A BEST MODEL ON MULTIPLE LINEAR REGRESSION

We study the selection variables (predictors) on multiple linear regression model (MLRM). The best subset (Cp Mallow) and the stepwise (forward selection and backward elimination) methods are used to identify a best model on the MLRM. A simulation study is conducted using secondary data on Y (percentage of poverty line) and four predictors X1 (jobless), X2 (population growth rate), X3 (life expectancy), and X4 (length of study). The result showed that the best model of the forward selection and backward elimination are similar, X1 and X3, are significant, with the model 0 1 1 3 3 ˆ ˆ ˆ ˆi y x x       , but the forward selection (with 2 steps) is more efficient than backward elimination (with 3 steps). Unfortunately, the best subset method has a different selection variables, here X2 is also significant, so the model is 0 1 1 2 2 3 3 ˆ ˆ ˆ ˆ ˆi y x x x         . Finally, we conclude that the best model is 0 1 1 3 3 ˆ ˆ ˆ ˆ . i y x x       This is due to the both variables X1 and X3 are available on the three tests, namely (1) correlation testing (partial test), (2) stepwise method, and (3) best subset methodFor your paper to be published in the conference proceedings, you must use this document as both an instruction set and as a template into which you can type your own text. If your paper does not conform to the required format, you will be asked to fix it. Keyword Best subset, Cp , MLRM, and stepwise.

. Finally, we conclude that the best model is This is due to the both variables X1 and X3 are available on the three tests, namely (1) correlation testing (partial test), (2) stepwise method, and (3) best subset methodFor your paper to be published in the conference proceedings, you must use this document as both an instruction set and as a template into which you can type your own text. If your paper does not conform to the required format, you will be asked to fix it. Keyword -Best subset, Cp , MLRM, and stepwise.

I. INTRODUCTION
Generally, inferences about population parameter can be drawn from sample, and increasing a number of sample will affect significantly to the quality of inferences (Soejoeti, 2010). More detail about this, Bancrof [21] already studied to improve the inferences population using non-sample prior information (NSPI) from trusted sources. Therefore, we need sampling to get the eligible and representative sample (n) from population (N), with small error on level of significance ( )  , 0.01, 0.05 and or 0.10.

 
Following, Bhatacharya and Johnson [12], Walpole and Myers [19] and Bluman [1], the sample size (n) is then formulated by x is a predictor and i y is a response. Note that the general model of the MLRM is given To ensure that the model is significant, we then test the significant model of theˆi y to be recommended to users. Moreover, the model (ˆi y ) is generally tested using three criteria, namely (1) the coefficient determination (R 2 ), (2) analysis of variance (Anova) with F test, and or (3) partial (Draper and Smith, [21]). In this case, some assumptions of the regression model are also tested, such as (1) normality test and random error of (2) autocorrelation of the error, (3) heteroscedasticity and (4) multicollinearity problem among predictors.
In this paper, the introduction is presented in Section 1. The regression model, best subset and stepwise methods are given in Section 2. A simulation is then obtained in Section 3. Section 4 described the conclusion of the research. document is a template.

Multiple Regression Model
For an n pair of observations on p independent variables (X 1 ,…,X p ) and one dependent variable (Y), (X ij ,Y i ), for i=1,2,...,n and j = 1,2,...,p, the multiple regression model (in matrix) is given by Here, is a ( dimensional column vector of unknown regression parameters, is vector of response variables, X is a matrix of know fixed values of the independent variables an e is the error term which is assumed to be identically and independently distributed as . Here, In is the identity matrix of order n and is the common variance of the error variables. Following Montgomery [5] and Graybil and Iyer [10], the estimate coefficient regression model in matrix is then given as (2) Furthermore, we test the hypothesis testing of the, H 0 : and it is presented in Table 1. Note that SSE is sum square error, SST is sum square total and MSE is mean square error. Following Montgomery [5], we must test the assumptions of the MLRM, such as (1) the multicollinearity is tested using variance inflaction factor (VIF), that is VIF > 10, or F test is significant but t is not. We guarantee that there is no autocorelation of the error term and there is no heteroscedasticity. Moreover, Gujarati [7] showed that if the plot of error versus i X going to be large (see Figure 1.), then i X and i Y must be divided by i X in order to get the eligible data.

Stepwise Regression Method
The stepwise regression method generally consists of forward selection and backward elimination (Soejoeti,[25]). Here, we want to choose a small subset from the larger set (large set of candidate predictor variables) so that the resulting regression model is good predictive ability. In this method, we enter and remove predictors until there is no justifiable reason to enter or remove more. First step, we fit each of the one-predictor models, that is, regress y on x 1 , regress y on x 2 ,…, regress y on x p . The first predictor is the predictor that has the smallest t-test (or high correlation (r) or significant in F test). Similarly, 2 nd step, we suppose x 1 was the "best" one predictor, then we fit each of the two-predictor models with x 1 in the model, that is, regress y on (x 1 , x 2 ), regress y on (x 1 , x 3 ),…, and y on (x 1 , x p ). Again, the second predictor is the predictor that has the smallest t-test (or high correlation (r) or significant in F test). But, we must consider to remove one of them if the model on the 2 nd step, , is not significant. For example, if the β 1 = 0 has become not significant, remove x 1 . The procedure is stopped when adding an additional predictor has no significant t-test more. Following, [20], the first step of the backward elimination procedure is to allow us to fit the full model of the MLRM. We then eliminate one by one using correlation (r) criteria (and or t or F test) in testing hypothesis of the coefficient regression parameters. Here, we need many steps in getting the eligible coefficient regression parameters model of the MLRM.

Best Subset Method
The best subset method is used to find the eligible predictors (X) in the MLRM model, with n > p. Here, we select the subset of predictors that do the best at meeting some objective criterion. Following Draper and Smith [21], we follow several steps to get the eligible predictor in the model, that are: (1) consider and choose the highest R 2 : From Table 2, it is clear that X 1 and X 3 have high correlation, so they are significant and eligible to the model. To make sure that model is really good, we then analysis it using both methods, namely forward and backward selection methods as follow.
The output of the procedure of the stepwise forward selection and backward elimination are presented in Table 3. and Table 4. (Suleman, [14]).
Here, we got two steps on the forward selection model but not in backward elimination model (3 steps). We then conclude that forward selection method is more efficient than backward elimination method. Moreover, we then analysis the data using the best subsets method using Minitab software, and the result is given in the Table 5. (Suleman,[14]). Here, we also presented the output of the best subset method in the case of poverty line (Y) and X 2 (population growth rate), X 3 (life expectancy), and X 4 (length of study) on different data (see Table 5., Sopanti, [9]).

IV. CONCLUSION
The paper studied the selection variables (predictors) in multiple regression model (MLRM) using best subset and stepwise regression methods. The result showed that the best model of the stepwise (forward and backward) method are similar (X 1 and X 3, are significant, and the model is , but the forward selection (with 2 steps) is more efficient than backward elimination (with 3 steps). Unfortunately, the best subset method has a different selection variables, where X 2 is also significant, so the model is