Question 1

(Requires Appendix material)The sample average of the OLS residuals is

Accepted Answer

A) some positive number since OLS uses squares. 
B) zero. 
C) unobservable since the population regression function is unknown. 
D) dependent on whether the explanatory variable is mostly positive or negative. 
A) some positive number since OLS uses squares. 
B) zero. 
C) unobservable since the population regression function is unknown. 
D) dependent on whether the explanatory variable is mostly positive or negative.

Question 2

In order to calculate the regression  $R ^ { 2 }$ you need the  TSS  and either the  SSR  or the  ESS . The TSS is fairly straightforward to calculate, being just the variation of  Y . However, if you had to calculate the  SSR  or  ESS  by hand (or in a spreadsheet), you would need all fitted values from the regression function and their deviations from the sample mean, or the residuals. Can you think of a quicker way to calculate the ESS simply using terms you have already used to calculate the slope coefficient?

Accepted Answer

The ESS is given by 11ec79c3_3966_6c70_9

Question 3

Assume that there is a change in the units of measurement on both  Y  and  X . The new variables are $Y ^ { * } = a Y$ and $X ^ { * } = b X$ What effect will this change have on the regression slope?

Accepted Answer

We now have the following samp

Question 4

The OLS residuals, $\hat { u } _ { i } ,$  are defined as follows:

Accepted Answer

A)  $\hat { Y } _ { i } - \widehat { \beta } _ { 0 } - \widehat { \beta } _ { 1 } X _ { i }$ 
B)  $Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i }$ 
C)  $Y _ { i } - \hat { Y } _ { i }$ 
D)  $\left( Y _ { i } - \bar { Y } \right) ^ { 2 }$ 
A)  $\hat { Y } _ { i } - \widehat { \beta } _ { 0 } - \widehat { \beta } _ { 1 } X _ { i }$ 
B)  $Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i }$ 
C)  $Y _ { i } - \hat { Y } _ { i }$ 
D)  $\left( Y _ { i } - \bar { Y } \right) ^ { 2 }$

Question 5

To obtain the slope estimator using the least squares principle, you divide the a. sample variance of $X$ by the sample variance of $Y$.
b. sample covariance of $X$ and $Y$ by the sample variance of $Y$.
c. sample covariance of $X$ and $Y$ by the sample variance of $X$.
d. sample variance of $X$ by the sample covariance of $X$ and $Y$.

Accepted Answer

The answer of To obtain the slope estimator using the...

Question 6

In the simple linear regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

Accepted Answer

A)  the intercept is typically small and unimportant. 
B)  $\beta _ { 0 } + \beta _ { 1 } X _ { i }$ represents the population regression function. 
C)  the absolute value of the slope is typically between 0 and 1 . 
D)  $\beta _ { 0 } + \beta _ { 1 } X _ { i }$ represents the sample regression function. 
A)  the intercept is typically small and unimportant. 
B)  $\beta _ { 0 } + \beta _ { 1 } X _ { i }$ represents the population regression function. 
C)  the absolute value of the slope is typically between 0 and 1 . 
D)  $\beta _ { 0 } + \beta _ { 1 } X _ { i }$ represents the sample regression function.

Question 7

Consider the sample regression function $$Y _ { i } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } X _ { i } + \hat { u } _ { i } .$$ First, take averages on both sides of the equation. Second, subtract the resulting equation from the above equation to write the sample regression function in deviations from means. (For simplicity, you may want to use small letters to indicate deviations from the mean, i.e., $z _ { i } = Z _ { i } - \bar { Z }$.) Finally, illustrate in a two-dimensional diagram with $S S R$ on the vertical axis and the regression slope on the horizontal axis how you could find the least squares estimator for the slope by varying its values through trial and error.

Accepted Answer

The answer of Consider the sample regression function \[Y _...

Question 8

Given the amount of money and effort that you have spent on your education, you wonder if it was (is) all worth it. You therefore collect data from the Current Population Survey (CPS) and estimate a linear relationship between earnings and the years of education of individuals. What would be the effect on your regression slope and intercept if you measured earnings in thousands of dollars rather than in dollars? Would the regression $R ^ { 2 }$  be affected? Should statistical inference be dependent on the scale of variables? Discuss.

Accepted Answer

It should be clear that interpretation o

Question 9

If the three least squares assumptions hold, then the large sample normal distribution of $\widehat { \beta } _ { 1 }$ is

