Semiparametric regression
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Methods
Many different semiparametric regression methods have been proposed and developed. The most popular methods are the partially linear, index and varying coefficient models.
Partially linear models
A partially linear model is given by
where is the dependent variable, is a vector of explanatory variables, is a vector of unknown parameters and . The parametric part of the partially linear model is given by the parameter vector while the nonparametric part is the unknown function . The data is assumed to be i.i.d. with and the model allows for a conditionally
This method is implemented by obtaining a consistent estimator of and then deriving an estimator of from the nonparametric regression of on using an appropriate nonparametric regression method.[1]
Index models
A single index model takes the form
where , and are defined as earlier and the error term satisfies . The single index model takes its name from the parametric part of the model which is a scalar single index. The nonparametric part is the unknown function .
Ichimura's method
The single index model method developed by Ichimura (1993) is as follows. Consider the situation in which is continuous. Given a known form for the function , could be estimated using the nonlinear least squares method to minimize the function
Since the functional form of is not known, we need to estimate it. For a given value for an estimate of the function
using kernel method. Ichimura (1993) proposes estimating with
the leave-one-out nonparametric kernel estimator of .
Klein and Spady's estimator
If the dependent variable is binary and and are assumed to be independent, Klein and Spady (1993) propose a technique for estimating using
where is the leave-one-out estimator.
Smooth coefficient/varying coefficient models
Hastie and Tibshirani (1993) propose a smooth coefficient model given by
where is a vector and is a vector of unspecified smooth functions of .
may be expressed as
See also
- Nonparametric regression
- Effective degree of freedom
Notes
- ^ See Li and Racine (2007) for an in-depth look at nonparametric regression methods.
References
- Robinson, P.M. (1988). "Root-n Consistent Semiparametric Regression". Econometrica. 56 (4). The Econometric Society: 931–954. JSTOR 1912705.
- Li, Qi; Racine, Jeffrey S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press. ISBN 978-0-691-12161-1.
- Racine, J.S.; Qui, L. (2007). "A Partially Linear Kernel Estimator for Categorical Data". Unpublished Manuscript, Mcmaster University.
- Ichimura, H. (1993). "Semiparametric Least Squares (SLS) and Weighted SLS Estimation of Single Index Models". Journal of Econometrics. 58 (1–2): 71–120. .
- Klein, R. W.; R. H. Spady (1993). "An Efficient Semiparametric Estimator for Binary Response Models". Econometrica. 61 (2). The Econometric Society: 387–421. JSTOR 2951556.
- Hastie, T.; R. Tibshirani (1993). "Varying-Coefficient Models". Journal of the Royal Statistical Society, Series B. 55: 757–796.