Analysis of Time-series Data

statistics
time-series
Published

May 16, 2024

On this project we’ll analyze a time-series data. Specifically, we’ll analyze the annual average temperature from Junin, Peru. The objective is build a model that can helps us predict the annual temprature for future years.

Code 
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline
from sklearn.linear_model import LinearRegression
from scipy.interpolate import interp1d
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from bokeh.io import output_notebook, show, export_png
# from bokeh.io import output_file
from bokeh.plotting import figure
from itertools import chain 
from sklearn.metrics import mean_squared_error
import datetime as dt
import statsmodels.api as sm
from statsmodels.graphics.tsaplots import plot_acf,plot_pacf
from statsmodels.tsa.ar_model import AutoReg
from statsmodels.tsa.arima.model import ARIMA
import pmdarima as pm
import seaborn as sns
import math
import pmdarima as pm
import statsmodels.api as sm
def mean_absolute_percentage_error(y_true, y_pred): 
     return np.mean(np.abs((y_true - y_pred) / y_true)) *100
output_notebook()
Loading BokehJS ...

Annual Average Temperature from Junin, Peru

First we need to load the data and create a single dataframe.

Code 
temp_df = pd.read_csv("observed-annual-average-mean-surface-air-temperature-of-junín,-peru-for-1901-2022.csv", sep=",")
temp_df = temp_df.rename(columns={"Category":"Year", "Annual Mean":"Temp_annual_mean"}).drop(columns=["5-yr smooth"])


prec_df = pd.read_csv("observed-annual-precipitation-of-junín,-peru-for-1901-2022.csv", sep=",")
prec_df = prec_df.rename(columns={"Category":"Year", "Annual Mean":"Prec_annual_mean"}).drop(columns=["5-yr smooth"])

climate_df = pd.merge(temp_df, prec_df, on="Year", how="left")
climate_df
Year Temp_annual_mean Prec_annual_mean
0 1901 14.17 1273.85
1 1902 14.16 1273.85
2 1903 13.85 1273.85
3 1904 13.70 1273.85
4 1905 14.07 1273.85
... ... ... ...
117 2018 14.65 1273.85
118 2019 14.84 1262.93
119 2020 15.00 1273.85
120 2021 14.97 1046.47
121 2022 14.84 782.31

122 rows × 3 columns

Due to some rare measures on the data, let’s subset the data from the year 1951 and plot it.

Code 
plt.figure(figsize=(6, 3), dpi=80)
subset_df = climate_df[climate_df.Year >= 1951] 

sns.lineplot(data=subset_df, x="Year", y="Temp_annual_mean")
plt.title("Average Temperature in Junin, Peru")
plt.ylabel('Temp (C)')
plt.show()
subset_df.head()

Year Temp_annual_mean Prec_annual_mean
50 1951 14.12 1183.29
51 1952 13.85 1287.30
52 1953 14.28 1345.91
53 1954 13.66 1328.92
54 1955 13.68 1008.42

We can see that there is an increasing trend across the years.

Removing the Trend

Linear Trend

First let’s define the functions to calculate the Akaike Information Criteria and the Bayesian Information Criteria to test the performance of our models.

Code 
from scipy.stats import norm 
def evaluate_AIC(k, residuals):
  """
  Finds the AIC given the number of parameters estimated and 
  the residuals of the model. Assumes residuals are distributed 
  Gaussian with unknown variance. 
  """
  standard_deviation = np.std(residuals)
  log_likelihood = norm.logpdf(residuals, 0, scale=standard_deviation)
  return 2 * k - 2 * np.sum(log_likelihood)

def evaluate_BIC(k, residuals):
  """
  Finds the AIC given the number of parameters estimated and 
  the residuals of the model. Assumes residuals are distributed 
  Gaussian with unknown variance. 
  """
  standard_deviation = np.std(residuals)
  log_likelihood = norm.logpdf(residuals, 0, scale=standard_deviation)
  return k * np.log(len(residuals)) - 2 * np.sum(log_likelihood)

Now, we can begin fitting a linear trend to the data.

Code 
X = subset_df["Year"].values.reshape(-1,1)
y = subset_df["Temp_annual_mean"].values.reshape(-1,1)


linear_model = LinearRegression()
linear_model.fit(X, y)

print(linear_model.coef_)

y_hat = linear_model.predict(X)

plt.plot(X, y, label='original data')
plt.plot(X, y_hat, 'r', label='fitted line')
plt.legend()

[[0.01386713]]

And plot the residuals of the model.

