Ecotrix Dummy

ECONOMETRICS:

MULTI VARIABLE REGRESSION ANALYSIS

Submitted by :

Shahbaz singh (521)

Aakash Choithani (552)

Charchit Bahl (563)

MASTER OF BUSINESS ECONOMICS

ACKNOWLEDGEMENTS

We wish to express our deep sense of gratitude in thanking Prof. S.C Aggarwal for his

able guidance, useful suggestions and for all his valuable assistance in this project work. We

are grateful to you Sir for your encouragement and cooperation in carrying out the project

work which helped us in completing this project in time.

We would also like to express our heartfelt thanks to our friends and classmates here at

MBE for their help, wishes and their pragmatic suggestions which played a pivotal part in

the successful completion of this project.


TABLE OF CONTENTS

Topic Page No.

1. Economic Theory 4

2. Description Of Data 5

3. Data Table 6

4. Procedure Adopted 7

5. Multivariate Model (without dummy variable) 8

6. Model With Dummy Variable 11

7. Test For Multicollinearity 15


ECONOMIC THEORY BEHIND THE DATA

In this Project, we have taken a family and have then analyzed their weekly consumption of

ice-cream for a period of 30 weeks. Certain factors that we think are really important in

shaping the demand of the ice-cream industry are explained below:

Price of the product

As we know, there exists an inverse relationship between price of the product and its

demand, unless it’s an inferior good. This was clearly observed during our study. As the price

of ice-cream varied, demand saw a movement in the opposite direction.

Income of the consumers (Dummy Variable)

Another important factor is income of the consumers. With a hike in their incomes,

consumers tend to consume more which reiterates the existence of a direct relationship

between the demand and the income.

Climatic conditions having an impact

Ice-cream is a seasonal product and obviously demand of the product increases during summers and shows a downward trend during the winter season.


DESCRIPTION OF DATA

In this particular data set, we have chosen ice-cream as our product and taken a family residing in

The United States. This data belongs to the year 1957 and is a weekly data and contains information

spread over a period of 30 weeks.

In this we have taken demand of ice-cream as the dependent variable and the various variables on

which it depends are Price, Income and climatic conditions that is the independent variables.

Both the dependent and the independent variables have been expressed in different units and they

are as follows:

Consumption- consumption of ice cream per head (in pints);

Price- price of ice cream (per pint) in US Dollars;

Income- average family income per week (in US Dollars);

Temperature- average temperature (in Fahrenheit);

Our source of information is:

Hildreth, C. and J. Lu (1960) Demand relations,Technical Bulletin No 2765, Michigan State University.


DATA TABLE

WeekConsumption per head (In Pints)

Price (Per Pint) (in US $)

Temperature

(In Fahrenhiet)Income (in US $)

1 0.386 0.27 41 less than 802 0.374 0.282 56 less than 803 0.393 0.277 63 more than 804 0.425 0.28 68 less than 805 0.406 0.272 69 less than 806 0.344 0.262 65 less than 807 0.327 0.275 61 less than 808 0.288 0.267 47 less than 809 0.269 0.265 32 less than 80

10 0.256 0.277 24 less than 8011 0.286 0.282 28 more than 8012 0.298 0.27 26 more than 8013 0.329 0.272 32 more than 8014 0.318 0.287 40 more than 8015 0.381 0.277 55 more than 8016 0.381 0.287 63 more than 8017 0.47 0.28 72 more than 8018 0.443 0.277 72 less than 8019 0.386 0.277 67 more than 8020 0.342 0.277 60 more than 8021 0.319 0.292 44 more than 8022 0.307 0.287 40 more than 8023 0.284 0.277 32 more than 8024 0.326 0.285 27 more than 8025 0.309 0.282 28 more than 8026 0.359 0.265 33 more than 8027 0.376 0.265 41 more than 8028 0.416 0.265 52 more than 8029 0.437 0.268 64 more than 8030 0.548 0.26 71 more than 80


PROCEDURE ADOPTED

1. At first we run the regression without the addition of income dummy. Thus we regress

the consumption (demand) on price & temperature

2. Interpretation of the coefficients. To interpret the result we take the partial slope

coefficients of X2 (Price) & X3 (Temperature).

3. Testing the level of significance. To test whether the partial slope coefficients and the

intercept are statistically significant we look at the t-statistic of each coefficient and will

test the null hypothesis that the coefficients are zero. We will follow the 2-t rule of

thumb as the DOF is greater than 20 and assuming the level of significance to be 5%.

4. Introducing the dummy variable. We introduce an income dummy in which we have

assigned the values 1 & 0 on the basis of the income of the household per week in USD

and further regress consumption on price, temperature & income (dummy) followed by

interpretation of coefficients where we will adopt the same procedure, the one which

we have used in the model without dummy .

5. After testing the level of significance as done earlier, we check the system for

multicollinearity using V.I.F. = 1/(1 – r²₂₃) ; Where r₂₃ is the coefficient of correlation

between X2 & X3.


MULTIVARIATE MODEL WITHOUT DUMMY VARIABLE

We will first run the regression without the addition of income dummy. So, we regress consumption

(demand) on price & temperature. After running the command “reg consumption price

temperature” in STATA we get the following regression equation:

Yi = 0.596 – 1.401X2 + .003X3

Se (0.258) (0.925) (0.001)

T 2.31 -1.52 6.45

R² = 0.6328

DOF = 27

F (2, 27) = 23.27

ESS = .0794

RSS = .0461

In the above model consumption is represented by Yi, whereas, X2 & X3 are price & Temperature

variables.


