PRACTITIONERS CORNER: On the Formulation of Wald Tests on Long‐Run Parameters
| Author | Peter Boswijk |
| DOI | http://doi.org/10.1111/j.1468-0084.1993.mp55001008.x |
| Published date | 01 February 1993 |
| Date | 01 February 1993 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 55,1(1993)
0305-9049 S3.00
PRACTITIONERS CORNER
On the Formulation of Wald Tests on Long-Run
Parameters
Peter Boswijk
I. INTRODUCTION*
The single-equation error correction model has become one of the most
important tools in the econometric analysis of time series. It provides a clear
distinction between short-run dynamic adjustments and long-run equilibrium
relations. The strong connection between cointegration and error correction
models that follows from the Granger representation theorem see Engle and
Granger (1987) has only increased its use. Because economic theory usually
concerns long-run properties, hypotheses of interest often lead to restrictions
on the long-run parameters. In this note, we shall analyse two Wald test statis-
tics of linear restrictions on these parameters. The first statistic utilizes an
estimated covariance matrix of the long-run parameter estimators, obtained
frOm a first-order Taylor series expansion. The second statistic uses a refor-
mulation of the null hypothesis, leading to a linear restriction on a model that
is linear in the parameters. It is argued that the latter statistic has to be
favoured because of its invariance properties, and because it is not affected
by the lack of moments of the long-run parameter estimators. The use of
asymptotic standard errors for constructing confidence intervals will also be
addressed. An application to the UK consumption function illustrates the
issues.
II. THE MODEL AND THE TEST STATISTICS
Consider the single-equation error corection model:
p-1 p-1 ßx,_1+1, t=1,..., T (1)
i1 j-O
* This paper was completed while the author was visiting Nuffield College, University of
Oxford. Helpful comments and suggestions from Anindya Banerjee, David Hendry, Jan Kiviet
and Michael Veal! are gratefully acknowledged.
137
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