One of the key assumptions of regression is that the variance of the errors is constant across observations. If the errors have constant variance, the errors are called homoscedastic. Typically, residuals are plotted to assess this assumption. Standard estimation methods are inefficient when the errors are heteroscedastic or have nonconstant variance.

For systems of equations, these tests are computed separately for the residuals of each equation. The residuals of an estimation are used to investigate the heteroscedasticity of the true disturbances. The statistic is asymptotically distributed as chi-squared with P—1 degrees of freedom, where P is the number of regressors in the regression, including the constant and n is the total number of observations. The null hypothesis for the modified Breusch-Pagan test is homosedasticity.

The null hypothesis of the Breusch-Pagan test is. This is a modified version of the Breusch-Pagan test, which is less sensitive to the assumption of normality than the original test Greenep. The statements in the following example produce the output in Figure There are two methods for improving the efficiency of the parameter estimation in the presence of heteroscedastic errors. If the error variance relationships are known, weighted regression can be used or an error model can be estimated.

For details about error model estimation, see the section Error Covariance Structure Specification. If the error variance relationship is unknown, GMM estimation can be used. Consider the following model, which has a heteroscedastic error term:. If this model is estimated with OLS, as shown in the following statements, the estimates shown in Figure If both sides of the model equation are multiplied bythe model has a homoscedastic error term. The weighted estimates are shown in Figure If a subset of the equations needs to be weighted, the residuals for each equation can be modified through the RESID.

These statements produce estimates of the parameters and standard errors that are identical to the weighted OLS estimates. Y variable must be done after Y is assigned; otherwise it would have no effect. Y is multiplied by. Here the multiplier is acting on the residual before it is squared.

If the form of the heteroscedasticity is unknown, generalized method of moments estimation GMM can be used.

GMM estimation generates estimates for the parameters shown in Figure Homoscedasticity is required for ordinary least squares regression estimates to be efficient. A nonconstant error variance, heteroscedasticity, causes the OLS estimates to be inefficient, and the usual OLS covariance matrix,is generally invalid:. Models that take into account the changing variance can make more efficient use of the data. When the variances,are known, generalized least squares GLS can be used and the estimator.

However, GLS is unavailable when the variances,are unknown. This estimator is considered somewhat unreliable in finite samples. Therefore, Davidson and MacKinnon propose three different modifications to estimating.

The first solution is to simply multiply bywhere n is the number of observations and df is the number of explanatory variables, so that. The first modification performed better. The second modification performed even better than the first, and the third modification performed the best. They concluded that the original HCCME should never be used in finite sample estimation, and that the second and third modifications should be used over the first modification if the diagonals of are available.In statisticsthe Breusch—Pagan testdeveloped in by Trevor Breusch and Adrian Pagan[1] is used to test for heteroskedasticity in a linear regression model.

It was independently suggested with some extension by R. In that case, heteroskedasticity is present. Ordinary least squares constrains these so that their mean is 0 and so, given the assumption that their variance does not depend on the independent variablesan estimate of this variance can be obtained from the average of the squared values of the residuals.

If the assumption is not held to be true, a simple model might be that the variance is linearly related to independent variables. Such a model can be examined by regressing the squared residuals on the independent variables, using an auxiliary regression equation of the form.

### SAS/ETS Examples

This is the basis of the Breusch—Pagan test. If the test statistic has a p-value below an appropriate threshold e. If the Breusch—Pagan test shows that there is conditional heteroskedasticity, one could either use weighted least squares if the source of heteroskedasticity is known or use heteroscedasticity-consistent standard errors.

Under the classical assumptions, ordinary least squares is the best linear unbiased estimator BLUEi. It remains unbiased under heteroskedasticity, but efficiency is lost. Before deciding upon an estimation method, one may conduct the Breusch—Pagan test to examine the presence of heteroskedasticity.

