【统计学】【2015】时间序列数据缺失值的多重填补

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本文为美国杜克大学（作者：Sohae Oh）的硕士论文，共48页。

由于各种原因，金融股票市场数据常常包含缺失的数值。其中一个原因是，由于市场因假日休市，所以并不总是观察每日股价，这就造成了信息上的空白，使得很难预测第二天的股价。在这种情况下，节日期间的信息可以从其他国家的股票市场“借来”，因为全球股票价格往往表现出类似的走势，事实上是高度相关的。

本研究的主要目的是结合全球不同市场的股指数据，利用不同时间序列间的「资讯分享」，开发一种计算个别股指缺失值的演算法。为了开发适应时间序列特征的插补算法，我们对时间序列和面板数据采用动态线性模型的多重插补方法。该算法采用了可忽略的丢失数据机制，主要针对由于假期而导致的数据丢失。利用蒙特卡罗马尔可夫链（MCMC）方法模拟了参数的后验分布，并利用Rubin组合规则对绘制集的估计值进行组合，给出了数据集的最终推断。具体地说，我们使用Gibbs取样器、前向滤波和后向采样（FFBS）来模拟联合后验分布和隐变量的后验预测分布等参数。利用均方根误差（RMSE）和归一化均方根误差（NRMSE）两种基于误差的测量方法对算法的有效性和性能进行了仿真研究。我们比较了输入时间序列与完整数据集的总体趋势，并以最终值结转法（LVCF）为基准检验了算法的不充分可预测性。将该算法应用于美国、日本、香港、英国和德国的实际股价指数数据，通过仿真和实际应用，我们得出结论：该插补算法能够很好地实现我们的原始目标，以预测节后开盘价的股价，其效果优于基准方法。我们相信这种多重插补算法可以应用于许多处理具有缺失值的时间序列应用，如金融和经济数据以及生物医学数据。

Financial stock market data, for variousreasons, frequently contain missing values. One reason for this is that,because the markets close for holidays, daily stock prices are not alwaysobserved. This creates gaps in information, making it difficult to predict thefollowing day’s stock prices. In this situation, information during the holidaycan be “borrowed” from other countries’ stock market, since global stock pricestend to show similar movements and are in fact highly correlated. The main goalof this study is to combine stock index data from various markets around theworld and develop an algorithm to impute the missing values in individual stockindex using “information sharing” between different time series. To developimputation algorithm that accommodate time series-specific features, we takemultiple imputation approach using dynamic linear model for time-series andpanel data. This algorithm assumes ignorable missing data mechanism, as which missingnessdue to holiday. The posterior distributions of parameters, including missingvalues, is simulated using Monte Carlo Markov Chain (MCMC) methods andestimates from sets of draws are then combined using Rubin’s combination rule,rendering final inference of the data set. Specifically, we use the Gibbssampler and Forward Filtering and Backward Sampling (FFBS) to simulate jointposterior distribution and posterior predictive distribution of latentvariables and other parameters. A simulation study is conducted to check thevalidity and the performance of the algorithm using two error-basedmeasurements: Root Mean Square Error (RMSE), and Normalized Root Mean SquareError (NRMSE). We compared the overall trend of imputed time series withcomplete data set, and inspected the insample predictability of the algorithmusing Last Value Carried Forward (LVCF) method as a bench mark. The algorithmis applied to real stock price index data from US, Japan, Hong Kong, UK andGermany. From both of the simulation and the application, we concluded that theimputation algorithm performs well enough to achieve our original goal,predicting the stock price for the opening price after a holiday, outperformingthe benchmark method. We believe this multiple imputation algorithm can be usedin many applications that deal with time series with missing values such asfinancial and economic data and biomedical data.