Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models

Abstract

This paper establishes consistency and asymptotic normality of the generalized quasi-maximum likelihood estimate (GQMLE) for a general class of periodic conditionally heteroskedastic time series models (PCH). In this class of models, the volatility is expressed as a measurable function of the infinite past of the observed process with periodically time-varying parameters, while the innovation is an independent and periodically distributed sequence. In contrast with the aperiodic case, the proposed GQMLE is rather based on S instrumental density functions where S is the period of the model while the corresponding asymptotic variance is in a “sandwich” form. Application to the periodic asymmetric power GARCH model is given. Moreover, we also discuss how to apply the GQMLE to the prediction of power problem in a one-step framework and to PCH models with complex periodic patterns such as high frequency seasonality and non-integer seasonality.

Appendices

Appendix A: Proofs

Proofs of Theorems 3.1 and Theorem 3.2 follow from similar arguments used in establishing asymptotics of the GQMLE for non-periodic CH models (Berkes and Horvàth 2004; Francq and Zakoïan 2004, 2013, 2015; El Ghourabi et al. 2016).

1.1 A.1. Proof of Theorem 3.1

Result (3.4) follows while establishing the following three lemmas.

Lemma A.1

Under A2, A3 and A5 we have

$$\begin{aligned} \sup _{\theta \in {\varTheta }}\left| \widetilde{L}_{NS,\underline{h}}\left( \theta \right) -L_{NS,\underline{h}}\left( \theta \right) \right| \overset{a.s.}{\underset{N\rightarrow \infty }{\rightarrow }}0, \end{aligned}$$

where $L_{T,\underline{h}}\left( \theta \right) =\frac{1}{T} \sum \limits _{n=0}^{N-1}\sum \limits _{v=1}^{S}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right)$.

Proof

In view of A2–A3 and (3.1)–(3.2), a Taylor expansion gives a.s.

$$\begin{aligned}&\sup _{\theta \in {\varTheta }}\left| \widetilde{L}_{NS,\underline{h}}\left( \theta \right) -L_{NS,\underline{h}}\left( \theta \right) \right| \nonumber \\&\quad \le \tfrac{1}{T}\sum \limits _{n=0}^{N-1}\sum \limits _{v=1}^{S}\sup _{\theta \in {\varTheta }}\left| g_{v1}\left( \epsilon _{nS+v},\sigma _{nS+v}^{*}\left( \theta \right) \right) \right| \left| \widetilde{\sigma }_{nS+v}\left( \theta \right) -\sigma _{nS+v}\left( \theta \right) \right| \nonumber \\&\quad \le \tfrac{1}{T}\sum \limits _{n=0}^{N-1}\sum \limits _{v=1}^{S}b_{nS+v} \left( \theta \right) \sup _{\theta \in {\varTheta }}\left| \tfrac{\epsilon _{t}}{\sigma _{nS+v}^{*2}\left( \theta \right) }\tfrac{h_{v1}^{\prime }}{ h_{v1}}\left( \tfrac{\epsilon _{nS+v}}{\sigma _{nS+v}^{*}\left( \theta \right) }\right) \right| +\tfrac{1}{TC}\sum \limits _{t=1}^{T}b_{t}\left( \theta \right) \nonumber \\&\quad \le \tfrac{1}{T}\sum \limits _{n=0}^{N-1}\sum \limits _{v=1}^{S}b_{nS+v} \left( \theta \right) \left| \epsilon _{nS+v}\right| ^{\delta _{v}}\sup _{\theta \in {\varTheta }}\left| \tfrac{1}{\sigma _{nS+v}^{*}\left( \theta \right) }\right| ^{1+\delta _{v}}+\tfrac{1}{TC} \sum \limits _{t=1}^{T}b_{t}\left( \theta \right) , \end{aligned}$$

(A.1)

where $\sigma _{nS+v}^{*}\left( \theta \right)$ is between $\widetilde{ \sigma }_{nS+v}\left( \theta \right)$ and $\sigma _{nS+v}\left( \theta \right)$ and

$$\begin{aligned} b_{t}\left( \theta \right) =\sup _{\theta \in {\varTheta }}\left| \widetilde{ \sigma }_{t}\left( \theta \right) -\sigma _{t}\left( \theta \right) \right| \text {.} \end{aligned}$$

Now from A5 and the Markov inequality it follows that for all $1\le v\le S$ and $\xi _{v}>0$

$$\begin{aligned} \sum \limits _{n=0}^{\infty }P\left( b_{nS+v}\left( \theta \right) \left| \epsilon _{nS+v}\right| ^{\delta _{v}}>\xi _{v}\right) \le \sum \limits _{n=0}^{\infty }\tfrac{CE\left( \left| \epsilon \right| ^{\tau _{v}}\right) \rho ^{\frac{\tau _{v}}{\delta _{v}}n}}{\xi _{v}^{\frac{ \tau _{v}}{\delta _{v}}}}, \end{aligned}$$

so by the Borel-Cantelli lemma

$$\begin{aligned} b_{nS+v}\left( \theta \right) \left| \epsilon _{nS+v}\right| ^{\delta _{v}}\overset{a.s.}{\underset{n\rightarrow \infty }{\rightarrow }}0 \quad \text { for any }\ 1\le v\le S. \end{aligned}$$

Thus, Lemma A.1 follows from (A.1) and the Césaro lemma. $\square$

Lemma A.2

Under A1, A2 and A4

$$\begin{aligned} E\left( \sum \limits _{v=1}^{S}g_{v}\left( \epsilon _{v},\sigma _{v}\left( \theta \right) \right) \right) <E\left( \sum \limits _{v=1}^{S}g_{v}\left( \epsilon _{v},\sigma _{v}\left( \theta _{0}\right) \right) \right) \ \quad \text {for all} \ \theta \ne \theta _{0}. \end{aligned}$$

(A.2)

Proof

Using A1, the fact that

$$\begin{aligned} g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) =g_{v}\left( \eta _{nS+v},\tfrac{\sigma _{nS+v}\left( \theta \right) }{ \sigma _{nS+v}\left( \theta _{0}\right) }\right) -\log \left( \sigma _{nS+v}\left( \theta _{0}\right) \right) , \end{aligned}$$

and A4 we have

$$\begin{aligned}&E\left( \sum \limits _{v=1}^{S}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) -\sum \limits _{v=1}^{S}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta _{0}\right) \right) \right) \\&\quad = \sum \limits _{v=1}^{S}E\left( g_{v}\left( \eta _{v}, \tfrac{\sigma _{v}\left( \theta \right) }{\sigma _{v}\left( \theta _{0}\right) }\right) -g_{v}\left( \eta _{v},1\right) \right) <0, \end{aligned}$$

with equality if and only if $\sigma _{nS+v}\left( \theta \right) =\sigma _{nS+v}\left( \theta _{0}\right)$ and by A2 if and only if $\theta =\theta _{0}$. $\square$

Lemma A.3

Under A1–A5, for all $\theta \ne \theta _{0}$ there is a neighborhood $V\left( \theta \right)$ such that

$$\begin{aligned} \underset{N\rightarrow \infty }{\lim \sup }\sup \limits _{\theta ^{*}\in V\left( \theta \right) }\widetilde{L}_{NS,\underline{h}}\left( \theta ^{*}\right) <\underset{N\rightarrow \infty }{ \lim \sup }\widetilde{L} _{NS,\underline{h}}\left( \theta _{0}\right) { \ \ }a.s. \end{aligned}$$

(A.3)