**Heteroscedasticity**

The following Lagrange multiplier LM yields the test statistic for the Breusch—Pagan test: [ citation needed ]. A variant of this test, robust in the case of a non- Gaussian error term, was proposed by Roger Koenker. As Koenker notespagewhile the revised statistic has correct asymptotic size its power "may be quite poor except under idealized Gaussian conditions. In Stata, one specifies the full regression, and then enters the command estat hettest followed by all independent variables.

In gretlthe command modtest --breusch-pagan can be applied following an OLS regression. From Wikipedia, the free encyclopedia. This article has multiple issues. Please help improve it or discuss these issues on the talk page.

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New York: Springer. Stata Manual. Colin; Trivedi, Pravin K. Microeconometrics Using Stata Revised ed. Stata Press. Retrieved Categories : Statistical tests Regression diagnostics.One of the key assumptions of regression is that the variance of the errors is constant across observations.

If the errors have constant variance, the errors are called homoscedastic. Typically, residuals are plotted to assess this assumption. Standard estimation methods are inefficient when the errors are heteroscedastic or have nonconstant variance.

For systems of equations, these tests are computed separately for the residuals of each equation. The residuals of an estimation are used to investigate the heteroscedasticity of the true disturbances.

The statistic is asymptotically distributed as chi-squared with P—1 degrees of freedom, where P is the number of regressors in the regression, including the constant and n is the total number of observations.

The null hypothesis for the modified Breusch-Pagan test is homosedasticity. The null hypothesis of the Breusch-Pagan test is. This is a modified version of the Breusch-Pagan test, which is less sensitive to the assumption of normality than the original test Greenep.

The statements in the following example produce the output in Figure Figure There are two methods for improving the efficiency of the parameter estimation in the presence of heteroscedastic errors.

If the error variance relationships are known, weighted regression can be used or an error model can be estimated. For details about error model estimation, see the section Error Covariance Structure Specification.

If the error variance relationship is unknown, GMM estimation can be used. Consider the following model, which has a heteroscedastic error term:. If this model is estimated with OLS, as shown in the following statements, the estimates shown in Figure If both sides of the model equation are multiplied bythe model has a homoscedastic error term. The weighted estimates are shown in Figure If a subset of the equations needs to be weighted, the residuals for each equation can be modified through the RESID.The White test is computed by finding nR 2 from a regression of e i 2 on all of the distinct variables inwhere X is the vector of dependent variables including a constant.

This statistic is asymptotically distributed as chi-square with k -1 degrees of freedom, where k is the number of regressors, excluding the constant term.

The Breusch-Pagan test is a Lagrange multiplier test of the hypothesis that the independent variables have no explanatory power on the e i 2 's. The sample consists of 51 observations of per capita expenditure on public schools and per capita income for each state and the District of Columbia in The following DATA step reads in the 51 observations, transforms the variable INC by multiplying it by for consistency with Greenecreates the variable INC2 as the square of income, and then deletes Wisconsin from the sample due to a missing value for expenditure.

Breusch, T. Greene, W. Koenkar, R. SAS Institute Inc. White, H. If this assumption is violated, the errors are said to be "heteroscedastic. For example, in analyzing public school spending, certain states may have greater variation in expenditure than others.

If heteroscedasticity is present and a regression of spending on per capita income by state and its square is computed, the parameter estimates are still consistent but they are no longer efficient. Thus, inferences from the standard errors are likely to be misleading. Testing for Heteroscedasticity There are several methods of testing for the presence of heteroscedasticity. This test involves looking for patterns in a plot of the residuals from a regression.

Correcting for Heteroscedasticity One way to correct for heteroscedasticity is to compute the weighted least squares WLS estimator using an hypothesized specification for the variance.

Often this specification is one of the regressors or its square. The following commands estimate the preceding model, perform two different tests for heteroscedasticity the White and the Breusch-Paganand output the residuals into a data set for further investigation. Notice, however, that both the White test This implies that the standard errors of the parameter estimates are incorrect and, thus, any inferences derived from them may be misleading.

