基于适应性函数系数ARCH-M模型的风险厌恶度量
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摘要
如何度量风险厌恶一直是经济学者和统计学者研究的课题。有些学者已经研究了风险厌恶度量问题,但是很少有人研究风险厌恶与单个经济变量的关系,特别是与多个经济变量的关系。本文就是要研究总体风险厌恶与多个经济变量的关系,由于出现了“维数祸根”问题,所以我们提出了适应性函数系数ARCH-M模型并用来度量风险厌恶。我们利用局部多项式估计方法分二步对参数进行估计,第一步利用局部极大似然估计法估计函数系数;第二步在函数系数已知的情况下,利用One-step Newton-Rephson寻找其它参数的最优估计并给出了窗宽选取方法和算法的执行过程。本文还研究了参数的相合性和渐进正态性,并在附录中给出了证明。最后,我们对所提出的模型及其估计方法做了数值模拟,发现其结果还是令人满意的。在此基础上,我们对美国股票市场做了实证研究,并对其结果做了相应的分析和解释。
How to measure the risk aversion is a task on which economists and statisticians always do research. Some scholars have studied the problem of measuring the risk aversion, however, few have conducted research on the relationship between the risk aversion and an economical variable, especially several economical variables. In the present thesis, we will study relationship between the aggregate risk aversion and economical variables. Due to the problem of "curse of dimensionality", we propose an adaptive Function-Coefficient ARCH-M model to measure the risk aversion. we apply local polynomial method to estimate the parameters in two steps, in which the first step estimate is based on local maximum likelihood method to estimate functional-coefficient, and the second step estimate, we employ a one-step Newton-Raphson estimation to search for some other parametrical estimations with the functional-coefficient fixed. We also give the ways to select the bandwidth and implementation of the arithmetic. The thesis also studies the consistency and asymptotic normality of the estimators. We give their proof in the appendix. Finally, we do simulation about the proposed model and find that the results of simulation performs well. Based on these, we apply the proposed model to analyze America stock market to measure risk aversion and explain the estimated results.
How to measure the risk aversion is a task on which economists and statisticians always do research. Some scholars have studied the problem of measuring the risk aversion, however, few have conducted research on the relationship between the risk aversion and an economical variable, especially several economical variables. In the present thesis, we will study relationship between the aggregate risk aversion and economical variables. Due to the problem of "curse of dimensionality", we propose an adaptive Function-Coefficient ARCH-M model to measure the risk aversion. we apply local polynomial method to estimate the parameters in two steps, in which the first step estimate is based on local maximum likelihood method to estimate functional-coefficient, and the second step estimate, we employ a one-step Newton-Raphson estimation to search for some other parametrical estimations with the functional-coefficient fixed. We also give the ways to select the bandwidth and implementation of the arithmetic. The thesis also studies the consistency and asymptotic normality of the estimators. We give their proof in the appendix. Finally, we do simulation about the proposed model and find that the results of simulation performs well. Based on these, we apply the proposed model to analyze America stock market to measure risk aversion and explain the estimated results.
引文
[1]黄奇辅,李兹森伯格.金融经济学基础[M].清华大学出版社,2004.
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[9]Chou,R.,Robert,F.,Engle,R.F.and Kane,A.Measuring risk aversion from excess return on a stock index[J].Journal of Econometrics,1992,52,201-224.
[10]Michael,W.B.,Kevin and Wang,Q.Time-varying risk aversion and unexpected inflation[J].Journal of Monetary Economics 2003,50,1457-1498.
[11]李元,张兴发.基于半参数ARCH-M模型的风险厌恶度量[J].广州大学学报,2008,3,1-5.
[12]Hastie,T.,Tibshirani,R.Generalized Additive Models[M].Chapman and Hall,London,1990.
[13]Green,P.J.,Silverman,B.W.Nonparametric Regression and Generalized Linear Models:a Roughness Penalty Approach[J].Chapman and Hall,London,1994.
[14]Wand,M.R,Jones,M.C.Kernel Smoothing[M].Chapman and Hall,London,1995.
[15]Fan,J.and Gijbels,I.Local Polynomial Modelling and Its Applications[M].Chapman and Hall,London,1996.
