Perfect multicollinearity within indicator variables

[Paul Shlapa]

Hello everyone,
I do a simple cross section on wage determinants among individuals. Part of my data includes dummies for regions of work (1-9) and part_time occupation (1 for part-time, 0 full). Problem occurs when trying to use them all at once, even if leaving one region out or using nocon specification (as region 1-6 = part-time only and reg 7-9 = full-time only). How could one construct a regression (if possible) that still allows to estimate difference between every region and additionally part_time influence on wages.

[Carlo Lazzaro]

Paul:
first of all, take a look at -fvvarlist- notation.
Second, please post what you typed and what Stata gave you back within CODE delimiters (as per FAQ). Thanks.

[Paul Shlapa]

Hope this works

* Example generated by -dataex-. To install: ssc install dataex
clear
input float(id nearc4 educ age reg661 reg662 reg663 reg664 reg665 reg666 reg667 reg668 reg669 part_time hisp union wage IQ married female services region pwe pwe2 lw female_educ female_hisp)
4171 0  9 32 0 0 0 0 0 0 0 0 1 1 0 0  635  74 1 0 0 9 -29 841 6.453625  0 0
2828 1 18 32 0 0 0 0 1 0 0 0 0 0 0 0  535 129 1 0 1 5 -20 400 6.282267  0 0
1430 1 16 26 0 0 1 0 0 0 0 0 0 1 0 1  300 106 0 0 0 3 -16 256 5.703783  0 0
1711 1 13 24 0 0 1 0 0 0 0 0 0 1 0 1  769 102 1 1 0 3 -17 289 6.645091 13 0
3993 1 15 26 0 0 0 0 0 0 0 0 1 1 0 1  263 101 0 0 0 9 -17 289 5.572154  0 0
1912 1 12 28 0 0 1 0 0 0 0 0 0 1 0 1  500  79 1 0 0 3 -22 484 6.214608  0 0
1559 1 12 34 0 0 1 0 0 0 0 0 0 1 0 1  962   . 1 0 0 3 -28 784 6.869014  0 0
 917 0 17 31 0 1 0 0 0 0 0 0 0 1 0 1  625  96 1 1 0 2 -20 400 6.437752 17 0
3671 1 13 25 0 0 0 0 0 0 0 0 1 1 0 1  845 115 1 1 0 9 -18 324 6.739336 13 0
2116 1 12 28 0 0 1 0 0 0 0 0 0 1 0 1 1251  94 1 0 0 3 -22 484 7.131699  0 0
 505 0 17 24 0 1 0 0 0 0 0 0 0 1 0 1  525   . 1 0 0 2 -13 169 6.263398  0 0
3729 1 18 27 0 0 0 0 0 0 0 0 1 1 0 1  643 118 0 0 0 9 -15 225 6.466145  0 0
2610 1 12 30 0 0 0 0 0 0 1 0 0 0 0 1  581   . 1 0 1 7 -24 576 6.364751  0 0
3079 0  8 26 0 0 0 0 1 0 0 0 0 0 1 1  285   . 1 0 1 5 -24 576 5.652489  0 0
4257 1 14 25 0 0 0 0 1 0 0 0 0 0 0 1  529 118 1 0 1 5 -17 289 6.270988  0 0
2091 1 11 32 0 0 1 0 0 0 0 0 0 1 0 1  726  93 1 0 0 3 -27 729  6.58755  0 0
1076 1 16 32 0 0 1 0 0 0 0 0 0 1 1 1  797   . 0 0 0 3 -22 484 6.680855  0 0
  30 1 15 32 1 0 0 0 0 0 0 0 0 1 0 0  726 111 1 0 0 1 -23 529  6.58755  0 0
1395 1 16 34 0 0 1 0 0 0 0 0 0 1 0 1 1442   . 1 0 0 3 -24 576 7.273787  0 0
 487 1 16 24 0 1 0 0 0 0 0 0 0 1 0 1  750 112 0 1 0 2 -14 196 6.620073 16 0
2315 0 12 24 0 0 1 0 0 0 0 0 0 1 0 1  160  83 0 0 0 3 -18 324 5.075174  0 0
2424 1 12 33 0 0 0 0 0 0 1 0 0 0 0 1 1100  89 1 0 1 7 -27 729 7.003066  0 0
4377 1 12 32 0 0 0 0 1 0 0 0 0 0 0 1  596  98 0 0 1 5 -26 676 6.390241  0 0
 283 1 16 24 0 1 0 0 0 0 0 0 0 1 0 0  385 118 0 1 0 2 -14 196 5.953243 16 0
