Question

Macroeconomics is the study of the economy as a whole. A macroeconomic variable is one that measures a characteristic of the whole economy or one of its large-scale sectors. In forecasting the sales of a product, market researchers frequently use macroeconomic variables in addition to marketing mix variables (marketing mix variables include product, price, place [or distribution] and promotion.)

A market researcher is analyzing an existing multiple regression model that predicts sales for different brands of digital cameras. The dependent variable is

y = monthly sales of specified digital camera (in thousands of dollars)

The independent variables are the following marketing mix variables:

x1 = ratings given by a popular digital photography magazine

x2 = average sale price (in dollars)

x3 = advertising spending for the given month (in thousands of dollars)

The estimated multiple regression equation using data with 33 observations is as follows:

ŷ = 2,350 + 204x1 - 194x2 + 373x3

The regression just given yields a multiple coefficient of determination of R2 = 0.49 and an adjusted multiple coefficient of determination of R2a = 0.44. The multiple coefficient of determination indicates the proportion of variability in the dependent variable that can be explained by the regression model.

The researcher would like to improve upon this model by including a macroeconomic variable that may affect sales. He decides to include the following variable:

x4 = most recent quarterly GDP growth rate

The estimated multiple regression equation with the additional independent variable is as follows:

ŷ = 2,020 + 268x1 - 139x2 + 301x3 + 54x4

The ANOVA table for the new regression model is shown as follows:

Analysis of Variance
Source of Variation Sum of Squares Degrees of Freedom Mean Square* F-value P-value
Regression 63,209 4 15,802 7.63 0.0003
Error 58,013 28 2,072
Total 121,222 32 3,788
*Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,788 is the total sum of squares divided by its degrees of freedom, or 121,222/32.

The multiple coefficient of determination, denoted R2, is the ratio of the _____________________________ A. sum of squares due to error B. total sum of squares C. sum of squares due to regression to the ____________________________ A. sum of squares due to error B. total sum of squares C. sum of squares due to regression . The R2 for the new regression is _______________ A. 0.52 B. 0.51 C. 0.48 D. 0.49 , indicating that the new estimated multiple regression equation explains __________________ A. 51% B. 52% C. 49% D. 48% of the variability of digital camera sales.

The sum of squares due to error divided by the total sum of squares is ________________ A. 0.45 B. 0.48 C. 0.52 D. 0.49 , and 1 minus this ratio is _______________ A. 0.48 B. 0.52 C. 0.55

The adjusted multiple coefficient of determination, denoted by R2a, for the new regression is _____________________ A. 0.55 B. 0.44 C. 0.45 D. 0.56

The mean square due to error divided by the total mean square is _________________ A. 0.55 B. 0.49 C. 0.45 D. 0.52, and 1 minus this ratio is ___________________ A. 0.52 B. 0.55 C. 0.45 D. 0.48

In general, adding independent variables to a multiple regression model reduces the ___________________________ A. total sum of squares B. sum of squares due to regression C. sum of squares due to error . The multiple coefficient of determination _____________________ A. could either increase or decrease B. decreses C. increases , and the adjusted multiple coefficient of determination ______________________ A. decreases B. increases C. could either increase or decrease.

Adding the independent variable x4 to the multiple regression model _________________ A. decreases B. increases the multiple coefficient of determination and _____________________ A. increases B. decreases the adjusted multiple coefficient of determination.

Answer 1

Check the explanation

Explanation:

The multiple coefficient of determination, denoted R2, is the ratio of the sum of squares due to regression to the total sum of squares.

The R2 for the new regression is 63209/121222=0.52 (A), indicating that the new estimated multiple regression equation explains 52% (B) of the variability of digital camera sales.

The sum of squares due to error divided by the total sum of squares is 58013/121222=0.4785=0.48 (B), and 1 minus this ratio is 1-0.48=0.52 (B).

The adjusted multiple coefficient of determination, denoted by R2a, for the new regression is 1-[(1-r^2)(n-1/n-k-1)]=0.45 (C).

The mean square due to error divided by the total mean square is 2072/3788=0.5469=0.55 (A) , and 1 minus this ratio is 1-0.55=0.45 (C).

In general, adding independent variables to a multiple regression model reduces the sum of squares due to error (C). The multiple coefficient of determination increases (C), and the adjusted multiple coefficient of determination could either increase or decrease (C).

Adding the independent variable x4 to the multiple regression model increases (B) the multiple coefficient of determination and increases (A) the adjusted multiple coefficient of determination.