Proof

For any $\theta \in {\varTheta }$ and any positive integer k, let $V_{k}\left( \theta \right)$ be the open ball of center $\theta$ and radius 1 / k. Using Lemma A.1 we have

$$\begin{aligned}&\underset{N\rightarrow \infty }{\lim \sup }\sup _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}\widetilde{L}_{NS,\underline{h} }\left( \theta ^{*}\right) \\&\quad \le \underset{N\rightarrow \infty }{ \lim \sup }\sup _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}L_{NS,\underline{h}}\left( \theta ^{*}\right) -\underset{N\rightarrow \infty }{\lim \sup } \sup _{\theta \in {\varTheta }}\left| L_{NS,\underline{h}}\left( \theta \right) -\widetilde{L}_{NS,\underline{h}}\left( \theta \right) \right| \\&\quad \le \underset{N\rightarrow \infty }{\lim \sup }\left( S^{-1}\sum _{v=1}^{S}N^{-1}\sum _{n=0}^{N-1}\sup _{\theta \in V_{k}\left( \theta \right) \cap {\varTheta }}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta ^{*}\right) \right) \right) {,\ \ }a.s. \end{aligned}$$

As the instrumental functions $\left( h_{1},\ldots ,h_{S}\right)$ are by A3 integrable and differentiable, they are bounded. Therefore, by A2

$$\begin{aligned} S^{-1}\sum _{v=1}^{S}E\left( \sup _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta ^{*}\right) \right) \right)<S^{-1}\sum _{v=1}^{S}\left( \tfrac{1}{ \underline{\omega }_{v}}+C\right) <\infty . \end{aligned}$$

(A.4)

Now since by A1 $\left\{ \epsilon _{t},t\in \mathbb {Z} \right\}$ is strictly periodically stationary and periodically ergodic, it follows that for all $1\le v\le S$, the sub-process $\left\{ \epsilon _{nS+v},n\in \mathbb {Z} \right\}$ is strictly stationary and ergodic. Hence, as

$$\begin{aligned} \sup \limits _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}\left( g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta ^{*}\right) \right) \right) , \end{aligned}$$

is a measurable function of the terms of $\left\{ \epsilon _{nS+v},n\in \mathbb {Z} \right\}$, it follows that the sequence

$$\begin{aligned} \left\{ \sup \limits _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta ^{*}\right) \right) ,n\in \mathbb {Z} \right\} , \end{aligned}$$

(A.5)

is strictly stationary and ergodic with

$$\begin{aligned} E\left( \sup _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta ^{*}\right) \right) \right) \in [-\infty ,+\infty ). \end{aligned}$$

For the process given by (A.5), applying the ergodic theorem for strictly stationary and ergodic sequences with possibly an infinite mean (cf. Billingsley 1995, p. 284, 495) and using $E_{\theta _{0}}\left( g_{v}^{-}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) \right) <\infty$ we get

$$\begin{aligned} \underset{N\rightarrow \infty }{\lim \sup }\sup _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}\widetilde{L}_{NS,\underline{h} }\left( \theta ^{*}\right) \le \tfrac{1}{S}\sum _{v=1}^{S}E\left( \sup _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta ^{*}\right) \right) \right) . \end{aligned}$$

By the Beppo-Levi theorem (e.g. Billingsley 1995 p. 219) and using (A.4), the sequence

$$\begin{aligned} \left( \tfrac{1}{S}\sum _{v=1}^{S}E\left( \sup _{\theta ^{*}\in V_{k}\left( \theta \right) \cap {\varTheta }}g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta ^{*}\right) \right) \right) \right) _{k\in \mathbb {N} ^{*}}, \end{aligned}$$

converges while decreasing to

$$\begin{aligned} \tfrac{1}{S}\sum _{v=1}^{S}E_{\theta _{0}}\left( g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) \right) , \end{aligned}$$

as $k\rightarrow \infty$. Thus, (A.3) follows from (A.2).

Proof of Theorem 3.1

To complete the proof of the theorem, we use a standard compactness argument and Lemmas A.1–A.3. We have shown from Lemmas A.1–A.3 that for any neighborhood $V\left( \theta _{0}\right)$ of $\theta _{0}$,

$$\begin{aligned} \underset{N\rightarrow \infty }{\lim \sup }\sup _{\theta ^{*}\in V\left( \theta _{0}\right) }\widetilde{L}_{NS,\underline{h}}\left( \theta ^{*}\right)\le & {} \underset{N\rightarrow \infty }{\lim }\widetilde{L} _{NS,\underline{h}}\left( \theta _{0}\right) =\underset{N\rightarrow \infty }{\lim }L_{NS,\underline{h}}\left( \theta _{0}\right) \nonumber \\= & {} \tfrac{1}{S}\sum _{v=1}^{S}E_{\theta _{0}}\left( g_{v}\left( \epsilon _{v},\sigma _{v}\left( \theta _{0}\right) \right) \right) . \end{aligned}$$

(A.6)

The compact ${\varTheta }$ is recovered by the union of any neighborhood $V\left( \theta _{0}\right)$ of $\theta _{0}$ and a set of neighborhoods $V\left( \theta \right)$, $\theta \in {\varTheta }\backslash V\left( \theta _{0}\right)$ , where $V\left( \theta \right)$ fulfills Lemma A.3. Therefore, there exists a finite sub-covering of ${\varTheta }$ by $V\left( \theta _{0}\right)$, $V\left( \theta _{1}\right) ,\ldots ,V\left( \theta _{k}\right)$ such that

$$\begin{aligned} \sup _{\theta \in {\varTheta }}\widetilde{L}_{NS,\underline{h}}\left( \theta \right) =\min _{i\in \left\{ 1,2,\ldots ,k\right\} }\sup _{\theta \in {\varTheta }\cap V\left( \theta _{i}\right) }\left( \widetilde{L}_{NS,\underline{h}}\left( \theta \right) \right) . \end{aligned}$$

From Lemmas A.3 and (A.6), the latter equality shows that $\widehat{\theta } _{NS,\underline{h}}\in V\left( \theta _{0}\right)$ for N sufficiently large, which completes the proof of the theorem. $\square$

1.2 A.2. Proof of Theorem 3.2

From A6 and the strong consistency of $\widehat{\theta }_{T, \underline{h}}$, a Taylor expansion gives

$$\begin{aligned} \sqrt{N}\tfrac{\partial }{\partial \theta }\widetilde{L}_{T,\underline{h} }\left( \widehat{\theta }_{T,\underline{h}}\right)= & {} 0 \\= & {} \sqrt{N}\tfrac{\partial }{\partial \theta }L_{T,\underline{h}}\left( \widehat{\theta }_{T,\underline{h}}\right) +\sqrt{N}\tfrac{\partial }{ \partial \theta }\widetilde{L}_{T,\underline{h}}\left( \widehat{\theta }_{T, \underline{h}}\right) -\sqrt{N}\tfrac{\partial }{\partial \theta }L_{T, \underline{h}}\left( \widehat{\theta }_{T,\underline{h}}\right) \\= & {} \sqrt{N}\tfrac{\partial }{\partial \theta }L_{T,\underline{h}}\left( \theta _{0}\right) +\tfrac{\partial ^{2}}{\partial \theta \partial \theta ^{\prime }}L_{T,\underline{h}}\left( \theta ^{*}\right) \sqrt{N}\left( \widehat{\theta }_{T,\underline{h}}-\theta _{0}\right) \\&+\,\sqrt{N}\left( \tfrac{\partial }{\partial \theta }\widetilde{L}_{T, \underline{h}}\left( \widehat{\theta }_{T,\underline{h}}\right) -\tfrac{ \partial }{\partial \theta }L_{T,\underline{h}}\left( \widehat{\theta }_{T, \underline{h}}\right) \right) , \end{aligned}$$

where $\theta ^{*}$ is between $\widehat{\theta }_{T,\underline{h}}$ and $\theta _{0}$. Therefore, the asymptotic normality result (3.6) follows while the three following lemmas are proved.

Lemma A.4

Under A1–A5, A7 and A9–A10,

$$\begin{aligned} \sup _{\theta \in V\left( \theta _{0}\right) }\sqrt{N}\left\| \tfrac{ \partial }{\partial \theta }L_{T,\underline{h}}\left( \theta \right) -\tfrac{ \partial }{\partial \theta }\widetilde{L}_{T,\underline{h}}\left( \theta \right) \right\| \overset{p}{\underset{N\rightarrow \infty }{\rightarrow } }0, \end{aligned}$$

for some neighborhood $V\left( \theta _{0}\right)$ of $\theta _{0}$.

Proof

We have

$$\begin{aligned}&\sup _{\theta \in V\left( \theta _{0}\right) }\sqrt{N}\left\| \tfrac{\partial }{\partial \theta }L_{T,\underline{h}}\left( \theta \right) - \tfrac{\partial }{\partial \theta }\widetilde{L}_{T,\underline{h}}\left( \theta \right) \right\| \nonumber \\&\quad =\sup _{\theta \in V\left( \theta _{0}\right) } \tfrac{1}{S\sqrt{N}}\left\| \sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\right. \nonumber \\&\qquad \left. \times \left[ g_{v1}\left( \varepsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) \tfrac{\partial \sigma _{nS+v}\left( \theta \right) }{ \partial \theta }\right. -g_{v1}\left( \varepsilon _{nS+v},\widetilde{\sigma }_{nS+v}\left( \theta \right) \right) \tfrac{\partial \widetilde{\sigma }_{nS+v}\left( \theta \right) }{\partial \theta }\right] \nonumber \\&\quad \le \sup _{\theta \in V\left( \theta _{0}\right) }\tfrac{1}{S\sqrt{N}} \sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\left| g_{v1}\left( \varepsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) -g_{v1}\left( \varepsilon _{nS+v},\widetilde{\sigma }_{nS+v}\left( \theta \right) \right) \right| \left\| \tfrac{\partial \sigma _{nS+v}\left( \theta \right) }{\partial \theta }\right\| \nonumber \\&\qquad +\sup _{\theta \in V\left( \theta _{0}\right) }\tfrac{1}{S\sqrt{N}} \sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\left| g_{v1}\left( \varepsilon _{nS+v},\widetilde{\sigma }_{nS+v}\left( \theta \right) \right) \right| \left\| \tfrac{\partial \sigma _{nS+v}\left( \theta \right) }{\partial \theta }-\tfrac{\partial \widetilde{\sigma }_{nS+v}\left( \theta \right) }{\partial \theta }\right\| . \end{aligned}$$

(A.7)

From A3 and A9, the second term in the right hand side of (A.7) is bounded by

$$\begin{aligned} \tfrac{C}{S\sqrt{N}}\sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\rho ^{n} \tfrac{1}{\underline{\omega }_{v}}\left( 1+K_{v}\left| \tfrac{ \varepsilon _{nS+v}}{\underline{\omega }_{v}}\right| ^{\delta _{v}}\right) , \end{aligned}$$

which is of order $O\left( T^{-1/2}\right) ~a.s$. For the first term in (A.7), using a Taylor expansion, assumptions A3, A5, A7, A10 and the Cauchy–Shwartz inequality, it follows that this term is bounded by

$$\begin{aligned}&\tfrac{C}{S\sqrt{N}}\sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\rho ^{n} \tfrac{1}{\underline{\omega }_{v}}\left| g_{v2}\left( \varepsilon _{nS+v},\sigma _{nS+v}^{*}\left( \theta \right) \right) \right| \left\| \tfrac{\partial \sigma _{nS+v}\left( \theta \right) }{\partial \theta }\right\| \\&\quad \le \tfrac{C}{S\sqrt{N}}\sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\rho ^{n}\tfrac{1}{\underline{\omega }_{v}}\left| 1+3K_{v}\left( 1+\left| \tfrac{\varepsilon _{nS+v}}{\underline{\omega }_{v}}\right| ^{\delta _{v}}\right) \right| \sup _{\theta \in V\left( \theta _{0}\right) }\left\| \tfrac{1}{\sigma _{nS+v}\left( \theta \right) }\tfrac{\partial \sigma _{nS+v}\left( \theta \right) }{\partial \theta }\right\| \\&\quad = o\left( 1\right) a.s.\text {,} \end{aligned}$$

where $\sigma _{nS+v}^{*}\left( \theta \right)$ is between $\widetilde{ \sigma }_{nS+v}\left( \theta \right)$ and $\sigma _{nS+v}\left( \theta \right)$. This completes the proof of the lemma. $\square$

Lemma A.5

Under A1–A10, for any $\theta ^{*}$ between $\widehat{\theta }_{T,\underline{h}}$ and $\theta _{0}$,

$$\begin{aligned} \tfrac{\partial ^{2}L_{T,\underline{h}}\left( \theta ^{*}\right) }{ \partial \theta \partial \theta ^{\prime }}\overset{p}{\underset{ N\rightarrow \infty }{\rightarrow }}\tfrac{1}{4}A_{\underline{h},\underline{f }}\left( \theta _{0}\right) . \end{aligned}$$

Proof

From A3 and A7 we have

$$\begin{aligned} \left\| \tfrac{\partial ^{2}L_{T,\underline{h}}\left( \theta \right) }{\partial \theta \partial \theta ^{\prime }}\right\|= & {} \left\| \tfrac{1}{NS}\sum \limits _{n=0}^{N-1}\sum \limits _{v=1}^{S}g_{v2}\left( \varepsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) \tfrac{ \partial \sigma _{nS+v}\left( \theta \right) }{\partial \theta }\tfrac{ \partial \sigma _{nS+v}\left( \theta \right) }{\partial \theta ^{\prime }} \right. \\&\left. +\,g_{v1}\left( \varepsilon _{nS+v},\sigma _{nS+v}\left( \theta \right) \right) \tfrac{\partial ^{2}\sigma _{nS+v}\left( \theta \right) }{ \partial \theta \partial \theta ^{\prime }}\right\| \le \tfrac{C}{NS} \sum \limits _{n=0}^{N-1}\sum \limits _{v=1}^{S}\left( 1+\left| \tfrac{ \sigma _{nS+v}\left( \theta _{0}\right) \eta _{nS+v}}{\sigma _{nS+v}\left( \theta \right) }\right| ^{\delta _{v}}\right) \\&\times \left( \left\| \tfrac{1}{\sigma _{nS+v}\left( \theta \right) }\tfrac{ \partial ^{2}\sigma _{nS+v}\left( \theta \right) }{\partial \theta \partial \theta ^{\prime }}\right\| +\left\| \tfrac{1}{\sigma _{nS+v}^{2}\left( \theta \right) }\tfrac{\partial \sigma _{nS+v}\left( \theta \right) }{ \partial \theta }\tfrac{\partial \sigma _{nS+v}\left( \theta \right) }{ \partial \theta ^{\prime }}\right\| \right) . \end{aligned}$$

