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自己相関

『Rによる計量経済学』第5章「自己相関」をPythonで実行する。
テキスト付属データセット(「k0501.csv」等)については出版社サイトよりダウンロードしてください。
また、以下の説明は本書の一部を要約したものですので、より詳しい説明は本書を参照してください。





In [1]:



    


%matplotlib inline



    











In [2]:



    


# -*- coding:utf-8 -*-
from __future__ import print_function
import numpy as np
import pandas as pd
import statsmodels.api as sm
from statsmodels.stats.stattools import durbin_watson
import matplotlib.pyplot as plt
import warnings
warnings.filterwarnings('ignore')

例題5-1





In [3]:



    


# データ読み込み
data = pd.read_csv('example/k0501.csv')
data



    


















    
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      Y
    

  
  
    

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      1
    

    

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In [4]:



    


# 説明変数設定
X = data[['X']]
X = sm.add_constant(X)
X



    


















    
Out[4]:










  
    

      
      const
      X
    

  
  
    

      0
      1
      1
    

    

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In [5]:



    


# 被説明変数設定
Y = data['Y']
Y



    


















    
Out[5]:








0     1
1     3
2     4
3     5
4     4
5     6
6     7
7     8
8    10
9    10
Name: Y, dtype: int64

















In [6]:



    


# OLSの実行(Ordinary Least Squares: 最小二乗法)
model = sm.OLS(Y,X)
results = model.fit()
print(results.summary())



    


















    






                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      Y   R-squared:                       0.950
Model:                            OLS   Adj. R-squared:                  0.944
Method:                 Least Squares   F-statistic:                     153.2
Date:                Sun, 19 Jul 2015   Prob (F-statistic):           1.69e-06
Time:                        04:02:41   Log-Likelihood:                -9.5470
No. Observations:                  10   AIC:                             23.09
Df Residuals:                       8   BIC:                             23.70
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
const          0.5333      0.480      1.111      0.299        -0.574     1.640
X              0.9576      0.077     12.376      0.000         0.779     1.136
==============================================================================
Omnibus:                        0.891   Durbin-Watson:                   2.029
Prob(Omnibus):                  0.641   Jarque-Bera (JB):                0.480
Skew:                          -0.492   Prob(JB):                        0.787
Kurtosis:                       2.571   Cond. No.                         13.7
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

















In [7]:



    


# グラフ生成
plt.plot(data["X"], data["Y"], 'o', label="data")
plt.plot(data["X"], results.fittedvalues, label="OLS")
plt.xlim(min(data["X"])-1, max(data["X"])+1)
plt.ylim(min(data["Y"])-1, max(data["Y"])+1)
plt.title('5-1: Auto Correlation')
plt.legend(loc=2)
plt.show()

Durbin-Watson: 2.029となり、2に近い値であり、自己相関がないと結論出来る。
DW統計量に対応するP値を求めることがPythonではおそらくできない(Excelでも出来ない)ため、ダービン=ワトソン統計量の分布表を利用して有意性の検定を行います。

この場合、標本の大きさ=10, 定数項を除いた説明変数の数=1なので、有意水準1%において、下限分布ではDW=0.604, 上限分布ではDW=1.001が有意となる境界、有意水準5%において、下限分布ではDW=0.879, 上限分布ではDW=1.320が有意となる境界です。(詳しくは本書の表を参考にしてください。)

例題5-2





In [8]:



    


# データ読み込み
data = pd.read_csv('example/k0502.csv')
data



    


















    
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      10.0
    

  




















In [9]:



    


# 説明変数設定
X = data[['X']]
X = sm.add_constant(X)
# 被説明変数設定
Y = data['Y']
# OLSの実行(Ordinary Least Squares: 最小二乗法)
model = sm.OLS(Y,X)
results = model.fit()
print(results.summary())
# グラフ生成
plt.plot(data["X"], data["Y"], 'o', label="data")
plt.plot(data["X"], results.fittedvalues, label="OLS")
plt.xlim(min(data["X"])-1, max(data["X"])+1)
plt.ylim(min(data["Y"])-1, max(data["Y"])+1)
plt.title('5-2: Auto Correlation')
plt.legend(loc=2)
plt.show()



    


















    






                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      Y   R-squared:                       0.945
Model:                            OLS   Adj. R-squared:                  0.938
Method:                 Least Squares   F-statistic:                     136.2
Date:                Sun, 19 Jul 2015   Prob (F-statistic):           2.65e-06
Time:                        04:02:41   Log-Likelihood:                -8.0714
No. Observations:                  10   AIC:                             20.14
Df Residuals:                       8   BIC:                             20.75
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
const          1.2739      0.398      3.197      0.013         0.355     2.193
X              0.7548      0.065     11.671      0.000         0.606     0.904
==============================================================================
Omnibus:                        1.970   Durbin-Watson:                   1.094
Prob(Omnibus):                  0.373   Jarque-Bera (JB):                0.741
Skew:                          -0.665   Prob(JB):                        0.690
Kurtosis:                       2.912   Cond. No.                         13.1
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

DW=1.094となり、上限分布において有意水準5%でも帰無仮説を棄却することができ、自己相関が存在すると結論することができる。

例題5-3

変数変換によって自己相関に対処する方法を考える。





In [10]:



    


data['dX'] = np.nan
data['dY'] = np.nan
for i in range(len(data)):
    if i == 0:
        data['dX'][i] = np.nan
        data['dY'][i] = np.nan
    else:
        data['dX'][i] = data['X'][i] - data['X'][i-1]
        data['dY'][i] = data['Y'][i] - data['Y'][i-1]
data



    


















    
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      dX
      dY
    

  
  
    

      0
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      1.0
      2.0
      NaN
      NaN
    

    

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In [11]:



    


# 説明変数設定
X = data['dX'][1:]
X = sm.add_constant(X)
# 被説明変数設定
Y = data['dY'][1:]
# OLSの実行(Ordinary Least Squares: 最小二乗法)
model = sm.OLS(Y,X)
results = model.fit()
print(results.summary())
# グラフ生成
plt.plot(data["dX"], data["dY"], 'o', label="data")
plt.plot(data["dX"][1:], results.fittedvalues, label="OLS")
plt.xlim(min(data["dX"][1:])-0.5, max(data["dX"][1:])+0.5)
plt.ylim(min(data["dY"][1:])-0.5, max(data["dY"][1:])+0.5)
plt.title('5-3: Auto Correlation')
plt.legend(loc=2)
plt.show()



    


















    






                            OLS Regression Results                            
==============================================================================
Dep. Variable:                     dY   R-squared:                       0.532
Model:                            OLS   Adj. R-squared:                  0.465
Method:                 Least Squares   F-statistic:                     7.945
Date:                Sun, 19 Jul 2015   Prob (F-statistic):             0.0258
Time:                        04:02:42   Log-Likelihood:                -7.8155
No. Observations:                   9   AIC:                             19.63
Df Residuals:                       7   BIC:                             20.03
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
const         -0.1194      0.419     -0.285      0.784        -1.110     0.871
dX             0.9552      0.339      2.819      0.026         0.154     1.757
==============================================================================
Omnibus:                        1.861   Durbin-Watson:                   1.694
Prob(Omnibus):                  0.394   Jarque-Bera (JB):                0.834
Skew:                          -0.259   Prob(JB):                        0.659
Kurtosis:                       1.602   Cond. No.                         3.66
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Content source: ogaway/Econometrics

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