The Bierens Test for Certain Nonstationary
Models
Ioannis Kasparis
University of Cyprus
25 October 2007*
Abstract
We adapt the Bierens (Econometrica, 1990) test to the I-regular models of
Park and Phillips (Econometrica, 2001). Bierens (1990) defines the test hypoth-
esis in terms of a conditional moment condition. Under the null hypothesis, the
moment condition holds with probability one. The probability measure used
is that induced by the variables in the model, that are assumed to be strictly
stationary. Our framework is nonstationary and this approach is not always ap-
plicable. We show that Lebesgue measure can be used instead in a meaningful
way. The resultant test is consistent against all I-regular alternatives.
1 Introduction
A series of consistent specification tests for parametric regression functions has been
initiated are by H. Bierens (e.g. 1982, 1984, 1987, 1988, 1990). The most appealing
one is that of Bierens (1990). Contrary to the other tests mentioned above, the latter
test has a tractable limit distribution. In addition, its consistency is not achieved
by randomisation of some test parameter. The test was originally developed for
i.i.d. data and was adapted to strictly stationary weakly dependent data by de Jong
(1996). To the best of our knowledge, there is no fully consistent test for some class of
nonstationary models. In this paper we propose a Bierens (1990) kind of test for the
I-regular family of Park and Phillips (2001) (P&P hereafter). This family comprises
models, where the regression function is some integrable transformation of a unit
root process. Kasparis (2004) and Marmer (2005) develop functional form tests for
I-regular models with a single covariate. The tests of the two aforementioned papers
*Earlier versions of this paper were presented at the University of York, June 2005, and at the
conference in the honour of P. Dhrymes, at Paphos, June 2007. I would like to thank Jon Levellen
for sharing his data on NYSE returns.