Stochastic Restricted Biased Estimators in misspecified regression model with incomplete prior information

In this article, the analysis of misspecification was extended to the recently introduced stochastic restricted biased estimators when multicollinearity exists among the explanatory variables. The Stochastic Restricted Ridge Estimator (SRRE), Stochastic Restricted Almost Unbiased Ridge Estimator (SRAURE), Stochastic Restricted Liu Estimator (SRLE), Stochastic Restricted Almost Unbiased Liu Estimator (SRAULE), Stochastic Restricted Principal Component Regression Estimator (SRPCR), Stochastic Restricted r-k class estimator (SRrk) and Stochastic Restricted r-d class estimator (SRrd) were examined in the misspecified regression model due to missing relevant explanatory variables when incomplete prior information of the regression coefficients is available. Further, the superiority conditions between estimators and their respective predictors were obtained in the mean square error matrix (MSEM) sense. Finally, a numerical example and a Monte Carlo simulation study were used to illustrate the theoretical findings.


Introduction
Misspecification due to left out relevant explanatory variables is very often when considering the linear regression model, which causes these variables to become a part of the error term. Consequently, the expected value of error term of the model will not be zero. Also, the omitted variables may be correlated with the variables in the model. Therefore, one or more assumptions of the linear regression model will be violated when the model is misspecified, and hence the estimators become biased and inconsistent. Further, it is well-known that the ordinary least squares estimator (OLSE) may not be very reliable if multicollinearity exists in the linear regression model. As a remedial measure to solve multicollinearity problem, biased estimators based on the sample model = + with prior information which can be exact or stochastic restrictions have received much attention in the statistical literature. The intention of this work is to examine the performance of the recently introduced stochastic restricted biased estimators in the misspecified regression model with incomplete prior knowledge about regression coefficients when there exists multicollinearity among explanatory variables.
When we consider the biased estimation in misspecified regression model without any restrictions on regression parameters, Sarkar (1989) discussed the consequences of exclusion of some important explanatory variables from a linear regression model when multicollinearity exists. Şiray (2015) and Wu (2016) examined the efficiency of the r-d class estimator and r-k class estimator over some existing estimators, respectively in the misspecified regression model. Chandra and Tyagi (2017) studied the effect of misspecification due to the omission of relevant variables on the dominance of the r-(k,d) class estimator. Recently, Kayanan and Wijekoon (2017) examined the performance of existing biased estimators and the respective predictors based on the sample information in a misspecified linear regression model without considering any prior information about regression coefficients. It is recognized that the mixed regression estimator (MRE) introduced by Theil and Goldberger (1961) outperform ordinary least squares estimator (OLSE) when the regression model is correctly specified. The biased estimation with stochastic linear restrictions in the misspecified regression model due to inclusion of an irrelevant variable with the incorrectly specified prior information was discussed by Teräsvirta (1980). Later Mittelhmmer (1981), Ohtani and Honda (1984), Kadiyala (1986) and Trenkler and Wijekoon (1989) discussed the efficiency of MRE under misspecified regression model due to exclusion of a relevant variable with correctly specified prior information. Further, the superiority of MRE over the OLSE under the misspecified regression model with incorrectly specified sample and prior information was discussed by Wijekoon and Trenkler (1989). Hubert and Wijekoon (2004) have considered the improvement of Liu estimator (LE) under a misspecified regression model with stochastic restrictions, and introduced the Stochastic Restricted Liu Estimator (SRLE).
