结构时间序列模型在季节调整中的理论分析与应用研究
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摘要
经济序列的季节调整问题,一直都是各国统计和计量经济领域关注的焦点。为了有效的考察经济数据变化规律,解释数据背后的经济意义,如何准确、完整的分解出实际经济序列的趋势信息和季节信息,成为季节调整工作的核心目标。从1931年Macauley提出用移动平均比率法进行季节调整以来,基于移动平均滤波的非参数季节调整方法和基于经典时间序列模型的参数信号提取季节调整方法不断发展和完善。但是,随着越来越多的季节调整问题被提出,原理方面的固有缺陷使二者在解决季节调整问题时显得越来越乏力。于是,基于状态空间方法的结构时间序列模型被引入季节调整问题的研究中。由于结构时间序列模型充分的灵活性,越来越多的季节调整问题都通过结构时间序列模型很好的拟合并解决。结构时间序列模型逐渐成为季节调整理论发展的新方向。本文对结构时间序列模型中经典的HS季节模型进行有针对性的改进,给出了能够拟合季节异方差和季节趋势的HS-SH模型和HS-ST模型的具体形式,并提出了这两个模型季节异方差和季节趋势存在性的检验方法,使结构时间序列模型在季节调整中的应用理论更加完备。
本文首先系统介绍了X11和SEATS两种季节调整方法的基本理论,并论述了各自的局限性；然后全面介绍了结构时间序列模型及其估计的基本理论；之后总结现有可用于季节调整的结构时间序列模型的具体理论基础和模型构成,并分析比较了各个模型的优点和不足；在此基础上,本文提出了可以拟合季节异方差和季节趋势问题的改进的HS模型,同时,提出基于改进的HS模型的季节异方差LR检验方法和季节趋势AIC检验方法,并研究了不同因素设定对季节异方差LR检验的检验尺度和检验功效的影响。最后,本文使用改进的HS模型对我国税收完成总额序列和发电总量序列进行了季节调整,得到两个序列在固有趋势、季节波动特征方面的实证结论,以及改进的HS模型季节调整方法比X11和SEATS方法更有效的结论。
在理论研究方面,本文的主要创新如下：
(1)对比现有各种可用于季节调整的结构时间序列模型,包括DS模型、TS模型、HS模型和SSLLT模型的具体理论基础和模型构成,并分析比较了各个模型的优缺点。在此基础上提出适用于拟合各类实际问题,尤其是季节异方差和季节趋势问题的改进的HS模型的具体形式,并就该模型如何进行估计、如何改进异常值及如何预测进行了详细的论述。
(2)针对改进的HS模型,提出检验季节异方差和季节趋势是否存在的方法,给出季节异方差检验的LR统计量的模拟分布,讨论影响分布的各种因素,并研究在不同参数设定下该统计检验的检验尺度和检验功效。
(3)在现有状态空间模型MATLAB程序包的基础上,设计针对改进的HS模型估计、检验等用途的程序模块,实现对改进的HS模型的估计和检验。
在实证研究方面,本文的主要创新如下：
(1)使用HS-SH模型对1991年1月至2011年12月我国税收完成总额月度序列进行季节调整研究。实证研究结果表明,我国税收收入具有稳定的、大于国内产出增长率的内在增长趋势,同时容易受到外部冲击的影响。此外,我国税收收入还具有明显的季节波动特征,受税收制度改革的影响,这种季节波动特征伴随着显著的波动形式变化和季节异方差性。
(2)使用HS-SH和HS-ST组合模型对1991年1月至2011年12月我国发电总量月度序列进行季节调整研究。实证研究结果表明,我国发电总量具有相对稳定的增长趋势,这一趋势的大小由我国宏观经济内生性发展对重工业的影响程度决定。同时,与其他宏观经济指标一样,我国发电总量也容易受到外部冲击的影响。此外,我国发电总量具有明显的季节波动特征,受春节假期影响,这种季节波动特征伴随着显著的季节异方差性,同时,受气候变暖和生活用电增加的影响,这种季节波动特征还伴随着显著的季节趋势。
Seasonal adjustment on economic series has always been one of the focuses in statistics and econometrics in many countries. In order to inspect the variety law and explain the economic significance of the data effectively, the seasonal adjustment must get the trend and season signals from the real economic series actually and entirely. Since Macauley advanced the ratio-to-moving average method for seasonal adjustment in1931, the nonparametric method based on moving average filters and the parametric method based on classic time series model signal extraction both have been improved well. However, as more and more problems have been raised in seasonal adjustment, the inherent default in fundamentals brings these two methods more and more weak in seasonal adjustment. Then, the structural time series models based on state space method have been introduced into the seasonal adjustment research. As the structural time series models are more flexible, more and more problems during seasonal adjustment can be fitted and solved. So the structural time series models have been new development direction of the theory and method of seasonal adjustment. This paper improved the HS seasonal model purposively, which is the classic one of the structural time series models, to give the HS-SH model and HS-ST model to fit seasonal heteroscedasticity and seasonal trend, and gave the ways to test the existence of the seasonal heteroscedasticity and the seasonal trend through these two models to make the structural time series model method more complete.
