What is Heteroskedasticity?
Heterogeneity is a term used to describe the variation in data. In the context of regression analysis, heteroskedasticity refers to the presence of non-constant variance in the residuals. This means that the variance of the residuals is not the same across the range of predicted values.
Why is Heteroskedasticity a Problem?
Heterogeneity can lead to problems in regression analysis, including:
- Inaccurate standard errors: The standard errors of the regression coefficients are not accurate when there is heteroskedasticity. This can lead to incorrect inferences about the significance of the coefficients.
- Biased estimates: The estimates of the regression coefficients can be biased when there is heteroskedasticity. This means that the estimates may not be accurate representations of the true underlying relationships.
- Reduced power: Heteroskedasticity can reduce the power of the regression model. This means that the model may be less likely to detect significant relationships.
How to deal with this problem
There are a number of ways to deal with heteroskedasticity. One way is to transform the data. For example, if the residuals are increasing as the predicted values increase, you could transform the data using a logarithmic transformation. Another way to deal with heteroskedasticity is to use a weighted least squares regression. This method gives more weight to the residuals that are more variable.
Heteroskedasticity use cases
Here are some examples of heteroskedasticity:
- The relationship between income and spending: The variance of spending is likely to be higher for people with higher incomes. This is because people with higher incomes have more discretionary income, which means that they can spend more on a wider variety of goods and services.
- The relationship between test scores and hours of study: The variance of test scores is likely to be higher for people who study for longer periods of time. This is because people who study for longer periods of time are more likely to be familiar with the material and to be able to perform well on the test.
- The relationship between sales and advertising expenditure: The variance of sales is likely to be higher for businesses that spend more on advertising. This is because businesses that spend more on advertising are more likely to reach a wider audience and to generate more sales.
Example with Python code to detect heteroskedasticity in regression model:
import numpy as np
import pandas as pd
from statsmodels.stats.diagnostic import het_breuschpagan
# Create a dataset
data = np.random.randint(0, 100, (100, 2))
# Fit a linear regression model
model = sm.OLS(data[:, 1], data[:, 0]).fit()
# Check for heteroskedasticity using the Breusch-Pagan test
bp_test = het_breuschpagan(model.resid, model.model.exog)
# Print the test results
print(bp_test)
This code will first create a dataset of 100 observations with two variables. The first variable will be the independent variable and the second variable will be the dependent variable. The independent variable will be randomly generated from a uniform distribution between 0 and 100. The dependent variable will be generated from a linear regression model with the independent variable as the only explanatory variable.
The code will then fit a linear regression model to the data. The Breusch-Pagan test will then be used to check for heteroskedasticity. The results of the test will be printed to the console.
To run this code, you will need to install the following Python packages:
- NumPy
- Pandas
- Statsmodels
Once you have installed the packages, you can run the code by saving it as a Python file and then running the file from the command line.
The output of the code will be a list of three values: the test statistic, the p-value, and the degrees of freedom. If the p-value is less than the chosen significance level (e.g. 0.05), then you can conclude that there is heteroskedasticity in the data.
Examining Heteroskedasticity in the Linear Regression Model of Brain Weight and Head Size Relationship
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.formula.api import ols
from statsmodels.stats.diagnostic import het_breuschpagan
# Load the data
data = pd.read_csv('/headbrain.csv')
print(data.head())
# Fit a linear regression model
model = ols('Brain_Weight ~ Head_Size', data=data).fit()
# Check for heteroskedasticity using the Breusch-Pagan test
bp_test = het_breuschpagan(model.resid, model.model.exog)
# Plot the residuals
plt.figure(figsize=(14, 8))
sns.residplot(x=data['Head_Size'], y=data['Brain_Weight'], color='blue')
plt.axhline(y=0, color='red')
plt.show()
# Print the results of the Breusch-Pagan test
print(bp_test)
Gender Age Range Head_Size Brain_Weight
0 1 1 4512 1530
1 1 1 3738 1297
2 1 1 4261 1335
3 1 1 3777 1282
4 1 1 4177 1590