Point Estimation

In point estimation procedure we make an attempt to compute a numerical value, from sample observations, which could be taken as an approximation to the parameter. The estimators, which are also referred to as statistics (plural of statistic), since based on observations which are random variables are themselves random variables. A number of estimation methods like method of least square, method of maximum likelihood, method of moments etc., are available with some specific properties.

Method of Least Square:

The method of least square is specifically used in regression analysis to estimate the regression coefficients. To understand the technique of estimation let us consider the following simple example. The reader may please note that a formal treatment of the least square method which involves the inclusion of the disturbance term has been avoided for simplicity.

Suppose that consumption expenditure Y is linearly related to only one variable, family income X, this can be written mathematically as Y = a + bX

In economics such a relation is termed as a consumption function, where a is a measure of the consumption expenditure at zero level of income and b is a measure of the marginal propensity to consume i.e., it gives a measure of how much will be consumed from each additional unit of income. The consumption function is in the parametric form, specifying a different relationship for different values of the parameters (a and b) the parameters (a, b)are not known and needs to be estimated on the basis of a sample. Let a random sample of n household is drawn from the population under study. The information about consumption and income is recorded as follows for each of these households.

Consumption Expenditure
Family Income

On the basis of these sample observations we wish to estimate the consumption function. Let the estimating equation is \widehat Y = \widehat a + \widehat bX, where \widehat Y (Y- hat), \widehat a (a- hat) and \widehat b (b- hat) are the estimates of Y, a and b respectively.

Since \widehat Y is an estimate of Y, it will be very lucky on our part to have a \widehat Y equals to Y otherwise they will be different. The different between an estimate value \widehat Y, and the observed value Y is denoted by e which is usually termed as “residual”, “deviation” or “error term”. These residual may be positive or negative.

\begin{gathered} e = Y - \widehat Y \\ e = Y - \widehat a - \widehat bX \\ \end{gathered}

The smaller the residual are, the closer would be the estimating equation \widehat Y = \widehat a + \widehat bX to the original model Y = a + bX. Hence, to have a closer estimating equation for Y = a + bX we should minimize the residuals. The residual are minimized according to the following principle, which states that

“Those values of \widehat a and \widehat b should be chosen which minimize the sum of squared residual”. This principle is known as the “principle of least squares”

Thus, the sum of squared residual may be written as \sum {e^2} = \sum {\left( {\widehat Y - \widehat a - \widehat bX} \right)^2}
In order to minimize the quantity \sum {e^2}, we will use the technique of differential calculus. Hence, differentiating \sum {e^2} = \sum {\left( {\widehat Y - \widehat a - \widehat bX} \right)^2}with respect to \widehat a and \widehat b equating the resulting derivatives to zero.

\begin{gathered} \frac{{\partial \sum {e^2}}}{{\partial \widehat a}} = - 2\sum \left( {Y - \widehat a - \widehat bX} \right) = 0 \\ \frac{{\partial \sum {e^2}}}{{\partial \widehat b}} = - 2\sum X\left( {Y - \widehat a - \widehat bX} \right) = 0 \\ \end{gathered}

Simplifying the above equations, we have

\begin{gathered} \sum Y = n\widehat a + \widehat b\sum X \\ \sum XY = \widehat a\sum X + \widehat b\sum {X^2} \\ \end{gathered}

These two equations are called the “Normal Equation” in which if we substitute the values \sum Y,\,\,\sum X,\,\,\sum {X^2},\,\,\sum XY and n from our sample observations the two estimates \widehat a and \widehat b of the unknown parameters a and b can be determined by solving the simultaneous equations.