Question

10 product. 2013 Koong Manufacturing has collected the following data on quantity sold and price. Period 1 2 3 4 5 6 7 8 9 Quantity. 280 330 240 330 360 260, 200 300 380 320 Price s 19 8 14 6 8 14 18 8 4 10 (a) Use the least squares regression technique to estimate the linear relationship between quantity and price. (b) Evaluate the strength of the relationship between quantity and price by computing the t- statistics and R'. (c) Koong plans to increase its price to $20. lise the estimated regression equation to predict the quantity that will be sold. Do you have any concerns about this prediction line

answer this question simple method and step by step

Answer 1

a)

We have the data on the quantity sold and the price in 10 different periods.

As we are asked to construct a linear relationship between quantity sold and price, we can write them in a generic fashion as follows:

$Q=\alpha + \beta P+u$

• where Q is the quantity sold (this is the dependent variable, depending on the price of the good. If the price is low, people will find the good to be cheap and hence the quantity sold will be high. On the other hand, if the price is high, people will find the good to be expensive and hence the only a few people will buy that. So, the quantity sold in that case would be low)
• $\alpha$ is the intercept term (this gives us the value of Q when X=0)
• $\beta$ is the slope coefficient (this gives us for a unit change in price, what would be the corresponding change in the quantity sold)
• P is the price charged of the good (remember that this is an independent variable. The price that is charged doesn't depend on any other factor. The firm is free to charge whatever price it feels like.)
• u refers to the unobserved and unaccounted factors other than P, that affects Q (this can include taste and preferences of people, income of people and so on) . The mean of u terms is taken to be zero.

Now, the idea of linear regression is to estimate the value of the dependent variable when we are given definite values of the independent variable. But before we can do that, we have to estimate the values of $\alpha$ and $\beta$, to derive their estimated counterparts, which we write as $\widehat{\alpha}$ and $\widehat{\beta}$.

Now these values of  $\widehat{\alpha}$ and $\widehat{\beta}$ would be then used in the estimated equation to derive the estimated relationship between Q and P, written as (in generic form) :

$\widehat{Q}=\widehat{\alpha}+\widehat{\beta}P$

Note that the $\widehat{Q}$ refers to the estimated value of the quantity sold, as we plug in different values of P. We will derive the values of $\widehat{\alpha}$ and $\widehat{\beta}$ from the information / data that we have gotten.

Now, the idea of least squares method which is applied in this regression tries to minimise the sum of squares of the estimated error    $\sum_{1}^{10}\widehat {u}^2$  

=

$\sum_{1}^{10}(Q-\widehat{Q})^2$

=

$\sum_{1}^{10}(Q-\widehat{\alpha}-\widehat{\beta}P)^2$

Taking partial derivatives of this above quantity with respect to $\widehat{\alpha}$ and $\widehat{\beta}$ , setting them 0 and solving for the values of $\widehat{\alpha}$ and $\widehat{\beta}$, yields the following OLS estimators :

$\widehat{\beta}=\frac{\sum_{i=1}^{10}(P_{i}-\overline{P})(Q_{i}-\overline{Q})}{\sum_{i=1}^{10}(P_{i}-\overline{P})^2}$

and,

$\widehat{\alpha}=\overline{Q}-\widehat{\beta}\overline{P}$

In order to obtain the values of $\widehat{\alpha}$ and $\widehat{\beta}$ , we have to use the data from the problem:

Qi	Pi	$(P_{i}-\overline{P})(Q_{i}-\overline{Q})$	$(P_{i}-\overline{P})^2$	$(Q_{i}-\overline{Q})^2$
280	10	(10-10)(280-300)=0	0	400
330	8	(8-10)(330-300)=-60	4	900
240	14	(14-10)(240-300)=-240	16	3600
330	6	(6-10)(330-300)=-120	16	900
360	8	(8-10)(360-300)=-120	4	3600
260	14	(14-10)(260-300)=-160	16	1600
200	18	(18-10)(200-300)=-800	64	10000
300	8	(8-10)(300-300)=0	4	0
380	4	(4-10)(380-300)=-480	36	6400
320	10	(10-10)(320-300) =0	0	400

Now,

$\overline{P}$ is gotten as

100/10 = 10

$\overline{Q}$ is gotten as

3000/10=300

Now,

$\sum_{i=1}^{10}(P_{i}-\overline{P})(Q_{i}-\overline{Q})$ = -1980

$\sum_{i=1}^{10}(P_{i}-\overline{P})^2$= 160

hence,

$\widehat{\beta}=\frac{\sum_{i=1}^{10}(P_{i}-\overline{P})(Q_{i}-\overline{Q})}{\sum_{i=1}^{10}(P_{i}-\overline{P})^2}$

= (-1980)/160 = -12.375

And,

$\widehat{\alpha}=\overline{Q}-\widehat{\beta}\overline{P}$

= 300-(-12.375)(10) = 423.75

Hence, the estimated model is given as:

$\widehat{Q}=423.75-12.375P$

This is the estimated linear relationship between Q and P.

b)

In order to derive the R² (which is coefficient of determination), we can simply find the correlation coefficient and find it's square term.

Correlation coefficient is given as :

$\frac{\sum_{i=1}^{10}(P_{i}-\overline{P})(Q_{i}-\overline{Q})}{\sqrt{\sum_{i=1}^{10}(P_{i}-\overline{P})^2}\sqrt{\sum_{i=1}^{10}(Q_{i}-\overline{Q})^2}}$

$\sum (Q_{i}-\overline{Q})^2$ = 27800 is gotten from the above table

So ,

$r=\frac{-1980}{\sqrt{27800*160}}$

or,

$r=\frac{-1980}{2109.028}$

or,

$r=-0.9388$

Thus,

$R^{2}=(-0.9388)^2$

$=0.8813$

Thus, R² is computed as 0.8813

To calculate the t statistic, we have to calculate the test statistic :

[$\widehat{\beta}$- E($\widehat{\beta}$) ] / standard error ($\widehat{\beta}$)

or,

[$\widehat{\beta}$- $\beta$ ] / standard error ($\widehat{\beta}$) ,

and under the null hypothesis, H₀ : $\beta$ = 0 ,

we can rewrite the test statistic as :

$\widehat{\beta}$/ standard error ($\widehat{\beta}$)

So, we have to find var ($\widehat{\beta}$) and find it's square root to get the denominator :

var ($\widehat{\beta}$) =

$\frac{\sum_{i=1}^{10}(P_{i}-\overline{P})^2 }{10}$

= 160/10 = 16

Thus, standard error is square root of 16, i.e. 4.

Hence, the required t statistic is :

$\widehat{\beta}$/ standard error ($\widehat{\beta}$) = -12.375 /4 = -3.09375

Now, we have to compare this value of test statistic obtained from the t table against 95% level at 8 degrees of freedom (i.e. critical value)

If the mod value of  test statistic is greater than the critical value, then we reject the null hypothesis, else we fail to reject it.

The critical value is = 2.306

As the (mod value) of test statistic is greater than the critical value thus obtained, thus we reject the null hypothesis. Thus, we conclude that $\widehat{\beta}$ is significantly different from 0. Or, in other words, P has a significant role in defining Q.

c)

If P=30, put in the estimated equation to get the corresponding Q.

Thus,

$\widehat{Q}=423.75-12.375P$

= 423.75 - 12.375 (30)

= 52.5