2925 1 13 32 0 0 0 0 1 0 0 0 0 0 0 1  962  94 1 0 1 5 -25 625 6.869014  0 0
2131 1 17 28 0 0 0 1 0 0 0 0 0 1 0 0  423   . 1 0 0 4 -17 289 6.047372  0 0
 561 1 13 32 0 1 0 0 0 0 0 0 0 1 0 1  389   . 1 0 0 2 -25 625 5.963579  0 0
4879 0 16 32 0 0 0 0 1 0 0 0 0 0 0 0  738 107 1 0 1 5 -22 484 6.603944  0 0
 516 1 12 24 1 0 0 0 0 0 0 0 0 1 0 1  488   . 1 0 0 1 -18 324 6.190315  0 0
3945 1 12 29 0 0 0 0 0 0 0 0 1 1 0 1 1200 100 1 0 0 9 -23 529 7.090077  0 0
3267 1 10 27 0 0 0 0 1 0 0 0 0 0 1 0  225   . 1 0 1 5 -23 529 5.416101  0 0
2764 1 12 24 0 0 0 0 0 0 1 0 0 0 0 1  397   . 1 0 1 7 -18 324 5.983936  0 0
5123 0 10 25 0 0 0 0 0 0 1 0 0 0 1 0  200  75 1 0 1 7 -21 441 5.298317  0 0
1155 1 16 25 0 0 1 0 0 0 0 0 0 1 0 1  243 121 0 0 0 3 -15 225 5.493062  0 0
3305 1 12 28 0 0 0 0 0 0 1 0 0 0 1 1  500  65 1 0 1 7 -22 484 6.214608  0 0
2143 1 16 27 0 0 0 1 0 0 0 0 0 1 0 0  396 118 1 0 0 4 -17 289 5.981414  0 0
2621 1 18 29 0 0 0 0 0 0 1 0 0 0 1 1  750  96 1 0 1 7 -17 289 6.620073  0 0
  11 1 12 29 0 1 0 0 0 0 0 0 0 1 0 1  515  97 1 0 0 2 -23 529 6.244167  0 0
3684 1 16 27 0 0 0 0 0 0 0 0 1 1 0 1  577 117 0 0 0 9 -17 289 6.357842  0 0
3329 1 12 25 0 0 0 0 1 0 0 0 0 0 0 1  346   . 1 1 1 5 -19 361 5.846439 12 0
4701 0 12 26 0 0 0 0 1 0 0 0 0 0 1 1  475  79 0 0 0 5 -20 400 6.163315  0 0
2803 1 16 24 0 0 0 0 0 1 0 0 0 0 0 1  400   . 1 0 1 6 -14 196 5.991465  0 0
 904 0 12 26 0 1 0 0 0 0 0 0 0 1 0 1  514   . 0 0 0 2 -20 400 6.242223  0 0
4164 0 10 34 0 0 0 0 0 0 0 0 1 1 0 0  575   . 1 0 0 9 -30 900  6.35437  0 0
   5 1 11 27 0 1 0 0 0 0 0 0 0 1 0 1  250  88 1 0 0 2 -22 484 5.521461  0 0
3206 1 16 27 0 0 0 0 0 0 1 0 0 0 0 1  361 112 0 0 1 7 -17 289 5.888878  0 0
4974 0 12 25 0 0 0 0 0 1 0 0 0 0 1 0  250   . 1 0 1 6 -19 361 5.521461  0 0
2146 1 13 31 0 0 0 1 0 0 0 0 0 1 0 0  450   . 1 0 0 4 -24 576 6.109248  0 0
2602 1 13 24 0 0 0 0 0 0 1 0 0 0 1 1  435   . 0 0 1 7 -17 289 6.075346  0 0
4659 0  9 30 0 0 0 0 1 0 0 0 0 0 1 1  350   . 0 0 1 5 -27 729 5.857933  0 0
2098 1 15 28 0 0 1 0 0 0 0 0 0 1 0 1  423 118 0 0 0 3 -19 361 6.047372  0 0
2176 1 12 28 0 0 1 0 0 0 0 0 0 1 0 1  230  74 0 0 0 3 -22 484 5.438079  0 0
2470 1 10 30 0 0 0 0 0 1 0 0 0 0 1 1  520   . 1 0 1 6 -26 676 6.253829  0 0
4342 1 15 24 0 0 0 0 0 0 1 0 0 0 1 1  400   . 0 0 1 7 -15 225 5.991465  0 0
4470 1 12 28 0 0 0 0 1 0 0 0 0 0 0 0  465 118 1 0 1 5 -22 484 6.142037  0 0
3040 1  8 29 0 0 0 0 1 0 0 0 0 0 0 0  528   . 1 0 1 5 -27 729 6.269096  0 0
2737 0 16 25 0 0 0 0 0 1 0 0 0 0 1 1  250  79 1 0 1 6 -15 225 5.521461  0 0
3646 1 16 29 0 0 0 0 0 0 0 0 1 1 0 1  824 106 0 0 0 9 -19 361  6.71417  0 0
1655 1 12 28 0 0 1 0 0 0 0 0 0 1 0 0  823  91 1 0 0 3 -22 484 6.712956  0 0
 838 1 14 33 0 1 0 0 0 0 0 0 0 1 0 1 1402   . 1 0 0 2 -25 625 7.245655  0 0
3346 1 13 34 0 0 0 0 1 0 0 0 0 0 0 1  519  98 1 0 1 5 -27 729 6.251904  0 0
   4 0 12 34 1 0 0 0 0 0 0 0 0 1 0 1  721 103 1 0 0 1 -28 784 6.580639  0 0
