Estimation of Nested and Zero-Inflated Ordered Probit Models

We introduce three new STATA commands, nop, ziop2 and ziop3, for the estimation of a three-part nested ordered probit model, the two-part zero-inflated ordered probit models of Harris and Zhao (2007, Journal of Econometrics 141: 1073-1099) and Brooks, Harris and Spencer (2012, Economics Letters 117: 683-686), and a three-part zero-inflated ordered probit model for ordinal outcomes, with both exogenous and endogenous switching. The three-part models allow the probabilities of positive, neutral (zero) and negative outcomes to be generated by distinct processes. The zero-inflated models address a preponderance of zeros and allow them to emerge in different latent regimes. We provide postestimation commands to compute probabilistic predictions and various measures of their accuracy, to access the goodness of fit, and to perform model comparison using the Vuong test (Vuong 1989, Econometrica 57: 307-333) with the corrections based on the Akaike and Schwarz information criteria. We investigate the finite-sample performance of the maximum likelihood estimators by Monte Carlo simulations, discuss the relations among the models, and illustrate the new commands with an empirical application to the U.S. federal funds rate target.


Introduction
We introduce the STATA commands, nop, ziop2 and ziop3, which estimate two-level nested and zero-in ‡ated ordered probit (OP) models for ordinal outcomes, including the zeroand middle-in ‡ated OP models of Harris and Zhao (2007), Bagozzi and Mukherjee (2012), Brooks, Harris and Spencer (2012) and Sirchenko (2013). The rationale behind the two-level nested decision process is standard in discrete-choice modeling when the set of alternatives faced by a decision-maker can be partitioned into subsets (or nests) with similar alternatives correlated due to common unobserved factors. The choice among the nests and the choice among the alternatives within each nest can be driven by di¤erent sets of observed and unobserved factors (and common factors can have di¤erent weights).
In unordered categorical data, in which choices can be grouped into the nests of similar options, the nested logit model is a popular method. Nested models for ordinal data are rare although the rationale behind them is similar: choosing among a negative response (decrease), a neutral response (no change) or a positive response (increase) is quite di¤erent from choosing the magnitude of a negative or positive response; and choosing the magnitude of a negative response can be driven by quite di¤erent determinants than choosing the magnitude of a positive response. This leads to three implicit decisions: an upper-level regime decision -a choice among the nests, and two lower-level outcome decisions -the choices of the magnitude of the negative and positive responses (see the top left panel of Figure 1).

Figure 1. Decision trees of nested and zero-in ‡ated ordered probit models
Notes: Decisionmakers are not assumed to choose sequentially. The tree diagrams simply represent a nesting structure of the system of OP models. Furthermore, it would be reasonable for the zero (no-change) alternative to be in three nests: its own, one with the negative responses, and one with the positive responses; hence, some zeros can be driven by similar factors as the negative or positive responses. This leads 2 Models

Notation and assumptions
The observed dependent variable y t , t = 1; 2; :::; T is assumed to take on a …nite number of ordinal values j coded as f J ; :::; 1; 0; 1; :::; J + g; where a potentially heterogeneous (and typically predominant) response is coded as zero. The latent unobserved (or only partially observed) variables are denoted by " ". Each model assumes an ordered-choice regime decision and ordered-choice outcome decisions conditional on the regime. The regime decision can be correlated with each outcome decision. We denote: by x t ; x t ; x + t and z t the t th rows of the observed data matrices (which in addition to the predetermined explanatory variables may also include the lags of y t ); by ; ; + and the vectors of slope parameters; by ; ; + and the vectors of threshold parameters; by ; and + the vectors of correlation coe¢ cients; by " t ; " t ; " + t and t the error terms that are independently and identically distributed (iid) across t with normal cumulative distribution function (CDF) , the zero means and the variances 2 ; 2 ; 2 + and 2 , respectively; and by 2 (g 1 ;g 2 ; 2 1 ; 2 2 ; ) the CDF of the bivariate normal distribution of the two random variables g 1 and g 2 with the zero means, the variances 2 1 and 2 2 and the correlation coe¢ cient : 2 (g 1 ;g 2 ; 2 1 ; 2 2 ; ) = 1 2 1 2 p 1 2 g 1 R 1 g 2 R 1 exp u 2 = 2 1 2 uw= 1 2 +w 2 = 2 2 2(1 2 ) dudw:

