Inference When a Nuisance Parameter
Is Not Identified Under the Null Hypothesis
Bruce E. Hansen
University of Rochester
and
The Rochester Center for Economic Research
Working Paper No. 296
January
1991
Revised: September
1991
I would like to thank the NSF for financial support.
1
ABSTRACT
It is not uncommon to find economists testing hypotheses in models where a
nuisance parameter is not identified under the null hypotheses. This paper studies the
asymptotic distribution theory for such problems. The asymptotic distributions of test
statistics are found to be functionals of chi-square processes. In general, the
distributions depend upon a large number of unknown parameters. A simulation
method is proposed which can calculate the asymptotic distribution. The testing
method is applied to the threshold autoregressive model for GNP growth rates
proposed by Potter (1991). We present formal statistical tests which (marginally)
support Potter's claim that there is a statistically significant threshold effect in a
univariate autoregression for U.S. GNP growth rates.
--- - - -
2
1. INTRODUCTION
This paper studies the problem of inference in the presence of nuisance
parameters which are not identified under the null hypothesis. The asymptotic
distributions of Wald, likelihood ratio (LR) and Lagrange multiplier-like (LM-like)
statistics are obtained for parametric econometric estimators under quite general
assumptions, allowing for simultaneous equations, stochastic regressors, heterogeneity,
and weak dependence. The asymptotic distributions are shown to be represented by
the supremum of a
chi-square
process,
a stochastic process which is a quadratic form
in a vector Gaussian process indexed by the nuisance parameter. This generalizes the
results of Davies (1977,
1987).
Unfortunately, these distributions appear to depend, in
general, upon the covariance function of the chi square process, which may depend in
complicated ways upon the model and data, precluding tabulation. As a proposed
remedy, we develop a simulation method which approximates the asymptotic null
distribution. This approximation is an improvement over the bounds of Davies (1977,
1987),
whose approximation error increases with sample size in many cases of interest.
This paper is organized
as
follows.
Section 2 gives several examples of
non-identified nuisance parameters. Section 3 introduces a distinction between global
estimates (where the structural and nuisance parameters are estimated jointly) and
pointwise estimates (where the structural parameters are estimated for fixed nuisance
parameters). Conditions for consistent pointwise estimation of the structural
parameters, uniformly in the nuisance parameter, are given. Section
4 develops a
theory for testing structural hypotheses when the nuisance parameter is not identified
under the null hypothesis. Likelihood ratio, Wald, Lagrange multiplier (LM), and
maximal pointwise Wald and LM tests statistics are examined. Section 5 develops an
asymptotic distribution theory for the test statistics. This distributions are represented
as functionals of
chi-square
processes,
which are quadratic forms in mean-zero
3
Gaussian processes. In the absence of heteroskedasticity and serial correlation, these
test statistics have the same asymptotic distribution. A new finding is
that
only the
maximal pointwise Wald and LM statistics have asymptotic distributions which are
robust to the presence of heteroskedasticity and serial correlation. The standard LR
and Wald statistics, for example, are not robust. Section 6 develops a simulation
method which can approximate the null asymptotic distribution. Section 7 extends the
results to
t-etatistics, Section 8
shows
how to apply the theory and techniques to
threshold models, and reports an application to a threshold autoregressive model of
GNP. All proofs are left to the appendix.
Throughout the paper
"~"
is used to denote weak convergence of probability
measures with respect to the uniform metric, and
11·11
denotes the Euclidean metric.
Sample size is
n.
4
2. EXAMPLES
It
may not be commonly understood
how
pervasive is the problem of
unidentified nuisance parameters. I list
below
a
few
examples taken from the recent
literature. In most of the
following
examples, the model has been parameterized so
that the null and alternative hypotheses are
H : e= 0
o
and the nuisance parameter I is not identified under H
o'
In this situation, an error commonly made in applied research is the unqualified
reporting of
t-statistics to measure the "significance" of the parameter estimate of e.
Since
the t-statistic is testing the hypothesis that e=
0,
under which I is not
identified, the normal approximation is not valid and inferences made from a
conventional interpretation of the
t-statistic may be misguided.
In the
following
examples, yt ' x
t
,and
e
t
are real-valued.
1. Additive non-linearity. Gallant,
(1987)
p.
139.
A simple example of this is
2. Box-Cox Transformation. Box and Cox (1964).
y11-1
x
l 2
- 1
= Q + e t + e
t
.
11 12
Originally introduced as a transformation of the dependent variable, the Box-Cox
transformation has been used by some authors, such as Heckman and Polachek (1974),
separately for each independent variable as
well.
In the above specification, neither
'1
nor
'2
is identified when e=
o.
