Does the test detect every kind of heteroscedasticity?

No; it targets the specified auxiliary relationship.

Is it the same as the modified Park test used in some generalized-model work?

Related procedures share a variance-power idea, but targets and implementation can differ.

AI Glossary / Statistics and probability / Intermediate

Park test

Definition

The Park test is a regression-based diagnostic for heteroscedasticity, using an auxiliary relationship between the logarithm of squared residuals and the logarithm of a positive variable thought to explain error variance. In its common form, a nonzero auxiliary slope suggests that error spread changes systematically with that variable under the assumed variance structure.

What it means

A regression can predict the average outcome reasonably while being much less precise for some observations than others. For example, prediction errors may become more dispersed as order size increases.
The Park approach investigates a particular power-law pattern in that dispersion. It does not test every possible form of unequal variance. It also does not prove that the mean model is correct or identify a causal reason for changing uncertainty.

How it works

First fit the proposed mean regression and obtain residuals e_i. Choose a positive variable Z_i on substantive grounds as a possible driver of error variance. It can be a predictor or another justified scale variable.
The motivating variance relationship is proportional to Z_i raised to a power gamma. Taking logarithms gives a linear form in log(Z_i). Because true errors and their variances are unobserved, the auxiliary regression uses log(e_i squared) as its response.
Fit log(e_i squared) = alpha + gamma log(Z_i) + v_i, then assess the hypothesis gamma = 0 with an inferential procedure appropriate to the auxiliary model. A positive slope indicates increasing log-squared residual magnitude; a negative slope indicates decreasing magnitude.
The use of estimated residuals and the distribution of their logarithms complicate inference. A routine regression t statistic should not automatically be described as an exact finite-sample test under all original-error distributions. Leverage, outliers, and misspecification can affect the diagnostic.
Logarithms require positive Z_i and nonzero squared residuals. Adding an arbitrary small constant changes the calculation; document any alternative handling and consider a more suitable diagnostic if zeros are material.
Inspect the original mean relationship and residual plots as well. Systematic residual patterns can result from an omitted nonlinear mean term rather than the proposed variance mechanism alone. Nonrejection can also reflect limited power or an incorrect choice of variance predictor.
If unequal variance is credible, distinguish remedies for inference from changes to estimation. Heteroscedasticity-consistent covariance estimates, a justified weighted model, or a revised mean model address different problems. A diagnostic result alone does not select the right remedy.

Three practical examples

Operational forecasting: analysts examine whether error dispersion increases with positive shipment size after fitting an average-time model.
AI regression evaluation: a team investigates a proposed relationship between residual magnitude and positive task scale, while checking model misspecification.
Cost modeling: researchers assess a power relationship between spread and a positive explanatory quantity before considering an explicit variance model.

Think of it this way

Think of drawing a prediction line surrounded by a corridor. The corridor may widen as you move along the horizontal axis. The Park diagnostic asks whether that widening follows a particular relationship on logarithmic scales, rather than assuming every point has the same uncertainty.

Worked example

Suppose a fictional auxiliary regression has been fitted to log-squared residuals using log(Z), with Z positive. For illustration, its reported slope is 1.2 and its standard error is 0.3. These are stipulated fitted-summary inputs, not results from an unreported dataset.
The slope-to-standard-error statistic for the zero-slope null is 1.2/0.3 = 4. A p-value would additionally require a justified reference distribution and relevant degrees of freedom or asymptotic calibration; none is invented here.
Under the proposed variance relationship Var(error given Z) = C times Z^1.2, doubling Z multiplies variance by 2^1.2, approximately 2.2974. The corresponding standard deviation multiplies by sqrt(2.2974), or 2^0.6, approximately 1.5157.
Thus the variance would increase by about 129.74%, while standard deviation would increase by about 51.57%. These are different quantities.
The auxiliary intercept should not simply be exponentiated and labeled the exact variance constant without accounting for the log-error transformation and model assumptions. The example illustrates the slope's power-law interpretation, not a validated variance forecast or a complete Park test result.

Illustrative case study

Illustrative fictional case: Cedar Forecasting notices larger residuals on larger transactions. A Park-style diagnostic suggests a positive relationship with transaction scale.
Before reweighting the model, analysts inspect residual plots and discover that a missing nonlinear mean term explains part of the pattern. They revise the mean specification, reassess variance behavior, and use an uncertainty calculation appropriate to the remaining structure. This fictional case illustrates why a diagnostic should trigger investigation rather than an automatic transformation.

Common misconceptions and mistakes

Do not confuse a model for squared errors with a model for the outcome mean. Do not interpret the power on variance as the same power on standard deviation. Avoid calling a nonsignificant auxiliary slope proof of constant variance.
Do not take logarithms of nonpositive predictors or zero residual squares without an explicit policy. Do not assume unequal variance by itself biases ordinary least-squares coefficients when the conditional mean and exogeneity assumptions hold; its effects on efficiency and conventional standard errors are separate issues.

Business use

The Park test can help businesses question whether a single uncertainty estimate is appropriate across different operating scales. Its results should be interpreted alongside model diagnostics and domain knowledge. Reports should identify the variance predictor, auxiliary form, handling of zeros, reference procedure, and the reason for any subsequent modeling change.

Check your understanding

What does a variance exponent of 1.2 imply when Z doubles?

Answer: Variance multiplies by about 2.2974.

Why is the standard-deviation multiplier smaller?

Answer: Standard deviation is the square root of variance.

Why is no p-value reported in the example?

Answer: The fitted summaries alone do not establish an appropriate reference calculation.

Related terms

White test
Weighted least squares
Heteroscedasticity
residual analysis
auxiliary regression
Breusch-Pagan test
robust standard errors
variance function

Learn next

Next compare targeted variance diagnostics with broader heteroscedasticity tests. Then separate mean-model misspecification, variance modeling, and robust uncertainty estimation, including their small-sample limitations.

Sources

R. E. Park, Estimation with Heteroscedastic Error Terms (1966): https://www.jstor.org/stable/1910108 — original publication record verified; full text unavailable through this access. S. W. Rheeder, Inference for linear regression models in the presence of heteroskedasticity, section 4.2.1: https://repository.nwu.ac.za/bitstreams/822bd612-6448-495b-b735-ef0ea213d2aa/download — author dissertation explaining the log variance structure. Original numerical interpretation and fictional case provided here.

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