Prediction (out of sample)¶
[1]:
%matplotlib inline
[2]:
import numpy as np
import matplotlib.pyplot as plt
import statsmodels.api as sm
plt.rc("figure", figsize=(16,8))
plt.rc("font", size=14)
Artificial data¶
[3]:
nsample = 50
sig = 0.25
x1 = np.linspace(0, 20, nsample)
X = np.column_stack((x1, np.sin(x1), (x1-5)**2))
X = sm.add_constant(X)
beta = [5., 0.5, 0.5, -0.02]
y_true = np.dot(X, beta)
y = y_true + sig * np.random.normal(size=nsample)
Estimation¶
[4]:
olsmod = sm.OLS(y, X)
olsres = olsmod.fit()
print(olsres.summary())
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.984
Model: OLS Adj. R-squared: 0.983
Method: Least Squares F-statistic: 970.0
Date: Thu, 05 Nov 2020 Prob (F-statistic): 1.43e-41
Time: 07:28:38 Log-Likelihood: 1.5386
No. Observations: 50 AIC: 4.923
Df Residuals: 46 BIC: 12.57
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 4.9649 0.083 59.546 0.000 4.797 5.133
x1 0.4954 0.013 38.528 0.000 0.470 0.521
x2 0.4615 0.051 9.129 0.000 0.360 0.563
x3 -0.0190 0.001 -16.796 0.000 -0.021 -0.017
==============================================================================
Omnibus: 4.447 Durbin-Watson: 2.049
Prob(Omnibus): 0.108 Jarque-Bera (JB): 3.703
Skew: 0.662 Prob(JB): 0.157
Kurtosis: 3.163 Cond. No. 221.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In-sample prediction¶
[5]:
ypred = olsres.predict(X)
print(ypred)
[ 4.49083841 4.95047481 5.3736972 5.73535453 6.01937263 6.22139516
6.34949933 6.42286885 6.46864209 6.51745333 6.59839963 6.73426028
6.93775419 7.20945026 7.53767386 7.900425 8.26899278 8.61267261
8.9038108 9.12234804 9.25911506 9.31733912 9.31211379 9.26791911
9.21459959 9.18246084 9.19729045 9.27612116 9.42443242 9.63525015
9.89029254 10.16297372 10.42277173 10.64024318 10.79186077 10.86387868
10.85459102 10.77461212 10.64513255 10.49443765 10.35326029 10.24973155
10.20475886 10.22859099 10.31913416 10.46229611 10.63430134 10.80559744
10.94571039 11.0282514 ]
Create a new sample of explanatory variables Xnew, predict and plot¶
[6]:
x1n = np.linspace(20.5,25, 10)
Xnew = np.column_stack((x1n, np.sin(x1n), (x1n-5)**2))
Xnew = sm.add_constant(Xnew)
ynewpred = olsres.predict(Xnew) # predict out of sample
print(ynewpred)
[11.02549655 10.90062849 10.67174529 10.38009018 10.07995374 9.8253817
9.65694264 9.59179541 9.61948802 9.70451667]
Plot comparison¶
[7]:
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
ax.plot(x1, y, 'o', label="Data")
ax.plot(x1, y_true, 'b-', label="True")
ax.plot(np.hstack((x1, x1n)), np.hstack((ypred, ynewpred)), 'r', label="OLS prediction")
ax.legend(loc="best");

Predicting with Formulas¶
Using formulas can make both estimation and prediction a lot easier
[8]:
from statsmodels.formula.api import ols
data = {"x1" : x1, "y" : y}
res = ols("y ~ x1 + np.sin(x1) + I((x1-5)**2)", data=data).fit()
We use the I
to indicate use of the Identity transform. Ie., we do not want any expansion magic from using **2
[9]:
res.params
[9]:
Intercept 4.964910
x1 0.495433
np.sin(x1) 0.461495
I((x1 - 5) ** 2) -0.018963
dtype: float64
Now we only have to pass the single variable and we get the transformed right-hand side variables automatically
[10]:
res.predict(exog=dict(x1=x1n))
[10]:
0 11.025497
1 10.900628
2 10.671745
3 10.380090
4 10.079954
5 9.825382
6 9.656943
7 9.591795
8 9.619488
9 9.704517
dtype: float64