FE-IV with quadratic endogenous variable

[Luc Nguyen]

Hi all,

I am currently looking at the effects of hours worked on physical health using panel data. There is reason to believe that this relationship is non-linear, i.e., working an x-amount of hours may improve your physical health, but at some point, it will take a toll on your physical health (inverted U-shaped relationship).

Imagine I ran the following code in STATA:

 xi: xtivreg2 PH q (HW HW^2 = z), fe

where:

> PH is a continuous variable measuring physical health, with a range of 0 (low functioning) to 100 (high functioning)
> q is a set of exogenous variables.
> HW / HW^2 are the variables denoting the No. of hours normally worked per week, which I suspect are endogenous.
> z is, what I think, a non-weak instrument, which is a dummy variable.

The issue here is that I am unable to find a second instrument for my regression. And since z is a dummy, I am unable to square it as you would do with a continuous variable.

My question is, is there an econometric technique to tackle the fact that two variables (hours worked, and hours worked squared) are endogenous with only one dummy instrument, or is it simply not possible?

Thanks for all the help in advance!

Luc

[George Ford]

I've seen this sort of thing done.

https://www.statalist.org/forums/forum/general-stata-discussion/general/1701746-2sls-with-endogenous-variable-z-fitted-values-from-1st-stage-instrument

You estimate equations for HW and HW^2 and use the predictions as instruments in the xtivreg2 command.

G

[Jeff Wooldridge]

George's solution might work, but it requires an extra x in the equation -- your q variable in your case -- that definitely appears in the first stage for HW. Even if it does, the resulting IV might be weak.

Another possibility is to use a control function approach.

xtreg HW q z, fe
predict vhat, e
xtreg PH HW c.HW#c.HW vhat q, fe vce(cluster id)

The standard errors in the second estimation are generally incorrect because they don't adjust for estimation of vhat. But it's easy to bootstrap both steps.

[Luc Nguyen]

Thank you both for your suggestions. I tried both, and the control function approach yielded "better" results with my instrument. Just to double-check, below are the results, I hope I did it right... How do I adjust the standard errors in Stata for the second stage?

As for the variables used;
> sf12pcs_dv represents the continuous variable measuring physical health
> jbhrs is the No. of hours normally worked per week
> nchunder16 is my instrument ("z")

xtreg jbhrs dvage married hhsize mh_dv1 i.qfhigh_dv i.jbrgsc_dv i.jbpl j2has urban_dv i.gor_dv i.year nchunder16 if sex_dv == 1 & employ == 1, fe

Fixed-effects (within) regression               Number of obs     =     60,089
Group variable: pidp                            Number of groups  =     13,034

R-squared:                                      Obs per group:
     Within  = 0.0304                                         min =          1
     Between = 0.0099                                         avg =        4.6
     Overall = 0.0015                                         max =         12

                                                F(42,47013)       =      35.07
corr(u_i, Xb) = -0.5363                         Prob > F          =     0.0000

