(Requires Appendix material from Chapters 4 and 5)Shortly before you are making a group presentation on the testscore/student-teacher ratio results, you realize that one of your peers forgot to type all the relevant information on one of your slides.Here is what you see: \[\begin{array} { r l r } \widehat { \text { TestScore } } = & 698.9 - \text { STR } & R ^ { 2 } = 0.051 , \text { SER } = 18.6 \\ & ( 9.47 ) ( 0.48 ) & \end{array}\] In addition, your group member explains that he ran the regression in a standard spreadsheet program, and that, as a result, the standard errors in parenthesis are homoskedasticity-only standard errors. (a)Find the value for the slope coefficient.

There is no theory that suggests the hom

Exam 5: Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Your textbook states that under certain restrictive conditions, the t- statistic has a Student t-distribution with n-2 degrees of freedom.The loss of two degrees of freedom is the result of OLS forcing two restrictions onto the data.What are these two conditions, and when did you impose them onto the data set in your derivation of the OLS estimator?

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Question 1

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Answer The two conditions are $\sum _ { i = 1 } ^ { n } \hat { u } _ { i } = 0$ and $\sum _ { i = 1 } ^ { n } \hat { u } _ { i } X _ { i } = 0$ . These were the result of minimizing the sum of the squared prediction errors, i.e., taking the derivative of the prediction mistakes and setting them to zero.

Explain carefully the relationship between a confidence interval, a one-sided hypothesis test, and a two-sided hypothesis test.What is the unit of measurement of the t-statistic?

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Question 2

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In the case of a two-sided hypothesis test, the relationship between the t-
statistic and the confidence interval is straightforward.The t-statistic
calculates the distance between the estimate and the hypothesized value
in standard deviations.If the distance is larger than 1.96 (size of the test:
5%), then the distance is large enough to reject the null hypothesis.The
confidence interval adds and subtracts 1.96 standard deviations in this
case, and asks whether or not the hypothesized value is contained within
the confidence interval.Hence the two concepts resemble the two sides
of a coin.They are simply different ways to look at the same problem.In
the case of the one-sided test, the relationship is more complex.Since
you are looking at a one-sided alternative, it does not really make sense
to construct a confidence interval.However, the confidence interval
results in the same conclusion as the t-test if the critical value from the
standard normal distribution is appropriately adjusted, e.g.to 10% rather
than 5%.The unit of measurement of the t-statistic is standard
deviations.
20

The homoskedastic normal regression assumptions are all of the following with the exception of:

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Question 3

Correct Answer:

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D

The 95% confidence interval for the predicted effect of a general change in X is a. $\quad \left( \beta _ { 1 } \Delta x - 1.96 \operatorname { SE } \left( \beta _ { 1 } \right) \times \Delta x , \beta _ { 1 } \Delta x + 1.96 \operatorname { SE } \left( \beta _ { 1 } \right) \times \Delta x \right)$ b. $\left( \widehat { \beta } _ { 1 } \Delta x - 1.645 \operatorname { SE } \left( \hat { \beta } _ { 1 } \right) \times \Delta x , \widehat { \beta } _ { 1 } \Delta x + 1.645 \operatorname { SE } \left( \hat { \beta } _ { 1 } \right) \times \Delta x \right)$ c. $\left( \hat { \beta } _ { 1 } \Delta x - 1.96 \operatorname { SE } \left( \hat { \beta } _ { 1 } \right) \times \Delta x , \widehat { \beta } _ { 1 } \Delta x + 1.96 \operatorname { SE } \left( \hat { \beta } _ { 1 } \right) \times \Delta x \right)$ d. $\left( \widehat { \beta } _ { 1 } \Delta x - 1.96 , \widehat { \beta } _ { 1 } \Delta x + 1.96 \right)$

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Question 4

In the regression through the origin model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$ the OLS estimator is $\widehat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } }$ Prove that the estimator is a linear function of $Y _ { 1 } , \ldots , Y _ { n }$ and prove that it is conditionally unbiased.

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Question 5

Below you are asked to decide on whether or not to use a one-sided alternative or a two-sided alternative hypothesis for the slope coefficient.Briefly justify your decision. (a) $\widehat { q _ { i } ^ { d } } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } p _ { i } \text {, where } q ^ { d } \text { is the quantity demanded for a good, and } p \text { is its price. }$

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Question 6

The construction of the t-statistic for a one- and a two-sided hypothesis

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Question 7

The p-value for a one-sided left-tail test a. is given by $\operatorname { Pr } \left( Z > t ^ { a c t } \right) = \phi \left( t ^ { a c t } \right)$ . b. is given by $\operatorname { Pr } \left( Z < t ^ { a c t } \right) = \phi \left( t ^ { a c t } \right)$ . c. is given by $\operatorname { Pr } \left( Z < t ^ { a c t } \right) < 1.645$ . d. cannot be calculated, since probabilities must always be positive.

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Question 8

If the errors are heteroskedastic, then

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Question 9

The error term is homoskedastic if a. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ is constant for $i = 1 , \ldots , n$ b. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ depends on $x$ c. $X _ { i }$ is normally distributed d. there are no outliers.

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Question 10

In the presence of heteroskedasticity, and assuming that the usual least squares assumptions hold, the OLS estimator is

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Question 11

(Requires Appendix material) Your textbook shows that OLS is a linear estimator $\hat { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } \hat { a } _ { i } Y _ { i } , \text { where } \hat { a } _ { i } = \frac { X _ { i } - \bar { X } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$ . For OLS to be conditionally unbiased, the following two conditions must hold: $\sum _ { i = 1 } ^ { n } \hat { a } _ { i } = 0 \text { and } \sum _ { i = 1 } ^ { n } \hat { a } _ { i } X _ { i } = 1$ Show that this is the case.

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Question 12

The confidence interval for the sample regression function slope

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Question 13

A binary variable is often called a

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Question 14

(Requires Appendix)(continuation from Chapter 4).At a recent county fair, you observed that at one stand people's weight was forecasted, and were surprised by the accuracy (within a range).Thinking about how the person could have predicted your weight fairly accurately (despite the fact that she did not know about your "heavy bones"), you think about how this could have been accomplished.You remember that medical charts for children contain 5%, 25%, 50%, 75% and 95% lines for a weight/height relationship and decide to conduct an experiment with 110 of your peers.You collect the data and calculate the following sums: =17,375,=7,665.5, =94,228.8,=1,248.9,=7,625.9 where the height is measured in inches and weight in pounds. (Small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ .) (a)Calculate the homoskedasticity-only standard errors and, using the resulting t- statistic, perform a test on the null hypothesis that there is no relationship between height and weight in the population of college students.

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Question 15

(Continuation from Chapter 4, number 5) You have learned in one of your economics courses that one of the determinants of per capita income (the "Wealth of Nations") is the population growth rate. Furthermore you also found out that the Penn World Tables contain income and population data for 104 countries of the world. To test this theory, you regress the GDP per worker (relative to the United States) in 1990 (RelPersInc) on the difference between the average population growth rate of that country $( n )$ to the U.S. average population growth rate $\left( n _ { u s } \right)$ for the years 1980 to 1990 . This results in the following regression output: = 0.518-18.831\times n- ,=0.522, SER =0.197 (0.056)(3.177) (a)Is there any reason to believe that the variance of the error terms is homoskedastic?

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Question 16

(Requires Appendix material from Chapters 4 and 5)Shortly before you are making a group presentation on the testscore/student-teacher ratio results, you realize that one of your peers forgot to type all the relevant information on one of your slides.Here is what you see: = 698.9- STR =0.051, SER =18.6 (9.47)(0.48) In addition, your group member explains that he ran the regression in a standard spreadsheet program, and that, as a result, the standard errors in parenthesis are homoskedasticity-only standard errors. (a)Find the value for the slope coefficient.

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Question 17

The 95 % confidence interval for $\beta _ { 0 }$ is the interval

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Question 18

When estimating a demand function for a good where quantity demanded is a linear function of the price, you should

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Question 19

Heteroskedasticity means that

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Question 20

Explain carefully the relationship between a confidence interval, a one-sided hypothesis test, and a two-sided hypothesis test.What is the unit of measurement of the t-statistic?

The homoskedastic normal regression assumptions are all of the following with the exception of:

The construction of the t-statistic for a one- and a two-sided hypothesis

If the errors are heteroskedastic, then

The error term is homoskedastic if a. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ is constant for $i = 1 , \ldots , n$ b. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ depends on $x$ c. $X _ { i }$ is normally distributed d. there are no outliers.

In the presence of heteroskedasticity, and assuming that the usual least squares assumptions hold, the OLS estimator is

The confidence interval for the sample regression function slope

A binary variable is often called a

The 95 % confidence interval for $\beta _ { 0 }$ is the interval

When estimating a demand function for a good where quantity demanded is a linear function of the price, you should

Heteroskedasticity means that

Economic Questions and Data

Review of Probability

Review of Statistics

Linear Regression With One Regressor

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

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Exam 5: Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Explain carefully the relationship between a confidence interval, a one-sided hypothesis test, and a two-sided hypothesis test.What is the unit of measurement of the t-statistic?

The homoskedastic normal regression assumptions are all of the following with the exception of:

The construction of the t-statistic for a one- and a two-sided hypothesis

If the errors are heteroskedastic, then

In the presence of heteroskedasticity, and assuming that the usual least squares assumptions hold, the OLS estimator is

The confidence interval for the sample regression function slope

A binary variable is often called a

The 95 % confidence interval for β0\beta _ { 0 }β0​ is the interval

When estimating a demand function for a good where quantity demanded is a linear function of the price, you should

Heteroskedasticity means that

Economic Questions and Data

Review of Probability

Review of Statistics

Linear Regression With One Regressor

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

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The 95 % confidence interval for $\beta _ { 0 }$ is the interval