The purpose of the present paper is to analyse a simple bubble model suggested by Blanchard and Watson. The model is de…ned by yt = styt 1 + t; = 1;:::;n; where st is an i.i.d. binary variable with p = P(st = 1); independent of t i.i.d. with mean zero and …nite variance. We take � > 1 so the process is explosive for a period and collapses when st = 0: We apply the drift criterion for non-linear time series to show that the process is geometrically ergodic when p < 1, because of the recurrent collapse. It has a …nite mean if p� < 1; and a …nite variance if p� 2 < 1: The question we discuss is whether a bubble model with in…nite variance can create the long swings, or persistence, which are observed in many macro variables. We say that a variable is persistent if its autoregressive coe¢ cient ^ �n of yt on yt 1; is close to one. We show that �n P ! �p if the variance is …nite, but if the variance of yt is in…nite, we prove the curious result that ^ �n P ! � 1 . The proof applies the notion of a tail index of sums of positive random variables with in…nite variance to …nd the order of magnitude of P n=1 y 2 1 and P n t=1 ytyt 1 and hence the limit of ^ �n: