In [1]:

    

# Versão da Linguagem Python
from platform import python_version
print('Versão da Linguagem Python Usada Neste Jupyter Notebook:', python_version())


    

In [2]:

    

# Para visualização de gráficos
from pylab import *
%matplotlib inline


    

In [3]:

    

import numpy as np
import pandas as pd
import statsmodels as st
import sys
import warnings
if not sys.warnoptions:
    warnings.simplefilter("ignore")
warnings.simplefilter(action='ignore', category=FutureWarning)
warnings.filterwarnings("ignore", category=FutureWarning)
import statsmodels.api as sm
from statsmodels.sandbox.regression.predstd import wls_prediction_std
np.random.seed(9876789)


    

In [4]:

    

np.__version__


    

    
Out[4]:





'1.18.2'

In [5]:

    

pd.__version__


    

    
Out[5]:





'1.0.3'

In [6]:

    

st.__version__


    

    
Out[6]:





'0.11.1'

In [7]:

    

# Criando dados artificiais
nsample = 100
x = np.linspace(0, 10, 100)
X = np.column_stack((x, x**2))
beta = np.array([1, 0.1, 10])
e = np.random.normal(size=nsample)


    

In [8]:

    

X = sm.add_constant(X)
y = np.dot(X, beta) + e


    

In [9]:

    

model = sm.OLS(y, X)
results = model.fit()
print(results.summary())


    

    




                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       1.000
Model:                            OLS   Adj. R-squared:                  1.000
Method:                 Least Squares   F-statistic:                 4.020e+06
Date:                Mon, 23 Mar 2020   Prob (F-statistic):          2.83e-239
Time:                        20:09:10   Log-Likelihood:                -146.51
No. Observations:                 100   AIC:                             299.0
Df Residuals:                      97   BIC:                             306.8
Df Model:                           2                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          1.3423      0.313      4.292      0.000       0.722       1.963
x1            -0.0402      0.145     -0.278      0.781      -0.327       0.247
x2            10.0103      0.014    715.745      0.000       9.982      10.038
==============================================================================
Omnibus:                        2.042   Durbin-Watson:                   2.274
Prob(Omnibus):                  0.360   Jarque-Bera (JB):                1.875
Skew:                           0.234   Prob(JB):                        0.392
Kurtosis:                       2.519   Cond. No.                         144.
==============================================================================

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

In [10]:

    

print('Parameters: ', results.params)
print('R2: ', results.rsquared)


    

    




Parameters:  [ 1.34233516 -0.04024948 10.01025357]
R2:  0.9999879365025871

In [11]:

    

nsample = 50
sig = 0.5
x = np.linspace(0, 20, nsample)
X = np.column_stack((x, np.sin(x), (x-5)**2, np.ones(nsample)))
beta = [0.5, 0.5, -0.02, 5.]

y_true = np.dot(X, beta)
y = y_true + sig * np.random.normal(size=nsample)

res = sm.OLS(y, X).fit()
print(res.summary())


    

    




                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.933
Model:                            OLS   Adj. R-squared:                  0.928
Method:                 Least Squares   F-statistic:                     211.8
Date:                Mon, 23 Mar 2020   Prob (F-statistic):           6.30e-27
Time:                        20:09:10   Log-Likelihood:                -34.438
No. Observations:                  50   AIC:                             76.88
Df Residuals:                      46   BIC:                             84.52
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
x1             0.4687      0.026     17.751      0.000       0.416       0.522
x2             0.4836      0.104      4.659      0.000       0.275       0.693
x3            -0.0174      0.002     -7.507      0.000      -0.022      -0.013
const          5.2058      0.171     30.405      0.000       4.861       5.550
==============================================================================
Omnibus:                        0.655   Durbin-Watson:                   2.896
Prob(Omnibus):                  0.721   Jarque-Bera (JB):                0.360
Skew:                           0.207   Prob(JB):                        0.835
Kurtosis:                       3.026   Cond. No.                         221.
==============================================================================

