Re: st: intreg cluster vs tobit cluster (one reports some missing std errors,

[jpitblado@stata.com (Jeff Pitblado, StataCorp LP)]

Leandro Brufman <leandrobrufman@gmail.com> wrote in about -intreg- reporting
missing standard error values compared to -tobit-.

Short reply
-----------------------------------------------------------------------------

-tobit- and -intreg- are fitting the same model in Leandro's case; the problem
is that the robust VCE is not full rank.

The reduced rank causes -intreg- to report a missing value in Leandro's
example, because -intreg- fits the model using a more convergent
parameterization of the model then transforms the result into the standard
regression parameters we are all familiar with.  The reduced rank of the
original -e(V)- yields a zero along the diagonal of the transformed -e(V)-.

Long reply
-----------------------------------------------------------------------------

Leandro's example was the following:

***** BEGIN:
. intreg amt_mt2 amt_mt3 ip_vsam_ipolate_w cridum ccc*, cluster(cricode)

Interval regression                               Number of obs   =       6757
                                                  Wald chi2(11)   =     996.00
Log pseudolikelihood = -1331.6745                 Prob > chi2     =     0.0000

                               (Std. Err. adjusted for 21 clusters in cricode)
------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ip_vsam_ip~w |  -.0263045          .        .       .            .           .
      cridum |  -156.2131    50.9088    -3.07   0.002    -255.9925   -56.43371
        ccc1 |   219.9064   70.44264     3.12   0.002     81.84139    357.9715
        ccc2 |   442.6939   147.2167     3.01   0.003     154.1545    731.2333
        ccc3 |   97.37137   32.76476     2.97   0.003     33.15362    161.5891
        ccc4 |   555.3243    185.262     3.00   0.003     192.2175    918.4311
        ccc5 |   421.7124   142.3647     2.96   0.003     142.6827    700.7422
        ccc6 |    269.146   87.67189     3.07   0.002     97.31224    440.9797
        ccc8 |    96.5494   37.97573     2.54   0.011     22.11834    170.9804
        ccc9 |   36.67169   13.64486     2.69   0.007     9.928247    63.41513
       ccc10 |   179.9544   60.99526     2.95   0.003     60.40586    299.5029
       ccc11 |   58.70021   28.58481     2.05   0.040      2.67502    114.7254
       _cons |  -300.8123   94.71781    -3.18   0.001    -486.4558   -115.1688
-------------+----------------------------------------------------------------
    /lnsigma |   4.989588   .3018269    16.53   0.000     4.398018    5.581157
-------------+----------------------------------------------------------------
       sigma |   146.8758   44.33108                      81.28957    265.3786
------------------------------------------------------------------------------

  Observation summary:      6613  left-censored observations
                             144     uncensored observations
                               0 right-censored observations
                               0       interval observations
***** END:

Leandro sent us a copy of the data.

The above model had 14 parameters, so it would seem that 21 clusters should be
enough to get a full rank VCE matrix; however, the rank of -e(V)- for the
above model is 13.

We compared the values of the 'ccc*' variables to the levels of 'cricode' and
found that each 'ccc*' variable was uniquely identifying one or two levels of
'cricode'.  Here was the pattern we discovered:

	cricode value		indicated by
	---------------------------------------------------------------------
	0-1			ccc1
	2-3			ccc2
	4-5			ccc3
	6-7			ccc4
	8-9			ccc5
	10-11			ccc6
	12-13			ccc7
	14-15			ccc8
	16-17			ccc9
	18			ccc10
	20-21			ccc11

This pattern leads to the reduced rank of the robust VCE.

-tobit- posted 14 for -e(rank)-, but that number comes from the model based
VCE; the recalculation of the rank of the robust -e(V)- from Leandro's -tobit-
model fit yields a rank of 13.  We will fix -tobit- in a future ado-file
update.

When specified without constraints, starting values, or the -het()- option,
-intreg- fits the model parameters according to the following
parameterization

(0)	b/sigma, 1/sigma

where 'b' is the vector of regression coefficients and 'sigma' is the standard
deviation of the errors.  After the model is fit, -intreg- transforms this
parameterization to

(1)	b, ln(sigma)

If V0 is the VCE for the original model fit, and V1 is the VCE for the
transformed model, then

	V1 = {L (V0)^-1 L'}^-1

Since V0 is not full rank, (V0)^-1 will be a generalized inverse, meaning it
will contain one or more zeros along the diagonal.  Zero variances associated
with nonzero coefficients are reported as missing values.

We will look into computing V1 in a manner that doesn't require us to
invert V0 when it is not full rank.

--Jeff
jpitblado@stata.com
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