Joint Detection of Random Coefficient Variability and ARCH Effects via Optimal Parametric and Pseudo-Gaussian Tests

Abstract

We propose parametric and pseudo-Gaussian tests for jointly detecting random coefficient variability and ARCH(1) effects in linear regression models. Both tests are derived from the ULAN property of the model and are locally asymptotically most powerful unbiased. The pseudo-Gaussian test is valid under a broad class of error distributions, not just the Gaussian. Existing procedures treat these two effects separately; our test handles them simultaneously, at the cost of a modest, quantifiable power loss relative to each marginal test at its own boundary alternative – the price of the equal-power (MSSMP) criterion underlying its construction. Monte Carlo simulations under normal, logistic, Student, and skew-normal errors show that the joint test consistently achieves the highest power when both effects are simultaneously present, at a modest power cost relative to the relevant marginal test when only one effect occurs; we confirm this with three empirical applications: housing prices, Bitcoin returns, and U.S. inflation/interest rate data.