Inference in Nonlinear Regression when the
Strength of Identi
f
cation is Unknown
∗
Timothy Cogley
†
and Richard Startz
‡
February 1, 2011
(Preliminary and Incomplete)
Abstract
We present a Bayesian approach to inference in the presence of weak iden
ti
f
cation. A number of important weak identi
f
cation problems can be cast
in the form of a nonlinear regression. We suggest an approach to computing
Bayesian posteriors for such problems. The
f
rst contribution of the paper is
a solution to what can be a di
ﬃ
cult numerical computation task. We then
analyze the simplest such weak identi
f
cation problem, the ratio of coe
ﬃ
cients
in a linear regression where the denominator is close to zero. This problem
is also of signi
f
cant independent interest. A Monte Carlo analysis shows that
the proposed method has good frequentist properties. The second contribution
of the paper is to show that the Bayesian approach, when used with weakly
informative priors, provides a useful frequentist estimator. An application to
the accelerationist Phillips curve is included.
1
Introduction
This paper reconsiders the problem of constructing con
f
dence intervals in a non
linear regression in which the strength of identi
f
cation is unknown a priori. We study
regressions of the form
=
(
)+
(1)
where
is the parameter of interest and
and
2
are nuisance parameters. The
variable
is strictly exogenous, and the innovation
is
(0
2
)
. The parameter
∗
For comments and suggestions, we are grateful to James Ramsey, Tao Zha, and seminar partic
ipants at U.C. Davis and U.C. Santa Barbara.
†
Department of Economics, New York University, 19 W. 4th St., 6FL, New York, NY 10012.
Email: [email protected]
‡
Department of Economics University of Washington Box 353330 Seattle, WA 98l953330. Email:
[email protected]
1
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is unidenti
f
ed when
=0
, and it is weakly identi
f
ed when
≈
0
We assume
only weak prior information about
so the strength of identi
f
cation is unknown a
priori. We want a procedure for estimating
and constructing con
f
dence regions
that works well regardless of whether
is strongly or weakly identi
f
ed. I.e., the
coverage probability should be close to the nominal size, and the intervals should not
be excessively long.
An example of equation (1) is a Phillips curve a la Staiger et. al. (1997),
∆
=
(
−
1
−
∗
)+
(2)
where
is in
F
ation,
is unemployment, and
∗
is the natural rate of unemployment,
which is assumed to be constant. The parameters of interest are the natural rate
∗
and the slope of the Phillips curve
The natural rate is unidenti
f
ed when
and it is weakly identi
f
ed when
≈
0
ForUSdatacover
ingtheper
iodo
ftheGreat
Moderation (1985.Q12008.Q4), the OLS estimate of
is 0.012 with an asymptotic
standard error of 0.1. The point estimate is close enough to zero and its standard error
is large enough to raise concerns that
∗
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 Bayesian probability, HPD, entire real line, ) intervals.

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