Three-part nested ordered probit (NOP) model
Despite the wide-spread use of nested logit models for unordered categorical responses, we are aware of only one example of the nested ordered probit model in the literature (Sirchenko 2013). The two-level NOP model can be described as Upper-level decision: r t = z t + t ; s t = 8 < : 1 if 2 < r t ; 0 if 1 < r t 2 ; 1 if r t 1 : Lower-level decisions: y t = x t + " t ; y + t = x + t + + " + t ; y t = 8 < : j(j > 0) if s t = 1 and + j 1 < y + t + j ; 0 if s t = 0; j(j < 0) if s t = 1 and j < y t j+1 ; where 1 = + 0 + 1 ::: ::: The probabilities of the outcome j in the NOP model are given by Pr(y t = jjz t ; x t ; x + t ) = I j<0 Pr(r t 1 and j < y t j+1 jz t ; x t ) +I j=0 Pr( 1 < r t 2 jz t ) + I j>0 Pr( 2 < r t and + j 1 < y + t + j jz t ; where I j<0 is an indicator function such that I j<0 = 1 if j < 0, and I j<0 = 0 if j 0 (analogously for I j=0 and I j>0 ).
In the case of exogenous switching (when = + = 0), the probabilities of the outcome j in the NOP can be computed as In the case of two or three outcome choices the NOP model degenerates to the conventional single-equation OP model.

Two-part zero-in ‡ated ordered probit (ZIOP-2) model
The ZIOP-2 model, which represents the zero-in ‡ated OP model of Brooks, Harris and Spencer (2012) and the middle-in ‡ated OP model of Bagozzi and Mukherjee (2012), can be described by the following system Regime decision: Outcome decision: Correlation among decisions: The probabilities of the outcome j in the ZIOP-2 model are given by Pr(y t = jjz t ; x t ) = I j=0 Pr(r t jz t ) + Pr( < r t and j 1 < y t j jz t ; x t ) = I j=0 Pr( t z t ) + Pr( z t < t and j 1 x t < " t j x t ) = I j=0 ( z t ; 2 ) + 2 ( + z t ; j x t ; 2 ; 2 ; ) 2 ( + z t ; j 1 x t ; 2 ; 2 ; ). (2) In the case of exogenous switching (when = 0), these probabilities can be computed as If y t 0 for 8t; the ZIOP-2 model becomes the model of Harris and Zhao (2007).

Three-part zero-in ‡ated ordered probit (ZIOP-3) model
The ZIOP-3 model developed by Sirchenko (2013) is a three-part generalization of the ZIOP-2 model, and can be described by the following system Regime decision: Outcome decisions: ::: ::: Correlation among decisions: The probabilities of the outcome j in the ZIOP-3 model are given by Pr(y t = jjz t ; x t ; x + t ) = I j 0 Pr(r t 1 and j < y t j+1 jz t ; x t ) +I j=0 Pr( 1 < r t 2 jz t ) + I j 0 Pr( 2 < r t and + j 1 < y + t + j jz t ; where I j 0 is an indicator function such that I j 0 = 1 if j 0, and I j 0 = 0 if j > 0 (analogously for I j 0 ).
In the case of exogenous switching (when = + = 0), these probabilities can be computed as . The in ‡ated outcome does not have to be in the very middle of the ordered choices. If it is located at the end of the ordered scale, i.e. if y t 0 for 8t; the ZIOP-3 model reduces to the ZIOP-2 model of Harris and Zhao (2007).

