Exam 15: Multiple Regression

A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies. She proceeds to randomly select 26 large corporations and record Information in millions of dollars. A statistical analyst discovers that capital spending by Corporations has a significant inverse relationship with wage spending. What should the Microeconomist who developed this multiple regression model be particularly concerned with?

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the model that includes $X _ { 1 } , X _ { 3 } , X _ { 5 } \text { and } X _ { 6 }$ should be among the appropriate models using the Mallow's $C _ { p }$ statistic.

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, what is the value of the variance inflationary factor of Head?

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the variable X5 should be dropped to remove collinearity?

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, the value of the t test statistic for testing whether the quadratic term for the number of laborers is statistically significant after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers is _____.

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the model that includes $X _ { 1 } , X _ { 5 } \text { and } X _ { 6 }$ should be among the appropriate models using the Mallow's $C _ { p }$ statistic.

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, the model that includes all three independent variables should be selected using the adjusted $r ^ { 2 }$ statistic.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, you should keep the quadratic term for land size at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers.

The parameter estimates are biased when collinearity is present in a multiple regression equation.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, what is your decision on testing whether the quadratic term for the number of laborers is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers?

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the model that includes $X _ { 1 } , X _ { 5 } \text { and } X _ { 6 }$ should be selected using the adjusted $r ^ { 2 }$ statistic.

SCENARIO 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use a t test to determine if the linear term is significant. The p-value of the test is ______.

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the variable X6 should be dropped to remove collinearity?

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, what is your decision on testing whether the quadratic term for the number of laborers is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers?

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, there is reason to suspect collinearity between some pairs of predictors based on the values of the variance inflationary factor.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, what is your decision on testing whether the quadratic term for land size is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers?

SCENARIO 15-7-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataB.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-B, the model that includes only $X _ { 1 } \text { and } X _ { 3 }$ should be selected using the adjusted $r ^ { 2 }$ statistic.

The stepwise regression approach takes into consideration all possible models.

SCENARIO 15-1 A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status owners believe they gain in obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model: $Y = \beta _ { 0 } + \beta _ { 1 } X + \beta _ { 2 } X ^ { 2 } + \varepsilon$ where Y = demand (in thousands) and X = retail price per carat. This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below: -Referring to Scenario 15-1, what is the value of the test statistic for testing whether there is an upward curvature in the response curve relating the demand (Y) and the price (X)?

SCENARIO 15-1 A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status owners believe they gain in obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model: $Y = \beta _ { 0 } + \beta _ { 1 } X + \beta _ { 2 } X ^ { 2 } + \varepsilon$ where Y = demand (in thousands) and X = retail price per carat. This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below: -Referring to Scenario 15-1, a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).