At first, this paper contrasted the X11and SEATS seasonal adjustment methods completely to summarize the merits and faults of them. Then, it presented the basic structure and estimation method of the structural time series models in detail. And then, it contrasted the theory fundamentals and basic structure of the structural time series models, which can be used in seasonal adjustment, to summarize the merits and faults of each one. And then, it advanced the improved HS model, which can fit the seasonal heteroscedasticity and seasonal trend, and presented the LR test on seasonal heteroscedasticity and the AIC test on seasonal trend based on the improved HS model. At last, this paper used the improved HS model to do seasonal adjustment on China's monthly tax revenues series and China's monthly electricity generation series to gain their features of trend and seasonal move, and at the same time, it contrasted the result of X11and SEATS method on these two series to find that the improved HS model method more effective.
In the theory research aspect, this paper's main innovation are as follows:
First, this paper contrasted almost every structural model's fundamentals and basic structure, likes DS, TS, HS and SSLLT model, which can be used for seasonal adjustment, to summarize the merits and faults of each one. Based on this, the improved HS model, which can solve actual problems such as seasonal heteroscedasticity and seasonal trend, and its estimation, outlier revision and forecast have been advanced.
Second, for the improved HS model, this paper gave methods for testing the existence of seasonal heteroscedasticity and seasonal trend. Moreover, it gave the distributions of the LR statistics for the test of seasonal heteroscedasticity, found all influencing factors of the distribution, and summarized the size and power of the test under many conditions with different values of the factors.
Third, this paper designed the programming module for the estimation and test of the improved HS model based on the MATLAB program package for state space model to make the improved HS model's estimation and test achievable.
In the empirical research aspect, this paper's main innovation are as follows:
First, this paper used the HS-SH model to do seasonal adjustment on China's monthly tax revenues series during1991January to2011December. The result indicated that China's tax revenues has a steady increase trend, whose increase rate is higher than the GDP's, and can be impacted by external crisis easily. Moreover, China's tax revenues have an obvious seasonal feature, which is accompanied with variety and heteroscedasticity because of the reform of the taxation system.
Second, this paper used the combined model, which is combined by the HS-SH model and the HS-ST model, to do seasonal adjustment on China's monthly electricity generation series during1991January to2011December. The result indicated that China's electricity generation has a relative steady increase trend, which is affected by the influence of China's macroeconomic endogenous increase to heavy industry. Like the tax revenues, China's electricity generation can be impacted by external crisis easily, too. Moreover, China's electricity generation have an obvious seasonal feature, which is accompanied with trend and heteroscedasticity because of the spring festival holiday, global warming, and home electricity increasing.