[16] Fan, J..Yao, Q., Cai, Z. Adaptive varying-coefficient linear models[J]. J. Roy. Statist. Soc. B 2003, 65, 57-80.
[17] Fan, J., and Chen, J. One-step local quasi-likelihood estimation[J]. J. Roy. Statist. Soc. B, 1999,61,927-943.
[18] Wahba, G. A survey of some smoothing problems and the method of generalized cross-validation for soving them [J]. In Applications of Statistics (ed. P. R. Krisnaiah), Amesterdam: North-Holland, 1977, 507-523.
[19] Craven, P. and Wahba, G. Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation [J]. Numerische Mathematik, 1979, 31, 377-403.
[20] Doukhan, P. Mixing[M]. Springer-Verlag, New York, 1994.
[21] Volkonskii,V.A.and Rozanov,Yu.A. Some limit theorems for random functions[J]. Theory Prob. Appl. 1959,4, 178-197.
[22] Shao Q and Yu H. Weak convergence for weighted empirical processes of dependent sequences[JJ. The Annals of Probability. 1996, 24, 2098-2127.
[2]Arrow,K.Essays in the Theory of Risk-Bearing[M].Amsterdam:North-Holland,1970.
[3]Pratt,J.Risk aversion in the small and in the large[J].Econometrica 1964,32,122-136.
[4]Engle,R.F.Autoregressive condictional heteroscedasticity with estimates of the variance of U.K.inflation[J].Econometrica,1982,50,987-1008.
[5]Bollerslev,T.Generalized autoregressive conditional heteroscedasticity[J].Journal of Econometrics,1986,31,307-327.
[6]Engle,R.F.,Lilien,D.and Robins,R.Estimating time varying risk premia in the term structure:the ARCH-M model[J].Econometrica,1987,55,391-407.
[7]Fama,E.and Macbeth G.Risk return and equilibrium:Empirical test[J].Journal of Political Economy,1973,81,607-636.
[8]Black,F.,M.Jenson,and M.Scholes.The capital asset pricing model:Some empirical test,reprinted in:M.Jensen,ed.,Studies in the theory of capital markets [J].Praeger,New York,NY,1972,79-124.
[9]Chou,R.,Robert,F.,Engle,R.F.and Kane,A.Measuring risk aversion from excess return on a stock index[J].Journal of Econometrics,1992,52,201-224.
[10]Michael,W.B.,Kevin and Wang,Q.Time-varying risk aversion and unexpected inflation[J].Journal of Monetary Economics 2003,50,1457-1498.
[11]李元,张兴发.基于半参数ARCH-M模型的风险厌恶度量[J].广州大学学报,2008,3,1-5.
[12]Hastie,T.,Tibshirani,R.Generalized Additive Models[M].Chapman and Hall,London,1990.
[13]Green,P.J.,Silverman,B.W.Nonparametric Regression and Generalized Linear Models:a Roughness Penalty Approach[J].Chapman and Hall,London,1994.
[14]Wand,M.R,Jones,M.C.Kernel Smoothing[M].Chapman and Hall,London,1995.
[15]Fan,J.and Gijbels,I.Local Polynomial Modelling and Its Applications[M].Chapman and Hall,London,1996.
[16] Fan, J..Yao, Q., Cai, Z. Adaptive varying-coefficient linear models[J]. J. Roy. Statist. Soc. B 2003, 65, 57-80.
[17] Fan, J., and Chen, J. One-step local quasi-likelihood estimation[J]. J. Roy. Statist. Soc. B, 1999,61,927-943.
[18] Wahba, G. A survey of some smoothing problems and the method of generalized cross-validation for soving them [J]. In Applications of Statistics (ed. P. R. Krisnaiah), Amesterdam: North-Holland, 1977, 507-523.
[19] Craven, P. and Wahba, G. Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation [J]. Numerische Mathematik, 1979, 31, 377-403.
[20] Doukhan, P. Mixing[M]. Springer-Verlag, New York, 1994.
[21] Volkonskii,V.A.and Rozanov,Yu.A. Some limit theorems for random functions[J]. Theory Prob. Appl. 1959,4, 178-197.
[22] Shao Q and Yu H. Weak convergence for weighted empirical processes of dependent sequences[JJ. The Annals of Probability. 1996, 24, 2098-2127.