Asymptotic theory for QMLE for power-transformed asymmetric double autoregressive models

Guobing Cui; Guobing Cui

doi:10.3934/math.2025419

AIMS Mathematics

2025, Volume 10, Issue 4: 9094-9121. doi: 10.3934/math.2025419

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Asymptotic theory for QMLE for power-transformed asymmetric double autoregressive models

Guobing Cui ^{1,2
,
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1.
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
2.
Department of Basic, Officers College of PAP, Chengdu, 610200, China

Received: 13 January 2025 Revised: 06 April 2025 Accepted: 09 April 2025 Published: 21 April 2025
MSC : 62F10, 62F12

To better capture asymmetry and heavy-tailedness, we have proposed a power-transformed asymmetric double autoregressive (PTADAR(p, q)) model. First we gave a sufficient condition for the existence of a strict stationarity solution of the PTADAR(p, q) model. Then we studied the quasi-maximum likelihood estimation (QMLE) of the model, and proved the consistency and asymptotic normality for the QMLE estimator. We set the power parameter $ \delta > 0 $, which includes $ \delta = 1, 2 $, and in empirical application, the power parameter $ \delta > 0 $ may be unknown. This could overcome the shortcomings of the double autoregressive (DAR(p, q)) model and asymmetry linear double autoregressive model, where the power parameter is only limited to 2 or 1. Based on QMLE, we proposed Akaike's information criterion (AIC) and Bayesian information criterion (BIC) for model selection. Illustrations and an empirical example show our model's usefulness, and we also compared it with other models.
- power-transformed asymmetric DAR model,
- power parameter,
- estimation,
- daily exchange rate
Citation: Guobing Cui. Asymptotic theory for QMLE for power-transformed asymmetric double autoregressive models[J]. AIMS Mathematics, 2025, 10(4): 9094-9121. doi: 10.3934/math.2025419

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Abstract

To better capture asymmetry and heavy-tailedness, we have proposed a power-transformed asymmetric double autoregressive (PTADAR(p, q)) model. First we gave a sufficient condition for the existence of a strict stationarity solution of the PTADAR(p, q) model. Then we studied the quasi-maximum likelihood estimation (QMLE) of the model, and proved the consistency and asymptotic normality for the QMLE estimator. We set the power parameter $ \delta > 0 $, which includes $ \delta = 1, 2 $, and in empirical application, the power parameter $ \delta > 0 $ may be unknown. This could overcome the shortcomings of the double autoregressive (DAR(p, q)) model and asymmetry linear double autoregressive model, where the power parameter is only limited to 2 or 1. Based on QMLE, we proposed Akaike's information criterion (AIC) and Bayesian information criterion (BIC) for model selection. Illustrations and an empirical example show our model's usefulness, and we also compared it with other models.

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