Question About Specifying Omitted Interaction Term

[David Beheshti]

I have state-by-year panel data, and I'm running a regression where I want to interact year fixed effects with with a continuous treatment variable which is constant within states. I'd like to normalize the 2009 interaction to be equal to zero, but am running into problems. The code below illustrates my two attempts:

* Example generated by -dataex-. To install: ssc install dataex
clear
input double STFIPS float year byte xvar float yvar
 2 2005 91     40.35
 2 2006 91      40.5
 2 2007 91     40.65
 2 2008 91      41.4
 2 2009 91      37.5
 2 2010 91     39.75
 2 2011 91     44.55
 2 2012 91      44.7
 5 2005 64   15.5847
 5 2006 64 24.054644
 5 2007 64  22.86885
 5 2008 64  18.59153
 5 2009 64  21.51366
 5 2010 64 22.572403
 5 2011 64 16.643442
 5 2012 64 15.669398
 6 2005 46  25.89734
 6 2006 46  24.80797
 6 2007 46  23.52053
 6 2008 46  23.37198
 6 2009 46 20.400965
 6 2010 46  19.65821
 6 2011 46 28.769323
 6 2012 46  29.90821
 8 2005 60  22.91542
 8 2006 60   20.4602
 8 2007 60  23.03234
 8 2008 60  27.35821
 8 2009 60  26.07214
 8 2010 60 26.422886
 8 2011 60  27.94279
 8 2012 60 31.274876
 9 2005 89  42.71393
 9 2006 89   48.4602
 9 2007 89  36.87189
 9 2008 89  52.96144
 9 2009 89  41.08582
 9 2010 89  39.55348
 9 2011 89 35.339554
 9 2012 89  38.59577
10 2005 69  49.96875
10 2006 69  49.75521
10 2007 69 32.565105
10 2008 69  38.33073
10 2009 69  41.21354
10 2010 69 36.622395
10 2011 69     30.75
10 2012 69  41.85417
11 2005 47   22.8701
11 2006 47   19.7549
11 2007 47  23.02206
11 2008 47 24.845587
11 2009 47  25.30147
11 2010 47  21.73039
11 2011 47  27.27696
11 2012 47  33.12745
12 2005 62 20.906746
12 2006 62 20.377975
12 2007 62  19.36111
12 2008 62 19.849205
12 2009 62  19.07639
12 2010 62 18.710318
12 2011 62 18.669641
12 2012 62 19.320436
13 2005 54  24.41026
13 2006 54  22.71795
13 2007 54 22.615385
13 2008 54  21.48718
13 2009 54 19.641026
13 2010 54        20
13 2011 54  22.66667
13 2012 54  25.02564
15 2005 55 36.111115
15 2006 55  21.01852
15 2007 55 21.481483
15 2008 55 22.685186
15 2009 55 21.481483
15 2010 55  21.38889
15 2011 55 18.796297
15 2012 55 18.055557
16 2005 58  52.45161
16 2006 58 34.258064
16 2007 58   29.6129
16 2008 58 34.258064
16 2009 58  34.45161
16 2010 58 29.709677
16 2011 58  28.16129
16 2012 58  21.87097
17 2005 36  31.25126
17 2006 36  31.45202
17 2007 36  28.84217
17 2008 36  30.64899
17 2009 36 32.790405
17 2010 36 32.857323
17 2011 36 36.872475
17 2012 36  41.75758
18 2005 63  26.03693
18 2006 63 18.958334
18 2007 63 22.035984
18 2008 63 25.482954
18 2009 63 31.207386
18 2010 63  28.99148
18 2011 63  29.79167
18 2012 63  30.89962

* Attempt 1
reg yvar ib2009.year#c.xvar i.STFIPS i.year, cluster(STFIPS)
end

This produces the following output, which omits the 2012 interaction instead of the 2009 interaction like I wanted, but I'm not sure why.

