You have estimated the following model for demand of grape juice using one
hundred observations.
Qg = 153.81 – 0.75Pg + 0.37Po + 0.65Y.
(0.04) (0.23) (0.02) (0.52) R-square = 0.87
Where
Qg = quantity demanded of grape juice bottles.
Pg = price of grape juice per unit.
Po = Price of orange juice.
Y = Income.
Standard errors are in parenthesis.
i) Interpret the results
ii) Which partial slope coefficients are statistically different from zero? Which test do you
use and why?
iii) Is the income coefficient equal to 1?
iv) Calculate cross-price elasticity when the mean values of price and quantity
demanded are respectively, 25 and 68.
v) Test the significance of the estimated R-square.
vi) Calculate the value of adjusted R-square.
i) Based on the estimated model, the intercept coefficient (153.81) suggests that when all other variables are held constant, the predicted quantity demanded of grape juice bottles is 153.81.
The coefficients for Pg, Po, and Y indicate the impact of these variables on the quantity demanded of grape juice bottles. Specifically, a decrease in the price of grape juice (Pg) by 1 unit is associated with a decrease in the quantity demanded by 0.75 units. Similarly, an increase in the price of orange juice (Po) by 1 unit is associated with an increase in the quantity demanded by 0.37 units. Lastly, an increase in income (Y) by 1 unit is associated with an increase in the quantity demanded by 0.65 units.
ii) To determine which partial slope coefficients are statistically different from zero, we can use a t-test. By comparing the magnitude of the coefficients to their respective standard errors, we can assess if they are significantly different from zero. If the absolute value of the coefficient divided by the standard error is greater than the critical value (e.g., 1.96 for a 95% confidence level), then the coefficient is statistically different from zero.
iii) To determine if the income coefficient is equal to 1, we can perform a hypothesis test. We would set up the null hypothesis as the income coefficient being equal to 1 (H0: βY = 1) and use a t-test to assess if the coefficient is significantly different from 1.
iv) The cross-price elasticity can be calculated by taking the derivative of the quantity demanded with respect to the price of orange juice (Po) and multiplying it by the mean value of Po divided by the mean value of Qg. This will give us the percentage change in the quantity demanded of grape juice bottles when the price of orange juice changes by 1 unit.
v) To test the significance of the estimated R-square, we can use an F-test. This test compares the explained variance of the model to the unexplained variance. If the F-statistic is greater than the critical value, we can conclude that the estimated R-square is statistically significant.
vi) The adjusted R-square can be calculated using 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. It provides an adjusted measure of how well the model fits the data, accounting for the number of variables included.
i) Interpretation of the results:
- The intercept term, 153.81, indicates the demand for grape juice bottles when all other variables are equal to zero. In this case, it suggests that even with no price, no orange juice price, and no income, there would still be a demand for grape juice bottles.
- The coefficient for Pg (-0.75) indicates that a one-unit increase in the price of grape juice per unit leads to a decrease in the quantity demanded of grape juice bottles by 0.75 units, holding all other variables constant.
- The coefficient for Po (0.37) suggests that a one-unit increase in the price of orange juice leads to an increase in the quantity demanded of grape juice bottles by 0.37 units, assuming all other variables remain constant.
- The coefficient for Y (0.65) indicates that a one-unit increase in income leads to an increase in the quantity demanded of grape juice bottles by 0.65 units, with all other variables held constant.
- The R-square value of 0.87 indicates that 87% of the variation in the quantity demanded of grape juice bottles can be explained by the independent variables in the model. This suggests a good fit of the model to the data.
ii) Testing the statistically significant coefficients:
To determine which partial slope coefficients are statistically different from zero, we can consult the standard errors provided in parenthesis.
