This paper discusses the problem of decision support systems in the organization. The procedure (linear combination) developed with the aim to combine some predicted results obtained with simulation of linear and nonlinear regression models (experts), multiple regression model, nonparametric regression model, and semi parametric regression model. This adjustment procedure enforce some statistical characteristics like the expected value of the gross production rate based on Cobb-Douglas production function is unbiased for the actual value, and the total weights(importance) of all models(experts) is equal to one. We used modeling and simulation techniques to generate our data and to apply the procedure.
In economics, the relationship between the production (outputs) and the
production elements (inputs) known as production function. Usually decision makers
in the organization and companies used some statistical and economics methods to
support the production policy. Cobb-Douglas (CD) production function is one of the
most useful tools to support the production policy, [
input variables, we choose these three types of models because we think there is linear relationship, non-linear relationship and mixture relationship between gross production and the predictor variables. It is based on our hypothesis that the linear combination method will give us more accurate predictive gross production since we believe this is a better way to take into account the results of multiple models accompanied by an appropriate weighting scheme. Furthermore, we prove that this method gives an unbiased estimator to the gross production. Our contribution is that we use linear combination method to pool three regression models for predicting the gross production.
Linear combination is linear pooling of some sets of ordered pairs (importance, gross production), we have explored the prediction associated with different regression model can be combined into one final model via weighted linear combination will give more accurate results. Mathematically linear combination of the sequence y 1 , y 2 ,..., y n each with mean is: Y
n a y
i i i 1 , where ai is the weight of y i ,and
n (Y ) ai ( y i ) a1 a2 ... an i 1 (1) n The (Y ) if ai 1
i 1
technique to predict the value of the response (dependent) variable given any value of
the predictor (independent) variable. A general regression model is [
linear regression model: E ( y i x i ) x i
y i x i ei (3) Where α is the intercept and is k 1 vector of parameters. Under Gauss-Markov assumptions, the estimators of α and are Best Linear Unbiased Estimators (BLUE), and can be estimate by using Ordinary Least Squares method (OLS).
If we do not know the data generating process, it is very unlikely that a linear
regression (parametric) model is exactly the appropriate model specification, so the
estimators α and are not BLUE (best linear unbiased estimator). Instead of
making assumptions for the functional form of E (yi│xi), nonparametric regression
methods do not require any presumptions about the underlying data generating
process [
y i m (x i ) ei (4)
Where m (.) is some function of unspecified functional form. Some basic assumptions about m (.) are commonly made. Many methods have been devised to estimate the regression function m (.), but we will just consider a simple but effective estimate known as the Kernel regression estimate. Suppose we have a random sample (x1, y1), (x2, y2) …, (xn,yn). The Kernel regression estimate of m(x) =E (y│X=x) is given by: n k ( x i x j d
)y i mˆ (x) = i 1 n k (
x i x j ) i 1 d
Here, K is a nonnegative symmetric function that is not increasing as its argument gets away from zero, and d is a parameter called the smoothing parameter that is selected by the user to control the amount of smoothing. The estimator in equation (4) is called the Local- Constant Least- Squares, which can be interpreted as a weighted average of yi, where: k ( x i x j )
d n k ( x i x j ) d
is the weight attached to yi It should be noted that the product Kernel function ( x i x j d of the Kernel function of all components of x. That is: ( x i x j ) k k (
x is x js ) d s 1 d S
Where xjs is the sth component of xj and ds the sth component of d. The Kernel function k (.) can take several forms. In this paper we used the Gaussian Kernel function is defined as: k ( x is x js ) 1 exp[ 1 x is x js )2 ]
( d S 2 2 d S
The smoothing parameter d is generally the most important factor when performing nonparametric regression. The bandwidth is chosen to obtain a desirable ) is the product (6) (5) (7) trade-off between the bias and the variance of estimation, so we need a method that could balance the bias and variance of the resulting estimate.
This method is depending on the principle of selecting bandwidth that minimizes the
sum squared error of the resulting estimates. We will try to find the minimum MSE,
mathematically; we are trying to minimize the following function [
n i 1
Where mˆ j (x j ) is the Leave-One-Out estimator of m (.) evaluated at xj. SAS software uses this method to find the optimal bandwidth values.
