On the ARCH model with stationary liquidity

Marko Voutilainen (), Pauliina Ilmonen, Soledad Torres, Ciprian Tudor and Lauri Viitasaari
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Marko Voutilainen: Aalto University School of Science
Pauliina Ilmonen: Aalto University School of Science
Soledad Torres: CIMFAV - Facultad de Ingeniería, Universidad de Valparaíso
Ciprian Tudor: UFR Mathématiques, Université de Lille 1
Lauri Viitasaari: University of Helsinki

Metrika: International Journal for Theoretical and Applied Statistics, 2021, vol. 84, issue 2, No 4, 195-224

Abstract: Abstract The classical ARCH model together with its extensions have been widely applied in the modeling of financial time series. We study a variant of the ARCH model that takes account of liquidity given by a positive stationary process. We provide minimal assumptions that ensure the existence and uniqueness of the stationary solution for this model. Moreover, we give necessary and sufficient conditions for the existence of the autocovariance function. After that, we derive an AR(1) characterization for the stationary solution yielding Yule–Walker type quadratic equations for the model parameters. In order to define a proper estimation method for the model, we first show that the autocovariance estimators of the stationary solution are consistent under relatively mild assumptions. Consequently, we prove that the natural estimators arising out of the quadratic equations inherit consistency from the autocovariance estimators. Finally, we illustrate our results with several examples and a simulation study.

Keywords: ARCH model; Stationarity; Estimation; Consistency; 60G10; 62M10; 62G05 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s00184-020-00779-x

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