SECTION 10.

QUALITATIVE DEPENDENT VARIABLE ESTIMATION

GQOPT contains several built-in function subroutines that permit maximization of the likelihood functions for several qualitative dependent variable models. For those described in Sections 10.1, 10.1.1, 10.2, 10.2.1, 10.3 and 10.3.1 the MAIN program should contain 

      COMMON/USERP1/DATA(n,k+1) 
      COMMON/USERP2/NDIM,KDIM

where

DATA = array containing observations on the variables by columns, with column 1 normally being reserved for the dependent variable
n = the actual number of observations
k = the actual number of independent variables stored (normally = NP)
NDIM = a variable set equal to n
KDIM = a variable set equal to k

SECTION 10.1 PROBIT ANALYSIS

See Amemiya, T., "Qualitative Response Models," Journal of Economic Literature, XIX (1981), 1483-1536.

1. The model is the following:

            DATAi1 = 1 if X1DATAi2 + ... +XNPDATAi,NP+1 + Ui > 0 
            DATAi1 = 0 otherwise

where Ui is assumed distributed iid as N(0,1). Estimation is by maximum likelihood. The MAIN program should contain, in addition to the standard cards (see Section 1),the followin:

      EXTERNAL PROB1A 
      COMMON/BPROB/IPROB,JPROB 
      NDIM=n 
      KDIM=k
       . . .
      IPROB=1
      CALL OPT(X,NP,F,METHOD,ITERL,MAX,IER,ACC,PROB1A,ALABEL)

where PROB1A is the name of the built-in loglikelihood function. Since this is a maximization problem, set MAX=1. If the probit equation contains a constant term, as is normally the case, the column DATA(I,2) would be filled by the user by the number 1.0 and X(1) would represent the constant term. If there is no constant term in the equation, the user should set JPROB=0 prior to calling OPT. If the user wishes to have first, or first and second partial derivatives evaluated analytically, he should include the following subroutines... 

      SUBROUTINE FP(X,NP,F,FPD,FUNC) 
      IMPLICIT REAL*8 (A-H,O-Z) 
      DIMENSION X(NP),FPD(NP) 
      EXTERNAL FUNC 
      CALL PRFP1A(X,NP,F,FPD,FUNC) 
      RETURN 
      END 
      SUBROUTINE SP(X,NP,F,FPD,SPD,FUNC) 
      IMPLICIT REAL*8 (A-H,O-Z) 
      DIMENSION X(NP),FPD(NP),SPD(NP,NP) 
      EXTERNAL FUNC 
      CALL PRSP1A(X,NP,F,FPD,SPD,FUNC) 
      RETURN 
      END

2. Further options.

      COMMON/BPROB1/WP,SSR,ER2,WSSR,SCC,FAD,JFLG,IFLG,KFLG,LFLG

where

WP = Fraction of wrong predictions
SSR = Sum of squared residuals
ER2 = Efron's R-squared
WSSR = Weighted sum of squared residuals
SCC = Squared correlation coefficient
FAD = McFadden's R-squared
JFLG = 1 if ER2 not computable, 0 otherwise
IFLG = 1 if WSSR not computable, 0 otherwise
KFLG = 1 if SCC not computable, 0 otherwise
LFLG = 1 if FAD not computable, 0 otherwise

If the user wishes to have these summary statistics computed and printed, he should insert in the MAIN program 

      CALL OPTOU3(X,NP,FU0,METHOD)

Warning: OPTOU3 must be called after OPTOUT or OPTMOV (see Section 1.2) because it alters some of the arrays used by the latter two subroutines.

Note: An optimization METHOD (e.g. GRADX) must be specified in the call to OPTOU3 because it must solve an optimization problem to compute McFadden's R-squared.

