Shrinkage estimators for semiparametric regression model

Semiparametric regression models are extensions of linear regression models to include a nonparametric function of some explanatory variables. In semiparametric regression model researchers often encounter the problem of multicollinearity. In the context of ridge estimator, the estimation of shrinkage parameter plays an important role in analyzing data. In this paper, numerous selection methods of the shrinkage parameter of ridge estimator are explored and investigated. Our Monte Carlo simulation results suggest that some estimators can bring significant improvement relative to others, in terms of mean squared error.


Introduction
Semiparametric regression models have received considerable attention in statics and econometrics, because of their flexibility in modeling events [1,2]. "Consider a semiparametric regression model given by i i x t [3].
Most of the approaches for the semiparametric regression model are based on different nonparametric regression procedures. The have been several approaches to estimating β and (.) f . An alternative approach to nonparametric procedure is differencing methodology.
This incoming ,used differences to remove the trend in the data that arises from the function (.) f and does not require an estimator of the function (.) f and often called difference-based procedure.
Provided that (.) f is differentiable and the t ordinates are closely spaced , it is possible to remove the effect of the function (.) f by differencing the data appropriately. In model (Eq.(1)), [5] [5] concentrated on estimation of the linear component and used difference-based estimation procedure is optimal in the sense that the estimator of the linear component is asymptotically efficient and the estimator of the nonparametric component is asymptotically minimax rate optimal for the semiparametric model used higher order differences for optimal efficiency in estimating the linear party by using special class of difference sequences. Now consider a semiparametric regression model in the presence of multicollinearity. The existence of multicollinearity may lead to wide confidence intervals for the individual parameters or linear combination of the parameters and signs. For our purpose we only employ the ridge regression concept due to Hoerl and ken nard (1970), to combat multicollinearity. There are a lot of work adopting ridge regression methodology to overcome the multicollinearity problem.
Note that with 1 m p = = from (2.2) we have We then estimate the linear regression coefficient β by the ordinary least-square estimators based on the differences. Then we obtain the least-squares estimate The role of constraints (Eq. (3)) is now evident. The first condition ensures that, as the t 's become close , the nonparametric effect is removed and the second one ensures that the variance of the sum of weighted residuals remains equals to 2 σ in Eq. (2). Now, we define the ( ) n m n − × differencing matrix D whose element satisfy Eq. (3) as This and related matrices are given, for example, in [4,[6][7][8][9].
Applying the differencing matrix to model (Eq. (2)) permits direct estimation of the parametric effect. As a result of development in Speck man (1988) it is know that the parameter vector β in (Eq. (1)) can be estimated with parametric efficiency. We now show the difference-based estimators that can be used for this purpose. Since the data have been ordered so that the values of the nonparametric variable(s) are close, the application of the differencing matrix D in model (Eq. (2) where (.) tr is the trace function for a squared matrix and p is the projection matrix defined as

Ridge Estimator
To overcome the effect of multicollinearity, ridge estimator is usually utilized. The ridge estimator for the semiparametric regression model (RE) is defined as

Simulation Results
A Mont Carlo simulation scheme to evaluate the performance of the estimating methods for the ridge estimator shrinkage parameter. "The explanatory variables   , and for all study sample sizes (small, medium and large), and that they improved the performance of the ridge estimator compared to other methods because they gave the lowest values for MSE.

5-As for the correlation coefficient
, it was noticed the superiority of the K1 method for all sizes, followed by the K 8 method in small and medium samples size, but in the case of large samples size, the HK 2 method came second.

6-
The results showed that, when a correlation coefficient 0.99 ρ = , the K 3 method was the best, and in the next rank was the K11 method at different sizes of the study samples.