Question

Refer to this regression equation:

(standard errors in parentheses.)

    Q = 8,400 - 10 P + 5 A + 4 Px + 0.05 I

(1,732) (2.29) (1.36) (1.75) (0.15)

• R2 = 0.65
• N = 120
• F = 35.25
• Standard error of estimate = 34.3
• Q = Quantity demanded
• P = Price = 1,000
• A = Advertising expenditures, in thousands = 40
• PX = price of competitor's good = 800
• I = average monthly income = 4,000

1) Calculate the elasticity for each variable and briefly comment on what information this gives you in each case.

2) Calculate t-statistics for each variable and explain what this tells you.

3) What information does the R2 give you?

4) How would you evaluate the quality of this equation overall? Do you have any concerns? Explain.

5) Should this firm be concerned if macroeconomic forecasters predict a recession? Explain.

Answer 1

1) The regression equation is provided in the problem as:

Q = 8,400 - 10 P + 5 A + 4 Px + 0.05 I

with P = 1,000
A = 40
Px = 800
I = 4000

Substituting the values in the equation, we get
Q = 8,400 - 10 (1000) + 5 (40) + 4 (800) + 0.05 (4000)
Q = 12000 - 10000 = 2,000

a. Price elasticity of demand: Ep = (dQ/dP)*(P/Q)
Here, dQ/dP is the first degree differential of the regression equation with respect to the variable P and is given by its coefficient, that is (-10).

Here, Ep = -5 tells us that with 1% change in the price of the good will result in a decrease in the quantity by 5%.

b. Price elasticity of demand of Competitor: Epx = (dQ/dPx)*(Px/Q)

Here, dQ/dPx is the first degree differential of the regression equation with respect to the variable Px and is given by its coefficient, that is (4).

Here, Epx = 1.6 tells us that with 1% change in the competitor's price, will result in an increase in the quantity of the firm by 1.6%, which is a greater ratio.

c. Income elasticity of demand: E_Y = (dQ/dI)*(I/Q)

Here, dQ/dI is the first degree differential of the regression equation with respect to the variable I and is given by its coefficient, that is (0.05).

Here, Ey = 0.1 tells us that with 1% change in the income of the consumer will result in an increase in the quantity demanded by 0.1%, which is not much ratio. this also implies that the good sold by the firm is a normal good.

d. Advertising elasticity of demand: Ead = (dQ/dA)*(A/Q)

Here, dQ/dA is the first degree differential of the regression equation with respect to the variable A and is given by its coefficient, that is (5).

Here, Ead = 0.1 tells us that with 1% change in the expenditure on advertisement will result in an increase in the quantity demanded by 0.1%, which shows that the ratio is good.

2) The t-statistic for a variable in a regression equation is calculated as the regression coefficient divided by its standard error. The t-statistic for each variable in the provided problem is calculated as:

For Price P, t = 10 / 2.29 = 4.37
For Advertising A, t = 1500 / 525 = 2.86
For Competitor's price P_x, t = 4/1.75 = 2.29
For income I, t = 2/1.5 = 1.33

Here, it can be seen that the independent variables Price, Advertising and Competitor's Price are statistically significant while income is not. This implies that the quantity demanded of the good is impacted by all variables except income of the consumer.

3) R² (R-squared) or coefficient of determination tells the difference between the data and fitted regression line. It shows the percentage of how well the dependent variable is explained by the multi variable linear regression model. It is calculated as:

R-squared = Explained variation / Total variation

Given that R-squared is 0.65, it shows that 65% of the variation in the dependent variable (quantity demanded) is explained by the variation in the independent variables (price of the good, price of the competitor, advertising and income) of the linear model.

4) The F-value of the provided problem is 35.25, which shows that the regression equation is significant. Also, the sample size N is very large. With R-squared as 65%, which shows that the variation in the dependent variable is well-defined by the variations in the independent variables. The presence of one insignificant variable is there, however, it is not affecting the overall OLS model much. This shows that there are no major concerns with the regression model.