4639 0 12 24 0 0 0 0 1 0 0 0 0 0 1 0  278   . 0 0 1 5 -18 324 5.627621  0 0
2237 0 11 32 0 0 1 0 0 0 0 0 0 1 0 0  433   . 1 0 0 3 -27 729 6.070738  0 0
2021 1 12 24 0 0 1 0 0 0 0 0 0 1 0 1  721 102 1 0 0 3 -18 324 6.580639  0 0
4906 0 12 26 0 0 0 0 1 0 0 0 0 0 0 0  600 102 1 0 1 5 -20 400  6.39693  0 0
 954 0 12 28 0 1 0 0 0 0 0 0 0 1 0 0  863 110 1 0 0 2 -22 484 6.760415  0 0
2358 1 12 34 0 0 0 1 0 0 0 0 0 1 0 1  400   . 1 0 0 4 -28 784 5.991465  0 0
   7 1 12 26 0 1 0 0 0 0 0 0 0 1 0 1  500  85 1 0 0 2 -20 400 6.214608  0 0
4698 0 15 24 0 0 0 0 1 0 0 0 0 0 0 0  300  96 0 0 1 5 -15 225 5.703783  0 0
 750 1 15 27 0 1 0 0 0 0 0 0 0 1 1 1  500  97 0 0 0 2 -18 324 6.214608  0 0
3943 1 18 33 0 0 0 0 0 0 0 0 1 1 0 1  423   . 1 0 0 9 -21 441 6.047372  0 0
2074 0 18 29 0 0 1 0 0 0 0 0 0 1 0 1  759 111 1 0 0 3 -17 289 6.632002  0 0
5076 0 17 34 0 0 0 0 1 0 0 0 0 0 0 1  635   . 1 0 1 5 -23 529 6.453625  0 0
 618 1 16 28 0 1 0 0 0 0 0 0 0 1 0 1 1410 102 1 0 0 2 -18 324 7.251345  0 0
1264 1 17 25 0 0 1 0 0 0 0 0 0 1 0 0  329 117 1 0 0 3 -14 196 5.796058  0 0
3837 1 15 28 0 0 0 0 0 0 0 0 1 1 0 1  250  87 1 0 0 9 -19 361 5.521461  0 0
 889 0 12 28 0 1 0 0 0 0 0 0 0 1 0 1  275  75 0 0 0 2 -22 484 5.616771  0 0
3931 1 13 29 0 0 0 0 0 0 0 0 1 1 0 1  675 121 1 0 0 9 -22 484 6.514713  0 0
 366 1 12 27 1 0 0 0 0 0 0 0 0 1 0 1  615 114 1 0 0 1 -21 441 6.421622  0 0
3658 1 17 31 0 0 0 0 0 0 0 0 1 1 0 0  865 119 0 0 0 9 -20 400  6.76273  0 0
1310 1 14 33 0 0 1 0 0 0 0 0 0 1 0 1 1055  98 1 0 0 3 -25 625 6.961296  0 0
1287 1 15 25 0 0 1 0 0 0 0 0 0 1 0 1  750   . 1 0 0 3 -16 256 6.620073  0 0
 324 1 16 25 1 0 0 0 0 0 0 0 0 1 0 1  250 146 0 0 0 1 -15 225 5.521461  0 0
3127 0 12 24 0 0 0 0 0 0 1 0 0 0 1 0  305   . 1 0 1 7 -18 324 5.720312  0 0
2821 1 14 24 0 0 0 0 1 0 0 0 0 0 0 1  452   . 1 0 1 5 -16 256 6.113682  0 0
2406 1 13 26 0 0 0 0 1 0 0 0 0 0 0 0  905 106 1 0 1 5 -19 361 6.807935  0 0
1081 1 12 28 0 0 1 0 0 0 0 0 0 1 1 0  503 109 1 0 1 3 -22 484  6.22059  0 0
2216 0 12 30 0 0 1 0 0 0 0 0 0 1 0 1  600  86 1 0 0 3 -24 576  6.39693  0 0
1033 0 16 33 1 0 0 0 0 0 0 0 0 1 0 0  692 105 1 0 0 1 -23 529 6.539586  0 0
2399 1 16 26 0 0 0 0 1 0 0 0 0 0 0 0  673 107 0 0 1 5 -16 256 6.511745  0 0
4591 0 12 32 0 0 0 0 0 1 0 0 0 0 0 1  848 102 1 0 1 6 -26 676 6.742881  0 0
2783 1 13 29 0 0 0 0 1 0 0 0 0 0 1 1  540   . 0 0 1 5 -22 484 6.291569  0 0
3012 0 16 25 0 0 0 0 0 1 0 0 0 0 0 0  909 124 0 0 1 6 -15 225 6.812345  0 0
4637 0 12 24 0 0 0 0 1 0 0 0 0 0 1 0  396   . 0 0 1 5 -18 324 5.981414  0 0
2169 0 15 30 0 0 0 1 0 0 0 0 0 1 0 1  346 104 0 0 0 4 -21 441 5.846439  0 0
1791 1 16 28 0 0 0 1 0 0 0 0 0 1 0 0  491  94 1 0 0 4 -18 324 6.196444  0 0
 368 1 10 32 1 0 0 0 0 0 0 0 0 1 0 1  670  73 0 0 0 1 -28 784 6.507277  0 0
3455 1 13 25 0 0 0 0 0 0 1 0 0 0 1 1  617  95 1 0 1 7 -18 324 6.424869  0 0
 147 1 16 28 0 1 0 0 0 0 0 0 0 1 0 1  788 103 1 0 0 2 -18 324 6.669498  0 0
end

