Semiparametric efficient adaptive estimation of the GJR-GARCH model

Nicola Ciccarelli

doi:10.1515/strm-2017-0015

Article

Semiparametric efficient adaptive estimation of the GJR-GARCH model

Nicola Ciccarelli

Published/Copyright: August 9, 2018

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From the journal Statistics & Risk Modeling Volume 35 Issue 3-4

Abstract

In this paper we derive a semiparametric efficient adaptive estimator for the GJR-GARCH(1,1) model. We first show that the quasi-maximum likelihood estimator is consistent and asymptotically normal for the model used in analysis, and we secondly derive a semiparametric estimator that is more efficient than the quasi-maximum likelihood estimator. Through Monte Carlo simulations, we show that the semiparametric estimator is adaptive for the parameters included in the conditional variance of the GJR-GARCH(1,1) model with respect to the unknown distribution of the innovation.

Keywords: Semiparametric estimation; nonparametric inference; asymmetric volatility; GJR-GARCH

MSC 2010: 62F35; 62G07; 62F40

A Analytical solution for the location-scale score

In this section we report the analytical solution for

ψt⁢(θ0)=-(1+ηt⁢(θ0)⁢f′⁢(ηt⁢(θ0))f⁢(ηt⁢(θ0)))

for each zero mean and unit variance innovation ηt Below 𝟙{⋅} denotes the indicator function.

Normal(0,1) innovation:

ψt⁢(θ0)=-(1-ηt2).

Laplace(0,1) innovation:

ψt⁢(θ0)=-(1-10.5⁢ηt2).

Gaussian mixture innovation:

ψt⁢(θ0)=-1-ηt⁢5⁢{[exp⁡(-0.5⁢(5⁢ηt-2)2)⁢[-(5⁢ηt-2)]+exp⁡(-0.5⁢(5⁢ηt+2)2)⁢[-(5⁢ηt+2)]]exp⁡(-0.5⁢(5⁢ηt-2)2)+exp⁡(-0.5⁢(5⁢ηt+2)2)}.

t5-student innovation:

ψt⁢(θ0)=-(1-6⁢ηt25+ηt2).

t7-student innovation:

ψt⁢(θ0)=-(1-8⁢ηt27+ηt2).

t9-student innovation:

ψt⁢(θ0)=-(1-10⁢ηt29+ηt2).

t11-student innovation:

ψt⁢(θ0)=-(1-12⁢ηt211+ηt2).

t13-student innovation:

ψt⁢(θ0)=-(1-14⁢ηt213+ηt2).

χ62 innovation:

ψt⁢(θ0)=-(1-12⁢ηt+6⁢ηt212⁢ηt+6⁢𝟙{12ηt+6≥0}).

χ122 innovation:

ψt⁢(θ0)=-(1-24⁢ηt+12⁢ηt224⁢ηt+12⁢𝟙{24ηt+12≥0}).

B Proof of the LAN Theorem 2.3

The reparametrizated PTTGARCH(1,1) model in (2.3)–(2.4) fits into the general time-series framework of [5] since it is a general location-scale model in which the location-scale parameters only depend on the past. Therefore, in order to prove the LAN Theorem 2.3, it suffices to verify conditions (2.3’), (A.1) and (2.4) of [5]. We also prove condition (3.3’) of [5], which we will need in the proof of Theorem 2.5. With the notation introduced in Section 2.4 (see (2.9), (2.12), (2.13)), and denoting the expectation under θ of the product ψ⁢(θ)⁢ψ⁢(θ)′ as Il⁢s⁢(f), we need to show, under θ0,

1n⁢∑t=1nWt⁢(θ0)⁢Il⁢s⁢(f)⁢Wt⁢(θ0)′⁢⟶𝑃⁢I⁢(θ0)>0,

(B.1)1n⁢∑t=1n|Wt⁢(θ0)|2⁢𝟙{n-12|Wt(θ0)|>δ}⁢⟶𝑃⁢0,

(B.2)1n⁢∑t=1nWt⁢(θ0)⁢⟶𝑃⁢W⁢(θ0),

(B.3)1n⁢∑t=1n|Wt⁢(θn)-Wt⁢(θ0)|2⁢⟶𝑃⁢0,

and, under θn,

(B.4)∑t=1n|n-12⁢(Mn⁢t,Sn⁢t)′-Wt⁢(θn)′⁢(θ~n-θn)|2⁢⟶𝑃⁢0

for some positive definitive matrix I⁢(θ0) and some random matrix W⁢(θ0). If equations (B.1)–(B.4) are satisfied, and using [5, Lemma A.1], we can show that the PTTGARCH(1,1) model satisfies the LAN property.