By the Hölder inequality, A7 and A10 it follows that

$$\begin{aligned} E\left( \sup _{\theta \in V\left( \theta _{0}\right) }\left\| \tfrac{ \partial ^{2}L_{T,\underline{h}}\left( \theta \right) }{\partial \theta \partial \theta ^{\prime }}\right\| \right) <\infty , \end{aligned}$$

so the ergodic theorem implies that

$$\begin{aligned}&\lim _{N\rightarrow \infty }\sup _{\theta \in V\left( \theta _{0}\right) }\left\| \tfrac{\partial ^{2}L_{T,\underline{h}}\left( \theta \right) }{\partial \theta \partial \theta ^{\prime }}-\tfrac{\partial ^{2} L_{T,\underline{h}}\left( \theta _{0}\right) }{\partial \theta \partial \theta ^{\prime }}\right\| \\&\quad \le E\left( \sup _{\theta \in V\left( \theta _{0}\right) }\left\| \sum \limits _{v=1}^{S}\left( \tfrac{\partial ^{2}g_{v2}\left( \varepsilon _{v},\sigma _{v}\left( \theta \right) \right) }{\partial \theta \partial \theta ^{\prime }}-\tfrac{\partial ^{2}g_{v2}\left( \varepsilon _{v},\sigma _{v}\left( \theta _{0}\right) \right) }{\partial \theta \partial \theta ^{\prime }}\right) \right\| \right) , \, {\textit{a.s.}} \end{aligned}$$

From the dominated convergence theorem, the latter expectation tends to zero when $V\left( \theta _{0}\right)$ tends to the singleton $\left\{ \theta _{0}\right\}$. Now since $\widehat{\theta }_{T,\underline{h}}$ is consistent then

$$\begin{aligned} \left\| \tfrac{\partial ^{2}L_{T,\underline{h}}\left( \theta ^{*}\right) }{\partial \theta \partial \theta ^{\prime }}-\tfrac{\partial ^{2}L_{T,\underline{h}}\left( \theta _{0}\right) }{\partial \theta \partial \theta ^{\prime }}\right\| \overset{a.s.}{\underset{N\rightarrow \infty }{ \rightarrow }}0. \end{aligned}$$

On the other hand since by A3

$$\begin{aligned} g_{v1}\left( x,\varsigma \right) =\tfrac{\partial g_{v}\left( x,\varsigma \right) }{\partial \varsigma }=-\tfrac{1}{\varsigma }-\tfrac{h_{v}^{\prime }\left( \frac{x}{\varsigma }\right) }{h_{v}\left( \frac{x}{\varsigma } \right) }\tfrac{x}{\varsigma ^{2}}, \quad 1\le v\le S, \end{aligned}$$

exists for all $\varsigma >0$ and $E\left( \sup _{\varsigma \in V\left( 1\right) }\left| g_{v}\left( \eta _{v},\varsigma \right) \right| \right) <\infty$, then A4 and the dominated convergence theorem entail the following moment conditions

$$\begin{aligned} E\left( \tfrac{h_{v}^{\prime }\left( \eta _{v}\right) }{h_{v}\left( \eta _{v}\right) }\eta _{v}\right) =-1,\quad \text { for all } \quad \ 1\le v\le S, \end{aligned}$$

which in turn imply that

$$\begin{aligned} E\left( g_{v1}\left( \varepsilon _{v},\sigma _{v}\left( \theta _{0}\right) \right) \tfrac{\partial ^{2}\sigma _{v}\left( \theta _{0}\right) }{\partial \theta \partial \theta ^{\prime }}\right) =0. \end{aligned}$$

Note that the following equality

$$\begin{aligned} g_{v2}\left( x,\varsigma \right) =\tfrac{\partial g_{v1}\left( x,\varsigma \right) }{\partial \varsigma }=\tfrac{1}{\varsigma ^{2}}\left[ 1+\tfrac{x}{ \varsigma }\left( 2\tfrac{h_{v}^{\prime }\left( \tfrac{x}{\varsigma }\right) }{h_{v}\left( \tfrac{x}{\varsigma }\right) }+\tfrac{x}{\varsigma }\left( \tfrac{h_{v}^{\prime }\left( \tfrac{x}{\varsigma }\right) }{h_{v}\left( \tfrac{x}{\varsigma }\right) }\right) ^{\prime }\right) \right] , \end{aligned}$$

gives

$$\begin{aligned} g_{v2}\left( \varepsilon _{nS+v},\sigma _{nS+v}\left( \theta _{0}\right) \right) =g_{v2}\left( \eta _{nS+v},1\right) \tfrac{\partial ^{2}\sigma _{v}\left( \theta _{0}\right) }{\partial \theta \partial \theta ^{\prime }}. \end{aligned}$$

Therefore, by the periodic ergodic theorem we finally get

$$\begin{aligned} \tfrac{\partial ^{2}L_{T,\underline{h}}\left( \theta _{0}\right) }{\partial \theta \partial \theta ^{\prime }}\overset{a.s.}{\underset{N\rightarrow \infty }{\rightarrow }}\tfrac{1}{4}A_{\underline{h},\underline{f}}\left( \theta _{0}\right) , \end{aligned}$$

which proves the lemma. $\square$

Lemma A.6

Under A1–A10

$$\begin{aligned} \sqrt{N}\tfrac{\partial L_{T,\underline{h}}\left( \theta _{0}\right) }{ \partial \theta }\overset{\mathcal {L}}{\underset{N\rightarrow \infty }{ \rightarrow }}N\left( 0,\tfrac{1}{4}B_{\underline{h},\underline{f}}\left( \theta _{0}\right) \right) , \end{aligned}$$

where $B_{\underline{h},\underline{f}}\left( \theta _{0}\right)$ given by (3.5b) is invertible under A8.

Proof

Note that

$$\begin{aligned} \sqrt{N}\tfrac{\partial L_{T,\underline{h}}\left( \theta _{0}\right) }{\partial \theta }= & {} \tfrac{1}{S\sqrt{N}}\sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\tfrac{\partial g_{v}\left( \epsilon _{nS+v}, \sigma _{nS+v}\left( \theta _{0}\right) \right) }{\partial \theta } \\= & {} \tfrac{1}{S}\sum \limits _{v=1}^{S}\tfrac{1}{\sqrt{N}}\sum \limits _{n=0}^{N-1}g_{v1}\left( \eta _{nS+v},1\right) \tfrac{1}{2\sigma _{nS+v}^{2}\left( \theta _{0}\right) }\tfrac{\partial \sigma _{nS+v}^{2}\left( \theta _{0}\right) }{\partial \theta }. \end{aligned}$$

Since by the periodic ergodic theorem we have

$$\begin{aligned}&\left. \sum \limits _{v=1}^{S}\sum \limits _{n=0}^{N-1}\left( \tfrac{1}{S\sqrt{ N}}\right) ^{2}\tfrac{\partial g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta _{0}\right) \right) }{\partial \theta }\tfrac{\partial g_{v}\left( \epsilon _{nS+v},\sigma _{nS+v}\left( \theta _{0}\right) \right) }{\partial \theta ^{\prime }}\right. \\&\quad =\sum \limits _{v=1}^{S}\tfrac{1}{N}\sum \limits _{n=0}^{N-1}\tfrac{ g_{v1}^{2}\left( \eta _{nS+v},1\right) }{4S^{2}\sigma _{nS+v}^{2}\left( \theta _{0}\right) }\tfrac{\partial \sigma _{nS+v}^{2}\left( \theta _{0}\right) }{\partial \theta }\tfrac{\partial \sigma _{nS+v}^{2}\left( \theta _{0}\right) }{\partial \theta ^{\prime }}\overset{a.s.}{\underset{ N\rightarrow \infty }{\rightarrow }}\tfrac{1}{4}B_{\underline{h},\underline{f }}\left( \theta _{0}\right) , \end{aligned}$$

then by the martingale central limit theorem (Billingsley 1961) we get (A.8).

Now we prove that $B_{\underline{h},\underline{f}}\left( \theta _{0}\right)$ is nonsingular under A8. If $B_{\underline{h},\underline{f}}\left( \theta _{0}\right)$ is singular, then there exists a non-null vector $\mathbf {x}\in \mathbb {R} ^{m}$ such that $\mathbf {x}^{\prime }B_{\underline{h},\underline{f}}\left( \theta _{0}\right) \mathbf {x}=0$. Note that

$$\begin{aligned} \mathbf {x}^{\prime }B_{\underline{h},\underline{f}}\left( \theta _{0}\right) \mathbf {x}= & {} \tfrac{1}{S^{2}}\sum \limits _{v=1}^{S}E\left( g_{v1}\left( \eta _{v},1\right) ^{2}\right) E\left( \mathbf {x}^{\prime }\tfrac{1}{\sigma _{v}^{4}\left( \theta _{0}\right) }\tfrac{\partial \sigma _{v}^{2}\left( \theta _{0}\right) }{\partial \theta }\tfrac{\partial \sigma _{v}^{2}\left( \theta _{0}\right) }{\partial \theta ^{\prime }}\mathbf {x}\right) \\= & {} \tfrac{1}{S^{2}}\sum \limits _{v=1}^{S}E\left( g_{v1}\left( \eta _{v},1\right) ^{2}\right) E\left( \tfrac{1}{\sigma _{v}^{4}\left( \theta _{0}\right) }\left( \mathbf {x}^{\prime }\tfrac{\partial \sigma _{v}^{2}\left( \theta _{0}\right) }{\partial \theta }\right) ^{2}\right) . \end{aligned}$$

Since by A8, $E\left( g_{v1}\left( \eta _{v},1\right) ^{2}\right) >0$ for any $v\in \left\{ 1,\ldots ,S\right\}$, it follows that $\mathbf {x} ^{\prime }B_{\underline{h},\underline{f}}\left( \theta _{0}\right) \mathbf {x} =0$ if and only if $E\left( \tfrac{1}{\sigma _{v}^{4}\left( \theta _{0}\right) }\left( \mathbf {x}^{\prime }\tfrac{\partial \sigma _{v}^{2}\left( \theta _{0}\right) }{\partial \theta }\right) ^{2}\right) =0$ , $\forall v\in \left\{ 1,\ldots ,S\right\}$, which holds if and only if

$$\begin{aligned} \tfrac{1}{\sigma _{v}^{2}\left( \theta _{0}\right) }\left( \mathbf {x}^{\prime }\tfrac{\partial \sigma _{v}^{2}\left( \theta _{0}\right) }{ \partial \theta }\right) ^{2}=0{\ \ }a.s. \forall v\in \left\{ 1,\ldots ,S\right\} \Leftrightarrow \mathbf {x}^{\prime }\tfrac{\partial \sigma _{v}^{2}\left( \theta _{0}\right) }{\partial \theta }=0{\ \ }a.s.\forall v\in \left\{ 1,\ldots ,S\right\} . \end{aligned}$$

By the last part of A8 this implies that $\mathbf {x}=0$ , which contradicts the fact that $\mathbf {x}\ne 0$ . Hence, $B_{ \underline{h},\underline{f}}\left( \theta _{0}\right)$ is nonsingular. $\square$

Appendix B: Some examples of the PCH model

The following examples illustrate the model (2.1) via some specific subclasses of it.

1.1 B.1. The infinite periodic $ARCH\left( \infty \right)$ model

An important case of (2.1) is the infinite periodic ARCH ($PARCH\left( \infty \right)$) model, which is defined by

$$\begin{aligned} \left\{ \begin{array}{l} \epsilon _{t}=\sigma _{t}\eta _{t}, \\ \sigma _{t}^{2}=\alpha _{0t,0}+\alpha _{0t,1}\epsilon _{t-1}^{2}+\alpha _{0t,2}\epsilon _{t-2}^{2}+... \end{array} \right. { \ \ \ \ }t\in \mathbb { \mathbb {Z} }, \end{aligned}$$

(B.1)

where $\left\{ \eta _{t},t\in \mathbb { \mathbb {Z} }\right\}$ is $ipd_{S}$ and the positive coefficients $\left( \alpha _{0t,j},t\in \mathbb {Z} \right)$ are S-periodic over t for all $j\in \mathbb {N} ,$ i.e. $\alpha _{0t,j}=\alpha _{0,t+kS,j}$, $k,t\in \mathbb {Z}$, $j\in \mathbb {N}$. A number of S identifiability conditions on $\left\{ \eta _{t},t\in \mathbb { \mathbb {Z} }\right\}$ are required. They are induced by the instrumental functions used in calculating the GQMLE we propose below and should also be compatible with other objectives of the model such as the prediction of powers of the observed process (cf. Francq and Zakoïan 2013 in the CH case). For model (B.1), the function $\varphi _{t}$ in (2.1) corresponds to $\varphi _{t}\left( x_{1},x_{2},\ldots \right) =\alpha _{0t,0}+\sum _{j=1}^{\infty }\alpha _{0t,j}x_{j}~\left( t\in \mathbb {Z} \right)$ while the corresponding parameter vector $\theta _{0}=\left( \theta _{01}^{\prime },\ldots ,\theta _{0S}^{\prime }\right) ^{\prime }\in {\varTheta }$ is obtained by parameterizing the coefficients $\left( \alpha _{0t,j},t\in \mathbb {Z} ,j\in \mathbb {N} \right)$. Specifically, for $1\le v\le S$ we assume that

$$\begin{aligned} \alpha _{0v,j}=\alpha _{v,j}\left( \theta _{0v}\right) , \end{aligned}$$

with known functions $\alpha _{v,j}\left( .\right) :$ ${\varTheta }_{v}\rightarrow [0,\infty )$, for some ${\varTheta }_{v}\subset \mathbb {R} ^{m_{v}}$. For instance, a simple $PARCH\left( \infty \right)$ model is obtained for the functions

$$\begin{aligned} \alpha _{v,j}\left( \theta _{0v}\right) =\left\{ \begin{array}{ll} \frac{b_{v}}{j^{d_{v}+1}} &{} \quad \text { if }j\ge 1 \\ 1 &{} \quad \text { if }j=0, \end{array} \right. \text {, }\quad 1\le v\le S, \end{aligned}$$

with $\theta _{0v}=\left( b_{v},d_{v}\right) ^{\prime }\in {\varTheta }_{v}=\left[ \underline{b}_{v},\overline{b}_{v}\right] \times \left[ \underline{d}_{v}, \overline{d}_{v}\right]$ and ${\varTheta }={\varTheta }_{1}\times \cdots \times {\varTheta }_{S}\subset \left( 0,\infty \right) ^{2S},$ where $0<\underline{b}_{v}< \overline{b}_{v}$ and $0<\underline{d}_{v}<\overline{d}_{v}~\left( 1\le v\le S\right)$ (see Francq and Zakoïan 2016 in the $ARCH\left( \infty \right)$ case).

