Quantile regression¶
This example page shows how to use statsmodels’ QuantReg class to replicate parts of the analysis published in
Koenker, Roger and Kevin F. Hallock. “Quantile Regression”. Journal of Economic Perspectives, Volume 15, Number 4, Fall 2001, Pages 143–156
We are interested in the relationship between income and expenditures on food for a sample of working class Belgian households in 1857 (the Engel data).
Setup¶
We first need to load some modules and to retrieve the data. Conveniently, the Engel dataset is shipped with statsmodels.
[1]:
%matplotlib inline
[2]:
import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
import matplotlib.pyplot as plt
data = sm.datasets.engel.load_pandas().data
data.head()
[2]:
| income | foodexp | |
|---|---|---|
| 0 | 420.157651 | 255.839425 |
| 1 | 541.411707 | 310.958667 |
| 2 | 901.157457 | 485.680014 |
| 3 | 639.080229 | 402.997356 |
| 4 | 750.875606 | 495.560775 |
Least Absolute Deviation¶
The LAD model is a special case of quantile regression where q=0.5
[3]:
mod = smf.quantreg("foodexp ~ income", data)
res = mod.fit(q=0.5)
print(res.summary())
QuantReg Regression Results
==============================================================================
Dep. Variable: foodexp Pseudo R-squared: 0.6206
Model: QuantReg Bandwidth: 64.51
Method: Least Squares Sparsity: 209.3
Date: Thu, 14 Nov 2024 No. Observations: 235
Time: 16:49:45 Df Residuals: 233
Df Model: 1
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 81.4823 14.634 5.568 0.000 52.649 110.315
income 0.5602 0.013 42.516 0.000 0.534 0.586
==============================================================================
The condition number is large, 2.38e+03. This might indicate that there are
strong multicollinearity or other numerical problems.
Visualizing the results¶
We estimate the quantile regression model for many quantiles between .05 and .95, and compare best fit line from each of these models to Ordinary Least Squares results.
Prepare data for plotting¶
For convenience, we place the quantile regression results in a Pandas DataFrame, and the OLS results in a dictionary.
[4]:
quantiles = np.arange(0.05, 0.96, 0.1)
def fit_model(q):
res = mod.fit(q=q)
return [q, res.params["Intercept"], res.params["income"]] + res.conf_int().loc[
"income"
].tolist()
models = [fit_model(x) for x in quantiles]
models = pd.DataFrame(models, columns=["q", "a", "b", "lb", "ub"])
ols = smf.ols("foodexp ~ income", data).fit()
ols_ci = ols.conf_int().loc["income"].tolist()
ols = dict(
a=ols.params["Intercept"], b=ols.params["income"], lb=ols_ci[0], ub=ols_ci[1]
)
print(models)
print(ols)
q a b lb ub
0 0.05 124.880096 0.343361 0.268632 0.418090
1 0.15 111.693660 0.423708 0.382780 0.464636
2 0.25 95.483539 0.474103 0.439900 0.508306
3 0.35 105.841294 0.488901 0.457759 0.520043
4 0.45 81.083647 0.552428 0.525021 0.579835
5 0.55 89.661370 0.565601 0.540955 0.590247
6 0.65 74.033434 0.604576 0.582169 0.626982
7 0.75 62.396584 0.644014 0.622411 0.665617
8 0.85 52.272216 0.677603 0.657383 0.697823
9 0.95 64.103964 0.709069 0.687831 0.730306
{'a': np.float64(147.47538852370573), 'b': np.float64(0.48517842367692354), 'lb': 0.4568738130184233, 'ub': 0.5134830343354237}
First plot¶
This plot compares best fit lines for 10 quantile regression models to the least squares fit. As Koenker and Hallock (2001) point out, we see that:
Food expenditure increases with income
The dispersion of food expenditure increases with income
The least squares estimates fit low income observations quite poorly (i.e. the OLS line passes over most low income households)
[5]:
x = np.arange(data.income.min(), data.income.max(), 50)
get_y = lambda a, b: a + b * x
fig, ax = plt.subplots(figsize=(8, 6))
for i in range(models.shape[0]):
y = get_y(models.a[i], models.b[i])
ax.plot(x, y, linestyle="dotted", color="grey")
y = get_y(ols["a"], ols["b"])
ax.plot(x, y, color="red", label="OLS")
ax.scatter(data.income, data.foodexp, alpha=0.2)
ax.set_xlim((240, 3000))
ax.set_ylim((240, 2000))
legend = ax.legend()
ax.set_xlabel("Income", fontsize=16)
ax.set_ylabel("Food expenditure", fontsize=16)
[5]:
Text(0, 0.5, 'Food expenditure')
Second plot¶
The dotted black lines form 95% point-wise confidence band around 10 quantile regression estimates (solid black line). The red lines represent OLS regression results along with their 95% confidence interval.
In most cases, the quantile regression point estimates lie outside the OLS confidence interval, which suggests that the effect of income on food expenditure may not be constant across the distribution.
[6]:
n = models.shape[0]
p1 = plt.plot(models.q, models.b, color="black", label="Quantile Reg.")
p2 = plt.plot(models.q, models.ub, linestyle="dotted", color="black")
p3 = plt.plot(models.q, models.lb, linestyle="dotted", color="black")
p4 = plt.plot(models.q, [ols["b"]] * n, color="red", label="OLS")
p5 = plt.plot(models.q, [ols["lb"]] * n, linestyle="dotted", color="red")
p6 = plt.plot(models.q, [ols["ub"]] * n, linestyle="dotted", color="red")
plt.ylabel(r"$\beta_{income}$")
plt.xlabel("Quantiles of the conditional food expenditure distribution")
plt.legend()
plt.show()
