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„Implementing Restricted Least Squares in Linear Models“

„Implementing Restricted Least Squares in Linear Models“ Dr. John P. Haisken-DeNew jhaiskendenew@rwi-essen.de. 1a. Background.

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„Implementing Restricted Least Squares in Linear Models“

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  1. „Implementing Restricted Least Squaresin Linear Models“ Dr. John P. Haisken-DeNewjhaiskendenew@rwi-essen.de Haisken-DeNew / Stata 2006 Mannheim

  2. 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 PointKrueger/Summers (1988) „Efficiency Wages and the Inter-Industry Wage Structure“, Econometrica, 56, p 259-93.- 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 Haisken-DeNew / Stata 2006 Mannheim

  3. 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. 517-21.- 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 <hds97.ado>- What does the syntax look like? Haisken-DeNew / Stata 2006 Mannheim

  4. 2a. RLS <hds97.ado> - One Dummy Set • Run a linear regressionreg/xtreg depvar indepvars • Standard Syntax (only ONE dummy set)hds97 indepvars [, options] options description refname( string ) a string containing the name of the "reference" categoryrealname( string ) a string containing a descriptive name for the set of dummy variablesweight( varname ) a string containing the name of the weighting variable Haisken-DeNew / Stata 2006 Mannheim

  5. 2b. RLS <hds97.ado> - Many Dummy Sets • Run a linear regressionreg/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_Xglobal hds97_2 Zvar_1 Zvar_2 Zvar_ref descriptive_name_for_Zglobal 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)] Haisken-DeNew / Stata 2006 Mannheim

  6. 2c. RLS <hds97.ado> • Output created by <hds97.ado>(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) Haisken-DeNew / Stata 2006 Mannheim

  7. 3. Illustrative Example (in Handout) • American Current Population Survey (CPS)- Use freely available January 2004 CPS sample- http://www.nber.org/morg/annual/morg04.dta • 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 <hds97.ado> to implement RLS Haisken-DeNew / Stata 2006 Mannheim

  8. 3. Sample Regression Output (in Handout) • . regress lhw age genderm raceb raceo msmar msdiv mssep Source | SS df MS Number of obs = 8417-------------+------------------------------ F( 7, 8409) = 181.36 Model | 242.712792 7 34.673256 Prob > F = 0.0000 Residual | 1607.68867 8409 .191186665 R-squared = 0.1312-------------+------------------------------ Adj R-squared = 0.1304 Total | 1850.40146 8416 .219867093 Root MSE = .43725------------------------------------------------------------------------------ lhw | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | .00861 .0004585 18.78 0.000 .0077112 .0095088 genderm | .1737988 .0095849 18.13 0.000 .1550101 .1925876 raceb | -.0730053 .0162526 -4.49 0.000 -.1048645 -.0411462 raceo | -.0131488 .0193254 -0.68 0.496 -.0510315 .0247338 msmar | .1365145 .0125807 10.85 0.000 .1118532 .1611758 msdiv | .1014927 .0180303 5.63 0.000 .0661489 .1368365 mssep | .0237369 .0341694 0.69 0.487 -.0432435 .0907174 _cons | 6.5783 .016593 396.45 0.000 6.545774 6.610826------------------------------------------------------------------------------ • . global hds97_1 genderm genderfgender. global hds97_2 raceb raceo racewrace. global hds97_3 msmar msdiv mssep mssglmarital. hds97 Name of reference description Haisken-DeNew / Stata 2006 Mannheim

  9. 3a. Gender (2-Way) Haisken-DeNew / Stata 2006 Mannheim

  10. 3b. Race (3-Way) Haisken-DeNew / Stata 2006 Mannheim

  11. 3c. Marital Status (4-Way) Haisken-DeNew / Stata 2006 Mannheim

  12. 3d. State of Residence (51-Way) Ref=Hi Haisken-DeNew / Stata 2006 Mannheim

  13. 3d. State of Residence (51-Way) Ref=Lo Haisken-DeNew / Stata 2006 Mannheim

  14. 3d. State of Residence (51-Way) Haisken-DeNew / Stata 2006 Mannheim

  15. 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 <hds97.ado>- Can be used afterregress 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… Haisken-DeNew / Stata 2006 Mannheim

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