note: 2012.year#c.xvar omitted because of collinearity

Linear regression                               Number of obs     =        104
                                                F(11, 12)         =          .
                                                Prob > F          =          .
                                                R-squared         =     0.7987
                                                Root MSE          =      4.689

                                (Std. Err. adjusted for 13 clusters in STFIPS)
------------------------------------------------------------------------------
             |               Robust
        yvar |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
 year#c.xvar |
       2005  |   .1098391   .1730562     0.63   0.538    -.2672179    .4868962
       2006  |   .2446919   .1539233     1.59   0.138    -.0906781    .5800619
       2007  |   .1050133   .0972725     1.08   0.302    -.1069253    .3169519
       2008  |   .2513997   .1661789     1.51   0.156    -.1106731    .6134725
       2009  |   .1133386   .1336829     0.85   0.413    -.1779315    .4046087
       2010  |   .1335541   .1034933     1.29   0.221    -.0919384    .3590466
       2011  |   .0121195   .0548488     0.22   0.829    -.1073857    .1316248
       2012  |          0  (omitted)
             |
      STFIPS |
          5  |  -18.21407   2.587279    -7.04   0.000    -23.85127   -12.57687
          6  |  -11.17718   4.312132    -2.59   0.024    -20.57251   -1.781851
          8  |  -11.73156    2.97058    -3.95   0.002     -18.2039   -5.259225
          9  |   1.015249   .1916503     5.30   0.000     .5976793     1.43282
         10  |   1.624867   2.108153     0.77   0.456    -2.968404    6.218139
         11  |  -11.09913   4.216307    -2.63   0.022    -20.28567   -1.912582
         12  |  -18.12493    2.77893    -6.52   0.000     -24.1797   -12.07016
         13  |  -14.36844   3.545531    -4.05   0.002    -22.09349    -6.64339
         15  |  -14.18288   3.449706    -4.11   0.001    -21.69914   -6.666618
         16  |  -4.077157    3.16223    -1.29   0.222    -10.96706     2.81275
         17  |  -1.197523   5.270384    -0.23   0.824     -12.6807    10.28566
         18  |  -11.10461   2.683104    -4.14   0.001    -16.95059   -5.258626
             |
        year |
       2005  |   3.270064   5.972014     0.55   0.594    -9.741837    16.28196
       2006  |  -7.650341   4.488798    -1.70   0.114    -17.43059     2.12991
       2007  |  -1.434293   5.019483    -0.29   0.780    -12.37081     9.50222
       2008  |  -7.698874   5.075604    -1.52   0.155    -18.75767    3.359917
       2010  |  -2.293811   2.640422    -0.87   0.402    -8.046797    3.459174
       2011  |   5.758627   8.278618     0.70   0.500    -12.27893    23.79619
       2012  |   8.485741   8.335906     1.02   0.329    -9.676639    26.64812
             |
       _cons |   30.33711   10.85311     2.80   0.016     6.690209    53.98401
------------------------------------------------------------------------------

Next, I tried the following regression:

* Attempt 2
reg yvar ib2009.year##c.xvar i.STFIPS, cluster(STFIPS)

which produces

reg yvar ib2009.year##c.xvar i.STFIPS, cluster(STFIPS)
note: 18.STFIPS omitted because of collinearity

Linear regression                               Number of obs     =        104
                                                F(11, 12)         =          .
                                                Prob > F          =          .
                                                R-squared         =     0.7987
                                                Root MSE          =      4.689

