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import numpy as np
import statsmodels.api as smTesting default example from statsmodel.
import numpy as np
import statsmodels.api as sm
spector_data = sm.datasets.spector.load(as_pandas=False)
spector_data.exog = sm.add_constant(spector_data.exog, prepend=False)
mod = sm.OLS(spector_data.endog, spector_data.exog)
res = mod.fit()
print(res.summary())## OLS Regression Results
## ==============================================================================
## Dep. Variable: y R-squared: 0.416
## Model: OLS Adj. R-squared: 0.353
## Method: Least Squares F-statistic: 6.646
## Date: Tue, 05 Jan 2021 Prob (F-statistic): 0.00157
## Time: 16:36:00 Log-Likelihood: -12.978
## No. Observations: 32 AIC: 33.96
## Df Residuals: 28 BIC: 39.82
## Df Model: 3
## Covariance Type: nonrobust
## ==============================================================================
## coef std err t P>|t| [0.025 0.975]
## ------------------------------------------------------------------------------
## x1 0.4639 0.162 2.864 0.008 0.132 0.796
## x2 0.0105 0.019 0.539 0.594 -0.029 0.050
## x3 0.3786 0.139 2.720 0.011 0.093 0.664
## const -1.4980 0.524 -2.859 0.008 -2.571 -0.425
## ==============================================================================
## Omnibus: 0.176 Durbin-Watson: 2.346
## Prob(Omnibus): 0.916 Jarque-Bera (JB): 0.167
## Skew: 0.141 Prob(JB): 0.920
## Kurtosis: 2.786 Cond. No. 176.
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.Testing the sm_OLS function where there are more observations and coefficients, as well as when there are less. Tests similar to the test on this page.
First, more observations than coefficients:
# Number of observations
it_sample = 1000000
# Vaues of the x variable
ar_x = np.linspace(0, 10, it_sample)
# generate matrix of inputs with polynomial expansion
mt_x = np.column_stack((ar_x, ar_x**2, ar_x**3, ar_x**4, ar_x**5, ar_x**6, ar_x**7, ar_x**8))
# model coefficients
ar_beta = np.array([100, 10, 1, 1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6])
# Generate the error term
ar_e = np.random.normal(size=it_sample)
# add constant
mt_x = sm.add_constant(mt_x)
# generate the outocome variable
ar_y = np.dot(mt_x, ar_beta) + ar_e
# regression result
ob_model = sm.OLS(ar_y, mt_x)
# Show results
ob_results = ob_model.fit()
print(ob_results.summary())## OLS Regression Results
## ==============================================================================
## Dep. Variable: y R-squared: 1.000
## Model: OLS Adj. R-squared: 1.000
## Method: Least Squares F-statistic: 4.991e+09
## Date: Tue, 05 Jan 2021 Prob (F-statistic): 0.00
## Time: 16:36:01 Log-Likelihood: -1.4183e+06
## No. Observations: 1000000 AIC: 2.837e+06
## Df Residuals: 999991 BIC: 2.837e+06
## Df Model: 8
## Covariance Type: nonrobust
## ==============================================================================
## coef std err t P>|t| [0.025 0.975]
## ------------------------------------------------------------------------------
## const 99.9866 0.009 1.11e+04 0.000 99.969 100.004
## x1 10.0545 0.042 242.037 0.000 9.973 10.136
## x2 0.9236 0.062 14.911 0.000 0.802 1.045
## x3 0.1481 0.042 3.536 0.000 0.066 0.230
## x4 -0.0059 0.015 -0.394 0.693 -0.035 0.023
## x5 0.0040 0.003 1.308 0.191 -0.002 0.010
## x6 -0.0002 0.000 -0.621 0.535 -0.001 0.000
## x7 2.797e-05 2.13e-05 1.316 0.188 -1.37e-05 6.96e-05
## x8 5.787e-07 5.3e-07 1.091 0.275 -4.61e-07 1.62e-06
## ==============================================================================
## Omnibus: 1.572 Durbin-Watson: 2.002
## Prob(Omnibus): 0.456 Jarque-Bera (JB): 1.568
## Skew: 0.002 Prob(JB): 0.456
## Kurtosis: 3.004 Cond. No. 2.10e+09
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## [2] The condition number is large, 2.1e+09. This might indicate that there are
## strong multicollinearity or other numerical problems.second, less observations than coefficients. Note that there are nine coefficients to estimates, so if we have only 5 observations, that is less than coefficients. Unlike Stata and many other packages, estimates are provided even when full rank is not possible. See here for more information. This is actually very useful for testing purposes. For models in very large parameter space, can test solution and estimation structure even when the number of observations are limited. See also Moore–Penrose inverse.
# Number of observations
it_sample = 5
# Vaues of the x variable
ar_x = np.linspace(0, 10, it_sample)
# generate matrix of inputs with polynomial expansion
mt_x = np.column_stack((ar_x, ar_x**2, ar_x**3, ar_x**4, ar_x**5, ar_x**6, ar_x**7, ar_x**8))
# model coefficients
ar_beta = np.array([100, 10, 1, 1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6])
# Generate the error term
ar_e = np.random.normal(size=it_sample)
# add constant
mt_x = sm.add_constant(mt_x)
# generate the outocome variable
ar_y = np.dot(mt_x, ar_beta) + ar_e
# regression result
ob_model = sm.OLS(ar_y, mt_x)
# Show results
ob_results = ob_model.fit()
print(ob_results.summary())## OLS Regression Results
## ==============================================================================
## Dep. Variable: y R-squared: 1.000
## Model: OLS Adj. R-squared: nan
## Method: Least Squares F-statistic: nan
## Date: Tue, 05 Jan 2021 Prob (F-statistic): nan
## Time: 16:36:01 Log-Likelihood: 95.245
## No. Observations: 5 AIC: -180.5
## Df Residuals: 0 BIC: -182.4
## Df Model: 4
## Covariance Type: nonrobust
## ==============================================================================
## coef std err t P>|t| [0.025 0.975]
## ------------------------------------------------------------------------------
## const 100.7089 inf 0 nan nan nan
## x1 0.0648 inf 0 nan nan nan
## x2 0.1482 inf 0 nan nan nan
## x3 0.3106 inf 0 nan nan nan
## x4 0.5401 inf 0 nan nan nan
## x5 0.5638 inf 0 nan nan nan
## x6 -0.2963 inf -0 nan nan nan
## x7 0.0427 inf 0 nan nan nan
## x8 -0.0019 inf -0 nan nan nan
## ==============================================================================
## Omnibus: nan Durbin-Watson: 1.088
## Prob(Omnibus): nan Jarque-Bera (JB): 1.682
## Skew: 1.419 Prob(JB): 0.431
## Kurtosis: 3.143 Cond. No. 1.01e+08
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## [2] The input rank is higher than the number of observations.
## [3] The condition number is large, 1.01e+08. This might indicate that there are
## strong multicollinearity or other numerical problems.
##
## C:\Users\fan\AppData\Roaming\Python\Python38\site-packages\statsmodels\stats\stattools.py:70: ValueWarning: omni_normtest is not valid with less than 8 observations; 5 samples were given.
## warn("omni_normtest is not valid with less than 8 observations; %i "