Fully Modified Least Squares for Multicointegrated Systems

Multicointegration is traditionally defined as a particular long run relationship among variables in a parametric vector autoregressive model that introduces links between these variables and partial sums of the equilibrium errors. This paper departs from the parametric model, using a semiparametric formulation that reveals the explicit role that singularity of the long run conditional covariance matrix plays in determining multicointegration. The semiparametric framework has the advantage that short run dynamics do not need to be modeled and estimation by standard techniques such as fully modified least squares (FM-OLS) on the original I(1) system is straightforward. The paper derives FM-OLS limit theory in the multicointegrated setting, showing how faster rates of convergence are achieved in the direction of singularity and that the limit distribution depends on the distribution of the conditional one-sided long run covariance estimator used in FM-OLS estimation. Wald tests of restrictions on the regression coefficients have nonstandard limit theory which depends on nuisance parameters in general. The usual tests are shown to be conservative when the restrictions are isolated to the directions of singularity and, under certain conditions, are invariant to singularity otherwise.  Simulations show that approximations derived in the paper work well in finite samples. We illustrate our findings by analyzing fiscal sustainability of the US  government over the post-war period.