3. Several probit models. At times, the user may wish to estimate several probit models, using subsets of the same basic DATA matrix. Section 1.11 is relevant. The number of columns in the DATA matrix may be much larger than the number of parameters to be estimated. NP will still be set to k, but the effective number of parameters to be estimated, NPE, will be smaller. (Presumably, some parameters will be set = 0 and "permed out".The user must set the array IPRM (as described in Section 1.11) so as to permute the variables with respect to which optimization is to be per- formed into the first NPE positions. Thus, if DATA is dimen- sioned to have 6 columns and the user wishes to estimate the probit equation 

                 DATAi1 = 1 if X1DATAi2+X5DATAi6+X4DATAi5+Ui>0 
                 DATAi1 = 0 otherwise,

he should set 

      NP=5
      NPE=3
      IPRM(1)=1 
      IPRM(2)=5
      IPRM(3)=4 
      IPRM(4)=2 
      IPRM(5)=3

All other conventions of Section 1.11 should also be observed; e.g., OPT should be called with PERM as the name of the subroutine that computes the function to be maximized, and PERM should call PERM1 with the subroutine name PROB1A. In particular, if analytic first and second partial derivatives are desired, it is important to make the following replacements.

a. In FP replace CALL PRFP1A(X,NP,F,FPD,FUNC) by the following: 

      DIMENSION FPD0(np) 
C where np is the actual value of NP 
      COMMON/BPRM/IPRM(np) 
      COMMON/BPRM2/XPRM(np) 
      DO I=1,NP 
      XPRM(IPRM(I))=X(I)
      END DO
      CALL PRFP1A(XPRM,np,F,FPD0,FUNC) 
      DO I=1,NP 
      FPD(I)=FPD0(IPRM(I)) 
      END DO

b. In SP replace CALL PRSP1A(X,NP,F,FPD,SPD,FUNC) by the following: 

      DIMENSION SPD0(np,np) 
      COMMON/BPRM/IPRM(np) 
      COMMON/BPRM2/XPRM(np) 
      DO I=1,NP 
      XPRM(IPRM(I))=X(I)
      END DO
      CALL PRSP1A(XPRM,np,F,FPD,SPD0,FUNC) 
      DO I=1,NP 
      DO J=1,NP 
      SPD(I,J)=SPD0(IPRM(I),IPRM(J)) 
      END DO
      END DO

SECTION 10.1.1 PROBIT JACKKNIFE CALCULATION

See Sunil K. Sapra, "A Jackknife Maximum Likelihood Estimator for the Probit Model," Applied Economics Letters, 9(2002), 73-74.

The jackknife estimator is computed by alternately omitting each one of the n observations and maximizing the likelihood function. If X(i) denotes the probit estimate with the ith observation omitted, X(.) the quantity (1/n)ΣX(i), and X the probit estimate based on all observations, the bias-corrected jackknife maximum likelihood estimate is

nX − (n-1)X(i).

In the MAIN program, include the statements 


     common/bprob/iprob,jprob
     common/btob1/itob
     common/userp3/icjack
and set iprob=1, itob=0 for probit. It should also contain 
      COMMON/USERP1/DATA(n,k+1) 
      COMMON/USERP2/NDIM,KDIM

where

DATA = array containing observations on the variables by columns, with column 1 normally being reserved for the dependent variable
n = the actual number of observations
k = the actual number of independent variables stored (normally = NP)
NDIM = a variable set equal to n
KDIM = a variable set equal to k

In SUBROUTINE PERM, PERM1 should be called with PROB1J in the calling sequence of PERM1 and should be declared in an EXTERNAL statement.

Then call the jackknife estimator as 

      call probjack(X,NP,FU0,METHOD,ITERL,MAX,IER,ACC,PERM,ALABEL,xful,xjack,xstd,xtob,n,jer)
where all symbols have the same meaning as in calling 
      SUBROUTINE OPT
except

xful = a REAL*8 array dimensioned NP, which will contain the probit estimates based on all observations
xjack = a REAL*8 array dimensioned NP, which will contain the jackknife estimates
xstd = a REAL*8 array dimensioned NP, which will contain the standard deviations of the estimates
xtob = a REAL*8 array dimensioned NP, which will contain the tobit jackknife estimates (see below)
n = the number of observations and the dimension of the
jer = an error return, =0 if no errors, =−3 if certain arrays could not be allocated, =−2 if optimization error using all observations, =−1 optimization error in jackknife optimization.

Warning: OPTOUT and OPTOU3 are ineffective and should not be called after optimization.

SECTION 10.2 TOBIT ANALYSIS

See Amemiya, T., "Qualitative Response Models," Journal of Economic Literature, XIX (1981),1483-1536 and Amemiya, T., "Tobit Models: A Survey," Rhodes Associates, CJRS 4, July 1982. 1.

