Conditional asymmetry in Power ARCH($\infty$) models

Royer, Julien (2021): Conditional asymmetry in Power ARCH($\infty$) models.

Abstract

We consider an extension of ARCH($\infty$) models to account for conditional asymmetry in the presence of high persistence. After stating existence and stationarity conditions, this paper develops the statistical inference of such models and proves the consistency and asymptotic distribution of a Quasi Maximum Likelihood estimator. Some particular specifications are studied and we introduce a Portmanteau test of goodness-of-fit. In addition, test procedures for asymmetry and GARCH validity are derived. Finally, we present an application on a set of equity indices to reexamine the preeminence of GARCH(1,1) specifications. We find strong evidence that the short memory feature of such models is not suitable for peripheral assets.

• Item Type: MPRA Paper
• Language: English
• Keywords: Quasi Maximum Likelihood Estimation, Moderate memory, Testing parameters on the boundary, Recursive design bootstrap
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

D. W. Andrews. Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica, 69(3):683–734, 2001.

J.-M. Bardet and O. Wintenberger. Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes. The Annals of Statistics, 37(5B):2730–2759, 2009.

I. Berkes, L. Horv\'ath, and P. Kokoszka. Asymptotics for GARCH squared residual correlations. Econometric Theory, pages 515–540, 2003.

I. Berkes, L. Horv\'ath, and P. Kokoszka. GARCH processes: Structure and estimation. Bernoulli, 9(2):201–227, 2003.

M. Bernardi and L. Catania. The model confidence set package for R. International Journal of Computational Economics and Econometrics, 8(2):144–158, 2018.

E. Beutner, A. Heinemann, and S. Smeekes. A residual bootstrap for conditional value-at-risk. arXiv preprint arXiv:1808.09125, 2018.

P. Billingsley. The Lindeberg-Lévy theorem for martingales. Proceedings of the American Mathematical Society, 12(5):788–792, 1961.

P. Billingsley. Probability and measure. John Wiley & Sons, 3rd edition, 1995.

T. Bollerslev and H. Ole Mikkelsen. Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73(1):151–184, 1996.

G. E. Box and D. A. Pierce. Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American statistical Association, 65(332):1509–1526, 1970.

M. Carbon and C. Francq. Portmanteau goodness-of-fit test for asymmetric power GARCH models. Austrian Journal of Statistics, 40(1&2):55–64, 2011.

G. Cavaliere, H. B. Nielsen, R. S. Pedersen, and A. Rahbek. Bootstrap inference on the boundary of the parameter space, with application to conditional volatility models. Journal of Econometrics, 2020.

Z. Ding, C. W. Granger, and R. F. Engle. A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1):83–106, 1993.

R. Douc, F. Roueff, and P. Soulier. On the existence of some ARCH($\infty$) processes. Stochastic Processes and their Applications, 118(5):755–761, 2008.

C. Francq and L. Q. Thieu. QML inference for volatility models with covariates. Econometric Theory, 35(1):37–72, 2019.

C. Francq and J.-M. Zako\"ian. Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli, 10(4):605–637, 2004.

C. Francq and J.-M. Zako\"ian. Quasi-maximum likelihood estimation in garch processes when some coefficients are equal to zero. Stochastic Processes and their Applications, 117(9):1265–1284, 2007.

C. Francq and J.-M. Zako\"ian. Testing the nullity of GARCH coefficients: correction of the standard tests and relative efficiency comparisons. Journal of the American Statistical Association, 104(485):313–324, 2009.

C. Francq and J.-M. Zako\"ian. GARCH models: structure, statistical inference and financial applications. John Wiley & Sons, 2nd edition, 2019.

R. Giacomini, D. N. Politis, and H. White. A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric theory, 29(3):567–589, 2013.

L. Giraitis, P. Kokoszka, and R. Leipus. Stationary ARCH models: Dependence structure and central limit theorem. Econometric Theory, 16(1):3–22, 2000.

L. Giraitis and P. M. Robinson. Whittle estimation of ARCH models. Econometric Theory, 17(3):608–631, 2001.

L. Giraitis, D. Surgailis, and A. Škarnulis. Stationary integrated ARCH($\infty$) and AR($\infty$) processes with finite variance. Econometric Theory, 34(6):1159–1179, 2018.

L. R. Glosten, R. Jagannathan, and D. E. Runkle. On the relation between the expected value and the volatility of the nominal excess return on stocks. The journal of finance, 48(5):1779–1801, 1993.

G. González-Rivera, T.-H. Lee, and S. Mishra. Forecasting volatility: A reality check based on option pricing, utility function, value-at-risk, and predictive likelihood. International Journal of forecasting, 20(4):629–645, 2004.

C. Gouriéroux and A. Monfort. Statistics and Econometric Models, volume 2 of Themes in Modern Econometrics. Cambridge University Press, 1995.

C. M. Hafner and A. Preminger. On asymptotic theory for ARCH ($\infty$) models. Journal of Time Series Analysis, 38(6):865–879, 2017.

T. Hamadeh and J.-M. Zako\"ian. Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes. Journal of Statistical Planning and Inference, 141(1):488–507, 2011.

P. R. Hansen, A. Lunde, and J. M. Nason. The model confidence set. Econometrica, 79(2):453–497, 2011.

J. Hidalgo and P. Zaffaroni. A goodness-of-fit test for ARCH($\infty$) models. Journal of econometrics, 141(2):973–1013, 2007.

V. Kazakevičius and R. Leipus. On stationarity in the ARCH($\infty$) model. Econometric Theory, 18(1):1–16, 2002.

P. Kupiec. Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 3(2), 1995.

W. K. Li and T. Mak. On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. Journal of Time Series Analysis, 15(6):627–636, 1994.

O. Linton and E. Mammen. Estimating semiparametric ARCH($\infty$) models by kernel smoothing methods. Econometrica, 73(3):771–836, 2005.

D. B. Nelson. Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2):347–370, 1991.

P. M. Robinson. Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics, 47(1):67–84, 1991.

P. M. Robinson and P. Zaffaroni. Pseudo-maximum likelihood estimation of ARCH($\infty$) models. The Annals of Statistics, 34(3):1049–1074, 2006.

P. Zaffaroni. Stationarity and memory of ARCH($\infty$) models. Econometric theory, pages 147–160, 2004.

P. Zaffaroni. Whittle estimation of EGARCH and other exponential volatility models. Journal of econometrics, 151(2):190–200, 2009.

J.-M. Zako\"ian. Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18(5):931–955, 1994.