How to Easily Perform White’s Test for Heteroscedasticity in Python

Understanding Heteroscedasticity and White’s Test

White’s test is a robust diagnostic tool specifically designed to determine if heteroscedasticity is present within the error terms of a regression model. Identifying this condition is vital because heteroscedasticity violates the core assumption of ordinary least squares (OLS) regression that the variance of the residuals is constant across all levels of the predictor variables.

Heteroscedasticity refers to the unequal scatter (non-constant variance) of residuals at different levels of the independent variables. If this condition exists, the standard errors calculated for the regression coefficients become unreliable, potentially leading to incorrect conclusions about the statistical significance of the predictors.

The following step-by-step example demonstrates how to implement White’s test in Python. We will use the powerful functionality of the statsmodels package to rigorously assess whether heteroscedasticity poses a problem in a sample regression analysis.

Step 1: Load Necessary Libraries and Data

To begin, we must load the necessary Python libraries, including statsmodels for the test itself, sklearn for general linear modeling tools, and pandas for efficient data handling. In this demonstration, we will fit a model using the well-known mtcars dataset, which is commonly used for statistical examples.

The code below illustrates how to import the required modules and load the mtcars dataset into a pandas DataFrame directly from a hosted URL. We then display the summary information to confirm the data structure and types.

from sklearn.linear_model import LinearRegression
from statsmodels.stats.diagnostic import het_white
import statsmodels.api as sm
import pandas as pd

#define URL where dataset is located
url = "https://raw.githubusercontent.com/arabpsychology/Python-Guides/main/mtcars.csv"

#read in data
data = pd.read_csv(url)

#view summary of data
data.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 32 entries, 0 to 31
Data columns (total 12 columns):
 #   Column  Non-Null Count  Dtype  
---  ------  --------------  -----  
 0   model   32 non-null     object 
 1   mpg     32 non-null     float64
 2   cyl     32 non-null     int64  
 3   disp    32 non-null     float64
 4   hp      32 non-null     int64  
 5   drat    32 non-null     float64
 6   wt      32 non-null     float64
 7   qsec    32 non-null     float64
 8   vs      32 non-null     int64  
 9   am      32 non-null     int64  
 10  gear    32 non-null     int64  
 11  carb    32 non-null     int64  
dtypes: float64(5), int64(6), object(1)

Step 2: Fit the OLS Regression Model

Once the data is prepared, the next step involves defining the model structure and performing the regression analysis. We will use the Miles Per Gallon (mpg) as our response variable (Y) and Displacement (disp) and Horsepower (hp) as the two predictor variables (X).

We use the OLS class from the statsmodels.api module. It is standard practice in OLS regression using this library to explicitly include a constant (intercept) term, which we achieve using sm.add_constant(x) before fitting the model.

The resulting model object contains all the necessary components, including the calculated residuals, which will be input directly into the White’s test function in the next step.

#define response variable
y = data['mpg']

#define predictor variables
x = data[['disp', 'hp']]

#add constant to predictor variables
x = sm.add_constant(x)

#fit regression model
model = sm.OLS(y, x).fit()

Step 3: Executing White’s Test in Python

With the OLS model fitted, we can now execute the diagnostic test. We utilize the het_white function, imported from statsmodels.stats.diagnostic, to perform White’s test. This function requires the model residuals (model.resid) and the original predictor matrix (model.model.exog) as inputs.

The function returns a tuple of results, which we then map to descriptive labels for clarity. These results include the Test Statistic (based on the Lagrange Multiplier, distributed as Chi-squared), its corresponding p-value, the F-Statistic, and its p-value. The output provides the critical metrics needed to make a statistical decision regarding the presence of heteroscedasticity.

#perform White's test
white_test = het_white(model.resid,  model.model.exog)

#define labels to use for output of White's test
labels = ['Test Statistic', 'Test Statistic p-value', 'F-Statistic', 'F-Test p-value']

#print results of White's test
print(dict(zip(labels, white_test)))

{'Test Statistic': 7.076620330416624, 'Test Statistic p-value': 0.21500404394263936,
 'F-Statistic': 1.4764621093131864, 'F-Test p-value': 0.23147065943879694}

Interpreting the White’s Test Results

The output provides two sets of test results: one based on the Chi-squared distribution (Test Statistic) and one based on the F-distribution (F-Statistic). In practice, the interpretation based on the p-value associated with the Test Statistic is most common.

Based on the execution above, the key results are:

• The Lagrange Multiplier Test Statistic (X²) is 7.0766.
• The corresponding p-value is 0.215.

To interpret these values, we must formally state the hypotheses used in White’s test:

• Null Hypothesis (H₀): Homoscedasticity is present (the variance of residuals is constant).
• Alternative Hypothesis (H_A): Heteroscedasticity is present (the variance of residuals is not constant).

Using a standard significance level ($alpha$) of 0.05, we compare the p-value (0.215) to this threshold. Since the p-value is greater than 0.05, we fail to reject the null hypothesis (H₀). This indicates that we do not have sufficient statistical evidence to conclude that heteroscedasticity is a significant issue in this specific linear regression model.

Remediation Strategies for Heteroscedasticity

If the White’s test results lead you to fail to reject the null hypothesis (as in our example), you can proceed confidently with interpreting the output of your original OLS method regression, as the assumption of homoscedasticity is met.

However, if the test yields a p-value less than $alpha$ (e.g., p < 0.05), you must reject the null hypothesis, confirming the presence of heteroscedasticity. In this scenario, the standard errors of the regression coefficients are biased and unreliable, potentially invalidating your hypothesis testing results. Two common strategies exist to address this statistical issue:

1. Transforming the Response Variable

One effective solution is to apply a mathematical transformation to the response variable. For example, taking the natural logarithm of the response variable often stabilizes the variance of the residuals, thereby causing heteroscedasticity to dissipate. Other transformations, such as the square root or inverse, may also be appropriate depending on the nature of the data.

2. Using Weighted Regression (WLS)

Another robust method is employing Weighted Least Squares (WLS) regression. WLS assigns a specific weight to each data point inversely proportional to the variance of its corresponding fitted value. Essentially, observations associated with higher variance (the source of the heteroscedasticity) are given smaller weights in the estimation process. When the proper weights are successfully determined and applied, WLS provides unbiased and efficient standard error estimates, effectively mitigating the problem caused by unequal residual variances.

Further Resources on Python Regression

For those looking to deepen their understanding of regression analysis and diagnostics in Python, the following tutorials offer additional valuable information: