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I should like to thank Lorenzo Figliuoli, Marcus Miller, Lars Svensson, and Bob Traa for useful discussions. Any errors remain my responsibility.
See, for example, the classic studies of stock return distributions by Fama (1965) and Mandelbrot (1963). More recent work by Akgiray and Booth (1988) on freely floating exchange rates, Marsh and Rosenfeld (1983) on bond prices and Jarrow and Rosenfeld (1984), Ball and Torous (1985) and Ho, Perraudin and Sørensen (1989) on stock returns suggest that mixed Brownian-Poisson processes provide a better statistical model of financial returns than does Brownian motion with drift.
This equation may be derived from a simple monetary model of exchange rate determination. Suppose that the logarithms of money demand in the domestic economy and the rest of the world are equal to: mt = pt + γ1yt + γ2it - υt and
For example, solving the Fokker-Plank equation for the conditional density of a process is often more difficult when the underlying forcing process has jump components.
See Boughton (1990). Boughton estimates a semi-elasticity of demand for UK M1, with respect to short term interest rates of 0.09. Given the use of the uncovered interest arbitrage formula in the derivation of the basic exchange rate equation in footnote 3 above, it is necessary to express interest rates so that, for example, a 10% rate equals 0.1. In this case, Boughton’s figure becomes 9
These results resemble those of other studies that have modeled financial return data with mixed Brownian-Poisson processes. Drift parameters are generally difficult to estimate precisely, while coefficients on Brownian terms tend to be large and highly significant (see Ho, Perraudin and Sørensen (1989) and the references cited therein).
Note that the correct distribution for the likelihood ratio statistic when parameters are not identified under the null is hard to determine (see Ho, Sørensen and Perraudin (1989) for a discussion of this issue in the context of ML estimation of mixed Brownian-Poisson processes). A reasonably safe expedient, however, is to take a X2 distribution with degrees of freedom equal to the total number of parameters associated with the jump components.