Question 1

(Requires Calculus)Consider the following model:
Yi = β1Xi + ui.
Derive the OLS estimator for β1.

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Yi = β1Xi...

Question 2

The OLS slope estimator is not defined if there is no variation in the data for the explanatory variable.You are interested in estimating a regression relating earnings to years of schooling.Imagine that you had collected data on earnings for different individuals,but that all these individuals had completed a college education (16 years of education).Sketch what the data would look like and explain intuitively why the OLS coefficient does not exist in this situation.

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The answer of The OLS slope estimator is not defined...

Question 3

Consider the sample regression function
Yi =  0 +  1Xi +  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. ,zi = Zi -  . )Finally,illustrate in a two-dimensional diagram with SSR 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.

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Yi = ...

Question 4

(Requires Appendix material)Consider the sample regression function  ,
where * indicates that the variable has been standardized.What are the units of measurement for the dependent and explanatory variable? Why would you want to transform both variables in this way? Show that the OLS estimator for the intercept equals zero.Next prove that the OLS estimator for the slope in this case is identical to the formula for the least squares estimator where the variables have not been standardized,times the ratio of the sample standard deviation of X and Y,i.e. ,  .

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Question 5

A peer of yours,who is a major in another social science,says he is not interested in the regression slope and/or intercept.Instead he only cares about correlations.For example,in the testscore/student-teacher ratio regression,he claims to get all the information he needs from the negative correlation coefficient corr(X,Y)=-0.226.What response might you have for your peer?

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Question 6

(Requires Appendix material)In deriving the OLS estimator,you minimize the sum of squared residuals with respect to the two parameters  0 and  1.The resulting two equations imply two restrictions that OLS places on the data,namely that  and  .Show that you get the same formula for the regression slope and the intercept if you impose these two conditions on the sample regression function.

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Question 7

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 R2 be affected? Should statistical inference be dependent on the scale of variables? Discuss.

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Question 8

In order to calculate the regression R2 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?

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The answer of In order to calculate the regression R2...

Question 9

Indicate in a scatterplot what the data for your dependent variable and your explanatory variable would look like in a regression with an R2 equal to zero.How would this change if the regression R2 was equal to one?

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Question 10

Prove that the regression R2 is identical to the square of the correlation coefficient between two variables Y and X.Regression functions are written in a form that suggests causation running from X to Y.Given your proof,does a high regression R2 present supportive evidence of a causal relationship? Can you think of some regression examples where the direction of causality is not clear? Is without a doubt?

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Question 11

(Requires Appendix material)Show that the two alternative formulae for the slope given in your textbook are identical.

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Question 12

Imagine that you had discovered a relationship that would generate a scatterplot very similar to the relationship Yi =  ,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?

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Question 13

At the Stock and Watson (http://www.pearsonhighered.com/stock_watson)website go to Student Resources and select the option "Datasets for Replicating Empirical Results." Then select the "California Test Score Data Used in Chapters 4-9" (caschool.xls)and open it in a spreadsheet program such as Excel.
In this exercise you will estimate various statistics of the Linear Regression Model with One Regressor through construction of various sums and ratio within a spreadsheet program.
Throughout this exercise,let Y correspond to Test Scores (testscore)and X to the Student Teacher Ratio (str).To generate answers to all exercises here,you will have to create seven columns and the sums of five of these.They are
(i)Yi, (ii)Xi, (iii)(Yi-  ), (iv)(Xi-  ), (v)(Yi-  )×(Xi-  ), (vi)(Xi-  )2, (vii)(Yi-  )2
Although neither the sum of (iii)or (iv)will be required for further calculations,you may want to generate these as a check (both have to sum to zero).
a.Use equation (4.7)and the sums of columns (v)and (vi)to generate the slope of the regression.
b.Use equation (4.8)to generate the intercept.
c.Display the regression line (4.9)and interpret the coefficients.
d.Use equation (4.16)and the sum of column (vii)to calculate the regression R2.
e.Use equation (4.19)to calculate the SER.
f.Use the "Regression" function in Excel to verify the results.

Accepted Answer

The answer of At the Stock and Watson (http://www.pearsonhighered.com/stock_watson)website go...

Question 14

The help function for a commonly used spreadsheet program gives the following definition for the regression slope it estimates:  Prove that this formula is the same as the one given in the textbook.

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Question 15

(Requires Calculus)Consider the following model:
Yi = β0 + ui.
Derive the OLS estimator for β0.

Accepted Answer

The answer of (Requires Calculus)Consider the following model:
Yi = β0...

Question 16

Show first that the regression R2 is the square of the sample correlation coefficient.Next,show that the slope of a simple regression of Y on X is only identical to the inverse of the regression slope of X on Y if the regression R2 equals one.

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The answer of Show first that the regression R2 is...

Question 17

(Requires Appendix material)A necessary and sufficient condition to derive the OLS estimator is that the following two conditions hold:  = 0 and  = 0.Show that these conditions imply that  = 0.

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Question 18

In order to calculate the slope,the intercept,and the regression R2 for a simple sample regression function,list the five sums of data that you need.

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Question 19

You have obtained a sample of 14,925 individuals from the Current Population Survey (CPS)and are interested in the relationship between average hourly earnings and years of education.The regression yields the following result:  = -4.58 + 1.71×educ ,R2 = 0.182,SER = 9.30
where ahe and educ are measured in dollars and years respectively.
a.Interpret the coefficients and the regression R2.
b.Is the effect of education on earnings large?
c.Why should education matter in the determination of earnings? Do the results suggest that there is a guarantee for average hourly earnings to rise for everyone as they receive an additional year of education? Do you think that the relationship between education and average hourly earnings is linear?
d.The average years of education in this sample is 13.5 years.What is mean of average hourly earnings in the sample?
e.Interpret the measure SER.What is its unit of measurement.

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The answer of You have obtained a sample of 14,925...

Question 20

You have analyzed the relationship between the weight and height of individuals.Although you are quite confident about the accuracy of your measurements,you feel that some of the observations are extreme,say,two standard deviations above and below the mean.Your therefore decide to disregard these individuals.What consequence will this have on the standard deviation of the OLS estimator of the slope?

Accepted Answer

The answer of You have analyzed the relationship between the...

Quiz 4: Linear Regression With One Regressor

(Requires Calculus)Consider the following model: Yi = β1Xi + ui. Derive the OLS estimator for β1.

Indicate in a scatterplot what the data for your dependent variable and your explanatory variable would look like in a regression with an R2 equal to zero.How would this change if the regression R2 was equal to one?

(Requires Appendix material)Show that the two alternative formulae for the slope given in your textbook are identical.

The help function for a commonly used spreadsheet program gives the following definition for the regression slope it estimates: Prove that this formula is the same as the one given in the textbook.

(Requires Calculus)Consider the following model: Yi = β0 + ui. Derive the OLS estimator for β0.

Show first that the regression R2 is the square of the sample correlation coefficient.Next,show that the slope of a simple regression of Y on X is only identical to the inverse of the regression slope of X on Y if the regression R2 equals one.

(Requires Appendix material)A necessary and sufficient condition to derive the OLS estimator is that the following two conditions hold: = 0 and = 0.Show that these conditions imply that = 0.

In order to calculate the slope,the intercept,and the regression R2 for a simple sample regression function,list the five sums of data that you need.