Question

5.The following results have been obtained from a simple of 11 observations on the values of sales (Y) of a firm and the corresponding prices (X).

Estimate the regression line of sale on price and interpret the results

What is the part of the variation in sales which is not explained by the regression line?  

Estimate the price elasticity of sales.

6. Suppose that a researcher estimates a consumptions function and obtains the following results: 

where C=Consumption, Yd=disposable income, and numbers in the parenthesis are the ‘standard errors’

Test the significant of Yd statistically using t-ratios

Determine the estimated standard deviations of the parameter estimates. 

Answer 1

5. a) Sales regression line on price is given by

$Y = a +bX$

where $b = \dfrac{[N (\sum XY) - (\sum X)(\sum Y)]}{ N (\sum X^2) - (\sum X)^2}$

and $a = \bar{Y} -b\bar{ X}$

N = 11

$\sum XY = 1296836 \\ \sum X = N \bar{X} \\ \sum Y = N \bar{Y} \\ \sum X^2 = 3134543$

Putting the values,

$b = \dfrac{(11 * 1296836 - 11*519.18 * 11* 217.82) }{ (11*3134543 - 11*519.18*11*519.18)}$

On solving $b = \dfrac{52870.3364 }{ 169516.4036}=0.31$

a = 217.82 - 0.31*519.18

a = 55.89

The sales regression on price is, therefore, Y = 55.89 + 0.31X

b) The part of sales that is explained by the sales regression is calculated by $R^2$

$R^2 = \dfrac{[N (\sum XY) - (\sum X)(\sum Y)]}{\sqrt{ N Σ(Χ) - ΣΧ)][ΝΣΥ) - ΣΥ)}}$

Putting the values we calculate R2

$R^2 = \dfrac{(11 * 1296836 - 11*519.18 * 11* 217.82) }{ [(11*3134543 - 11*519.18*11*519.18) (11*539512 - 11*217.82*11*217.82)]0.5}$

On solving $R^2 = \dfrac{52870.3364 }{ 169516.4036 }* 17610.9236$

$R^2 = 0.96764$

So, the part of sales that is not explained by price is $1- R^2 = 1- 0.96764 = 0.03235$

Hence. 3.24% of sales is not explained by the price

c) Price elasticity against sales

$= (\dfrac{dY}{dX}) * (\dfrac{X}{Y})$

$= b * (\dfrac{X}{Y})$

$= \dfrac{0.31 X}{ Y}$

6.Data is not given in this question.