Testing for Explosiveness in Panels: A Self-Normalized Test for Stationarity in Cointegrating Residuals

We propose a panel test for whether a long-run cointegrating relationship holds throughout the sample or temporarily breaks down through a transient explosive episode before reverting. Existing stationarity-null tests are designed to detect permanent nonstationarity and have low power against temporary explosiveness, while recursive explosive-root tests require a known normalized cointegrating vector and are not applicable when the long-run relationship involves multiple estimated fundamentals. Our procedure fills this gap by first estimating the cointegrating relationship using pooled Dynamic OLS and then applying a rolling self-normalized stationarity test to the estimated equilibrium errors. Forward and backward rolling supremum statistics are combined to detect explosive episodes regardless of timing. Local-to-unity pre-filtering improves size control under highly persistent errors, while bootstrap critical values provide inference. Monte Carlo simulations indicate good size and satisfactory power of the test. Applied to a quarterly panel of 50 U.S. states from 1975 to 2025, the test identifies a transient explosive departure from the house-price-to-fundamentals equilibrium during the mid-2000s housing boom, followed by reversion rather than permanent breakdown of the long-run relationship.