本文首先系统介绍了X11和SEATS两种季节调整方法的基本理论,并论述了各自的局限性；然后全面介绍了结构时间序列模型及其估计的基本理论；之后总结现有可用于季节调整的结构时间序列模型的具体理论基础和模型构成,并分析比较了各个模型的优点和不足；在此基础上,本文提出了可以拟合季节异方差和季节趋势问题的改进的HS模型,同时,提出基于改进的HS模型的季节异方差LR检验方法和季节趋势AIC检验方法,并研究了不同因素设定对季节异方差LR检验的检验尺度和检验功效的影响。最后,本文使用改进的HS模型对我国税收完成总额序列和发电总量序列进行了季节调整,得到两个序列在固有趋势、季节波动特征方面的实证结论,以及改进的HS模型季节调整方法比X11和SEATS方法更有效的结论。
在理论研究方面,本文的主要创新如下：
(1)对比现有各种可用于季节调整的结构时间序列模型,包括DS模型、TS模型、HS模型和SSLLT模型的具体理论基础和模型构成,并分析比较了各个模型的优缺点。在此基础上提出适用于拟合各类实际问题,尤其是季节异方差和季节趋势问题的改进的HS模型的具体形式,并就该模型如何进行估计、如何改进异常值及如何预测进行了详细的论述。
(2)针对改进的HS模型,提出检验季节异方差和季节趋势是否存在的方法,给出季节异方差检验的LR统计量的模拟分布,讨论影响分布的各种因素,并研究在不同参数设定下该统计检验的检验尺度和检验功效。
(3)在现有状态空间模型MATLAB程序包的基础上,设计针对改进的HS模型估计、检验等用途的程序模块,实现对改进的HS模型的估计和检验。
在实证研究方面,本文的主要创新如下：
(1)使用HS-SH模型对1991年1月至2011年12月我国税收完成总额月度序列进行季节调整研究。实证研究结果表明,我国税收收入具有稳定的、大于国内产出增长率的内在增长趋势,同时容易受到外部冲击的影响。此外,我国税收收入还具有明显的季节波动特征,受税收制度改革的影响,这种季节波动特征伴随着显著的波动形式变化和季节异方差性。
(2)使用HS-SH和HS-ST组合模型对1991年1月至2011年12月我国发电总量月度序列进行季节调整研究。实证研究结果表明,我国发电总量具有相对稳定的增长趋势,这一趋势的大小由我国宏观经济内生性发展对重工业的影响程度决定。同时,与其他宏观经济指标一样,我国发电总量也容易受到外部冲击的影响。此外,我国发电总量具有明显的季节波动特征,受春节假期影响,这种季节波动特征伴随着显著的季节异方差性,同时,受气候变暖和生活用电增加的影响,这种季节波动特征还伴随着显著的季节趋势。
Seasonal adjustment on economic series has always been one of the focuses in statistics and econometrics in many countries. In order to inspect the variety law and explain the economic significance of the data effectively, the seasonal adjustment must get the trend and season signals from the real economic series actually and entirely. Since Macauley advanced the ratio-to-moving average method for seasonal adjustment in1931, the nonparametric method based on moving average filters and the parametric method based on classic time series model signal extraction both have been improved well. However, as more and more problems have been raised in seasonal adjustment, the inherent default in fundamentals brings these two methods more and more weak in seasonal adjustment. Then, the structural time series models based on state space method have been introduced into the seasonal adjustment research. As the structural time series models are more flexible, more and more problems during seasonal adjustment can be fitted and solved. So the structural time series models have been new development direction of the theory and method of seasonal adjustment. This paper improved the HS seasonal model purposively, which is the classic one of the structural time series models, to give the HS-SH model and HS-ST model to fit seasonal heteroscedasticity and seasonal trend, and gave the ways to test the existence of the seasonal heteroscedasticity and the seasonal trend through these two models to make the structural time series model method more complete.
At first, this paper contrasted the X11and SEATS seasonal adjustment methods completely to summarize the merits and faults of them. Then, it presented the basic structure and estimation method of the structural time series models in detail. And then, it contrasted the theory fundamentals and basic structure of the structural time series models, which can be used in seasonal adjustment, to summarize the merits and faults of each one. And then, it advanced the improved HS model, which can fit the seasonal heteroscedasticity and seasonal trend, and presented the LR test on seasonal heteroscedasticity and the AIC test on seasonal trend based on the improved HS model. At last, this paper used the improved HS model to do seasonal adjustment on China's monthly tax revenues series and China's monthly electricity generation series to gain their features of trend and seasonal move, and at the same time, it contrasted the result of X11and SEATS method on these two series to find that the improved HS model method more effective.
In the theory research aspect, this paper's main innovation are as follows:
First, this paper contrasted almost every structural model's fundamentals and basic structure, likes DS, TS, HS and SSLLT model, which can be used for seasonal adjustment, to summarize the merits and faults of each one. Based on this, the improved HS model, which can solve actual problems such as seasonal heteroscedasticity and seasonal trend, and its estimation, outlier revision and forecast have been advanced.
