查看“︁HP滤波”︁的源代码

'''HP滤波'''（{{lang-en|Hodrick–Prescott filter}}或{{lang|en|Hodrick–Prescott decomposition}}）是[[宏观经济学]]中用到的[[时间序列]]分析方法，尤其在{{tsl|en|real business cycle theory|实际经济周期理论}}中较为常用。HP滤波可以从原始数据中分离出周期性的部分，并得到一条平滑的曲线来表述整个时间序列，即把对短期波动更敏感的数据转成了对长期波动更敏感的表示方式，改变乘数<math>\lambda</math>可以调整其敏感程度。20世纪90年代，两位经济学家{{tsl|en|Robert J. Hodrick|罗伯特·霍德里克}}和[[诺贝尔经济学奖|诺贝尔奖得主]][[爱德华·普雷斯科特]]发表了这种方法并受到学界欢迎。<ref name="hp">{{cite journal |last=Hodrick |first=Robert |first2=Edward C. |last2=Prescott |year=1997 |title=Postwar U.S. Business Cycles: An Empirical Investigation |url=https://archive.org/details/sim_journal-of-money-credit-and-banking_1997-02_29_1/page/n4 |journal=Journal of Money, Credit, and Banking |volume=29 |issue=1 |pages=1–16 |jstor=2953682 }}</ref>不过实际上早在1923年{{tsl|en|E. T. Whittaker|惠特克}}就首次发表了该方法<ref>{{cite journal |last=Whittaker |first=E. T. |year=1923 |title=On a New Method of Graduation |journal=Proceedings of the Edinburgh Mathematical Association |volume=41 |issue= |pages=63–75 |doi=10.1017/S001309150000359X }} - as quoted in [http://cowles.econ.yale.edu/P/cd/d17b/d1771.pdf Philips 2010] {{Wayback|url=http://cowles.econ.yale.edu/P/cd/d17b/d1771.pdf |date=20120417132021 }}</ref>。

== 数学表述 ==
这个方法的思想类似于{{tsl|en|decomposition of time series|时间序列分解}}。令<math>\{y_t\},\ t = 1, 2, ..., T\,</math>表示一组[[时间序列]]变量的[[对数]]，则<math>\{y_t\}</math>由一系列趋势项<math>\tau_t</math>、周期项<math>c_t</math>和误差项<math>\epsilon_t</math>组成，即<math>y_t\ = \tau_t\ + c_t\ + \epsilon_t</math>。<ref>Kim, Hyeongwoo. "[http://www.auburn.edu/~hzk0001/hpfilter.pdf Hodrick–Prescott Filter] {{Wayback|url=http://www.auburn.edu/~hzk0001/hpfilter.pdf |date=20160311083655 }}" March 12, 2004</ref>给定合适的正数<math>\lambda</math>，存在一个趋势项满足
: <math>\min_{\tau}\left(\sum_{t = 1}^T {(y_t - \tau _t )^2 } + \lambda \sum_{t = 2}^{T - 1} {[(\tau _{t+1} - \tau _t) - (\tau _t - \tau _{t - 1} )]^2 }\right)</math>。

上式第一项表示变量偏离趋势项的误差<math>d_t=y_t-\tau_t</math>的平方和，从而控制了周期项的大小；第二项用乘子<math>\lambda</math>乘上趋势项二阶[[差分]]的平方和，从而控制了趋势项变化的剧烈程度。<math>\lambda</math>越大，后者的控制就越强。霍德里克和普雷斯科特建议季度数据<math>\lambda</math>取为1600，若单位不是季度则<math>\lambda</math>正比于每单位所含季度数的平方，即年度数据取100、月度数据取14,400。<ref name="hp" />雷文（Ravn）和乌利希（Uhlig）则在2002年发表的文章中提出<math>\lambda</math>应该正比于数据每单位所含季度数的四次方，即年度数据<math>\lambda</math>应取6.25、月度数据取129,600。<ref>{{cite journal
  | last1=Ravn
  | first1=Morten
  | last2=Uhlig
  | first2=Harald
  | title=On adjusting the Hodrick–Prescott filter for the frequency of observations
  | journal=The Review of Economics and Statistics
  | year=2002
  | number=2
  | volume=84
  | pages=371
  | doi=10.1162/003465302317411604
  | url=https://www.econstor.eu/bitstream/10419/75742/1/cesifo_wp479.pdf
  | access-date=2019-07-06
  | archive-date=2019-03-29
  | archive-url=https://web.archive.org/web/20190329093302/https://www.econstor.eu/bitstream/10419/75742/1/cesifo_wp479.pdf
  | dead-url=no
  }}</ref>

麦克尔罗伊的一篇论文中给出了双侧HP滤波[[谱分解]]的精确数学表达式<ref>{{cite journal
  | last1= McElroy
  | title=Exact Formulas for the Hodrick-Prescott Filter
  | journal=Econometrics Journal
  | year=2008
  | volume=11
  | pages = 209–217
  | doi=10.1111/j.1368-423x.2008.00230.x
}}</ref>。