The model is the following:

                  DATAi1 = X1DATAi2 +...+ XNP-1DATAi,NP + Ui 
                                       if the right hand side is positive,
                  DATAi1 = 0 otherwise

where Ui is assumed distributed iid as N(0,S2). Estimation is by maximum likelihood. The MAIN program should contain, in addition to the standard cards (see Section 1), the following:

      
      COMMON/BTOB1/ITOB 
      COMMON/BPRM3/MP
      EXTERNAL TOB2A 
       . . . 
      NP=np 
C (the actual value of np) 
      MP=NP 
      ITOB=1 
      CALL OPT(X,NP,F,METHOD,ITERL,MAX,IER,ACC,TOB2A,ALABEL)

where TOB2A is the name of the built-in likelihood function. (See also Section 10.1). GQOPT actually estimates the tobit model in a transformed version. In this version, instead of S2, we estimate H=1./S and instead of X(J), J=1,...,NP-1 we estimate H*X(J). If the user reads into X(1),...X(NP-1) the slope coefficients appropriate to this normalization and reads into X(NP) the proper value of H, he/she may directly call OPT. If, however, he/she finds it desirable to read in the parameter values correspon- ding to the form of the model given at the beginning of this section, he/she should CALL ENSCRM before and CALL DESCRM after calling OPT: 

      CALL ENSCRM(X,NP) 
      CALL OPT(X,NP,FU0,METHOD,ITERL,MAX,IER,ACC,TOB2A,ALABEL) 
      CALL DESCRM(X,NP)

The CALL DESCRM should not precede a call to OPTOUT or OPTMOV. If the user wishes to have first, or first and second partial derivatives evaluated analytically, he/she should include the following subroutines:

      SUBROUTINE FP(X,NP,F,FPD,FUNC) 
      IMPLICIT REAL*8 (A-H,O-Z) 
      DIMENSION X(NP),FPD(NP) 
      EXTERNAL FUNC 
      CALL TOFP2A(X,NP,F,FPD,FUNC) 
      RETURN 
      END 
      SUBROUTINE SP(X,NP,F,FPD,SPD,FUNC) 
      IMPLICIT REAL*8 (A-H,O-Z)
      DIMENSION X(NP),FPD(NP),SPD(NP,NP)
      EXTERNAL FUNC 
      CALL TOSP2A(X,NP,F, FPD,SPD,FUNC) 
      RETURN 
      END

2. Further options. At times the user may wish to estimate several tobit models, using subsets of the same basic DATA matrix. This can be accomplished in the same manner as in the probit case. The reader should consult Section 10.1.3 and Section 1.11. It is further to be noted that SUBROUTINE PRMCHK should be called AFTER the call to ENSCRM and that before calling DESCRM (after optimization is finished), the user should "unperm" the variable array by

      DO I=1,NP 
      XPRM(IPRM(I)=X(I)
      END DO

Thus, the sequence of commands will be 

      CALL ENSCRM(X,NP) 
      CALL PRMCHK(NP,X,ALABEL,iflag,&nn) 
      CALL OPT(X,NPE,FU0,METHOD,ITERL,MAX,IER,ACC,PERM,ALABEL)
      CALL OPTOUT(0) 
      DO I=1,NP 
      XPRM(IPRM(I))=X(I) 
      END DO
      CALL DESCRM(X,NP)

SECTION 10.2.1 TOBIT JACKKNIFE CALCULATION

See Sunil K. Sapra, "A Jackknife Maximum Likelihood Estimator for the Probit Model," Applied Economics Letters, 9(2002), 73-74.

The jackknife estimator is computed by alternately omitting each one of the n observations and maximizing the likelihood function. If X(i) denotes the tobit estimate with the ith observation omitted, X(.) the quantity (1/n)ΣX(i), and X the probit estimate based on all observations, the bias-corrected jackknife maximum likelihood estimate is

nX − (n-1)X(i).

