Question 1

In calculating the standard error of the estimate,S_YX = $$\sqrt { \mathrm { MSE } }$$ there are n * k * 1 degrees of freedom,where n is the sample size and k represents the number of independent variables in the model.

Accepted Answer

This statement is true. The formula for calculating the standard error of the estimate (also known as the residual standard error) is based on the sum of squared errors (SS1) and the degrees of freedom, which is equal to n-k-1. Therefore, knowing the sample size (n) and the number of independent variables (k) is essential in calculating the standard error of the estimate.

Question 2

In a multiple regression model,which of the following is correct regarding the value of the adjusted r²? A) It has to be larger than the coefficient of multiple determination. B) It can be larger than 1. C) It can be negative. D) It has to be positive.

Accepted Answer

The adjusted R-squared can be negative if the chosen model fits worse than a horizontal line (or mean value). This occurs when the model's explanatory power is essentially non-existent, leading to worse predictions than simply using the mean of the observed outcomes.

Question 3

The variation attribuInstruction to factors other than the relationship between the independent variables and the explained variable in a regression analysis is represented by
A) regression mean squares.
B) total sum of squares.
C) regression sum of squares.
D) error sum of squares.

Accepted Answer

The error sum of squares in regression analysis represents the variation attributed to factors other than the relationship between the independent variables and the explained variable. It quantifies the discrepancy between the observed values and the values predicted by the regression model.

Question 4

From the coefficient of multiple determination,you cannot detect the strength of the relationship between Y and any individual independent variable.

Accepted Answer

The coefficient of multiple determination (R-squared) only indicates the overall strength of the relationship between the dependent variable and all the independent variables combined, but it doesn't tell us about the strength of the relationship of any individual independent variable with the dependent variable.

Question 5

Instruction 13.15 An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross domestic product ($ billions) and aggregate price (consumer price index). The Microsoft Excel output of this regression is partially reproduced below. OUTPUT SUMMARY Regression Statistics $\begin{array} { l l } \text { MultipleR } & 0.991 \ \text { R Square } & 0.982 \ \text { Adj. R Square } & 0.976 \ \text { Std. Error } & 0.299 \ \text { Observations } & 10 \end{array}$ ANOVA $\begin{array} { l l l l l l } & d f & \text { SS } & \text { MS } & F & \text { Signiff } \ \text { Regression } & 2 & 33.4163 & 16.7082 & 186.325 & 0.0001 \ \text { Residual } & 7 & 0.6277 & 0.0897 & & \ \text { Total } & 9 & 34.0440 & & & \ & & & & & \ & \text { Coeff } & \text { StdError } & t \text { Stat } & p \text { value } & \ \text { Intercept } & - 1.6335 & 0.5674 & - 0.152 & 0.8837 & \ \text { GDP } & 0.7654 & 0.0574 & 13.340 & 0.0001 & \ \text { Price } & - 0.0006 & 0.0028 & - 0.219 & 0.8330 & \end{array}$ Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.15,when the economist used a simple linear regression model with consumption as the dependent variable and GDP as the independent variable,he obtained an r² value of 0.971.What additional percentage of the total variation of consumption has been explained by including aggregate prices in the multiple regression? A) 98.2 B) 1.1 C) 11.1 D) 2.8

Accepted Answer

The answer of Instruction 13.15
An economist is interested to see...

Question 6

Instruction 13.15
An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross domestic product ($ billions) and aggregate price (consumer price index). The Microsoft Excel output of this regression is partially reproduced below.
 OUTPUT
SUMMARY
Regression Statistics
$\begin{array} { l l } \text { MultipleR } & 0.991 \ \text { R Square } & 0.982 \ \text { Adj. R Square } & 0.976 \ \text { Std. Error } & 0.299 \ \text { Observations } & 10 \end{array}$

ANOVA
$\begin{array} { l l l l l l } & d f & \text { SS } & \text { MS } & F & \text { Signiff } \ \text { Regression } & 2 & 33.4163 & 16.7082 & 186.325 & 0.0001 \ \text { Residual } & 7 & 0.6277 & 0.0897 & & \ \text { Total } & 9 & 34.0440 & & & \ & & & & & \ & \text { Coeff } & \text { StdError } & t \text { Stat } & p \text { value } & \ \text { Intercept } & - 1.6335 & 0.5674 & - 0.152 & 0.8837 & \ \text { GDP } & 0.7654 & 0.0574 & 13.340 & 0.0001 & \ \text { Price } & - 0.0006 & 0.0028 & - 0.219 & 0.8330 & \end{array}$ Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error
-Referring to Instruction 13.15,the p-value for GDP is
A) 0.01.
B) 0.05.
C) 0.001.
D) None of the above.

