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help for goprobit
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Maximum-Likelihood Estimation of Generalized Ordered Probit Models

        goprobit depvar [indepvars] [weight] [if exp] [in range] [, pl
               pl(varlist) npl npl(varlist) constraints(clist) robust
               cluster(varname) level(#) score(newvarlist|stub*)
               maximize_options ]

    goprobit shares the features of all estimation commands; see help est.
    goprobit typed without arguments redisplays previous results. fweights,
    iweights, and pweights are allowed; see help weights.

    The syntax of predict following goprobit is

        predict [type] newvarname(s) [if exp] [in range] [, statistic
                outcome(outcome) ]

    where statistic is

        p        probability (specify one new variable and outcome() option,
                     or specify k new variables, k = # of outcomes); the
                     default

        xb       linear prediction (outcome() option required)

        stdp     S.E. of linear prediction (outcome() option required)

        stddp    S.E. of difference in linear predictions (outcome() option
                     is outcome(outcome1,outcome2))

    Note that you specify one new variable with xb, stdp, and stddp and
    specify either one or k new variables with p.

    These statistics are available both in and out of sample; type "predict
    ... if e(sample) ..." if wanted only for the estimation sample.



Description

    goprobit is a user-written program that estimates generalized ordered
    probit models. The actual values taken on by the dependent variable are
    irrelevant except that larger values are assumed to correspond to
    "higher" outcomes. This model relaxes the parallel regression assumption
    of the standard ordered probit model; see below and help oprobit.
    goprobit supports linear constraints and allows the user to partially
    relax equal coefficients by specifying variables in npl() or pl().

    goprobit is a modified version of Vincent Kang Fu's gologit and
    particularly Richard Williams' gologit2 programs. The current version of
    gologit2 allows to estimate the generalized ordered probit model using
    the link(probit) option and therefore produces results equivalent to
    goprobit.  goprobit was written for Stata 8 and many of the references in
    this help file are for Stata 8 manuals and commands.



Options

    pl, npl, npl(), pl() provide alternative means for imposing or relaxing
        equal coefficients. Only one may be specified at a time.

        pl specified without parameters constrains all independent variables
            to meet the parallel regression assumption. It will produce
            results that are equivalent to oprobit.

        npl specified without parameters relaxes the parallel regression
            assumption for all explanatory variables. This is the default
            option.

        pl(varlist) constrains the specified explanatory variables to meet
            the parallel regression assumption. All other variables do not
            need to meet the assumption. The variables specified must be a
            subset of the explanatory variables.

        npl(varlist) frees the specified explanatory variables from meeting
            the parallel regression assumption. All other explanatory
            variables are constrained to meet the assumption. The variables
            specified must be a subset of the explanatory variables.

    constraints(clist) specifies the linear constraints to be applied during
        estimation. The default is to perform unconstrained estimation.
        Constraints are defined with the constraint command.  constraints(1)
        specifies that the model is to be constrained according to constraint
        1; constraints(1-4) specifies constraints 1 through 4;
        constraints(1-4,8) specifies 1 through 4 and 8. Keep in mind that the
        pl and npl options work by generating across-equation constraints,
        which may affect how any additional constraints should be specified.
        When using the constraint command, refer to equations by their
        equation #, e.g. #1, #2, etc.

    robust specifies that the Huber/White/sandwich estimator of variance is
        to be used in place of the traditional calculation; see [U] 23.14
        Obtaining robust variance estimates.  robust combined with cluster()
        allows observations which are not independent within cluster
        (although they must be independent between clusters).  If you specify
        pweights, robust is implied.

    cluster(varname) specifies that the observations are independent across
        groups (clusters) but not necessarily within groups.  varname
        specifies to which group each observation belongs; e.g.,
        cluster(personid) in data with repeated observations on individuals.
        cluster() affects the estimated standard errors and
        variance-covariance matrix of the estimators (VCE), but not the
        estimated coefficients.  cluster() can be used with pweights to
        produce estimates for unstratified cluster-sampled data.

    level(#) specifies the confidence level in percent for the confidence
        intervals of the coefficients; see help level.

    score(newvarlist|stub*) creates J-1 new variables, where J is the number
        of observed outcomes. Each new variable contains the contributions to
        the scores for an equation in the model; see [U] 23.15 Obtaining
        scores.

        If score(newvarlist) is specified, J-1 new variables must be
        provided.

        If score(stub*) is specified, then variables stub1, stub2, ...,
        stubJ-1 will be created.

        The first variable contains d(ln L_i)/d(x_i B_1); the second variable
        contains d(ln L_i)/d(x_i B_2); and so on.

    maximize_options control the maximization process; see help maximize. You
        should never have to specify them.



Options for predict

    p, the default, calculates predicted probabilities.

        If you do not specify the outcome() option, you must specify k new
        variables. For instance, say you fitted your model by typing
        "goprobit happy income health" and that happy takes on three values.
        Then you could type "predict p1 p2 p3, p" to obtain all three
        predicted probabilities.

