Estimation and Testing of Cointegrated Systems by an Autoregressive Approximation

This paper studies the estimation and testing of general cointegrated systems by using an autoregressive approximation. Simple estimators for both the cointegration vectors and their weight matrix in the autoregressive error correction model representation of the system are developed. Since these estimators assume that the number of cointegration vectors and their normalization are fixed in advance, convenient specification tests for checking the validity of these assumptions are also provided. The asymptotic distributions of the estimators and test statistics are derived by assuming that the order of the auto-regressive approximation increases with the sample size at a suitable rate. This generalizes some previous results derived for finite-order autoregressions as no assumption of a finite-parameter data-generating process is imposed. The estimators and tests of the paper are interpreted in terms of autoregressive spectral density estimators at the zero frequency and, in the special case of a finite-order Gaussian autoregression, their relation to maximum likelihood procedures is discussed. All estimators of the paper can be applied with simple least-squares techniques and used to construct conventional Wald tests with asymptotic chi-square distributions under the null hypothesis. The limit theory of the specification tests is nonstandard, similar to that in univariate unit root tests.