Exam 6: Linear Regression With Multiple Regressors

Under the least squares assumptions for the multiple regression problem (zero conditional mean for the error term,all Xi and Yi being i.i.d. ,all Xi and ui having finite fourth moments,no perfect multicollinearity),the OLS estimators for the slopes and intercept

The cost of attending your college has once again gone up.Although you have been told that education is investment in human capital,which carries a return of roughly 10% a year,you (and your parents)are not pleased.One of the administrators at your university/college does not make the situation better by telling you that you pay more because the reputation of your institution is better than that of others.To investigate this hypothesis,you collect data randomly for 100 national universities and liberal arts colleges from the 2000-2001 U.S.News and World Report annual rankings.Next you perform the following regression = 7,311.17 + 3,985.20 × Reputation - 0.20 × Size + 8,406.79 × Dpriv - 416.38 × Dlibart - 2,376.51 × Dreligion R2=0.72,SER = 3,773.35 where Cost is Tuition,Fees,Room and Board in dollars,Reputation is the index used in U.S.News and World Report (based on a survey of university presidents and chief academic officers),which ranges from 1 ("marginal")to 5 ("distinguished"),Size is the number of undergraduate students,and Dpriv,Dlibart,and Dreligion are binary variables indicating whether the institution is private,a liberal arts college,and has a religious affiliation. (a)Interpret the results.Do the coefficients have the expected sign? (b)What is the forecasted cost for a liberal arts college,which has no religious affiliation,a size of 1,500 students and a reputation level of 4.5? (All liberal arts colleges are private. ) (c)To save money,you are willing to switch from a private university to a public university,which has a ranking of 0.5 less and 10,000 more students.What is the effect on your cost? Is it substantial? (d)Eliminating the Size and Dlibart variables from your regression,the estimation regression becomes = 5,450.35 + 3,538.84 × Reputation + 10,935.70 × Dpriv - 2,783.31 × Dreligion; =0.72,SER = 3,792.68 Why do you think that the effect of attending a private institution has increased now? (e)What can you say about causation in the above relationship? Is it possible that Cost affects Reputation rather than the other way around?

Give at least three examples from macroeconomics and three from microeconomics that involve specified equations in a multiple regression analysis framework.Indicate in each case what the expected signs of the coefficients would be and if theory gives you an indication about the likely size of the coefficients.

The main advantage of using multiple regression analysis over differences in means testing is that the regression technique

In the multiple regression model,the SER is given by

The OLS formula for the slope coefficients in the multiple regression model become increasingly more complicated,using the "sums" expressions,as you add more regressors.For example,in the regression with a single explanatory variable,the formula is whereas this formula for the slope of the first explanatory variable is (small letters refer to deviations from means as in ) in the case of two explanatory variables.Give an intuitive explanations as to why this is the case.

The population multiple regression model when there are two regressors,X1i and X2i can be written as follows,with the exception of:

The sample regression line estimated by OLS

The dummy variable trap is an example of

In the multiple regression model,the least squares estimator is derived by

The probability limit of the OLS estimator in the case of omitted variables is given in your text by the following formula: Give an intuitive explanation for two conditions under which the bias will be small.

Your textbook extends the simple regression analysis of Chapters 4 and 5 by adding an additional explanatory variable,the percent of English learners in school districts (PctEl).The results are as follows: = 698.9 - 2.28 × STR and = 698.0 - 1.10 × STR - 0.65 × PctEL Explain why you think the coefficient on the student-teacher ratio has changed so dramatically (been more than halved).

Consider the following multiple regression models (a)to (d)below.DFemme = 1 if the individual is a female,and is zero otherwise;DMale is a binary variable which takes on the value one if the individual is male,and is zero otherwise;DMarried is a binary variable which is unity for married individuals and is zero otherwise,and DSingle is (1-DMarried).Regressing weekly earnings (Earn)on a set of explanatory variables,you will experience perfect multicollinearity in the following cases unless:

Attendance at sports events depends on various factors.Teams typically do not change ticket prices from game to game to attract more spectators to less attractive games.However,there are other marketing tools used,such as fireworks,free hats,etc. ,for this purpose.You work as a consultant for a sports team,the Los Angeles Dodgers,to help them forecast attendance,so that they can potentially devise strategies for price discrimination.After collecting data over two years for every one of the 162 home games of the 2000 and 2001 season,you run the following regression: = 15,005 + 201 × Temperat + 465 × DodgNetWin + 82 × OppNetWin + 9647 × DFSaSu + 1328 × Drain + 1609 × D150m + 271 × DDiv - 978 × D2001; =0.416,SER = 6983 where Attend is announced stadium attendance,Temperat it the average temperature on game day,DodgNetWin are the net wins of the Dodgers before the game (wins-losses),OppNetWin is the opposing team's net wins at the end of the previous season,and DFSaSu,Drain,D150m,Ddiv,and D2001 are binary variables,taking a value of 1 if the game was played on a weekend,it rained during that day,the opposing team was within a 150 mile radius,the opposing team plays in the same division as the Dodgers,and the game was played during 2001,respectively. (a)Interpret the regression results.Do the coefficients have the expected signs? (b)Excluding the last four binary variables results in the following regression result: = 14,838 + 202 × Temperat + 435 × DodgNetWin + 90 × OppNetWin + 10,472 × DFSaSu, =0.410,SER = 6925 According to this regression,what is your forecast of the change in attendance if the temperature increases by 30 degrees? Is it likely that people attend more games if the temperature increases? Is it possible that Temperat picks up the effect of an omitted variable? (c)Assuming that ticket sales depend on prices,what would your policy advice be for the Dodgers to increase attendance? (d)Dodger stadium is large and is not often sold out.The Boston Red Sox play in a much smaller stadium,Fenway Park,which often reaches capacity.If you did the same analysis for the Red Sox,what problems would you foresee in your analysis?

Consider the multiple regression model with two regressors X1 and X2,where both variables are determinants of the dependent variable.You first regress Y on X1 only and find no relationship.However when regressing Y on X1 and X2,the slope coefficient changes by a large amount.This suggests that your first regression suffers from

The following OLS assumption is most likely violated by omitted variables bias:

(Requires Calculus)For the case of the multiple regression problem with two explanatory variables,show that minimizing the sum of squared residuals results in three conditions:

In the case of perfect multicollinearity,OLS is unable to calculate the coefficients for the explanatory variables,because it is impossible to change one variable while holding all other variables constant.To see why this is the case,consider the coefficient for the first explanatory variable in the case of a multiple regression model with two explanatory variables: (small letters refer to deviations from means as in ). Divide each of the four terms by to derive an expression in terms of regression coefficients from the simple (one explanatory variable)regression model.In case of perfect multicollinearity,what would be R2 from the regression of on ? As a result,what would be the value of the denominator in the above expression for ?

If you had a two regressor regression model,then omitting one variable which is relevant

It is not hard,but tedious,to derive the OLS formulae for the slope coefficient in the multiple regression case with two explanatory variables.The formula for the first regression slope is (small letters refer to deviations from means as in ). Show that this formula reduces to the slope coefficient for the linear regression model with one regressor if the sample correlation between the two explanatory variables is zero.Given this result,what can you say about the effect of omitting the second explanatory variable from the regression?