Multilateral exchange rates: A multivariate regression framework

Currencies must be priced in terms of a numéraire when they are included in a regression model. The numéraire can be either a single-currency numéraire or a multicurrency numéraire: a weighted basket of numéraire currencies. Pricing currencies in terms of a multicurrency numéraire results in a system of multilateral exchange rates. A no-arbitrage condition enforces the movements in the system of multilateral exchange rates associated with the numéraire currencies to be a singular system, where the covariance matrix is singular and its ordinary inverse does not exist. Singular systems pose a methodological challenge in a multivariate regression model. This paper provides a solution to overcome this methodological challenge by imposing implicit restrictions on both the explanatory variables and the regression coefficients. In addition, the generalized least squares estimator is modified by replacing the ordinary inverse with the generalized inverse. The proposed solution provides a consistent multivariate regression model to explain the observed heterogeneity in the relative currency market.