You have estimated the following model for demand of grape juice using one
hundred observations.
Qg= 150.83–0.75Pg+ 0.37Po+0.65Y.
(0.04) (0.23) (0.02) (0.52)R-square=0.78
Where, Qg=quantitydemandedofgrapejuicebottles.
Pg=price ofgrapejuiceperunit.
Po=Priceoforangejuice.
Y=Income.
Standarderrorsareinparenthesis.
i)Interpret the results.
ii)Which partial slope coefficients are statistically different fromzero? Which test do
you use and why?
iii)Is the income coefficient equal to 1?
iv)Do income and price of orange juice have any significant effect on the demand
for grape juice?
v)Calculate cross-price elasticity when the mean values of price and quantity
demanded are respectively, 25 and 68.
vi)Test the significance of the estimated R-square.
vii)Calculate thevalueofadjustedR-square.
i) Interpret the results:
The estimated model for demand of grape juice is given by Qg = 150.83 - 0.75Pg + 0.37Po + 0.65Y. This means that the quantity demanded of grape juice bottles (Qg) is affected by the price of grape juice (Pg), the price of orange juice (Po), and income (Y).
ii) To determine which partial slope coefficients are statistically different from zero, you need to perform hypothesis tests. A common test is to use t-tests for each coefficient. The test compares the estimated coefficient to the standard error and checks if it is significantly different from zero. If the absolute value of the t-statistic is greater than the critical value (usually determined at a desired significance level, such as 0.05 or 0.01), then the coefficient is statistically different from zero.
iii) To test if the income coefficient is equal to 1, you would formulate the null hypothesis as "The income coefficient is equal to 1" and use a t-test to test this hypothesis. If the null hypothesis is rejected, it implies that the income coefficient is significantly different from 1.
iv) To determine if income and the price of orange juice have a significant effect on the demand for grape juice, you would examine the coefficients of these variables. If their coefficients are statistically different from zero (as determined by the t-tests mentioned in (ii)), then they have a significant effect on the demand for grape juice.
v) To calculate the cross-price elasticity when the mean values of price and quantity demanded are respectively 25 and 68, you would use the formula for cross-price elasticity:
Cross-price elasticity = (Change in Quantity Demanded / Mean Quantity Demanded) / (Change in Price / Mean Price)
So, for example, if the price increases from 25 to 30 and the quantity demanded changes from 68 to some other value, you can plug in the values into the formula to calculate the cross-price elasticity.
vi) To test the significance of the estimated R-square, you would use a hypothesis test. The null hypothesis would be that R-square is equal to zero (no linear relationship between the independent variables and the dependent variable). If the test rejects the null hypothesis, it implies that the R-square value is significantly different from zero, indicating a significant linear relationship between the variables.
vii) To calculate the adjusted R-square value, you would use the formula:
Adjusted R-square = 1 - [(1 - R-square) * ((N - 1) / (N - k - 1))]
where N is the number of observations and k is the number of independent variables in the model.