.
. reg lw educ pwe pwe2 female female_educ ib(none).region hisp union married services female_hisp part_time, nocon
note: part_time omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =     3,000
-------------+----------------------------------   F(19, 2981)     =  48897.71
       Model |  117928.977        19  6206.78825   Prob > F        =    0.0000
    Residual |  378.390633     2,981  .126934127   R-squared       =    0.9968
-------------+----------------------------------   Adj R-squared   =    0.9968
       Total |  118307.367     3,000  39.4357891   Root MSE        =    .35628

------------------------------------------------------------------------------
          lw |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        educ |   .0762595   .0034969    21.81   0.000     .0694028    .0831162
         pwe |  -.1124042   .0143021    -7.86   0.000    -.1404472   -.0843611
        pwe2 |  -.0017198   .0003231    -5.32   0.000    -.0023533   -.0010864
      female |   .2465791   .1817885     1.36   0.175    -.1098645    .6030228
 female_educ |  -.0260866   .0121901    -2.14   0.032    -.0499885   -.0021847
             |
      region |
          1  |   3.400629   .1730487    19.65   0.000     3.061322    3.739936
          2  |   3.529841   .1705954    20.69   0.000     3.195344    3.864338
          3  |   3.583971   .1709269    20.97   0.000     3.248825    3.919118
          4  |   3.501226   .1713533    20.43   0.000     3.165243    3.837209
          5  |   3.486608   .1713174    20.35   0.000     3.150695     3.82252
          6  |   3.523101   .1713432    20.56   0.000     3.187138    3.859064
          7  |    3.46762   .1725126    20.10   0.000     3.129364    3.805875
          8  |   3.342965   .1749211    19.11   0.000     2.999987    3.685944
          9  |   3.585189   .1716193    20.89   0.000     3.248685    3.921694
             |
        hisp |  -.1602946   .0185927    -8.62   0.000    -.1967505   -.1238387
       union |   .1888312   .0152923    12.35   0.000     .1588467    .2188157
     married |   .1602008   .0148949    10.76   0.000     .1309954    .1894061
    services |  -.0795046   .0252685    -3.15   0.002      -.12905   -.0299592
 female_hisp |   .1371725   .0580395     2.36   0.018     .0233709    .2509741
   part_time |          0  (omitted)
------------------------------------------------------------------------------

Part_time is ommited, as one sees

[Carlo Lazzaro]

