Some Econometric Results for the Blanchard-Watson Bubble Model

Soren Johansen; Theis Lange

Some Econometric Results for the Blanchard-Watson Bubble Model

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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)¿y(t-1)+e(t), t=1,…,n, where s(t) is an i.i.d. binary variable with p=P(s(t)=1), independent of e(t) i.i.d. with mean zero and finite variance. We take ¿>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 p¿<1, and a finite variance if p¿²<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 ¿(n) of y(t) on y(t-1), is close to one. We show that the estimator of ¿(n) converges to ¿p, if the variance is finite, but if the variance of y(t) is infinite, we prove the curious result that the estimator converges to ¿¿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).

Originalsprog	Engelsk
Udgiver	Department of Economics, University of Copenhagen
Antal sider	9
Status	Udgivet - 2011

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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)¿y(t-1)+e(t), t=1,…,n, where s(t) is an i.i.d. binary variable with p=P(s(t)=1), independent of e(t) i.i.d. with mean zero and finite variance. We take ¿>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 p¿<1, and a finite variance if p¿²<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 ¿(n) of y(t) on y(t-1), is close to one. We show that the estimator of ¿(n) converges to ¿p, if the variance is finite, but if the variance of y(t) is infinite, we prove the curious result that the estimator converges to ¿¿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 = "Faculty of Social Sciences",

author = "Soren Johansen and Theis Lange",

note = "JEL Classification: C32",

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N2 - 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)¿y(t-1)+e(t), t=1,…,n, where s(t) is an i.i.d. binary variable with p=P(s(t)=1), independent of e(t) i.i.d. with mean zero and finite variance. We take ¿>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 p¿<1, and a finite variance if p¿²<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 ¿(n) of y(t) on y(t-1), is close to one. We show that the estimator of ¿(n) converges to ¿p, if the variance is finite, but if the variance of y(t) is infinite, we prove the curious result that the estimator converges to ¿¿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).

AB - 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)¿y(t-1)+e(t), t=1,…,n, where s(t) is an i.i.d. binary variable with p=P(s(t)=1), independent of e(t) i.i.d. with mean zero and finite variance. We take ¿>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 p¿<1, and a finite variance if p¿²<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 ¿(n) of y(t) on y(t-1), is close to one. We show that the estimator of ¿(n) converges to ¿p, if the variance is finite, but if the variance of y(t) is infinite, we prove the curious result that the estimator converges to ¿¿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).

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BT - Some Econometric Results for the Blanchard-Watson Bubble Model

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