642—Chapter 21. Discrete and Limited Dependent Variable Models
Estimation Problems
Most of the previous discussion of estimation problems for binary models (page 627) also holds for ordered models. In general, these models are well-behaved and will require little intervention.
There are cases, however, where problems will arise. First, EViews currently has a limit of 750 total coefficients in an ordered dependent variable model. Thus, if you have 25 righthand side variables, and a dependent variable with 726 distinct values, you will be unable to estimate your model using EViews.
Second, you may run into identification problems and estimation difficulties if you have some groups where there are very few observations. If necessary, you may choose to combine adjacent groups and re-estimate the model.
EViews may stop estimation with the message “Parameter estimates for limit points are non-ascending”, most likely on the first iteration. This error indicates that parameter values for the limit points were invalid, and that EViews was unable to adjust these values to make them valid. Make certain that if you are using user defined parameters, the limit points are strictly increasing. Better yet, we recommend that you employ the EViews starting values since they are based on a consistent first-stage estimation procedure, and should therefore be quite well-behaved.
Views of Ordered Equations
EViews provides you with several views of an ordered equation. As with other equations, you can examine the specification and estimated covariance matrix as well as perform Wald and likelihood ratio tests on coefficients of the model. In addition, there are several views that are specialized for the ordered model:
•Dependent Variable Frequencies — computes a one-way frequency table for the ordered dependent variable for the observations in the estimation sample. EViews presents both the frequency table and the cumulative frequency table in levels and percentages.
•Expectation-Prediction Table — classifies observations on the basis of the predicted response. EViews performs the classification on the basis of maximum predicted probability as well as the expected probability.
Procedures for Ordered Equations—643
Dependent Variable: DANGER
Method: ML - Ordered Probit
Date: 09/13/97 Time: 10:00
Sample(adjusted): 1 61
Included observations: 58
Excluded observations: 3 after adjusting endpoints
Prediction table for ordered dependent variable
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Count of obs |
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Sum of all |
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Value |
Count |
with Max Prob |
Error |
Probabilities |
Error |
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1 |
18 |
27 |
-9 |
18.571 |
-0.571 |
2 |
14 |
16 |
-2 |
13.417 |
0.583 |
3 |
10 |
0 |
10 |
9.163 |
0.837 |
4 |
9 |
8 |
1 |
8.940 |
0.060 |
5 |
7 |
7 |
0 |
7.909 |
-0.909 |
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|
|
|
|
There are two columns labeled “Error”. The first measures the difference between the observed count and the number of observations where the probability of that response is highest. For example, 18 individuals reported a value of 1 for DANGER, while 27 individuals had predicted probabilities that were highest for this value. The actual count minus the predicted is –9. The second error column measures the difference between the actual number of individuals reporting the value, and the sum of all of the individual probabilities for that value.
Procedures for Ordered Equations
Make Ordered Limit Vector/Matrix
The full set of coefficients and the covariance matrix may be obtained from the estimated equation in the usual fashion (see “Working With Equation Statistics” on page 454). In some circumstances, however, you may wish to perform inference using only the estimates of the γ coefficients and the associated covariances.
The Make Ordered Limit Vector and Make Ordered Limit Covariance Matrix procedures provide a shortcut method of obtaining the estimates associated with the γ coefficients. The first procedure creates a vector (using the next unused name of the form LIMITS01, LIMITS02, etc.) containing the estimated γ coefficients. The latter procedure creates a symmetric matrix containing the estimated covariance matrix of the γ . The matrix will be given an unused name of the form VLIMITS01, VLIMITS02, etc., where the “V” is used to indicate that these are the variances of the estimated limit points.
Forecasting using Models
You cannot forecast directly from an estimated ordered model since the dependent variable represents categorical or rank data. EViews does, however, allow you to forecast the probability associated with each category. To forecast these probabilities, you must first create a
644—Chapter 21. Discrete and Limited Dependent Variable Models
model. Choose Proc/Make Model and EViews will open an untitled model window containing a system of equations, with a separate equation for the probability of each ordered response value.
To forecast from this model, simply click the Solve button in the model window toolbar. If you select Scenario 1 as your solution scenario, the default settings will save your results in a set of named series with “_1” appended to the end of the each underlying name. See Chapter 26, “Models”, beginning on page 777 for additional detail on modifying and solving models.
For this example, the series I_DANGER_1 will contain the fitted linear index i′ ˆ . The fit- x β
ted probability of falling in category 1 will be stored as a series named DANGER_1_1, the fitted probability of falling in category 2 will be stored as a series named DANGER_2_1, and so on. Note that for each observation, the fitted probability of falling in each of the categories sums up to one.
