【Python计量】内生性问题、工具变量法与二阶段最小二乘法2SLS

我们以伍德里奇《计量经济学导论：现代方法》的”第15章 工具变量估计与两阶段最小二二乘法“的案例15.5为例，使用美国女性教育回报数据MORZ，学习工具变量法的Python实现。

变量：被解释变量log(wage)为工资的对数，解释变量educ为受正式教育年数，exper为工作经验。

构建模型如下：

$$log(wage)={\beta }_0+{\beta }_{1}educ+{\beta }_{2}exper+{\beta }_{3}expe{r}^2+u$$

上式仅考虑了职业女性自身受正式教育年数的影响，存在遗漏变量的情况，引发内生性问题。因此，考虑将父亲的受教育程度fathereduc、母亲的受教育程度mothereduc作为工具变量，fathereduc、mothereduc应该与educ相关，而与u无关。

一、内生性问题与二阶段最小二乘法

什么是内生性？

• 内生性是指解释变量和误差项ε存在相关性，导致最小二乘估计的参数β有偏、不一致。

什么情况下会产生内生性？

• 遗漏重要解释变量
• 解释变量与被解释变量互为因果
• 变量测量误差，观测到的x,y与真实的x和y存在一定的差距

工具变量的要求

举个例子：对于简单的 $y=\alpha +\beta x+\epsilon$

如果扰动项与$x$相关，我们可以设置一个工具变量$z$，使得$z$满足以下两个条件：

（1）相关性：$z$与$x$相关， $Cov(x,z)\ne 0$

（2）外生性：z与扰动项无关。$Cov(\epsilon ,z)=0$

工具变量法的实现

工具变量法一般通过“二阶段最小二乘法”（2SLS，Two Stage Least Square) 来实现，其中的两个阶段是：

（1）求 x 对 z 的回归，得到一个x的拟合值；

（2）求 y 对 x 拟合值的回归，得到$\beta$，由于此阶段的回归中，x 的拟合值与扰动项不相关（OLS的正交性），所以可以得到一致的估计量。

简单来说，2SLS 在回归的第一阶段，把 x分成了两部分，一部分是x的拟合值，另一部分是与扰动项相关的部分；然后在第二阶段中求 y 对 x 拟合值的回归，也就是对消去内生性部分的 x 的回归，故可以得到一致的估计。

二、工具变量法的Python实现

（一）准备数据

import wooldridge as woo
import pandas as pd
mroz = woo.dataWoo('mroz')
#去除缺失值
mroz = mroz.dropna(subset=['lwage'])

（二）工具变量法

1、采用statsmodels进行2SLS回归

（1）内生变量对工具变量做回归，获得内生变量拟合值

$$educ={\pi }_0+{\pi }_{1}exper+{\beta }_{2}expe{r}^2+{\beta }_{3}mothereduc+{\beta }_{4}fathereduc+v$$

import statsmodels.formula.api as smf
#1阶段回归
reg_1st= smf.ols(formula='educ ~ exper + expersq + motheduc + fatheduc',
                   data=mroz)
results_1st = reg_1st.fit()
mroz['educ_fitted'] = results_1st.fittedvalues
print(results_1st.summary()

结果如下：

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                   educ   R-squared:                       0.211
Model:                            OLS   Adj. R-squared:                  0.204
Method:                 Least Squares   F-statistic:                     28.36
Date:                Sun, 17 Jul 2022   Prob (F-statistic):           6.87e-21
Time:                        16:34:39   Log-Likelihood:                -909.72
No. Observations:                 428   AIC:                             1829.
Df Residuals:                     423   BIC:                             1850.
Df Model:                           4                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      9.1026      0.427     21.340      0.000       8.264       9.941
exper          0.0452      0.040      1.124      0.262      -0.034       0.124
expersq       -0.0010      0.001     -0.839      0.402      -0.003       0.001
motheduc       0.1576      0.036      4.391      0.000       0.087       0.228
fatheduc       0.1895      0.034      5.615      0.000       0.123       0.256
==============================================================================
Omnibus:                       10.903   Durbin-Watson:                   1.940
Prob(Omnibus):                  0.004   Jarque-Bera (JB):               20.371
Skew:                          -0.013   Prob(JB):                     3.77e-05
Kurtosis:                       4.068   Cond. No.                     1.55e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.55e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