In the MAIN program, include the statements 

 
     common/bprob/iprob,jprob
     common/btob1/itob
     common/userp3/icjack
and set iprob=0, itob = 1 tobit. It should also contain 
      COMMON/USERP1/DATA(n,k+1) 
      COMMON/USERP2/NDIM,KDIM

where

DATA = array containing observations on the variables by columns, with column 1 normally being reserved for the dependent variable
n = the actual number of observations
k = the actual number of independent variables stored (normally = NP)
NDIM = a variable set equal to n
KDIM = a variable set equal to k

SUBROUTINE PERM should be called with TOB2AJ in the calling sequence of PERM1 and should be declared in an EXTERNAL statement.

Then call the jackknife estimator as 

      call probjack(X,NP,FU0,METHOD,ITERL,MAX,IER,ACC,PERM,ALABEL,xful,xjack,xstd,xtob,n,jer)
where all symbols have the same meaning as in calling 
      SUBROUTINE OPT
except

xful = a REAL*8 array dimensioned NP, which will contain the probit estimates based on all observations
xjack = a REAL*8 array dimensioned NP, which will contain the jackknife estimates
xstd = a REAL*8 array dimensioned NP, which will contain the standard deviations of the estimates
xtob = a REAL*8 array dimensioned NP, which will contain the tobit jackknife estimates (see below)
n = the number of observations
jer = an error return, =0 if no errors, =−3 if certain arrays could not be allocated, =−2 if optimization error using all observations, =−1 optimization error in jackknife optimization.

Warning: OPTOUT and OPTOU3 are ineffective and should not be called after optimization.

SECTION 10.3 LOGIT ANALYSIS

See Amemiya, T., "Qualitative Response Models," Journal of Economic Literature, XIX (1981), 1483-1536.

1. The model is the same as in Section 10.1 except that the error term is assumed to have logistic distribution. It is also assumed that there is no constant term in the equation. The MAIN program should contain, in addition to the standard statements (see Section 1), the following:

      EXTERNAL PROB2A 
      COMMON/BPROB/IPROB,JPROB
      NDIM=n 
      KDIM=k
      . . . 
      IPROB=2 
      CALL OPT(X,NP,F,METHOD,ITERL,MAX,IER,ACC,PROB2A,ALABEL)

where PROB2A is the name of the built-in loglikelihood function. Since this is a maximization problems, set MAX=1. Since it is assumed that there is no constant term in the equation, JPROB will automatically be set = 0. For analytic derivatives, see Section 10.1.1 with the following EXCEPTION: in SUBROUTINES FP and SP the user should call SUBROUTINES PRFP2A and PRSP2A respectively.

2. For further options see Section 10.1.2 with the following NOTE: Although in the call to OPTOU3 and optimization method needs to be specified because the way the call is structured, no optimization will actually be done because there is no constant term; hence under the null hypothesis the loglikeli- hood can be computed directly by setting X(J)=0.0 for all J. 3. For estimating several logit models, using the same basic DATA matrix, which normally involves suboptimization, see Section 10.1.3.

SECTION 10.3.1 LOGIT JACKKNIFE CALCULATION

See Sunil K. Sapra, "A Jackknife Maximum Likelihood Estimator for the Probit Model," Applied Economics Letters, 9(2002), 73-74.

The jackknife estimator is computed by alternately omitting each one of the n observations and maximizing the likelihood function. If X(i) denotes the tobit estimate with the ith observation omitted, X(.) the quantity (1/n)ΣX(i), and X the probit estimate based on all observations, the bias-corrected jackknife maximum likelihood estimate is

nX − (n-1)X(i).

In the MAIN program, include the statements 

 
     common/bprob/iprob,jprob
     common/userp3/icjack
and set iprob=2. It should also contain 
      COMMON/USERP1/DATA(n,k+1) 
      COMMON/USERP2/NDIM,KDIM

where

DATA = array containing observations on the variables by columns, with column 1 normally being reserved for the dependent variable
n = the actual number of observations
k = the actual number of independent variables stored (normally = NP)
NDIM = a variable set equal to n
KDIM = a variable set equal to k

SUBROUTINE PERM should be called with PROB2AJ in the calling sequence of PERM1 and should be declared in an EXTERNAL statement.