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

In [12]:

    

print('Parameters: ', res.params)
print('Standard errors: ', res.bse)
print('Predicted values: ', res.predict())


    

    




Parameters:  [ 0.46872448  0.48360119 -0.01740479  5.20584496]
Standard errors:  [0.02640602 0.10380518 0.00231847 0.17121765]
Predicted values:  [ 4.77072516  5.22213464  5.63620761  5.98658823  6.25643234  6.44117491
  6.54928009  6.60085051  6.62432454  6.6518039   6.71377946  6.83412169
  7.02615877  7.29048685  7.61487206  7.97626054  8.34456611  8.68761335
  8.97642389  9.18997755  9.31866582  9.36587056  9.34740836  9.28893189
  9.22171529  9.17751587  9.1833565   9.25708583  9.40444579  9.61812821
  9.87897556 10.15912843 10.42660281 10.65054491 10.8063004  10.87946503
 10.86825119 10.78378163 10.64826203 10.49133265 10.34519853 10.23933827
 10.19566084 10.22490593 10.32487947 10.48081414 10.66779556 10.85485568
 11.01006072 11.10575781]

In [13]:

    

prstd, iv_l, iv_u = wls_prediction_std(res)

fig, ax = plt.subplots(figsize=(8,6))

ax.plot(x, y, 'o', label="data")
ax.plot(x, y_true, 'b-', label="True")
ax.plot(x, res.fittedvalues, 'r--.', label="OLS")
ax.plot(x, iv_u, 'r--')
ax.plot(x, iv_l, 'r--')
ax.legend(loc='best')


    

    
Out[13]:





<matplotlib.legend.Legend at 0x7fc4d1028fd0>






    

In [14]:

    

from statsmodels.tsa.arima_process import arma_generate_sample


    

In [15]:

    

# Gerando dados
np.random.seed(12345)
arparams = np.array([.75, -.25])
maparams = np.array([.65, .35])


    

In [16]:

    

# Parâmetros
arparams = np.r_[1, -arparams]
maparam = np.r_[1, maparams]
nobs = 250
y = arma_generate_sample(arparams, maparams, nobs)


    

In [17]:

    

dates = sm.tsa.datetools.dates_from_range('1980m1', length=nobs)
y = pd.Series(y, index=dates)
arma_mod = sm.tsa.ARMA(y, order=(2,2))
arma_res = arma_mod.fit(trend='nc', disp=-1)


    

    




/Users/dmpm/opt/anaconda3/lib/python3.7/site-packages/statsmodels/tsa/base/tsa_model.py:162: ValueWarning: No frequency information was provided, so inferred frequency M will be used.
  % freq, ValueWarning)

In [18]:

    

print(arma_res.summary())


    

    




                              ARMA Model Results                              
==============================================================================
Dep. Variable:                      y   No. Observations:                  250
Model:                     ARMA(2, 2)   Log Likelihood                -245.887
Method:                       css-mle   S.D. of innovations              0.645
Date:                Mon, 23 Mar 2020   AIC                            501.773
Time:                        20:09:11   BIC                            519.381
Sample:                    01-31-1980   HQIC                           508.860
                         - 10-31-2000                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1.y        0.8411      0.403      2.089      0.037       0.052       1.630
ar.L2.y       -0.2693      0.247     -1.092      0.275      -0.753       0.214
ma.L1.y        0.5352      0.412      1.299      0.194      -0.273       1.343
ma.L2.y        0.0157      0.306      0.051      0.959      -0.585       0.616
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1            1.5618           -1.1289j            1.9271           -0.0996
AR.2            1.5618           +1.1289j            1.9271            0.0996
MA.1           -1.9835           +0.0000j            1.9835            0.5000
MA.2          -32.1793           +0.0000j           32.1793            0.5000
-----------------------------------------------------------------------------