Maximum likelihood (ML) estimation
The probabilities in each OP equation can be consistently estimated under fairly general conditions by an asymptotically normal ML estimator (Basu and de Jong 2007). The simultaneous estimation of the OP equations in the NOP, ZIOP-2 and ZIOP-3 models can be also performed using an ML estimator of the vector of the parameters that solves where I tj is an indicator function such that I tj = 1 if y t = j and I tj = 0 otherwise; includes ; ; ; + ; ; + ; and + for the NOP and ZIOP-3 models, and ; ; ; and for the ZIOP-2 model; is a parameter space; x all t is a vector that contains the values of all independent variables in the model; and Pr(y t = jjx all t ; ) are the probabilities from either (1) or (2) or (3). The asymptotic standard errors of b can be computed from the Hessian matrix.
The intercept components of ; ; + and are identi…ed up to scale and location, that is, only jointly with the corresponding threshold parameters ; ; + and and variances 2 ; 2 ; 2 + ; and 2 . As is common in the identi…cation of discrete-choice models, the variances 2 ; 2 ; 2 + ; and 2 are …xed to one, and the intercept components of ; ; + and are …xed to zero. The probabilities in (1), (2) and (3) are invariant to these (arbitrary) identifying assumptions: up to scale and location, we can identify all parameters in because of the non-linearity of OP equations, i.e. via the functional form (Heckman 1978;Wilde 2000). However, since the normal CDF is approximately linear in the middle of its support, the simultaneous estimation of two or three equations may experience a weak identi…cation problem if the regime and outcome equations contain the same set of independent variables. To enhance the precision of parameter estimates we may impose exclusion restrictions on the speci…cation of the independent variables in each equation.
The three regimes (nests) in the NOP model are fully observable, contrary to the latent (only partially observed) regimes in the ZIOP-2 and ZIOP-3 models. The likelihood function of the NOP model -again in contrast with the ZIOP-2 and ZIOP-3 models -is separable with respect to the parameters in the three equations. Thus, solving (4) for the NOP model is equivalent to maximizing separately the likelihoods of the three OP models representing the upper-and lower-level decisions. 2

Marginal e¤ects (ME)
The marginal e¤ects of a continuous independent variable k (the k th element of x all t ) on the probability of each discrete outcome j are computed for the ZIOP-3 model as where f is the probability density function of the standard normal distribution, and all k , or +all k is zero if the k th independent variable in x all t is not included into the corresponding equation). For a discrete-valued independent variable, the ME can be computed as the change in the probabilities when this independent variable changes by one increment and all other independent variables are …xed.
The MEs for the NOP model are computed by replacing I j 0 in the above formula with I j>0 and I j 0 with I j<0 .
The MEs for the ZIOP-2 model are computed as where all k is the coe¢ cient on the k th independent variable in x all t in the outcome equation ( all k is zero if the k th independent variable in x all t is not included into the outcome equation). The asymptotic standard errors of the MEs are computed using the Delta method as the square roots of the diagonal elements of

Relations among the models and their comparison
We now discuss the choice of a formal statistical test to compare the NOP, ZIOP-2, ZIOP-3 and conventional OP models. The choice depends on whether the models are nested in each other.
The exogenous-switching version of each model is nested in its endogenous-switching version as its uncorrelated special case; their comparison can be performed using any classical likelihood-based test for nested hypotheses, such as the likelihood ratio (LR) test.
The OP is not nested either in the NOP or ZIOP-3 model. We can compare the OP model with them using a likelihood-based test for non-nested models, such as the Vuong test (Vuong 1989). The OP model is however nested in the ZIOP-2 model. The latter reduces to the former if ! 1; hence, Pr(y t = 0jx t ; s t = 1) ! 0. Therefore, the Vuong test for non-nested hypothesis cannot be used to compare the OP and ZIOP-2 model: for nested hypothesis, the Voung test reduces to the LR test. However, the critical values of the classical LR test are invalid in this case since some of the standard regularity conditions of the classical LR test fail to hold (Andrews 2001;Andrews and Cheng 2012). In particular, the value of in the null hypothesis is not an interior point of the parameter space; hence, the asymptotic distribution of the LR statistics is not standard. 3 The NOP model is nested in the ZIOP-3 model. The latter becomes the former if 1 ! 1 and + 1 ! 1; therefore, Pr(y t = 0jx + t ; s t = 1) ! 0 and Pr(y t = 0jx t ; s t = 1) ! 0. The values of 1 and + 1 in the null hypothesis are not the interior points of the parameter space; thus, the asymptotic distribution of the LR statistics is not standard. The comparison of the NOP and ZIOP-3 models can also be performed using the LR test with simulated adjusted critical values (Andrews 2001;Andrews and Cheng 2012).
Generally, the ZIOP-2 model is not a special case of the ZIOP-3 model, and vice versa. We can compare them using the Vuong test. A special case when the ZIOP-3 model nests the ZIOP-2 model emerges under certain restrictions on the parameters as explained below. In this case, the selection between the ZIOP-3 and ZIOP-2 models can be performed using any classical likelihood-based test for nested hypotheses such as the LR test.
The special case emerges if y t takes on only three discrete values j 2 f 1; 0; 1g, the regressors in x t and x + t in the outcome equations of the ZIOP-3 model contain all the regressors in the ZIOP-2 regime equation (denoted below by z 2t with the parameter vector 2 ), and the regressors in the regime equation of the ZIOP-3 model (denoted below by z 3t with the parameter vector 3 ) include all the regressors in the x t in the ZIOP-2 outcome equation. According to (2) the probabilities of the outcome j in the ZIOP-2 model are given by since 2 (x; y; ) = (x) 2 (x; y; ). Similarly, according to (3) the probabilities of the outcome j in the ZIOP-3 model are Suppose the regressors in x t and x + t in the ZIOP-3 outcome equations are identical to the regressors in z 2t in the ZIOP-2 regime equation, the regressors in z 3t in the ZIOP-3 regime equation are identical to the regressors in the x t in the ZIOP-2 outcome equation, and the parameters are restricted as follows: , the probabilities for the ZIOP-3 model can be written as which are identical to the probabilities for the ZIOP-2 model in (5). Notice that the restrictions = + = 2 and 0 = + 0 = impose a sort of symmetry in the ZIOP-3 model, because they imply that the conditional probability of a positive response is equal to the conditional probability of a negative response: In general, if x t and x + t are not identical to z 2t but contain all the regressors in z 2t , and if z 3t is not identical to x t but contains all the regressors in x t , the ZIOP-2 model is still nested in the ZIOP-3 model with the additional zero restrictions for the coe¢ cients on all the extra regressors in x t , x + t and z 3t .

