VAR模型¶
简介¶
向量自回归（VARs）于20世纪80年代被引入计量经济学文献，用于描述多个变量之间的（线性）依赖关系。对于一个\(K\) x \(1\)的向量\(y_{t}\)，我们可以将VAR(p)模型定义为：
\[y_{t} = c + A_{1}y_{t-1} + ... + A_{p}y_{t-p} + e_{t}\]
这些模型可以通过OLS快速估计。但当因变量数量较多时，待估参数的数量会急剧增加。关于处理这类模型的替代方法，请参阅贝叶斯向量自回归的笔记本。
示例¶
我们将为美国银行业股票运行一个VAR模型。
import numpy as np
import pyflux as pf
from pandas_datareader import DataReader
from datetime import datetime
import matplotlib.pyplot as plt
%matplotlib inline
ibm = DataReader(['JPM','GS','BAC','C','WFC','MS'], 'yahoo', datetime(2012,1,1), datetime(2016,6,28))
opening_prices = np.log(ibm['Open'])
plt.figure(figsize=(15,5));
plt.plot(opening_prices.index,opening_prices);
plt.legend(opening_prices.columns.values,loc=3);
plt.title("Logged opening price");
这里我们指定一个任意的VAR(2)模型，通过OLS方法进行拟合：
model = pf.VAR(data=opening_prices, lags=2, integ=1)
接下来我们估计潜在变量。对于这个例子，我们将使用OLS估计\(z^{OLS}\)：
x = model.fit()
x.summary()
VAR(2)
======================================== =================================================
Dependent Variable: Differenced BAC Method: OLS
Start Date: 2012-01-05 00:00:00 Log Likelihood: 21547.2578
End Date: 2016-06-28 00:00:00 AIC: -42896.5156
Number of observations: 1126 BIC: -42398.8994
==========================================================================================
Latent Variable Estimate Std Error z P>|z| 95% C.I.
========================= ========== ========== ======== ======== ========================
Diff BAC Constant 0.0007 0.0006 1.2007 0.2299 (-0.0004 | 0.0018)
Diff BAC AR(1) -0.0525 0.0005 -97.5672 0.0 (-0.0535 | -0.0514)
Diff C to Diff BAC AR(1) -0.0616 0.0004 -143.365 0.0 (-0.0625 | -0.0608)
Diff GS to Diff BAC AR(1) 0.0595 0.0004 132.6638 0.0 (0.0587 | 0.0604)
Diff JPM to Diff BAC AR(1)0.0296 0.0006 49.3563 0.0 (0.0284 | 0.0308)
Diff MS to Diff BAC AR(1) -0.0231 0.0004 -62.6218 0.0 (-0.0239 | -0.0224)
Diff WFC to Diff BAC AR(1)-0.0417 0.0598 -0.6968 0.4859 (-0.159 | 0.0756)
Diff BAC AR(2) 0.1171 0.0555 2.1087 0.035 (0.0083 | 0.226)
Diff C to Diff BAC AR(2) -0.1266 0.0444 -2.8528 0.0043 (-0.2136 | -0.0396)
Diff GS to Diff BAC AR(2) 0.1698 0.0464 3.6618 0.0003 (0.0789 | 0.2606)
Diff JPM to Diff BAC AR(2)-0.0959 0.062 -1.5472 0.1218 (-0.2174 | 0.0256)
Diff MS to Diff BAC AR(2) -0.0213 0.0381 -0.557 0.5775 (-0.096 | 0.0535)
Diff WFC to Diff BAC AR(2)-0.001 0.0701 -0.0149 0.9881 (-0.1384 | 0.1363)
Diff C Constant 0.0003 0.065 0.0047 0.9962 (-0.1272 | 0.1278)
Diff C AR(1) 0.0193 0.052 0.3706 0.7109 (-0.0826 | 0.1211)
Diff BAC to Diff C AR(1) -0.0576 0.0543 -1.0613 0.2886 (-0.164 | 0.0488)