Make Residual Series
The generalized residuals of the ordered model are the derivatives of the log likelihood with respect to a hypothetical unit- x variable. These residuals are defined to be uncorrelated with the explanatory variables of the model (see Chesher and Irish (1987), and Gourieroux, Monfort, Renault and Trognon (1987) for details), and thus may be used in a variety of specification tests.
To create a series containing the generalized residuals, select View/Make Residual Series…, enter a name or accept the default name, and click OK. The generalized residuals for an ordered model are given by:
egi =
where γ0 = −∞ , and γM + 1
ˆ |
ˆ |
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f( γg − xi′ β) − f( γg − 1 |
− xi′β) |
, |
(21.23) |
--------------------------------------------------------------------------ˆ |
ˆ - |
F( γg − xi′ β) − F( γg − 1 |
− xi′β) |
|
= ∞ .
Censored Regression Models
In some settings, the dependent variable is only partially observed. For example, in survey data, data on incomes above a specified level are often top-coded to protect confidentiality. Similarly desired consumption on durable goods may be censored at a small positive or zero value. EViews provides tools to perform maximum likelihood estimation of these models and to use the results for further analysis.
Theory
Consider the following latent variable regression model:
yi = xi′β + σ i , |
(21.24) |
Estimating Censored Models in EViews—645
where σ is a scale parameter. The scale parameter σ is identified in censored and truncated regression models, and will be estimated along with the β .
In the canonical censored regression model, known as the tobit (when there are normally distributed errors), the observed data y are given by:
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if yi ≤ 0 |
yi |
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0 |
= |
y |
(21.25) |
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if y > 0 |
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i |
i |
In other words, all negative values of yi are coded as 0. We say that these data are left censored at 0. Note that this situation differs from a truncated regression model where negative values of yi are dropped from the sample. More generally, EViews allows for both left and right censoring at arbitrary limit points so that:
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≤ c |
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c |
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if y |
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i |
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i |
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i |
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yi |
= |
y |
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if c |
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< y |
≤ c |
(21.26) |
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i |
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i |
i |
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i |
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c |
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if c |
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< y |
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i |
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i |
i |
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where ci , ci are fixed numbers representing the censoring points. If there is no left cen- |
soring, then we can set ci |
= −∞ . If there is no right censoring, then ci |
= ∞ . The |
canonical tobit model is a special case with ci = 0 and ci = ∞ . |
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The parameters β , σ are estimated by maximizing the log likelihood function: |
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N |
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l( β, σ) = |
Σ log f( ( yi − xi′β) ⁄ σ) 1( ci < yi < ci) |
(21.27) |
i = 1
+log ( F( (ci − xi′β) ⁄ σ) ) 1( yi = ci)
+log ( 1− F( ( ci − xi′ β) ⁄ σ)) 1( yi = ci)
where f , F are the density and cumulative distribution functions of , respectively.
Estimating Censored Models in EViews
Consider the model: |
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HRSi = β1 + β2AGEi + β3EDUi + β4KID1i + i , |
(21.28) |
where hours worked (HRS) is left censored at zero. To estimate this model, select Quick/ Estimate Equation… from the main menu. Then from the Equation Estimation dialog,
646—Chapter 21. Discrete and Limited Dependent Variable Models
select the CENSORED estimation method. The dialog will change to provide a number of different input options.
Specifying the Regression Equation
In the Equation specification field, enter the name of the censored dependent variable followed by a list of regressors. In our example, you will enter:
hrs c age edu kid1
Censored estimation only supports specification by list so you may not enter an explicit equation.
Next, select one of the three distributions for the error term. EViews allows you three possible choices for the distribution of :
Standard normal |
E( ) = 0 , var( ) = 1 |
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Logistic |
E( ) = 0 , var( ) = π2 ⁄ 3 |
Extreme value (Type I) |
E( ) ≈ 0.5772 (Euler’s constant), |
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var( ) = π2 ⁄ 6 |
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Specifying the Censoring Points
You must also provide information about the censoring points of the dependent variable. There are two cases to consider: (1) where the limit points are known for all individuals,
Estimating Censored Models in EViews—647
and (2) where the censoring is by indicator and the limit points are known only for individuals with censored observations.
Limit Points Known
You should enter expressions for the left and right censoring points in the edit fields as required. Note that if you leave an edit field blank, EViews will assume that there is no censoring of observations of that type.
For example, in the canonical tobit model the data are censored on the left at zero, and are uncensored on the right. This case may be specified as:
Left edit field: |
0 |
Right edit field: |
[blank] |
Similarly, top-coded censored data may be specified as, |
Left edit field: |
[blank] |
Right edit field: |
20000 |
while the more general case of left and right censoring is given by:
Left edit field: |
10000 |
Right edit field: |
20000 |
EViews also allows more general specifications where the censoring points are known to differ across observations. Simply enter the name of the series or auto-series containing the censoring points in the appropriate edit field. For example:
Left edit field: |
lowinc |
Right edit field: |
vcens1+10 |
specifies a model with LOWINC censoring on the left-hand side, and right censoring at the value of VCENS1+10.
