Lesson M06.L02: Testing for Heteroskedasticity

Field	Detail
Module	06 — Heteroskedasticity
Level	Graduate
Duration	30–35 min
Learning Objective	Execute and interpret the Breusch-Pagan LM test and the White test for heteroskedasticity; understand the intuition behind regressing squared residuals; know the limitations of formal tests.
Provenance	Wooldridge Introductory Econometrics, Ch 8.3

Explanation

Why Test Formally?

Residual plots (M06.L01) provide informal visual evidence of heteroskedasticity. Formal tests provide a decision rule with a known significance level. However, both tools have limitations: formal tests depend on functional form assumptions, and all tests have low power in small samples.

The Breusch-Pagan (BP) Test

The Breusch-Pagan test is the most widely used formal test for heteroskedasticity. It tests whether the error variance is a linear function of the regressors.

Null and Alternative Hypotheses

\[H_0: \text{Var}(u_i \mid \mathbf{x}_i) = \sigma^2 \quad \text{(homoskedasticity)}$$ $$H_1: \text{Var}(u_i \mid \mathbf{x}_i) = \sigma^2 h(\delta_0 + \delta_1 x_{i1} + \cdots + \delta_k x_{ik})\]

The test is equivalent to:

\[H_0: \delta_1 = \delta_2 = \cdots = \delta_k = 0\]

Under $H_0$, the variance of $u_i$ does not depend on any of the $x$'s.

Steps

Step 1: Estimate the original model by OLS; save the residuals $\hat{u}_i$.

\[y_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_k x_{ik} + u_i\]

Step 2: Compute squared residuals $\hat{u}_i^2$ and regress them on all regressors:

\[\hat{u}_i^2 = \delta_0 + \delta_1 x_{i1} + \cdots + \delta_k x_{ik} + v_i\]

Step 3: Compute the LM statistic (Lagrange Multiplier):

\[LM = n \cdot R_{\hat{u}^2}^2\]

where $R_{\hat{u}^2}^2$ is the $R^2$ from the auxiliary regression in Step 2.

Step 4: Under $H_0$, the LM statistic has an asymptotic chi-squared distribution:

\[LM \sim \chi^2_k\]

where $k$ is the number of slope regressors (the number of $x$'s) in the original model.

Step 5: Reject $H_0$ if $LM > \chi^2_{k,\alpha}$ (the critical value at significance level $\alpha$).

Intuition

The key idea: if the error variance is homoskedastic ($\sigma_i^2 = \sigma^2$ for all $i$), then knowing the values of $x_{i1}, \ldots, x_{ik}$ tells you nothing about $\hat{u}_i^2$. The auxiliary regression would have $R^2 \approx 0$ and $n \cdot R^2 \approx 0$. If $x$'s predict $\hat{u}_i^2$ well (high $R^2$), it suggests the variance depends on the $x$'s — evidence against homoskedasticity.

The White Test

The White test is a more general alternative to BP. Instead of regressing $\hat{u}_i^2$ only on the original $x$'s, it also includes squares and cross-products of the regressors.

Auxiliary Regression (for $k = 2$ regressors, $x_1$ and $x_2$):

\[\hat{u}_i^2 = \delta_0 + \delta_1 x_{i1} + \delta_2 x_{i2} + \delta_3 x_{i1}^2 + \delta_4 x_{i2}^2 + \delta_5 x_{i1} x_{i2} + v_i\]

Test Statistic

\[LM = n \cdot R_{\hat{u}^2}^2 \sim \chi^2_{p}\]

where $p$ is the number of regressors in the auxiliary regression (here $p = 5$).

BP vs White

Feature	Breusch-Pagan	White
Null	Variance linear in $x$'s	Any form of heteroskedasticity
Auxiliary regressors	$x_1, \ldots, x_k$	$x$'s, $x^2$'s, cross-products
Degrees of freedom	$k$	Up to $k(k+3)/2$
Power against linear het.	Higher	Lower (loses df)
Power against non-linear het.	May miss it	Higher
Preferred when	Model for het. is known	You want a general check

Practical rule: Run BP first as a quick check. If you suspect non-linear heteroskedasticity or have no prior model, follow with White.