                                (Std. Err. adjusted for 13 clusters in STFIPS)
------------------------------------------------------------------------------
             |               Robust
        yvar |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        year |
       2005  |   3.270064   5.972014     0.55   0.594    -9.741837    16.28196
       2006  |  -7.650341   4.488798    -1.70   0.114    -17.43059     2.12991
       2007  |  -1.434293   5.019483    -0.29   0.780    -12.37081     9.50222
       2008  |  -7.698874   5.075604    -1.52   0.155    -18.75767    3.359917
       2010  |  -2.293811   2.640422    -0.87   0.402    -8.046797    3.459174
       2011  |   5.758627   8.278618     0.70   0.500    -12.27893    23.79619
       2012  |   8.485741   8.335906     1.02   0.329    -9.676639    26.64812
             |
        xvar |   .5099317   .0480172    10.62   0.000     .4053113    .6145522
             |
 year#c.xvar |
       2005  |  -.0034994   .0750769    -0.05   0.964     -.167078    .1600792
       2006  |   .1313533   .0691458     1.90   0.082    -.0193025    .2820091
       2007  |  -.0083253   .0848945    -0.10   0.923    -.1932945     .176644
       2008  |   .1380611   .0915993     1.51   0.158    -.0615166    .3376388
       2010  |   .0202155   .0441306     0.46   0.655    -.0759368    .1163678
       2011  |  -.1012191   .1464932    -0.69   0.503    -.4204003    .2179622
       2012  |  -.1133386   .1336829    -0.85   0.413    -.4046087    .1779315
             |
      STFIPS |
          5  |  -7.506054   2.25e-14 -3.3e+14   0.000    -7.506054   -7.506054
          6  |   6.669513   3.88e-14  1.7e+14   0.000     6.669513    6.669513
          8  |   .5628257   2.57e-14  2.2e+13   0.000     .5628257    .5628257
          9  |   1.808436   1.93e-14  9.3e+13   0.000     1.808436    1.808436
         10  |   10.34992   1.89e-14  5.5e+14   0.000     10.34992    10.34992
         11  |   6.350973   3.80e-14  1.7e+14   0.000     6.350973    6.350973
         12  |  -6.623729   2.41e-14 -2.7e+14   0.000    -6.623729   -6.623729
         13  |   .3055085   3.11e-14  9.8e+12   0.000     .3055085    .3055085
         15  |   .0944733   3.01e-14  3.1e+12   0.000     .0944733    .0944733
         16  |   9.010417   2.73e-14  3.3e+14   0.000     9.010417    9.010417
         17  |    20.6151   4.91e-14  4.2e+14   0.000      20.6151     20.6151
         18  |          0  (omitted)
             |
       _cons |  -5.752869   2.974407    -1.93   0.077    -12.23355    .7278081
------------------------------------------------------------------------------

This does almost exactly what I want, except now it includes "xvar" by itself in the regression. I think that this is probably just changing the interpretation of the STFIPS fixed effects, but I'm not quite sure. Is it possible to adjust either attempt 1 to exclude the 2009 interaction or force attempt 2 to not include xvar by itself in the regression?

[Clyde Schechter]

Well, when you include year and STFIPS effects along with xvar, you have several colinearities that Stata needs to break by omitting something. Stata has its own way of deciding which things to omit and you have some control over it. In some instances, you exert that control by the order in which you specify the predictor variables. The following will get you what you are looking for:

reg yvar i.STFIPS ib2009.year##c.xvar, vce(cluster STFIPS)

By putting STFIPS first, you induce Stata to remove xvar, and your stipulation of 2009 as the base year is respected.

But why do you care? The model predictions are exactly the same regardless of how Stata breaks the colinearity. Remember that none of the coefficients for the variables involved in the colinearity are meaningful except in relation to the omitted categories. No matter which way you break the colinearity, the predicted values for yvar conditional on any choice of year and STFIPS will be exactly the same. I can perhaps understand that you want to focus on the year 2009 as reference for substantive reasons. But what difference does it make whether xvar gets omitted or an additional STFIPS does?

[David Beheshti]

Hi Clyde, this is perfect. Thanks! My main concern was getting 2009 to be the reference year for the sake of interpretation given the specifics of my research question. I was concerned that the inclusion of xvar may have changed the interpretation of my coefficients in a way I didn't understand, but I see now that it doesn't matter.

Thanks again for the helpful feedback.