Nonparametric cointegration analysis

Introduction

The concept of cointegration was introduced by Granger (1981) and elaborated further
by Engle and Granger (1987), Engle (1987), Engle and Yoo (1987), Stock and Watson
(1988), Phillips and Ouliaris (1990), Park (1990), Phillips (1991), Boswijk (1993,1994),
Perron and Campbell (1993), Johansen (1988, 1991, 1994), and Harris (1995), among others.
The basic idea behind cointegration is that if all the components of a vector time series
process z_t have a unit root there may exist linear combinations ξ^Tz_t without a unit root. These
linear combinations may then be interpreted as long term relations between the components of
^zt^.

In a recent series of influential papers, Johansen (1988, 1991, 1994) and Johansen and
Juselius (1990) propose an ingenious and practical full maximum likelihood estimation and
testing approach, based on a Gaussian Error Correction Model (ECM). This ECM is based on
the Engle-Granger (1987) error correction representation theorem for cointegrated systems,
and the asymptotic inference involved is related to the work of Sims, Stock and Watson
(1990). By stepwise concentrating all the parameter matrices in the likelihood function out,
except the matrix of cointegrating vectors, Johansen shows that the ML estimators of the
cointegrating vectors can be derived from the eigenvectors of a generalized eigenvalue
problem, and LR tests of the number of cointegrating vectors from the eigenvalues. This
approach has become the standard tool in macroeconometrics for analyzing long term
economic relations.

All cointegration approaches in the literature require consistent estimation of nuisance
and/or structural parameters. In this paper we propose consistent cointegration tests that do
not need specification of the data-generating process, apart from some mild regularity
conditions, or estimation of (nuisance) parameters. Thus these tests are completely
nonparametric. Our tests are conducted analogously to Johansen’s tests, inclusive the test for
parametric restrictions on the cointegrating vectors, namely on the basis of the ordered
solutions of a generalized eigenvalue problem. Moreover, similarly to Johansen’s approach we
can consistently estimate a basis of the space of cointegrating vectors, using the eigenvectors
of the generalized eigenvalue problem involved. However, in our case the two matrices
involved are constructed independently of the data-generating process, and we can use the

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