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: 0.985
Model: OLS Adj. R-squared: 0.984
Method: Least Squares F-statistic: 1018.
Date: Thu, 14 Dec 2023 Prob (F-statistic): 4.80e-42
Time: 14:39:40 Log-Likelihood: 3.0746
No. Observations: 50 AIC: 1.851
Df Residuals: 46 BIC: 9.499
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 4.9317 0.081 60.993 0.000 4.769 5.094
x1 0.5066 0.012 40.625 0.000 0.482 0.532
x2 0.5504 0.049 11.227 0.000 0.452 0.649
x3 -0.0206 0.001 -18.810 0.000 -0.023 -0.018
==============================================================================
Omnibus: 2.839 Durbin-Watson: 2.081
Prob(Omnibus): 0.242 Jarque-Bera (JB): 2.094
Skew: -0.492 Prob(JB): 0.351
Kurtosis: 3.190 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)
[ 4.41682151 4.92268515 5.38579485 5.77615583 6.07459828 6.27592688
6.38977441 6.43901906 6.45602567 6.47732824 6.53762749 6.66408968
6.8718832 7.1616865 7.51957661 7.91931663 8.32666638 8.705008
9.02136211 9.25180625 9.3854053 9.42600804 9.39161484 9.31142024
9.22101636 9.15654525 9.14876081 9.21797603 9.37072562 9.59869241
9.88007377 10.18316377 10.47156237 10.71015582 10.87088578 10.9373592
10.90754167 10.79409166 10.62228076 10.42584165 10.2414259 10.10258223
10.03424454 10.04863535 10.14325767 10.30130525 10.49442381 10.68736953
10.84379957 10.93224304]
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)
[10.91763232 10.75840089 10.47613227 10.12001251 9.75478779 9.44491225
9.23876741 9.15681664 9.18659505 9.28576134]
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 0x7f2614e341c0>
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 4.931706
x1 0.506602
np.sin(x1) 0.550372
I((x1 - 5) ** 2) -0.020595
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 10.917632
1 10.758401
2 10.476132
3 10.120013
4 9.754788
5 9.444912
6 9.238767
7 9.156817
8 9.186595
9 9.285761
dtype: float64
Last update:
Dec 14, 2023