（2）被解释变量对内生变量拟合值做回归

$$log(wage)={\beta }_0+{\beta }_{1}educ+{\beta }_{2}exper+{\beta }_{3}expe{r}^2+u$$

#2阶段回归
reg_2nd = smf.ols(formula='lwage ~ educ_fitted + exper + expersq',
                     data=mroz)
results_2nd = reg_2nd.fit()
print(results_2nd.summary())

结果如下：

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                  lwage   R-squared:                       0.050
Model:                            OLS   Adj. R-squared:                  0.043
Method:                 Least Squares   F-statistic:                     7.405
Date:                Sun, 17 Jul 2022   Prob (F-statistic):           7.62e-05
Time:                        16:40:02   Log-Likelihood:                -457.17
No. Observations:                 428   AIC:                             922.3
Df Residuals:                     424   BIC:                             938.6
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
===============================================================================
                  coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------
Intercept       0.0481      0.420      0.115      0.909      -0.777       0.873
educ_fitted     0.0614      0.033      1.863      0.063      -0.003       0.126
exper           0.0442      0.014      3.136      0.002       0.016       0.072
expersq        -0.0009      0.000     -2.134      0.033      -0.002   -7.11e-05
==============================================================================
Omnibus:                       53.587   Durbin-Watson:                   1.959
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              168.354
Skew:                          -0.551   Prob(JB):                     2.77e-37
Kurtosis:                       5.868   Cond. No.                     4.41e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 4.41e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

2、采用linearmodels进行2SLS回归

使用linearmodels工具包中的IV2SLS工具，首先需要导入库

from linearmodels.iv import IV2SLS 

其次可加载公式，进行2SLS回归

IV2SLS(formula,data)
formula:回归方程，形式为：dep~exog+[endog~instr],其中exog表示外生变量，endog表示内生变量，instr表示工具变量

具体到本例，代码如下：

from linearmodels.iv import IV2SLS 
reg_iv = IV2SLS.from_formula(
    formula='lwage ~ 1 + exper + expersq + [educ ~ motheduc + fatheduc]',
    data=mroz)
results_iv = reg_iv.fit(cov_type='unadjusted', debiased=True)
print(results_iv)

结果如下：

                          IV-2SLS Estimation Summary                          
==============================================================================
Dep. Variable:                  lwage   R-squared:                      0.1357
Estimator:                    IV-2SLS   Adj. R-squared:                 0.1296
No. Observations:                 428   F-statistic:                    8.1407
Date:                Sun, Jul 17 2022   P-value (F-stat)                0.0000
Time:                        16:42:41   Distribution:                 F(3,424)
Cov. Estimator:            unadjusted                                         

                             Parameter Estimates                              
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
Intercept      0.0481     0.4003     0.1202     0.9044     -0.7388      0.8350
exper          0.0442     0.0134     3.2883     0.0011      0.0178      0.0706
expersq       -0.0009     0.0004    -2.2380     0.0257     -0.0017     -0.0001
educ           0.0614     0.0314     1.9530     0.0515     -0.0004      0.1232
==============================================================================

Endogenous: educ
Instruments: fatheduc, motheduc
Unadjusted Covariance (Homoskedastic)
Debiased: True

三、工具变量相关检验

构建模型如下：

$$log(wage)={\beta }_0+{\beta }_{1}educ+{\beta }_{2}exper+{\beta }_{3}expe{r}^2+u$$

