Regression Analysis for Demand Estimation

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Demand Estimation by Regression Method – Some Statistical Concepts for application ( All the formulae marked in red for remembering. The rest is for your concept)

In case of demand estimation working with data on sales and prices for a period of say 10 years may lead to the problem of identification. In such a case the different variables that may have changed over time other than price, may have an impact on demand more rather than price. In order to void this problem of identification what we adopt is the techniques of demand estimation through regression process in order to distinguish the effects of different variables on demand. In order to understand the basic working and application of the model, let us start with two variable model

Two-variable Regression model
To find out the relation between two variables X & Y, usually a linear relation is estimated. If it is non-linear one then we convert it into log-linear to estimate the equation. Among the scatter of points in plane X-Y, we try to fit in the best line that can estimate the relationship. Here Y is the dependent variable and X is the independent variable. Let us take the example given in Salvatore. Let demand be the function of advertisement expenditure by the particular firm. Then the scatter diagram will show as the ad. Exp. Increases the sales volume will rise. In order to estimate the relationship of Sales (Y), on ad. Exp. (X), we regress the following equation,

In order to establish this relation we need to estimate a and b with the help of the data set on Y and X. we use a technique called ordinary least squares technique in order to find out the best fitted line. In order to do so, we minimize the sum of squared errors (measure of overall variation of estimated sales from observed sales), assuming that the sum of error is equal to zero. Thus the error is given by,

Thus we need to minimize the above in such a way that the estimated values minimize the above error variance. Minimizing the above with respect to a and b we get the following two equations to obtain the estimated values of a and b as follows, and

From the above the estimated b is (to be remembered for your calculation purpose)

And estimated a is (to be remembered)

Apart from calculating in this way there are several computer statistical packages that will give us directly the estimated results once we directly provide the Y input and X input. However there are other parameters the output box provides us.

Test of Significance of b value that implies how significant is the impact of the variation in the explanatory variable on variation caused for dependent variable. For this we test the null hypothesis b =0. for that we use a test statistic that follows the t- distribution with degrees of freedom n-k, where nis the number of observations and k is the number of parameters estimated In this case n=10 and k=2. therefore d.f=8. the test statistic t is defined as, as b=0 under null hypothesis and S.E. is the standard error of the estimated b. The S.E of estimated b is given by

(to be remembered). This means that as standard error of estimated b is high the variation due to unexplained variation is relatively hgher as compared with the variation explained by explanatory variable. Thus significance of b will be less as t value will be small. T value is compared with the tabulated value of t with degrees of freedom 8 and level of significance to be equal to 5% (level of significance is the region where we may commit Type I error – Rejecting Null Hypothesis when it is true). If the calculated value is greater than the tabulated value we reject null hypothesis b=0 but accept the hypothesis that b is significant.

Test of Goodness of fit ( Coefficient of Determination)
In order to see whether the overall regression has been a good fit or not, we take the help of Coefficient of Determination, which is given as follows

(to be remembered) where the...
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