11 where P = prices D = dividends X = a linear term on Dividends or a time trend. We included a quadratic time trend also.

i, i are the positive and negative roots, respectively of the dynamic system

**_Al2_** _a2_ /2 + _2u_ - _r_ = 0 described above. The estimation of the nonlinear equation was done restricting the values of these roots.

This model, besides allowing the quantification of the divergence of observed values from the predicted values, is useful to verify the existence of bubbles. The null hypothesis of no bubbles implies that cl = 0, while the alternative implies cl > 0. The estimations (Appendix C) lead to rejection of the "no bubble" hypothesis in all the cases except Argentina. In all cases (Graph 8), deviations of the observed price from the predicted one are persistent, frequent, and significant in size. Argentina is the only country that does not have a bubble after 1992. These results are similar to those of the simpler model. The timing of the bubbles tends to coincide in both approximations.<sup>22</sup> In Argentina, there is evidence of bubbles during the periods of hyperinflation from 1985 to 1989. Hall, Psaradakis, and Sola (1999) identify bubbles for money supply, exchange rate, and prices in 1985 and 1989, years for which we also find bubbles in the stock market.

Following the same procedure, we estimate the intrinsic bubble model for the U.S. and obtain interesting results (Graph 9). First, we cannot reject the hypothesis of bubble behavior, similar to Froot and Obstfeld's result and to the more recent ones of Bonds and Cummins (2000). Second, the deviations of the observed price from the predicted one show two periods of significant stock price overvaluation. One begins in 1986 and ends in 1987 (the crash of 1987). The second one occurs in the second half of the 1990s. The significant deviation of late 1997 was (temporarily) corrected in 1998, while the overvaluation in 2000 had not been corrected by the end of the year.<sup>23</sup> These results are very similar to Reinhart's (1998) quantification of overvalued stock prices in U.S.<sup>24</sup> Having quantified the over/under valuation of assets in the Latin American countries and the U.S.,25 we examine the correlation of these measures across time. For the period 22 The residuals are the difference between the observed price and the predicted price. Since the predicted price already has a bubble component, this approximation underestimates the size of the bubble and might lead us to overlook a bubble episode when it actually existed. This residual can be considered as the innovation in the bubble.

23 The data come from Robert Shiller's book, _Irrational Exuberance,_ and can be found on his webpage. He has dividend data until December 2000, and hence the model was estimated until that date.

> 24 Reinhart uses a cointegration approach to verify the equilibrium relationship between the price-toearnings ratio and fundamentals for the period 1980-96. His results of departures from the predicted model are summarized in Figure I of his paper.

**25** We tried to quantify for the East Asian countries the degree of departure from the benchmark cases, and we did not obtain any significant departures after 1995. This is a strange result, given that we reject, as Sarno and Taylor do, the no-bubbles hypothesis. The only explanation seems to be that all the bubbles happened before 1995. Future versions of this paper will present this, together with the crashes in East Asian countries.