A plot of the residuals shows more variance in the errors of higher income states. The significance of the estimates is greatly reduced, obscuring the individual effects of the explanatory variables.

## How to Perform a Breusch-Pagan Test in Stata

The White test 9. All of the preceding calculations can be found in Greenechapter References Breusch, T. Ordinary Least Squares. DF Model. DF Error. Root MSE.

Adj R-Sq. Approx Std Err. Number of Observations.In the case of heteroscedasticity, if the regression data are from a simple random sample, then Whiteshowed that matrix. MacKinnon and White introduced three alternative heteroscedasticity-consistent covariance matrix estimators that are all asymptotically equivalent to the estimator but that typically have better small sample behavior. These estimators labeled, and are defined as follows:. Long and Ervin studied the performance of these estimators and recommend using the estimator if the sample size is less than The ACOV option in the MODEL statement displays the heteroscedasticity-consistent covariance matrix estimator in effect and adds heteroscedasticity-consistent standard errors, also known as White standard errors, to the parameter estimates table.

The SPEC option performs a model specification test. The null hypothesis for this test maintains that the errors are homoscedastic and independent of the regressors and that several technical assumptions about the model specification are valid.

For details, see theorem 2 and assumptions 1—7 of White When the model is correctly specified and the errors are independent of the regressors, the rejection of this null hypothesis is evidence of heteroscedasticity. In implementing this test, an estimator of the average covariance matrix Whitep. The nonsingularity of this matrix is one of the assumptions in the null hypothesis about the model specification. When PROC REG determines this matrix to be numerically singular, a generalized inverse is used and a note to this effect is written to the log.

In such cases, care should be taken in interpreting the results of this test. Tests performed with the consistent covariance matrix are asymptotic. For more information, refer to White All rights reserved.

Previous Page Next Page.Deepanshu founded ListenData with a simple objective - Make analytics easy to understand and follow. He has over 10 years of experience in data science. During his tenure, he has worked with global clients in various domains like Banking, Insurance, Private Equity, Telecom and Human Resource. In a linear regression model, there should be homogeneity of variance of the residuals.

In other words, the variance of residuals are approximately equal for all predicted dependent variable values. Consequences of Heteroscedasticity The regression prediction remains unbiased and consistent but inefficient. The hypothesis tests t-test and F-test are no longer valid.

How to check Homoscedasticity. Breusch-Pagan test 3. P-value greater than. Statistics Tutorials : 50 Statistics Tutorials.

About Author: Deepanshu founded ListenData with a simple objective - Make analytics easy to understand and follow. Chandra Shekher 24 August at Omkar N 29 June at Anonymous 28 November at Newer Post Older Post Home.

## Checking Homoscedasticity with SAS

Subscribe to: Post Comments Atom. Love this Post? Spread the Word! Share Share Tweet Subscribe.This test produces a Chi-Square test statistic and a corresponding p-value. If the p-value is below a certain threshold common choices are 0. This tutorial explains how to perform a Breusch-Pagan Test in Stata.

Step 2: Perform multiple linear regression. In this case, it is In this case, it is 0. Since this value is less than 0. In this case, the standard errors that are shown in the output table of the regression are unreliable. There are several ways that you can fix this issue, including:. Transform the response variable.

You can try performing a transformation on the response variable. For example, you could use log price instead of price as the response variable. Another common transformation is to use the square root of the response variable. Use weighted regression. This type of regression assigns a weight to each data point based on the variance of its fitted value. Essentially, this gives small weights to data points that have higher variances, which shrinks their squared residuals.

When the proper weights are used, this can eliminate the problem of heteroscedasticity. Use robust standard errors. Check out this tutorial to learn about how to use robust standard errors in regression in Stata. Your email address will not be published. Skip to content Menu. Posted on March 20, by Zach.

Step 1: Load and view the data. First, use the following command to load the data: sysuse auto Then, view the raw data by using the following command: br Step 2: Perform multiple linear regression. There are several ways that you can fix this issue, including: 1. Published by Zach.

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