What Is Glejser Test?

A detailed examination of the Glejser Test, a statistical method to detect heteroscedasticity by regressing the absolute values of residuals on independent variables.

The Glejser Test is a statistical method employed to detect heteroscedasticity, a condition in which the size of the random error or variance of residuals in a regression model changes proportionally with changes in one or more independent (exogenous) variables. This article delves into the historical context, methodology, interpretation, significance, and examples of the Glejser Test.

Historical Context

The Glejser Test was introduced by Herbert Glejser in 1969 as a robust method to address heteroscedasticity, a common issue in regression models where the variability of residuals is not constant across levels of an explanatory variable. The standard assumption in ordinary least squares (OLS) regression is homoscedasticity, where residuals have constant variance. Heteroscedasticity violates this assumption, leading to inefficiencies in estimations and potentially invalid statistical inferences.

Methodology

The Glejser Test involves the following steps:

1. Conduct the OLS Regression: Begin by running the ordinary least squares regression on your primary model to obtain the residuals.
2. Compute Absolute Residuals: Calculate the absolute values of the residuals from the OLS regression.
3. Regress Absolute Residuals on Exogenous Variables: Perform a regression of these absolute residuals on the independent variables suspected to cause heteroscedasticity.
4. Evaluate the Test Statistic: Under the null hypothesis of homoscedasticity, the test statistic \(NR^2\) follows an asymptotic chi-square distribution with \(h\) degrees of freedom, where \(N\) is the sample size and \(h\) is the number of independent variables.

Formula

The formula for the test statistic is:

$$ \text{Test Statistic} = N \times R^2 $$

where \(R^2\) is the coefficient of determination from the auxiliary regression.

Key Events and Applications

• Introduction by Herbert Glejser: The test was first introduced in a 1969 paper by Glejser, who aimed to provide a straightforward method for diagnosing heteroscedasticity.
• Widespread Use in Econometrics: Over the years, the Glejser Test has become a standard diagnostic tool in econometric analyses and empirical research.

Importance and Applicability

Heteroscedasticity, if left undiagnosed and uncorrected, can lead to biased and inefficient estimates. The Glejser Test is particularly valuable for:

• Ensuring more reliable statistical inferences
• Improving the accuracy of confidence intervals
• Enhancing the precision of forecast models

Examples and Interpretations

Example Calculation

1. Suppose a regression model
   $$ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i $$
   is fitted, and the residuals \( \epsilon_i \) are obtained.
2. Compute the absolute values of these residuals: \( | \epsilon_i | \).
3. Regress \( | \epsilon_i | \) on \( X_i \):
   $$ | \epsilon_i | = \alpha_0 + \alpha_1 X_i + u_i $$
4. Calculate \( R^2 \) from this regression.
5. Determine the test statistic: \( N \times R^2 \).
6. Compare with the critical chi-square value with \(h\) degrees of freedom to decide whether heteroscedasticity is present.

Considerations

• Symmetry Assumption: The Glejser Test is valid under the assumption of symmetrically distributed errors. For skewed errors, modified versions of the test should be used.
• Sample Size: A larger sample size provides more reliable test results due to the asymptotic nature of the chi-square distribution.

Related Terms

• Homoscedasticity: The condition where the variance of residuals is constant across levels of an explanatory variable.
• OLS Residuals: The differences between observed values and predicted values from an OLS regression model.
• Chi-Square Distribution: A probability distribution commonly used in hypothesis testing for categorical data and goodness-of-fit tests.

Comparisons

• Breusch-Pagan Test: Another test for heteroscedasticity, which involves regressing the squared residuals on the independent variables and comparing the test statistic to a chi-square distribution.
• White Test: A more general test for heteroscedasticity that does not rely on the normality assumption of the errors.

Interesting Facts

• The Glejser Test was developed as a simpler alternative to more complex tests for heteroscedasticity, making it more accessible to practitioners.

FAQs

Q: What happens if heteroscedasticity is detected?

A: If heteroscedasticity is detected, remedial measures such as transforming the dependent variable, using heteroscedasticity-robust standard errors, or employing generalized least squares (GLS) can be taken.

Q: Can the Glejser Test detect non-linear relationships?

A: The primary purpose of the Glejser Test is to detect heteroscedasticity, not non-linearity. However, heteroscedasticity may sometimes signal the presence of non-linear relationships.

References

• Glejser, H. (1969). “A New Test for Heteroskedasticity.” Journal of the American Statistical Association, 64(325), 316-323.
• Greene, W. H. (2018). Econometric Analysis. 8th ed. Pearson.

Final Summary

The Glejser Test is an invaluable tool in econometrics and statistics for diagnosing heteroscedasticity, thereby ensuring the reliability and efficiency of regression analyses. By regressing the absolute residuals on the exogenous variables, researchers can detect and correct for variance inequalities, leading to more robust and accurate models.