Accepted Answer

The p-value for GDP is not given in the information provided. Therefore, none of the above options are the correct choice.

Question 7

The total sum of squares (SST)in a regression model will never exceed the regression sum of squares (SSR).

Accepted Answer

The regression sum of squares (SSR) is the sum of the squared differences between the predicted values and the mean of the dependent variable. The total sum of squares (SST) is the sum of the squared differences between the actual values and the mean of the dependent variable. Therefore, the SST will always be greater than or equal to the SSR.

Question 8

Instruction 13.16
A real estate builder wishes to determine how house size (House) is influenced by family income (Income), family size (Size) and education of the head of household (School). House size is measured in hundreds of square metres, income is measured in thousands of dollars and education is in years. The builder randomly selected 50 families and ran the multiple regression. Microsoft Excel output is provided below:
 OUTPUT
SUMMARY
Regression Statistics
$\begin{array} { l l } \text { Multiple R } & 0.865 \ \text { R Square } & 0.748 \ \text { Adj. R Square } & 0.726 \ \text { Std. Error } & 5.195 \ \text { Observations } & 50 \end{array}$

ANOVA
$\begin{array} { l l l l l l } & d f & \text { SS } & \text { MS } & F & \text { Signiff } \ \text { Regression } & & 3605.7736 & 901.4434 & & 0.0001 \ \text { Residual } & & 1214.2264 & 26.9828 & & \ \text { Total } & 49 & 4820.0000 & & & \ & & & & & \ & \text { Coeff } & \text { StdError } & t \text { Stat } & p \text { value } & \ \text { Intercept } & - 1.6335 & 5.8078 & - 0.281 & 0.7798 & \ \text { Income } & 0.4485 & 0.1137 & 3.9545 & 0.0003 & \ \text { Size } & 4.2615 & 0.8062 & 5.286 & 0.0001 & \ \text { School } & - 0.6517 & 0.4319 & - 1.509 & 0.1383 & \end{array}$ Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error
-Referring to Instruction 13.16,what fraction of the variability in house size is explained by income,size of family and education?
A) 74.8%
B) 33.4%
C) 27.0%
D) 86.5%

Accepted Answer

The fraction of the variability in house size explained by income, size of family, and education is represented by the R Square value, which is 0.748 or 74.8%.

Question 9

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 90.0% of the variation in the dependent variable is explained by the independent variables in the regression.

Accepted Answer

The determination coefficient (R-squared) is calculated by dividing the explained variation (SST-SSE) by the total variation (SST). Therefore,

R-squared = (SST - SSE) / SST

= (102.55 - 82.04) / 102.55

= 0.199

This means that 19.9% of the variation in the dependent variable is explained by the independent variables in the regression. Therefore, it cannot be said that 90.0% of the variation is explained.

Question 10

When an explanatory variable is dropped from a multiple regression model,the coefficient of multiple determination can increase.

Accepted Answer

The coefficient of multiple determination, denoted as R^2, measures the proportion of the variance in the dependent variable that is predictable from the independent variables. Dropping an explanatory variable from a model typically reduces the amount of variance explained by the model, thus decreasing R^2, not increasing it.

Question 11

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 20.0% of the variation in the dependent variable is explained by the independent variables in the regression.

Accepted Answer

To calculate the percentage of variation explained by the independent variables we need to use the coefficient of determination (R-squared), which is equal to (SST-SSE)/SST. 
In this case, R-squared = (102.55-82.04)/102.55 = 0.2 or 20.0%, which means that 20.0% of the variation in the dependent variable is explained by the independent variables in the regression. Therefore, the answer is A.

Question 12

The coefficient of multiple determination is calculated by taking the ratio of the regression sum of squares over the total sum of squares (SSR/SST)and subtracting that value from 1.

Accepted Answer

The coefficient of multiple determination is calculated by taking the ratio of the regression sum of squares over the total sum of squares (SSR/SST), not by subtracting that value from 1. The formula described is for calculating the coefficient of determination (R^2) in simple linear regression, not the coefficient of multiple determination.

Question 13

A regression had the following results: SST = 82.55,SSE = 29.85.It can be said that 63.84% of the variation in the dependent variable is explained by the independent variables in the regression.

