White's Test: Test of Homoscedasticity

White's Test is used to test the null hypothesis of homoscedasticity against the alternative of heteroscedasticity in a regression model.

White’s Test is a statistical test used to check for the presence of heteroscedasticity (non-constant variance) in a regression model. The presence of heteroscedasticity can lead to inefficient estimators and biased statistical inference.

Historical Context

White’s Test was developed by economist Halbert White in 1980 as a means of diagnosing heteroscedasticity in regression models. Halbert White’s influential work has significantly impacted econometrics and the methodology of statistical inference.

Types/Categories

Classical White’s Test

This version involves regressing the squared residuals on the independent variables, their squares, and their cross-products.

Robust White’s Test

An extension that includes a correction for small sample sizes or models with many variables.

Key Events

• 1980: Introduction of White’s Test by Halbert White.
• 1982: Adoption and widespread use of White’s Test in econometric software packages.

Detailed Explanation

Methodology

White’s Test starts with the null hypothesis of homoscedasticity (constant variance of errors):

$$ H_0: \ \sigma^2_i = \sigma^2 \ \forall \ i $$

The alternative hypothesis is heteroscedasticity (non-constant variance):

$$ H_a: \ \sigma^2_i = f(x_i) $$

Steps involved:

1. Run the original regression:

   $$ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + \epsilon $$

2. Obtain the residuals (\( \hat{\epsilon} \)) and squared residuals (\( \hat{\epsilon}^2 \)).

3. Regress the squared residuals (\( \hat{\epsilon}^2 \)) on all explanatory variables, their squares, cross-products, and a constant:

   $$ \hat{\epsilon}^2 = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + ... + \alpha_k x_k + \alpha_{k+1} x_1^2 + ... + \alpha_{k+k'} x_k x_j + \nu $$

4. Calculate the test statistic (\( NR^2 \)):

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

   Where \( N \) is the sample size and \( R^2 \) is the coefficient of determination from the auxiliary regression.

5. Compare this statistic with the chi-square distribution with \( k’ \) degrees of freedom, where \( k’ \) is the number of regressors in the test regression.

Mathematical Formulas

$$ NR^2 \sim \chi^2 (df = k') $$

Where \( k’ \) is the number of regressors excluding the constant term.

Charts and Diagrams

Here is a flowchart to visualize White’s Test methodology:

    flowchart TD
	    A[Start] --> B[Run Original Regression]
	    B --> C[Obtain Residuals (\hat{\epsilon}) and Squared Residuals (\hat{\epsilon}^2)]
	    C --> D[Regress \hat{\epsilon}^2 on Explanatory Variables]
	    D --> E[Compute Test Statistic NR^2]
	    E --> F[Compare with \chi^2 Distribution]
	    F --> G{Reject or Fail to Reject H0}
	    G --> H[End]

Importance

White’s Test is crucial for identifying issues related to heteroscedasticity, ensuring more accurate inference in regression analysis. It maintains the validity of hypothesis tests and confidence intervals.

Applicability

• Economics: Examining economic models for heteroscedasticity.
• Finance: Validating asset pricing models.
• Social Sciences: Assessing models where variance could change with demographics.

Examples

1. Economic Growth Model:

   $$ GDP = \beta_0 + \beta_1 \text{Investment} + \beta_2 \text{Education} + \epsilon $$

   Use White’s Test to check if variance changes with the level of investment or education.

2. Stock Return Volatility: Regress stock returns and check for constant variance using White’s Test.

Considerations

• The test can be sensitive to model specifications.
• Large models with many explanatory variables can complicate the auxiliary regression.
• Outliers can impact the results.

Related Terms

• Homoscedasticity: Constant variance of the error terms in a regression model.
• Heteroscedasticity: Non-constant variance of the error terms.
• Breusch-Pagan Test: Another test for heteroscedasticity.

Comparisons

• White’s Test vs. Breusch-Pagan Test:
  • White’s Test uses auxiliary regression with squares and cross-products.
  • Breusch-Pagan Test uses simpler auxiliary regression with original variables.

Interesting Facts

• Halbert White’s original paper has been cited thousands of times.
• The test is integral to many econometric software packages, including R and Stata.

Inspirational Stories

Halbert White developed his test during his Ph.D. and it became a fundamental tool in econometrics, showcasing the significant impact of academic research.

Famous Quotes

“Statistics is the grammar of science.” — Karl Pearson

Proverbs and Clichés

• “You can’t manage what you can’t measure.” — Focus on the importance of diagnosing model issues like heteroscedasticity.

Expressions, Jargon, and Slang

• OLS Residuals: Errors from the Ordinary Least Squares regression.
• Auxiliary Regression: A secondary regression used for diagnostic testing.

FAQs

What is the primary purpose of White's Test?

To detect heteroscedasticity in a regression model.

How do you interpret White's Test results?

Compare the test statistic to the chi-square distribution with the appropriate degrees of freedom.

Can White's Test be used in small samples?

Yes, but results should be interpreted cautiously.

References

• White, H. (1980). A Heteroscedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroscedasticity. Econometrica.
• Greene, W. H. (2018). Econometric Analysis.

Summary

White’s Test is a robust tool in regression analysis, diagnosing whether the variance of the errors changes with the level of the explanatory variables. It is widely used in various fields to ensure more reliable statistical inferences.

Ensuring your regression models satisfy the homoscedasticity assumption is crucial for the validity of hypothesis tests and confidence intervals, making White’s Test a fundamental procedure in econometrics and statistical analysis.