Second, for the improved HS model, this paper gave methods for testing the existence of seasonal heteroscedasticity and seasonal trend. Moreover, it gave the distributions of the LR statistics for the test of seasonal heteroscedasticity, found all influencing factors of the distribution, and summarized the size and power of the test under many conditions with different values of the factors.
Third, this paper designed the programming module for the estimation and test of the improved HS model based on the MATLAB program package for state space model to make the improved HS model's estimation and test achievable.
In the empirical research aspect, this paper's main innovation are as follows:
First, this paper used the HS-SH model to do seasonal adjustment on China's monthly tax revenues series during1991January to2011December. The result indicated that China's tax revenues has a steady increase trend, whose increase rate is higher than the GDP's, and can be impacted by external crisis easily. Moreover, China's tax revenues have an obvious seasonal feature, which is accompanied with variety and heteroscedasticity because of the reform of the taxation system.
Second, this paper used the combined model, which is combined by the HS-SH model and the HS-ST model, to do seasonal adjustment on China's monthly electricity generation series during1991January to2011December. The result indicated that China's electricity generation has a relative steady increase trend, which is affected by the influence of China's macroeconomic endogenous increase to heavy industry. Like the tax revenues, China's electricity generation can be impacted by external crisis easily, too. Moreover, China's electricity generation have an obvious seasonal feature, which is accompanied with trend and heteroscedasticity because of the spring festival holiday, global warming, and home electricity increasing.
引文
① 频率响应函数，对于N项移动平均滤波器而言,其频响函数为H(f)=sin(πfNsin(πf),f为频率。
① 这一系数表示方式来源于Kendall, M. Time Series, London:Charles Griffin & Co.,1973.
① 中国人民银行调查统计司,《时间序列X-12-ARIMA季节调整——原理与方法》,P73。
① 中国人民银行调查统计司,《时间序列X-12-ARIMA季节调整——原理与方法》,P78-82。
① Burman, J.P. Seasonal Adjustment by Signal Extraction. Journal of the Royal Statistical Society A,1980(143): 321-337
② Maravall, A. On Minimum Mean Squared Error Estimation of the Noise in Unobserved Component Models. Journal of Business and Economic Statistics,1987(5):115-120
③ 预估计,指在已知序列总长度为T的情况下,使用T-k时刻及之前的序列样本对t时刻进行预测得到的t时刻序列的估计值。这里,t>T-k。
① 本文只讨论初始值服从正态分布的简单情况。有关非正态分布的讨论,请详见J.Durbin and S.J..Koopman, Time Series Analysis by State Space Methods, Oxford University Press,2001, chapter 10.
① Hardy, G H. A Course of Pure Mathematics, Ninth edition, Cambridge University Press,1946:379-380.
② A. Zygmund. Trigonometric Series, Vol.1-2, Cambridge Univ.Press, Cambridge,1959
① Proietti,T. Seasonal Heteroscedasticity and Trends.Journal of Forecasting,1998(17):8
① Tanaka, K. Time Series Analysis:Nonstationary and Noninvertible Distribution Theory, NewYork:JohnWiley, 1996, section 8.7.
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[39]Box, Hillmer and Tiao. Analysis and modeling of time series. Seasonal Analysis of Economic Time Series (A. Zellner, Ed.), US. Department of Commerce, Bureau of the Census, Washington, DC,1978:309-333.
[40]Burman, J. P. Seasonal adjustment by signal extraction. Journal of the Royal Statistical Society A,1980,143:321-337.
[41]Burman, J. P. and Otto, M. C. Outliers in Time Series. Research Report 88/14, Bureau of Census, Washington,1988
[42]Burridge, P. and Wallis, K. F. Seasonal adjustment and kalman filtering:extension to periodic variances. Journal of Forecasting,1998,9:109-118.
[43]Burridge, P. and K. F. Wallis. Seasonal adjustment and kalman filtering:extension to periodic variances. Journal of Forecasting,1988,9:109-118.
[44]Chen, C. and Liu, L. M. Joint estimation of model parameters and outlier effects in time series. Journal of the American Statistical Association,1993,88:284-297.
[45]Cleveland, W. P. Analysis and Forecasting of Seasonal Time Series. University of Wisconsin (Unpublished Ph.D. Thesis)
[46]Cleveland, W. P. and Tiao, G.C. Decomposition of seasonal time series:a model for the Census X-11 program. J. Amer. Statist. Ass,1976,71:581-587.