Nonparametric techniques are very flexible because they do not need any
assumptions about functional form. However, there are cases in which the
relationship between the dependent variable and some independent variables is known
to be linear and the relation between the dependent variable and other independent
variables remain undetermined. In this situation semi parametric models are
developed to solve this problem. Semi parametric models have both parametric and
nonparametric components [
y i x i m (z i ) ei
Where is a k 1 vector of parameters, zi is a q 1 vector of independent
variables and ei is an additive error, is assumed to be uncorrelated with xi and zi. is
the parametric part of the model and the unknown function m (.) is the nonparametric
part. [
n n ˆ ( xi xi )1 xi y i i 1
xi xi- E (xi│ zi) and
y i =yi-E (yi│ zi)
Once we obtain ˆ , the nonparametric part m (zi) is easy to estimate. From equation (8), we get: m (zi) = y i xi ei
Then we can get the estimator of m (zi) as follow:
n ( y i xi )k ( z i d z j ) mˆ (z i ) i 1 n k ( z i z j ) i 1 d (9)
(9) (10)
(11)
variables: public capital (PUC), private capital (PRC), the employment rate (EMP), and the unemployment rate (UERP). The Cobb-Douglas (CD) production function is, y i f (PUC i , PRC i , EMPi ,UERPi ) (12) Where y i =gross product, PUC i =public capital, PRC i =private capital, EMPi =employment rate (labor) UERPi =unemployment rate.
( y i ) 0 PUC iB1 PRC iB2 EM Pi B3UER PIB4
i y i 0 1PUC i 2PRC i 3EMPi 4UERPi i (13)
1- If 1 2 3 4 =1, the product function has constant returns to scale. 2- If 1 2 3 4 <1, returns to scale are decreasing. 3- If 1 2 3 4 >1, returns to scale are increasing.
-2.010 (1.50) 4 -0.642 (0.28) 1 -0.211 (0.37) 1 -0.459 (0.470) 2
3.2
We will use the estimated models (experts), multiple parametric regression, nonparametric regression, and Semiparametric regression to predict o f the log-gross production, for sample of size equal to 10 observations, and by using the values of the columns(Parametric y 1 ,Nonparametric y 2 ,Semiparametric y 3 ) in the following table, we can estimates the weight corresponding to each expert(model) with SPSS software by using neural networks procedure and estimate the log-gross production Y as follow:
Y 0.194y 1 0.595y 2 0.255y 3 Correlation is a measure of the association between two variables; it is a very important part of statistics. One of the most fundamental concepts in many applications is the concept of correlation. If two variables are correlated, this means that you can use information about one variable to predict the values of the other variable. The correlation matrixes between the predicted values in the table 4 as follows: i
From Table 5 we can find that the correlation coefficient between the predicted values of the linear combination(Y) and the actual variable of gross production is (0.954) that means the linear combination method is more associated with the actual values than the expert1, expert2, and expert3.this strongest association supports the suggestion that the predicted value of the gross production which associated with the different experts that combined is better than taking the predictive values of each expert individually. The average difference is (0.261, 0.959, 0.146, 0.103) of the three regression models (parametric, non-parametric, and semi-parametric) respectively and linear combination procedure. The proposed method (linear combination) is more accurate than regression models because the difference (actual and Y) in table 4 is more less than the other models, so this means that the proposed method give us unbiased estimator. 4. Conclusion
between the linear combination method, parametric model, nonparametric model and semi-parametric and the actual values of the gross production are (0.954, 0.457, 0.935, and -0.015), because we believe the public capital (PUC) is linearly related to gross production, so we choose it as parametric variable and that is why the correlation between y3 and actual value is -0.015. Linear combination method is more associated with the actual value than the others experts (regression model), i.e. linear combination method reflects the experts views to find the most fitted predicted value to the actual value. Finally as forecasters often wish to provide an accurate predicted value, so we will use the linear combination method to predict of the gross production.