Code 
linear_residuals = y - y_hat
plt.plot(X, linear_residuals,'o')
sns.regplot(x=X, y=linear_residuals)

print("MSE with linear fit:", np.mean((linear_residuals)**2))
print("AIC:", evaluate_AIC(1, linear_residuals))
print("BIC:", evaluate_BIC(1, linear_residuals))
MSE with linear fit: 0.12615193027776006
AIC: 57.2678309969092
BIC: 59.54449711592525

The plot shows a good distribution of the residuals.

Let’s check the autocorrelation and partial autocorralation of the residuals.

Code 
sm.graphics.tsa.plot_acf(linear_residuals, lags=20)
plt.show()
sm.graphics.tsa.plot_pacf(linear_residuals, lags=20)
plt.show()

It seems there is an autocorrelation structure on the data.

Quadratic Trend

To check that the linear trend is a good choice, let’sw compare it with a quadratic fit.

Code 
degree=2
quadratic_model=LinearRegression()

new_x = np.hstack((X, X **2))

quadratic_model.fit(new_x, y)
quadratic = quadratic_model.predict(new_x)

print(quadratic_model.coef_)

y_hat_quad = quadratic_model.predict(new_x)

plt.plot(X, y, label='original data')
plt.plot(X, y_hat_quad, 'r', label='fitted line')
plt.legend()
[[ 5.74495264e-01 -1.41109523e-04]]

The residuals:

Code 
quadratic_residuals = y - y_hat_quad
plt.plot(X, quadratic_residuals,'o')
sns.regplot(x=X, y=quadratic_residuals)

print("MSE with linear fit:", np.mean((quadratic_residuals)**2))
print("AIC:", evaluate_AIC(2, quadratic_residuals))
print("BIC:", evaluate_BIC(2, quadratic_residuals))
MSE with linear fit: 0.12318196679493859
AIC: 57.55248071646069
BIC: 62.105812954492805

And the autocorrelation plots.

Code 
sm.graphics.tsa.plot_acf(quadratic_residuals, lags=20)
plt.show()
sm.graphics.tsa.plot_pacf(quadratic_residuals, lags=20)
plt.show()

These plots (ACF and PACF) shows that there might be a autocorrelation of order 16 (AR(16)) and possibly a moving average of order 13. Let’s make a grid search over possible values to corroborate this.

Code 
def grid_search_ARIMA(data, AR_range, MA_range, verbose=False):
  min_aic = np.inf 
  min_bic = np.inf
  min_aic_index = None
  min_bic_index = None 
  aic_matrix = np.zeros((len(AR_range), len(MA_range)))
  bic_matrix = np.zeros((len(AR_range), len(MA_range)))
  for AR_order in AR_range:
    for MA_order in MA_range:
      arma = ARIMA(data, order=(AR_order, 0, MA_order)).fit()
      aic_matrix[AR_order, MA_order] = arma.aic
      bic_matrix[AR_order, MA_order] = arma.bic
      if arma.aic < min_aic:
        min_aic = arma.aic
        min_aic_index = (AR_order, 0, MA_order)
      if arma.bic < min_bic:
        min_bic = arma.bic
        min_bic_index = (AR_order, 0, MA_order)
  if verbose:
    print("Minimizing AIC order: ", min_aic_index)
    print("Minimizing BIC order: ", min_bic_index )
    print("matrix of AIC", aic_matrix)
    print("Matrix of BIC", bic_matrix)
  return min_aic_index, min_bic_index, aic_matrix, bic_matrix

Luckily, statsmodels give us the BIC and AIC scores for each models we are building. We are using these scores to get the best model in our grid search.