[47]Depoutot, R. and Planas, C. Comparing seasonal adjustment and trend extraction filters with application to a model-based selection of X11 linear filters. Eurostat Working Papers, No.9, Eurostat, Luxembourg,1998
[48]Dominique Ladirary and Benoit Quenneville. Seasonal Adjustment with the X-11 Method. Springer,2001
[49]Eisenpress, H. Regression techniques applied to seasonal corrections and adjustments for calendar shifts. Journal of the American Statistical Association.1956, Vol.51: 615-620.
[50]Engle, R. F. Estimating structural models of seasonality (A. Zellner Ed.). Seasonal Analysis of Economic Time Series. Washington D.C.:Dept. of Commerce, Bureau of the Census, 1978:281-297.
[51]Engle, R. Wald, likelihood ratio and lagrange multiplier tests in econometrics. Handbook of Econometrics, Volume 2,1984, Amsterdam:North Holland
[52]Findley, D. F., Monsell, B. C., Shulman, H. B. and Pugh, M. G. Sliding spans diagnostics for seasonal and related adjustments. Journal of the American Statistical Association,1990,85: 345-355.
[53]Findley, D. and Soukup, R. J. On the Spectrum Diagnostics Used by X-12-ARIMA to Indicate the Presence of Trading Day Effects after Modeling or Adjustment. Bureau of Census, Washington,1999
[54]Findley, D. F. Some recent developments and directions in seasonal adjustment. Journal of Official Statistics,2005,21
[55]Fischer, B. Decomposition of Time Series-Comparing Different Methods in Theory and Practice. Luxembourg,1995
[56]Gomez, V. and Maravall, A. Time Series Regression with ARIMA Noise and Missing Observations-Program TRAMO. EUI Working Paper ECO,1992, No.92/81
[57]Gomez, V. and Maravall, A. Programs TRAMO and SEATS; Instructions for the User. Working Paper 9628. Servicio de Estudios. Banco de Espana.1997
[58]Gomez, V., Maravall, A. and Pena, D. Missing observations in ARIMA models:skipping approach versus additive outlier approach. Journal of Econometrics,1999,88:341-364.
[59]Hannan, E. J. and Rissanen, J. Recursive estimation of mixed autoregressive-moving average order. Biometrika,1982,69:81-94.
[60]Harvey, A.C. and Todd, P. H. J. Forecasting economic time series with structural and Box-Jenkins models, a case Study. Journal of Business and Economic statistics.1983, Vol.4: 299-306.
[61]Harvey, A. C. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press,1989
[62]Harrison, P. J. and Stevens, C. F. Bayesian Forecasting. Journal of Royal Statistical Society, Series B,1976,38:205-247.
[63]Hardy, G. H. A Course of Pure Mathematics. Ninth edition, Cambridge University Press, 1946:379-380.
[64]Henderson, R. Note on graduation by adjusted average. Transactions of the Actuarial Society of America,1916,17:43-48.
[65]Henderson, R. A new method of graduation. Transactions of the Actuarial Society of America, 1924,25:29-40.
[66]Hillmer, S. C. and Tiao, G. C. An ARIMA-model-based approach to seasonal adjustment. Journal of the American Statistical Association,1982,77:63-70.
[67]Jazwinski, A. H. Stochastic Processed and Filtering Theory. New York:Academic Press, 1970
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[69]Jong, p. and Penzer, J. Diagnosing shocks in time series. J. American Statistical Association, 1998,93:796-806.
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[71]Peng, J. Y. and Aston, J.A.D. State Space Models Manual.2011, http://ftp.jaist.ac.jp/pub/ sourceforge/s/project/ss/ssmodels/ssm/1.0/ssm_manual.pdf
[72].Kendall, M. Time Series. London:Charles Griffin & Co.,1973
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[75]Kim, C. J. and Nelson, C. R. State Space Models with Regime Switching. Cambridge, Massachusetts:MIT Press,1999
[76]Kitagawa, G. and Gersch, W. A smoothness priors-state space modeling on time series with trend and seasonality. Journal of the American Statistical Association,1984,79:378-389.
[77]McDonald, K. M., and Hood, C. C. H. Outlier Selection for RegARIMA Models. Bureau of Census, Washington,2001
[78]Musgrave J. A Set of End Weights to End all End Weights. Working paper, US Bureau of the Census, Washington,1964
[79]M. Morf and T. Kailath. Square-root algorithms for linear least-squares estimation. IEEE Transaction on Automattic Control,1975,8:487-497.