Accepted Answer

The proportion of the variation explained by the regression is equal to (SST-SSE)/SST. Plugging in the given values, we get (82.55-29.85)/82.55 = 0.6384, which is 63.84% as a percentage. Therefore, it is true that 63.84% of the variation in the dependent variable is explained by the independent variables in the regression.

Question 14

You have just computed a regression in which the value of coefficient of multiple determination is 0.57.To determine if this indicates that the independent variables explain a significant portion of the variation in the dependent variable,you would perform an F test.

Accepted Answer

The coefficient of multiple determination (R-squared) measures the proportion of the variation in the dependent variable that is explained by the independent variables in a regression model. However, it does not indicate whether the explained variation is statistically significant. To determine if the independent variables have a significant effect, an F test is typically used to compare the regression model to a null model (e.g., one that only includes an intercept term). Thus, the statement that an F test should be performed to determine if the independent variables explain a significant portion of the variation in the dependent variable is true.

Question 15

When an additional explanatory variable is introduced into a multiple regression model,the adjusted r² can never decrease.

Accepted Answer

The adjusted R-squared can decrease when an additional explanatory variable is introduced if the new variable does not improve the model's explanatory power enough to offset the penalty for adding another variable.

Question 16

The coefficient of multiple determination measures the proportion of the total variation in the dependent variable that is explained by a set of independent variables.

Accepted Answer

The coefficient of multiple determination (R-squared) is defined as the proportion of the variation in the dependent variable that is explained by the independent variables in a multiple regression model. Therefore, the statement is true.

Question 17

The coefficient of multiple determination r²_Y_.12 A) measures the proportion of variation in Y that is explained by X₁ and X₂. B) measures the proportion of variation in Y that is explained by X₁ holding X₂ constant. C) will have the same sign as b₁. D) measures the variation around the predicted regression equation.

Accepted Answer

The coefficient of multiple determination, denoted by r^2, measures the proportion of variation in the dependent variable (Y) that is explained by the independent variables (X₁ and X₂) included in the regression model. Therefore, choice A is correct as it accurately describes the meaning of r^2. On the other hand, choice B is not entirely accurate as r^2 measures the variation explained by all independent variables, not just one while holding the other constant. Choice C is incorrect as there is no direct relationship between the sign of b₁ and r^2. Finally, choice D is also incorrect as it refers to the residual variation that is not explained by the regression equation, rather than the proportion of variation explained by the independent variables.

Question 18

When an explanatory variable is dropped from a multiple regression model,the adjusted r²can increase.

Accepted Answer

The adjusted R-squared can increase when an explanatory variable is dropped from a multiple regression model because it adjusts for the number of variables in the model, penalizing models with too many variables. Removing a variable can sometimes improve the model fit and increase the adjusted R-squared. However, it is important to carefully consider the reasons for dropping a variable and whether it makes sense from a theoretical perspective.

Question 19

Instruction 13.15
An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross domestic product ($ billions) and aggregate price (consumer price index). The Microsoft Excel output of this regression is partially reproduced below.
 OUTPUT
SUMMARY
Regression Statistics
$\begin{array} { l l } \text { MultipleR } & 0.991 \ \text { R Square } & 0.982 \ \text { Adj. R Square } & 0.976 \ \text { Std. Error } & 0.299 \ \text { Observations } & 10 \end{array}$

ANOVA
$\begin{array} { l l l l l l } & d f & \text { SS } & \text { MS } & F & \text { Signiff } \ \text { Regression } & 2 & 33.4163 & 16.7082 & 186.325 & 0.0001 \ \text { Residual } & 7 & 0.6277 & 0.0897 & & \ \text { Total } & 9 & 34.0440 & & & \ & & & & & \ & \text { Coeff } & \text { StdError } & t \text { Stat } & p \text { value } & \ \text { Intercept } & - 1.6335 & 0.5674 & - 0.152 & 0.8837 & \ \text { GDP } & 0.7654 & 0.0574 & 13.340 & 0.0001 & \ \text { Price } & - 0.0006 & 0.0028 & - 0.219 & 0.8330 & \end{array}$ Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error
-Referring to Instruction 13.15,the p-value for the aggregated price index is
A) 0.001.
B) 0.05.
C) 0.01.
D) None of the above.

Accepted Answer

The p-value for the aggregated price index is 0.8330, which does not match any of the provided options (A, B, C).