Code 
min_aic_index, min_bic_index, _, _ = grid_search_ARIMA(linear_residuals, range(20), range(20), verbose=True)
if min_aic_index == min_bic_index:
  arma = ARIMA(linear_residuals, order=min_bic_index).fit()
  print(arma.summary())
  arma_predictions = arma.predict()
  arma_residuals = linear_residuals - arma_predictions
  arma_residuals = arma_residuals # Fitting AR 1 model means removing one observation
  plt.plot(linear_residuals, label='Residuals from fitted quadratic line')
  plt.plot(arma_predictions, 'r', label='fitted ARMA process')
  plt.legend()
  plt.show()
  plt.plot(arma_residuals, 'o')
  plt.show()
  print("Automatic selection finds model with AR {0}, MA {2}".format(*min_aic_index))
  print("MSE with selected model:", np.mean(arma_residuals**2))
else:
  print("AIC, BIC do not agree.")
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:978: UserWarning: Non-invertible starting MA parameters found. Using zeros as starting parameters.
  warn('Non-invertible starting MA parameters found.'
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
Minimizing AIC order:  (2, 0, 9)
Minimizing BIC order:  (0, 0, 0)
matrix of AIC [[59.26783104 58.54992334 56.88970575 57.79945966 59.13858623 61.06132125
  60.10357216 56.70724639 58.53031492 58.79141029 56.50215466 53.83056416
  55.53360818 55.23182903 56.25011347 58.29607542 59.22370895 60.69622205
  62.22647076 64.27980123]
 [60.01838137 59.01196539 54.78314438 58.68975059 58.21367941 58.11925325
  59.0577748  59.45027904 60.73655131 59.9252463  57.17792233 55.7365912
  57.5279835  56.6732098  58.58560271 59.49982214 61.03576455 62.8378866
  64.5385872  66.80960379]
 [55.60204822 57.44397806 55.93590971 55.28516924 59.26120688 56.7676285
  56.48409178 57.10386856 56.64832604 52.86203264 54.153882   54.80353584
  54.92519604 57.89801211 59.59621413 61.05672236 62.98633782 64.98255721
  64.98545754 68.24167112]
 [57.3687478  59.32047137 60.25882172 56.45016306 54.39221592 54.65960306
  64.22394424 60.57746518 57.70426024 54.70458015 55.95799305 56.65596753
  57.6468255  58.81023372 60.48193877 61.72053194 65.92771418 67.04341717
  70.63721548 71.18084068]
 [59.14942574 60.16696732 55.43512106 54.60174117 59.66358505 56.90480476
  59.81984521 61.72126301 53.78122065 54.47597726 56.43000237 58.33943936
  59.79851906 60.74759539 61.9397838  62.61877092 67.76229664 69.2277279
  72.87708691 72.62363139]
 [60.40981264 62.39919653 59.42642447 61.54330408 57.78042654 58.96114785
  61.09524713 61.63167538 53.60862951 54.66052224 55.9339392  60.1296563
  62.18164813 62.99282536 66.0525596  65.26549479 69.14330159 70.87240071
  68.02871147 72.67504957]
 [62.40170224 64.39927888 60.62995123 63.6825648  59.7051804  61.33200669
  59.13415    62.02401831 57.66308719 61.11610053 60.06519047 60.26112098
  64.19789296 61.2653695  64.85471283 65.67339349 69.7437752  72.95251941
  69.57430322 74.94677503]
 [64.34447628 60.05935422 62.3969647  63.67303358 64.07209862 65.60416731
  62.05504141 64.51656351 66.05892528 59.65529841 61.98584232 61.35293391
  65.47289344 64.66384057 64.41897783 62.29931871 69.36109345 72.72596842
  68.78138574 69.50693126]
 [64.41545455 61.71415914 63.54795097 65.66333692 60.10808786 59.90569828
  62.10370484 58.32770199 61.4240911  59.83238655 63.13851866 64.06893123
  62.06925577 64.08994125 66.73659766 69.73560144 72.76526793 74.67639407
  72.32997535 72.5167611 ]
 [58.88763152 60.62032348 56.32174956 58.37109165 57.18127799 57.9979968
  60.13371869 61.8217613  62.45246858 63.26705539 63.28433601 65.33991341
  64.15416778 65.80591849 70.40735139 71.09551673 73.70837078 76.31432766
  74.15964268 74.54904043]
 [60.46185714 62.33316031 57.97652193 59.98767802 60.01699452 59.91525829
  62.02879216 62.3272774  61.43014887 63.45995718 64.57083238 66.79380038
  66.72756485 62.78569633 65.79729765 69.1037027  74.05271602 73.91032018