In this paper, the performance of the recently introduced stochastic restricted estimators namely the Stochastic Restricted Ridge Estimator (SRRE) proposed by Li and Yang (2010), Stochastic Restricted Almost Unbiased Ridge Estimator (SRAURE) and Stochastic Restricted Almost Unbiased Liu Estimator (SRAULE) proposed by Wu and Yang (2014), Stochastic Restricted Principal Component Regression Estimator (SRPCR) proposed by He and Wu (2014), Stochastic Restricted r-k class estimator (SRrk) and Stochastic Restricted r-d class estimator (SRrd) proposed by Jibo Wu (2014) were examined in the misspecified regression model when multicollinearity exists among explanatory variables. Further, a generalized form to represent these estimators is also proposed.
The rest of this article is organized as follows. The model specification and the estimators are written in section 2. In section 3, the Mean Square Error Matrix (MSEM) comparison between two estimators and respective predictors are considered. In section 4, a numerical example and a Monte Carlo simulation study are given to illustrate the theoretical results in Scalar Mean Square Error (SMSE) criterion. Finally, some concluding remarks are mentioned in section 5. The references and appendixes are given at the end of the paper.
Since all these estimators can be written by incorporating ̂, now we write a generalized form to represent SRRE, SRAURE, SRLE, SRAULE, SRPCR, SRrk and SRrd as given below: and * , and it is non-negative definite matrix if it stands for ℎ , ℎ and ℎ . Now the expectation vector, bias vector, the dispersion matrix and the mean square error matrix can be written as where = ( + 2 ) −1 and = ( * ′ + * ′ −1 ).
Based on 2.27 to 2.30, the respective bias vector, dispersion matrix and MSEM of the MRE, SRRE, SRAURE, SRLE, SRAULE, SRPCR, SRrk and SRrd can easily be obtained, and given in Table B1 in Appendix B.
By using the approach of Kadiyala (1986) and equations (2.3) and (2.4), the generalized prediction function can be defined as follows: (2.32) where 0 is the actual value and ̂( ) is the corresponding predictor. The MSEM of the generalized predictor is given by Note that the predictors based on the MRE, SRRE, SRAURE, SRLE, SRAULE, SRPCR, SRrk and SRrd are denoted by ̂,̂,̂,̂,̂,̂,̂ and ̂ respectively.
This completes the proof.
The following theorem can be stated for the superiority of ̂( ) over ̂( ) with respect to the MSEM criterion.
Based on Theorem 1 and Theorem 2 we can define Corollaries C1-C28, written in the Appendix C, for the superiority conditions between two selected estimators and for the respective predictors by substituting the relevant expressions for (̂( ) ), (̂( ) ), (̂( ) ) and (̂( ) ) given in Table B1 in Appendix B.
Note that when ( , ) = (4, 0) the model is correctly specified, when ( , ) = (3, 1 ) one variable is omitted from the model and when ( , ) = (2, 2 ) two variables are omitted from the model. For simplicity we choose shrinkage parameter values k and d in the range (0, 1).
Then the following set up is considered to investigate the effects of different degrees of multicollinearity on the estimators: Three different sets of observations are considered by selecting ( , ) = (5, 0), ( , ) = (4, 1) and ( , ) = (3, 2) when = 50, where denotes the number of variable in the model and denotes the number of misspecified variables. Note that when ( , ) = (5, 0) the model is correctly specified, when ( , ) = (4, 1 ) one variable is omitted from the model and when ( , ) = (3, 2 ) two variables are omitted from the model. For simplicity, we select values and in the range(0,1).
The simulation is repeated 2000 times by generating new pseudo random numbers and the simulated SMSE values of the estimators and predictors are obtained using the following equations: respectively.
From Table B5,  The results in Table B6

Conclusion
Theorem 1 and Theorem 2 give the common form of superiority conditions to compare the estimators (MRE, SRRE, SRAURE, SRLE, SRAULE, SRPCR, SRrk and SRrd) and their respective predictors in MSEM criterion in the misspecified linear regression model when the prior information of the regression coefficients is incomplete, and the multicollinearity exists among the explanatory variables.
From the simulation study, it can be identified the superior estimators and predictors over the others when the conditions are different. The results obtained in this research will produce significant improvements in the parameter estimation in misspecified regression models with incomplete prior information, and the results are applicable to real-world applications.

Conflicts of Interest
The authors declare that they have no conflicts of interest. Lemma A3: (Baksalary and Kala, 1983) Let B≥ 0 of type × matrix, is a × 1 vector and is a positive real number. Then the following conditions are equivalent. i. − ′ ≥ 0 ii.