Exam 6: Linear Regression With Multiple Regressors

One of the least squares assumptions in the multiple regression model is that you have random variables which are "i.i.d." This stands for

The intercept in the multiple regression model

Under imperfect multicollinearity

In the multiple regression problem with k explanatory variable,it would be quite tedious to derive the formulas for the slope coefficients without knowledge of linear algebra.The formulas certainly do not resemble the formula for the slope coefficient in the simple linear regression model with a single explanatory variable.However,it can be shown that the following three step procedure results in the same formula for slope coefficient of the first explanatory variable, : Step 1: regress Y on a constant and all other explanatory variables other than ,and calculate the residual (Res1). Step 2: regress on a constant and all other explanatory variables,and calculate the residual (Res2). Step 3: regress Res1 on a constant and Res2. Can you give an intuitive explanation to this procedure?

For this question,use the California Testscore Data Set and your regression package (a spreadsheet program if necessary).First perform a multiple regression of testscores on a constant,the student-teacher ratio,and the percent of English learners.Record the coefficients.Next,do the following three step procedure instead: first,regress the testscore on a constant and the percent of English learners.Calculate the residuals and store them under the name resYX2.Second,regress the student-teacher ratio on a constant and the percent of English learners.Calculate the residuals from this regression and store these under the name resX1X2.Finally regress resYX2 on resX1X2 (and a constant,if you wish).Explain intuitively why the simple regression coefficient in the last regression is identical to the regression coefficient on the student-teacher ratio in the multiple regression.

In multiple regression,the R2 increases whenever a regressor is

In the multiple regression model,the adjusted R2, 2

You have to worry about perfect multicollinearity in the multiple regression model because

In the multiple regression model with two regressors,the formula for the slope of the first explanatory variable is (small letters refer to deviations from means as in ). An alternative way to derive the OLS estimator is given through the following three step procedure. Step 1: regress Y on a constant and ,and calculate the residual (Res1). Step 2: regress on a constant and ,and calculate the residual (Res2). Step 3: regress Res1 on a constant and Res2. Prove that the slope of the regression in Step 3 is identical to the above formula.

A subsample from the Current Population Survey is taken,on weekly earnings of individuals,their age,and their gender.You have read in the news that women make 70 cents to the $1 that men earn.To test this hypothesis,you first regress earnings on a constant and a binary variable,which takes on a value of 1 for females and is 0 otherwise.The results were: = 570.70 - 170.72 × Female, =0.084,SER = 282.12. (a)There are 850 females in your sample and 894 males.What are the mean earnings of males and females in this sample? What is the percentage of average female income to male income? (b)You decide to control for age (in years)in your regression results because older people,up to a point,earn more on average than younger people.This regression output is as follows: = 323.70 - 169.78 × Female + 5.15 × Age, =0.135,SER = 274.45. Interpret these results carefully.How much,on average,does a 40-year-old female make per year in your sample? What about a 20-year-old male? Does this represent stronger evidence of discrimination against females?

In the multiple regression model with two explanatory variables the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in zi = Zi - ): You have collected data for 104 countries of the world from the Penn World Tables and want to estimate the effect of the population growth rate ( )and the saving rate ( )(average investment share of GDP from 1980 to 1990)on GDP per worker (relative to the U.S. )in 1990.The various sums needed to calculate the OLS estimates are given below: = 33.33; = 2.025; = 17.313 = 8.3103; = .0122; = 0.6422 = -0.2304; = 1.5676; = -0.0520 (a)What are your expected signs for the regression coefficient? Calculate the coefficients and see if their signs correspond to your intuition. (b)Find the regression ,and interpret it.What other factors can you think of that might have an influence on productivity?

(Requires some Calculus)Consider the sample regression function . .Take the total derivative.Next show that the partial derivative is obtained by holding constant,or controlling for .

