A combined estimation functions method for autoregressive model with time-varying variance
-
摘要:
针对参数估计问题,利用联合估计函数方法对带有时变方差的自回归模型参数进行统计研究。介绍了带有时变方差自回归模型和联合估计函数理论的研究现状,利用联合估计函数理论,给出带有时变方差自回归模型的参数估计量,证明该参数联合估计量渐近收敛于正态分布。对提出的参数统计量进行数值模拟对比分析,模拟结果表明,与伪极大似然估计量、最小二乘估计量进行对比,提出的参数联合估计量略优于伪极大似然估计量,同时该统计量受误差项分布函数影响较小。
Abstract:With regard to the problem of parameter estimation, the combined estimation functions method is used to carry out statistical research on the parameter of autoregressive model with time-varying variance. The research status of the autoregressive model with time-varying variance and the combined estimation functions theory is reported. The combined estimation functions theory is used to obtain the parameter estimators of the autoregressive model with time-varying variance, and it is proved that the parameter estimators of the combined estimation functions method asymptotically converge to normal distribution. The numerical simulation is carried out for the comparative analysis of the proposed parameters. The simulation results show that, compared with the quasi maximum likelihood estimators and the least squares estimators, the proposed parameter estimators of combined estimation functions are slightly better than those of quasi maximum likelihood estimation, and the statistic is less affected by the distribution function of error terms.
-
表 1 θ0=0.35时的估计结果
Table 1. Estimation results at θ0=0.35
指标 CEF方法 QMLE方法 LS方法 n=100 n=200 n=100 n=200 n=100 n=200 均值 0.346 3 0.345 8 0.344 8 0.345 9 0.309 1 0.319 0 偏差 0.003 7 0.004 2 0.005 2 0.004 1 0.041 8 0.031 5 方差 0.008 5 0.004 4 0.008 4 0.004 6 0.021 5 0.014 6 表 2 θ0=0.75时的估计结果
Table 2. Estimation results at θ0=0.75
指标 CEF方法 QMLE方法 LS方法 n=100 n=200 n=100 n=200 n=100 n=200 均值 0.739 5 0.743 9 0.737 4 0.743 7 0.577 8 0.631 7 偏差 0.010 5 0.006 1 0.012 6 0.006 3 0.172 2 0.118 3 方差 0.005 1 0.002 2 0.005 2 0.002 1 0.106 8 0.077 0 表 3 误差项服从不同分布时的估计结果
Table 3. Estimation results when error terms follow different distributions
指标 标准正态分布 指数分布 伽马分布 CEF方法 QMLE方法 CEF方法 QMLE方法 CEF方法 QMLE方法 均值 0.441 8 0.439 5 0.445 7 0.443 6 0.448 5 0.445 7 偏差 0.008 2 0.010 5 0.004 3 0.006 4 0.001 5 0.004 3 方差 0.006 5 0.006 4 0.004 0 0.007 3 0.005 9 0.008 2 表 4 θ0=0.35,σ02=1,σ12=2时的估计结果(基于式(5))
Table 4. Estimation results at θ0=0.35, σ02=1, σ12=2 (based on Formula (5))
指标 τ=0.2 τ=0.5 τ=0.8 CEF方法 QMLE方法 CEF方法 QMLE方法 CEF方法 QMLE方法 均值 0.346 9 0.345 2 0.344 9 0.344 6 0.341 7 0.339 9 偏差 0.003 1 0.004 8 0.005 1 0.005 4 0.008 3 0.010 1 方差 0.007 8 0.009 5 0.009 0 0.009 8 0.008 2 0.009 2 表 5 θ0=0.75,σ02=1,σ12=5时的估计结果(基于式(5))
Table 5. Estimation results at θ0=0.75, σ02=1, σ12=5 (based on Formula (5))
指标 τ=0.2 τ=0.5 τ=0.8 CEF方法 QMLE方法 CEF方法 QMLE方法 CEF方法 QMLE方法 均值 0.739 2 0.736 5 0.734 5 0.734 3 0.734 4 0.733 6 偏差 0.010 8 0.013 5 0.015 5 0.015 7 0.015 6 0.016 4 方差 0.006 0 0.005 0 0.004 9 0.004 7 0.004 9 0.005 0 表 6 θ0=0.35,σ02=1,σ12=2时的估计结果(基于式(6))
Table 6. Estimation results at θ0=0.35, σ02=1, σ12=2 (based on Formula (6))
指标 τ=0.2 τ=0.5 τ=0.8 CEF方法 QMLE方法 CEF方法 QMLE方法 CEF方法 QMLE方法 均值 0.344 2 0.343 0 0.345 1 0.343 9 0.346 1 0.344 8 偏差 0.005 8 0.007 0 0.004 9 0.006 1 0.003 9 0.005 2 方差 0.008 3 0.008 4 0.008 5 0.008 4 0.007 9 0.008 7 表 7 θ0=0.75,σ02=1,σ12=5时的估计结果(基于式(6))
