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?
To interpret the results of the estimated model for demand of grape juice, let's analyze the coefficients and their respective standard errors:
i) Interpretation of the coefficients:
- Intercept (150.83): This represents the estimated quantity demanded of grape juice when all other variables (price of grape juice, price of orange juice, and income) are held constant. In this case, if all other variables are zero, the estimated quantity demanded would be 150.83 bottles of grape juice.
- Pg (-0.75): This coefficient indicates the relationship between the price of grape juice and the quantity demanded, assuming all other variables remain constant. A negative coefficient suggests an inverse relationship, meaning that as the price of grape juice increases by one unit, the quantity demanded decreases by 0.75 units, and vice versa.
- Po (0.37): This coefficient represents the relationship between the price of orange juice and the quantity demanded of grape juice, assuming all other variables remain constant. A positive coefficient suggests a positive relationship, meaning that as the price of orange juice increases by one unit, the quantity demanded of grape juice increases by 0.37 units, and vice versa.
- Y (0.65): This coefficient indicates the relationship between income and the quantity demanded of grape juice, assuming all other variables remain constant. A positive coefficient suggests a positive relationship, meaning that as income increases by one unit, the quantity demanded of grape juice increases by 0.65 units, and vice versa.
ii) Test for statistically significant coefficients:
To determine which partial slope coefficients are statistically different from zero, we need to conduct a hypothesis test. The standard errors provided in parentheses are crucial for this analysis.
The most common hypothesis test is to use a t-test, where we compare the estimated coefficient to zero. If the estimated coefficient is significantly different from zero, it suggests that the variable has a statistically significant impact on the quantity demanded of grape juice.
In this model, we have four coefficients: Pg, Po, Y, and the intercept. To determine which coefficients are statistically different from zero, we compare the absolute value of the coefficient divided by its standard error (i.e., t-statistic) to a critical value from the t-distribution table (e.g., at a specific significance level, such as 0.05).
If the absolute value of the t-statistic is greater than the critical value, we can conclude that the coefficient is statistically different from zero at that significance level.
For example, if the t-statistic for Pg is greater than the critical value, we would say that the price of grape juice coefficient is statistically different from zero, indicating that it has a significant impact on the quantity demanded of grape juice. The same procedure should be applied to other coefficients to determine their statistical significance.