Diff GS to Diff C AR(1) 0.0579 0.0726 0.7979 0.4249 (-0.0844 | 0.2002)
Diff JPM to Diff C AR(1) 0.0831 0.0447 1.8595 0.063 (-0.0045 | 0.1706)
Diff MS to Diff C AR(1) -0.037 0.0794 -0.4657 0.6414 (-0.1925 | 0.1186)
Diff WFC to Diff C AR(1) -0.1785 0.0737 -2.4235 0.0154 (-0.3229 | -0.0341)
Diff C AR(2) 0.1612 0.0589 2.7379 0.0062 (0.0458 | 0.2765)
Diff BAC to Diff C AR(2) -0.1021 0.0615 -1.6598 0.0969 (-0.2226 | 0.0185)
Diff GS to Diff C AR(2) 0.1109 0.0822 1.3483 0.1776 (-0.0503 | 0.272)
Diff JPM to Diff C AR(2) -0.0453 0.0506 -0.8946 0.371 (-0.1444 | 0.0539)
Diff MS to Diff C AR(2) 0.0127 0.0775 0.1643 0.8695 (-0.1391 | 0.1646)
Diff WFC to Diff C AR(2) -0.1313 0.0719 -1.8261 0.0678 (-0.2723 | 0.0096)
Diff GS Constant 0.0003 0.0575 0.006 0.9952 (-0.1123 | 0.113)
Diff GS AR(1) -0.016 0.06 -0.266 0.7903 (-0.1336 | 0.1017)
Diff BAC to Diff GS AR(1) 0.0051 0.0803 0.0633 0.9495 (-0.1523 | 0.1624)
Diff C to Diff GS AR(1) -0.0785 0.0494 -1.5891 0.112 (-0.1753 | 0.0183)
Diff JPM to Diff GS AR(1) 0.0507 0.0575 0.8814 0.3781 (-0.062 | 0.1633)
Diff MS to Diff GS AR(1) 0.0425 0.0534 0.7961 0.4259 (-0.0621 | 0.1471)
Diff WFC to Diff GS AR(1) -0.0613 0.0426 -1.4376 0.1505 (-0.1449 | 0.0223)
Diff GS AR(2) 0.0865 0.0445 1.9422 0.0521 (-0.0008 | 0.1738)
Diff BAC to Diff GS AR(2) -0.1896 0.0596 -3.1832 0.0015 (-0.3064 | -0.0729)
Diff C to Diff GS AR(2) 0.0423 0.0367 1.1553 0.248 (-0.0295 | 0.1142)
Diff JPM to Diff GS AR(2) 0.0667 0.0769 0.8664 0.3863 (-0.0841 | 0.2174)
Diff MS to Diff GS AR(2) 0.0433 0.0714 0.6067 0.5441 (-0.0966 | 0.1833)
Diff WFC to Diff GS AR(2) -0.0362 0.0571 -0.6347 0.5256 (-0.1481 | 0.0756)
Diff JPM Constant 0.0005 0.0596 0.0082 0.9934 (-0.1163 | 0.1173)
Diff JPM AR(1) -0.0304 0.0797 -0.3813 0.703 (-0.1866 | 0.1258)
Diff BAC to Diff JPM AR(1)-0.0281 0.049 -0.5738 0.5661 (-0.1243 | 0.068)
Diff C to Diff JPM AR(1) 0.0695 0.0594 1.1698 0.2421 (-0.047 | 0.186)
Diff GS to Diff JPM AR(1) -0.0106 0.0552 -0.1924 0.8474 (-0.1187 | 0.0975)
Diff MS to Diff JPM AR(1) -0.0338 0.0441 -0.7675 0.4428 (-0.1202 | 0.0526)
Diff WFC to Diff JPM AR(1)-0.0725 0.046 -1.5744 0.1154 (-0.1627 | 0.0178)
Diff JPM AR(2) 0.096 0.0616 1.559 0.119 (-0.0247 | 0.2167)
Diff BAC to Diff JPM AR(2)-0.1246 0.0379 -3.2883 0.001 (-0.1989 | -0.0503)
Diff C to Diff JPM AR(2) 0.0229 0.0696 0.3284 0.7426 (-0.1136 | 0.1593)
Diff GS to Diff JPM AR(2) -0.0084 0.0646 -0.1301 0.8965 (-0.1351 | 0.1182)
Diff MS to Diff JPM AR(2) 0.0319 0.0516 0.6182 0.5364 (-0.0693 | 0.1332)
Diff WFC to Diff JPM AR(2)-0.0117 0.0539 -0.2161 0.8289 (-0.1174 | 0.0941)
