Solved – Why is Stata automatically converting regressors to instrumental variables in ivprobit model

instrumental-variablesstata

I am running ivprobit in Stata to look at the determinants of enrolment in health insurance (cbhi). I have several exogenous regressors and one endogenous regressor (consumption). I am using wealthindex as an intrumental variable for consumption. However, when I run the ivprobit model all my exogenous regressors appear in the "instruments" list. Could someone please tell me how to prevent this from happening? I am copying the stata code and results below.

Thank you in advance for your help.

. # delimit;
delimiter now ;
. ivprobit cbhi age_hhead age2 edu_hh2 edu_hh3 edu_hh4 edu_hh5 mar_hh hhmem_cat2 hhmem_cat3
> hhhealth_poor hh_chronic hh_diffic risk pregnancy wra any_oldmem j_hh2 j_hh3 j_hh4
> hlthseek_mod campaign qual_percep urban door_to_door chief_mem
> pharm_in_vil drugsel_in_vil hlthcent_in_vil privclin_in_vil young edu_chief num_camp_vill home_collection
> qual2 qual3 size_ln name_hosp1 name_hosp3 name_hosp4 name_hosp5 name_hosp6 (cons_pcm=wealthindex) ;

Fitting exogenous probit model

Iteration 0:   log likelihood = -1909.5425
Iteration 1:   log likelihood =  -1687.946
Iteration 2:   log likelihood = -1681.6413
Iteration 3:   log likelihood = -1681.6142
Iteration 4:   log likelihood = -1681.6142

Fitting full model

Iteration 0:   log likelihood = -42769.776 
Iteration 1:   log likelihood = -42766.676 
Iteration 2:   log likelihood = -42736.028 
Iteration 3:   log likelihood = -42734.629 
Iteration 4:   log likelihood = -42734.269 
Iteration 5:   log likelihood = -42734.266 
Iteration 6:   log likelihood = -42734.266 

Probit model with endogenous regressors           Number of obs   =       3000
                                                  Wald chi2(42)   =    1113.82
Log likelihood = -42734.266                       Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    cons_pcm |   3.59e-06   2.15e-07    16.68   0.000     3.17e-06    4.01e-06
   age_hhead |   .0320042   .0129973     2.46   0.014       .00653    .0574784
        age2 |  -.0001774   .0001228    -1.45   0.148    -.0004181    .0000632
     edu_hh2 |    .051426   .0879334     0.58   0.559    -.1209203    .2237723
     edu_hh3 |  -.0075595   .1023748    -0.07   0.941    -.2082106    .1930915
     edu_hh4 |   -.000683   .1285689    -0.01   0.996    -.2526735    .2513075
     edu_hh5 |  -.2986249   .1713749    -1.74   0.081    -.6345136    .0372638
      mar_hh |  -.0539035    .067056    -0.80   0.421    -.1853308    .0775237
  hhmem_cat2 |   .4222092   .0675468     6.25   0.000     .2898198    .5545985
  hhmem_cat3 |   .7099401   .0822271     8.63   0.000     .5487778    .8711023
hhhealth_p~r |   .1550617   .0699122     2.22   0.027     .0180362    .2920872
  hh_chronic |    .159681   .0718297     2.22   0.026     .0188973    .3004647
   hh_diffic |   .2613713   .0741209     3.53   0.000      .116097    .4066456
        risk |  -.0482252   .0467953    -1.03   0.303    -.1399423    .0434919
   pregnancy |   .1200344   .1214774     0.99   0.323    -.1180569    .3581257
         wra |   .1033809   .0289192     3.57   0.000     .0467004    .1600614
  any_oldmem |   .0792428   .0680778     1.16   0.244    -.0541872    .2126729
       j_hh2 |   .0512404   .0815692     0.63   0.530    -.1086323     .211113
       j_hh3 |   -.349126   .0806268    -4.33   0.000    -.5071516   -.1911004
       j_hh4 |   .1928496   .0843208     2.29   0.022     .0275838    .3581154
hlthseek_mod |  -.0359996   .0969011    -0.37   0.710    -.2259223     .153923
    campaign |   .5068671   .0781293     6.49   0.000     .3537365    .6599977
 qual_percep |   .1840314   .0514758     3.58   0.000     .0831408    .2849221
       urban |  -.0999875   .0590944    -1.69   0.091    -.2158103    .0158353
door_to_door |   .0564977   .0719369     0.79   0.432     -.084496    .1974914
   chief_mem |   .0811248   .0599964     1.35   0.176     -.036466    .1987156
pharm_in_vil |   .0269211   .0582651     0.46   0.644    -.0872763    .1411185
drugsel_in~l |   .0693485   .0586169     1.18   0.237    -.0455385    .1842355
hlthcent_i~l |   .0089279   .0823509     0.11   0.914    -.1524769    .1703327
privclin_i~l |  -.0547559   .0857815    -0.64   0.523    -.2228846    .1133728
       young |   .0899074   .0715818     1.26   0.209    -.0503903    .2302051
   edu_chief |  -.0888131   .0528856    -1.68   0.093    -.1924671    .0148408
num_camp_v~l |   .0655667   .0548026     1.20   0.232    -.0418444    .1729779
home_colle~n |   .1944581   .0606159     3.21   0.001     .0756531    .3132631
       qual2 |  -.1778623   .0896638    -1.98   0.047    -.3536001   -.0021245
       qual3 |  -.2049839   .1081036    -1.90   0.058     -.416863    .0068952
     size_ln |  -.0059746    .042326    -0.14   0.888    -.0889321    .0769829
  name_hosp1 |   .1476783   .1168295     1.26   0.206    -.0813032    .3766598
  name_hosp3 |    .536909   .1350044     3.98   0.000     .2723053    .8015128
  name_hosp4 |   .4347936   .1069197     4.07   0.000     .2252348    .6443523
  name_hosp5 |    .228802   .1274495     1.80   0.073    -.0209945    .4785985
  name_hosp6 |   .9189969   .1301651     7.06   0.000     .6638779    1.174116
       _cons |   -4.82205   .4438201   -10.86   0.000    -5.691922   -3.952179
-------------+----------------------------------------------------------------
     /athrho |  -.8556093   .0998439    -8.57   0.000      -1.0513   -.6599188
    /lnsigma |   12.27712     .01291   950.98   0.000     12.25181    12.30242
-------------+----------------------------------------------------------------
         rho |  -.6939885   .0517571                     -.7823111   -.5783094
       sigma |   214725.5   2772.097                      209360.5    220228.1
------------------------------------------------------------------------------
Instrumented:  cons_pcm
Instruments:   age_hhead age2 edu_hh2 edu_hh3 edu_hh4 edu_hh5 mar_hh
               hhmem_cat2 hhmem_cat3 hhhealth_poor hh_chronic hh_diffic risk
               pregnancy wra any_oldmem j_hh2 j_hh3 j_hh4 hlthseek_mod
               campaign qual_percep urban door_to_door chief_mem
               pharm_in_vil drugsel_in_vil hlthcent_in_vil privclin_in_vil
               young edu_chief num_camp_vill home_collection qual2 qual3
               size_ln name_hosp1 name_hosp3 name_hosp4 name_hosp5
               name_hosp6 wealthindex
------------------------------------------------------------------------------
Wald test of exogeneity (/athrho = 0): chi2(1) =    73.44 Prob > chi2 = 0.0000