== 评价 ==
HP滤波很容易实现，不过它也存在一定缺陷，只在以下严苛条件下才能做出最优估计：<ref>{{cite journal |last=French |first=Mark W. |title=Estimating Changes in Trend Growth of Total Factor Productivity: Kalman and H-P Filters versus a Markov-Switching Framework |work=FEDS Working Paper No. 2001-44 |year=2001 |ssrn=293105 }}</ref>
* 时间序列是{{tsl|en|Order of integration|二阶整合}}的<ref>{{cite journal |author1=Carvalho V, Harvey A, Trimbur T |title=A note on common cycles, common trends, and convergence |journal=Journal of Business & Economic Statistics |date=2007 |volume=25 |issue=1 |page=12-20 |doi=10.1198/073500106000000431 |url=https://www.cass.city.ac.uk/__data/assets/pdf_file/0007/36565/WP-CEA-04-2006.pdf |access-date=2019-07-06 |archive-date=2019-07-06 |archive-url=https://web.archive.org/web/20190706045052/https://www.cass.city.ac.uk/__data/assets/pdf_file/0007/36565/WP-CEA-04-2006.pdf |dead-url=no }}</ref>，否则HP滤波会得到偏离实际情况的趋势项。
** 如果发生了单次的永久性冲击（permanent shock）或存在稳定的趋势增长率，HP滤波得到的周期项也会扭曲。
* 样本中的周期项是[[白噪音]]，或者趋势项和周期项中的随机变化机制相同。

标准的双侧HP滤波不应该用来估计基于递归状态空间表达的DSGE模型，这是因为HP滤波使用未来的观测<math>t+i, i>0 </math>去构造当前时间点<math>t</math>的结果，但递归状态空间要求当前的观测仅基于当前和过去的状态。要解决这个问题，可以使用单侧HP滤波。<ref>{{cite journal
  | last1= Stock
  | last2=Watson
  | title=Forecasting Inflation
  | url= https://archive.org/details/sim_journal-of-monetary-economics_1999-10_44_2/page/293
  | journal=Journal of Monetary Economics
  | year=1999
  | volume=44
  | pages = 293–335
  | doi=10.1016/s0304-3932(99)00027-6
}}</ref>

== 参见 ==
* [[带通滤波器]]
* [[卡尔曼滤波]]

== 参考文献 ==
{{Reflist}}

== 拓展阅读 ==
* {{cite book |last=Enders |first=Walter |chapter=Trends and Univariate Decompositions |title=Applied Econometric Time Series |url=https://archive.org/details/appliedeconometr0000ende |location=New York |publisher=Wiley |year=2010 |edition=Third |isbn=978-0470-50539-7 |pages=[https://archive.org/details/appliedeconometr0000ende/page/247 247]–7 }}
* {{cite book |last=Favero |first=Carlo A. |title=Applied Macroeconometrics |url=https://books.google.com/books?id=PBToVOKpyL4C&pg=PA54 |location=New York |publisher=Oxford University Press |year=2001 |isbn=0-19-829685-1 |pages=54–5 |access-date=2019-07-06 |archive-date=2020-09-22 |archive-url=https://web.archive.org/web/20200922111944/https://books.google.com/books?id=PBToVOKpyL4C&pg=PA54 |dead-url=no }}
* {{cite book |last=Mills |first=Terence C. |chapter=Filtering Economic Time Series |title=Modelling Trends and Cycles in Economic Time Series |url=https://archive.org/details/modellingtrendsc0000mill |location=New York |publisher=Palgrave McMillan |edition= |year=2003 |isbn=1-4039-0209-7 |pages=[https://archive.org/details/modellingtrendsc0000mill/page/75 75]–102 }}

== 外部链接  ==
*[http://dge.repec.org/codes/prescott/hpfilter.for 普雷斯科特给出的Fortran代码]{{Wayback|url=http://dge.repec.org/codes/prescott/hpfilter.for |date=20160306023050 }}
*[http://www.mathworks.com/matlabcentral/fileexchange/3972-hodrick-prescott-filter Matlab实现]{{Wayback|url=http://www.mathworks.com/matlabcentral/fileexchange/3972-hodrick-prescott-filter |date=20190329093311 }}
*[https://ideas.repec.org/c/dge/qmrbcd/181.html 单侧HP滤波的Matlab实现]{{Wayback|url=https://ideas.repec.org/c/dge/qmrbcd/181.html |date=20190329093300 }}
*[https://cran.r-project.org/web/packages/mFilter/mFilter.pdf R语言实现的HP滤波，包名mFilter]{{Wayback|url=https://cran.r-project.org/web/packages/mFilter/mFilter.pdf |date=20190329093306 }}
*[http://jaac.wz.cz/hponlineapp.php 在线进行HP滤波]{{Wayback|url=http://jaac.wz.cz/hponlineapp.php |date=20190727140154 }}

[[Category:时间序列]]