Diff MS Constant 0.0004 0.0721 0.005 0.996 (-0.141 | 0.1417)
Diff MS AR(1) 0.0249 0.0444 0.5605 0.5752 (-0.0621 | 0.1119)
Diff BAC to Diff MS AR(1) 0.0456 0.0783 0.5833 0.5597 (-0.1077 | 0.199)
Diff C to Diff MS AR(1) 0.0083 0.0726 0.1148 0.9086 (-0.134 | 0.1507)
Diff GS to Diff MS AR(1) 0.1319 0.0581 2.2717 0.0231 (0.0181 | 0.2457)
Diff JPM to Diff MS AR(1) -0.1771 0.0606 -2.9213 0.0035 (-0.296 | -0.0583)
Diff WFC to Diff MS AR(1) -0.151 0.0811 -1.8629 0.0625 (-0.31 | 0.0079)
Diff MS AR(2) 0.1512 0.0499 3.0308 0.0024 (0.0534 | 0.249)
Diff BAC to Diff MS AR(2) -0.2173 0.0772 -2.8157 0.0049 (-0.3686 | -0.066)
Diff C to Diff MS AR(2) 0.1827 0.0716 2.5499 0.0108 (0.0423 | 0.3231)
Diff GS to Diff MS AR(2) -0.0107 0.0573 -0.1873 0.8514 (-0.1229 | 0.1015)
Diff JPM to Diff MS AR(2) 0.0004 0.0598 0.0066 0.9947 (-0.1168 | 0.1176)
Diff WFC to Diff MS AR(2) -0.0697 0.08 -0.8711 0.3837 (-0.2264 | 0.0871)
Diff WFC Constant 0.0005 0.0492 0.0095 0.9924 (-0.096 | 0.0969)
Diff WFC AR(1) 0.0092 0.0574 0.1611 0.872 (-0.1032 | 0.1217)
Diff BAC to Diff WFC AR(1)-0.0059 0.0532 -0.1113 0.9114 (-0.1103 | 0.0984)
Diff C to Diff WFC AR(1) 0.0062 0.0425 0.1448 0.8848 (-0.0772 | 0.0896)
Diff GS to Diff WFC AR(1) 0.0525 0.0444 1.1811 0.2376 (-0.0346 | 0.1396)
Diff JPM to Diff WFC AR(1)-0.0047 0.0594 -0.0792 0.9368 (-0.1212 | 0.1118)
Diff MS to Diff WFC AR(1) -0.1996 0.0366 -5.4578 0.0 (-0.2713 | -0.1279)
Diff WFC AR(2) 0.0291 0.0773 0.3759 0.707 (-0.1225 | 0.1806)
Diff BAC to Diff WFC AR(2)-0.0509 0.0718 -0.7087 0.4785 (-0.1915 | 0.0898)
Diff C to Diff WFC AR(2) 0.0255 0.0574 0.4444 0.6567 (-0.0869 | 0.1379)
Diff GS to Diff WFC AR(2) 0.0235 0.0599 0.3922 0.6949 (-0.0939 | 0.1409)
Diff JPM to Diff WFC AR(2)0.015 0.0801 0.1878 0.851 (-0.142 | 0.1721)
Diff MS to Diff WFC AR(2) -0.0556 0.0493 -1.1276 0.2595 (-0.1522 | 0.041)
==========================================================================================
我们可以使用plot_z()方法来绘制潜在变量：
model.plot_z(list(range(0,6)),figsize=(15,5))
我们可以使用plot_fit()绘制样本内拟合图：
model.plot_fit(figsize=(15,5))
我们可以使用plot_predict()来用我们的模型进行前向预测：
model.plot_predict(past_values=19, h=5, figsize=(15,5))
我们的模型表现如何？我们可以通过进行滚动样本内预测来了解——plot_predict_is()：用于绘制图表：
model.plot_predict_is(h=30, figsize=((15,5)))
类描述¶
-
class
VAR(data, lags, integ, target, use_ols_covariance)¶ 向量自回归模型(VAR)。
参数 类型 描述 data pd.DataFrame or np.ndarray 包含单变量时间序列 lags int 自回归滞后项的数量 integ int 数据差分次数 (默认: 0) target string or int 指定使用DataFrame/array中的哪一列。 use_ols_covariance boolean 是否使用固定的OLS协方差 属性
