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(1)(1)(2)(3)(6)Public Affairs 854 Menzie D. ChinnSpring 2007 Social Sciences 7418University of Wisconsin-MadisonBubblesSuppose there is a stable money demand equation:where the absence of y indicates that income has been held constant, and its effect subsumed intoa constant (suppressed for convenience). Notice further that this notation differs from previous,in that the interest rate for period t is denoted by it+1. By the Fisher relation, this conventionalmoney demand equation can be re-written as a Cagan money demand equation:where once again, the rt term has been subsumed into a constant, and suppressed for notationalease. Set:that is, there is money market equilibrium and money supply is set exogenously. Further assumethe process defining money demand is nonstochastic (for the moment):Solving for pt yields:Notice that this is a recursive expression; it can be rewritten for the price level, lead byone period, viz.1 A similar paper which undertakes testing for bubbles is Woo (1987).(7)(8)(11)(1)Substituting this latter equation into equation (6) yields:this leads, via repeated substitution to:Clearly, equation (7) differs from the expression derived in Lecture 4 for the present value formof the flex-price monetary model. Only if the "no-bubble" restriction, is imposed will the two expressions be the same. The problem is that in general, this restrictioncannot be imposed, so that one cannot in turn rule out the following expression for the pricelevel:where b0 is an initial deviation of p0 from the value implied by the fundamentals, 2. Meese's Tests for Foreign Exchange Bubbles 12.1. A Joint Test for BubblesOnce again, assume a money demand equation:where now m and p are expressed in relative (to foreign country) terms. Assume uncovered(2)(3)(4)(5)(6)(7)interest parity (UIP), which is perfect capital substitutability.The object on the right-hand side of the equation is "expected depreciation", which is modeled asthe mathematical expectation of the log spot exchange rate at time t, based on time t informationset (Mt) minus the time t log-spot exchange rate. The next relation indicates that deviations from purchasing power parity (PPP) in log-levels follows a random walk: where the , is a white noise error term. This is a slightly odd assumption, as it implies the realexchange rate follows a random walk. However, Meese motivates it as an approximation to slowmean reversion to PPP (as would occur in the Dornbusch-Frankel model with very sticky prices). Substituting (2) and (3) into (1) yields:Defining b = a2/(1 + a2), 0 < b < 1, then equation (4) can be rewritten as:Since there is evidence of unit roots in nominal exchange rates, Meese first-differencesthe series being examined.Let the "fundamentals" be defined asIn other words, the fundamentals follow an AR(1) process. If the no-bubbles, or transversality, condition is imposed, then one obtains the usualpresent value expression for the exchange rate.(12)(13)(19)On the other hand, if the transversality condition in (9) is violated then any solution of the form:will be satisfied. Meese exploits the fact that McCallum's procedure, as applied to equation (6), yieldsconsistent estimates, while maximum likelihood applied to equation (12) will in general yieldinconsistent estimates if the bubble term is correlated with the RHS variables. Then a Hausmanspecification error test can then be applied, where the test statistic is given by:which is distributed P2 with one degree of freedom. Such an approach does not impose a specificform on the bubble, and as such is a more general test for bubbles. Another difference betweenMeese and other tests is that instead of estimating the income elasticity of money demand, hesearches over a grid of plausible parameter values.Note that in order to implement estimation of (12), )y and )m must be represented bylag polynomials of the same order, and that )x must be strongly exogenous with respect to )s.Examining the DM/$, £/$ and ¥/$ exchange rates, he in general he finds that the bubbletest statistic always rejects the null of identical coefficients. Since his coefficient estimates andthe diagnostic statistics for the US-Japan regressions are so poor, he considers these test resultsweak, and hence does not report them. This omission reflects the joint nature of the test,discussed further


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UW-Madison PA 854 - Bubbles

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