Examining the Conditions that will strengthen the Success of the Iterative Stein-Rule Estimator of The Disturbance Variance in Proxy Model

The aim of this study is to give the conditions, in a linear regression model with proxy variables, when is the difference of variances of two estimators getting closer to each other. One of the mentioned estimators is the iterative Stein-rule estimator (ISRE) of the disturbance variance which is obtained by taking the Stein-rule estimator of the parameters in the estimator of the disturbance variance; one is the usual ordinary least squares (OLS) estimator of the disturbance variance. For that purpose the theoretical difference of variances is derived and the numerical analysis is handled to see the pattern of given theoretical difference.


Introduction:
In some applied statistical analysis, some variables may not be observed by the researcher. In this situation, the researcher may ignore the variable or take approximate information for this unobservable variable(s). New variables obtained by taking the approximate information for the unobservable variables can be defined as Proxy variable.
In proxy variable case, the problem is how to treat this proxy information. There have been some studies about the performance of the Proxy variables in a linear regression model.
For instance, Wickens and McCallum showed that the use of proxy variable yields smaller bias (Wickens and McCallum 1972). Aigner showed that the use of a proxy variable does not necessarily lead to smaller mean squared error (MSE) (Aigner 1974). Namba and Ohtani derived the explicit formula of the predictive mean squared error of the Stein-rule estimator and the positive part Stein-rule estimator for the regression coefficients when the proxy variables are used (Namba and Ohtani 2006).
Ohtani considered the estimator of the disturbance variance in a linear regression when the Stein-rule estimator is used in place of the ordinary least squares (OLS) estimator and called as the iterative Stein-rule estimator (ISRE) of the disturbance variance (Ohtani 2006). Ünal and Akdeniz defined an iterative positive part Stein-rule estimator with proxy variables of the disturbance variance in a linear regression model with proxy variables. In that paper, they defined a pre-test estimator to get the MSE of the iterative positive part Stein-rule estimator (Ünal and Akdeniz 2006). Also they used incomplete beta functions and partial derivatives to analyse the performance of MSE of this estimator.
The ISRE of the disturbance variance in proxy models was also compared to OLS estimator of the disturbance variance in a linear regression model with proxy variables with respect to MSE criterion (Ünal 2010). Ünal gave the conditions to dominate the variance of the ISRE of the disturbance variance by usual OLS estimator of the disturbance variance theoretically in a linear regression model (Ünal 2007).
In this study, the variance formula for the iterative Stein-rule estimator of the disturbance variance in proxy models is given. Also the difference of variances of the ISRE and the usual OLS estimator of the disturbance variance in proxy model is demonstrated theoretically. And the conditions for which the difference of variances getting closer to each other, are taking into consideration. For these purposes: In Section 2, the model and the estimators are constituted. In Section 3, the theoretical formula for the variance of the ISRE of the disturbance variance is obtained. In Section 4, theoretical difference between variance of the ISRE and the variance of the usual OLS estimator of the disturbance variance is given. In Section 5, numerical analysis is handled to see the pattern of given theoretical difference in section 4, and observe how it changes with changing values of parameters. And the last part concludes the paper by giving overall results of paper.

The Proxy Model
Let us first consider the partitioned linear regression model Also The usual OLS estimator of the disturbance variance in a linear regression model with proxy variables can be given as, Let us take the iterative estimator of the disturbance variance in the model with proxy variables (Ünal 2010). This iterative disturbance variance is constituted by using * SRP b instead of * b in the model with proxy variables: . It is readily shown that equation (5) Then the iterative proxy variance will be   For two independently distributed non central chi-squared variables, and , the following explicit formula, ** 12 ** 12 ( ) 1 00 2  (10) to go one step further (Ünal 2006

Numerical Analysis
In view of the complexity of the theoretical expression obtained in previous section, this section devoted to numerical evaluations and illustrations to overcome the sophisticate nature of expression (13) and to make an inference on it. That is, we can see the result by numerical evaluations to get clear insight.
For purposes of analysis, some different values which are frequently used in literature (Ohtani 1986 2)] a k n k    . The evaluations are made using Mathematica programming, which allow us to perform some complicated and tedious algebraic calculation on a computer, as well as help us to find new exact solutions and some representative results are illustrated in Table 1  Similarly, in Table 2, the values of     , and vice verse.
The main findings discovered from numerical analysis can be given by generalization to contribute to the literature in Conclusion.

Conclusion
In this study, in proxy model, given ISRE of the disturbance variance, *2 IP  , and the usual OLS estimator of the disturbance variance, *2 s are taken into consideration. Also, their superiority of each other is examined by using variance criterion. Using the obtained theoretical formula for , the performance of variances owing to numerical computations of equations, by using mathematica code, are interpreted numerically. Especially, the conditions of the difference of variances getting closer to each other, are taking into consideration.
As noticed before, in numerical evaluations some various values are chosen for parameters which are frequently used in the literature and some represented results are given in the tables. This means that the tendency of the results is not changed when some other values are taken for the parameters.
If our aim of this study is to analyse the alteration of , then the general findings can be given as follows:  Increase in value of is the most effective factor in decrease of variance differences.


Small values of * 1  and * 2  , approaching to the central chi-squared distribution, have a small effect on the decreasing difference in proxy models.
 Whatever the magnitudes of other parameters, small values of parameter have an effect on decrease of variance differences also.

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In general, as gets bigger, gets smaller and non-centrality parameters get closer to 0, then variances of two estimators get closer to each other.


One broad feature which emerges from the results is that regardless of the values of , whenever distribution of 1 v and 2 v tends to central chi-squared distribution, the difference will decrease for . In other words, it is observed that, whenever , neither estimator strictly dominates the other, in all parts of the parameter space. These results are compatible with the result of the paper which concerned with the similarly proposed estimator of the disturbance variance in non-proxy cases and compared them by using MSE criterion (Ohtani 1987). In that paper Ohtani showed that, ISRE of the disturbance variance is dominated by the usual estimator of the disturbance variance based on the OLS estimator under the MSE criterion, if the number of regressors, is greater than or equal to five. Then it may convince us that ignoring biasness or facing with proxyness are still taking us similar results having it even more difficult and having it in strict sense (Ohtani 1987).
In many of situations that we have considered, the former estimator seems to have the edge on the latter for smaller (exactly, ) values with respect to variance criterion. Given these findings, we faced with, for the optimal choice of the parameters (as gets bigger, gets smaller and non-centrality parameters get closer to 0), the   Vs in proxy models.

Appendix for Tables
Some selected results are given in the following tables.