-
latent_variables¶ 一个包含模型潜在变量信息的pf.LatentVariables()对象，包括先验设置、任何拟合值、初始值和其他潜在变量信息。当模型被拟合时，这里就是潜在变量被更新/存储的地方。有关此对象内属性的信息以及访问潜在变量信息的方法，请参阅潜在变量文档。
方法
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adjust_prior(index, prior)¶ 调整模型潜在变量的先验分布。潜在变量及其索引可以通过打印附加到模型实例的
latent_variables属性来查看。参数 类型 描述 index int 要更改的潜变量索引 prior pf.Family instance Prior distribution, e.g. pf.Normal()返回: void - 修改模型的
latent_variables属性
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fit(method, **kwargs)¶ 估计模型的潜在变量。用户选择一个推断选项，该方法会返回一个结果对象，同时更新模型的
latent_variables属性。参数 类型 描述 method str 推断选项：例如 'M-H' 或 'MLE' 请参阅文档中的贝叶斯推断和经典推断部分，了解完整的推断选项列表。可以输入与所选特定推断模式相关的可选参数。
返回: 包含估计潜在变量信息的pf.Results实例
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plot_fit(**kwargs)¶ 绘制模型对数据的拟合情况。可选参数包括figsize，即绘图图形的尺寸。
返回 : void - 显示一个matplotlib绘图
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plot_predict(h, past_values, intervals, **kwargs)¶ 绘制模型的预测结果，并附带置信区间。
参数 类型 描述 h int 预测向前多少步 past_values int 要绘制的历史数据点数量 intervals boolean 是否绘制区间 可选参数包括figsize - 图表绘制的尺寸。请注意 如果您使用最大似然估计或变分推断，显示的区间将不会 反映潜在变量的不确定性。只有Metropolis-Hastings方法能提供完全贝叶斯 预测区间。由于平均场推断的局限性（无法考虑后验相关性）， 变分推断的贝叶斯区间不予显示。
返回 : void - 显示一个matplotlib绘图
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plot_predict_is(h, fit_once, fit_method, **kwargs)¶ 绘制模型在样本内的滚动预测。这意味着用户假装数据的最后一部分是样本外的，并在每个时间段后进行预测并评估其表现。用户可以选择是在开始时一次性拟合参数，还是在每个时间步都进行拟合。
参数 类型 描述 h int 使用多少个先前的时间步 fit_once boolean 是否只拟合一次，还是每个时间步都拟合 fit_method str 选择哪种推断方法，例如'MLE' 可选参数包括figsize - 要绘制的图形尺寸。h是一个整数，表示要模拟性能的前几步。
返回 : void - 显示一个matplotlib绘图
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plot_z(indices, figsize)¶ 返回潜在变量及其相关不确定性的绘图。
参数 类型 描述 indices int or list 要绘制的潜变量索引 figsize tuple matplotlib图形的大小 返回 : void - 显示一个matplotlib绘图
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predict(h)¶ 返回模型预测结果的DataFrame。
参数 类型 描述 h int 预测向前多少步 返回 : pd.DataFrame - 模型预测结果
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predict_is(h, fit_once, fit_method)¶ 返回模型样本内滚动预测的DataFrame。
参数 类型 描述 h int 使用多少个先前的时间步 fit_once boolean 是否只拟合一次，还是每个时间步都拟合 fit_method str 选择哪种推断方法，例如'MLE' 返回 : pd.DataFrame - 模型预测结果
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simulation_smoother(beta)¶ 返回从Durbin和Koopman(2002)模拟平滑器中抽取的数据的np.ndarray数组。
参数 类型 描述 beta np.array 潜在变量的np.array数组 如果已经拟合了模型，建议直接使用model.latent_variables.get_z_values()作为beta输入。
返回 : np.ndarray - 来自模拟平滑器的样本
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参考文献¶
Lütkepohl, H. 和 Kraetzig, M. (2004). 《应用时间序列计量经济学》. 剑桥大学出版社, 剑桥.