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In statistics, semiparametric regression includes regression models that combine parametric and nonparametric models. They are often used in situations where the fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. Semiparametric regression models are a particular type of semiparametric modelling and, since semiparametric models contain a parametric component, they rely on parametric assumptions and may be misspecified and inconsistent, just like a fully parametric model.
Many different semiparametric regression methods have been proposed and developed. The most popular methods are the partially linear, index and varying coefficient 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 heteroskedastic error process of unknown form. This type of model was proposed by Robinson (1988) and extended to handle categorical covariates by Racine and Li (2007).
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.
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 .
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 .
If the dependent variable is binary and and are assumed to be independent, Klein and Spady (1993) propose a technique for estimating using maximum likelihood methods. The log-likelihood function is given by
where is the leave-one-out estimator.
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
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In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.
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In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function.
In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.
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In statistics and econometrics, the maximum score estimator is a nonparametric estimator for discrete choice models developed by Charles Manski in 1975. Unlike the multinomial probit and multinomial logit estimators, it makes no assumptions about the distribution of the unobservable part of utility. However, its statistical properties are more complicated than the multinomial probit and logit models, making statistical inference difficult. To address these issues, Joel Horowitz proposed a variant, called the smoothed maximum score estimator.
In statistics and econometrics, optimal instruments are a technique for improving the efficiency of estimators in conditional moment models, a class of semiparametric models that generate conditional expectation functions. To estimate parameters of a conditional moment model, the statistician can derive an expectation function and use the generalized method of moments (GMM). However, there are infinitely many moment conditions that can be generated from a single model; optimal instruments provide the most efficient moment conditions.
A partially linear model is a form of semiparametric model, since it contains parametric and nonparametric elements. Application of the least squares estimators is available to partially linear model, if the hypothesis of the known of nonparametric element is valid. Partially linear equations were first used in the analysis of the relationship between temperature and usage of electricity by Engle, Granger, Rice and Weiss (1986). Typical application of partially linear model in the field of Microeconomics is presented by Tripathi in the case of profitability of firm’s production in 1997. Also, partially linear model applied successfully in some other academic field. In 1994, Zeger and Diggle introduced partially linear model into biometrics. In environmental science, Parda-Sanchez et al used partially linear model to analysis collected data in 2000. So far, partially linear model was optimized in many other statistic methods. In 1988, Robinson applied Nadaraya-Waston kernel estimator to test the nonparametric element to build a least-squares estimator After that, in 1997, local linear method was found by Truong.