aligned}$$

and

$$\begin{aligned}&\lambda _{\max }(\Delta P_{(m)3})=\lambda _{\max }(\Delta P_{(m)4}) \\&\quad =\lambda _{\max }^{1/2}\big ((\hat{X}_{(m)}-X_{(m)}^{*})(\hat{X} _{(m)}^{\prime }\hat{X}_{(m)})^{-1}X_{(m)}^{*\prime }X_{(m)}^{*}( \hat{X}_{(m)}^{\prime }\hat{X}_{(m)})^{-1}(\hat{X}_{(m)}-X_{(m)}^{*})^{\prime }\big ) \\&\quad \le \lambda _{\max }\left[ \left( \frac{\hat{X}_{(m)}^{\prime }\hat{X}_{(m)}}{ n}\right) ^{-1}\right] \lambda _{\max }^{1/2}\left( \frac{X_{(m)}^{*\prime }X_{(m)}^{*}}{n}\right) \nonumber \\&\quad \quad \quad \lambda _{\max }^{1/2}\left( \frac{(\hat{X} _{(m)}-X_{(m)}^{*})^{\prime }(\hat{X}_{(m)}-X_{(m)}^{*})}{n}\right) \\&\quad =O_{p}(n^{-\rho /2}). \end{aligned}$$

Therefore,

$$\begin{aligned} \lambda _{\max }(P_{(m)}^{*}-\hat{P}_{(m)})\le & {} \lambda _{\max }(\Delta P_{(m)1})+\lambda _{\max }(\Delta P_{(m)2})\\&+\lambda _{\max }(\Delta P_{(m)3})+\lambda _{\max }(\Delta P_{(m)4})\\= & {} O_{p}(n^{-\rho /2}). \end{aligned}$$

Thus, (58) holds under Condition (12). The proof of Theorem 2 is completed. $\square$

Proof of Theorem 3

Let $A(w)=I_{n}-P(w)$. From Lemma 1, we need only to verify that

$$\begin{aligned} \underset{w\in \widetilde{{\mathcal {H}}}_{n}}{\sup }|\mu ^{\prime }A(w)e|/ \widetilde{R}_{n}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(64)

$$\begin{aligned} \underset{w\in \widetilde{{\mathcal {H}}}_{n}}{\sup }|e^{\prime }P(w)e-\sigma ^{2}trP(w)|/\widetilde{R}_{n}(w)\overset{p}{\longrightarrow }0, \end{aligned}$$

(65)

$$\begin{aligned} \underset{w\in \widetilde{{\mathcal {H}}}_{n}}{\sup }|\widetilde{L}_{n}(w)/ \widetilde{R}_{n}(w)-1|\overset{p}{\longrightarrow }0, \end{aligned}$$

(66)

and

$$\begin{aligned} \underset{w\in \widetilde{{\mathcal {H}}}_{n}}{\sup }|trP(w)(\Vert A(w)Y\Vert ^{2}/n-\sigma ^{2})|/\widetilde{R}_{n}(w)\overset{p}{\longrightarrow }0. \end{aligned}$$

(67)

We obtain (64)–(66) by following the proof of Theorem $\text {1}^{\prime }$ of Wan et al. (2010), while (67) is valid from the proof of (29). $\square$