Prediction (out of sample)¶
[1]:
%matplotlib inline
[2]:
import numpy as np
import statsmodels.api as sm
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: 950.4
Date: Mon, 25 Apr 2022 Prob (F-statistic): 2.26e-41
Time: 14:03:04 Log-Likelihood: 3.0632
No. Observations: 50 AIC: 1.874
Df Residuals: 46 BIC: 9.522
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 5.1691 0.081 63.914 0.000 5.006 5.332
x1 0.4801 0.012 38.493 0.000 0.455 0.505
x2 0.4910 0.049 10.013 0.000 0.392 0.590
x3 -0.0187 0.001 -17.115 0.000 -0.021 -0.017
==============================================================================
Omnibus: 0.338 Durbin-Watson: 1.657
Prob(Omnibus): 0.844 Jarque-Bera (JB): 0.468
Skew: -0.171 Prob(JB): 0.791
Kurtosis: 2.672 Cond. No. 221.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In-sample prediction¶
[5]:
ypred = olsres.predict(X)
print(ypred)
[ 4.70052172 5.16475425 5.59072261 5.9516683 6.23048981 6.42255237
6.53644942 6.59259076 6.61984923 6.65081704 6.7164509 6.84098599
7.03795412 7.30796055 7.6385845 8.00641989 8.38092085 8.72942037
9.02249734 9.23881042 9.36860429 9.4153125 9.39499327 9.33369123
9.26315831 9.21563705 9.21856319 9.29005806 9.43595133 9.64882317
9.90922336 10.18886683 10.4552809 10.6771404 10.82941429 10.89747826
10.87951739 10.78682437 10.64194414 10.47496983 10.3185985 10.20275918
10.14969555 10.17031138 10.2623791 10.41090575 10.59059631 10.77000957
10.91672385 11.0026638 ]
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)
[10.99808449 10.86430345 10.62057544 10.31077966 9.99267653 9.72376598
9.54720944 9.48126241 9.5148049 9.61006397]
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 5.169090
x1 0.480122
np.sin(x1) 0.490990
I((x1 - 5) ** 2) -0.018743
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 10.998084
1 10.864303
2 10.620575
3 10.310780
4 9.992677
5 9.723766
6 9.547209
7 9.481262
8 9.514805
9 9.610064
dtype: float64