Question 1

Suppose the variable  has been omitted from the following regression equation,   is the estimator obtained when  is omitted from the equation. The bias in  is negative if _____.
A)  >0 and  and  are positively correlated
B)  <0 and  and  are positively correlated
C)  =0 and  and  are negatively correlated
D)  =0 and  and  are negatively correlated

Accepted Answer

In regression analysis, the bias of an estimator is the difference between its expected value and the true value of the parameter being estimated. If a variable that should be included in a regression equation is omitted, this can lead to biased estimates of the remaining coefficients. The direction and magnitude of this bias depends on the relationship between the omitted variable and the included variables. In the given options, only option B states that the bias is negative and the included variables are positively correlated. This implies that the omitted variable should have a negative correlation with the included variables, and its exclusion leads to an overestimate of the effect of the included variables on the dependent variable. Therefore, B is the best choice.

Question 2

High (but not perfect) correlation between two or more independent variables is called _____.
A)heteroskedasticty
B)homoskedasticty
C)multicollinearity
D)micronumerosity

Accepted Answer

The correct term for high correlation between independent variables is multicollinearity. Homoskedasticty refers to the assumption in regression analysis that the variance of residuals is constant across all levels of the predictor variables. Heteroskedasticty refers to violation of the homoskedasticity assumption, where the variance of residuals is not constant across all levels of the predictor variable. Micronumerosity is not a term used in statistics.

Question 3

The value of R² always _____. A)lies below 0 B)lies above 1 C)lies between 0 and 1 D)lies between 1 and 1.5

Accepted Answer

The value of R² is a probability value and probabilities always lie between 0 and 1. Therefore, option C is the best choice.

Question 4

The term "linear" in a multiple linear regression model means that the equation is linear in parameters.

Accepted Answer

In multiple linear regression, the equation is linear in the parameters (also known as the coefficients), which means that the relationship between the independent variables and the dependent variable is a straight line. However, the relationship between the independent variables themselves can be non-linear.

Question 5

The term _____ refers to the problem of small sample size.
A)micronumerosity
B)multicollinearity
C)homoskedasticity
D)heteroskedasticity

Accepted Answer

The answer of The term _____ refers to the problem...

Question 6

Find the degrees of freedom in a regression model that has 10 observations and 7 independent variables.
A)17
B)2
C)3
D)4

Accepted Answer

The degrees of freedom in a regression model are calculated as the number of observations minus the number of independent variables minus one. Therefore, with 10 observations and 7 independent variables, the degrees of freedom are 10 - 7 - 1 = 2.

Question 7

The key assumption for the general multiple regression model is that all factors in the unobserved error term be correlated with the explanatory variables.

Accepted Answer

The key assumption for the multiple regression model is that the errors are independently and identically distributed with mean zero and constant variance, not that they are correlated with the explanatory variables.

Question 8

The Gauss-Markov theorem will not hold if _____.
A)the error term has the same variance given any values of the explanatory variables
B)the error term has an expected value of zero given any values of the independent variables
C)the independent variables have exact linear relationships among them
D)the regression model relies on the method of random sampling for collection of data

Accepted Answer

The Gauss-Markov theorem assumes that the explanatory variables are not perfectly correlated with each other, which is violated if there are exact linear relationships among them.

Question 9

The assumption that there are no exact linear relationships among the independent variables in a multiple linear regression model fails if _____, where n is the sample size and k is the number of parameters.
A)n > 2
B)n = k + 1
C)n > k
D)n < k + 1

Accepted Answer

The assumption fails when $n < k + 1$, because in that case, there are not enough observations to estimate the parameters of the model uniquely, leading to perfect multicollinearity.

Question 10

In the equation,  is a(n) _____.
A)independent variable
B)dependent variable
C)slope parameter
D)intercept parameter

Accepted Answer

The answer of In the equation,  is a(n) _____.
A)independent...

Question 11

In econometrics, the general partialling out result is usually called the _____.​
A)​Gauss-Markov assumption
B)​Best linear unbiased estimator
C)​Frisch-Waugh theorem
D)​Gauss-Markov theorem

Accepted Answer

The general partialling out result is commonly known as the Frisch-Waugh theorem. This theorem is used in econometrics to isolate the effect of one independent variable on a dependent variable by removing the effects of other independent variables through regression analysis. The Gauss-Markov assumption and theorem and the best linear unbiased estimator are also important concepts in econometrics, but they are not directly related to the general partialling out result.

Question 12

Exclusion of a relevant variable from a multiple linear regression model leads to the problem of _____.
A)misspecification of the model
B)multicollinearity
C)perfect collinearity
D)homoskedasticity

Accepted Answer

Exclusion of a relevant variable from a multiple linear regression model leads to misspecification of the model, also known as omitted variable bias. This can lead to incorrect estimates of the coefficients of other variables in the model and distorted predictions.

Question 13

Suppose the variable  has been omitted from the following regression equation,   is the estimator obtained when  is omitted from the equation. If  is said to _____.
A)have an upward bias
B)have a downward bias
C)be unbiased
D)be biased toward zero

Accepted Answer

The given regression equation is incomplete and it is not clear which variable is omitted from the equation. However, the fact that the estimator obtained by omitting a variable is mentioned suggests that the omitted variable bias (OVB) may be present. OVB occurs when a relevant variable is omitted from a regression equation, leading to biased and inconsistent estimators for other coefficients in the equation. The direction of bias depends on the correlation between the omitted variable and other variables in the equation. OVB is said to have an upward bias when the omitted variable is positively correlated with the dependent variable and at least one of the included independent variables. Therefore, given the lack of information, we can only infer that the variable 11eb6b8a_02cb_edfe_bf83_397d9a56b27c_TB8272_11 is missing and that its omission is likely to create an upward bias.