When $\left\{ \eta _{t},t\in \mathbb { \mathbb {Z} }\right\}$ is iid, model (B.1) is a particular case of the periodic nonlinear $AR\left( \infty \right)$-$ARCH\left( \infty \right)$ model proposed by Ziel (2015, 2016) where conditions on its existence are provided. It is a periodic version of the infinite $ARCH\left( \infty \right)$ introduced by Robinson (1991) and is recommended for representing strong persistence and periodicity in volatility. It is also a generalization of the most widely used periodic GARCH model (Bollerslev and Ghysels 1996). Indeed, consider the following $PGARCH\left( 1,1\right)$ model given by

$$\begin{aligned} \left\{ \begin{array}{l} \epsilon _{nS+v}=\sigma _{nS+v}\eta _{nS+v}, \\ \sigma _{nS+v}^{2}=\omega _{0v}+\alpha _{0v}\epsilon _{nS+v-1}^{2}+\beta _{0v}\sigma _{nS+v-1}^{2} \end{array} \right. { \ \ \ \ }1\le v\le S,n\in \mathbb { \mathbb {Z} }, \end{aligned}$$

(B.2)

where $\left\{ \eta _{t},t\in \mathbb { \mathbb {Z} }\right\}$ is $ipd_{S}$ with $\sup _{1\le v\le S}E\left( \log \left( \eta _{v}^{2}\right) \right) <\infty$ and $\omega _{0v}>0,\alpha _{0v}\ge 0,\beta _{0v}\ge 0$. Under the stability condition $\prod \limits _{v=1}^{S} \beta _{0v}<1$ which in turn is implied by the strict periodic stationarity condition

$$\begin{aligned} \sum _{v=1}^{S}E\left( \log \left( \alpha _{0v}\eta _{v-1}^{2}+\beta _{0v}\right) \right) <0, \end{aligned}$$

(cf. Aknouche and Bibi 2009, Corollary 1) we have

$$\begin{aligned} \sigma _{t}^{2}=\sum _{j=0}^{\infty }\prod \limits _{i=0}^{j-1}\beta _{0,t-i}\left( \omega _{0,t-j}+\alpha _{0,t-j}\epsilon _{t-1-j}^{2}\right) . \end{aligned}$$

So (B.2) is a particular case of (B.1).

1.2 B.2. The periodic asymmetric power $GARCH\left( 1,1\right)$ model

We revisit model (4.4) for the explicit case $S=5$ where the parameter space is clarified. Consider the 5-periodic PAP-$GARCH\left( 1,1\right)$ given by

$$\begin{aligned} \left\{ \begin{array}{l} \epsilon _{5n+v}=\sigma _{5n+v}\eta _{5n+v},{ \ \ \ \ \ \ }1\le v\le 5 {,\ }\quad n\in \mathbb {Z}, \\ \sigma _{5n+v}^{\delta _{v}}=\omega _{0v}+\alpha _{0v+}(\epsilon _{5n+v-1}^{+})^{\delta _{v-1}} +\alpha _{0v-}(\epsilon _{5n+v-1}^{-})^{\delta _{v-1}}+\beta _{0v} \sigma _{5n+v-1}^{\delta _{v-1}}, \end{array} \right. \end{aligned}$$

(B.3)

where $\omega _{0v}>0$, $\alpha _{0v+}\ge 0$, $\alpha _{0v-}\ge 0,$ $\beta _{0v}\ge 0$, $\delta _{v}>0~\left( 1\le v\le 5\right) ,$ $\delta _{0}:=\delta _{5}$, $x^{+}=\max (x,0)$ and $x^{-}=-\min (x,0)$. Assuming $\delta _{v}$ known ($1\le v\le 5$), the unknown parameter of the model is denoted by $\theta _{0}=\left( \theta _{01}^{\prime },\cdots ,\theta _{05}^{\prime }\right) ^{\prime }\in {\varTheta }\subset (0,\infty )^{20}$ with $\theta _{0v}^{\prime }=\left( \omega _{0v},\alpha _{0v+},\alpha _{0v-},\beta _{0v}\right) \in {\varTheta }_{v}\subset (0,\infty )^{4}$, $1\le v\le 5$, where ${\varTheta }={\varTheta }_{1}\times \cdots \times {\varTheta }_{5}$ is a compact parameter space. Moreover, parameter restriction on the model may be considered when having prior information on the marginal distributions of the returns. For example, model (B.3) may be used to represent daily stock returns where each trading day of the week $v\in \left\{ 1,2,\ldots ,5\right\}$ has a proper marginal distribution. If for a given trading day $w\in \left\{ 1,2,\ldots ,5\right\}$ we admit that the asymmetry (or leverage effect) of the model is insignificant then one may assume that $\alpha _{0w+}=\alpha _{0w-}:=\alpha _{0w}$ so that the corresponding parameter reduces to $\theta _{0w}=\left( \omega _{0w},\alpha _{0w},\beta _{0w}\right) ^{\prime }\in (0,\infty )^{3}$. It is also possible to consider the powers $\delta _{v}$ ($1\le v\le 5$) as unknown parameters to be jointly estimated with $\theta _{0}$. In that case, the parameter of the model is denoted by $\psi _{0}=\left( \delta ^{\prime },\theta _{0}^{\prime }\right) ^{\prime }$ with $\delta =\left( \delta _{1},\delta _{2},\ldots ,\delta _{5}\right) ^{\prime }\in (0,\infty )^{5}$. Note also that the powers may be considered constant, i.e. $\delta _{1}=\cdots =\delta _{5}$.

1.3 B.3. Mixed specifications

The PCH model (2.1) also allows different specifications along seasons. For $S=5$, consider the following model

$$\begin{aligned} \left\{ \begin{array}{l} \epsilon _{5n+v}=\sigma _{5n+v}\eta _{5n+v}{, \ \ \ \ \ \ \ \ \ \ } 1\le v\le 5,n\in \mathbb {N} \\ \sigma _{5n+1}^{2}=\alpha _{1,0}\left( \theta _{01}\right) +\sum \limits _{j=1}^{\infty }\alpha _{1,j}\left( \theta _{01}\right) \epsilon _{5n+1-j}^{2} \\ \log \left( \sigma _{5n+2}^{2}\right) =\omega _{02}+\alpha _{02}\epsilon _{5n+1}^{2}+\beta _{02}\log \left( \sigma _{5n+1}^{2}\right) \\ \sigma _{5n+3}=\omega _{03}+\alpha _{03+}(\epsilon _{5n+2}^{+})+\alpha _{03-}(\epsilon _{5n+2}^{-})+\beta _{03}\sigma _{5n+2} \\ \sigma _{5n+4}^{\delta _{4}}=\omega _{04}+\alpha _{04}\left| \epsilon _{5n+3}\right| +\beta _{04}\sigma _{5n+3} \\ \sigma _{5n+5}^{\delta _{5}}=\omega _{05}+\alpha _{05+}(\epsilon _{5n+4}^{+})^{\delta _{4}}+\alpha _{05-}(\epsilon _{5n+4}^{-})^{\delta _{4}}+\beta _{05}\sigma _{5n+4}^{\delta _{4}} \end{array} \right. , \end{aligned}$$

(B.4)

where the parameter is denoted by $\theta _{0}=\left( \theta _{01}^{\prime },\cdots ,\theta _{05}^{\prime }\right) ^{\prime }\in {\varTheta }\subset \left( 0,\infty \right) ^{m_{1}+14}$ with $\theta _{01}\in {\varTheta }_{1}\subset \mathbb {R} ^{m_{1}}$ for some $m_{1}\in \mathbb {N} ^{*}$, $\theta _{02}=\left( \omega _{02},\alpha _{02},\beta _{02}\right) ^{\prime }\in {\varTheta }_{2}\subset \left( 0,\infty \right) ^{3}$, $\theta _{03}=\left( \omega _{03},\alpha _{03+},\alpha _{03-},\beta _{03}\right) ^{\prime }\in {\varTheta }_{3}\subset \left( 0,\infty \right) ^{4}$, $\theta _{04}=\left( \omega _{04},\alpha _{04},\beta _{04}\right) ^{\prime }\in {\varTheta }_{4}\subset \left( 0,\infty \right) ^{3}$ and $\theta _{05}=\left( \omega _{05},\alpha _{05+},\alpha _{05-},\beta _{05}\right) ^{\prime }\in {\varTheta }_{5}\subset \left( 0,\infty \right) ^{4}$. All parameter spaces ${\varTheta }_{1},\ldots ,{\varTheta }_{5}$ and ${\varTheta }={\varTheta }_{1}\times \cdots \times {\varTheta }_{5}$ are assumed compact while the powers $\delta _{4}>0$ and $\delta _{5}>0$ are known. Note that $\theta _{01}$ is a parametrization of the coefficients $\alpha _{1,j}~\left( j\in \mathbb {N} \right)$ in (B.4). As in the previous examples, the innovation sequence $\left\{ \eta _{t},t\in \mathbb {Z} \right\}$ satisfies certain identifiability assumptions depending on the chosen instrumental functions used in computing the GQMLE. For $v=1$, it is clear that $\sigma _{5n+1}$ has a similar form as (2.1). By successive replacement in (B.4), it can be seen that $\sigma _{5n+v}$ $\left( 2\le v\le 5\right)$ may be cast in the form (2.1) with some conditions on the $\alpha _{1,j}\left( \theta _{01}\right)$ ($j\in \mathbb {N}$) for the volatility to exist, but without any requirement on $\beta _{0v}~\left( 2\le v\le 5\right)$. So (B.4) is a particular case of (2.1). In fact, model (B.4) combines the infinite $ARCH\left( \infty \right)$ for $v=1$, the Exponential $GARCH\left( 1,1\right)$ ($EGARCH\left( 1,1\right)$) for $v=2$, the threshold $GARCH\left( 1,1\right)$ for $v=3$, the power $GARCH\left( 1,1\right)$ for $v=4$ and the asymmetric power GARCH(1, 1) for $v=5$.

In this illustrative model, various specifications across seasons are permitted. In practice, many seasonal volatility series may show certain stylized facts on a given season and not on another. For example, in daily return series, which generally show the day-of-the-week effect (Tsiakas 2006; Berument et al. 2007; Osborn et al. 2008), the “Monday” series may be characterized by a stronger persistence compared to other trading days. Also, a certain trading day may have a distribution with tails heaver than those of the other trading days (e.g. Boynton et al. 2009; Bidarkota et al. 2009). And this may be true for the asymmetry property (e.g. Balaban et al. 2001; Charles 2010). Thus, one can consider a periodic volatility model with different specifications each of which is adapted to the specific stylized facts marking the season in question. Of course, this is only a schematic and hypothetical example and other mixed specifications are conceivable. However, they should be preceded by preliminary theories (financial for example) and confirmed and reinforced by applications. $\square$

Appendix C: GQMLE for PCH models with complex periodic patterns

1.1 C.1. GQMLE and reduction of the number of parameters in high frequency PCH models

Though periodic CH models have been successfully applied to low frequency seasonal series like daily series (e.g. Bollerslev and Ghysels 1996; Franses and Paap 2004; Osborn et al. 2008), a potential drawback with these models is that they involve a very large number of parameters when the period S tends to be large, like in intraday series. For instance, for half-hourly series (e.g. Taylor 2006), which imply a period of $S=48$, the unrestricted PAP-GARCH(1, 1) model (4.4) requires $4S=192$ parameters, making their estimation and interpretation extremely challenging. To overcome this problem, several solutions have been suggested to reduce the number of implied parameters in high frequency periodic models. An ad hoc device is to restrict some parameters to reduce the parameter space. For example, in model (4.4) one might take $\beta _{v}=\beta _{1}$ ($2\le v\le S$) as already done by Franses and Paap (2000) for the PGARCH model. However, the most usual approach is to use some basis functions like Fourier approximation (Jones and Brelsford 1967; Bollerslev et al. 2000; Taylor 2006; Anderson et al. 2007; Tesfaye et al. 2011; Franses and Paap 2011; Rossi and Fantazani 2015), periodic B-splines (Ziel et al. 2015) or periodic wavelets (see also Ziel et al. 2016; Ambach and Croonenbroeck 2015; Ambach and Schmid 2015). In this Subsection we will see how the GQMLE may be adapted when model (2.1) is reparameterized to reduce the parameter space in high frequency PCH models. We follow here the approach of Jones and Brelsford (1967), which is based on the following reparametrization

$$\begin{aligned} \left\{ \begin{array}{l} \theta _{0v}=\left( \theta _{0v,1},\ldots ,\theta _{0v,m_{v}}\right) ^{\prime } \\ \theta _{0v,j}=\theta _{0j}^{*}+\theta _{0j}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\theta _{0j}^{*}\right) \text {, \ }1\le j\le m_{v} \end{array} \right. ,\quad 1\le v\le S, \end{aligned}$$

(C.1)

where for identifiability reasons we assume that $\theta _{0j}^{*}\in (0,1)$ for all j as $\cos \left( x+n\pi \right) =\left( -1\right) ^{n}\cos \left( x\right)$ (see also Rossi and Fantazani (2015) for the periodic long memory EGARCH model and Franses and Paap (2011) for the periodic autoregression). In lieu of $m=\sum \nolimits _{v=1}^{S}m_{v}$ parameters, the new reparametrization (C.1) only involves a number of $m^{*}=3\max _{1\le v\le S}\left( m_{v}\right)$ parameters to be estimated. For example, for the PAP-$GARCH\left( 1,1\right)$ model (4.4), representation (C.1) reduces to

$$\begin{aligned} \left\{ \begin{array}{l} \omega _{0v}=\omega _{01}^{*}+\omega _{02}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\omega _{03}^{*}\right) \\ \alpha _{0v+}=\alpha _{01+}^{*}+\alpha _{02+}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\alpha _{03+}^{*}\right) \\ \alpha _{0v-}=\alpha _{01-}^{*}+\alpha _{02-}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\alpha _{03-}^{*}\right) \\ \beta _{0v}=\beta _{01}^{*}+\beta _{02}^{*}\cos \left( \tfrac{2\pi v }{S}-\beta _{03}^{*}\right) \end{array} \right. ,\quad 1\le v\le S, \end{aligned}$$

(C.2)

where the parameter of the model is now denoted by $\theta _{0}^{*}=\left( \omega _{0}^{*\prime },\alpha _{0+}^{*\prime },\alpha _{0-}^{*\prime },\beta _{0}^{*\prime }\right) ^{\prime }$ with $\omega _{0}^{*}=\left( \omega _{01}^{*},\omega _{02}^{*},\omega _{03}^{*}\right) ^{\prime }$, $\alpha _{0+}^{*}=\left( \alpha _{01+}^{*},\alpha _{01+}^{*},\alpha _{01+}^{*}\right) ^{\prime }$, $\alpha _{0-}^{*}=\left( \alpha _{01-}^{*},\alpha _{01-}^{*},\alpha _{01-}^{*}\right) ^{\prime }$, $\beta _{0}^{*}=\left( \beta _{01}^{*},\beta _{02}^{*},\beta _{03}^{*}\right) ^{\prime }$. and $\left( \omega _{03}^{*},\alpha _{03+}^{*},\alpha _{03-}^{*},\beta _{03}^{*}\right) ^{\prime }\in \left( 0,1\right) ^{4}$. Note that the number of parameters in (C.2) does not depend on S and is reduced for large S from 4S to 12.