  68.89454402 75.9429185 ]
 [61.97041927 63.91206891 59.64344611 61.37400935 59.57250674 61.58339273
  62.7480419  63.38369221 63.97563916 64.35159438 64.63662196 68.0292894
  69.88978561 65.25591315 65.13319861 67.43781895 75.88506935 75.93701685
  75.14617187 76.36974229]
 [63.63597058 62.35808653 60.84634222 63.04798246 61.54304906 63.55419996
  64.79509909 62.54784339 65.39007171 65.64030002 65.31843543 70.37120629
  70.72790305 68.9481488  72.08035365 75.70498209 72.73480547 75.58563942
  73.68754741 84.29162461]
 [62.98285503 64.80495154 65.32919463 60.45838337 62.96135175 62.08647299
  64.51344513 64.89266037 66.59345037 67.44881901 66.7480505  66.31208872
  70.37463005 72.42160464 72.37227087 76.39744907 77.08760943 75.30260452
  75.91255923 76.75954076]
 [64.76611924 66.75592971 64.61713126 62.71140589 65.36156186 67.48855684
  65.42719093 67.81803452 68.75615495 69.4207174  69.759826   70.60916283
  72.37711488 74.19570454 73.67777429 78.21254447 77.36334509 79.6251962
  77.72553906 78.9407086 ]
 [66.66293554 67.77980308 65.4544313  64.58883831 67.92109901 66.55639664
  69.36610887 70.43391218 71.42742349 70.20057968 70.55349007 73.33025119
  74.52506171 71.74668773 73.83554345 78.81700438 75.63752456 84.31020079
  78.81331696 80.25713868]
 [64.05594172 65.94136223 67.58710883 69.04508423 65.47736499 71.29129912
  69.14330891 69.09748835 69.53539203 69.88838457 71.5360232  73.17185656
  72.45100046 72.30233633 78.24445035 75.79644228 82.82074378 79.51804671
  79.82388504 79.81846245]
 [65.88340344 67.31991573 69.24121591 71.03278703 68.52191482 70.5365999
  70.43184559 70.42099772 71.33343091 71.80401553 73.80889981 74.70363714
  72.4465451  75.79010158 80.5398399  77.62198717 84.90464036 81.8517004
  81.86401746 80.84358572]
 [67.46843461 68.93356711 71.14867163 71.50327634 69.79037073 71.63167747
  70.42909044 72.42562339 72.59689936 72.7441992  73.6218613  75.96098587
  78.51198994 79.27065894 82.11453555 78.47673277 86.52803398 82.67852269
  83.09122525 84.83274228]
 [68.49571987 70.38710021 72.13463923 74.21073296 69.51714758 70.5919664
  74.10239936 74.12004659 74.02198301 74.31395149 75.73698092 78.07651132
  80.45214899 78.70575615 83.66879418 80.77171565 82.83606246 84.96232135
  85.25326334 87.06597314]]
Matrix of BIC [[ 63.82116328  65.3799217   65.99637023  69.18279025  72.79858294
   76.99798408  78.31690112  77.19724146  81.29697612  83.8347376
   83.82214809  83.4272237   87.40693385  89.38182081  92.67677138
   96.99939945 100.20369909 103.95287831 107.75979314 112.08978973]
 [ 66.84837973  68.11862987  66.16647497  72.3497473   74.15034224
   76.3325822   79.54776987  82.21694023  85.77987862  87.24523973
   86.77458187  87.60991686  91.67797528  93.09986771  97.28892673
  100.47981228 104.29242081 108.37120898 112.3485757  116.8962584 ]
 [ 64.7087127   68.82730865  69.59590642  71.22183207  77.47453583
   77.25762357  79.25075297  82.14719587  83.96831946  82.45869219
   86.02720766  88.95352763  91.35185395  96.60133613 100.57620427
  104.31337862 108.5196602  112.79254571 115.07211216 120.60499185]
 [ 68.75207839  72.98046809  76.19548456  74.66349201  74.88221099
   77.42626425  89.26727155  87.89745861  87.30091979  86.57790582
   90.10798483  93.08262544  96.35014952  99.79022387 103.73859503
  107.25385432 113.73770268 117.13007179 123.00053622 125.82082754]
 [ 72.80942245  76.10363015  73.64845002  75.09173625  82.43024624
   81.94813207  87.13983863  91.31792256  85.65454631  88.62596904
   92.85666027  97.04276338 100.7785092  104.00425165 107.47310618
  110.42875942 117.84895126 121.59104863 127.51707377 129.54028437]
 [ 76.34647547  80.61252548  79.91641954  84.30996527  82.82375385
   86.28114128  90.69190667  93.50500105  87.7586213   91.08718014
   94.63726322 101.10964645 105.43830439 108.52614774 113.8625481
  115.35214941 121.50662232 125.51238756 124.94536444 131.86836866]