Table 7. Estimation results at θ0=0.75, σ02=1, σ12=5 (based on Formula (6))
指标 τ=0.2 τ=0.5 τ=0.8 CEF方法 QMLE方法 CEF方法 QMLE方法 CEF方法 QMLE方法 均值 0.740 7 0.739 4 0.739 1 0.737 9 0.739 7 0.738 6 偏差 0.009 3 0.010 6 0.010 9 0.012 1 0.010 3 0.011 4 方差 0.005 0 0.007 8 0.004 7 0.004 9 0.004 8 0.004 9 -
[1] CAVALIERE G, TAYLOR A M R. Testing for unit roots in time series models with non-stationary volatility[J]. Journal of Econometrics, 2007, 140(2): 919-947. doi: 10.1016/j.jeconom.2006.07.019 [2] PHILLIPS P C B, XU K L. Inference in autoregression under heteroskedasticity[J]. Journal of Time, 2010, 27(2): 289-308. [3] XU K L, PHILLIPS P C B. Adaptive estimation of autoregressive models with time-varying variances[J]. Journal of Econometrics, 2008, 142(1): 265-280. doi: 10.1016/j.jeconom.2007.06.001 [4] VALENTIN P, HAMDI R. Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean[J]. Journal of Statistical Planning and Inference, 2012, 142(11): 2891-2912. doi: 10.1016/j.jspi.2012.04.005 [5] GODAMBE V P. An optimum property of regular maximum likelihood estimation[J]. Annals of Mathematical Stats, 1960, 31(4): 1208-1211. doi: 10.1214/aoms/1177705693 [6] GODAMBE V P. The foundations of finite sample estimation in stochastic processes[J]. Biometrika, 1985, 72(2): 419-428. doi: 10.1093/biomet/72.2.419 [7] THAVANESWARAN A, HEYDE C C. Prediction via estimating functions[J]. Journal of Statistical Planning and Inference, 1999, 77(1): 89-101. doi: 10.1016/S0378-3758(98)00179-7 [8] NG K H, PEIRIS S, RICHARD G. Estimation and forecasting with logarithmic autoregressive conditional duration models: A comparative study with an application[J]. Expert Systems with Applications, 2014, 41(7): 3323-3332. doi: 10.1016/j.eswa.2013.11.024 [9] GHAHRAMANI M, THAVANESWARAN A. Combining estimating functions for volatility[J]. Journal of Statistical Planning and Inference, 2008, 139(4): 1449-1461. [10] LIANG Y, THAVANESWARAN A, ABRAHAM B. Joint estimation using quadratic estimating function[J]. Journal of Probability and Statistics, 2011, 2011: 372512. [11] THAVANESWARAN A, LIANG Y, FRANK J. Inference for random coefficient volatility models[J]. Statistics and Probability Letters, 2012, 82(12): 2086-2090. doi: 10.1016/j.spl.2012.07.008 [12] THAVANESWARAN A, RAVISHANKER N, LIANG Y. Generalized duration models and optimal estimation using estimating functions[J]. Annals of the Institute of Statistical Mathematics, 2015, 67(1): 129-156. doi: 10.1007/s10463-013-0442-9 [13] GHAHRAMANI M, THAVANESWARAN A. Nonlinear recursive estimation of volatility via estimating functions[J]. Journal of Statistical Planning and Inference, 2012, 142(1): 171-180. doi: 10.1016/j.jspi.2011.07.006 [14] ZHANG Y, ZOU J, RAVISHANKER N, et al. Modeling financial durations using penalized estimating functions[J]. Computational Statistics & Data Analysis, 2019, 131(C): 145-158. [15] RAMANATHAN T, ANUJ M, BOVAS A. Estimation, filtering and smoothing in the stochastic conditional duration model: An estimating function approach[J]. Stat, 2016, 5(1): 11-21. doi: 10.1002/sta4.101