INTERPRETATION OF COEFFICIENTS

Now, to interpret the result we will look at the partial slope coefficients of X2 (Price) & X3

(Temperature).

So, from the equation it is clear that the price have a negative relation with the consumption

which is depicted by the negative sign. The coefficient of -1.401 shows that if the price of ice

cream increases by 1 USD per pint the consumption decreases by 1.401 pints keeping other

variables constant.

Similarly, for temperature if it rises by 1 degree Fahrenheit the consumption increases by

0.003 pints keeping other variables constant.

The intercept 0.596 shows the estimated value of Yi (consumption) if both the X variables

are equal to zero.

R² for the model is 0.6328 which shows that 63.28% variation in consumption is explained by

price & temperature.


TEST OF SIGNIFICANCE

Now, to test whether the partial slope coefficients and the intercept are statistically significant we

look at the t-statistic of each coefficient and will test the null hypothesis that the coefficients are

zero. We will follow the 2-t rule of thumb as the DOF is greater than 20 and assuming the level of

significance 5%.

So, it is clear from the equation that the price variable (X2) is not statistically significant as

the t-statistic is less than 2 and the p value is .141 which is greater than .05. But the

intercept and X3 variable comes out to be statistically significant.

We also obtain the F-statistic from the STATA for the model which is 23.27, which is highly

significant showing that the overall model is significant.


MODEL WITH THE DUMMY VARIABLE

We are introducing an income dummy in which we have assigned the values 1 & 0 on the basis of

the income of the household per week in USD. The dummy generated is as follows:

D = 1 if Income is Less than 80 USD

= 0 otherwise (If Income is greater than 80 USD)

For generating the dummy we will run the command “tab Income, gen(dum)”

Now, after adding the dummy variable in the previous model we regress consumption on price,

temperature & income (dummy). So, we run the command “reg Consumption Price Temperature

Income_Dummy” & obtained the following regression equation & results:


Yi = 0.7305 – 1.8756X2 + 0.0032X3 – 0.0400Di

Se (0.239) (0.854) (0.0004) (.015)

T 3.06 -2.19 7.49 -2.66

R² = 0.7114

DOF = 26

F (3, 26) = 21.36

ESS = .0893

RSS = .0362

INTERPRETATION OF COEFFICIENTS

In the above equation Yi represents Ice cream consumption, whereas, X2 & X3 represents price &

temperature variables respectively. Di represents the dummy variable (Income Dummy). We will

approach the interpretation part the same way we did in the model without dummy.

By looking at the partial slope coefficients X2 & X3 we can find out the relationship between

consumption & price as well as consumption & temperature.

The negative sign of the partial slope coefficient for price variable (X2) again shows that

there is a negative relationship between price & consumption (demand). The slope

coefficient is -1.8756 which shows that if the price of ice cream increases by 1 USD per pint

the consumption for ice cream decreases by 1.8756 pints keeping other variables constant.

For temperature (X3) the partial slope coefficient is 0.0032, which clearly shows that if the

temperature increases by 1 degree Fahrenheit the consumption of ice cream increases by

0.0032 pints.

The R² value of 0.7114 shows that 71.14% of variation in consumption is explained by

explainatory variables.


THE DUMMY VARIABLE

The dummy we have used is for the weekly income of the concerned family in USD, with dummy

taking the value 1 if the income is less than 80 USD & 0 if the income is more than 80 USD. Other

thing to be noted is that the dummy which we have used is a simple dummy (not interaction

dummy), so, the impact of dummy is directly on the intercept not on the Xs variables.

Now, from the coefficient value of -0.0400 for the Di it is clear that if the dummy will take the value 1

(i.e. the income is less than 80 USD) it will reduce the intercept by 0.0400, which shows that there is

a positive relation between income & consumption as the estimated consumption of ice cream (Yi) is

greater for the family when they have an income of more than 80 USD than for those weeks when

their income is less than 80 USD.


The regression equation when the income of the family is more than 80 USD:

Yi = 0.7305 – 1.8756X2 + 0.0032X3

The regression equation when the income of the family is less than 80 USD:

Yi = 0.6905 – 1.8756X2 + 0.0032X3

TEST OF SIGNIFICANCE

To test whether the partial slope coefficients and the intercept are statistically significant we look at

the t-statistic of each coefficient and will test the null hypothesis that the coefficients are zero. We

will follow the 2-t rule of thumb as the DOF is greater than 20 and assuming the level of significance

5%.

The t-statistic for each of the partial slope coefficient is more than 2. So, by the rule of

thumb & also by looking at their p-value we can say that at 5% level of significance the

variables are statistically significant.

The F statistic of 21.36 also shows that the overall model is highly significant with DOF 3 &

26.


TEST FOR MULTICOLLINEARITY

To test whether the X variables i.e. X2 & X3 are correlated to each other or not, we will check the VIF (Variation inflating factor). The formula for the same is as follows:

V.I.F. = 1/(1 – r²₂₃)

Where r₂₃ is the coefficient of correlation between X2 & X3.

If the VIF has a Value of more than 10 then there is a serious problem of multicollinearity. VIF equal to 1 signifies that there is no multicollinearity.

Now, to find out the correlation coefficient between the two variables we run the command “correlate Price Temperature”. The result of the command is:

So the coefficient of correlation is -0.1082. Now, we can find out the VIF:

VIF = 1/(1 - r²₂₃)

= 1/(1 – 0.012)

= 1/.988

= 1.012

With a V.I.F. as small as 1.012 we can conclude that our model does not suffer from the problem of multicollinearity.


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