为什么我要从格兰杰因果学家那里得到“线性误差：奇异矩阵”？

Question

我试图在两个时间序列上运行grangercausalitytests：

import numpy as np
import pandas as pd

from statsmodels.tsa.stattools import grangercausalitytests

n = 1000
ls = np.linspace(0, 2*np.pi, n)

df1 = pd.DataFrame(np.sin(ls))
df2 = pd.DataFrame(2*np.sin(1+ls))

df = pd.concat([df1, df2], axis=1)

df.plot()

grangercausalitytests(df, maxlag=20)

但是，我得到了

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=272078066917221398041264652288.0000, p=0.0000  , df_denom=996, df_num=1
ssr based chi2 test:   chi2=272897579166972095424217743360.0000, p=0.0000  , df=1
likelihood ratio test: chi2=60811.2671, p=0.0000  , df=1
parameter F test:         F=272078066917220553616334520320.0000, p=0.0000  , df_denom=996, df_num=1

Granger Causality
number of lags (no zero) 2
ssr based F test:         F=7296.6976, p=0.0000  , df_denom=995, df_num=2
ssr based chi2 test:   chi2=14637.3954, p=0.0000  , df=2
likelihood ratio test: chi2=2746.0362, p=0.0000  , df=2
parameter F test:         F=13296850090491009488285469769728.0000, p=0.0000  , df_denom=995, df_num=2
...
/usr/local/lib/python3.5/dist-packages/numpy/linalg/linalg.py in _raise_linalgerror_singular(err, flag)
     88 
     89 def _raise_linalgerror_singular(err, flag):
---> 90     raise LinAlgError("Singular matrix")
     91 
     92 def _raise_linalgerror_nonposdef(err, flag):

LinAlgError: Singular matrix

我不知道为什么会这样。

Answer 1

这个问题是由于数据中两个序列之间的完美相关性而产生的。从回溯中可以看到，内部使用wald测试来计算滞后时间序列参数的最大似然估计。为此，需要估计参数协方差矩阵（然后接近于零）及其逆矩阵（您也可以在回溯中的invcov = np.linalg.inv(cov_p)行中看到）。对于某些最大滞后数（>；=5），这个接近零的矩阵现在是奇异的，因此测试崩溃。如果只在数据中添加一点噪声，则错误将消失：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.stattools import grangercausalitytests

n = 1000
ls = np.linspace(0, 2*np.pi, n)
df1Clean = pd.DataFrame(np.sin(ls))
df2Clean = pd.DataFrame(2*np.sin(ls+1))
dfClean = pd.concat([df1Clean, df2Clean], axis=1)
dfDirty = dfClean+0.00001*np.random.rand(n, 2)

grangercausalitytests(dfClean, maxlag=20, verbose=False)    # Raises LinAlgError
grangercausalitytests(dfDirty, maxlag=20, verbose=False)    # Runs fine