本文为美国加利福尼亚大学（作者：Berninger, Jordan）的硕士论文，共95页。

信息技术（IT）产业、智能手机和个人电脑的指数级增长已经定义了21世纪的内容。苹果公司（Apple Inc.）在这一增长中扮演了重要角色，苹果产品被广泛认为是IT革命的象征。苹果公司（Apple Inc.）的健康状况是该行业和依赖它的行业健康状况的一个预测因素。苹果股价是苹果健康状况的一个指标。因此，需要一个好的模型来预测这场IT革命的标志性公司的股价是特别有意义的。统计时间序列分析是至关重要的。

本文比较了单变量和多变量时间序列模型对苹果股票月首日开盘价的预测效果。在著名的单变量模型中，我们考虑了自回归综合移动平均（ARIMA）模型、具有广义自回归条件异方差（GARCH）和指数平滑ARIMA模型，所有这些模型的预测开盘价都依赖于其自身的过去值，而不依赖于其他模型。在时间序列分析的已知多变量方法中，我们考虑了向量自回归（VAR），将所有考虑的变量内生化，以及带有ARIMA残差的经典线性回归。我们将这些模型拟合到1990年1月至2016年9月历史开盘价的“样本内”（训练集），并用它们预测2016年10月至2017年9月12个月的“样本外”（测试集）。在多元模型中，预测苹果开盘价的因素是：标准普尔500指数、微软和德州仪器的股价。我们假设苹果的股票与标准普尔500指数呈正相关，与竞争对手微软呈负相关，与苹果的供应商德州仪器呈正相关。我们将每个模型的预测值与“样本外”时间序列（测试集）的实际值进行比较，根据均方根误差（RMSE）评估预测性能。我们发现，由所有这些模型预测组成的平均预测值，在预测行业中被称为“共识预测”，在12个月的测试集中具有最低的RMSE。这是一个重要的结果，不仅因为它解释了几十年来将许多模型的预测平均值视为任何相关经济变量“预测”的做法，而且还因为它强调了整合多种经典模型以获得良好预测性能的好处。

The 21st Century has been defined by the exponential growth of the information technology (IT) industry, the smart phone and the personal computer. Apple Inc. has played a major role in that growth, with Apple products widely considered emblematic of the IT revolution. The health of Apple Inc. is a predictor of the health of the industry and industries that depend on it. Apple’s stock price is an indicator of Apple’s health. Therefore, it is of particular interest to have good models to predict the stock price of such a hallmark company of this IT revolution. Statistical time series analysis is of paramount importance to do that. This thesis compares the forecasting performance of univariate and multivariate time series models of Apple stock’s opening price for the first day of each month. Among well-known univariate models, we consider Autoregressive Integrated Moving Average (ARIMA) models, ARIMA with Generalized Autoregressive Conditional Heteroscedascity (GARCH) and Exponential Smoothing, all models where the predicted opening price depends on past values of itself and nothing else. Among multivariate methods known in time series analysis, we consider Vector Autoregression (VAR), which endogenizes all the variables considered, and classical linear regression with ARIMA residuals. We fit these models to an “in-sample“ (training set) of historical opening price from January 1990 to September 2016, and use them to forecast 12 months “out of sample“ (test set) October 2016 to September 2017. In the multivariate models, the predictors of Apple’s opening price are: the stock price of the S&P500, Microsoft and Texas Instruments. We hypothesize a positive correlation of Apple’s stock with the S&P500 stock, a negative correlation with its competitor Microsoft and a positive correlation with Apple’s supplier, Texas Instruments. We compare the forecasts from each model with the actual values of the “out of sample“ time series (the test set) to assess forecasting performance according to the Root Mean Square Error (RMSE). We find that an average forecast consisting of the average of all those models’ forecasts, what is known in the forecasting industry as ”the consensus forecast,” has the lowest RMSE for the 12-month test set. This is an important result, not only because it explains the decades’ old practice of considering averages of forecasts from many models as “ the forecast“ of any relevant economic variable, but also because it highlights the benefits of integrating multiple classical models to obtain a good predictive performance.

1.       引言

2. 数据

3. 数据的时间序列分析

4. ARIMA模型

5. 指数平滑

6. 向量自回归

7. 时间序列回归

8. 结论

附录A 图片与表格

附录B R代码

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