Best Answer

In general all exogenous variables are always included as instruments. Usually instruments are picked for variables which are endogenous, but we can think (it follows from the mathematical derivation of instrumental variable estimation) that we need to choose the instruments for all the variables. Instruments for exogenous variables then are naturally themselves.

Related Solutions

Solved – How to do an instrumental variables regression with an instrumented interaction term in Stata

This is a question which appears sometimes in the Statalist. Let me write $x_{1}$ and $x_{2}$ instead of $x$ and $z$ (in the literature $z$ is usually reserved for instruments rather than endogenous variables) and let $x_3 = x_1 \cdot x_2$. Your model then becomes: $$y = ax_1 + bx_2 + cx_3 + e$$ which has three endogenous variables. Assuming that you have two variables $z_1$ and $z_2$ which are valid instruments for $x_1$ and $x_2$, then a valid instrument for $x_3$ is $z_3 = z_1 \cdot z_2$. In Stata it is straightforward to generate the corresponding interactions and to use them in the appropriate estimation command like ivreg2, for instance.

Note though that models with more than one endogenous variable can be difficult to interpret and also you might be confronted with the question why you are tackling two causal questions at the same time. This issue is discussed on the Mostly Harmless Econometrics blog by Angrist and Pischke.

Your second problem is similar for the case where you interact an endogenous ($x$) and an exogenous variable ($w$) in a model of the type $$y = ax + bw + c(x\cdot w) + e$$ If $z$ is a valid instrument for $x$, then a valid instrument for $(x\cdot w)$ is $(z\cdot w)$. This procedure was suggested in the Statalist. I just provide one link but there are many more discussions about this (most of which will pop up on Google when searching for: interaction of "two endogenous variables").

Solved – Why report r-squared in Instrumental Variables Estimation

It's true that $R^2$ in instrumental variables regressions is not useful. Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ into $\beta^2 Var(x) + Var(\epsilon)$, so the obtained $R^2$ neither has a natural interpretation nor can it be used for computation of F-tests for joint rejection. Also $R^2$ in instrumental variables regression can be negative and for this point it makes not difference for whether you use $$R^2 = \frac{MSS}{TSS} \quad \text{or} \quad R^2 = 1- \frac{RSS}{TSS}$$ because when $RSS>TSS$, then we also have that $MSS = TSS - RSS < 0$. In general the two expressions are the same so there should be no reason for why one would be more popular than the other. The issue is discussed in more length on the Stata website resources and support FAQs (link).

[edit] to address the additional question in the comment
When you instrument the endogenous variable $x$ with your instrument $z$ as $$x = \alpha + \pi z + \eta$$ you use the predicted values $\widehat{x}$ in the second stage $$y = a + \beta \widehat{x} + \epsilon$$ and if you do this procedure by hand in Stata like

reg x z
predict x_hat, xb
reg y x_hat

the standard errors will be calculated as $y - \widehat{x}\beta$ but these standard errors will be wrong. They are wrong because $\widehat{x}$ is an estimated quantity and not a random variable. The property of these standard errors though is that $RSS < TSS$ and there would be no negative $R^2$ and $\widehat{x}\beta$ is going to be a better predictor of $y$ than $\overline{y}$.

To calculate the corrected standard errors you use the actual values of the endogenous variable $x$ and not its fitted values when computing $e = y − x\beta$. The issue with this is that in this case you are computing the $RSS$ from a different set of regressors than those that are used to actually fit the model from which we take the $TSS$. For this reason it can happen that $x\beta$ is a worse predictor for $y$ than $\overline{y}$.

Best Answer

Related Solutions

Solved – How to do an instrumental variables regression with an instrumented interaction term in Stata

Solved – Why report r-squared in Instrumental Variables Estimation

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