Post on 06-Apr-2016
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1March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
„Implementing Restricted Least Squaresin Linear Models“
Dr. John P. Haisken-DeNewjhaiskendenew@rwi-essen.de
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2March 31, 2006 Haisken-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 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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3March 31, 2006 Haisken-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. 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?
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4March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
2a. RLS <hds97.ado> - 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
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5March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
2b. RLS <hds97.ado> - 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)]
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6March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
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)
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7March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
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
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8March 31, 2006 Haisken-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 = 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 genderf gender. global hds97_2 raceb raceo racew race. global hds97_3 msmar msdiv mssep mssgl marital
. hds97 descriptionName of reference
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9March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
3a. Gender (2-Way)
Gender Wage Differentials2-Way (SD=0.0867)
-0,20
-0,15
-0,10
-0,05
0,00
0,05
0,10
0,15
0,20
male female female male
Ref=Female Ref=Male Restricted Least Squares
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10March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
3b. Race (3-Way)
Race Wage Differentials3-Way (SD=0.0205)
whiteother
black
other
white
black
white
black
other
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Ref=Black Ref=White Ref=Other Restricted Least Squares
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11March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
3c. Marital Status (4-Way)
Marital Status Wage Differentials4-Way (SD=0.0609)
married
divorced
separated
divorced
separatedsingle
married
separatedsingle
divorced
separatedsingle
divorced
separatedsingle
married
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Ref=Single Ref=Married Ref=Divorced Ref=Seprated R-L-S
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12March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
3d. State of Residence (51-Way) Ref=Hi
State Wage Differential51-Way (Reference=Alaska)
AL
AR
AZCA
CO
CT
DCDE
FLGAHI
IA ID
IL
INKS
KY
LA
MA
MDMEMI
MN
MO
MS
MTNC
NDNE
NHNJ
NM
NVNY
OH
OKOR
PARISCSD
TNTXUT
VAVT
WAWI
WV
WY
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
American States (Ordinary Least Squares)
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13March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
3d. State of Residence (51-Way) Ref=Lo
State Wage Differential51-Way (Reference=Arkansas)
ALAZCA
CO
CT
DCDE
FLGAHI
IA ID
IL
INKS
KY
LA
MA
MDMEMI
MN
MO
MS
MTNC
NDNE
NHNJ
NM
NVNYOH
OKOR
PARISCSD
TNTXUT
VAVT
WAWI
WV
WY
AK
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
American States (Ordinary Least Squares)
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14March 31, 2006 Haisken-DeNew / Stata 2006 Mannheim
3d. State of Residence (51-Way)
State Wage Differentials51-Way (SD=0.0684)
AL
AK
AR
AZCA
CO
CT
DCDE
FLGAHI
IA ID
IL
INKS
KY
LA
MA
MDMEMI
MN
MO
MS
MTNC
NDNE
NHNJ
NM
NVNYOH
OKOR
PARISCSD
TNTXUT
VAVT
WAWI
WV
WY
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
American States (Restricted Least Squares)
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15March 31, 2006 Haisken-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 <hds97.ado>- 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…