A puzzle about -reg-

[Yugen Chen]

Dear Statalists,

I have one question regarding the -reg-. It returns a result that seems counterintuitive to me. I would appreciate it if someone could point out something that I missed here.

Outcome variable	Y	at the US tract level
Regressor of interests	X	at the US state level
State name variable	v_StateAbb

I confirm that X has no variations within a state

tab v_StateAbb , su(X)

            |  Summary of AnnualTrendinCVSshares
      State |        Mean   Std. dev.       Freq.
------------+------------------------------------
         AK |   .37829286           0          53
         AL |  -.38800469           0         690
         AR |  -1.5852451           0         403
         AZ |    .5369724           0         755
         CA |   1.9787552           0       3,646
         CO |  -.00790215           0         583
         CT |   .87569064           0         400
         DC |   .19462891           0         104
         DE |   2.1682057           0         123
         FL |   .34600583           0       2,674
         GA |  -1.1858838           0       1,250
         HI |  -.63364446           0         121
         IA |   .51249278           0         437
         ID |  -.11874523           0         193
         IL |   1.1629089           0       1,394
         IN |  -.40260735           0         699
         KS |   .46364346           0         407
         KY |   .07007198           0         585
         LA |  -.29478988           0         631
         MA |   .65337485           0         752
         MD |   .01431732           0         678
         ME |   .87541354           0         169
         MI |  -.23156051           0       1,422
         MN |   .26225701           0         572
         MO |  -.43804073           0         747
         MS |  -.88524896           0         367
         MT |   -.0210947           0         147
         NC |   .22245559           0       1,135
         ND |   1.3994192           0         119
         NE |  -.11140289           0         279
         NH |   .49829119           0         141
         NJ |   .27578011           0       1,226
         NM |   .15972424           0         198
         NV |   1.8037741           0         319
         NY |  -.87026596           0       2,791
         OH |   .60460913           0       1,366
         OK |  -.18099365           0         522
         OR |   .32661819           0         384
         PA |  -.10615063           0       1,660
         RI |   .07889557           0         118
         SC |   .01788273           0         592
         SD |   .08945541           0         111
         TN |  -.31160012           0         753
         TX |  -.18431321           0       3,156
         UT |  -.60220224           0         329
         VA |   .14057216           0         899
         VT |  -1.0188156           0          77
         WA |   .33728275           0         692
         WI |   .02243576           0         631
         WV |  -.49865133           0         274
         WY |   .51382083           0          68
------------+------------------------------------
      Total |   .18485678   .81794065      37,842

When I include X and state dummies in a regression, Stata -reg- should not give me an estimate on X as suggested by the FWL theorem, but I actually have the following results.

 reg Y X i.v_StateAbb
note: 58.v_StateAbb omitted because of collinearity.

      Source |       SS           df       MS      Number of obs   =     9,655
-------------+----------------------------------   F(50, 9604)     =      3.48
       Model |  4.90390359        50  .098078072   Prob > F        =    0.0000
    Residual |  270.859535     9,604  .028202784   R-squared       =    0.0178
-------------+----------------------------------   Adj R-squared   =    0.0127
       Total |  275.763439     9,654  .028564682   Root MSE        =    .16794

