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This article uses a Bayesian unit-root test in Unobserved-ARCH models. This time series of interest is the volatility that is unobservable. The unit root testing is based on the posterior odds ratio, which is approximated by Markov Chain Monte Carlo methods. Simulations show that the testing procedure is efficient for moderate sample size. The unit-root hypothesis is rejected in the daily exchange rate of the Germany marc (DEM) with respect to the Greek Drachma.