Rheinisch-Westfälisches Institut für Wirtschaftsforschung 1March 31, 2006Haisken-DeNew / Stata 2006 Mannheim „Implementing Restricted Least Squares in Linear Models“ Dr. John P. Haisken-DeNew

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 2March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 1a. Background  Inter-Industry Wage Differentials - Why do secretaries in the steel industry make more money than otherwise observably identical secretaries in the services industry? - Calculating „wage differentials“: Wages in steel > services ? - Dummy Variables: 0 or 1  Starting Point Krueger/Summers (1988) „Efficiency Wages and the Inter-Industry Wage Structure“, Econometrica, 56, p Would like to interpret differentials as deviations from a weighted average - Remove arbitrary selection of reference category - Excellent seminal paper, however technical problems … - Attempt to implement Restricted Least Squares (RLS) but.. - Incorrect standard errors: t-values systematically biased downward - Incorrect overall inference: Variation systematically biased downward

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 3March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 1b. Background  Technical Contribution (in Handout) Haisken-DeNew/Schmidt (1997) „Inter-Industry and Inter-Regional Differentials: Mechanics and Interpretation“, Review of Economics and Statistics, 79(3), p How to implement Restricted Least Squares (RLS) correctly - How to implement RLS after any linear model (OLS, FE, RE…) - RLS was implemented in GAUSS, LIMDEP and Stata (crudely)  Now RLS is implemented in Stata in a flexible Ado - What does the syntax look like?

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 4March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 2a. RLS - One Dummy Set  Run a linear regression reg/xtreg depvar indepvars  Standard Syntax (only ONE dummy set) hds97 indepvars [, options] options description refname( string ) a string containing the name of the "reference" category realname( string ) a string containing a descriptive name for the set of dummy variables weight( varname ) a string containing the name of the weighting variable

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 5March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 2b. RLS - Many Dummy Sets  Run a linear regression reg/xtreg depvar x* Xvar_1 Zvar_1 Zvar_2 Dvar_* XXLvar_*  Advanced Syntax (MANY dummy variable sets) global hds97_1 Xvar_1 Xvar_ref descriptive_name_for_X global hds97_2 Zvar_1 Zvar_2 Zvar_ref descriptive_name_for_Z global hds97_3 Dvar_* Dvar_ref descriptive_name_for_D... global hds97_50 XXLvar_* XXLvar_ref descriptive_name_for_XXL (up to 50 globals/constraints can be set) Xvar_1 is a regressor used in regress or xtreg previously Xvar_ref is a text name for the reference category descriptive_name is a descriptive text name of the dummy set hds97 [, weight(wgt_var_name)]

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 6March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 2c. RLS  Output created by (A) Original Regression (OLS, RE, FE etc) repeated (B) Each Dummy Variable Group using RLS is calculated - From “k-1” Dummy Variables: “k” Coefficients reported (C) Weighted Standard Deviation (Sampling Corrected) of RLS Betas - Measure of overall variation (D) F-Tests of Joint Significance - Are the dummy variables as a group significant (E) Sample Shares of each Dummy - What were the sample shares used to create the weighted average - From the weighted average, the deviations are calculated (see B)

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 7March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3. Illustrative Example (in Handout)  American Current Population Survey (CPS) - Use freely available January 2004 CPS sample -  Run simple wage regression (age 18-65) - log hourly wages = f (age, gender, race, marital status, state)  Dummy Indicators - gender: male, female - race: white, black, other - marital status: married, divorced, separated, single - states: AK, AL… WY  Selecting arbitrary dummy variable as reference - Which one? Makes no difference in the calculation, just in interpretation  With RLS, interpret the dummy variables as deviations from a weighted average as opposed to an arbitrary reference category  If logged wages, then interpretation: %-point deviations from average  Use to implement RLS

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 8March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3. Sample Regression Output (in Handout) . regress lhw age genderm raceb raceo msmar msdiv mssep Source | SS df MS Number of obs = F( 7, 8409) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = lhw | Coef. Std. Err. t P>|t| [95% Conf. Interval] age | genderm | raceb | raceo | msmar | msdiv | mssep | _cons | . global hds97_1 genderm genderf gender. global hds97_2 raceb raceo racew race. global hds97_3 msmar msdiv mssep mssgl marital. hds97 descriptionName of reference

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 9March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3a. Gender (2-Way)

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 10March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3b. Race (3-Way)

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 11March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3c. Marital Status (4-Way)

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 12March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3d. State of Residence (51-Way) Ref=Hi

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 13March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3d. State of Residence (51-Way) Ref=Lo

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 14March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 3d. State of Residence (51-Way)

Rheinisch-Westfälisches Institut für Wirtschaftsforschung 15March 31, 2006Haisken-DeNew / Stata 2006 Mannheim 4. Conclusions  RLS: Interpretation of Dummy Variables - Even with a small dimension, RLS intuitive interpretation - Remove arbitrariness of reference category - Allow for importance weighting of each category  Easily Implemented with - Can be used after regress or xtreg and coefficients calculated - Useful additional statistics calculated  Flexible use - Transform a single set of dummy variables - Transform up to 50 sets of dummy variables at once  Areas of Application - Wage Differentials by: Region, Industry, Occupation, Education, Marital Status, Race, etc…