[Paper Review] Local asymptotically optimal test in ARCH model

Tewfik Lounis|arXiv (Cornell University)|Jun 7, 2012

Financial Risk and Volatility Modeling12 references3 citations

TL;DR

This paper develops a locally asymptotically optimal (LAO) test for conditional heteroskedasticity in ARCH models by establishing local asymptotic normality (LAN) and constructing an optimal test under known parameters. It introduces a modified discrete estimator to maintain optimality when parameters are unknown, proving asymptotic efficiency under general stationarity and ergodicity conditions.

ABSTRACT

This work is an extension in Arch models of the theorem of S.Y. Hwang and I.V. Basawa Hwang and Basawa (2001) which was used before in nonlinear time series contiguous to AR(1) processes. Our results are established under some general assumptions and stationarity and ergodicity conditions. Local asymptotic normality (LAN) for the log likelihood ratio was established.An optimal test was constructed when the parameter is assumed known. Also the optimality of our test was proved when the parameter is unspecified. The method is based on the introducing of a new estimator.

Motivation & Objective

• To extend the LAO testing framework from AR(1) processes to general ARCH models.
• To establish local asymptotic normality (LAN) for the log-likelihood ratio in ARCH models under general regularity conditions.
• To construct an optimal test when the true parameter is known.
• To prove the optimality of the test when the parameter is unknown by introducing a modified discrete estimator.
• To ensure asymptotic efficiency of the test under stationarity, ergodicity, and moment conditions.

Proposed method

• Establishes LAN for the log-likelihood ratio in a general class of time series models including ARCH.
• Uses the Neyman-Pearson lemma to construct an optimal test based on the log-likelihood ratio under known parameters.
• Applies discrete estimation techniques from Le Cam (1960) and Kreiss (1987) to handle unknown parameters.
• Introduces a modified estimator that absorbs estimation error in the central sequence to preserve asymptotic optimality.
• Employs Le Cam’s third lemma and contiguity theory to analyze asymptotic power and efficiency.
• Derives the asymptotic distribution of the test under local alternatives using the central sequence approximation.

Experimental results

Research questions

• RQ1 Can the LAO testing framework be extended from AR(1) processes to general ARCH models?
• RQ2 Under what conditions does the log-likelihood ratio in ARCH models satisfy local asymptotic normality (LAN)?
• RQ3 Is the optimal test constructed under known parameters still asymptotically optimal when the parameter is estimated?
• RQ4 How can estimation error in the central sequence be corrected to preserve test optimality?
• RQ5 What is the asymptotic power of the proposed test under local alternatives?

Key findings

• The paper establishes local asymptotic normality (LAN) for the log-likelihood ratio in ARCH models under general stationarity and ergodicity conditions.
• An optimal test is constructed when the true parameter (ρ₀, θ₀) is known, based on the central sequence derived from the LAN property.
• The test maintains asymptotic optimality even when parameters are unknown, thanks to a novel modified discrete estimator that corrects for estimation-induced bias in the central sequence.
• The asymptotic power of the test is derived and shown to be optimal under local alternatives of the form H₁⁽ⁿ⁾.
• For the Student-t distribution with degrees of freedom l > 3, the score functions ˙Mf, ¨Mf, and x¨Mf are bounded, ensuring regularity for the LAN derivation.
• The method is general and applies to a wide class of time series models, including AR, ARMA, SETAR, and β-ARCH, under mild moment and smoothness conditions.