Then call the jackknife estimator as 

      call logjack(X,NP,FU0,METHOD,ITERL,MAX,IER,ACC,PERM,ALABEL,xful,xjack,xstd,n,jer)
where all symbols have the same meaning as in calling 
      SUBROUTINE OPT
except

xful = a REAL*8 array dimensioned NP, which will contain the probit estimates based on all observations
xjack = a REAL*8 array dimensioned NP, which will contain the jackknife estimates
xstd = a REAL*8 array dimensioned NP, which will contain the standard deviations of the estimates
n = the number of observations
jer = an error return, =0 if no errors, =−3 if certain arrays could not be allocated, =−2 if optimization error using all observations, =−1 optimization error in jackknife optimization.

Warning: OPTOUT and OPTOU3 are ineffective and should not be called after optimization.

SECTION 10.4 THE SIMPLE DISEQUILIBRIUM MODEL

See Quandt, R.E., The Econometrics of Disequilibrium, Blackwell, 1988.

1. The model is 

               Di = B1X1i1 + . . . + BK1X1i,K1 + U1i
               Si = BK1+1X2i1 + . . . + BK1+K2X2i,K2 + U2i 
               Qi = MIN(Di,Si)

where Qi,X1ij,X2ij represent data, B1,...BK1+K2 are coefficients in the equations, BK1+K2+1 and BK1+K2+2 are the variances of the the error terms in the equations. K1 is the number of parameters in the demand, K2 the number in the supply equation. N is the number of observations. The error terms are assumed to be distributed normally with zero means,zero covariance, and are independent over time. The exogenous variables should be stored in X1 and X2 by columns. The MAIN program should contain, in addition to the usual standard card images (see Section 1), the following:

      COMMON/USER1/Q(n) 
C where n is the number of observations 
      COMMON/USER2/X1(n,k1) 
C k1 the number of demand parameters 
      COMMON/USER3/X2(n,k2)
C k2 the number of supply parameters 
      COMMON/USER4/N,K1,K2 
      EXTERNAL DISEQ1
      N= ... 
      K1=... 
      K2=... 
      NP=K1+K2+2 
      . . . 
      CALL OPT(A,NP,F,METHOD,ITERL,MAX,IER,ACC,DISEQ1,ALABEL)

where DISEQ1 is the name of the built-in likelihood function. If the user wishes to have first, or first and second derivatives evaluated analytically, he/she should include the following subroutines:

      SUBROUTINE FP(X,NP,F,FPD,FUNC) 
      IMPLICIT REAL*8 (A-H,O-Z)
      COMMON/BOPT2/ACC,R ,PM1,IVAL,ITERL,ITERC,MX,IER 
      DIMENSION X(NP),FPD(NP) 
      EXTERNAL FUNC 
      CALL DISEFP(X,NP,F,FPD,FUNC,&10) 
      RETURN 
10    IER=-2 
      RETURN 
      END 
      SUBROUTINE SP(X,NP,F,FPD,SPD,FUNC) 
      IMPLICIT REAL*8 (A-H,O-Z) 
      COMMON/BOPT2/ACC,R ,PM1,IVAL,ITERL,ITERC,MX,IER 
      DIMENSION X(NP),FPD(NP),SPD(NP,NP) 
      EXTERNAL FUNC 
      CALL DISESP(X,NP,F,FPD,SPD,FUNC,&10)
      RETURN 
10    IER=-2 
      RETURN
      END

2. Further options. At times the user may wish to estimate several disequilibrium models, using subsets of the same basic X1 and X2 matrices. This can be accomplished in the same manner as in the probit case. The reader should consult Section 10.1.3 and Section 1.11.

SECTION 10.5 PRICE ADJUSTING DISEQUILIBRIUM MODEL

See R. Quandt, The Econometrics of Disequilibrium, Blackwell, 1988, pp. 23, 33-35.

1. The model is the following:

           Di = B1Pi + B2X1i1 + . . . + BK1X1i,K1-1 + U1i 
           Si = BK1+1Pi + BK1+2X2i1 + . . . + BK1+K2X2i,K2-1 + + U2i 
           Pi = Pi-1 + BK1+K2+1(Di-Si) + BK1+K2+2X3i1+ . . . + BK1+K2+K3X3i,K3-1 + U3i 
           Qi = min(Di,Si)

where Qi, Pi, X1ij, X2ij, X3ij represent data, B1, ...,BK1+K2+K3 are coefficients in the equations, and BK1+K2+K3+1, BK1+K2+K3+2, BK1+K2+K3+3 represent the variances of the error terms U1i, U2i, U3i respectively. The U's are assumed to be normally distributed with mean zero, independent over time, and with zero contemporaneous covariances. The exogenous variables should be stored in the X-arrays by columns. The MAIN program should contain, in addition to the usual standard statements (see Section 1), the following:

