Heteroskedasticity, a violation of constant variance in linear regression, can lead to biased standard errors and unreliable inferences. Detecting it is crucial for accurate econometric analysis. Visual methods and statistical tests like the White test help identify heteroskedasticity in regression models.

The White test is a popular method for detecting heteroskedasticity without specifying its functional form. It involves regressing squared residuals on explanatory variables and their cross-products. The test statistic follows a chi-square distribution, allowing for hypothesis testing to determine the presence of heteroskedasticity.

Heteroskedasticity

• Heteroskedasticity refers to a violation of the constant variance assumption in linear regression models where the variance of the error terms is not constant across observations
• Presence of heteroskedasticity can lead to biased and inconsistent standard errors, affecting the validity of hypothesis tests and confidence intervals in econometric analysis

Consequences of heteroskedasticity

• Ordinary least squares (OLS) estimators remain unbiased but are no longer the best linear unbiased estimators (BLUE) in the presence of heteroskedasticity
• Standard errors of the coefficients are biased, leading to incorrect inferences about the significance of the explanatory variables
• Confidence intervals and hypothesis tests based on t-statistics and F-statistics become unreliable due to the biased standard errors
• Predictions and forecasts based on the model may be less accurate and precise

Detecting heteroskedasticity

• Several methods are available to detect the presence of heteroskedasticity in linear regression models
• Visual inspection of residual plots can provide initial insights into the pattern of heteroskedasticity
• Formal statistical tests, such as the White test, Breusch-Pagan test, and Goldfeld-Quandt test, can be employed to assess the presence of heteroskedasticity

Visual methods

• Residual plots, such as a scatter plot of residuals against the predicted values or explanatory variables, can reveal patterns of heteroskedasticity
  • If the spread of residuals increases or decreases with the predicted values or explanatory variables, it suggests the presence of heteroskedasticity
• Scale-location plots, which plot the square root of the absolute residuals against the predicted values, can also highlight heteroskedasticity
• Residual plots with a funnel-shaped or fan-shaped pattern indicate the presence of heteroskedasticity

Statistical tests

• Formal statistical tests provide a more objective and rigorous approach to detect heteroskedasticity
• The White test is a widely used test for heteroskedasticity that does not require specifying the functional form of heteroskedasticity
• The Breusch-Pagan test is another popular test that assumes a specific functional form of heteroskedasticity (e.g., variance proportional to the square of an explanatory variable)
• The Goldfeld-Quandt test compares the variances of subsamples of the data to assess the presence of heteroskedasticity

White test

• The White test is a general test for heteroskedasticity that does not require specifying the functional form of heteroskedasticity
• It is based on the idea that if the model is correctly specified and homoskedastic, the squared residuals should be uncorrelated with the explanatory variables and their cross-products

Purpose of White test

• The primary purpose of the White test is to detect the presence of heteroskedasticity in linear regression models
• It helps in assessing whether the constant variance assumption of the error terms is violated
• The test provides evidence on the reliability of the standard errors and the validity of the inference based on the regression results

Null vs alternative hypothesis

• The null hypothesis ($H_0$) of the White test states that there is no heteroskedasticity in the model, i.e., the variance of the error terms is constant
• The alternative hypothesis ($H_1$) suggests the presence of heteroskedasticity, indicating that the variance of the error terms is not constant across observations

Test statistic calculation

• The White test involves regressing the squared residuals ($\hat{u}_i^2$) on the original explanatory variables, their squares, and cross-products
• The test statistic is calculated as $n \times R^2$, where $n$ is the sample size and $R^2$ is the coefficient of determination from the auxiliary regression
• The test statistic follows a chi-square distribution under the null hypothesis of homoskedasticity

Chi-square distribution

• The White test statistic follows a chi-square distribution with degrees of freedom equal to the number of regressors in the auxiliary regression minus one
• The chi-square distribution is a probability distribution that arises from the sum of squared standard normal random variables
• It is commonly used in hypothesis testing and assessing the goodness of fit of models

Degrees of freedom

• The degrees of freedom for the White test are determined by the number of regressors in the auxiliary regression minus one
• The number of regressors includes the original explanatory variables, their squares, and cross-products
• The degrees of freedom represent the number of independent pieces of information used to estimate the parameters in the auxiliary regression

Critical values

• The critical values for the White test are obtained from the chi-square distribution table based on the chosen significance level and the degrees of freedom
• Common significance levels are 1%, 5%, and 10%, denoted as $\alpha = 0.01$, $\alpha = 0.05$, and $\alpha = 0.10$, respectively
• If the calculated test statistic exceeds the critical value at the chosen significance level, the null hypothesis of homoskedasticity is rejected

P-value interpretation

• The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
• In the context of the White test, a small p-value (typically less than the chosen significance level) provides evidence against the null hypothesis of homoskedasticity
• If the p-value is smaller than the significance level, the null hypothesis is rejected, indicating the presence of heteroskedasticity

Advantages of White test

• The White test is a general test that does not require specifying the functional form of heteroskedasticity
• It can detect various forms of heteroskedasticity, including non-linear patterns and interactions between explanatory variables
• The test is relatively easy to implement and interpret, making it a popular choice among practitioners

Limitations of White test

• The White test may have low power in small samples, meaning it may fail to detect heteroskedasticity when it is actually present
• The test can be sensitive to model misspecification, such as omitted variables or incorrect functional form
• In the presence of outliers or influential observations, the White test may produce misleading results

Correcting for heteroskedasticity

• Once heteroskedasticity is detected in a linear regression model, it is important to address it to obtain reliable standard errors and valid inferences
• Several methods are available to correct for heteroskedasticity, depending on the nature and severity of the problem

Heteroskedasticity-consistent standard errors

• Heteroskedasticity-consistent standard errors, also known as robust standard errors, provide a way to obtain valid inferences in the presence of heteroskedasticity
• These standard errors are calculated using a different formula that accounts for the heteroskedasticity in the error terms
• The most common types of heteroskedasticity-consistent standard errors are White standard errors and Huber-White standard errors

White standard errors

• White standard errors, named after Halbert White, are a specific type of heteroskedasticity-consistent standard errors
• They are obtained by estimating the variance-covariance matrix of the coefficient estimates using the squared residuals from the original regression
• White standard errors are consistent and provide asymptotically valid inferences even in the presence of heteroskedasticity

Robust regression methods

• Robust regression methods are designed to be less sensitive to outliers and heteroskedasticity compared to ordinary least squares (OLS)
• These methods aim to minimize the impact of influential observations and provide more stable coefficient estimates
• Examples of robust regression methods include least absolute deviations (LAD) regression, M-estimation, and S-estimation

Weighted least squares

• Weighted least squares (WLS) is a regression technique that assigns different weights to each observation based on the variance of the error terms
• The weights are typically inversely proportional to the variance of the error terms, giving less weight to observations with higher variability
• WLS can be used to correct for heteroskedasticity by estimating the weights based on the estimated variance of the error terms from the original OLS regression
• The WLS estimator provides efficient and consistent estimates in the presence of heteroskedasticity