The nop, ziop2 and ziopcommands in Stata
The accompanying software includes the three new commands, the postestimation commands and the supporting help …les.

Syntax
The following commands estimate, respectively, the NOP, ZIOP-2 and ZIOP-3 models for discrete ordinal outcomes: An ordinal dependent variable depvar is assumed to take on at least …ve discrete ordinal values in the NOP model, at least two in the ZIOP-2 model, and at least three in the ZIOP-3 model. A list of the independent variables in the regime equation indepvars may be di¤erent from the lists of the independent variables in the outcome equations.
Options posindepvars(varlist) speci…es a list of the independent variables in the outcome equation, conditional on the regime s t = 1 for non-negative outcomes in the NOP and ZIOP-3 models; by default, it is identical to indepvars, the list of the independent variables in the regime equation.
negindepvars(varlist) speci…es a list of the independent variables in the outcome equation, conditional on the regime s t = 1 for non-positive outcomes in the NOP and ZIOP-3 models; by default, it is identical to indepvars, the list of the independent variables in the regime equation.
outindepvars(varlist) speci…es a list of the independent variables in the outcome equation of the ZIOP-2 model; by default, it is identical to indepvars, the list of the independent variables in the regime equation.
infcat(choice) is the value of the dependent variable in the regime s t = 0 that should be modeled as in ‡ated in the ZIOP-2 and ZIOP-3 models, and as neutral in the NOP model; by default, choice equals 0.
endoswitch speci…es that endogenous regime switching is to be used instead of default exogenous switching. Regime switching is endogenous if the unobserved random term in the regime equation is correlated with the unobserved random terms in the outcome equations, and exogenous otherwise.
robust speci…es that a robust sandwich estimator of variance is to be used; the default estimator is based on the observed information matrix.
cluster(varname) speci…es a clustering variable for the clustered robust sandwich estimator of variance.
initial(string) speci…es a space-delimited list string of the starting values of the parameters in the following order: ; ; + ; + ; ; ; and + for the NOP and ZIOP-3 models, and ; ; ; and for the ZIOP-2 model. nolog suppresses the iteration log and preliminary results.

Stored results
The descriptions of the stored results can be found in the help …les.

Postestimation commands
The following postestimation commands are available after nop, ziop2 and ziop3: The predict command

zeros regimes output(string)]
This command computes the predicted probabilities of the discrete choices (by default), the regimes and the types of zeros conditional on the regime, and the predicted outcomes and the expected values of the dependent variable for all observed values of the independent variables in the sample. The command creates (J + J + + 1) new variables under the names with a newvar pre…x. The following options are available: regimes indicates that the probabilities of the regimes s t 2 f 1; 0; 1g must be predicted instead of the choice probabilities. This option is ignored if the zeros option is used.
zeros indicates that the probabilities of the di¤erent types of zeros (the outcomes in the in ‡ated category infcat(choice) in the ZIOP-2 and ZIOP-3 models), conditional on di¤erent regimes, must be predicted instead of the choice probabilities.
output(string) speci…es the di¤erent types of predictions. The possible values of string are: choice for reporting the predicted outcome (the choice with the largest predicted probability); mean for reporting the expected value of the dependent variable computed as P i i Pr(y t = i); and cum for predicting the cumulative choice probabilities: Pr(y t <= J ), Pr(y t <= J + 1), ..., Pr(y t <= J + ). If string is not speci…ed, the usual choice probabilities Pr(y t = J ), Pr(y t = J + 1), ..., Pr(y t = J + ) are predicted and saved into the new variables with the newvar pre…x.