In general, we can use the t-test to test the null hypothesis that the coefficient is equal to zero. If the absolute value of the coefficient divided by the standard error is greater than the critical value from the t-distribution, we can conclude that the coefficient is statistically different from zero at a certain level of significance (typically 5%).
iii) Testing the income coefficient:
To determine if the income coefficient is equal to one, we can conduct a t-test with the null hypothesis that the coefficient is equal to one. If the absolute value of the coefficient divided by the standard error is greater than the critical value from the t-distribution, we can reject the null hypothesis and conclude that the income coefficient is statistically different from one.
iv) Calculating cross-price elasticity:
Cross-price elasticity measures the responsiveness of the quantity demanded of one good to the change in the price of another good. To calculate the cross-price elasticity in this case, we can use the formula:
Cross-price elasticity = (Percentage change in quantity demanded / Percentage change in price of orange juice)
First, we need to calculate the percentage change in quantity demanded and the percentage change in the price of orange juice based on the mean values given (Pg = 25, Qg = 68). Then, we can substitute these values into the formula to find the cross-price elasticity.
v) Testing the significance of the estimated R-square:
To test the significance of the estimated R-square, we can use an F-test. The null hypothesis states that all coefficients (except the intercept) in the model are equal to zero, indicating no relationship between the independent variables and the dependent variable. If the calculated F-statistic is larger than the critical value from the F-distribution, we can reject the null hypothesis and conclude that the estimated R-square is statistically significant.
vi) Calculating the value of adjusted R-square:
To calculate the adjusted R-square, we can 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 this case, the adjusted R-square value will provide a measure of the goodness of fit while penalizing for the number of variables in the model.
i) Interpretation of the results:
- The constant term (153.81) represents the estimated quantity demanded of grape juice when all the independent variables are zero.
- The coefficient of Pg (-0.75) indicates that a one-unit increase in the price of grape juice is associated with a decrease of 0.75 units in the quantity demanded, assuming other variables remain constant.
- The coefficient of Po (0.37) suggests that a one-unit increase in the price of orange juice is associated with an increase of 0.37 units in the quantity demanded of grape juice, assuming other variables remain constant.
- The coefficient of Y (0.65) indicates that a one-unit increase in income is associated with an increase of 0.65 units in the quantity demanded of grape juice, assuming other variables remain constant.
- The R-square value of 0.87 represents the percentage of the variation in the quantity demanded explained by the independent variables.
ii) Statistically significant coefficients:
To determine which partial slope coefficients are statistically different from zero, the standard errors given in parenthesis are used. If the absolute value of a coefficient is greater than twice its standard error, it can be considered statistically different from zero. Therefore, in this case:
- The coefficient of Pg (-0.75) is statistically different from zero because |(-0.75)| > 2*(0.23).
- The coefficient of Po (0.37) is statistically different from zero because |0.37| > 2*(0.02).
- The coefficient of Y (0.65) is statistically different from zero because |0.65| > 2*(0.52).
iii) Testing the income coefficient:
To determine if the income coefficient is equal to 1, we can conduct a hypothesis test. We compare the estimated coefficient (0.65) to the hypothesized value of 1. If the estimated coefficient is statistically different from 1, then we can conclude that the income coefficient is not equal to 1.
iv) Calculation of cross-price elasticity:
Cross-price elasticity measures the responsiveness of the quantity demanded of one good (grape juice in this case) to a change in the price of another good (orange juice in this case). The formula is:
Cross-price elasticity = (Partial slope coefficient of Po / Pg) * (Mean value of Pg / Mean value of Qg)
Using the given values, we have:
Cross-price elasticity = (0.37 / -0.75) * (25 / 68)
v) Significance of the estimated R-square:
To test the significance of the estimated R-square, we can perform an F-test. The null hypothesis is that all the coefficients (excluding the constant term) are equal to zero. If the F-statistic is statistically significant, then we can reject the null hypothesis and conclude that the estimated R-square is significant.
vi) Calculation of adjusted R-square:
The adjusted R-square takes into account the number of independent variables in the model. It is calculated using 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 (excluding the constant term).