Multicointegration is traditionally defined as a particular long run relationship among variables in a parametric vector autoregressive model that introduces additional coin-tegrating 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 multicointe-grated 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. The findings are illustrated empirically in an analysis of fiscal sustainability of the US government over the post-war period.
A semiparametric triangular systems approach shows how multicointegration can occur naturally in an I(1) cointegrated regression model. The framework reveals the source of multicointegration as singularity of the long run error covariance matrix in an I(1) system, a feature noted but little explored in earlier work. Under such singularity, cointegrated I(1) systems embody a multicointegrated structure and may be analyzed and estimated without appealing to the associated I(2) system but with consequential asymptotic properties that can introduce asymptotic bias into conventional methods of cointegrating regression. The present paper shows how estimation of such systems may be accomplished under multicointegration without losing the nice properties that hold under simple cointegration, including mixed normality and pivotal inference. The approach uses an extended version of high-dimensional trend IV estimation with deterministic orthonormal instruments that leads to mixed normal limit theory and pivotal inference in singular multicointegrated systems in addition to standard cointegrated I(1) systems. Wald tests of general linear restrictions are constructed using a fixed-b long run variance estimator that leads to robust pivotal HAR inference in both cointegrated and multicointegrated cases. Simulations show the properties of the estimation and inferential procedures in finite samples, contrasting the cointegration and multicointegration cases. An empirical illustration to housing stocks, starts and completions is provided.