# Some Econometric Results for the Blanchard-Watson Bubble Model

10 Pages Posted: 18 May 2011 Last revised: 18 Jun 2011

Date Written: May 9, 2011

### Abstract

The purpose of the present paper is to analyse a simple bubble model suggested by Blanchard and Watson. The model is defined by y(t) =s(t)ρy(t-1) ε(t), t=1,…,n, where s(t) is an i.i.d. binary variable with p=P(s(t)=1), independent of ε(t) i.i.d. with mean zero and finite variance. We take ρ>1 so the process is explosive for a period and collapses when s(t)=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 finite mean if pρ<1, and a finite variance if pρ²<1. The question we discuss is whether a bubble model with infinite 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 coefficient ρ(n) of y(t) on y(t-1), is close to one. We show that the estimator of ρ(n) converges to ρp, if the variance is finite, but if the variance of y(t) is infinite, we prove the curious result that the estimator converges to ρ⁻¹. The proof applies the notion of a tail index of sums of positive random variables with infinite variance to find the order of magnitude of the product moments of y(t).

**Keywords:** time series, explosive processes, bubble models

**JEL Classification:** C32

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