[Paper Review] Parameter stability and semiparametric inference in time-varying ARCH models

Lionel Truquet|arXiv (Cornell University)|Jun 9, 2015

Financial Risk and Volatility Modeling21 references3 citations

TL;DR

This paper develops a semiparametric methodology for detecting time-varying versus time-invariant parameters in ARCH models using kernel estimation. It establishes that non-time-varying parameters can be estimated at the parametric rate, and for Gaussian noise, achieves semiparametric efficiency; two tests are proposed for parameter stability and second-order dynamics, supported by a lag selection criterion and real data validation.

ABSTRACT

In this paper, we develop a complete methodology for detecting time-varying/non time-varying parameters in ARCH processes. For this purpose, we estimate and test various semiparametric versions of the time-varying ARCH model (tv-ARCH) which include two well known non stationary ARCH type models introduced in the econometric literature. Using kernel estimation, we show that non time-varying parameters can be estimated at the usual parametric rate of convergence and for a Gaussian noise, we construct estimates that are asymptotically efficient in a semiparametric sense. Then we introduce two statistical tests which can be used for detecting non time-varying parameters or for testing the second order dynamic. An information criterion for selecting the number of lags is also provided. We illustrate our methodology with several real data sets.

Motivation & Objective

• To develop a comprehensive statistical framework for detecting whether parameters in ARCH processes are time-varying or time-invariant.
• To estimate non-time-varying parameters in semiparametric time-varying ARCH models (tv-ARCH) with optimal convergence rates.
• To construct asymptotically efficient estimators under Gaussian noise assumptions in a semiparametric setting.
• To propose two formal statistical tests for parameter stability and second-order dynamics in tv-ARCH models.
• To provide an information criterion for selecting the optimal number of lags in the model specification.

Proposed method

• Kernel-based nonparametric estimation is used to model time-varying coefficients in the ARCH process.
• The method allows for consistent estimation of non-time-varying parameters at the standard parametric rate of convergence, despite semiparametric modeling.
• For Gaussian noise, the proposed estimator achieves semiparametric efficiency by minimizing asymptotic variance within the semiparametric model class.
• Two test statistics are derived: one for testing whether a parameter is time-invariant, and another for assessing the second-order dynamic structure.
• A model selection criterion based on information theory is introduced to determine the optimal number of lags in the conditional variance equation.
• The methodology is applied to real financial time series data to demonstrate empirical relevance and robustness.

Experimental results

Research questions

• RQ1 Can non-time-varying parameters in tv-ARCH models be estimated at the parametric rate of convergence using semiparametric techniques?
• RQ2 Under what conditions can semiparametric estimators in tv-ARCH models achieve asymptotic efficiency?
• RQ3 How can one formally test whether a parameter in a tv-ARCH model is time-invariant?
• RQ4 What is an effective method for testing the second-order dynamic structure of a tv-ARCH process?
• RQ5 How should the number of lags be selected in a tv-ARCH model to balance model fit and complexity?

Key findings

• Non-time-varying parameters in the tv-ARCH model can be estimated at the standard parametric rate of convergence using kernel smoothing.
• For Gaussian noise, the proposed estimator achieves semiparametric efficiency, meaning it attains the lowest possible asymptotic variance in the semiparametric model class.
• The proposed test for parameter stability is asymptotically valid and can detect time-invariant parameters with correct size and power under regularity conditions.
• The second-order dynamic test effectively distinguishes between different conditional variance dynamics in tv-ARCH models.
• The information criterion for lag selection performs well in finite samples and helps avoid overfitting.
• Empirical applications on real data sets confirm the practical utility and robustness of the proposed methodology.