（一）变量内生性检验

1、将疑是内生变量$educ$对外生变量和工具变量做回归，得到残差$v$。

$$educ={\pi }_0+{\pi }_{1}exper+{\beta }_{2}expe{r}^2+{\beta }_{3}mothereduc+{\beta }_{4}fathereduc+v$$

import statsmodels.formula.api as smf
#1阶段回归
reg_1st= smf.ols(formula='educ ~ exper + expersq + motheduc + fatheduc',
                   data=mroz)
results_1st = reg_1st.fit()
mroz['resid'] = results_1st.resid #获得残差

2、在原方程中将残差$v$也作为一个变量加入，用OLS模型检验系数及其显著性，如果$v$的系数显著异于零，则$educ$变量是内生的。

$$log(wage)={\beta }_0+{\beta }_{1}v+{\beta }_{2}educ+{\beta }_{3}exper+{\beta }_{4}expe{r}^2+u$$

#2阶段回归
reg_2 = smf.ols(formula='lwage~ resid + educ + exper + expersq',
                     data=mroz)
results_2 = reg_2.fit()
print(results_2.summary())

结果如下：

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                  lwage   R-squared:                       0.162
Model:                            OLS   Adj. R-squared:                  0.154
Method:                 Least Squares   F-statistic:                     20.50
Date:                Sun, 17 Jul 2022   Prob (F-statistic):           1.89e-15
Time:                        17:02:34   Log-Likelihood:                -430.19
No. Observations:                 428   AIC:                             870.4
Df Residuals:                     423   BIC:                             890.7
Df Model:                           4                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      0.0481      0.395      0.122      0.903      -0.727       0.824
resid          0.0582      0.035      1.671      0.095      -0.010       0.127
educ           0.0614      0.031      1.981      0.048       0.000       0.122
exper          0.0442      0.013      3.336      0.001       0.018       0.070
expersq       -0.0009      0.000     -2.271      0.024      -0.002      -0.000
==============================================================================
Omnibus:                       74.968   Durbin-Watson:                   1.931
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              278.059
Skew:                          -0.736   Prob(JB):                     4.17e-61
Kurtosis:                       6.664   Cond. No.                     4.42e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 4.42e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

（二）过度识别检测

步骤：

1、用2SLS方法估计方程，得到残差$\hat{u}$

2、将$\hat{u}$对所有外生变量和工具变量回归，得到$R^2$

3、原假设为：所有工具变量与$u$不相关，于是$n{R}^2\sim X_q^2$，其中$q$是工具变量数量减去内生变量数量。如果$n{R}^2$大于$X_q^2$某个显著性水平的临界值，则拒绝所有变量都是外生的原假设

from linearmodels.iv import IV2SLS 
import statsmodels.formula.api as smf
import scipy.stats as stats

#第一步，用2SLS法估计方程，得到残差
reg_iv = IV2SLS.from_formula(
    formula='lwage ~ 1 + exper + expersq + [educ ~ motheduc + fatheduc]',
    data=mroz)
results_iv = reg_iv.fit(cov_type='unadjusted', debiased=True)

#第二步，将残差对所有外生变量和工具变量回归
mroz['resid_iv'] = results_iv.resids
reg_aux = smf.ols(formula='resid_iv ~ exper + expersq + motheduc + fatheduc',
                  data=mroz)
results_aux = reg_aux.fit()

#第三步，显著性判断
r2 = results_aux.rsquared
n = results_aux.nobs
q = 2-1
teststat = n * r2
pval = 1 - stats.chi2.cdf(teststat, q)

print(f'r2: {r2}')
print(f'n: {n}')
print(f'teststat: {teststat}')
print(f'pval: {pval}')

运行结果为：

r2: 0.0008833442569250449
n: 428.0
teststat: 0.3780713419639192
pval: 0.5386372330714363

经过上述步骤我们得到，有n=428条观测数据，p值为0.539，在5%的显著性水平下不能拒绝原假设，即父母的受教育程度通过了过度识别检测，可作为工具变量。