We apply semiparametric efficient estimation procedures for a seemingly unrelated regression model where the multivariate error density is elliptically symmetric to study the efficiency of the foreign exchange market. We consider both cointegrating regressions and standard stationary regressions. The elliptical symmetry assumption allows us to avoid the curse of dimensionality problem that typically arises in multivariate semiparametric estimation procedures, because the multivariate elliptically symmetric density function can be written as a function of a scalar transformation of the observed multivariate data. We test the unbiasedness hypothesis on both weekly and daily exchange rate data and strongly reject unbiasedness at the weekly horizon, but fail to reject the unbiasedness hypothesis on the daily data. Estimates of the semiparametric procedure in some cases differ substantially from traditional OLS estimates.