python - statsmodel 线性回归(ols)的鲁棒性问题 - Python

Question

我正在使用统计模型测试一些基本类别回归:我建立了一个确定性模型

Y = X + Z

其中 X 可以取 3 个值(a、b 或 c)，而 Z 只能取 2 个值(d 或 e)。在那个阶段，模型是纯粹确定性的，我如下设置每个变量的权重

a's weight=1

b's weight=2

c's weight=3

d's weight=1

e's weight=2

因此，如果 X=a，则 1(X=a) 为 1，否则为 0，模型很简单:

Y = 1(X=a) + 2*1(X=b) + 3*1(X=c) + 1(Z=d) + 2*1(Z=e)

使用以下代码生成不同的变量并运行回归

from statsmodels.formula.api import ols
nbData = 1000
rand1 = np.random.uniform(size=nbData)
rand2 = np.random.uniform(size=nbData)
a = 1 * (rand1 <= (1.0/3.0))
b = 1 * (((1.0/3.0)< rand1) & (rand1< (4/5.0)))
c = 1-b-a
d = 1 * (rand2 <= (3.0/5.0))
e = 1-d
weigths = [1,2,3,1,2]
y = a+2*b+3*c+4*d+5*e
df = pd.DataFrame({'y':y, 'a':a, 'b':b, 'c':c, 'd':d, 'e':e})

mod = ols(formula='y ~ a + b + c + d + e - 1', data=df)
res = mod.fit()
print(res.summary())

我最终得到了正确的结果(必须查看 coef 之间的区别而不是 coef 本身)

                           OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       1.000
Model:                            OLS   Adj. R-squared:                  1.000
Method:                 Least Squares   F-statistic:                 1.006e+30
Date:                Wed, 16 Sep 2015   Prob (F-statistic):               0.00
Time:                        03:05:40   Log-Likelihood:                 3156.8
No. Observations:                 100   AIC:                            -6306.
Df Residuals:                      96   BIC:                            -6295.
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
a              1.6000   7.47e-16   2.14e+15      0.000         1.600     1.600
b              2.6000   6.11e-16   4.25e+15      0.000         2.600     2.600
c              3.6000   9.61e-16   3.74e+15      0.000         3.600     3.600
d              3.4000   5.21e-16   6.52e+15      0.000         3.400     3.400
e              4.4000   6.85e-16   6.42e+15      0.000         4.400     4.400
==============================================================================
Omnibus:                       11.299   Durbin-Watson:                   0.833
Prob(Omnibus):                  0.004   Jarque-Bera (JB):                5.720
Skew:                          -0.381   Prob(JB):                       0.0573
Kurtosis:                       2.110   Cond. No.                     2.46e+15
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 1.67e-29. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.

但是当我将数据点的数量增加到(比如说)600 时，回归产生了非常糟糕的结果。我在 Excel 和 R 中尝试过类似的回归，无论数据点的数量有多少，它们都会产生一致的结果。有谁知道 statsmodel ols 是否对解释此类行为有一些限制，或者我是否遗漏了什么？

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.167
Model:                            OLS   Adj. R-squared:                  0.161
Method:                 Least Squares   F-statistic:                     29.83
Date:                Wed, 16 Sep 2015   Prob (F-statistic):           1.23e-22
Time:                        03:08:04   Log-Likelihood:                -701.02
No. Observations:                 600   AIC:                             1412.
Df Residuals:                     595   BIC:                             1434.
Df Model:                           4                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
a              5.8070   1.15e+13   5.05e-13      1.000     -2.26e+13  2.26e+13
b              6.4951   1.15e+13   5.65e-13      1.000     -2.26e+13  2.26e+13
c              6.9033   1.15e+13   6.01e-13      1.000     -2.26e+13  2.26e+13
d             -1.1927   1.15e+13  -1.04e-13      1.000     -2.26e+13  2.26e+13
e             -0.1685   1.15e+13  -1.47e-14      1.000     -2.26e+13  2.26e+13
==============================================================================
Omnibus:                       67.153   Durbin-Watson:                   0.328
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               70.964
Skew:                           0.791   Prob(JB):                     3.89e-16
Kurtosis:                       2.419   Cond. No.                     7.70e+14
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 9.25e-28. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.

Answer 1

看来，正如 F 先生所提到的，主要问题是在这种情况下，statsmodel OLS 似乎无法像处理 Excel/R 那样处理共线性 pb，但是如果不是为每个定义一个变量>a, b, c, d and e，一个定义一个变量X，一个Z可以等于a, b or c 和 d 或 e resp，然后回归工作正常。即更新代码:

df['X'] = ['c']*len(df)
df.X[df.b!=0] = 'b'
df.X[df.a!=0] = 'a'
df['Z'] = ['e']*len(df)
df.Z[df.d!=0] = 'd'
mod = ols(formula='y ~ X + Z - 1', data=df)

导致预期的结果

                           OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       1.000
Model:                            OLS   Adj. R-squared:                  1.000
Method:                 Least Squares   F-statistic:                 2.684e+27
Date:                Thu, 17 Sep 2015   Prob (F-statistic):               0.00
Time:                        06:22:43   Log-Likelihood:             2.5096e+06
No. Observations:              100000   AIC:                        -5.019e+06
Df Residuals:                   99996   BIC:                        -5.019e+06
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
X[a]           5.0000   1.85e-14    2.7e+14      0.000         5.000     5.000
X[b]           6.0000   1.62e-14   3.71e+14      0.000         6.000     6.000
X[c]           7.0000   2.31e-14   3.04e+14      0.000         7.000     7.000
Z[T.e]         1.0000   1.97e-14   5.08e+13      0.000         1.000     1.000
==============================================================================
Omnibus:                      145.367   Durbin-Watson:                   1.353
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             9729.487
Skew:                          -0.094   Prob(JB):                         0.00
Kurtosis:                       1.483   Cond. No.                         2.29
==============================================================================

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