Limit Points Not Known
In some cases, the hypothetical censoring point is unknown for some individuals ( ci and ci are not observed for all observations). This situation often occurs with data where censoring is indicated with a zero-one dummy variable, but no additional information is provided about potential censoring points.
EViews provides you an alternative method of describing data censoring that matches this format. Simply select the Field is zero/one indicator of censoring option in the estimation dialog, and enter the series expression for the censoring indicator(s) in the appropriate edit field(s). Observations with a censoring indicator of one are assumed to be censored while those with a value of zero are assumed to be actual responses.
648—Chapter 21. Discrete and Limited Dependent Variable Models
For example, suppose that we have observations on the length of time that an individual has been unemployed (U), but that some of these observations represent ongoing unemployment at the time the sample is taken. These latter observations may be treated as right censored at the reported value. If the variable RCENS is a dummy variable representing censoring, you can click on the Field is zero/one indicator of censoring setting and enter:
Left edit field: |
[blank] |
Right edit field: |
rcens |
in the edit fields. If the data are censored on both the left and the right, use separate binary indicators for each form of censoring:
Left edit field: |
lcens |
Right edit field: |
rcens |
where LCENS is also a binary indicator.
Once you have specified the model, click OK. EViews will estimate the parameters of the model using appropriate iterative techniques.
A Comparison of Censoring Methods
An alternative to specifying index censoring is to enter a very large positive or negative value for the censoring limit for non-censored observations. For example, you could enter “1e-100” and “1e100” as the censoring limits for an observation on a completed unemployment spell. In fact, any limit point that is “outside” the observed data will suffice.
While this latter approach will yield the same likelihood function and therefore the same parameter values and coefficient covariance matrix, there is a drawback to the artificial limit approach. The presence of a censoring value implies that it is possible to evaluate the conditional mean of the observed dependent variable, as well as the ordinary and standardized residuals. All of the calculations that use residuals will, however, be based upon the arbitrary artificial data and will be invalid.
If you specify your censoring by index, you are informing EViews that you do not have information about the censoring for those observations that are not censored. Similarly, if an observation is left censored, you may not have information about the right censoring limit. In these circumstances, you should specify your censoring by index so that EViews will prevent you from computing the conditional mean of the dependent variable and the associated residuals.
Interpreting the Output
If your model converges, EViews will display the estimation results in the equation window. The first part of the table presents the usual header information, including informa-
Estimating Censored Models in EViews—649
tion about the assumed error distribution, estimation sample, estimation algorithms, and number of iterations required for convergence.
EViews also provides information about the specification for the censoring. If the estimated model is the canonical tobit with left-censoring at zero, EViews will label the method as a TOBIT. For all other censoring methods, EViews will display detailed information about form of the left and/or right censoring.
Here, we have the header output from a left censored model where the censoring is specified by value:
Dependent Variable: Y_PT
Method: ML - Censored Normal (TOBIT) Date: 09/14/97 Time: 08:27
Sample: 1 601
Included observations: 601 Convergence achieved after 8 iterations
Covariance matrix computed using second derivatives
Below the header are the usual results for the coefficients, including the asymptotic standard errors, z-statistics, and significance levels. As in other limited dependent variable models, the estimated coefficients do not have a direct interpretation as the marginal effect of the associated regressor j for individual i , xij . In censored regression models, a change in xij has two effects: an effect on the mean of y , given that it is observed, and an effect on the probability of y being observed (see McDonald and Moffitt, 1980).
In addition to results for the regression coefficients, EViews reports an additional coefficient named SCALE, which is the estimated scale factor σ . This scale factor may be used to estimate the standard deviation of the residual, using the known variance of the assumed distribution. For example, if the estimated SCALE has a value of 0.446 for a model with extreme value errors, the implied standard error of the error term is
0.5977 = 0.466π ⁄ 
6 .
Most of the other output is self-explanatory. As in the binary and ordered models above, EViews reports summary statistics for the dependent variable and likelihood based statistics. The regression statistics at the bottom of the table are computed in the usual fashion,
ˆ |
= |
yi − E( yi |
|
ˆ ˆ |
|
using the residuals i |
|
xi, β, σ) from the observed y . |
Views of Censored Equations
Most of the views that are available for a censored regression are familiar from other settings. The residuals used in the calculations are defined below.
The one new view is the Categorical Regressor Stats view, which presents means and standard deviations for the dependent and independent variables for the estimation sample. EViews provides statistics computed over the entire sample, as well as for the left censored, right censored and non-censored individuals.