 [ 80.61503119  84.88927395  83.39661242  88.72589211  87.02517383
   90.92866623  91.00747567  96.17401009  94.08974509  99.81942455
  101.04518061 103.51777724 109.73121534 109.075358   114.94136745
  118.03671422 124.38376205 129.86917238 128.76762232 136.41676025]
 [ 84.83447135  82.82601541  87.44029201  90.993027    93.66875817
   97.47749297  96.2050332  100.94322142 104.7622493  100.63528855
  105.24249858 106.88625629 113.28288194 114.75049518 116.78229856
  116.93930556 126.27774643 131.91928751 130.25137096 133.25358259]
 [ 87.18211574  86.75748645  90.8679444   95.25999647  91.98141352
   94.05569006  98.53036274  97.03102601 102.40408124 103.08904281
  108.67184104 111.87891973 112.15591039 116.45326198 121.37658451
  126.65225441 131.95858702 136.14637929 136.07662668 138.54007855]
 [ 83.93095883  87.94031691  85.91840911  90.24441732  91.33126978
   94.4246547   98.83704271 102.80175144 105.70912484 108.80037777
  111.09432451 115.42656803 116.51748852 120.44590535 127.32400437
  130.28883582 135.17835599 140.060979   140.18296014 142.849024  ]
 [ 87.78185057  91.92981986  89.8498476   94.13766981  96.44365243
   98.61858231 103.00878231 105.58393366 106.96347125 111.26994568
  114.657487   119.15712111 121.36755171 119.70234931 124.99061675
  130.57368792 137.79936735 139.93363763 137.19452759 146.51956819]
 [ 91.56707882  95.78539457  93.7934379   97.80066725  98.27583076
  102.56338287 106.00469816 108.91701459 111.78562766 114.43824899
  116.99994269 122.66927626 126.80643859 124.44923224 126.60318382
  131.18447028 141.9083868  144.23700042 145.72282155 149.2230581 ]
 [ 95.50929625  96.50807831  97.27300013 101.75130648 102.5230392
  106.81085622 110.32842147 110.35783189 115.47672633 118.00362076
  119.95842229 127.28785927 129.92122214 130.41813401 135.82700499
  141.72829954 141.03478904 146.16228911 146.54086321 159.42160654]
 [ 97.13284681 101.23160944 104.03251865 101.43837351 106.21800801
  107.61979537 112.32343363 114.97931498 118.95677111 122.08880587
  123.66470348 125.50540782 131.84461527 136.16825597 138.39558832
  144.69743264 147.66425912 148.15592033 151.04254115 154.1661888 ]
 [101.19277714 105.45925373 105.5971214  105.96806216 110.89488424
  115.29854534 115.51384555 120.18135525 123.39614181 126.33737037
  128.9531451  132.07914804 136.12376621 140.21902199 141.97775786
  148.78919416 150.2166609  154.75517812 155.1321871  158.62402277]
 [105.36625956 108.75979323 108.71108756 110.12216069 115.73108751
  116.64305126 121.72942961 125.07389904 128.34407646 129.39389877
  132.02347528 137.07690252 140.54837916 140.0466713  144.41219314
  151.67032019 150.76750649 161.71684884 158.49663112 162.21711897]
 [105.03593186 109.19801849 113.12043121 116.85507273 115.56401961
  123.65461985 123.78329577 126.01414133 128.72871113 131.35836978
  135.28267454 139.19517401 140.75098403 142.87898602 151.09776615
  150.92642421 160.22739182 159.20136088 161.78386532 164.05510885]
 [109.1400597  112.85323811 117.05120441 121.11944165 120.88523556
  125.17658676 127.34849856 129.61431681 132.80341612 135.55066687
  139.83221727 143.00362071 143.02319479 148.64341738 155.66982183
  155.02863522 164.58795452 163.81168069 166.10066386 167.35689825]
 [113.00175699 116.74355561 121.23532625 123.86659708 124.43035759
  128.54833044 129.62240953 133.89560861 136.34355069 138.76751665
  141.92184487 146.53763555 151.36530575 154.40064086 159.52118359
  158.16004693 168.48801427 166.91516909 169.60453777 173.62272092]
 [116.30570837 120.47375483 124.49795997 128.85071981 126.43380055
  129.78528549 135.57238458 137.86669792 140.04530046 142.61393506
  146.31363061 150.92982713 155.58213092 156.1124042  163.35210835
  162.73169593 167.07270886 171.47563387 174.04324199 178.1326179 ]]
AIC, BIC do not agree.
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "

We see that the grid search output an ARMA(2,9) model. Let’s work with this model and form forecasts.

To evaluate the model we will use a rolling window cross-validation. This will help us view how the model improve as we include more time series data. The window we are gonna use is 20 years and we’ll forecast 20 years.

Code 
window_size = 20

for i in range(20,72,window_size):
    
    train_temp, test_temp = quadratic_residuals[:i], quadratic_residuals[i:i+window_size]
    train_date, test_date = X[:i], X[i:i+window_size]
    arma = ARIMA(train_temp, order=min_aic_index).fit()
    # print(arma.summary())
    fig, ax = plt.subplots(figsize=(15, 5))
    fcast = arma.get_forecast(len(test_temp)).summary_frame()

    arma_predictions = arma.predict()
    ax.plot(X, quadratic_residuals, label='annual mean T°C, converted to stationary time series')
    predicted_values = arma_predictions.reshape(-1,1)
    ax.plot(train_date, predicted_values, 'r', label='fitted line')
    forecast_means = fcast['mean'].values.reshape(-1,1)
    test_set_mse = np.mean((forecast_means.reshape(test_temp.shape) - test_temp)**2)
    ax.plot(test_date, forecast_means, 'k--', label='mean forecast')
    ax.fill_between(test_date.flatten(), fcast['mean_ci_lower'], fcast['mean_ci_upper'], color='k', alpha=0.1);
    plt.legend();
    print("Test set mean squared error: ", test_set_mse)
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:866: UserWarning: Too few observations to estimate starting parameters for ARMA and trend. All parameters except for variances will be set to zeros.
  warn('Too few observations to estimate starting parameters%s.'
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:978: UserWarning: Non-invertible starting MA parameters found. Using zeros as starting parameters.
  warn('Non-invertible starting MA parameters found.'
Test set mean squared error:  0.20590924950605077
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:978: UserWarning: Non-invertible starting MA parameters found. Using zeros as starting parameters.
  warn('Non-invertible starting MA parameters found.'
Test set mean squared error:  0.09063865739126532
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
Test set mean squared error:  0.039150953597273795

In the first plot, we can see that only using 20 values for training, the forecast converge to the mean of the series. And, as we continue including data, the forcast improve.

Let’s see if using an external variable (in this case the annual mean precipitation) improve our model.

Code 
window_size = 20

for i in range(20,72,window_size):
    
    train_temp, test_temp = quadratic_residuals[:i], quadratic_residuals[i:i+window_size]
    train_date, test_date = X[:i], X[i:i+window_size]

    train_prec, test_prec = subset_df["Prec_annual_mean"].values.reshape(-1,1)[:i], subset_df["Prec_annual_mean"].values.reshape(-1,1)[i:i+window_size]


    arma = ARIMA(train_temp, order=min_aic_index, exog=train_prec).fit()
    # print(arma.summary())
    fig, ax = plt.subplots(figsize=(15, 5))
    fcast = arma.get_forecast(len(test_temp), exog=test_prec).summary_frame()

    arma_predictions = arma.predict()
    ax.plot(X, quadratic_residuals, label='annual mean T°C, converted to stationary time series')
    predicted_values = arma_predictions.reshape(-1,1)
    ax.plot(train_date, predicted_values, 'r', label='fitted line')
    forecast_means = fcast['mean'].values.reshape(-1,1)
    test_set_mse = np.mean((forecast_means.reshape(test_temp.shape) - test_temp)**2)
    ax.plot(test_date, forecast_means, 'k--', label='mean forecast')
    ax.fill_between(test_date.flatten(), fcast['mean_ci_lower'], fcast['mean_ci_upper'], color='k', alpha=0.1);
    plt.legend();
    print("Test set mean squared error: ", test_set_mse)
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:866: UserWarning: Too few observations to estimate starting parameters for ARMA and trend. All parameters except for variances will be set to zeros.
  warn('Too few observations to estimate starting parameters%s.'
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:978: UserWarning: Non-invertible starting MA parameters found. Using zeros as starting parameters.
  warn('Non-invertible starting MA parameters found.'
Test set mean squared error:  0.18613318748268368
Test set mean squared error:  0.11609711371821685
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:978: UserWarning: Non-invertible starting MA parameters found. Using zeros as starting parameters.
  warn('Non-invertible starting MA parameters found.'
c:\Users\LENOVO\miniconda3\envs\piero\lib\site-packages\statsmodels\base\model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
Test set mean squared error:  0.07317563428347647

We can see that including the precipitation doesn’t improve the predictions of temperature on this ARMA(2,9) model. This analysis use the autoregressive models to get forecasts on time series data but, nowadays this methods can be extenden with deep learning models like recurrent neural networks or transformers to have better predictions. Also, is possible to make a temporal-spatial analysis using gaussian processes plus deep learning models.