Paul:
your model has serious quasi-extreme multicollinearity issue (as it is also implicitly highlighgted by your sky-rocketing R-sq):

. estat vif, uncentered

    Variable |       VIF       1/VIF 
-------------+----------------------
        educ |     69.61    0.014366
         pwe |   2969.67    0.000337
        pwe2 |    819.65    0.001220
      female |     77.61    0.012885
 female_educ |     78.61    0.012720
      region |
          1  |     60.39    0.016558
          2  |    130.30    0.007675
          3  |    180.23    0.005549
          4  |     53.31    0.018759
          5  |    199.58    0.005011
          6  |     51.67    0.019355
          7  |     98.67    0.010135
          9  |    104.31    0.009587
        hisp |      2.04    0.489131
       union |      4.26    0.234881
     married |      4.31    0.232039
    services |     21.99    0.045482
-------------+----------------------
    Mean VIF |    289.78

You should probably think about a different specification.

[Paul Shlapa]

Here is another specification, using regions as binary, not categorical variables and without constant. An issue with high multicollinearity is not of a concern. What i need is to figure out if it is possible to do a regression and still include part_time as an explanatory variable, but thank you anyways for taking a look at it.

. reg lw educ pwe pwe2 female female_educ reg661-reg669 hisp union married services female_hisp part_time, robust
note: reg661 omitted because of collinearity
note: reg667 omitted because of collinearity

Linear regression                               Number of obs     =      3,000
                                                F(18, 2981)       =      90.22
                                                Prob > F          =     0.0000
                                                R-squared         =     0.3438
                                                Root MSE          =     .35628

------------------------------------------------------------------------------
             |               Robust
          lw |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        educ |   .0762595   .0035812    21.29   0.000     .0692375    .0832815
         pwe |  -.1124042   .0143322    -7.84   0.000    -.1405061   -.0843022
        pwe2 |  -.0017198   .0003233    -5.32   0.000    -.0023537    -.001086
      female |   .2465791   .1922923     1.28   0.200    -.1304599    .6236182
 female_educ |  -.0260866   .0136013    -1.92   0.055    -.0527555    .0005823
      reg661 |          0  (omitted)
      reg662 |   .1292123   .0338639     3.82   0.000     .0628134    .1956113
      reg663 |   .1833427   .0331685     5.53   0.000     .1183073    .2483781
      reg664 |   .1005972   .0386748     2.60   0.009     .0247653    .1764291
      reg665 |   .0189879   .0242325     0.78   0.433    -.0285263    .0665021
      reg666 |   .0554813   .0299734     1.85   0.064    -.0032894     .114252
      reg667 |          0  (omitted)
      reg668 |  -.0576635   .0481434    -1.20   0.231     -.152061    .0367341
      reg669 |   .1845606   .0376765     4.90   0.000     .1106859    .2584352
        hisp |  -.1602946   .0187619    -8.54   0.000    -.1970822    -.123507
       union |   .1888312   .0145794    12.95   0.000     .1602445    .2174179
     married |   .1602008   .0154207    10.39   0.000     .1299645    .1904371
    services |  -.0795046   .0269647    -2.95   0.003    -.1323759   -.0266333
 female_hisp |   .1371725   .0475706     2.88   0.004     .0438979    .2304471
   part_time |   -.066991   .0442425    -1.51   0.130      -.15374     .019758
       _cons |    3.46762    .174092    19.92   0.000     3.126267    3.808972
------------------------------------------------------------------------------

[Clyde Schechter]

If you cross-tabulate part_time and region you will see:

           |                                         region
 part_time |         1          2          3          4          5          6          7          9 |     Total
-----------+----------------------------------------------------------------------------------------+----------
         0 |         0          0          0          0         22          6         11          0 |        39 
         1 |         7         15         21          6          0          0          0         12 |        61 
-----------+----------------------------------------------------------------------------------------+----------
     Total |         7         15         21          6         22          6         11         12 |       100

In other words, all of your part_time observations come from regions 1 through 4 and 9, and all of your non-part-time come from 5 through 7. So, mathematically, your part_time variable is actually an indicator ("dummy") for a subset of your regions. In that case, it is, inevitably, colinear with the region indicators themselves.

Consequently, it is mathematically impossible to include all of the regions (or even all of the regions leaving out one as a reference category) and include a subset indicator as well. So you can keep i.part_time in your model if you like, but you will have to sacrifice a region indicator to do that. If you re-arrange the order of the variables in your regression so that i.part_time comes before i.region, then part_time will be retained, and Stata will drop an additional region.

[Paul Shlapa]

Thank you all for your replies and time consumed, very very helpful