      COMMON/USSER0/Q(n) 
C where n is the number of observations 
      COMMON/USSER1/P(n)
      COMMON/USSER2/X1(n,k1-1) 
      COMMON/USSER3/X2(1)
      COMMON/USSER4/X3(1) 
C k1, k2, k3 are the (total) number of parameters in demand, supply,
C and price adjustment equations 
      COMMON/USSER5/N,K1,K2,K3 
      EXTERNAL DISEQ2 
      N= ... 
      K1= ...
      K2= ... 
      K3= ... 
      NP=K1+K2+K3+3
      . . . 
      CALL OPT(B,NP,F,METHOD,ITERL,MAX,IER,ACC,DISEQ2,ALABEL)

where DISEQ2 is the name of the built-in likelihood function. Warning: This model frequently exhibits an unbounded likelihood function. When this is the case, computations will usually terminate with the message "CONTINUAL STRAYING INTO FORBIDDEN REGION" and one or more of the variances is likely to be very small (e.g., 1.D-8).

SECTION 10.6 GINSBURGH-TISHLER-ZANG DISEQUILIBRIUM MODEL

See R. Quandt, The Econometrics of Disequilibrium, Blackwell, 1988, formula (4.2.9) on p. 112.

1. The model is the following: 

                       Di = B1X1i1 + . . . + BK1X1iK1 
                       Si = BK1+1X2i1 + . . . + BK1+K2X2iK2
                       Qi = min(Di,Si) + Ui

where Qi, X1ij, X2ij represent data, B1,...,BK1+K2 are coefficients in the equations, and BK1+K2+1 represents the variance of the error term Ui. K1 is the number of parameters in the demand, K2 the number of parameters in the supply equation. N is the number of observations. The error is assumed to be distributed normally with mean zero and is independent over time. The exogenous variables should be stored in X1 and X2 by columns. The MAIN program should contain, in addition to the usual standard statements (see Section 1), the following:

      COMMON/USER1/Q(n)
C  where n is the number of observations 
      COMMON/USER2/X1(n,k1) 
C  k1 is the number of demand parameters 
      COMMON/USER2/X2(n,k2) 
C  k2 is the number of supply parameters 
      COMMON/USER4/N,K1,K2 
      EXTERNAL DISEQ3 
      N= ... 
      K1=... 
      K2=... 
      NP=K1+K2+1
      . . . 
      CALL OPT(B,NP,F,METHOD,ITERL,MAX,IER,ACC,DISEQ3,ALABEL)

where DISEQ3 is the name of the built-in likelihood function. Warning: This likelihood function can easily have several or even many local optima.

SECTION 10.7 POLYCHOTOMOUS RESPONSE MODELS (POLYCHO)

See Maddala, G. S., Limited-dependent and Qualitative Variables in Econometrics, Cambridge Univ. Press, 1983, pp. 46-51.

The model is the following:

Yi = b'xi + ui

where Yi is not observed, but we observe the the category into which Yi falls, i.e., it falls in the jth category if

aj-1 < Yi < aj, (j = 1,. . .m)

where ui is assumed to be N(0,1). The MAIN program should contain the following:

      
      IMPLICIT REAL*8 (A-H,O-Z)
      DIMENSION C(k+ncat-1)
      COMMON/USERQ1/DATA(N,K)
      COMMON/USERQ2/IDATA(N)
      COMMON/USERQ3/N,K,NCAT
      EXTERNAL POLYCHO
      N=...
      K=...
      NCAT=...
      CALL OPT(C,NP,F,METHOD,ITERL,MAX,IER,ACC,POLYCHO,ALABEL)

where

DATA(N,K) contains the observations on the independent variables
IDATA(N) contains the (integer) category of each observation
N = the number of observations
K = the number of elements in the b-vector
NCAT = the number of categories
C = the vector to be estimated; its first k components are the elements of the b-vector and the next ncat-1 elements are the a's (with the convention that a0 = -infinity and ancat=+infinity).

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