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Stata 6: How can I estimate a fixed-effects regression with instrumental variables?

Question

Is anyone aware of a routine in Stata to estimate instrumental variable regression for the fixed-effects model? I cannot see that it is possible to do it directly in Stata.

Answer

If we don’t have too many fixed-effects, that is to say the total number of fixed-effects and other covariates is less than Stata's maximum matrix size of 800, and then we can just use indicator variables for the fixed effects. This approach is simple, direct, and always right.

For example, using the auto dataset and rep78 as the panel variable (with missing values dropped) we could estimate a fixed-effects model of mpg on weight and displacement. Let's assume displacement is endogenous and we have gear_ratio and headroom as instruments.

Solution 1

First, generate indicator variables named dr1-dr5, then use ivreg to perform the estimation.

. tab rep78, gen(dr)

(output omitted)

. ivreg mpg weight dr* (displ = gear_ratio headroom)
 
 Instrumental variables (2SLS) regression
 

      Source 
 
       SS           df       MS   
   Number of obs   =        69
 
 
 
   F(6, 62)        =     20.25
       Model 
 
  1536.20001         6  256.033334
   Prob > F        =    0.0000
    Residual 
 
  804.002893        62  12.9677886
   R-squared       =    0.6564
 
 
 
   Adj R-squared   =    0.6232
       Total 
 
   2340.2029        68  34.4147485
   Root MSE        =    3.6011




         mpg 
 
 Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
 
 
 
displacement 
 
  -.0158046   .0281621    -0.56   0.577    -.0720999    .0404907
      weight 
 
  -.0038296   .0030457    -1.26   0.213    -.0099178    .0022587
         dr1 
 
    .093266   2.932757     0.03   0.975    -5.769231    5.955763
         dr2 
 
  (dropped)                                                     
         dr3 
 
   -.094414   1.444669    -0.07   0.948    -2.982265    2.793437
         dr4 
 
  -.3131534   1.596628    -0.20   0.845    -3.504767    2.878461
         dr5 
 
   2.217366   1.895149     1.17   0.246    -1.570984    6.005716
       _cons 
 
   35.79702   4.005494     8.94   0.000     27.79015    43.80389

Instrumented:  displacement
Instruments:   weight dr1 dr2 dr3 dr4 dr5 gear_ratio headroom

In real examples, you will probably have to increase matsize before estimating the model. You will see

. ivreg ...
 matsize too small; type -help matsize-
 r(908)

You can set matsize to any number up to 800, assuming you have sufficient memory:

. set matsize 800

Solution 2

What if we have too many panels to estimate the model directly? In that case, xtdata can be used to transform the data to mean differences, and ivreg can then be used to estimate the fixed-effects model on the transformed data; thus you will not have to reset matsize at all.

We can estimate the same model as above by typing

. drop if rep78==.
(5 observations deleted)
 
. xtdata mpg weight displ gear_ratio headroom, i(rep78) fe clear
 
. ivreg mpg weight (displ = gear_ratio headroom)
 
 Instrumental variables (2SLS) regression
 

      Source 
 
       SS           df       MS   
   Number of obs   =        69
 
 
 
   F(6, 62)        =     20.25
       Model 
 
  986.784228         2  493.392114
   Prob > F        =    0.0000
    Residual 
 
  804.002893        66   12.181862
   R-squared       =    0.5510
 
 
 
   Adj R-squared   =    0.5374
       Total 
 
  1790.78712        68  26.3351047
   Root MSE        =    3.4903




         mpg 
 
 Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
 
 
 
displacement 
 
  -.0158046   .0272954    -0.58   0.565    -.0703016    .0386924
      weight 
 
  -.0038296    .002952    -1.30   0.199    -.0097233    .0020642
       _cons 
 
   36.03048   3.843726     9.37   0.000     28.35623    43.70472

Instrumented:  displacement
Instruments:   weight gear_ratio headroom

Comparing results

 
        Solution 1      
        Solution 2      
     mpg 
      Coef.   Std. Err. 
      Coef.  Std. Err. 
   displ 
  -.0158046   .0281621 
  -.0158046   .0272954 
  weight 
  -.0038296   .0030457 
  -.0038296    .002952 
   _cons 
   35.79702   4.005494 
   36.03048   3.843726 

Comparing the estimates, we observe

1. The coefficients for displ and weight are identical.
2. The standard errors differ slightly.
3. The intercepts differ.

The standard errors (SEs) differ by a scale factor and that is easily fixed. The intercept differs because of an unimportant difference in interpretation.

The SEs differ by a scale factor because our estimate of the residual variance, RMSE, also differs between the solutions. In Solution 2, the SEs are not adjusted for the fact that we estimated the fixed effects. We can adjust them:

    SE(soln. 1) = SE(soln. 2) * sqrt( e(dfr) / (e(dfr)-M+1) ) 

where M is the number of panels and M=5 in this case. For instance, the standard error for _b[displ] can be obtained by typing

. display _se[displ] * sqrt(e(dfr)/(e(dfr)-5+1))
 .02816214

You do not have to adjust the standard errors—the reported SEs are asymptotically equivalent. Solution 1’s SEs have been adjusted for finite samples, and many researchers prefer that the adjustment be made. With reasonable sample sizes, however, the adjustment will not amount to much.

The intercept differs because of difference in interpretation.

In Solution 1, we did not take any care in computing the intercept and let the indicator for rep78==2 drop out of the equation. Thus the intercept in that equation is the intercept for the case rep78==2.

In Solution 2, xtdata mean-differenced the data and then added back in the overall mean. Thus the reported intercept is essentially the overall intercept for all the data; see the FAQ at stata.com/support/faqs/statistics/intercept-in-fixed-effects-model for a discussion.

Both solutions are equivalent.