Deck 13: Multiple Regression Analysis

Question 1

Three predictor variables are being considered for use in a linear regression model. Given the correlation matrix above, does it appear that multicollinearity could be a problem?
______________
Explain.
________________________________________________________

Answer 1

Yes; It appears that multicollinearity could be a problem because x[3] is highly correlated with both x[1] and x[2].

Question 2

When two or more of the predictor variables are highly correlated with one another, adding or deleting a predictor variable may cause significant changes in the values of the other regression coefficients.

Answer 2

True

Question 3

Multicollinearity is present if the dependent variable is linearly related to one of the explanatory variables.

Answer 3

False

Question 4

Qualitative predictor variables are entered into a regression model through dummy variables.

Question 5

One of the consequences of multicollinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

Question 6

If a multiple regression model includes 10 or more predictor variables, it is almost certain that changes in the predictor variables cause changes in the response variable y.

Question 7

When the independent variables are correlated with one another in a multiple regression analysis, this condition is called:

A) heteroscedasticity
B) homoscedasticity
C) multicollinearity
D) causality
E) collinearity

Question 8

The problem of multicollinearity arises when:

A) the dependent variables are highly correlated with one another
B) the independent variables are highly correlated with one another
C) the independent variables are highly correlated with the dependent variable
D) the independent variables are not correlated with each other
E) none of these

Question 9

Discuss some of the signals for the presence of multicollinearity.

Question 10

If multicollinearity exists among the independent variables included in a multiple regression model, then:

A) regression coefficients will be difficult to interpret
B) standard errors of the regression coefficients for the correlated independent variables will increase
C) multiple coefficient of determination will assume a value close to zero
D) regression coefficients will be difficult to interpret and standard errors of the regression coefficients for the correlated independent variables will increase
E) none of these

Question 11

Multicollinearity will result in excessively low standard errors of the parameter estimates reported in the regression output.

Question 12

When multicollinearity is present, the estimated regression coefficients will have large standard error, causing imprecision in confidence and prediction intervals.

Question 13

Which of the following statements regarding multicollinearity is not true?

A) It exists in virtually all multiple regression models.
B) It is also called collinearity and intercorrelation.
C) It is a condition that exists when the independent variables are highly correlated with the dependent variable.
D) It does not affect the F-test of the analysis of variance.
E) It exists in virtually all multiple regression models and it is also called collinearity and intercorrelation.

Question 14

Multicollinearity is a situation in which two or more of the independent variables are highly correlated with each other.

Question 15

In multiple regression analysis, which of the following is a clue that multicollinearity is present?

A) The value ofis large, indicting a good fit, but the individual t - tests are nonsignificant.
B) The signs of the regression coefficients are contrary to what we would intuitively expect the contributions of those variables would be.
C) A matrix of correlations, generated by computer, shows which predictor variables are highly correlated with each other and with the response variable y.
D) All of these.
E) None of these.

Question 16

In regression analysis, multicollinearity refers to:

A) the response variables being highly correlated with one another
B) the predictor variables being highly correlated with one another
C) the response variable and the predictor variables are highly correlated with one another
D) the response variables are highly correlated over time
E) the predictor variables are highly correlated over time

Question 17

What is the effect of multicollinearity on the estimated regression coefficients?

Question 18

Multicollinearity is present when there is a high degree of correlation between the dependent variable and all the independent variables included in the model.

Question 19

Discuss briefly what is meant by multicollinearity.

Question 20

Typical symptoms of the presence of multicollinearity include:

A) the estimated regression coefficients to vary substantially from sample to sample ; this fact raises their standard errors; hence, theis unlikely to be greater than 2, or statistically significant
B) the estimated regression coefficients change greatly in value as independent variables are dropped from or added to the regression equation
C) the signs of the estimated regression coefficients are nonsensical; they are negative when common sense suggests positive signs and vice versa
D) all of these and more
E) none of these

Question 21

The t-distribution with df = n - 2 is used for testing a specific set of regression coefficients,e.g.,
.

Question 22

In stepwise regression procedure, if two independent variables are highly correlated, then:

A) both variables will enter the equation
B) only one variable will enter the equation
C) neither variable will enter the equation
D) the regression is equal to zero
E) Not enough information is given to answer this question.

Question 23

Quantitative predictor variables are entered into a regression model through indicator variables.

Question 24

If a stepwise regression procedure is used to enter, one at a time, three variables into a regression model, the resulting regression equation may differ from the regression equation that occurs when all three variables are entered at one step.

Question 25

In order to incorporate the marital status variable into a multiple regression model, there are four possible categories for this variable: married, single, divorced, or widow. Based on this information, four indictor variables will need to be created (one for each category) and incorporated into the regression model.

