Question

2. Multiple coefficient of determination Aa Aa E 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: ymonthly sales of specified digital camera (in thousands of dollars) The independent variables are the following marketing mix variables: x1ratings given by a popular digital photography magazine x2average sale price (in dollars) x3advertising spending for the given month (in thousands of dollars) The estimated multiple regression equation using data with 29 observations is as follows: 2,130 + 210x1-199x2 + 352x3 y The regression just given yields a multiple coefficient of determination of R2 0.43 and an adjusted multiple coefficient of determination of R2a 0.39. 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: 1,826 + 243X1-124X2 + 271X3 + 49X4 y The ANOVA table for the new regression model is shown as follows: Analysis of Variance Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 2,471 is the total sum of squares divided by its degrees of freedom, on 69,185/28 Sum of Squares 30,420 38,765 69,185 Degrees of Freedom 4 24 28 Mean Square* 7,605 1,615 2,471 F-value P-value 4.71 0.0060 The multiple coefficient of determination, denoted R2, is the ratio of the the estimated multiple regression equation explains The R2 for the new regression is , indicating that the new of the variability of digital camera sales. and 1 minus this ratio is The sum of squares due to error divided by the total sum of squares IS The adjusted multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is , and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the . The multiple coefficient of determination could either increase or decrease _, and the adjusted multiple coefficient of determination could either increase or decrease. Adding the independent variable x4 to the multiple regression model determination and the multiple coefficient of the adjusted multiple coefficient of determination decreases increases

Answer 1