Exam 6: Linear Regression With Multiple Regressors

The sample regression line estimated by OLS

You have collected data on individuals and their attributes. Consequently you have generated several binary variables, which take on a value of "1" if the individual has that characteristic and are "0" otherwise. One example is the binary variable DMarr which is "1" for married individuals and "0" for non-married variables. If you run the following regression: ahe_i= β₀ + β₁×educ_i + β₂×DMarr_i + u_i a. What is the interpretation for β₂? b. You are interested in directly observing the effect that being non-married ("single")has on earnings, controlling for years of education. Instead of recording all observations such that they are "1" for a not married individual and "0" for a married person, how can you generate such a variable (DSingle)through a simple command in your regression program?

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

You have collected data for 104 countries to address the difficult questions of the determinants for differences in the standard of living among the countries of the world. You recall from your macroeconomics lectures that the neoclassical growth model suggests that output per worker (per capita income)levels are determined by, among others, the saving rate and population growth rate. To test the predictions of this growth model, you run the following regression: $\widehat { \text { RelPersinc } }$ = 0.339 - 12.894 × n + 1.397 × S_K, $R ^ { 2 }$ =0.621, SER = 0.177 where RelPersInc is GDP per worker relative to the United States, n is the average population growth rate, 1980-1990, and S_K is the average investment share of GDP from 1960 to 1990 (remember investment equals saving). (a)Interpret the results. Do the signs correspond to what you expected them to be? Explain. (b)You remember that human capital in addition to physical capital also plays a role in determining the standard of living of a country. You therefore collect additional data on the average educational attainment in years for 1985, and add this variable (Educ)to the above regression. This results in the modified regression output: $\widehat { \text { RelPersinc } }$ = 0.046 - 5.869 × n + 0.738 × S_K + 0.055 × Educ, $R ^ { 2 }$ =0.775, SER = 0.1377 How has the inclusion of Educ affected your previous results? (c)Upon checking the regression output, you realize that there are only 86 observations, since data for Educ is not available for all 104 countries in your sample. Do you have to modify some of your statements in (d)? (d)Brazil has the following values in your sample: RelPersInc = 0.30, n = 0.021, S_K = 0.169, Educ = 3.5. Does your equation overpredict or underpredict the relative GDP per worker? What would happen to this result if Brazil managed to double the average educational attainment?

Consider the following earnings function: ahe_i= β₀ + β₁×DFemme_i + β₂×educ_i+...+ u_i versus the alternative specification ahe_i= γ₀ × DMale + γ₁×DFemme_i + γ₂×educ_i+...+ u_i where ahe is average hourly earnings, DFemme is a binary variable which takes on the value of "1" if the individual is a female and is "0" otherwise, educ measures the years of education, and DMale is a binary variable which takes on the value of "1" if the individual is a male and is "0" otherwise. There may be additional explanatory variables in the equation. a. How do the βs and γs compare? Putting it differently, having estimated the coefficients in the first equation, can you derive the coefficients in the second equation without re-estimating the regression? b. Will the goodness of fit measures, such as the regression R², differ between the two equations? c. What is the reason why economists typically prefer the second specification over the first?

In the multiple regression model with two regressors, the formula for the slope of the first explanatory variable is $\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$ (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ). An alternative way to derive the OLS estimator is given through the following three step procedure. Step 1: regress Y on a constant and $X _ { 2 }$ , and calculate the residual (Res1). Step 2: regress $X _ { 1 }$ on a constant and $X _ { 2 }$ , 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.

In the multiple regression model with two explanatory variables $Y _ { i } = \mathcal { P } _ { 0 } + \mathcal { p } _ { 1 } X _ { 1 i } + \mathcal { \rho } _ { 2 } X _ { 2 i } + u _ { i }$ the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in z_i = Z_i - $\bar { Z }$ ): $\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 }$ $\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$ $\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 23 } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$ 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 ( $X _ { 1 i }$ )and the saving rate ( $X _ { 2 i }$ )(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: $\sum _ { i = 1 } ^ { n } Y _ { i }$ = 33.33; $\sum _ { i = 1 } ^ { n } X _ { 1 i }$ = 2.025; $\sum _ { i = 1 } ^ { n } X _ { 2 i }$ = 17.313 $\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 }$ = 8.3103; $\sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 }$ = .0122; $\sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 }$ = 0.6422 $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i }$ = -0.2304; $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i }$ = 1.5676; $\sum _ { i = 1 } ^ { n } x _ { 1 \mathrm { i } } x _ { 2 i }$ = -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 $R ^ { 2 }$ , and interpret it. What other factors can you think of that might have an influence on productivity?

You try to establish that there is a positive relationship between the use of a fertilizer and the growth of a certain plant. Set up the design of an experiment to establish the relationship, paying particular attention to relevant control variables. Discuss in this context the effect of omitted variable bias.

Your econometrics textbook stated that there will be omitted variable bias in the OLS estimator unless the included regressor, X, is uncorrelated with the omitted variable or the omitted variable is not a determinant of the dependent variable, Y. Give an intuitive explanation for these two conditions.

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

Imperfect multicollinearity

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

In the multiple regression model, the adjusted R², $\bar { R }$ ²

The probability limit of the OLS estimator in the case of omitted variables is given in your text by the following formula: $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \hat { \beta } _ { 1 } + \rho _ { \mathrm { X } } \frac { \sigma _ { u } } { \sigma _ { \mathrm { X } } }$ Give an intuitive explanation for two conditions under which the bias will be small.

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 (S_K - remember investment equals savings), and (iv)educational attainment in years for 1985 (Educ). The results for close to 100 countries is as follows: $\widehat { g 6090 }$ = 0.004 - 0.172 × n + 0.133 × S_K + 0.002 × Educ - 0.044 × RelProd₆₀, $R ^ { 2 }$ =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?

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= $\mid ( 50 \% - \% \text { of Male Students at Table } ) \mid$ Where " $| Z |$ " 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: $\widehat{\text { GenderComp }}$ = 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, $R ^ { 2 }$ =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?

You would like to find the effect of gender and marital status on earnings. As a result, you consider running the following regression: ahe_i= β₀ + β₁×DFemme_i + β₂×DMarr_i + β₃×DSingle_i + β₄×educ_i+...+ u_i Where ahe is average hourly earnings, DFemme is a binary variable which takes on the value of "1" if the individual is a female and is "0" otherwise, DMarr is a binary variable which takes on the value of "1" if the individual is married and is "0" otherwise, DSingle takes on the value of "1" if the individual is not married and is "0" otherwise. The regression program which you are using either returns a message that the equation cannot be estimated or drops one of the coefficients. Why do you think that is?

In the multiple regression model, the SER is given by

(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: $\sum _ { i = 1 } ^ { n } { \hat u_ { i } } = 0 ; \sum _ { i = 1 } ^ { n } \hat { u _ { i } } X _ { 1 i } = 0 ; \sum _ { i = 1 } ^ { n } \hat { u _ { i } } X _ { 2 i } = 0$

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