Question 14

Which of the following is true of R²? A)R² is also called the standard error of regression. B)A low R² indicates that the Ordinary Least Squares line fits the data well. C)R² usually decreases with an increase in the number of independent variables in a regression. D)R² shows what percentage of the total variation in the dependent variable, Y, is explained by the explanatory variables.

Accepted Answer

R² indicates the proportion of the variance in the dependent variable that is predictable from the independent variables.

Question 15

The coefficient of determination (R²) decreases when an independent variable is added to a multiple regression model.

Accepted Answer

The coefficient of determination (R-squared) is a measure of how much of the variation in the dependent variable can be explained by the independent variables in the regression model. When an independent variable is added to a multiple regression model, the R-squared value is likely to increase because the new variable has the potential to explain additional variation in the dependent variable. Therefore, the statement is false.

Question 16

If an independent variable in a multiple linear regression model is an exact linear combination of other independent variables, the model suffers from the problem of _____.
A)perfect collinearity
B)homoskedasticity
C)heteroskedasticty
D)omitted variable bias

Accepted Answer

Perfect collinearity occurs when one or more independent variables in a multiple linear regression model are an exact linear combination of other independent variables. This causes the model to lack a unique solution, making it impossible to estimate the effects of individual independent variables on the dependent variable. Therefore, the correct answer is A. Homoskedasticity, heteroskedasticity, and omitted variable bias are separate issues that can affect the accuracy of a multiple linear regression model, but they are not directly related to perfect collinearity.

Question 17

If the explained sum of squares is 35 and the total sum of squares is 49, what is the residual sum of squares?
A)10
B)12
C)18
D)14

Accepted Answer

The residual sum of squares can be found by subtracting the explained sum of squares from the total sum of squares: 49 - 35 = 14.

Question 18

Which of the following is true of BLUE?​
A)​It is a rule that can be applied to any one value of the data to produce an estimate.
B)​An estimator  is an unbiased estimator of  if  for any  .
C)​An estimator is linear if and only if it can be expressed as a linear function of the data on the dependent variable.
D)​It is the best linear uniform estimator.

Accepted Answer

Choice C defines a linear estimator, while choices A, B, and D do not pertain to the definition of BLUE.

Question 19

Suppose the variable x2 has been omitted from the following regression equation, is the estimator obtained when is omitted from the equation. The bias in is positive if _____. A) >0 and and are positively correlated B) <0 and and are positively correlated C) >0 and and are negatively correlated D) = 0 and and are negatively correlated

Accepted Answer

If the omitted variable x2 is positively correlated with both the included variable x1 and the dependent variable y, then the bias in the coefficient estimate of x1 will be positive. This is because the omitted variable captures some of the variation in y that is also explained by x1, leading to an overestimate of the effect of x1 on y. In this scenario, A is the only choice in which the first condition is satisfied.

Question 20

Consider the following regression equation:  . What does  imply?
A)  measures the ceteris paribus effect of  on  .
B)  measures the ceteris paribus effect of  on  .
C)  measures the ceteris paribus effect of  on  .
D)  measures the ceteris paribus effect of  on  .

Accepted Answer

The answer of Consider the following regression equation:  ....

Quiz 3: Multiple Regression Analysis Estimation

Suppose the variable has been omitted from the following regression equation, is the estimator obtained when is omitted from the equation. The bias in is negative if _____.

High (but not perfect) correlation between two or more independent variables is called _____.

The value of R2 always _____.

The term "linear" in a multiple linear regression model means that the equation is linear in parameters.

The term _____ refers to the problem of small sample size.

Find the degrees of freedom in a regression model that has 10 observations and 7 independent variables.

The key assumption for the general multiple regression model is that all factors in the unobserved error term be correlated with the explanatory variables.

The Gauss-Markov theorem will not hold if _____.

The assumption that there are no exact linear relationships among the independent variables in a multiple linear regression model fails if _____, where n is the sample size and k is the number of parameters.

In the equation, is a(n) _____.

In econometrics, the general partialling out result is usually called the _____.​

Exclusion of a relevant variable from a multiple linear regression model leads to the problem of _____.

Suppose the variable has been omitted from the following regression equation, is the estimator obtained when is omitted from the equation. If is said to _____.

Which of the following is true of R2?

The coefficient of determination (R2) decreases when an independent variable is added to a multiple regression model.

If an independent variable in a multiple linear regression model is an exact linear combination of other independent variables, the model suffers from the problem of _____.

If the explained sum of squares is 35 and the total sum of squares is 49, what is the residual sum of squares?

Which of the following is true of BLUE?​

Suppose the variable x2 has been omitted from the following regression equation, is the estimator obtained when is omitted from the equation. The bias in is positive if _____.

Consider the following regression equation: . What does imply?

The value of R² always _____.

In econometrics, the general partialling out result is usually called the _____.

Which of the following is true of R²?

The coefficient of determination (R²) decreases when an independent variable is added to a multiple regression model.

Which of the following is true of BLUE?