In estimating a null/intercept only model using longitudinal panel data in
person-year format (i.e., long, occasions, repeated measures) and a
difference score as the dependent variable



(Yit - Yit-1) = a + Ui + Eit



the model fails to converge using stata’s xtmixed (“flat and continuous
region”) and using lmer in R the between individual variation (Ui) is
essentially zero. When I include Yit-1 as an independent variable:



(Yit - Yit-1) = a + Yit-1 + Ui + Eit



the model converges just fine and reports what seem to be reasonable results
(and the results in Stata and in R correspond).  Below is an example using
Stata. The dependent variable is the difference in family income from t-1 to
t in thousands of dollars and is grand mean centered.





*. xtsum dinc*



Variable         |      Mean   Std. Dev.       Min        Max |
Observations

-----------------+--------------------------------------------+----------------

dinc     overall | -.1507094   47.16423  -2447.117   2749.135 |     N =
68075

         between |             24.94159   -675.392   638.0056 |     n =
14869

         within  |             44.31102  -2029.263   2437.635 | T-bar =
4.57832



* *

*. xtmixed dinc ||id: *



Performing EM optimization:



Performing gradient-based optimization:



Iteration 0:   log restricted-likelihood = -359298.64

numerical derivatives are approximate

flat or discontinuous region encountered

Iteration 1:   log restricted-likelihood = -358930.79

numerical derivatives are approximate

flat or discontinuous region encountered………………………………………………………………………………….





*. xtmixed dinc laginc ||id:  *



Performing EM optimization:



Performing gradient-based optimization:



Iteration 0:   log restricted-likelihood = -353006.12

Iteration 1:   log restricted-likelihood = -353004.36

Iteration 2:   log restricted-likelihood = -353004.35



Computing standard errors:



Mixed-effects REML regression                   Number of obs      =
68075

Group variable: id                              Number of groups   =
14869



                                                Obs per group: min =
1

                                                               avg =
4.6

                                                               max =
11





                                                Wald chi2(1)       =
19885.84

Log restricted-likelihood = -353004.35          Prob > chi2        =
0.0000



------------------------------------------------------------------------------

        dinc |      Coef.   Std. Err.      z    P>|z|     [95% Conf.
Interval]

-------------+----------------------------------------------------------------

      laginc |  -.4772225   .0033841  -141.02   0.000    -.4838553
-.4705897

       _cons |  -1.546629   .2300348    -6.72   0.000    -1.997489
-1.095769

------------------------------------------------------------------------------



------------------------------------------------------------------------------

  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf.
Interval]

-----------------------------+------------------------------------------------

id: Identity                 |

                   sd(_cons) |   17.95496   .2854801      17.40406
18.5233

-----------------------------+------------------------------------------------

                sd(Residual) |   40.56819   .1264724      40.32107
40.81683

------------------------------------------------------------------------------

LR test vs. linear regression: chibar2(01) =  1940.19 Prob >= chibar2 =
0.0000





I’ve also replicated this pattern using other data (still in person-year
format) and other dependent variables.



I must be missing something obvious here. Does taking the first difference
in a null model essentially condition out the between individual variation
(like a fixed-effects estimator Xit-Xibar, even though there are more than
two occasions with repeated measures)? And why then does the inclusion of a
lagged variable seem to resolve the issue? Or does it?



Any insight or suggested literature on this would be greatly appreciated.



Jeremy


-- 
Jeremy Pais
Doctoral Student
Department of Sociology
University at Albany, SUNY

jeremy.pais01@albany.edu
jpais.albany@gmail.com

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