[80]Maravall, A. On minimum mean squared error estimation of the noise in unobserved component models. Journal of Business and Economic Statistics,1987,5:115-120.
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[82]Macauley, F.R. The Smoothing of Time Series. New York:National Bureau of Economic Research,1931
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[86]Osborn, D. R. The implications of periodically time varying coefficients for seasonal time series processes, Journal of Econometrics,1991,55,373-384.
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① 这一系数表示方式来源于Kendall, M. Time Series, London:Charles Griffin & Co.,1973.
① 中国人民银行调查统计司,《时间序列X-12-ARIMA季节调整——原理与方法》,P73。
① 中国人民银行调查统计司,《时间序列X-12-ARIMA季节调整——原理与方法》,P78-82。
① Burman, J.P. Seasonal Adjustment by Signal Extraction. Journal of the Royal Statistical Society A,1980(143): 321-337
② Maravall, A. On Minimum Mean Squared Error Estimation of the Noise in Unobserved Component Models. Journal of Business and Economic Statistics,1987(5):115-120
③ 预估计,指在已知序列总长度为T的情况下,使用T-k时刻及之前的序列样本对t时刻进行预测得到的t时刻序列的估计值。这里,t>T-k。
① 本文只讨论初始值服从正态分布的简单情况。有关非正态分布的讨论,请详见J.Durbin and S.J..Koopman, Time Series Analysis by State Space Methods, Oxford University Press,2001, chapter 10.
① Hardy, G H. A Course of Pure Mathematics, Ninth edition, Cambridge University Press,1946:379-380.
② A. Zygmund. Trigonometric Series, Vol.1-2, Cambridge Univ.Press, Cambridge,1959
① Proietti,T. Seasonal Heteroscedasticity and Trends.Journal of Forecasting,1998(17):8
① Tanaka, K. Time Series Analysis:Nonstationary and Noninvertible Distribution Theory, NewYork:JohnWiley, 1996, section 8.7.
① Peng,J.-Y and Aston,J.A.D.State Space Models Manual.2011
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[45]Cleveland, W. P. Analysis and Forecasting of Seasonal Time Series. University of Wisconsin (Unpublished Ph.D. Thesis)
[46]Cleveland, W. P. and Tiao, G.C. Decomposition of seasonal time series:a model for the Census X-11 program. J. Amer. Statist. Ass,1976,71:581-587.
[47]Depoutot, R. and Planas, C. Comparing seasonal adjustment and trend extraction filters with application to a model-based selection of X11 linear filters. Eurostat Working Papers, No.9, Eurostat, Luxembourg,1998
[48]Dominique Ladirary and Benoit Quenneville. Seasonal Adjustment with the X-11 Method. Springer,2001
[49]Eisenpress, H. Regression techniques applied to seasonal corrections and adjustments for calendar shifts. Journal of the American Statistical Association.1956, Vol.51: 615-620.
[50]Engle, R. F. Estimating structural models of seasonality (A. Zellner Ed.). Seasonal Analysis of Economic Time Series. Washington D.C.:Dept. of Commerce, Bureau of the Census, 1978:281-297.
[51]Engle, R. Wald, likelihood ratio and lagrange multiplier tests in econometrics. Handbook of Econometrics, Volume 2,1984, Amsterdam:North Holland
[52]Findley, D. F., Monsell, B. C., Shulman, H. B. and Pugh, M. G. Sliding spans diagnostics for seasonal and related adjustments. Journal of the American Statistical Association,1990,85: 345-355.
[53]Findley, D. and Soukup, R. J. On the Spectrum Diagnostics Used by X-12-ARIMA to Indicate the Presence of Trading Day Effects after Modeling or Adjustment. Bureau of Census, Washington,1999
[54]Findley, D. F. Some recent developments and directions in seasonal adjustment. Journal of Official Statistics,2005,21
[55]Fischer, B. Decomposition of Time Series-Comparing Different Methods in Theory and Practice. Luxembourg,1995
[56]Gomez, V. and Maravall, A. Time Series Regression with ARIMA Noise and Missing Observations-Program TRAMO. EUI Working Paper ECO,1992, No.92/81
[57]Gomez, V. and Maravall, A. Programs TRAMO and SEATS; Instructions for the User. Working Paper 9628. Servicio de Estudios. Banco de Espana.1997
[58]Gomez, V., Maravall, A. and Pena, D. Missing observations in ARIMA models:skipping approach versus additive outlier approach. Journal of Econometrics,1999,88:341-364.
