This paper continues the work of Saikkonen (2001,
Econometric Theory 17, 296–326) and develops
an asymptotic theory of statistical inference in cointegrated
vector autoregressive models with nonlinear time trends
in cointegrating relations and general nonlinear parameter
restrictions. Inference on parameters in cointegrating
relations and short-run dynamics is studied separately.
It is shown that Gaussian maximum likelihood estimators
of parameters in cointegrating relations have mixed normal
limiting distributions and that related Wald, Lagrange
multiplier, and likelihood ratio tests for general nonlinear
hypotheses have usual asymptotic chi-square distributions.
These results are shown to hold even if parameters in the
short-run dynamics are not identified. In that case suitable
estimators of the information matrix have to be used to
justify the application of Wald and Lagrange multiplier
tests, whereas the likelihood ratio test is free of this
difficulty. Similar results are also obtained when inference
on parameters in the short-run dynamics is studied, although
then Gaussian maximum likelihood estimators have usual
normal limiting distributions. All results of the paper
are proved without assuming existence of second partial
derivatives of the likelihood function, and in some cases
even differentiability with respect to nuisance parameters
is not required.
Principles of Statistical Inference: Likelihood and the Bayesian Paradigm