        If you also specify the outcome() option, then you specify one new
        variable. Say that happy took on values 1, 2, and 3. Then typing
        "predict p1, p outcome(1)" would produce the same p1 as above,
        "predict p2, p outcome(2)" the same p2 as above, etc. If happy took
        on values 7, 22, and 93, you would specify outcome(7), outcome(22),
        and outcome(93). Alternatively, you could specify the outcomes by
        referring to the equation number (outcome(#1), outcome(#2), and
        outcome(#3).

    xb calculates the linear prediction. You must also specify the outcome()
        option.

    stdp calculates the standard error of the linear prediction. You must
        specify option outcome().

    stddp calculates the standard error of the difference in two linear
        predictions. You must specify option outcome(), in this case with two
        particular outcomes of interest inside the parentheses; for example,
        "predict sed, stdp outcome(1,3)".

    outcome() specifies for which outcome the statistic is to be calculated.
        equation() is a synonym for outcome(): it does not matter which one
        you use. outcome() and equation() can be specified using (1) #1, #2,
        ..., with #1 meaning the first category of the dependent variable, #2
        the second category, etc.; or (2) values of the dependent variable.


Remarks

    The oprobit command included with Stata imposes what is called the
    parallel regression assumption. By default, goprobit relaxes the parallel
    regression assumption and allows the effects of the explanatory variables
    to vary with the point at which the categories of the dependent variable
    are dichotomized. However, if the pl option is specified, goprobit
    estimates the standard ordered probit model, e.g. the commands oprobit y
    x1 x2 x3 and goprobit y x1 x2 x3, pl will produce equivalent results.

    In practice, the parallel regression assumption is often violated by the
    data.  Standard advice in such situations is to go to a non-ordinal
    model, such as mlogit. Unfortunately, such models do not take into
    account the ordinal nature of the dependent variable and therefore cannot
    be efficient.  goprobit provides an alternative generalized model
    introduced by Maddala (1983:46) and Terza (1985). This model possibly
    relaxes the parallel regression assumption for some explanatory variables
    while being maintained for others.  For example, the command goprobit y
    x1 x2 x3, npl(x1) would relax the parallel regression assumption for x1
    while maintaining it for x2 and x3.  An equivalent command is goprobit y
    x1 x2 x3, pl(x2 x3) which forces x2 and x3 to meet the parallel
    regression assumption while not imposing it on x1.

    More formally, suppose we have an ordinal dependent variable Y which
    takes on the values 1, 2, ..., J.  The generalized ordered probit model
    estimates a set of coefficients (including one for the constant) for each
    of the J - 1 points at which the dependent variable can be dichotomized.
    The probabilities that Y will take on each of the values 1, ..., J is
    equal to

        P( Y = 1 ) = F( -XB_1 )
        P( Y = j ) = F( -XB_j ) - F( -XB_(j-1) )        j = 2, ..., J - 1
        P( Y = m ) = 1 - F( -XB_(J-1) )

    The generalized ordered probit model uses the normal distribution as the
    cumulative distribution F(.), although other distributions may also be
    used; see help gologit and help gologit2.

    The standard ordered probit model (estimated by Stata's oprobit command
    and by goprobit with the pl option) restricts the B_j coefficients to be
    the same for every dividing point j = 1, ..., J-1. The generalized
    ordered probit model (estimated in goprobit via the npl() and pl()
    options) restricts some B_j coefficients to be the same for every
    dividing point while others are free to vary.

    Note that the generalized ordered probit model imposes explicit
    restrictions on the parameters. Since probabilities are by definition
    constrained to be in the range [0,1], valid combinations must satisfy the
    following inequalities:

        XB_1 >= XB_2 >= XB_3 ... >= XB_J-1

    The current version of goprobit does not impose these restrictions during
    the maximization process. After fitting the model, the user should verify
    the validity of the model by calculating predicted probabilities.  See
    help gologit2 and http://www.nd.edu/~rwilliam/gologit2/ for further
    discussion on this topic.

    A panel data version of goprobit with random effects can be estimated by
    regoprob; see help regoprob if installed.


Examples

        . goprobit happy linc unempl health if male == 1, robust

        . goprobit happy linc unempl health if male == 1, robust npl(linc)

        . goprobit, level(99)

        . predict xb1, xb outcome(#1)



Author

     Stefan Boes
     Socioeconomic Institute
     Statistics and Empirical Economics Research Group
     University of Zurich
     boes@sts.unizh.ch
     http://www.unizh.ch/sts/



Acknowledgements

     Richard Williams of the Notre Dame Department of Sociology wrote gologit2.
     Richard Williams kindly gave me permission to use his code of gologit2 and
     I adapted much of it when programming goprobit. For a more detailed
     description of gologit2 and its features, see the reference below or help 
     gologit2.



References

     Boes, S. and R. Winkelmann (2006) "Ordered Response Models." Allgemeines
     Statistisches Archiv 90: 165-179.

     Fu, V.K. (1998) "Estimating Generalized Ordered Logit Models." Stata
     Technical Bulletin 8: 160-164.

     Long, J.S and J. Freese (2003) "Regression Models for Categorical
     Dependent Variables Using Stata", revised edition, Stata Press.

     Maddala, G. (1983) "Limited-Dependent and Qualitative Variables in
     Econometrics." Cambridge University Press: Cambridge.

     Terza, J. (1985) "Ordered Probit: A Generalization." Communications in
     Statistics – A. Theory and Methods 14: 1–11.

     Williams, R. (2006) "Generalized Ordered Logit/ Partial Proportional Odds
     Models for Ordinal Dependent Variables." The Stata Journal 6(1): 58-82. A
     pre-publication version is available at 
     http://www.nd.edu/~rwilliam/gologit2/gologit2.pdf.

     Winkelmann, R. and S. Boes (2006) "Analysis of Microdata." Springer:
     Berlin.


Also see

    Manual:  [U] 23 Estimation and post-estimation commands,
             [U] 29 Overview of Stata estimation commands

    Online:  help for estcom, postest, constraint, oprobit, ologit, gologit, 
             gologit2, regoprob