weak instruments

[john ghi]

I'm using the command ivregress 2sls, with clusters (each cluster is a school) and with pweights.
I have two endogenous variables (x1, x2) and 4 instruments. I'm trying to test my model and check if my instruments are not weak.
I used the estat firstatage command and I'm not sure how to interpret the result.

. estat firststage, all forcenonrobust
First-stage regression summary statistics
--------------------------------------------------------------------------
| Adjusted Partial Robust
Variable | R-sq. R-sq. R-sq. F(4,155) Prob > F
-------------+------------------------------------------------------------
X1 | 0.9640 0.9636 0.4335 27.2534 0.0000
X2| 0.9822 0.9820 0.0520 1.34133 0.2571
--------------------------------------------------------------------------
(F statistics adjusted for 156 clusters in school)
Shea's partial R-squared
--------------------------------------------------
| Shea's Shea's
Variable | Partial R-sq. Adj. Partial R-sq.
-------------+------------------------------------
X1 | 0.4119 0.4048
X2 | 0.0494 0.0379
--------------------------------------------------

[ALKEBSEE RADWAN]

John, that the critical values do not appear is a good thing, shows that the folks at Stata Corp know what they are doing here. These critical values are valid only under homoskedasticity. As you are calculating clustered variances these critical values are invalid.

I think in your case of two endogenous variables and cluster robust variance there is no theory developed to tell you whether the instruments are weak.

But you do not even need this. Your regression has problems regarding X2 being unidentified without any weak instruments considerations. The weak instruments considerations only rise the critical values, in some cases by a lot. Your X2 cannot pass even the standard critical values for testing the significance of the excluded instruments in explaining X2.

Hello Joro Kolev
Actually I have a problem with interpreting the outcomes of this command

estat firststage, all forcenonrobust

I have run this code after running 2SLS model
I got the following

.  estat firststage, all forcenonrobust

  First-stage regression summary statistics
  --------------------------------------------------------------------------
               |            Adjusted      Partial       Robust
      Variable |   R-sq.       R-sq.        R-sq.     F(1,7456)   Prob > F
  -------------+------------------------------------------------------------
        RRR |  0.1946      0.1903       0.0170       96.1701    0.0000
  --------------------------------------------------------------------------


  Shea's partial R-squared
  --------------------------------------------------
               |     Shea's             Shea's
      Variable |  Partial R-sq.   Adj. Partial R-sq.
  -------------+------------------------------------
       RRR |     0.0170             0.0113
  --------------------------------------------------


  Minimum eigenvalue statistic = 128.679     

  Critical Values                      # of endogenous regressors:    1
  Ho: Instruments are weak             # of excluded instruments:     1
  ---------------------------------------------------------------------
                                     |    5%     10%     20%     30%
  2SLS relative bias                 |         (not available)
  -----------------------------------+---------------------------------
                                     |   10%     15%     20%     25%
  2SLS Size of nominal 5% Wald test  |  16.38    8.96    6.66    5.53
  LIML Size of nominal 5% Wald test  |  16.38    8.96    6.66    5.53
  ---------------------------------------------------------------------

Can I ague that my IV is relevant and robust ?

[Joro Kolev]

John, that the critical values do not appear is a good thing, shows that the folks at Stata Corp know what they are doing here. These critical values are valid only under homoskedasticity. As you are calculating clustered variances these critical values are invalid.

I think in your case of two endogenous variables and cluster robust variance there is no theory developed to tell you whether the instruments are weak.

But you do not even need this. Your regression has problems regarding X2 being unidentified without any weak instruments considerations. The weak instruments considerations only rise the critical values, in some cases by a lot. Your X2 cannot pass even the standard critical values for testing the significance of the excluded instruments in explaining X2.

Joro Kolev Thank you very much for your comprehensive answer!
Actually, I noticed that the output is partial, but that's all the output that was shown in stata. The critical values are indeed don't appears there when estimating with clustering (and pweights)

It would indeed be clearer to me if I had the critical values to compare against. However, since I don't have them, is there another way to make a clear determination whether or not the instruments are weak? Would a significant p-value be enough to make such a call (that the instruments are not weak)?


Thank you again.

[john ghi]

Joro Kolev Thank you very much for your comprehensive answer!
Actually, I noticed that the output is partial, but that's all the output that was shown in stata. The critical values are indeed don't appears there when estimating with clustering (and pweights)

It would indeed be clearer to me if I had the critical values to compare against. However, since I don't have them, is there another way to make a clear determination whether or not the instruments are weak? Would a significant p-value be enough to make such a call (that the instruments are not weak)?


Thank you again.

[Joro Kolev]

You seem to have shown only partial output of the command.

You are ahead of theory, I think. You have two endogenous variables, and you estimate robust variance--I do not think there is theory developed for this case which displays the critical values for weak instruments.

Otherwise you interpret the output you have shown as follows:

The R-squares and adjusted R-squares are what you would get in the first stage regression of the given endogenous variable on all exogenous variables (both included and excluded). This is not a very interesting quantity in this context.

The partial R-squares and adjusted partial R-squares are what you would get in the first stage regression of the given endogenous variable on only excluded exogenous variables, after you have partialled out the included exogenous variables from both the endogenous and the excluded exogenous. This is an interesting quantity in this context. For your first endogenous variable X1 partial R-squares are very high, for the second X2 they are very low.

The F-statistic is for a test that the slopes on the excluded instruments are jointly 0, in the first stage regression of the respective endogenous variable on all exogenous variables.

Even thought we do not know the appropriate critical values for weak instruments for your case, we clearly see that your ivregression has serious problems as the second endogenous variable X2 is no identified, e.g., you cannot reject that the excluded instruments have no explanatory power for X2 at any reasonable level, p-value = 0.2571. Another way to see this lack of explanatory power is via the Shea's adjusted partial R-squared = 0.0379.

[john ghi]

Joro Kolev