Linear regression has a long history in its way of development from the very begging of eighteenth century till today. A lot of literatures are available in this area, these literatures involves the estimation of regression coefficients and constant by Ordinary Least Square (OLS) method i.e. by minimizing the sum of square of the vertical distances between the observed points and the assumed regression line, and estimate the regression coefficients traditionally known as OLS estimation procedure.

M. F. Hossain and G. Khalaf, (2009) showed that OLS method does not minimize actual distance from the observed point to the fitted regression line. They have suggested minimum perpendicular distance square (MPDS) Method estimation for simple linear regression in case of homoscedasticity which boils down the traditional OLS method. But regression disturbances whose variances are not constant across observations are heteroscedastic. Heteroscedasticity arises in numerous applications, in both cross-section and time-series data. For example, even after accounting for firm sizes, we expect to observe greater variation in the profits of large firms than in those of small ones. The variance of profits might also depend on product diversification, research and development expenditure, and industry characteristics and therefore might also vary across firms of similar sizes. When analyzing family spending patterns, we observe greater variation in expenditure on certain commodity groups among high-income families than low ones due to the greater discretion allowed by higher incomes [1]. MPDS method is not suitable for this type of heteroscedasticity situation because this method was established only for homoscedasticity cases.

In this paper we have considered minimum perpendicular distance square method in case of heteroscedasticity which we called Generalized Minimum Perpendicular Distance Square (GMPDS) method.

Suppose the simple linear regression model is

where the response variable is related to the explanatory variable through the regression coefficient, constant intercept and random disturbance term. We assume that the disturbance terms follow all assumptions of classical linear regression model.

The estimation procedure of regression coefficient by Ordinary Least Square (OLS) method and Generalized Least Square (GLS) method is actually minimizing the sum of square of the vertical distances from the observed points to the assumed regression line.

The OLS estimators are:

and

The important assumption for applying OLS method is that the variance of each disturbance term, conditional on the chosen values of the explanatory variables, is some constant number (is called homoscedasticity assumption). If the data violet this homoscedasticity assumption that is the variance of each disturbance term conditional on the chosen values of the explanatory variables is random (say) then we can not apply OLS and in this case we apply GLS estimation procedure for estimating parameters [2].

The GLS estimators are:

where,

The problem of OLS and GLS estimation is that, actually they don’t minimize real distance from the observed point to the fitted regression line rather they minimize the vertical distance from the observe point to the fitted regression line. For this reason we have the well known theorem is

.

where is the estimated regression coefficient of on and is the estimated regression coefficient of on. If OLS and GLS minimize real distance (error) then should be unity that is. But in OLS and GLS methods, it only occurs if data are perfectly correlated, that is. In real life problem this type of perfect correlation occurs in rare case.

The Minimum Perpendicular Distance Square Method suggested by Hossain and Khalaf (2009) produced the estimator which gives for all cases and it indicates that the errors are really minimized and gives more accurate result than that of OLS [3].

The real distance of the assumed regression line from the points are not the vertical distances or height of the point minus height of regression line i.e..

In fact the actual distances from the line to the points are the perpendicular distances’s (as indicated in Figure 1). These perpendicular distances would also be positive and negative according to is above the line or below the line. Also assuming that

. Hence estimating and by minimizing sum of the squares of these perpendicular distances will produce the closest fitted regression line from the points which may be used for more accurate prediction purposes.

Let us consider two-variable linear regression function is

which for ease of algebraic simplification we write as

where for each and the response variable is related to the explanatory variable through the regression coefficient, constant intercept and random disturbance term. We know that one of the important assumptions of the classical linear regression model is that the variance of each disturbance term, conditional on the chosen values of the explanatory variables is some constant number equal to. This is the assumption of homoscedasticity. Symbolically,

Now if the conditional variance of are not same for each of the. i.e., heteroscedasticity. Symbolically,

and suppose the heteroscedastic variance are known. Then dividing (1) by both sides, we get

which for ease of exposition we write

where the transformed variables are the original variables divided by (the known). We use the notation and, the parameters of the transformed model, to distinguish them from the usual MPDS parameters and. Now we see

which is a constant. That is, the variance of the transformed disturbance term is now homoscedastic.

This procedure of transforming the original variables is done in such a way that the transformed variables satisfy the assumptions of the classical model. Now applying MPDS method to this transformed model to estimate parameter we call Generalized Minimum Perpendicular Distance Squares Method (GMPDSM). In short, GMPDS is MPDS on the transformed variables that satisfy the classical regression assumptions. The estimators thus obtained are knows as GMPDSM estimators.

The method of MPDS estimation actually minimize real distances from the observed points to the fitted regression line but OLS and GLS method fail to do that by using vertical distance from the observe points to the fitted regression line. But one of the crucial assumptions of MPDS method and also for traditional OLS method is that the variance of each disturbance terms remains some constant number. So we can not apply MPDS method when this assumption is violated. That is, in presence of heteroscedasticity OLS and MPDS is not suitable. In this paper our main focus is on minimum perpendicular deviations in case of heteroscedasticity, and we have shown in mathematically that GMPDS method gives an estimator that the error term is really minimized. Hence we propose GMPDS method in case of heteroscedasticity.