Accepted Answer

A)  $N \left( 0 , \frac { 1 } { n } \frac { \left. \operatorname { var } \left[ X _ { i } - \mu _ { X } \right) u _ { i } \right] } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$ 
B)  $N \left( \beta _ { 1 } , \frac { 1 } { n } \frac { \left. \operatorname { var } \left( u _ { i } \right) \right] ^ { 2 } } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$ 
C)  $N \left( \beta _ { 1 } , \frac { \sigma _ { u } ^ { 2 } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } } \right.$ 
D)  $N \left( \beta _ { 1 } , \frac { 1 } { n } \frac { \left. \operatorname { var } \left( u _ { i } \right) \right] } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$ 
A)  $N \left( 0 , \frac { 1 } { n } \frac { \left. \operatorname { var } \left[ X _ { i } - \mu _ { X } \right) u _ { i } \right] } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$ 
B)  $N \left( \beta _ { 1 } , \frac { 1 } { n } \frac { \left. \operatorname { var } \left( u _ { i } \right) \right] ^ { 2 } } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$ 
C)  $N \left( \beta _ { 1 } , \frac { \sigma _ { u } ^ { 2 } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } } \right.$ 
D)  $N \left( \beta _ { 1 } , \frac { 1 } { n } \frac { \left. \operatorname { var } \left( u _ { i } \right) \right] } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$

Question 10

Imagine that you had discovered a relationship that would generate a scatterplot very similar to the relationship $Y _ { i } = X _ { i } ^ { 2 }$ , and that you would try to fit a linear regression through your data points. What do you expect the slope coefficient to be? What do you think the value of your regression  R2  is in this situation? What are the implications from your answers in terms of fitting a linear regression through a non-linear relationship?

Accepted Answer

You would expect the slope to be a strai

Question 11

The reason why estimators have a sampling distribution is that

Accepted Answer

A) economics is not a precise science. 
B) individuals respond differently to incentives. 
C) in real life you typically get to sample many times. 
D) the values of the explanatory variable and the error term differ across samples. 
A) economics is not a precise science. 
B) individuals respond differently to incentives. 
C) in real life you typically get to sample many times. 
D) the values of the explanatory variable and the error term differ across samples.

Question 12

The following are all least squares assumptions with the exception of: a. The conditional distribution of $u _ { i }$ given $X _ { i }$ has a mean of zero.
b. The explanatory variable in regression model is normally distributed.
c. $\left( X _ { i } , Y _ { i } \right) , i = 1 , \ldots , n$ are independently and identically distributed.
d. Large outliers are unlikely.

Accepted Answer

The answer of The following are all least squares assumptions...

Question 13

In the simple linear regression model, the regression slope

Accepted Answer

A) indicates by how many percent Y increases, given a one percent increase in X. 
B) when multiplied with the explanatory variable will give you the predicted Y. 
C) indicates by how many units Y increases, given a one unit increase in X. 
D) represents the elasticity of Y on X. 
A) indicates by how many percent Y increases, given a one percent increase in X. 
B) when multiplied with the explanatory variable will give you the predicted Y. 
C) indicates by how many units Y increases, given a one unit increase in X. 
D) represents the elasticity of Y on X.

Question 14

Your textbook presented you with the following regression output: $$\begin{array} { l } 
\text { TestScore } = 698.9 - 2.28 \times S T R \
n = 420 , R ^ { 2 } = 0.051 , S E R = 18.6
\end{array}$$ (a)How would the slope coefficient change, if you decided one day to measure testscores in
100s, i.e., a testscore of 650 became 6.5? Would this have an effect on your
interpretation?

Accepted Answer

Since statistical inference will depend

(Requires Appendix material)The sample average of the OLS residuals is

Assume that there is a change in the units of measurement on both Y and X . The new variables are $Y ^ { * } = a Y$ and $X ^ { * } = b X$ What effect will this change have on the regression slope?

The OLS residuals, $\hat { u } _ { i } ,$ are defined as follows:

In the simple linear regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

If the three least squares assumptions hold, then the large sample normal distribution of $\widehat { \beta } _ { 1 }$ is

The reason why estimators have a sampling distribution is that

In the simple linear regression model, the regression slope

Economic Questions and Data

Review of Probability

Review of Statistics

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Linear Regression With One Regressor

The Theory of Multiple Regression

Filters

Exam 4: Linear Regression With One Regressor

(Requires Appendix material)The sample average of the OLS residuals is

Assume that there is a change in the units of measurement on both Y and X . The new variables are Y∗=aYY ^ { * } = a YY∗=aY and X∗=bXX ^ { * } = b XX∗=bX What effect will this change have on the regression slope?

The OLS residuals, u^i,\hat { u } _ { i } ,u^i​, are defined as follows:

In the simple linear regression model Yi=β0+β1Xi+uiY _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }Yi​=β0​+β1​Xi​+ui​

If the three least squares assumptions hold, then the large sample normal distribution of β^1\widehat { \beta } _ { 1 }β​1​ is

The reason why estimators have a sampling distribution is that

In the simple linear regression model, the regression slope

Economic Questions and Data

Review of Probability

Review of Statistics

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Linear Regression With One Regressor

The Theory of Multiple Regression

Filters

Assume that there is a change in the units of measurement on both Y and X . The new variables are $Y ^ { * } = a Y$ and $X ^ { * } = b X$ What effect will this change have on the regression slope?

The OLS residuals, $\hat { u } _ { i } ,$ are defined as follows:

In the simple linear regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

If the three least squares assumptions hold, then the large sample normal distribution of $\widehat { \beta } _ { 1 }$ is