The ziopprobabilities command
ziopprobabilities [, at(string) zeros regimes] This command shows the predicted probabilities estimated at the speci…ed values of the independent variables along with the standard errors. The options zeros and regimes are speci…ed as in predict. The option at() is speci…ed as follows: at(string) speci…es for which values of the independent variables to estimate the predictions.
If at(string) is used (string is a list of varname = value expressions, separated by commas), the predictions are estimated at these values and displayed without saving to the dataset. If some independent variable names are not speci…ed, their median values are taken instead. If at() is not used, by default the predictions are estimated at the median values of the independent variables.

The ziopcontrasts command
ziopcontrasts [, at(string) to (string) zeros regimes] This command shows the di¤erences in the predicted probabilities, estimated …rst at the values of the independent variables in at() and then at the values in to(), along with the standard errors. The options zeros, regimes and at() are speci…ed as in ziopprobabilities. The options to() is speci…ed analogously to at().

The ziopmargins command
ziopmargins [, at(string) zeros regimes] This command shows the marginal e¤ects of each independent variable on the predicted probabilities estimated at the speci…ed values of the independent variables along with the standard errors. The options zeros, regimes and at() are speci…ed as in ziopprobabilities.

The ziopclassi…cation command ziopclassification [if ] [in]
This command shows the classi…cation table (or confusion matrix); the percentage of correct predictions; the two strictly proper scores -the probability, or Brier, score (Brier 1950) and the ranked probability score (Epstein 1969); the precisions, the hit rates (or recalls) and the adjusted noise-to-signal ratios (Kaminsky and Reinhart 1999).
The classi…cation table reports the predicted choices (the ones with the highest predicted probability) in columns, the actual choices in rows, and the number of (mis)classi…cations in each cell.
The Brier probability score is computed as 1 T P T t=1 P J + j= J [Pr(y t = j) I jt ] 2 , where indicator I jt = 1 if y t = j and I jt = 0 otherwise. The ranked probability score is computed as 1 The better the prediction, the smaller both score values. Both scores have a minimum value of zero when all the actual outcomes are predicted with a unit probability.
The precision, the hit rate (or recall) and the adjusted noise-to-signal ratios are de…ned as follows. Let TP denote a true positive event, that is, the outcome was predicted and occurred; let FP denote a false positive event, that is, the outcome was predicted but did not occur; let FN denote a false positive event, that is, the outcome was not predicted but did occur; and let TN denote a true negative event, that is, the outcome was not predicted and did not occur. The desirable outcomes fall into categories TP and TN, while the noisy ones fall into categories FP and FN. A perfect prediction has no entries in FP and FN, while a noisy prediction has many entries in This command performs the Vuong test for non-nested hypotheses, which compares the closeness of two models to the true data distribution using the di¤erences in the pointwise log likelihoods of the two models. The arguments modelspec 1 and modelspec 2 are the names under which the estimation results are saved using the estimates store command. Any model that stores the vector e(ll_obs) of observation-wise log-likelihood can technically be used to perform the test. The command provides the three Vuong test statistics (z-scores): the standard one and two adjusted ones with corrections to address the comparison of models with di¤erent numbers of parameters based on AIC and BIC. They can be used to test the hypothesis that one of the models explains the data better than the other. A signi…cant positive z-score indicates a preference for the …rst model, while a signi…cant negative value of the z-score indicates a preference for the second model. An insigni…cant z-score implies no preference for either model.

Monte Carlo simulations
We conducted extensive Monte Carlo experiments to illustrate the …nite sample performance of the ML estimators of each model.

