我正在使用Stats模型测试一些基本的类别回归: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)， Z只能取2个值(d或e)。在这个阶段，模型是完全确定的，我为每个变量设置权重，如下所示
的体重= 1
b的体重= 2
c的体重= 3
d的体重= 1
e的体重= 2
因此，当X=a时1(X=a)为1，否则为0，则模型简单为:
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(人们必须看系数之间的区别，而不是系数本身)
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，无论数据点的数量如何，它们都能产生一致的结果。有人知道statsmodel ols解释这种行为是否有一些限制吗?还是我错过了什么?
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.
看来，正如F先生所提到的，主要问题是，在这种情况下，statsmodel OLS似乎不像Excel/R那样处理共线性pb，但如果不是为每个a, b, c, d and e定义一个变量，而是定义一个变量X和一个Z，它可以等于a, b or c和d or e resp，那么回归工作得很好。我用:
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.