Empirical Methods for Microeconomic Applications

William GreeneDepartment of EconomicsStern School of Business

Lab 2. Binary Choice

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Model Commands Generic form:

Model name ; Lhs = dependent variable ; Rhs = independent variables $

Almost all models require ;Lhs and ;Rhs. Rhs should generally include ONE to request a

constant term. Models have different other required specifications Many optional specifications.

Probit Model Command

Text Editor

Load healthcare.lpj Command Builder

Command Builder

Go button in command builder

Partial Effects for Interactions2

1 2 3 4

1 3 4

2 4

Prob[ 1| ] [ ] [ ]Partial Effects?

[ ]( 2 )

[ ]( )

Compute without extensive additional computation ofe

y x x z x xzA

P A x zxP A xz

xtra variables, etc.

Partial Effects• Build the interactions into the model

statement PROBIT ; Lhs = Doctor

; Rhs = one,age,educ,age^2,age*educ $

• Built in computation for partial effects PARTIALS ; Effects:

Age & Educ = 8(2)20 ; Plot(ci) $

Average Partial Effects---------------------------------------------------------------------Partial Effects Analysis for Probit Probability Function---------------------------------------------------------------------Partial effects on function with respect to AGEPartial effects are computed by average over sample observationsPartial effects for continuous variable by differentiationPartial effect is computed as derivative = df(.)/dx---------------------------------------------------------------------df/dAGE Partial Standard(Delta method) Effect Error |t| 95% Confidence Interval---------------------------------------------------------------------Partial effect .00441 .00059 7.47 .00325 .00557EDUC = 8.00 .00485 .00101 4.80 .00287 .00683EDUC = 10.00 .00463 .00068 6.80 .00329 .00596EDUC = 12.00 .00439 .00061 7.18 .00319 .00558EDUC = 14.00 .00412 .00091 4.53 .00234 .00591EDUC = 16.00 .00384 .00138 2.78 .00113 .00655EDUC = 18.00 .00354 .00192 1.84 -.00023 .00731EDUC = 20.00 .00322 .00250 1.29 -.00168 .00813

Useful Plot

More Elaborate Partial Effects

• PROBIT ; Lhs = Doctor ; Rhs = one,age,educ,age^2,age*educ, female,female*educ,income $

• PARTIAL ; Effects: income @ female = 0,1 ? Do for each subsample | educ = 12,16,20 ? Set 3 fixed values & age = 20(10)50 ? APE for each setting

Constructed Partial Effects

Predictions List and keep predictions Add ; List ; Prob = PFIT to the probit or logit command (Tip: Do not use ;LIST with large samples!)

Sample ; 1-100 $PROBIT ; Lhs=doctor ; Rhs=… ; List ; Prob=Pfit $DSTAT ; Rhs = Doctor,PFIT $

Testing Restrictions

Testing a Hypothesis – LM Test

PROBIT ; LHS = doctor ; RHS = one,age,educ $PROBIT ; LHS = doctor ; RHS = one,age,educ,female,married ; Start = b,0,0 ; MAXIT = 0 $

ˆ ˆ

ˆ

ˆ

0 0 -1 0

0

0

LM = ( ) [Est.Hessian ] ( )

=MLE with restrictions imposed

Hessian is computed at .

Lagrange Multiplier Test

g β g β

β

β

Results of an LM test

Note: LM equaled 485.27.

Wald Test

Wald Test Using Matrix Algebra

Note: LM equaled 485.27.

Likelihood Ratio Test

LR = 2[LogL(unrestricted) -Logl(restricted)]

Note: LM equaled 485.27. Wald equaled 484.55

Using Matrix AlgebraNamelists with the current sample serve 2 major functions:

(1) Define lists of variables for model estimation (2) Define the columns of matrices built from the data.

NAMELIST ; X = a list ; Z = a list … $

Set the sample any way you like. Observations are now the rows of all matrices. When the sample changes, the matrices change.

Lists may be anything, may contain ONE, may overlap (some or all variables) and may contain the same variable(s) more than once

Matrix FunctionsMatrix Product: MATRIX ; XZ = X’Z $Moments and Inverse MATRIX ; XPX = X’X

; InvXPX = <X’X> $Moments with individual specific weights in variable w. Σi wi xixi’ = X’[w]X. [Σi wi xixi’ ]-1 = <X’[w]X>Unweighted Sum of Rows in a Matrix Σi xi = 1’XColumn of Sample Means (1/n) Σi xi = 1/n * X’1 or MEAN(X) (Matrix function. There are over 100 others.)Weighted Sum of rows in matrix Σi wi xi = 1’[w]X

LM Test for Normality in Probit

i i

i i i i

i i i3 i42

i3 i i4 i

Testing for normality in the probit model: RHS variables. y = LHS variable

Prob[y 1| ] ( ), Normal CDF. ( ) density[ ,z ,z ],

z -(1/3)[( ) 1], z (1/4){( )[3

xx βx βx

z xβx βx

2i

ii i i i

i i

1n n n 2i i i i i i i ii=1 i=1 i=1

( ) ]}( ) e y ( ), d

( )[1 ( )]Lagrange Multiplier Statistic. ̂= compute at MLE of

ˆ ˆ ˆˆ ˆˆ ˆ ˆ LM= (ed) d (ed)

βxβxx

βx βxβ

z ' z zThanks to Joachim Wilde, Univ. Halle, Germany for suggesting this.