Question 26

A dummy or indicator variable is a dependent variable whose values are either 0.0 or 1.0.

Question 27

Stepwise regression is a statistical technique that is always implemented when developing a regression model to fit a nonlinear relationship between the dependent and potential independent variables.

Question 28

Stepwise regression analysis is a procedure that is implemented by computer and is available in most statistical packages. It is mainly used to determine which of a large number of independent variables should be included in the model.

Question 29

In stepwise regression procedure, the independent variable with the largest F-statistic, or equally with the smallest p-value, is chosen as the first entering variable. The standard, also called the F-to-enter, is usually set at F equals:

A) 4
B) 2
C) 5
D) 0
E) 1

Question 30

Plots of the residuals against or against the individual independent variables often indicate departures from the assumptions required for an analysis of variance, and they also may suggest changes in the underlying model.

Question 31

Stepwise regression is an iterative procedure that:

A) adds one independent variable at a time
B) deletes one independent variable at a time
C) adds one dependent variable at a time
D) either adds one independent variable at a time or deletes one independent variable at a time
E) both adds one independent variable at a time and deletes one independent variable at a time

Question 32

Three qualitative variables need to be incorporated into a regression model. The first variable has 5 possible categories, the second one has 3 possible categories, and the third one has 2 possible categories. Based on this information, ten dummy variables need to be included in the regression model.

Question 33

In order to incorporate qualitative variables into a regression model, one or more dummy variables are needed.

Question 34

In constructing a multiple regression model with two independent variables and it was known that the correlation between and y is .75, and the correlation between and y is .55. Based on this information, the regression model containing both independent variable will explain 65% of the variation in the dependent variable y.

Question 35

Stepwise regression is especially useful when there are:

A) a great many independent variables
B) few independent variables
C) a great many dependent variables
D) few dependent variables
E) two independent variables

Question 36

The stepwise regression analysis is best used as a preliminary tool for identifying which of a large number of variables should be considered in the model.

Question 37

If you wish to develop a multiple regression model that includes a qualitative variable; education status, in which the following categories exist: no degree, high school diploma, junior college degree, bachelor degree, and graduate degree, you need to code the categories as 1, 2, 3, 4, and 5.

Question 38

Stepwise regression analysis is most useful when it is anticipated that there are curvilinear relationships between the dependent variable and the potential independent variables.

Question 39

In a multiple regression analysis, the regression equation is obtained. The variable is quantitative variable, and the variable is a dummy variable with values 0 and 1. Given this information, we can interpret the slope coefficient (-3) on variable as follows: Holding constant, if the value of is changed from 0 to 1, the average value of y will decrease by 3 units.

Question 40

If stepwise procedure is used, a variable selected at an earlier step can be removed from the model if, in the presence of other variables, it no longer contributes significantly to explaining the variation in the dependent variable y.

Question 41

In multiple regression, the prediction equation is the line that minimizes SSE, the sum of squares of the deviations of the observed values y from the predicted values .

Question 42

Which of the following is an advantage of using stepwise regression compared to just entering all the independent variables at one time?

A) There are no advantages of using stepwise regression over entering all the variables at one time.
B) Stepwise regression allows us to observe the effects of multicollinearity more early than when all variables are entered at one time.
C) Stepwise regression will generally produce a model with larger value for the coefficient of determination.
D) All of these.
E) None of these.

Question 43

If you wish to develop a regression model in which the High School Class standing is a qualitative variable with 4 possible levels of response, then you will need to include how many dummy variables?

A) 6
B) 5
C) 4
D) 3
E) none of these

Question 44

Having a large number of predictors in a regression model guarantees that the model fit is good.

Question 45

Which of the following statements is true?

A) It is appropriate to compute a correlation coefficient for the relationship between a dummy variable and a dependent variable.
B) If a qualitative variable has m categories, you should use m - 1 dummy variables to incorporate the qualitative variable into a regression model.
C) Dummy variables are used to incorporate qualitative variables into a regression model.
D) All of these.
E) None of these.

Question 46

Same statistical packages print a second statistic, called the adjusted coefficient of determination, which has been adjusted for the degrees of freedom to take into account the sample size and the number of predictor variables.

Question 47

Assume that a company is tracking their advertising expenditures as they relate to television ( ) and radio advertising ( ). The owner of the company believes that it would improve the regression model to add a third variable that represents the sum of the advertising on radio and television ( = + ). This assessment is generally correct.

Question 48

The coefficient of determination R² represents the proportion of the total variability in y that can be explained by the regression of y on x. When transformed to a percentage, it represents the percentage reduction in the sum of the squares of the error that can be accomplished by using the model to predict the dependent variable as opposed to just using the sample mean of the dependent variable.