Question 20

In a multiple regression model,the value of the coefficient of multiple determination
A) can fall between any pair of real numbers.
B) has to fall between -1 and +1.
C) has to fall between -1 and 0.
D) has to fall between 0 and +1.

Accepted Answer

The coefficient of multiple determination (R-squared) is defined as the proportion of total variation in the dependent variable that is explained by the independent variables in the model. Since the total variation can be no greater than 1 and the unexplained variation (residuals) can be no less than 0, the value of R-squared must fall between 0 and +1. Therefore, the correct choice is D.

Quiz 13: Introduction to Multiple Regression

In calculating the standard error of the estimate,S_YX = $\sqrt { \mathrm { MSE } }$ there are n * k * 1 degrees of freedom,where n is the sample size and k represents the number of independent variables in the model.

In a multiple regression model,which of the following is correct regarding the value of the adjusted r²?

The variation attribuInstruction to factors other than the relationship between the independent variables and the explained variable in a regression analysis is represented by

From the coefficient of multiple determination,you cannot detect the strength of the relationship between Y and any individual independent variable.

The total sum of squares (SST)in a regression model will never exceed the regression sum of squares (SSR).

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 90.0% of the variation in the dependent variable is explained by the independent variables in the regression.

When an explanatory variable is dropped from a multiple regression model,the coefficient of multiple determination can increase.

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 20.0% of the variation in the dependent variable is explained by the independent variables in the regression.

The coefficient of multiple determination is calculated by taking the ratio of the regression sum of squares over the total sum of squares (SSR/SST)and subtracting that value from 1.

A regression had the following results: SST = 82.55,SSE = 29.85.It can be said that 63.84% of the variation in the dependent variable is explained by the independent variables in the regression.

You have just computed a regression in which the value of coefficient of multiple determination is 0.57.To determine if this indicates that the independent variables explain a significant portion of the variation in the dependent variable,you would perform an F test.

When an additional explanatory variable is introduced into a multiple regression model,the adjusted r² can never decrease.

The coefficient of multiple determination measures the proportion of the total variation in the dependent variable that is explained by a set of independent variables.

The coefficient of multiple determination r²_Y_.12

When an explanatory variable is dropped from a multiple regression model,the adjusted r²can increase.

In a multiple regression model,the value of the coefficient of multiple determination

Quiz 13: Introduction to Multiple Regression

In calculating the standard error of the estimate,SYX = MSE\sqrt { \mathrm { MSE } }MSE​ there are n * k * 1 degrees of freedom,where n is the sample size and k represents the number of independent variables in the model.

In a multiple regression model,which of the following is correct regarding the value of the adjusted r2?

The variation attribuInstruction to factors other than the relationship between the independent variables and the explained variable in a regression analysis is represented by

From the coefficient of multiple determination,you cannot detect the strength of the relationship between Y and any individual independent variable.

The total sum of squares (SST)in a regression model will never exceed the regression sum of squares (SSR).

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 90.0% of the variation in the dependent variable is explained by the independent variables in the regression.

When an explanatory variable is dropped from a multiple regression model,the coefficient of multiple determination can increase.

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 20.0% of the variation in the dependent variable is explained by the independent variables in the regression.

The coefficient of multiple determination is calculated by taking the ratio of the regression sum of squares over the total sum of squares (SSR/SST)and subtracting that value from 1.

A regression had the following results: SST = 82.55,SSE = 29.85.It can be said that 63.84% of the variation in the dependent variable is explained by the independent variables in the regression.

You have just computed a regression in which the value of coefficient of multiple determination is 0.57.To determine if this indicates that the independent variables explain a significant portion of the variation in the dependent variable,you would perform an F test.

When an additional explanatory variable is introduced into a multiple regression model,the adjusted r2 can never decrease.

The coefficient of multiple determination measures the proportion of the total variation in the dependent variable that is explained by a set of independent variables.

The coefficient of multiple determination r2Y.12

When an explanatory variable is dropped from a multiple regression model,the adjusted r2 can increase.

In a multiple regression model,the value of the coefficient of multiple determination

In calculating the standard error of the estimate,S_YX = $\sqrt { \mathrm { MSE } }$ there are n * k * 1 degrees of freedom,where n is the sample size and k represents the number of independent variables in the model.

In a multiple regression model,which of the following is correct regarding the value of the adjusted r²?

When an additional explanatory variable is introduced into a multiple regression model,the adjusted r² can never decrease.

The coefficient of multiple determination r²_Y_.12

When an explanatory variable is dropped from a multiple regression model,the adjusted r²can increase.