Normality Test for ProbitNAMELIST ; XI = One,... $CREATE ; yi = the dependent variable $PROBIT ; Lhs = yi ; Rhs = Xi ; Prob = Pfi $CREATE ; bxi = b'Xi ; fi = N01(bxi) $CREATE ; zi3 = -1/2*(bxi^2 - 1) ; zi4 = 1/4*(bxi*(bxi^2+3)) $NAMELIST ; Zi = Xi,zi3,zi4 $CREATE ; di = fi/sqr(pfi*(1-pfi)) ; ei = yi - pfi

; eidi = ei*di ; di2 = di*di $MATRIX ; List ; LM = 1'[eidi]Zi * <ZI'[di2]Zi> * Zi'[eidi]1 $

Endogenous Variable in Probit ModelPROBIT ; Lhs = y1, y2 ; Rh1 = rhs for the probit model,y2 ; Rh2 = exogenous variables for y2 $

SAMPLE ; All $CREATE ; GoodHlth = Hsat > 5 $PROBIT ; Lhs = GoodHlth,Hhninc ; Rh1 = One,Female,Hhninc ; Rh2 = One,Age,Educ $

Binary Choice Models with Panel Data

Global Setting for Panels

SETPANEL ; Group = the name of the ID variable ; PDS = the name of the groupsize variable to create $

Subsequent model commands state ;PANEL with no other specifications requred to set the panel.Some other specifications usually required for thespecific model – e.g., fixed vs. random effects.

Panel Data

Unbalanced Panel Data Set

Examine Distribution of Group Sizes

Sample ; all$Setpanel ; Group = id ; Pds = ti $

Create ; t = ndx(id,1) $Histogram ; if[t=1] ; rhs = ti $

Group Sizes

Cluster CorrectionPROBIT ; Lhs = doctor ; Rhs = one,age,female,educ,married,working ; Cluster = ID $

Fixed Effects Models

? Fixed Effects Probit. ? Looks like an incidental parameters problem.Namelist ; X = age,educ,married $Probit ; Lhs = doctor; Rhs = X,one ; FEM ; Marginal ; Panel $Probit ; Lhs = doctor ; Rhs = X,one ; Marginal $

Computed Fixed Effects Parameters

Logit Fixed Effects Models Conditional and Unconditional FE

? Logit, conditional vs. unconditionalLogit ; Lhs = doctor ; Rhs = X ; Panel $ (Conditional)Logit ; Lhs = doctor ; Rhs = X ; Panel ; Fixed $

A Hausman Test for Fixed Effects

Random Effects and Random Constant

Bivariate Probit ModelTwo equation modelGeneral usage of

• LHS = the set of dependent variables• RH1 = one set of independent variables• RH2 = a second set of variables

Economical use of namelists is useful here

Namelist ; x1=one,age,female,educ,married,working $Namelist ; x2=one,age,female,hhninc,hhkids $BivariateProbit ;lhs=doctor,hospital

;rh1=x1 ;rh2=x2;marginal effects $

BivariateProbit ; lhs=doctor,hospital ; rh1=one,age,female,educ,married,working ; rh2=one,age,female,hhninc,hhkids ; rst = beta1,beta2,beta3,be,bm,bw, beta1,beta2,beta3,bi,bk, 0.4 $

--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Index equation for DOCTORConstant| -1.69181*** .08938 -18.928 .0000 AGE| .01244*** .00167 7.440 .0000 44.3352 FEMALE| .38543*** .03157 12.209 .0000 .42277 EDUC| .08144*** .00457 17.834 .0000 10.9409 MARRIED| .42021*** .03987 10.541 .0000 .84539 WORKING| .03310 .03910 .847 .3972 .73941 |Index equation for HOSPITALConstant| -1.69181*** .08938 -18.928 .0000 AGE| .01244*** .00167 7.440 .0000 44.3352 FEMALE| .38543*** .03157 12.209 .0000 .42277 HHNINC| -.98617*** .08917 -11.060 .0000 .34930 HHKIDS| -.09406** .04600 -2.045 .0409 .45482 |Disturbance correlationRHO(1,2)| .40000 ......(Fixed Parameter)......--------+-------------------------------------------------------------

Endogenous Variable in Probit Model“Instrumental Variable Probit”FIML Estimator

PROBIT ; Lhs = y1, y2 ; Rh1 = rhs for the probit model,y2 ; Rh2 = exogenous variables for y2 $

PROBIT ; Lhs = healthy,income ; Rh1 = one,age,educ,female,income ; Rh2 = one,age,educ,married $