Question 49

In testing the significance of a multiple regression model in which there are three independent variables, the null hypothesis is .

Question 50

Residuals are the deviations between the observed values of y and their predicted values .

Question 51

The more predictors that are added to a regression model, the larger the coefficient of determination R² value will be.

Question 52

In order to test the usefulness of a multiple regression model involving 5 predictor variables and 25 observations, the numerator and denominator degrees of freedom for the critical value of F are 4 and 24, respectively.

Question 53

Which of the following statements is correct?

A) The number of dummy variables that must be added to a regression model is one less than the number of categories for a qualitative independent variable.
B) A dummy variable is incorporated into a regression model if the dependent variable is qualitative.
C) Including a dummy variable into a regression model will simplify the regression results and help people to interpret the meaning of the regression parameters.
D) The number of dummy variables that must be added to a regression model is one more than the number of categories for a qualitative independent variable.
E) All of these.

Question 54

Suppose that one equation has 3 explanatory variables and an F-ratio of 52. Another equation has 5 explanatory variables and an F-ratio of 40. The first equation will always be considered a better model.

Question 55

In reference to the equation , the value 0.63 is the average change in y per unit change in , regardless of the value of .

Question 56

Let be the least squares estimate of the population coefficient . If the regression assumptions hold true, the test statistic given by has an F distribution with k and n-k-1 degrees of freedom, where n is the number of observations and k is the number of predictor variables.

Question 57

A multiple regression equation includes 5 predictor variables, and the coefficient of multiple determination is 0.7921. The percentage of the variation in y that is explained by the regression equation is 89%.

Question 58

Assume you are considering including two additional qualitative variables into a regression model. The first variable has 4 categories, and the second variable has 4 categories as well. Given this information, how many indicator variables will be incorporated into the model?

A) 8
B) 7
C) 6
D) 5
E) 4

Question 59

A multiple regression model has the form . The coefficient is interpreted as the change in y per unit change in .

Question 60

In multiple regression analysis, which procedure permits variables to enter and leave the model at different stages of its development?

A) forward selection
B) residual analysis
C) backward elimination
D) stepwise regression
E) chi-square test

Question 61

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is . For this model, SST = 800 and SSE = 245. Then, the value of the F statistic for testing the significance of the model is 15.102.

Question 62

If we want to relate a random variable y to two-independent variables and , a regression hyperplane is the three-dimensional equivalent of a regression line that minimizes the sum of the squared vertical deviations between the sample points suspended in y vs. vs. space and their associated multiple regression estimates, all of which lie on this hyperplane.

Question 63

A multiple regression model involves 40 observations and 4 independent variables produces SST = 2,000 and SSR = 1,608. The value of MSE is 11.2.

Question 64

In multiple regression, the descriptor "multiple" refers to more than one dependent variable.

Question 65

In regression analysis, a p-value provides the probability (judged by the t-value associated with an estimated regression coefficient) of being true, given the claim : The true regression coefficient equals 0.

Question 66

In a multiple regression problem, the regression equation is . The estimated value for y when and is 48.

Question 67

For each x term in the multiple regression equation, the corresponding is referred to as a partial regression coefficient.

Question 68

A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.81. Then, the percentage of the variation in y that is explained by the regression equation is 90%.

Question 69

Multiple regression is the process of using several independent variables to predict a number of dependent variables.

Question 70

A coefficient of multiple correlation is a measure of how well an estimated regression plane (or hyperplane) fits the sample data on which it is based.

Question 71

An estimated partial-regression coefficient gives the partial change in y for a unit change in that independent variable, while holding other independent variables constant.

Question 72

Multiple regression analysis is a type of regression analysis in which several independent variables are used to estimate the value of an unknown dependent variable; hence, each of these predictor variables explains part of the total variation of the dependent variable.

Question 73

In reference to the equation: , the value -0.80 is the y-intercept.

Question 74

In reference to the equation , the value -0.75 is the intercept.

Question 75

A coefficient of multiple correlation is denoted by and equals the proportion of the total variation in the values of the dependent variable, y that is explained by the estimated multiple regression of y on , , and possibly additional independent variable ( , and so on).

Question 76

In testing the significance of a multiple regression model in which there are three independent variables, the null hypothesis is .

Question 77

An estimated partial-regression coefficient is the coefficient of a dependent variable in an estimated multiple-regression equation.

Question 78

In multiple regression analysis, the adjusted multiple coefficient of determination is adjusted for the number of independent variables and the sample size.

Question 79

A multiple regression analysis includes 25 data points and 4 independent variables produces SST = 400 and SSR = 300. Then, the multiple standard error of estimate is 5.

Question 80

Multiple linear regression is an extension of simple linear regression to allow for more than one dependent variable.