------------------------------------------------------------------------------
           Y | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
           X |   1.844638   .9464016     1.95   0.051    -.0105091    3.699785
             |
  v_StateAbb |
         AL  |   1.452647    .801136     1.81   0.070    -.1177481    3.023043
         AR  |   3.622016   1.932963     1.87   0.061    -.1669999    7.411032
         AZ  |  -.2874431   .1004394    -2.86   0.004    -.4843256   -.0905606
         CA  |  -2.916875   1.442792    -2.02   0.043    -5.745052   -.0886981
         CO  |   .8158381   .4439045     1.84   0.066    -.0543083    1.685985
         CT  |  -.9001777   .4026835    -2.24   0.025    -1.689522   -.1108329
         DC  |    .377255    .256196     1.47   0.141    -.1249433    .8794533
         DE  |  -3.206503   1.622339    -1.98   0.048     -6.38663   -.0263755
         FL  |   .0838297   .1218526     0.69   0.491     -.155027    .3226865
         GA  |    2.89117   1.554955     1.86   0.063    -.1568707    5.939211
         HI  |   1.912112   1.033152     1.85   0.064    -.1130829    3.937307
         IA  |  -.2475502   .0877354    -2.82   0.005    -.4195302   -.0755703
         ID  |   1.166855   .5537868     2.11   0.035     .0813161    2.252394
         IL  |   -1.41285   .6723271    -2.10   0.036    -2.730753   -.0949466
         IN  |    1.45249   .8148669     1.78   0.075    -.1448211    3.049801
         KS  |  -.1574409   .0678766    -2.32   0.020    -.2904933   -.0243886
         KY  |   .5938723    .370954     1.60   0.109    -.1332758     1.32102
         LA  |   1.249468   .7132864     1.75   0.080     -.148724     2.64766
         MA  |  -.4566652   .1977022    -2.31   0.021    -.8442032   -.0691272
         MD  |   .7202919   .4227036     1.70   0.088    -.1082963     1.54888
         ME  |   -.878546   .4035683    -2.18   0.030    -1.669625    -.087467
         MI  |   1.135163   .6536229     1.74   0.082    -.1460762    2.416401
         MN  |   .2140441   .1943763     1.10   0.271    -.1669744    .5950626
         MO  |   1.521465   .8483378     1.79   0.073    -.1414564    3.184386
         MS  |   2.330777   1.270893     1.83   0.067    -.1604409    4.821995
         MT  |   .7367254   .4576467     1.61   0.107    -.1603587    1.633809
         NC  |   .3071262    .229894     1.34   0.182    -.1435145    .7577669
         ND  |  -1.883608   .8979758    -2.10   0.036     -3.64383   -.1233861
         NE  |   .9033112   .5413516     1.67   0.095    -.1578521    1.964475
         NH  |  -.1324646   .0792314    -1.67   0.095    -.2877748    .0228456
         NJ  |   .2406748   .1819778     1.32   0.186      -.11604    .5973897
         NM  |   .4031799   .2892051     1.39   0.163    -.1637232     .970083
         NV  |  -2.629496   1.277481    -2.06   0.040    -5.133628   -.1253645
         NY  |   2.360282   1.256556     1.88   0.060    -.1028333    4.823396
         OH  |  -.4121096   .1547749    -2.66   0.008     -.715501   -.1087182
         OK  |   1.031681   .6062353     1.70   0.089    -.1566682     2.22003
         OR  |   .3022176   .1411878     2.14   0.032     .0254598    .5789754
         PA  |    .915221   .5356034     1.71   0.088    -.1346748    1.965117
         RI  |   .5681525   .3628655     1.57   0.117    -.1431404    1.279446
         SC  |   .6933975   .4193797     1.65   0.098    -.1286752     1.51547
         SD  |   .5328004   .3655079     1.46   0.145    -.1836722    1.249273
         TN  |   1.296993   .7290961     1.78   0.075    -.1321894    2.726175
         TX  |   1.066651   .6090893     1.75   0.080     -.127293    2.260594
         UT  |    1.92987   1.003672     1.92   0.055    -.0375378    3.897278
         VA  |   .4708615   .3050879     1.54   0.123    -.1271752    1.068898
         VT  |   2.577159   1.398083     1.84   0.065    -.1633792    5.317697
         WA  |   .0756488    .131186     0.58   0.564    -.1815034     .332801
         WI  |   .6684756     .41537     1.61   0.108    -.1457372    1.482688
         WV  |   1.717644   .9057221     1.90   0.058    -.0577621    3.493051
         WY  |          0  (omitted)
             |
       _cons |  -.6978133   .4359562    -1.60   0.109    -1.552379    .1567529
------------------------------------------------------------------------------

[Clyde Schechter]

There is a colinearity among X, the constant term, and the state variables that leaves the model unidentified. To produce some results, (at least) one of those variable has to go. You were expecting that Stata would identify the model by removing X. Instead, it chose to remove WY. And, no, that's not the usual removal of one from a group of "dummy variables," because if you look carefully at your output, AK is also absent. Removal of AK is the standard removal of one from a group of "dummies"; removal of WY is the additional variable removed to identify the model.

Remember that it doesn't matter which way Stata breaks the colinearity for the purpose of identifying the model. The resulting models are always just different parameterizations of the same model and all model-level results, and variable-level results for the variables not involved in the colinearity, are the same regardless. Crucially, you must also bear in mind that regardless of which variable is omitted to break colinearity (or if the colinearity is broken by imposing some other linear constraint(s)), the results for the variables involved in the colinearity are meaningless and should be disregarded.

If you want Stata to remove X instead of WY, rewrite the command as -reg Y i.v_StateAbb X-, because Stata usually breaks colinearities by removing the involved variable closest to the end of the regressor variable list in the command. But I can't emphasize enough that there is no statistical reason to care which variable gets removed--it is purely a matter of taste because the effects of the retained uninvolved variables are not estimable anyway and the "results" for them are just artifacts of the particular way the colinearity got broken..

[Yugen Chen]

There is a colinearity among X, the constant term, and the state variables that leaves the model unidentified. To produce some results, (at least) one of those variable has to go. You were expecting that Stata would identify the model by removing X. Instead, it chose to remove WY. And, no, that's not the usual removal of one from a group of "dummy variables," because if you look carefully at your output, AK is also absent. Removal of AK is the standard removal of one from a group of "dummies"; removal of WY is the additional variable removed to identify the model.

Remember that it doesn't matter which way Stata breaks the colinearity for the purpose of identifying the model. The resulting models are always just different parameterizations of the same model and all model-level results, and variable-level results for the variables not involved in the colinearity, are the same regardless. Crucially, you must also bear in mind that regardless of which variable is omitted to break colinearity (or if the colinearity is broken by imposing some other linear constraint(s)), the results for the variables involved in the colinearity are meaningless and should be disregarded.

If you want Stata to remove X instead of WY, rewrite the command as -reg Y i.v_StateAbb X-, because Stata usually breaks colinearities by removing the involved variable closest to the end of the regressor variable list in the command. But I can't emphasize enough that there is no statistical reason to care which variable gets removed--it is purely a matter of taste because the effects of the retained uninvolved variables are not estimable anyway and the "results" for them are just artifacts of the particular way the colinearity got broken..

I'm grateful for this detailed explanation. Thank you, Professor Schechter.

[Carlo Lazzaro]

Yugen:
possibly OOT, but why not clustering your SEs at US States level?

[Yugen Chen]

Yugen:
possibly OOT, but why not clustering your SEs at US States level?

Thank you, Dr Lazzaro! I wanted to keep my regression as simple as possible in the post. My question was about the coefficient estimation, so I was sloppy on the estimation for the variance matrix. But yes, a serious practice should use the cluster robust variance estimates at the state level.