Types of Heteroskedasticity

Oct 13, 2025 by miodeya.com 63 views

Introduction

In regression analysis, heteroskedasticity is a common issue that can lead to biased and inefficient estimates of model parameters. It occurs when the variance of the error term is not constant across all levels of the independent variables. In this article, we will examine a given regression model and determine whether it is suffering from heteroskedasticity.

The Model

The given model is:

y = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_1 x_2^3 + \mu

where $y$ is the dependent variable, $x_1$ and $x_2$ are independent variables, $\theta_0$ , $\theta_1$ , $\theta_2$ , and $\theta_3$ are model parameters, and $\mu$ is the error term.

What is Heteroskedasticity?

Heteroskedasticity occurs when the variance of the error term is not constant across all levels of the independent variables. This can be visualized as a situation where the spread of the data points increases or decreases as the values of the independent variables change.

Types of Heteroskedasticity

There are two main types of heteroskedasticity:

Homoscedasticity: The variance of the error term is constant across all levels of the independent variables.
Heteroscedasticity: The variance of the error term is not constant across all levels of the independent variables.

Testing for Heteroskedasticity

To determine whether the given model is suffering from heteroskedasticity, we can use several tests, including:

Breusch-Pagan Test: This test is used to detect the presence of heteroskedasticity in the error term.
White Test: This test is used to detect the presence of heteroskedasticity in the error term.
Goldfeld-Quandt Test: This test is used to detect the presence of heteroskedasticity in the error term.

Breusch-Pagan Test

The Breusch-Pagan test is a popular test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

where $\hat{\sigma}^2$ is the estimated variance of the error term, $y_i$ is the observed value of the dependent variable, $\hat{y}_i$ is the predicted value of the dependent variable, $n$ is the number of observations, and $k$ is the number of model parameters.

The test statistic is calculated as:

BP = n \cdot \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

where $x_i$ is the value of the independent variable, $\bar{x}$ is the mean of the independent variable, and $\hat{\sigma}^2$ is the estimated variance of the error term.

White Test

The White test is another popular test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

W = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

where $x_i$ is the value of the independent variable, $\bar{x}$ is the mean of the independent variable, and $\hat{\sigma}^2$ is the estimated variance of the error term.

Goldfeld-Quandt Test

The Goldfeld-Quandt test is a test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

GQ = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

where $x_i$ is the value of the independent variable, $\bar{x}$ is the mean of the independent variable, and $\hat{\sigma}^2$ is the estimated variance of the error term.

Conclusion

In conclusion, the given model is:

y = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_1 x_2^3 + \mu

To determine whether this model is suffering from heteroskedasticity, we can use several tests, including the Breusch-Pagan test, the White test, and the Goldfeld-Quandt test.

The Breusch-Pagan test is a popular test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

BP = n \cdot \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

The White test is another popular test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

W = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

The Goldfeld-Quandt test is a test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

GQ = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

By using these tests, we can determine whether the given model is suffering from heteroskedasticity.

References

Breusch, T. S., & Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica, 47(5), 1287-1294.
White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817-838.
Goldfeld, S. M., & Quandt, R. E. (1965). Some tests for homoscedasticity. Journal of the American Statistical Association, 60(312), 927-936.
Q&A: Heteroskedasticity in Regression Models =====================================================

Q: What is heteroskedasticity?

A: Heteroskedasticity is a condition in which the variance of the error term in a regression model is not constant across all levels of the independent variables. This can lead to biased and inefficient estimates of model parameters.

Q: What are the consequences of heteroskedasticity?

A: The consequences of heteroskedasticity include:

Biased estimates of model parameters
Inefficient estimates of model parameters
Incorrect predictions of the dependent variable
Incorrect interpretation of the results

Q: How can I detect heteroskedasticity in my regression model?

A: There are several tests that can be used to detect heteroskedasticity in a regression model, including:

Breusch-Pagan test
White test
Goldfeld-Quandt test

Q: What is the Breusch-Pagan test?

A: The Breusch-Pagan test is a popular test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

BP = n \cdot \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

Q: What is the White test?

A: The White test is another popular test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

W = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

Q: What is the Goldfeld-Quandt test?

A: The Goldfeld-Quandt test is a test for detecting heteroskedasticity in the error term. The test is based on the following equation:

\hat{\sigma}^2 = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-k}

The test statistic is calculated as:

GQ = \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2 \cdot (x_i - \bar{x})^2}{\hat{\sigma}^2 \cdot \sum_{i=1}^n (x_i - \bar{x})^2}

Q: How can I correct for heteroskedasticity in my regression model?

A: There are several ways to correct for heteroskedasticity in a regression model, including:

Using a heteroskedasticity-consistent covariance matrix estimator
Using a weighted least squares estimator
Using a generalized least squares estimator

Q: What is a heteroskedasticity-consistent covariance matrix estimator?

A: A heteroskedasticity-consistent covariance matrix estimator is a type of estimator that is designed to handle heteroskedasticity in the error term. The most common type of heteroskedasticity-consistent covariance matrix estimator is the Huber-White sandwich estimator.

Q: What is a weighted least squares estimator?

A: A weighted least squares estimator is a type of estimator that is designed to handle heteroskedasticity in the error term. The weighted least squares estimator assigns different weights to each observation based on the variance of the error term.

Q: What is a generalized least squares estimator?

A: A generalized least squares estimator is a type of estimator that is designed to handle heteroskedasticity in the error term. The generalized least squares estimator uses a weighted least squares approach to estimate the model parameters.

Conclusion

In conclusion, heteroskedasticity is a common issue in regression analysis that can lead to biased and inefficient estimates of model parameters. There are several tests that can be used to detect heteroskedasticity, including the Breusch-Pagan test, the White test, and the Goldfeld-Quandt test. There are also several ways to correct for heteroskedasticity, including using a heteroskedasticity-consistent covariance matrix estimator, a weighted least squares estimator, or a generalized least squares estimator.