-------------------------------------------------------------------------------------
              jbhrs | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------------+----------------------------------------------------------------
              dvage |   .1594898   .1052088     1.52   0.130     -.046721    .3657006
            married |   .4855547   .1375986     3.53   0.000     .2158595    .7552499
             hhsize |  -.3339338   .0502108    -6.65   0.000    -.4323476     -.23552
             mh_dv1 |   .0259449   .0907773     0.29   0.775      -.15198    .2038697
                    |
          qfhigh_dv |
1st degree or eq..  |  -1.000311   .4381971    -2.28   0.022    -1.859183    -.141438
           A level  |  -7.328987   .5122484   -14.31   0.000    -8.333001   -6.324972
      GCSE/O level  |  -9.917486   .5463722   -18.15   0.000    -10.98838   -8.846588
 None of the above  |  -8.308353   .4704644   -17.66   0.000     -9.23047   -7.386236
                    |
          jbrgsc_dv |
managerial & tec..  |   .1061291   .2332947     0.45   0.649    -.3511318      .56339
skilled non-manual  |  -2.146714   .2609277    -8.23   0.000    -2.658137   -1.635292
    skilled manual  |  -1.120323   .2698386    -4.15   0.000     -1.64921    -.591435
partly skilled o..  |   -3.13389   .2799163   -11.20   0.000     -3.68253    -2.58525
unskilled occupa..  |  -2.567031    .336432    -7.63   0.000    -3.226442   -1.907619
                    |
               jbpl |
At your employer..  |   .3104018   .1711999     1.81   0.070    -.0251525    .6459561
Driving or trave..  |   .5808678   .1999707     2.90   0.004     .1889224    .9728133
Or at one or mor..  |   .6329742     .19781     3.20   0.001     .2452637    1.020685
 SPONTANEOUS Other  |    1.07363    .394049     2.72   0.006     .3012884    1.845972
                    |
              j2has |   1.795575   .1419171    12.65   0.000     1.517415    2.073734
           urban_dv |  -.4086814   .1947922    -2.10   0.036    -.7904769   -.0268859
                    |
             gor_dv |
        North West  |  -1.878361   1.090768    -1.72   0.085    -4.016282    .2595602
Yorkshire and th..  |  -1.986552   1.034807    -1.92   0.055    -4.014788    .0416842
     East Midlands  |  -2.838112   1.156183    -2.45   0.014    -5.104248   -.5719754
     West Midlands  |  -2.251281   1.134119    -1.99   0.047    -4.474171   -.0283919
   East of England  |  -1.133398   1.138457    -1.00   0.319    -3.364789    1.097994
            London  |  -1.088907   1.109957    -0.98   0.327     -3.26444    1.086626
        South East  |  -1.144854   1.106345    -1.03   0.301    -3.313307    1.023598
        South West  |  -1.580609   1.158272    -1.36   0.172     -3.85084    .6896219
             Wales  |  -4.338598    1.35419    -3.20   0.001     -6.99283   -1.684366
          Scotland  |  -5.279508   1.194328    -4.42   0.000    -7.620408   -2.938609
  Northern Ireland  |   -2.47921    1.77812    -1.39   0.163    -5.964351    1.005931
                    |
               year |
              2010  |   -.408918    .152431    -2.68   0.007     -.707685   -.1101511
              2011  |  -.6925551    .240089    -2.88   0.004    -1.163133   -.2219772
              2012  |  -.9540243   .3387319    -2.82   0.005    -1.617944   -.2901049
              2013  |  -1.161776   .4398307    -2.64   0.008    -2.023851   -.2997015
              2014  |  -1.419908   .5440084    -2.61   0.009    -2.486173   -.3536441
              2015  |  -1.617174   .6476885    -2.50   0.013    -2.886653    -.347695
              2016  |  -1.872816   .7492509    -2.50   0.012    -3.341359   -.4042737
              2017  |  -2.155606   .8536827    -2.53   0.012    -3.828836   -.4823754
              2018  |  -2.149697   .9562561    -2.25   0.025    -4.023973   -.2754209
              2019  |  -2.641438   1.059903    -2.49   0.013    -4.718863   -.5640127
              2020  |  -2.820414   1.162619    -2.43   0.015    -5.099165   -.5416634
                    |
         nchunder16 |   .2149353   .0660266     3.26   0.001     .0855222    .3443484
              _cons |   38.24314   4.133731     9.25   0.000     30.14097    46.34532
--------------------+----------------------------------------------------------------
            sigma_u |  10.508088
            sigma_e |  5.7963907
                rho |   .7667088   (fraction of variance due to u_i)
-------------------------------------------------------------------------------------
F test that all u_i=0: F(13033, 47013) = 7.58                Prob > F = 0.0000


predict vhat, e
(275,002 missing values generated)


xtreg sf12pcs_dv jbhrs c.jbhrs#c.jbhrs vhat dvage married hhsize mh_dv1 i.qfhigh_dv i.jbrgsc_dv i.jbpl j2has urban_dv i.gor_dv i.year, fe vce(cluster pidp)

Fixed-effects (within) regression               Number of obs     =     59,583
Group variable: pidp                            Number of groups  =     12,990

R-squared:                                      Obs per group:
     Within  = 0.0115                                         min =          1
     Between = 0.0059                                         avg =        4.6
     Overall = 0.0034                                         max =         12

                                                F(44,12989)       =       8.05
corr(u_i, Xb) = -0.7119                         Prob > F          =     0.0000

                                     (Std. err. adjusted for 12,990 clusters in pidp)
-------------------------------------------------------------------------------------
                    |               Robust
         sf12pcs_dv | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------------+----------------------------------------------------------------
              jbhrs |   .6934251   .3199868     2.17   0.030     .0662041    1.320646
                    |
    c.jbhrs#c.jbhrs |  -.0004355   .0001683    -2.59   0.010    -.0007654   -.0001055
                    |
               vhat |  -.6492582   .3200159    -2.03   0.042    -1.276536     -.02198
              dvage |  -.1652542   .1103911    -1.50   0.134     -.381637    .0511286
            married |  -.4674809   .2233065    -2.09   0.036    -.9051944   -.0297674
             hhsize |    .219635   .0944863     2.32   0.020     .0344281    .4048419
             mh_dv1 |   .7362578   .1146355     6.42   0.000     .5115554    .9609601
                    |
          qfhigh_dv |
1st degree or eq..  |   .8378768   .5037108     1.66   0.096    -.1494703    1.825224
           A level  |   3.994628   2.399016     1.67   0.096    -.7077952    8.697051
      GCSE/O level  |   6.073666   3.234546     1.88   0.060    -.2665187    12.41385
 None of the above  |   4.717639    2.70648     1.74   0.081    -.5874584    10.02274
                    |
          jbrgsc_dv |
managerial & tec..  |  -.2221832    .210739    -1.05   0.292    -.6352626    .1908961
skilled non-manual  |   1.422117   .7382663     1.93   0.054    -.0249934    2.869227
    skilled manual  |   .6542066   .4496283     1.45   0.146    -.2271307    1.535544
partly skilled o..  |    1.92226   1.054441     1.82   0.068    -.1445993     3.98912
unskilled occupa..  |    1.56837    .913718     1.72   0.086    -.2226514    3.359391
                    |
               jbpl |
At your employer..  |  -.3902609   .1893624    -2.06   0.039    -.7614391   -.0190828
Driving or trave..  |  -.2434599   .2702937    -0.90   0.368    -.7732753    .2863554
Or at one or mor..  |  -.4334897   .2786316    -1.56   0.120    -.9796485     .112669
 SPONTANEOUS Other  |  -.4848773   .5070672    -0.96   0.339    -1.478803    .5090488
                    |
              j2has |  -1.259948   .5924512    -2.13   0.033    -2.421239   -.0986567
           urban_dv |   .6597886   .2272884     2.90   0.004     .2142701    1.105307
                    |
             gor_dv |
        North West  |   2.123956   1.374238     1.55   0.122    -.5697514    4.817664
Yorkshire and th..  |   2.743164   1.258908     2.18   0.029     .2755188    5.210809
     East Midlands  |   3.239816   1.593356     2.03   0.042     .1166043    6.363028
     West Midlands  |   3.111558   1.668454     1.86   0.062    -.1588568    6.381973
   East of England  |    2.88427   1.401706     2.06   0.040     .1367195     5.63182
            London  |   2.903864    1.33502     2.18   0.030     .2870289    5.520699
        South East  |   2.565106   1.349864     1.90   0.057    -.0808264    5.211038
        South West  |   3.102943   1.455778     2.13   0.033     .2494043    5.956482
             Wales  |    6.14632    2.15404     2.85   0.004     1.924085    10.36855
          Scotland  |    4.10806   2.074958     1.98   0.048     .0408368    8.175283
  Northern Ireland  |   3.956089    1.67747     2.36   0.018     .6680024    7.244176
                    |
               year |
              2010  |   .4030869   .1948526     2.07   0.039     .0211472    .7850266
              2011  |   .4939884   .3228141     1.53   0.126    -.1387747    1.126751
              2012  |   .6036291   .4449722     1.36   0.175    -.2685817     1.47584
              2013  |   .5190652   .5613394     0.92   0.355    -.5812424    1.619373
              2014  |   .8332941   .6908966     1.21   0.228    -.5209645    2.187553
              2015  |   .6435381   .8056211     0.80   0.424    -.9355975    2.222674
              2016  |    .886837   .9271781     0.96   0.339     -.930568    2.704242
              2017  |   .7902299   1.063425     0.74   0.457    -1.294238    2.874698
              2018  |   .4420441   1.136793     0.39   0.697    -1.786237    2.670325
              2019  |   .9367959   1.312854     0.71   0.476     -1.63659    3.510182
              2020  |    1.01104   1.425143     0.71   0.478    -1.782449    3.804529
                    |
              _cons |   28.75708   12.64283     2.27   0.023     3.975269    53.53888
--------------------+----------------------------------------------------------------
            sigma_u |   9.283354
            sigma_e |  5.2296585
                rho |  .75910041   (fraction of variance due to u_i)
-------------------------------------------------------------------------------------

.
end of do-file

[Jeff Wooldridge]

Use the panel bootstrap applied to both steps of the estimation. That should run fairly quickly. If you want a reference for the unbalanced case, it’s in a paper of mine with a former student, Riju Joshi. Annals of Economics and Statistics.