How to Test for Heteroscedasticity with the Breusch-Pagan Test in SPSS

The Breusch-Pagan test is designed to formally assess whether the variance of the errors in a regression is dependent on the values of the independent variables. This procedure involves running an auxiliary regression using the squared residuals obtained from the primary model.

The following comprehensive, step-by-step example illustrates the correct process for executing the Breusch-Pagan test in SPSS, ensuring the validity of your regression findings.

Step 1: Setting up the Data and Initial Model Formulation

To demonstrate the procedure, we will use a hypothetical scenario involving a multiple linear regression analysis. Suppose our objective is to fit a linear regression model that utilizes two predictor variables—the number of hours spent studying and the number of prep exams taken—to predict the final exam score of students.

The theoretical structure of this model is defined as:

Exam Score = β₀ + β₁(hours) +β₂(prep exams) + ε

Our initial task is to input the relevant dataset into SPSS. This dataset contains the scores and predictor information for 20 unique students, which will serve as the foundation for our analysis:

Step 2: Fitting the Primary Linear Regression Model

Once the data is correctly structured in SPSS, the next critical step is to fit the primary multiple linear regression model. Furthermore, we must instruct SPSS to save the essential components required for the Breusch-Pagan calculation: the unstandardized predicted values and the unstandardized residuals.

To initiate the regression, navigate through the menu bar: Click the Analyze tab, hover over Regression, and then click Linear. This action opens the Linear Regression dialogue box:

In the dialogue box that appears, specify the model variables: Drag the score variable into the Dependent panel (as it is the variable we are predicting), and then drag both hours and prep_exams into the Independent(s) panel (as they are the predictors):

After specifying the variables, click the Save button located within the Linear Regression dialogue box. This opens the submenu for saving regression output variables to the dataset. Under the Predicted Values section, check the box next to Unstandardized. Similarly, under the Residuals section, check the box next to Unstandardized. These two new variables, often labeled PRE_1 and RES_1, are crucial for the subsequent steps of the Breusch-Pagan test:

After checking the necessary boxes, click Continue to close the Save submenu, and then click OK in the main Linear Regression dialogue box to run the analysis. SPSS will generate the standard regression output and add two new columns to your data sheet (predicted values and residuals).

Step 3: Calculating the Squared Residuals for the Test

The core of the Breusch-Pagan test is to determine if the variability of the errors is systematically related to the independent variables. To measure this variability, we must calculate the squared values of the unstandardized residuals (RES_1) saved in the previous step. This calculated variable will serve as the dependent variable in our auxiliary regression.

To create this new variable, click the Transform tab in the menu bar and then click Compute Variable:

In the Compute Variable dialogue box, assign a suitable name to the new variable, such as res_squared, in the Target Variable field. For the calculation, type the formula RES_1*RES_1 in the Numeric Expression box. This expression instructs SPSS to multiply the unstandardized residual variable by itself, effectively squaring each residual value:

Click OK to execute the computation. SPSS will generate a new column in your dataset named res_squared, which contains the necessary values for the final test stage. This variable represents the magnitude of the errors for each observation:

Step 4: Executing the Auxiliary Regression and Interpreting Results

The final step of the Breusch-Pagan test involves running a second, or auxiliary, regression model. In this model, the squared residuals (the res_squared variable) are used as the dependent variable, while the original predictor variables (hours and prep_exams) are used as the independent variables.

Reopen the Linear Regression dialogue box: Click Analyze, then Regression, then Linear. Now, reset the variables. Drag res_squared into the Dependent panel. Retain hours and prep_exams in the Independent(s) box:

Click OK. The resulting output provides the statistics needed for the Breusch-Pagan calculation, particularly focusing on the ANOVA table from this auxiliary regression:

The crucial value for the Breusch-Pagan test is the p-value found under the Sig. column of the ANOVA table for this auxiliary regression. This p-value relates directly to the F-statistic of the auxiliary model. In this specific example, we observe that the p-value is 0.085.

Step 5: Statistical Interpretation of the Breusch-Pagan Test

The interpretation of the Breusch-Pagan test hinges on the comparison between the calculated p-value and a predetermined significance level (alpha, typically α = 0.05). The test’s null hypothesis (H₀) states that homoscedasticity is present (i.e., the variance of the errors is constant).

In our scenario, the p-value (0.085) is greater than the standard alpha level of 0.05. Consequently, we must fail to reject the null hypothesis of the test. This outcome suggests that there is insufficient statistical evidence to conclude that heteroscedasticity is present within the original regression model. If the p-value had been less than 0.05, we would reject H₀ and conclude that heteroscedasticity is a significant issue.

Since we failed to reject the null, the assumption of homoscedasticity holds true for this data, and it is safe to interpret the standard errors of the coefficient estimates in the primary regression summary table without concern for severe bias.

Remedial Actions if Heteroscedasticity is Detected

If the Breusch-Pagan test indicates a rejection of the null hypothesis (p < 0.05), this means significant heteroscedasticity is present in the data. When this occurs, the OLS estimates of the coefficients remain unbiased, but the reported standard errors are unreliable, invalidating t-tests and overall model inference. Several common techniques can be employed to correct this issue:

• 1. Transform the Response Variable.

A frequent and effective approach is to apply a mathematical transformation to the response variable (the dependent variable). For instance, using the logarithm (log) of the response variable instead of its original scale often stabilizes the variance and effectively mitigates heteroscedasticity. Alternatively, using the square root of the response variable is another common variance-stabilizing transformation.

• 2. Employ Robust Standard Errors.

A more modern and generally preferred approach, which does not require changing the model specification or transforming variables, is the use of Heteroscedasticity-Consistent Standard Errors (HCSEs), often referred to as “robust standard errors.” These methods adjust the calculation of the standard errors to account for the non-constant variance without altering the coefficient estimates themselves. While SPSS does not provide the robust standard error option directly in the standard Linear Regression dialogue box, it can be implemented through syntax or specialized add-ons.

• 3. Use Weighted Regression (Weighted Least Squares).

This specialized type of regression model assigns a specific weight to each data point based on the estimated variance of its error term. Observations associated with higher error variance (where the errors are spread out widely) are given smaller weights, effectively shrinking their influence on the squared residuals and the overall fit. When appropriate weights are accurately estimated, weighted regression can successfully eliminate the problem of heteroscedasticity and restore the reliability of the standard errors.