Although Wt⁢(θ0) may not be strictly stationary and ergodic under θ0, the following proposition shows that these variables can be approximated by a strictly stationary and ergodic sequence, and hence equations (B.1), (B.2) and (B.3) hold in general.

Proposition B.1.

Let ht⁢(θ), Ht⁢(θ) and Wt⁢(θ) be given by (2.8), (2.11), (2.12), respectively, and let hst⁢(θ), Hst⁢(θ) and Wst⁢(θ) be their corresponding stationary solutions under θ, i.e.

hst⁢(θ)=∑j=0∞∏k=1j{β+α+⁢(ξt-k+)2+α-⁢(ξt-k-)2},

Hst⁢(θ)=∑i=0∞βi⁢hs,t-1-i⁢(θ)⁢((ξt-1-i+)2(ξt-1-i-)21),

Wst⁢(θ)=σ-1⁢(12⁢hst-1⁢(θ)⁢Hst⁢(θ)⁢(μ,σ)I2).

Then, under θ0,

1n⁢∑t=1n|Wt⁢(θ0)-Wst⁢(θ0)|2→0 (a.s.) as ⁢n→∞.

Proof.

The proof follows along the same lines as [4, proof of Proposition A.1].^[27]∎

In the following proposition, which is parallel to [4, Proposition A.2], we show that slight perturbations of the parameters yield solutions of equations (2.3) and (2.4) that are close.

Proposition B.2.

Let ht⁢(θ) and Ht⁢(θ) be given by (2.8) and (2.11), respectively, and define

Qt⁢(θ)=Ht⁢(θ)ht⁢(θ)=∑i=0t-2βiht⁢(θ)⁢((Yt-1-i+)2(Yt-1-i-)2ht-1-i⁢(θ))=∑i=0t-2βi⁢ht-1-i⁢(θ)ht⁢(θ)⁢((ξt-1-i+⁢(θ))2(ξt-1-i-⁢(θ))21)

and

Rt⁢(θ,θ~)=ht⁢(θ~)ht⁢(θ)-1-(α~+-α,α~--α,β~-β)⁢Qt⁢(θ).

Let θn and θ~n satisfy the conditions just above (2.6). Put Qn⁢t=Qt⁢(θn) and Rn⁢t=Rt⁢(θn,θ~n). Then, under θn,

1n⁢∑t=1n|Qn⁢t|2=Op⁢(1),1n⁢∑t=1n|Qn⁢t|2⁢𝟙{n-1/2|Qn⁢t|>δ}→0 (a.s.) as ⁢n→∞

and

∑t=1nRn⁢t2→0 (a.s.) as ⁢n→∞.

Proof.

The proof follows along the same lines as [4, proof of Proposition A.2].^[28] ∎

Using Propositions B.1–B.2, and following the same line of reasoning reported in [4, p. 218], it is straightforward to prove (B.1), (B.2) and (B.4). Finally, we have to prove (B.3). Note that

|Wt⁢(θn)-Wt⁢(θ0)|2≤C⁢|Qt⁢(θn)-Qt⁢(θ0)|2+C⁢|Qt⁢(θ0)|2⁢|θn-θ0|2,

and we obtain contiguity of Pθn and Pθ0 from (B.1) and (B.4), and [5, Theorem 2.1]. Then the required result is obtained from

Qt⁢(θ~)-Qt⁢(θ)=∑i=0t-2(β~i-βi)⁢ht-1-i⁢(θ~)ht⁢(θ~)⁢((ξt-1-i+⁢(θ~))2(ξt-1-i-⁢(θ~))21)+Qt⁢(θ)⁢{(θ1-θ~1)′⁢Qt⁢(θ~)+Rt⁢(θ~,θ)}

-(00∑i=0t-2β~i⁢ht-1-i⁢(θ~)ht⁢(θ~)⁢{(θ1-θ~1)′⁢Qt-1-i⁢(θ~)+Rt-1-i⁢(θ~,θ)})

along the lines of [4, proofs of Propositions A.1–A.2]. This completes the proofs of the theorems in Section 2.4.

C Proof of Theorem 2.5

See [4, proof of Theorem 3.1].^[29]

Acknowledgements

We would like to thank the anonymous reviewers whose comments vastly improved this manuscript.

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Received: 2017-04-28

Revised: 2018-06-11

Accepted: 2018-07-19

Published Online: 2018-08-09

Published in Print: 2018-12-01

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Articles in the same Issue

https://doi.org/10.1515/strm-2017-0015

Keywords for this article

Semiparametric estimation; nonparametric inference; asymmetric volatility; GJR-GARCH