Monte Carlo design
We simulated six processes generated by the NOP, ZIOP-2 and ZIOP-3 models, each of them with both exogenous and endogenous switching. Repeated samples with 200, 500 and 1,000 observations were independently generated and then estimated by the true model. There were 10,000 replications in each experiment.
Three independent variables w 1 , w 2 and w 3 were drawn in each replication as w 1 iid N (0; 1) + 2, w 2 iid N (0; 1), and w 3 = 1 if u 0:3, 0 if 0:3 < u 0:7, or 1 if u > 0:7, where u iid U[0; 1]. The repeated samples were generated for the NOP and ZIOP-3 models with Z = (w 1 ; w 2 ), X = (w 1 ; w 3 ), X + = (w 2 ; w 3 ), and for the ZIOP-2 model with Z = (w 1 ; w 3 ), X = (w 2 ; w 3 ). The dependent variable y was generated with …ve values: -2, -1, 0, 1 and 2. The parameters were calibrated to yield, on average, the following frequencies of the above outcomes: 7%, 14%, 58%, 14% and 7%, respectively. To avoid the divergence of ML estimates due to the problem of complete separation (perfect prediction), which could happen if the actual number of observations in any outcome category is very low, the samples with any outcome category frequency lower than 6% were re-generated. The variances of the error terms in all equations were …xed to one. The true values of all other parameters in the simulations are shown in Table A1 in Appendix. The starting values for the slope and threshold parameters were obtained using the independent OP estimations of each equation.
The starting values for , and + were obtained by maximizing the likelihood functions of the endogenous-switching models holding the other parameters …xed at their estimates in the corresponding exogenous-switching model. The values of the choice probabilities, which depend on the values of the regressors, are computed at the population medians of the regressors. Table 1 reports the measures of the accuracy for the estimates of the choice probabilities. The results for the estimates of the parameters and MEs are qualitatively and quantitatively similar. The simulations show that the ML estimators are consistent and reliable even in samples with only 200 observations: the biases of choice probability estimates are smaller than …ve percent and the asymptotic coverage rates di¤er from the nominal 0.95 level by less than one percent. For each model, the bias and RMSE decrease as the sample size increases. The RMSE decreases, in most cases, faster than the asymptotic rate p n. This may be caused by a small number of large deviations in the parameter estimation in small samples. For all models and sample sizes, the bias and RMSE are, as expected, slightly higher for a more complex endogenous-switching version. The standard error estimates, on average, correspond to the actual standard errors; however, large deviations make standard error estimates biased in small samples, but do not move the coverage rates from the nominal level by more than one percent even with only 200 observations. The accuracy in the NOP models is, as expected, higher than in the zero-in ‡ated OP models. Notes: Bias -the absolute di¤erence between the estimated and true values, divided by the true value; RMSE -the absolute root mean square error of the estimates; Coverage rate -the percentage of times the estimated asymptotic 95% con…dence intervals cover the true values; Bias of standard error estimates -the absolute di¤erence between the average of the estimated asymptotic standard errors of the estimates and the standard deviation of the estimates in all replications. The above measures are averaged across …ve outcome categories.

