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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…