S. Kirk Elwood

Contents: I. Introduction. – II. Theory. – III. Econometric Methodology. –

We wish to acknowledge useful inputs from Kenneth Rogoff, Andrew K. Rose, and Jamshed K. Uppal.

**I. Introduction**
The literature on speculative bubbles in foreign exchange rates is voluminous, with much of it failing to reject the presence of bubbles in many exchange markets.^{1} ^{ }Serious testing of this issue began with the work of Meese (1986), Evans (1986), and Woo (1987). Each used a different approach, and each found evidence failing to reject the presence of bubbles in at least some exchange markets.

A bubble is a sustained price movement away from a fundamental. Thus, thee central empirical problem in considering whether or not a given phenomenon represents a bubble is properly specifying the fundamental. That most tests cannot separate a bubble test from a model test is a deep difficulty as initially noted by Flood and Garber (1980).^{2} This problem is especially difficult for studying foreign exchange markets as there is disagreement on what is the true theoretical fundamental for a foreign exchange rate, much less being able to identify it in a given empirical situation. This paper avoids problems of misspecifying exchange rate fundamentals by observing deviations from uncovered interest parity. Regardless of the fundamentals of exchange rates and interest rates, efficient markets and rational expectations assure that the fundamental of deviations from uncovered interest parity is zero at all times.

We use state-space analysis employing the Kalman filter to estimate models that specify rational speculative bubbles as an unobserved component to test for their presence. This was first used to study the German hyperinflation (Burmeister and Wall, 1982) and more recently used by Wu (1995) to study foreign exchange rate bubbles. In contrast with most literature, Wu rejected continuous stochastic rational bubble activity in a set of recent exchange rate series using state-space analysis. Stochastic rational bubbles crash but at a time known to rational investors only by a probability for which they must be compensated with a risk premium. Thus, such a bubble rises faster than the rate of interest (Blanchard and Watson, 1982), the rate that a perfect foresight bubble that never crashes rises (Brock, 1974).

A main innovation of our approach is to use a backward induction method whereby the time series is examined backwards in time and the unobserved component is a decaying first order autoregressive component from a shock (or innovation). When the data is reversed back to its proper order, this shock has the interpretation of being the crash of a steadily increasing rational bubble component. If this rate of increase exceeds the rate of interest, then this could be a stochastic rational bubble.

Another innovation is that while using the uncovered interest parity model, we eschew analyzing forward exchange rates and stick instead with spot rate changes as our measure of expectations, thereby following fully the assumption of rational expectations. If rational expectations hold, then we avoid the misspecified fundamentals problem, although there are studies casting doubts on rational expectations in foreign exchange markets motivated by apparently empirically perverse relationships between forward rate changes and interest rate changes and survey data showing agents in foreign exchange markets having substantial heterogeneity of expectations (Froot and Frankel, 1989; Ito, 1990; MacDonald, 1990; Liu and Maddala, 1992).

After conducting Monte Carlo experiments to examine the effectiveness of the state-space technique, we then use it to test the series representing the deviations from uncovered interest parity by Japanese and German exchange and interest rates on a monthly basis between January of 1980 and August, 1995. Inspecting the whole series using standard diagnostic procedures finds no significant serial correlation with values appearing normally distributed around zero, apparently white noise as predicted by uncovered interest parity. However, we find one short span of the series where a bubble or some other changing intervention apparently occurred. It is estimated to have grown at a rate exceeding the rate of interest and is thus consistent with stochastic rational bubble theory. Furthermore, the timing of the apparent bubble is consistent with evidence of other possible bubble phenomena in German and Japanese financial markets.