Consider the multiple regression model with two regressors X1 and X2,where both variables are determinants of the dependent variable.When omitting X2 from the regression,then there will be omitted variable bias for

(Requires Calculus)In the multiple regression model you estimate the effect on Yi of a unit change in one of the Xi while holding all other regressors constant.This

The administration of your university/college is thinking about implementing a policy of coed floors only in dormitories.Currently there are only single gender floors.One reason behind such a policy might be to generate an atmosphere of better "understanding" between the sexes.The Dean of Students (DoS)has decided to investigate if such a behavior results in more "togetherness" by attempting to find the determinants of the gender composition at the dinner table in your main dining hall,and in that of a neighboring university,which only allows for coed floors in their dorms.The survey includes 176 students,63 from your university/college,and 113 from a neighboring institution. (a)The Dean's first problem is how to define gender composition.To begin with,the survey excludes single persons' tables,since the study is to focus on group behavior.The Dean also eliminates sports teams from the analysis,since a large number of single-gender students will sit at the same table.Finally,the Dean decides to only analyze tables with three or more students,since she worries about "couples" distorting the results.The Dean finally settles for the following specification of the dependent variable: GenderComp= Where " " stands for absolute value of Z.The variable can take on values from zero to fifty.Briefly analyze some of the possible values.What are the implications for gender composition as more female students join a given number of males at the table? Why would you choose the absolute value here? Discuss some other possible specifications for the dependent variable. (b)After considering various explanatory variables,the Dean settles for an initial list of eight,and estimates the following relationship: = 30.90 - 3.78 × Size - 8.81 × DCoed + 2.28 × DFemme + 2.06 × DRoommate - 0.17 × DAthlete + 1.49 × DCons - 0.81 SAT + 1.74 × SibOther, =0.24,SER = 15.50 where Size is the number of persons at the table minus 3,DCoed is a binary variable,which takes on the value of 1 if you live on a coed floor,DFemme is a binary variable,which is 1 for females and zero otherwise,DRoommate is a binary variable which equals 1 if the person at the table has a roommate and is zero otherwise,DAthlete is a binary variable which is 1 if the person at the table is a member of an athletic varsity team,DCons is a variable which measures the political tendency of the person at the table on a seven-point scale,ranging from 1 being "liberal" to 7 being "conservative," SAT is the SAT score of the person at the table measured on a seven-point scale,ranging from 1 for the category "900-1000" to 7 for the category "1510 and above," and increasing by one for 100 point increases,and SibOther is the number of siblings from the opposite gender in the family the person at the table grew up with. Interpret the above equation carefully,justifying the inclusion of the explanatory variables along the way.Does it make sense to interpret the constant in the above regression? (c)Had the Dean used the number of people sitting at the table instead of Number-3,what effect would that have had on the above specification? (d)If you believe that going down the hallway and knocking on doors is one of the major determinants of who goes to eat with whom,then why would it not be a good idea to survey students at lunch tables?

Imagine you regressed earnings of individuals on a constant,a binary variable ("Male")which takes on the value 1 for males and is 0 otherwise,and another binary variable ("Female")which takes on the value 1 for females and is 0 otherwise.Because females typically earn less than males,you would expect

The Solow growth model suggests that countries with identical saving rates and population growth rates should converge to the same per capita income level.This result has been extended to include investment in human capital (education)as well as investment in physical capital.This hypothesis is referred to as the "conditional convergence hypothesis," since the convergence is dependent on countries obtaining the same values in the driving variables.To test the hypothesis,you collect data from the Penn World Tables on the average annual growth rate of GDP per worker (g6090)for the 1960-1990 sample period,and regress it on the (i)initial starting level of GDP per worker relative to the United States in 1960 (RelProd60), (ii)average population growth rate of the country (n), (iii)average investment share of GDP from 1960 to 1990 (SK - remember investment equals savings),and (iv)educational attainment in years for 1985 (Educ).The results for close to 100 countries is as follows: = 0.004 - 0.172 × n + 0.133 × SK + 0.002 × Educ - 0.044 × RelProd60, =0.537,SER = 0.011 (a)Interpret the results.Do the coefficients have the expected signs? Why does a negative coefficient on the initial level of per capita income indicate conditional convergence ("beta-convergence")? (b)Equations of the above type have been labeled "determinants of growth" equations in the literature.You recall from your intermediate macroeconomics course that growth in the Solow growth model is determined by technological progress.Yet the above equation does not contain technological progress.Is that inconsistent?

When you have an omitted variable problem,the assumption that E(ui Xi)= 0 is violated.This implies that

(Requires Appendix material)Consider the following population regression function model with two explanatory variables: .It is easy but tedious to show that SE( )is given by the following formula: .Sketch how SE( )increases with the correlation between and .

The OLS residuals in the multiple regression model