Testing for Discrimination

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1 Testing for Discrimination Spring 2010 Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Relevant Readings BFW Appendix 7A (pgs ) Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

2 Testing for Discrimination For each theory or model of discrimination, I will first present the basic concepts and then provide empirical studies that test the hypothesis of the model/theory. Two main ways to test for discrimination: 1 2 Audit Studies Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Technique used to analyze data consisting of: Dependent variable or response variable One or more independent variables or explanatory variable Dependent variable is modeled as function of: 1 Independent variables 2 Fixed coefficients known as parameters 3 Error term that captures unexplained variation and treated as random Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

3 cont. Parameter values estimated to provide the best fit to the data Most common method is least squares: Parameter values chosen to minimize squared difference between true and fitted values summed over all observations Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Univariate Analysis Univariate Analysis Simplest form of regression analysis One independent variable Assuming linear relationship, the general equation can be written as: W i = α + βx i + ɛ i where W i is the dependent variable value for person i X i is the independent variable value for person i α and β are parameter values ɛ i is the random error term Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

4 Univariate Analysis Univariate Analysis W i = α + βx i + ɛ i α is called the intercept or the (expected) value of W when X = 0 β is called the slope or the change in W when X increases by one Since there is only one independent variable, regression analysis estimates the parameters that provide the best fitting line when the dependent variable is graphed on the vertical axis and independent variable is on the horizontal axis. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Univariate Analysis Univariate Example Suppose we have the following data on wages and education levels: Individual Wage Education A 8 11 B 9 11 C D E 9 13 F G H I J K L Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

5 Univariate Analysis Univariate Example II We want to estimate the relationship between wages and education. Using the data above, we can estimate the following equation Wage i = α + βeducation i + ɛ i Goal: Find estimates ˆα and ˆβ for α and β that minimize the squared difference between the true and fitted value for all individuals or minimize Σ i (W i Ŵ i ) 2 Σ i (W i ˆα ˆβEducation i ) 2 where Ŵ i is the fitted wage value for person i. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Univariate Analysis Univariate Example III Graph the data with wage on the vertical axis and education on the horizontal axis. The parameters, ˆα and ˆβ, are the values that provide the best fitting line given the data. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

6 Univariate Analysis Univariate Example IV In this case, the best fitting line slopes upward implying a positive relationship between education and wages. The best fitting line does not need to go through all or even any of the data points. The best fitted line graphed on the previous slide corresponds to the following equation: Ŵ i = Ed i (1.26)(0.09) Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Univariate Analysis Univariate Example V The estimate of the intercept, ˆα, is 3 implying that a person with no education has an estimated wage of $3 per hour. The estimate of the slope parameter, ˆβ, is 0.50 implying an additional year of education corresponds to an estimated hourly wage increase of $0.50. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

7 Multivariate Analysis Multivariate Analysis Even though univariate analysis is useful to demonstrate basic regression analysis, economic theory typically suggests there exists multiple factors that influence the variable of interest. Other factors that may influence wage include experience, age, location, job type, sex, race, etc. Economic theory is used to decided which variables should be included and excluded from the model and also suggests the direction of causation. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Multivariate Analysis Multivariate Analysis Suppose we want to model wages Economic theory suggests we include education, experience, sex and race into the model Suppose the available data set includes only whites and blacks. Previous economic research suggests that wages will be increasing in education, increasing in experience, higher for males, and higher for whites. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

8 Multivariate Analysis Multivariate Analysis We can estimate the following equation, W i = α + β 1 Ed i + β 2 Exp i + β 3 Male i + β 4 White i + ɛ i where Male i and White i are dummy variables that take the value of 1 if the person has the characteristic and 0 otherwise. Parameters (α, β 1, β 2, β 3, β 4 ) can be estimated using multiple regression analysis. Multiple regression analysis is analogous to the univariate analysis Estimated parameter values are those that construct the best straight-line relationship between the dependent variable and the SET of independent variables. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Multivariate Analysis Handout 3: Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

9 Common Transformations in : Level-Level Log-Log Log-Level Level-Log Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Level-Level Both dependent and independent variable of interest are in level form. For example: W i = α + β 1 Ed i + β 2 Age i + β 3 Exp i + ɛ i Goal: Interpret coefficients (α, β s) Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

10 Level-Level In this case (level-level), the coefficient on the variable of interest can be interpreted as the marginal effect. The marginal effect is the effect on the dependent variable from a one-unit change or state change for dummy variables in the corresponding independent variable, holding all other independent variables constant. W i Ed i = Change in wage from a one year increase in education = β 1 Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Ceteris Paribus A common phrase used in economics is ceteris paribus, which is Latin for with other things the same. For example: ˆβ 1 is the estimated effect on hourly wage of an additional year of eduction, ceteris paribus. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

11 Level-Level Example Suppose the fitted equation is Ŵ i = Ed i Age i Exp i + ɛ i Based on the data used for this regression: An additional year of education corresponds to an increase in hourly wages of $0.75, holding all other variable constant. Ceteris paribus, an additional year of experience is associated with a $0.30 per hour wage increase. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Log-Log Dependent and independent variable of interest are both in log form. Coefficients can no longer be interpreted as marginal effects. Example: Suppose economic theory suggests estimation of our wage equation with the dependent variable in log form and inclusion of the independent variable hours of community volunteer per week (Comm) also in log form. log(w i ) = α + β 1 Ed i + β 2 Age i + β 3 Exp i + β 4 log(comm i ) + ɛ i We would like to interpret the coefficient on the community volunteer variable (β 4 ). Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

