%matplotlib inline

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

import statsmodels.api as sm
from statsmodels.sandbox.regression.predstd import wls_prediction_std

np.random.seed(9876789)

"""
Explanation: Ordinary Least Squares
End of explanation
"""


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

"""
Explanation: OLS estimation
Artificial data:
End of explanation
"""


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

"""
Explanation: Our model needs an intercept so we add a column of 1s:
End of explanation
"""


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

"""
Explanation: Fit and summary:
End of explanation
"""


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

"""
Explanation: Quantities of interest can be extracted directly from the fitted model. Type dir(results) for a full list. Here are some examples:
End of explanation
"""


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

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

"""
Explanation: OLS non-linear curve but linear in parameters
We simulate artificial data with a non-linear relationship between x and y:
End of explanation
"""


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

"""
Explanation: Fit and summary:
End of explanation
"""


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

"""
Explanation: Extract other quantities of interest:
End of explanation
"""


prstd, iv_l, iv_u = wls_prediction_std(res)

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

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

"""
Explanation: Draw a plot to compare the true relationship to OLS predictions. Confidence intervals around the predictions are built using the wls_prediction_std command.
End of explanation
"""


nsample = 50
groups = np.zeros(nsample, int)
groups[20:40] = 1
groups[40:] = 2
#dummy = (groups[:,None] == np.unique(groups)).astype(float)

dummy = pd.get_dummies(groups).values
x = np.linspace(0, 20, nsample)
# drop reference category
X = np.column_stack((x, dummy[:,1:]))
X = sm.add_constant(X, prepend=False)

beta = [1., 3, -3, 10]
y_true = np.dot(X, beta)
e = np.random.normal(size=nsample)
y = y_true + e

"""
Explanation: OLS with dummy variables
We generate some artificial data. There are 3 groups which will be modelled using dummy variables. Group 0 is the omitted/benchmark category.
End of explanation
"""


print(X[:5,:])
print(y[:5])
print(groups)
print(dummy[:5,:])

"""
Explanation: Inspect the data:
End of explanation
"""


res2 = sm.OLS(y, X).fit()
print(res2.summary())

"""
Explanation: Fit and summary:
End of explanation
"""


prstd, iv_l, iv_u = wls_prediction_std(res2)

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

ax.plot(x, y, 'o', label="Data")
ax.plot(x, y_true, 'b-', label="True")
ax.plot(x, res2.fittedvalues, 'r--.', label="Predicted")
ax.plot(x, iv_u, 'r--')
ax.plot(x, iv_l, 'r--')
legend = ax.legend(loc="best")

"""
Explanation: Draw a plot to compare the true relationship to OLS predictions:
End of explanation
"""


R = [[0, 1, 0, 0], [0, 0, 1, 0]]
print(np.array(R))
print(res2.f_test(R))

"""
Explanation: Joint hypothesis test
F test
We want to test the hypothesis that both coefficients on the dummy variables are equal to zero, that is, $R \times \beta = 0$. An F test leads us to strongly reject the null hypothesis of identical constant in the 3 groups:
End of explanation
"""


print(res2.f_test("x2 = x3 = 0"))

"""
Explanation: You can also use formula-like syntax to test hypotheses
End of explanation
"""


beta = [1., 0.3, -0.0, 10]
y_true = np.dot(X, beta)
y = y_true + np.random.normal(size=nsample)

res3 = sm.OLS(y, X).fit()

print(res3.f_test(R))

print(res3.f_test("x2 = x3 = 0"))

"""
Explanation: Small group effects
If we generate artificial data with smaller group effects, the T test can no longer reject the Null hypothesis:
End of explanation
"""


from statsmodels.datasets.longley import load_pandas
y = load_pandas().endog
X = load_pandas().exog
X = sm.add_constant(X)

"""
Explanation: Multicollinearity
The Longley dataset is well known to have high multicollinearity. That is, the exogenous predictors are highly correlated. This is problematic because it can affect the stability of our coefficient estimates as we make minor changes to model specification.
End of explanation
"""


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

"""
Explanation: Fit and summary:
End of explanation
"""


norm_x = X.values
for i, name in enumerate(X):
    if name == "const":
        continue
    norm_x[:,i] = X[name]/np.linalg.norm(X[name])
norm_xtx = np.dot(norm_x.T,norm_x)

"""
Explanation: Condition number
One way to assess multicollinearity is to compute the condition number. Values over 20 are worrisome (see Greene 4.9). The first step is to normalize the independent variables to have unit length:
End of explanation
"""


eigs = np.linalg.eigvals(norm_xtx)
condition_number = np.sqrt(eigs.max() / eigs.min())
print(condition_number)

"""
Explanation: Then, we take the square root of the ratio of the biggest to the smallest eigen values.
End of explanation
"""


ols_results2 = sm.OLS(y.iloc[:14], X.iloc[:14]).fit()
print("Percentage change %4.2f%%\n"*7 % tuple([i for i in (ols_results2.params - ols_results.params)/ols_results.params*100]))

"""
Explanation: Dropping an observation
Greene also points out that dropping a single observation can have a dramatic effect on the coefficient estimates:
End of explanation
"""


infl = ols_results.get_influence()

"""
Explanation: We can also look at formal statistics for this such as the DFBETAS -- a standardized measure of how much each coefficient changes when that observation is left out.
End of explanation
"""


2./len(X)**.5

print(infl.summary_frame().filter(regex="dfb"))

"""
Explanation: In general we may consider DBETAS in absolute value greater than $2/\sqrt{N}$ to be influential observations
End of explanation
"""