Examples
The new commands are applied to a real-world time-series sample of all decisions of the U.S. Federal Open Market Committee (FOMC) on the federal funds rate target made at scheduled and unscheduled meetings during the 9/1987 -9/2008 period.
The dependent variable, the change to the rate target, is classi…ed into …ve ordered categories: "-0.5" (a cut of 0.5% or more), "-0.25" (a cut less than 0.5% but more than 0.0625%), "0" (no change or change by no more than 0.0625%), "0.25" (a hike more than 0.0625% but less than 0.5%) and "0.5" (a hike of 0.5% or more). The FOMC decisions are aligned with the real-time values of the explanatory variables as they were truly available to the public on the previous day before each FOMC meeting. The explanatory variables include: spread (the di¤erence between the one-year treasury constant maturity rate and the e¤ective federal funds rate, …ve-business-day moving average; data source: ALFRED 4 ); pb (the trichotomous indicator that we constructed from the "policy bias"statements at the previous FOMC meeting: it equals 1 if the statement was asymmetric toward tightening, 0 if the statement was symmetric, and -1 if the statement was asymmetric toward easing; data source: FOMC statements and minutes 5 ); houst (the Greenbook projection for the current quarter of the total number of new privately owned housing units started; data source: RTDSM 6 ); gdp (the Greenbook projection for the current quarter of quarterly growth in the nominal gross domestic (before 1992: national) product, annualized percentage points; data source: RTDSM).
We start by estimating the conventional OP model using the oprobit command: . oprobit rate_change spread pb houst gdp, nolog   We now allow the negative, zero and positive changes to the rate target to be generated by di¤erent processes, and estimate the three-part NOP model. The nop command yields the following results: . nop rate_change spread pb houst gdp, neg(spread gdp) pos(spread pb) inf (0)      The NOP model provides a substantial improvement of the likelihood, and is preferred to the standard OP model according to AIC and the Vuong test (the p-value is 0.01). However, the Vuong tests with the corrections based on AIC and BIC are indi¤erent between the two models. Endogenous switching does not signi…cantly improve the likelihood of the NOP model (the log likelihood with endogenous switching is -150.2, the p-value of the LR test of the null of exogenous switching is 0.48), the correlation coe¢ cients and + are not signi…cant, and both AIC and BIC favor the NOP model with exogenous switching.
Next we allow for an in ‡ation of zero outcomes and estimate the three-part ZIOP-3 model. The ziop3 command with exogenous switching yields the following results: . ziop3 rate_change spread pb houst gdp, neg(spread gdp) pos(spread pb) inf (0)      The empirical evidence in favor of zero in ‡ation is convincing: with only two extra parameters, the ZIOP-3 model has a much higher likelihood than the NOP model (-139.6 vs. -151.0), and is clearly preferred by both AIC and BIC to the NOP and OP models. The Vuong tests for zero in ‡ation (the standard one and one with the correction based on AIC) favor the ZIOP-3 model over the OP model at the 0.001 and 0.01 level, respectively. Endogenous switching does not signi…cantly improve the likelihood of the ZIOP-3 model either (the p-value of the LR test of exogenous switching is 0.30, and both AIC and BIC prefer the exogenous switching).
In contrast, the likelihood of the two-part ZIOP-2 model is even lower than that of the NOP model. According both to AIC and BIC, the ZIOP-2 model is inferior to all the above models, including the OP one. The ziop2 command yields the following results: . ziop2 rate_change spread pb houst gdp, out(spread pb houst gdp ) infcat (0) The Vuong tests prefer the ZIOP-3 model to the ZIOP-2 model at the 0.01 signi…cance level using the standard test statistic, and at the 0.02 and 0.03 levels using the corrected statistics based, respectively, on AIC and BIC: . quietly ziop3 rate_change pb spread houst gdp, neg(spread gdp ) pos(pb spread) inf(0) (output omitted) . est store ziop3_model . quietly ziop2 rate_change spread pb houst gdp, out(spread pb houst gdp) inf (0) . est store ziop2_model . Now we report the selected output of the postestimation commands, performed for the ZIOP-3 model. The predicted probabilities of the three latent regimes s t 2 f 1; 0; 1g or the probabilities of the three types of zeros conditional on each regime can be estimated for each sample observation using the command predict with the option zeros or regimes, respectively: . predict p_zero, zeros . predict p_reg, regimes . tabstat p_zero* p_reg*, stat(mean) The average predicted probabilities of the regimes s t = 1, s t = 0 and s t = 1 in the sample are 0.40, 0.39 and 0.21, respectively. However, the average probability of zeros conditional on the regime s t = 1 (0.15) is much higher than on the regime s t = 1 (0.00).

Measure of fit
Notes: The NOP, ZIOP-2 and ZIOP-3 models are estimated with exogenous switching.

Concluding remarks
This article describes the ML estimation of the nested and cross-nested zero-in ‡ated ordered probit models using the new STATA commands nop, ziop2 and ziop3. Such models can be applied to a variety of data sets in which the discrete ordinal outcomes can be divided into groups (nests) of similar choices, for example, the decisions to reduce, leave unchanged, or increase the choice variable (monetary policy interest rates, rankings, prices, consumption levels), or the negative, neutral, or positive attitudes to survey questions. The choice among the nests is driven by an ordered-choice switching mechanism that can be either exogenous or endogenous to the outcome decisions, which are also naturally ordered (large or small increase/decrease; disagree or strongly disagree; etc.). The models allow the probabilities of choices from di¤erent nests (e.g., no change and an increase) to be driven by distinct mechanisms. Moreover, the cross-nested zero-in ‡ated models allow the often abundant no-change or neutral outcomes to belong to all nests and be in ‡ated by several di¤erent processes. The results of Monte Carlo simulations indicate that the proposed ML estimators are consistent and perform well in small samples.