Now with $\theta _{0}^{*}$ in place of $\theta _{0}$, model (2.1) may be rewritten as follows

$$\begin{aligned} \left\{ \begin{array}{l} \epsilon _{nS+v}=\sigma _{nS+v}\eta _{nS+v}, \\ \sigma _{nS+v}=\varphi _{v}^{*}\left( \epsilon _{nS+v-1},\epsilon _{nS+v-2},\ldots ;\theta _{0v}^{*}\right) :=\sigma _{nS+v}^{*}\left( \theta _{0}^{*}\right) , \end{array} \right. {,}1\le v\le S, \end{aligned}$$

(C.3)

where the function $\varphi _{v}^{*}$ is obtained from $\varphi _{v}$ by rearrangement while replacing $\theta _{0}$ by $\theta _{0}^{*}$. We assume that $\theta _{0}^{*}\in {\varTheta }^{*}\subset \mathbb {R} ^{m^{*}}$ for some compact parameter space ${\varTheta }^{*}$. Of course, the stability and positivity constraints on $\theta _{0}$ in (2.1) are directly translated in terms of $\theta _{0}^{*}$ through (C.1). Like model (2.1), we define $\sigma _{nS+v}^{*}\left( \theta ^{*}\right)$, $\widetilde{\sigma }_{nS+v}^{*}\left( \theta ^{*}\right)$ and $\widetilde{L}_{T,\underline{h}}^{*}\left( \theta \right)$ as in (3.0), (3.1) and (3.2), respectively for some instrumental functions $\underline{h}:=\left( h_{1},\ldots ,h_{S}\right) ^{\prime }$, i.e.

$$\begin{aligned}&\left. \begin{array}{l} \sigma _{nS+v}^{*}\left( \theta ^{*}\right) =\varphi _{v}^{*}\left( \epsilon _{nS+v-1},\epsilon _{nS+v-2},\ldots ;\theta _{v}^{*}\right) , \\ \widetilde{\sigma }_{nS+v}^{*}\left( \theta ^{*}\right) =\varphi _{v}^{*}\left( \epsilon _{nS+v-1},\epsilon _{nS+v-2},\ldots ,\widetilde{ \epsilon }_{0},\widetilde{\epsilon }_{-1},\ldots ;\theta _{v}^{*}\right) , \end{array} \right. \left. \begin{array}{l} 1\le v\le S, \\ n\in \mathbb {Z} , \end{array} \right. \\&\left. \widetilde{L}_{T,\underline{h}}^{*}\left( \theta ^{*}\right) =\tfrac{1}{NS}\sum \limits _{n=0}^{N-1}\sum \limits _{v=1}^{S}g_{v} \left( \epsilon _{nS+v},\widetilde{\sigma }_{nS+v}^{*}\left( \theta ^{*}\right) \right) ,\right. \end{aligned}$$

where $g_{v}~\left( 1\le v\le S\right)$ is defined as above and $\widetilde{\epsilon }_{0},\widetilde{\epsilon }_{-1},\ldots$ are fixed initial values. The GQMLE of $\theta ^{*}$ is then given by

$$\begin{aligned} \widehat{\theta }_{T,\underline{h}}^{*}=\arg \max _{\theta ^{*}\in {\varTheta }^{*}}\widetilde{L}_{T,\underline{h}}^{*}\left( \theta ^{*}\right) . \end{aligned}$$

Note finally that consistency and asymptotic normality of $\widehat{\theta } _{T,\underline{h}}^{*}$ are established in the same way as $\widehat{ \theta }_{T,\underline{h}}$ under the same assumptions A1– A10 with an appropriate adaptation considering $\theta ^{*}$ in place of $\theta$.

1.2 C.2. GQMLE for PCH models when the period S is non-integer

Next to high frequency seasonality, another well-observed case of complex periodic patterns is seasonality with a non-integer period. For example, many weekly series have an annual seasonal pattern with period $365.25/7\approx 52.179$ (e.g. De Livera et al. 2011). When a periodic model like (2.1) is fitted to a series characterized by a non-integer period $S\in \left( 1,\infty \right)$, one usually takes (by a naive approximation) the period to be the integer part of S, which is denoted by $\left[ S\right]$, where $\left[ S\right] =n\in \mathbb {N} ^{*}$ with $n\le S<n+1$. In doing so, the proposed $\left[ S\right]$ -periodic model in which

$$\begin{aligned} \theta _{0t}=\theta _{0,t+\left[ S\right] },\quad t\in \mathbb {Z} , \end{aligned}$$

will not reflect the actual S-periodicity of the series and will induce a kind of “shift” between the $\left[ S\right]$-seasonal series it generates and the actual S-seasonal series to which it is devoted to represent. Thus, a $\left[ S\right]$-periodic model will be inadequate. At first glance, it seems not possible to envisage a periodic model with non-integer S since the period actually represents the number of model parameters and hence it cannot take a priori non-integer values. However, we can exploit a variation of the trigonometric approximation (C.1) dealing with non-integer S. Indeed, in the framework of the PCH model (2.1) consider the following generalization of (C.1) given by

$$\begin{aligned} \left\{ \begin{array}{l} \theta _{0v}=\left( \theta _{0v,1},\ldots ,\theta _{0v,m_{v}}\right) ^{\prime } \\ \theta _{0v,j}=\theta _{0j}^{*}+\theta _{0j}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\theta _{0j}^{*}\right) , \quad 1\le j\le m_{v} \end{array} \right. ,\quad 1\le v\le \left[ S\right] , \end{aligned}$$

(C.4)

where S is now assumed a positive real number. In particular, for the PAP -$GARCH\left( 1,1\right)$ model (4.4), the corresponding “augmented” specification of (C.2) with non-integer period is

$$\begin{aligned} \left\{ \begin{array}{l} \omega _{0v}=\omega _{01}^{*}+\omega _{02}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\omega _{03}^{*}\right) \\ \alpha _{0v+}=\alpha _{01+}^{*}+\alpha _{02+}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\alpha _{03+}^{*}\right) \\ \alpha _{0v-}=\alpha _{01-}^{*}+\alpha _{02-}^{*}\cos \left( \tfrac{ 2\pi v}{S}-\alpha _{03-}^{*}\right) \\ \beta _{0v}=\beta _{01}^{*}+\beta _{02}^{*}\cos \left( \tfrac{2\pi v }{S}-\beta _{03}^{*}\right) \end{array} \right. \text {, }\quad 1\le v\le \left[ S\right] . \end{aligned}$$

(C.5)

A similar approach has been introduced by De Livera et al. (2011) in the case of seasonal (but non-periodic) exponential smoothing ${\textit{TBATS}}$ models (The acronym ${\textit{TBATS}}$ refers to: Trigonometric Box-Cox transform, ${\textit{ARMA}}$ errors, Trend, and Seasonal components). But in contrast with seasonal models, the period S in a periodic model is generally interpreted as the number of model parameters, making the adaptation of periodic models to non-integer periods more challenging. Note that if S is non-integer then model (C.4) (and hence model (C.5)) is not $\left[ S\right]$-periodic over v, since for example

$$\begin{aligned} \omega _{0,v+\left[ S\right] }= & {} \omega _{01}^{*}+\omega _{02}^{*}\cos \left( \tfrac{2\pi v}{S}-\omega _{03}^{*}+2\pi \tfrac{\left[ S \right] }{S}\right) \\\ne & {} \omega _{0v}, \end{aligned}$$

and so on. So specification (C.4) avoids inducing the aforementioned shift in modeling like model (2.1) and then it would be more suitable in representing non-integer periodicity.

Now with specification (C.4), model (2.1) may be reparametrized as in (C.3) to deal with non-integer periods, giving the following variation of (C.3) for a positive real period $S>0,$

$$\begin{aligned} \left\{ \begin{array}{l} \epsilon _{nS+v}=\sigma _{nS+v}\eta _{nS+v}, \\ \sigma _{nS+v}=\varphi _{v}^{*}\left( \epsilon _{nS+v-1},\epsilon _{nS+v-2},\ldots ;\theta _{0v}^{*}\right) :=\sigma _{nS+v}^{*}\left( \theta _{0}^{*}\right) , \end{array} \right. ,\quad 1\le v\le \left[ S\right] , \end{aligned}$$

(C.6)

where the function $\varphi _{v}^{*}$ and $\theta _{0}^{*}$ are defined as in (C.3). The GQMLE of (C.6) is defined in the same way as (C.3) and its properties are established under similar assumptions.

Note finally that other trigonometric (or more generally Fourier) approximations can be considered in place of (C.4) (see e.g. Tesfaye et al. 2011; Franses and Paap 2011).