Simple Estimators for Monotone Index Models
University College London,
James L. Powell
University of California, Berkeley
In this paper, estimation of the coe¢cients in a “single-index” regression model is considered
under the assumption that the regression function is a smooth and strictly monotonic function
of the index. The estimation method follows a “two-step” approach, where the …rst step uses a
nonparametric regression estimator for the dependent variable, and the second step estimates the
unknown index coe¢cients (up to scale) by an eigenvector of a matrix de…ned in terms of this …rst-
step estimator. The paper gives conditions under which the proposed estimator is root-n-consistent
and asymptotically normal.
JEL Classi…cation: C24, C14, C13.
This research was supported by the National Science Foundation. Hyungtaik Ahn’s research
was supported by Dongguk Research Fund. We are grateful to Bo Honoré, Ekaterini Kyriazidou,
Robin Lumsdaine, Thomas Rothenberg, Paul Ruud, and Mark Watson for their helpful comments.
Estimation of the unknown coe¢cients ?0in the single index regression model
E(yijxi) = G(x0
where yiand xiare observable and G(?) is an unknown function, has been investigated in a number
of papers in the econometric literature on semiparametric estimation. (A survey of these estima-
tors is given in Powell (1994).) Some estimation methods, like the “average derivative” approach
of Härdle and Stoker (1989) and Powell, Stock, and Stoker (1989) and the “density-weighted least
squares” estimator of Ruud (1986) and Newey and Ruud (1991) exploit an assumption of smooth-
ness (continuity and di¤erentiability) of the unknown function G, but require all components of the
regressor vector x to be jointly continuously distributed, which rarely applies in practice. Härdle
and Horowitz (1996) has extended the average derivative estimator to allow for discrete regressors
at the expense of introducing four additional nuisance parameters to be chosen by users of their
estimator in addition to the standard smoothing parameter choice required in all nonparametric
estimators. Other estimation methods which assume smoothness of G include the “single-index
regression” estimators of Ichimura (1993a), Ichimura and Lee (1991), and, for the special case of
a binary dependent variable, Klein and Spady (1993); these estimation methods permit general
distributions of the regressors, but can be computationally burdensome, since they involve min-
imization problems with nonparametric estimators of G whose solutions cannot be written in a
simple closed form. Still other estimators of were proposed for the “generalized regression model”
proposed by Han (1987),
where the unknown transformation T(?) is assumed to be monotonic in its …rst argument, and where
the unobservable error term "iis assumed to be independent of xi. The assumed monotonicity of T,
which implies monotonicity of G in (1.1), is fundamental for the consistency of the “maximum rank
correlation” estimator of Han (1987) and the related monotonicity-based estimators of Cavanagh
and Sherman (1991); like the “single index regression” estimators, computation of the “monotonic-
ity” estimators is typically formidable, since it requires minimization of a criterion which may be
discontinuous and involves a double sum over the data.
In this paper, which combines the results of Ahn (1995) and Ichimura and Powell (1996), both
“smoothness” and monotonicity the nuisance function G are imposed – more speci…cally, it is as-
sumed to be di¤erentiable (up to a high order) and invertible in its argument. Simple “two-step”
estimators are proposed under these restrictions; the …rst step obtains a nonparametric estimator
of the conditional mean giof yigiven xiusing a standard (kernel) method, while the second step
extracts an estimator of ?0from a matrix de…ned using this …rst-step estimator. One estimator
of the unknown coe¢cients is based upon the “eigenvector” approach that was used in a di¤erent
context by Ichimura (1993b), and the corresponding second-step matrix estimator was considered
(again in a di¤erent context) by Ahn and Powell (1993). An alternative, closed-form estimator
of ?0is also proposed; the relation of the “eigenvector” to the “closed form” estimation approach
is analogous to the relation of limited information maximum likelihood (LIML) to two-stage least
squares (2SLS) for simultaneous equations models. These estimators are computationally simple
(since the second-step matrix estimator can be written in closed form), and do not require that all
components of the regressor vector xiare jointly continuously distributed. And, as shown below,
they are root-n consistent (where n is the sample size) and asymptotically normal under regular-
ity conditions that have been imposed elsewhere in the econometric literature on semiparametric
2. The Model and Estimator
Rewriting the single-index regression model (1.1) as
yi? gi+ ui? G(x0
i?0) + ui; (2.1)
i?0) ? E[yijxi] (2.2)
is the conditional mean of the (scalar) dependent variable yi given the p-dimensional vector of
regressors xi (so the unobservable ui has E[uijxi] = 0), the maintained assumption that G is
?(gi) ? G?1(gi) = x0
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