Worked Example

Dataset: Wooldridge WAGE1, $n = 526$. Original model:

\[\log(wage_i) = \beta_0 + \beta_1\, educ_i + \beta_2\, exper_i + \beta_3\, exper_i^2 + u_i\]

Step 1: Run OLS, obtain residuals $\hat{u}_i$.

Step 2: Compute $\hat{u}_i^2$. Run auxiliary regression:

\[\hat{u}_i^2 = \delta_0 + \delta_1\, educ_i + \delta_2\, exper_i + \delta_3\, exper_i^2 + v_i\]

Results: $R^2 = 0.0286$ (the $x$'s explain 2.86% of variation in squared residuals).

Step 3: Compute LM statistic:

\[LM = 526 \times 0.0286 = 15.04\]

Step 4: Compare to $\chi^2_3$ (three slope regressors: $educ$, $exper$, $exper^2$). Critical value at 5%: $\chi^2_{3,0.05} = 7.815$.

Step 5: Since $15.04 > 7.815$, reject $H_0$. There is statistically significant evidence of heteroskedasticity in the log-wage equation.

Interpretation: The squared residuals are partially explained by the regressors, indicating that the error variance varies with education and/or experience. The OLS standard errors reported by default software are not reliable; use heteroskedasticity-robust standard errors (M06.L03).

Common Misconception

"If the Breusch-Pagan test fails to reject, homoskedasticity holds and I can use OLS standard errors."

Correction: Failing to reject $H_0$ means we lack sufficient statistical evidence of heteroskedasticity — it does not prove homoskedasticity. The test may lack power in small samples or against non-linear forms of heteroskedasticity. There is no cost to using heteroskedasticity-robust standard errors regardless of test results — they are valid under both homoskedasticity and heteroskedasticity. The modern consensus in applied econometrics is to always use robust standard errors in cross-sectional work, without relying on a pre-test. If you only use robust SEs after a significant BP test, you introduce a pre-test bias: your choice of SE depends on a noisy inference step.

Practice Prompts

Prompt 1 — BP test computation A researcher estimates a model with $k = 3$ regressors, $n = 200$ observations. The auxiliary regression of $\hat{u}_i^2$ on the three regressors yields $R^2 = 0.052$.

(a) Compute the Breusch-Pagan LM statistic. (b) What is the chi-squared critical value at the 5% significance level? (c) Do you reject or fail to reject homoskedasticity? What should the researcher do next?

Prompt 2 — White test vs BP A researcher suspects that the error variance in a firm-level dataset depends on the square of firm revenue, not just its level. The BP test (using revenue levels) fails to reject at 5%. The White test (adding revenue² and other cross products) rejects at 1%.

(a) Why might BP fail to detect this heteroskedasticity? (b) What does the White test result imply? (c) What is the researcher's recommended next step?

Prompt 3 — Interpretation of auxiliary R² In the Breusch-Pagan auxiliary regression, a researcher obtains $R^2 = 0.001$ with $n = 1000$ and $k = 5$.

(a) Compute the LM statistic. Is it significant at 5%? (b) Does a low $R^2$ in the auxiliary regression mean heteroskedasticity is not economically important? Explain. (c) Why can even a very small $R^2$ lead to rejection with a large sample?

Further Resources

Wooldridge, Introductory Econometrics (7th ed.), §8.3
Stata: estat hettest (Breusch-Pagan after regress); estat imtest, white (White test)
R: bptest(lm_object) from the lmtest package; bptest(lm_object, studentize = FALSE) for classical BP
Original paper: Breusch & Pagan (1979), "A Simple Test for Heteroscedasticity and Random Coefficient Variation", Econometrica, 47(5), 1287–1294
White (1980), "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity", Econometrica, 48(4), 817–838

Feature	Breusch-Pagan	White
Null	Variance linear in \(x\)'s	Any form of heteroskedasticity
Auxiliary regressors	\(x_1, \ldots, x_k\)	\(x\)'s, \(x^2\)'s, cross-products
Degrees of freedom	\(k\)	Up to \(k(k+3)/2\)
Power against linear het.	Higher	Lower (loses df)
Power against non-linear het.	May miss it	Higher
Preferred when	Model for het. is known	You want a general check