How do I estimate recursive systems using a subset of available instruments?

Normally, we fit models requiring instrumental variables with ivregress, but sometimes we may want to perform the two-step computations for the instrumental variable estimator instead of using ivregress. For example, we may want to do this when a simultaneous equation system is recursive (sometimes called triangular), but there is some theoretical support for the hypothesis that the error terms are correlated across equations. The estimates from ivregress would still be consistent for such models, but we might prefer to exclude some unnecessary instruments.

Another approach that also leads to recursive systems is directed acyclical graphs (DAGs); see Pearl (2000) and Brito and Pearl (2002). In the figure below, the straight arrows correspond to direct causal links between each pair of variables, whereas the bidirected arc represents correlated errors in the data-generating process for X and Y. The equation for Y would require having Z as an instrument for X. We should not include W in the first-stage equation for X because, according to the DAG, there is not a causal link from W to X.

The correct variance–covariance matrix for the second stage of the instrumental variable estimator must take into account that one of the regressors has been predicted from a previous (first stage) regression. To obtain the adjusted standard errors, we must compute the residuals from the second-stage equation by using the parameter estimates obtained with regress but substituting the instrumented variable (the predicted values of the endogenous variable) for the original values of that variable. Greene (2011, chap. 8) explains the approach and provides the formula for the estimated asymptotic covariance matrix.

Warning: Instrumental variables are commonly used to fit simultaneous systems models. What follows is not appropriate for such models. For a discussion, see Must I use all of my exogenous variables as instruments when estimating instrumental variables regression?

Let’s assume we are interested in the parameter estimates of the following recursive model:

trunk = delta₀ + delta₁headroom + epsilon

price = Beta₀ + Beta₁trunk + Beta₂displacement + mu

where trunk is endogenous. In Stata, you can fit the second equation of this model by using ivregress as follows:

 . sysuse auto
 (1978 Automobile Data)

 . ivregress 2sls price displacement (trunk=headroom), small

 Instrumental variables (2SLS) regression

       Source |       SS       df       MS              Number of obs =      74
 -------------+------------------------------           F(  2,    71) =   11.29
        Model |   108641939     2  54320969.4           Prob > F      =  0.0001
     Residual |   526423457    71  7414414.89           R-squared     =  0.1711
 -------------+------------------------------           Adj R-squared =  0.1477
        Total |   635065396    73  8699525.97           Root MSE      =  2722.9
 
 ------------------------------------------------------------------------------
        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
        trunk |  -222.3396   175.7292    -1.27   0.210    -572.7338    128.0545
 displacement |   22.19871   6.071088     3.66   0.000      10.0933    34.30411
        _cons |   4844.184   1620.835     2.99   0.004     1612.331    8076.037
 ------------------------------------------------------------------------------
 Instrumented:  trunk
 Instruments:   displacement headroom

We used the small option to obtain small-sample statistics because our dataset has only 74 observations. The instruments reported at the bottom of the output correspond to the two exogenous variables in the system. If you need to fit the model with headroom as the only instrument, you can use regress twice and compute the standard errors accounting for the inclusion of a predicted regressor through the following five steps.

Step 1

First, fit the model for the endogenous variable as a function of headroom:

 . regress trunk headroom

       Source |       SS       df       MS              Number of obs =      74
 -------------+------------------------------           F(  1,    72) =   56.17
        Model |  585.347842     1  585.347842           Prob > F      =  0.0000
     Residual |   750.27378    72  10.4204692           R-squared     =  0.4383
 -------------+------------------------------           Adj R-squared =  0.4305
        Total |  1335.62162    73  18.2961866           Root MSE      =  3.2281

 ------------------------------------------------------------------------------
        trunk |      Coef.  Std. Err.     t    P>|t|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
     headroom |   3.347171   .4465957     7.49   0.000     2.456899    4.237443
        _cons |    3.73786   1.388442     2.69   0.009      .970052    6.505667
 ------------------------------------------------------------------------------

Step 2

Next, predict trunk and fit the second-stage regression, substituting trunk with its predicted values:

 . predict double trunk_hat
 (option xb assumed; fitted values)

 . regress price trunk_hat displacement 

       Source |       SS       df       MS              Number of obs =      74
 -------------+------------------------------           F(  2,    71) =   12.71
        Model |   167440536     2  83720268.2           Prob > F      =  0.0000
     Residual |   467624860    71  6586265.63           R-squared     =  0.2637
 -------------+------------------------------           Adj R-squared =  0.2429
        Total |   635065396    73  8699525.97           Root MSE      =  2566.4

 ------------------------------------------------------------------------------
        price |      Coef.  Std. Err.     t    P>|t|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
    trunk_hat |  -161.7683    120.504    -1.34   0.184    -402.0465    78.50998
 displacement |   18.26261   3.715596     4.92   0.000     10.85392     25.6713
        _cons |     4787.5   1490.392     3.21   0.002     1815.743    7759.256
 ------------------------------------------------------------------------------

The point estimates for this regression correspond to the instrumental variable estimation. However, the standard errors do not take into account that trunk was predicted in a previous regression.

Step 3

To compute the correct standard errors, obtain the estimated variance of the residuals, using trunk instead of trunk_hat to get the corresponding residuals:

 . local dof=e(df_r) /* Where dof corresponds to the degrees
                        of freedom of the residuals from the
                        previous regression */

 . replace trunk_hat=trunk
 (74 real changes made)

 . predict double e_hat,residuals

 . generate double ee_hat=e_hat^2

 . summarize ee_hat, meanonly

 . scalar sigsq_hat_pr= r(sum)/`dof'  

Step 4

Get the inverse of the instrumented regressors, W ' W, by removing the mean squared error from the VCE of the second stage.

 . matrix V=e(V)

 . matrix xx_1=e(V)/(e(rmse)^2) 

where e(V) and e(rmse) are the covariance matrix and the root mean squared error from the regression in step 2.

Step 5

Finally, compute the covariance matrix of the IV estimator, and post and display the results:

 . matrix V=sigsq_hat_pr*xx_1

 . matrix b=e(b) /* Where e(b) contains the point estimates
                   from the second stage regression */

 . ereturn post b V, dof(`dof')

 . ereturn display

 ------------------------------------------------------------------------------
              |      Coef.  Std. Err.     t    P>|t|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
    trunk_hat |  -161.7683   125.6518    -1.29   0.202    -412.3109    88.77442
 displacement |   18.26261   3.874322     4.71   0.000     10.53743    25.98779
        _cons |     4787.5    1554.06     3.08   0.003     1688.793    7886.207
 ------------------------------------------------------------------------------

For a different perspective on the same problem, see Must I use all of my exogenous variables as instruments when estimating instrumental variables regression?

References

Brito, C., and J. Pearl. 2002.

Generalized instrumental variables. In Uncertainty in Artificial Intelligence, Proceedings of the Eighteenth Conference, 85–93. San Francisco: Morgan Kaufmann.

Greene, W. H. 2011.

Econometric Analysis. 7th ed. Upper Saddle River, NJ: Prentice Hall.

Pearl, J. 2000.

Causality. Cambridge: Cambridge University Press.