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Implementing Restricted Least Squaresin Linear
Models Dr. John P. Haisken-DeNewjhaiskendenew_at_r
wi-essen.de
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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 gt 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

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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
  lthds97.adogt- What does the syntax look like?

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2a. RLS lthds97.adogt - 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

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2b. RLS lthds97.adogt - 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)

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2c. RLS lthds97.adogt

• Output created by lthds97.adogt(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)

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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 lthds97.adogt to implement RLS

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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 gt 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 Pgtt 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 genderf
  gender. global hds97_2 raceb raceo racew
  race. global hds97_3 msmar msdiv
  mssep mssgl marital. hds97

description
Name of reference
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3a. Gender (2-Way)
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3b. Race (3-Way)
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3c. Marital Status (4-Way)
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3d. State of Residence (51-Way) RefHi
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3d. State of Residence (51-Way) RefLo
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3d. State of Residence (51-Way)
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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 lthds97.adogt- 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