12 Log-Log Consider taking the differential of the equation holding all independent variables constant except Comm i. since d[log(x )] = 1 X d(x ) d[log(w i )] = d[log(comm i )]β dw i = dcomm i β 4 W i Comm i Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Log-Log Rearranging the last equation, 100 dw i W i 100 dcomm = β 4 i Comm i where the left hand side is the (partial) elasticity of W with respect to Comm. Elasticity is the ratio of the percent change in one variable to the percent change in another variable. The coefficient in a regression is a partial elasticity since all other variables in the equation are held constant. β 4 can be interpreted as the percent change in hourly wages from a one percent increase in community volunteer hours per week holding education, age, and experience constant. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

13 Log-Log Example Consider the following fitted equation: ˆ log(w i ) = Ed i Age i Exp i log(comm i ) A one percent increase in community volunteer hours per week is associated with a 1.2% increase in hourly wages. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Log-Level Dependent variable in log form and independent variable of interest in level form. Consider the following wage equation: log(w i ) = α + β 1 Ed i + β 2 Age i + β 3 Exp i + β 4 log(comm i ) + ɛ i We want to interpret the coefficients on education (β 1 ), age (β 2 ), and experience (β 3 ). Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

14 Log-Level Take the differential holding all other independent variables constant. d[log W i ] = ded i β 1 dw i W i = ded i β 1 Multiply both sides by 100 and rearrange, 100dW i W i = 100dEd i β β 1 = 100dW i W i ded i = % W i unit Ed i Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Log-Level Since, 100 β 1 = 100dW i W i ded i = % W i unit Ed i we can interpret 100 β 1 as the percentage change in W i from a unit increase in Ed i, holding all other independent variables constant. Similar derivations can derive the interpretation for the coefficients on age and experience. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

15 Log-Level Example Consider the following fitted equation: ˆ log(w i ) = Ed i Age i Exp i log(comm i ) Holding all other independent variables constant, an additional year of schooling is associated with a 24% increase in hourly wages. An additional year of experience is associated with a 16% increase in hourly wages, ceteris paribus. What is the interpretation for the coefficient on Age? Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Level-Log Dependent variable in level form and the independent variable of interest in log form. For example, consider the following equation: W i = α + β 1 Ed i + β 2 Age i + β 3 Exp i + β 4 log(comm i ) + ɛ i We would like to interpret the coefficient on community volunteer hours (β 4 ). Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

16 Level-Log Take the differential on both sides, holding all independent variables constant except community volunteer hours: dw i = β 4 d[log(comm i )] dw i = β 4 1 Comm i dcomm i Divide both sides by 100 and rearrange, β = dw i 100dComm i Comm i = unit W i % Comm i β can be interpreted as the increase in hourly wages from a one percent increase in community volunteer hours per week. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Level-Log Example Consider the following fitted equation: Ŵ i = Ed i Age i Exp i log(comm i ) Holding education, age, and experience constant, a one percent increase in community volunteer hours per week is associated with a $0.132 increase in hourly wages. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

17 Statistical Significance Statistical Significance Estimated parameter values are only estimates of the true relationship Uncertainty associated with each estimated parameter measured by its standard error or the estimated standard deviation of the coefficient Standard errors are commonly reported in the parentheses below the parameter estimate. Ŵ i = Ed i (1.26)(0.09) Standard errors useful to test hypotheses about estimated parameter values Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Statistical Significance Testing Hypothesis We can test the hypothesis that the slope coefficient is positive against the hypothesis that the coefficient is zero (i.e. no relationship between education and wages). Convenient way to test whether a parameter estimate is different from zero is to calculate the t-statistic. The t-statistic is the absolute value of the parameter estimate divided by the standard error of the parameter estimate t-statistic = Coefficient Standard Error Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

18 Statistical Significance Testing Hypothesis Rule of thumb: if the t-statistic is greater than 2, we can reject the hypothesis that the parameter estimate is equal to zero. When this holds, we say that the estimated parameter is statistically different than zero or statistically significant. In the previous example, the t-statistic for for the coefficient on ED is = 5.4 > We are very confident that the true relationship between education and wages is positive. Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Statistical Significance Handout 4: and Statistical Significance Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

19 Statistical Significance Testing for Discrimination Regress some measure of consumption outcome (ex: wages) on group membership and other controls Determine whether the coefficient on group membership is significant Wage = β X + α Group + ɛ X includes variables such as human capital measures, location measures, etc. Evaluate sign and significance of α Issues: Indirect measure of discrimination and bias Trying to isolate impacts of discrimination without directly observing discriminatory behavior Excluded variables Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Audit Studies Audit Studies Use equally qualified individuals from two different groups in real-world situations Examples: Real-estate, car sales, job interviews, etc. Evaluate if treatment different Important to make pairs as similar as possible Benefit: If set up correctly, can minimize biases Issues: Need to interpret results with care Not necessarily result faced by average member of group Cannot do audit study in all steps in transaction Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

20 Economic and Math Symbols Frequently Used Economic and Math Symbols Symbol π f (x) Σ Π C MP i U MRS Economic/Math Term Profit Production Function Sum Product Cost Partial Change Marginal Product of i Utility Marginal Rate of Substitution Alicia Rosburg (ISU) Testing for Discrimination Spring / 40 Economic and Math Symbols Readings for next lecture(s) Articles: Arrow (1971) Black and Brainerd (2004) Bertrand and Mullainathan (2004) Background: BFW, Chapter 7 ( ) ES, Chapter 12 Lang, Chapter 10 Alicia Rosburg (ISU) Testing for Discrimination Spring / 40

Research Methods in Political Science I

Research Methods in Political Science I 6. Linear Regression (1) Yuki Yanai School of Law and Graduate School of Law November 11, 2015 1 / 25 Today s Menu 1 Introduction What Is Linear Regression? Some