一种识别动态因子模型的秩条件——基于Gaussian似然函数的视角
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摘要
为了提高高维动态因子模型识别的有效性,借鉴SVAR模型的结构分析方法,提出了通过隐性因子的新息ε_t来推断和识别动态因子模型正交结构冲击的分析过程.并且,根据动态因子模型的似然函数表示,通过信息矩阵推导出了动态因子模型识别的秩条件.秩条件仅仅依赖于因子的个数以及约束条件,而不依赖于数据的维数,易于在实际应用中验证动态因子模型的识别性.
In order to improve the identification effectiveness of high-dimensional dynamic factor model, analogously to structural VAR analysis, this paper proposes to infer and identify the dynamic factors structural shocks using the innovations of the factor variables. And according to the likelihood function of the dynamic factor models, we establish a kind of identification rank condition through the information matrix. The rank condition depends only on the number of factors and constraints of models without depending on the dimension of the data, and easy to verify the identification of the dynamic factor model in practical applications.
In order to improve the identification effectiveness of high-dimensional dynamic factor model, analogously to structural VAR analysis, this paper proposes to infer and identify the dynamic factors structural shocks using the innovations of the factor variables. And according to the likelihood function of the dynamic factor models, we establish a kind of identification rank condition through the information matrix. The rank condition depends only on the number of factors and constraints of models without depending on the dimension of the data, and easy to verify the identification of the dynamic factor model in practical applications.
引文
[1] Forni M, Hallin M, Lippi M, Reichlin L, The generaized dynamic factor model:identification and estimation[J]. Rev Econ Statist, 2000, 82(4):540-554.
[2] Anderson T W, Rubin H. Statistical inference in factor analysis[J]. In:Neyman J(Ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Univ. of California Press, 1956(5):111-150.
[3] Bekker P A. A note on the identification of restricted factor loading matrices[J]. Psychometrika,1986, 51(4):607-611.
[4] Lawley D N, Maxwell A E. Factor Anlysis as Statistical Method, 1971.
[5] Bekker P A, and Pollock D S G. Identification of Linear Stochastic Models with Covariance Restrictions[J]. Journal of Econometrics, 1986, 31:179-208.
[6] Neudecker H. On the identification of restricted factor loading matrices:an alternative condition[J].J Math Psychol, 1990, 34:237-241.
[7] Bai J, Ng S. Principal components estimation and identification of static factors[J]. J Econometrics,2013, 176:18-29.
[8] Bai J, and Wang P. Identification theory for high dimensional static and dynamic factor models[J].J Econometrics, 2014, 178:794-804.
[9] Geweke J. The dynamic factor analysis of economic time series. In Latent Variables in Socio Economic Models, Aigner D.J. and Goldberger A.S.(eds). North Holland, Amsterdam, 1977.
[10] Sargent T J, and Sims C A. Business Cycle Modeling Without Pretending to have too much a priori economic theory. FRB Working Paper#55, 1977.
[11] Doz C, Giannone D, and Reichlin L.. A quasi maximum likelihood approach for large approximate dynamic factor models[J]. Review of Economics and Statistics, 2012, 94, 4, 1014-1024.
[12] Stock J H, and Watson M W, Implications of dynamic factor models for VAR analysis. NBER Working Paper 2005:11467.
[13] Bai J. Inferential theory for factor models of large dimensions[J]. Econometrica, 2003, 71(1):135-172.
[14] Sims C A. Are Forecasting Models Usable for Policy Analysis?, Quarterly Review of the Federal Reserve Bank of Minneapolis, winter, 1986:2-16.
[15] Rothenberg T J. Identification in parametric models[J]. Econometrica, 1971, 39(3):577-591.
[17] Magnus J R. Linear Structures[J]. Griffin, London, 1988.
[18] Magnus J R, Neudecker H. Matrix Differential Calculus. Wiley, Chichester, 1999.
1.根据Stock&Watson(2005)、Bai&Ng(2002)、Bai(2003)以及Forni et al.(2000)等人的研究结论,这一结论是合理的.
2.在SVAR模型中的K-模型的结构识别假设和本文的假设1具有相同的形式.
3.对x为任-q(q-1)/2维向量,D_qx=vecW, W为某一斜对称矩阵,即D_qx为某一斜对称矩阵的向量化形式.由于N_q=(I_q2+K_q2),且K_q2变换矩阵,容易验证D_qx为N_qy=0的通解.
[2] Anderson T W, Rubin H. Statistical inference in factor analysis[J]. In:Neyman J(Ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Univ. of California Press, 1956(5):111-150.
[3] Bekker P A. A note on the identification of restricted factor loading matrices[J]. Psychometrika,1986, 51(4):607-611.
[4] Lawley D N, Maxwell A E. Factor Anlysis as Statistical Method, 1971.
[5] Bekker P A, and Pollock D S G. Identification of Linear Stochastic Models with Covariance Restrictions[J]. Journal of Econometrics, 1986, 31:179-208.
[6] Neudecker H. On the identification of restricted factor loading matrices:an alternative condition[J].J Math Psychol, 1990, 34:237-241.
[7] Bai J, Ng S. Principal components estimation and identification of static factors[J]. J Econometrics,2013, 176:18-29.
[8] Bai J, and Wang P. Identification theory for high dimensional static and dynamic factor models[J].J Econometrics, 2014, 178:794-804.
[9] Geweke J. The dynamic factor analysis of economic time series. In Latent Variables in Socio Economic Models, Aigner D.J. and Goldberger A.S.(eds). North Holland, Amsterdam, 1977.
[10] Sargent T J, and Sims C A. Business Cycle Modeling Without Pretending to have too much a priori economic theory. FRB Working Paper#55, 1977.
[11] Doz C, Giannone D, and Reichlin L.. A quasi maximum likelihood approach for large approximate dynamic factor models[J]. Review of Economics and Statistics, 2012, 94, 4, 1014-1024.
[12] Stock J H, and Watson M W, Implications of dynamic factor models for VAR analysis. NBER Working Paper 2005:11467.
[13] Bai J. Inferential theory for factor models of large dimensions[J]. Econometrica, 2003, 71(1):135-172.
[14] Sims C A. Are Forecasting Models Usable for Policy Analysis?, Quarterly Review of the Federal Reserve Bank of Minneapolis, winter, 1986:2-16.
[15] Rothenberg T J. Identification in parametric models[J]. Econometrica, 1971, 39(3):577-591.
[17] Magnus J R. Linear Structures[J]. Griffin, London, 1988.
[18] Magnus J R, Neudecker H. Matrix Differential Calculus. Wiley, Chichester, 1999.
1.根据Stock&Watson(2005)、Bai&Ng(2002)、Bai(2003)以及Forni et al.(2000)等人的研究结论,这一结论是合理的.
2.在SVAR模型中的K-模型的结构识别假设和本文的假设1具有相同的形式.
3.对x为任-q(q-1)/2维向量,D_qx=vecW, W为某一斜对称矩阵,即D_qx为某一斜对称矩阵的向量化形式.由于N_q=(I_q2+K_q2),且K_q2变换矩阵,容易验证D_qx为N_qy=0的通解.