[59]Hannan, E. J. and Rissanen, J. Recursive estimation of mixed autoregressive-moving average order. Biometrika,1982,69:81-94.
[60]Harvey, A.C. and Todd, P. H. J. Forecasting economic time series with structural and Box-Jenkins models, a case Study. Journal of Business and Economic statistics.1983, Vol.4: 299-306.
[61]Harvey, A. C. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press,1989
[62]Harrison, P. J. and Stevens, C. F. Bayesian Forecasting. Journal of Royal Statistical Society, Series B,1976,38:205-247.
[63]Hardy, G. H. A Course of Pure Mathematics. Ninth edition, Cambridge University Press, 1946:379-380.
[64]Henderson, R. Note on graduation by adjusted average. Transactions of the Actuarial Society of America,1916,17:43-48.
[65]Henderson, R. A new method of graduation. Transactions of the Actuarial Society of America, 1924,25:29-40.
[66]Hillmer, S. C. and Tiao, G. C. An ARIMA-model-based approach to seasonal adjustment. Journal of the American Statistical Association,1982,77:63-70.
[67]Jazwinski, A. H. Stochastic Processed and Filtering Theory. New York:Academic Press, 1970
[68]J. Durbin and S. J. Koopman. Time Series Analysis by State Space Methods. Oxford University Press,2001
[69]Jong, p. and Penzer, J. Diagnosing shocks in time series. J. American Statistical Association, 1998,93:796-806.
[70]Peng, J. Y. and Aston, J.A.D. The state space models toolbox for MATLAB. Journal of Statistical Software,2011,41(6):1-26.
[71]Peng, J. Y. and Aston, J.A.D. State Space Models Manual.2011, http://ftp.jaist.ac.jp/pub/ sourceforge/s/project/ss/ssmodels/ssm/1.0/ssm_manual.pdf
[72].Kendall, M. Time Series. London:Charles Griffin & Co.,1973
[73]K. E., Kalman. A new approach to linear filtering and prediction problem. Transactions of the ASME-Journal of Basic Engineering, No.82 (Series D),1960:35-45.
[74]Kitagawa, G and Gerseh, W. Smoothness Priors Analysis of Time Series. New York: Springer-Verlag,1996
[75]Kim, C. J. and Nelson, C. R. State Space Models with Regime Switching. Cambridge, Massachusetts:MIT Press,1999
[76]Kitagawa, G. and Gersch, W. A smoothness priors-state space modeling on time series with trend and seasonality. Journal of the American Statistical Association,1984,79:378-389.
[77]McDonald, K. M., and Hood, C. C. H. Outlier Selection for RegARIMA Models. Bureau of Census, Washington,2001
[78]Musgrave J. A Set of End Weights to End all End Weights. Working paper, US Bureau of the Census, Washington,1964
[79]M. Morf and T. Kailath. Square-root algorithms for linear least-squares estimation. IEEE Transaction on Automattic Control,1975,8:487-497.
[80]Maravall, A. On minimum mean squared error estimation of the noise in unobserved component models. Journal of Business and Economic Statistics,1987,5:115-120.
[81]Maravall, A. A class of diagnostics in the ARIMA-model-based decomposition of a time series. Seasonal Adjustment, ECB,2003:23-36.
[82]Macauley, F.R. The Smoothing of Time Series. New York:National Bureau of Economic Research,1931
[83]Masanao Aok. State Space Modeling of Time Series. Berlin Heidelberg Germany:Springer-Verlag,1987:287-298.
[84]Miron, J. A. The Economics of Seasonal Cycles. Cambridge MA:The MIT Press,1996
[85]Miron, J. A. and Beaulieu, J. What have macroeconomists learned about business cycles from the study of processes. Journal of Econometrics,1996,55:373-384.
[86]Osborn, D. R. The implications of periodically time varying coefficients for seasonal time series processes, Journal of Econometrics,1991,55,373-384.
[87]Persons, W. M. Indices of business conditions. Review of Economics and Statistics, 1919, Vol.1:5-107.
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