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, 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: 923.4
Date: Wed, 30 Nov 2022 Prob (F-statistic): 4.34e-41
Time: 21:29:37 Log-Likelihood: 1.1397
No. Observations: 50 AIC: 5.721
Df Residuals: 46 BIC: 13.37
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 5.0182 0.084 59.707 0.000 4.849 5.187
x1 0.4945 0.013 38.153 0.000 0.468 0.521
x2 0.4081 0.051 8.009 0.000 0.306 0.511
x3 -0.0195 0.001 -17.163 0.000 -0.022 -0.017
==============================================================================
Omnibus: 0.829 Durbin-Watson: 1.771
Prob(Omnibus): 0.661 Jarque-Bera (JB): 0.865
Skew: -0.282 Prob(JB): 0.649
Kurtosis: 2.689 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)
[ 4.52991477 4.9702252 5.37741426 5.72924135 6.01149236 6.22031506
6.36285198 6.45606693 6.52395788 6.59361416 6.6907658 6.83555609
7.03923216 7.30229705 7.61442711 7.95616814 8.30213162 8.62516602
8.90081769 9.11134854 9.24865036 9.31557694 9.32547527 9.2999926
9.26551988 9.24885584 9.27280396 9.35242598 9.4925673 9.68706086
9.91974037 10.16709645 10.40213905 10.59883195 10.73637034 10.80259907
10.79600966 10.72598813 10.61127295 10.47687634 10.34997481 10.25544416
10.21177247 10.22802252 10.30234284 10.42227188 10.56678542 10.70975077
10.82422032 10.88685894]
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)
[10.87048771 10.74475174 10.52565482 10.24966759 9.96479824 9.71883843
9.54766222 9.46644288 9.46593789 9.51475189]
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")
[7]:
<matplotlib.legend.Legend at 0x7f3b50e96c20>

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 5.018240
x1 0.494549
np.sin(x1) 0.408091
I((x1 - 5) ** 2) -0.019533
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 10.870488
1 10.744752
2 10.525655
3 10.249668
4 9.964798
5 9.718838
6 9.547662
7 9.466443
8 9.465938
9 9.514752
dtype: float64