Interpreting weakiv results

[Ale Rossi]

Dear all,
I am running an instrumental variable regression.
My model has 3 endogenous variables and 9 instruments and I fear I face problems with weak instruments.
I am using xtivreg2 with a fixed-effects estimator and a continuously updated estimator.

xtivreg2 y (x1 x2 x3=$instruments) $controls_lag yeardum*,fe robust bw(1) small cue

My post estimation controls provide the following results

Underidentification test (Kleibergen-Paap rk LM statistic):             18.633
                                                   Chi-sq(7) P-val =    0.0094
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):                1.800
                         (Kleibergen-Paap rk Wald F statistic):          2.063
Stock-Yogo weak ID test critical values:                       <not available>
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         9.458
                                                   Chi-sq(6) P-val =    0.1494
------------------------------------------------------------------------------

My question is the following
1) Should I be worried about weak instruments using a cue? I have read that cue is able to tame the bias coming from weak instruments. Moreover , why the Stock-Yogo weak ID test critical values are not available?

Since my k-p statistic is very low, I further explore whether I have weak instruments using the weakiv command. But is am not sure how I could interpret these results.

weakiv, estadd ci usegrid

Weak instrument robust tests for linear IV
H0: beta[y:x1 x2 x3] = [0 0 0]
----------------------------------------
 Test |       Statistic         p-value
------+---------------------------------
  CLR | stat(.)   =    11.81     0.0070
    K | chi2(3)   =     8.70     0.0335
    J | chi2(6)   =    12.75     0.0472
  K-J |        <n.a.>            0.0416
   AR | chi2(9)   =    21.45     0.0108
------+---------------------------------
 Wald | chi2(3)   =    17.09     0.0007
----------------------------------------
Confidence sets estimated for (11 x 11 x 11) points in [-.409409, 3.93645], [-12.1167, 2.06556],
[-3.54564, .986649].
CLR distribution and p-values obtained by simulation (10000 draws).
Number of obs N = 2530.  Number of groups = 142; avg obs per group = 17.8.  Warning - singleton
groups detected.  4 observation(s) not used.
Method = lagrange multiplier (LM).  Weight on K in K-J test = 0.800.
Tests robust to heteroskedasticity. Tests robust to autocorrelation: kernel=Bartlett, bw=1. Small
sample adjustments were used.
Wald statistic in last row is based on xtivreg2 estimation and is not robust to weak instruments.

I am aware the help file for weakiv provides some guidance but I am still struggling to interpret the results
My questions are the following:
1) Am I correct to believe the results of the CLR and the K test (H0:beta=b0) suggest that we cannot reject the possibility that my estimated coefficient is different from 0?
2) that the J test (H0:E(Zu)=0) suggest that the instruments are different from 0 (are not weak(?))
3) the AR and the Wald test suggests that my instruments are exogenous or that the coefficient of the endogenous regression are different from 0
4) In sum, do these test suggests that my model is correctly specified?

I am sorry if these are trivial questions, but hopefully they will also be useful to others.
In any case, if anyone has suggestions alternative or more appropriate test I could run, I would be happy to hear any suggestions.


I thank you in advance for your help

Best

[Joro Kolev]

From the first set of results you show, you do not seem to have weak instruments problem, Kleibergen-Paap rejects the underidentification null.

Note that the Stock-Yogo critical values assume iid, and are not valid in your case.

From reading the help file of the user contributed -weakiv-, you can reject the null that your endogenous variables are jointly 0 at 5% level when using the robust to weak identification procedures.

[Ale Rossi]

Dear Prof. Kolev thanks a lot for your helpful reply, just to make sure I correctly understand:
the Kleibergen-Paap rk LM statistic of the xtivreg allow me to reject my under-identification test,
I should not be concerned with the Kleibergen-Paap rk Wald F statistic and the Stock-Yogo critical values because they assume iid.
The results from the weakiv, suggest that my instruments correlated to the endogenous regressor (not weak) and are exogenous (uncorrelated with the error term) at the 5% level?

Is this a correct interpretation?

A sincere thanks for your help

Best Regards

[Joro Kolev]

the Kleibergen-Paap rk LM statistic of the xtivreg allow me to reject my under-identification test:

Yes, and it is robust to whatever departure from iid you have specified in your regression, in your case heteroskedasticity.

the Kleibergen-Paap rk Wald F statistic :

I do not know whether this one is robust or not, you should check yourself if you are interested.

the Stock-Yogo critical values because they assume iid:

No, this is a different test, a test of weak identification, as opposed to under identification. So not that you should not be concerned, it is just that the test is invalid if error is not iid. I think that there is no generally accepted procedure for weak identification for non iid errors.

Overall, you are rejecting under identification null, and there is no known generally accepted way to test for weak identification. If you want you can worry about this, or not, but there is not much that can be done. If I were you I would not dig any further into the issue.


The results from the weakiv, suggest that my instruments correlated to the endogenous regressor (not weak) and are exogenous (uncorrelated with the error term) at the 5% level?

No, from what I understand in the help file of -weakiv-, it tests the null that your endogenous regressors are jointly 0, and it tests this null in a fashion robust to weak instruments. It does not test that your instruments are weak. It tests whether endogenous regressors are jointly 0, and uses a procedure which is valid if you have identification, under identification, or weak identification. So you can simply employ this test, and disregard any testing of the strength of your instruments.

Dear Prof. Kolev thanks a lot for your helpful reply, just to make sure I correctly understand:
the Kleibergen-Paap rk LM statistic of the xtivreg allow me to reject my under-identification test,
I should not be concerned with the Kleibergen-Paap rk Wald F statistic and the Stock-Yogo critical values because they assume iid.
The results from the weakiv, suggest that my instruments correlated to the endogenous regressor (not weak) and are exogenous (uncorrelated with the error term) at the 5% level?

Is this a correct interpretation?

A sincere thanks for your help

Best Regards

[Ale Rossi]

Dear Prof. Kolev,
Thanks a lot for your thorough reply, this is extremely helpful and useful also it is welcomed news . I sincerely thank you for your help

Best regards

[Eric de Souza]

From my lecture notes for the heteroscedastic and/or serially correlated case:
When the residuals are heteroscedastic and/or serially correlated, the above tests for weak instruments are not applicable: there is no formal justification in this case for comparing the weak identification statistic with the Stock-Yogo critical values.
Recourse must be had to the test devised by J.L. Montiel Olea and Carolin Pflueger. This test has been coded in the user-written weakivtest programme in Stata. The test is only valid when there is one endogenous regressor. Research is still being undertaken concerning what to do if there is more than one endogenous regressor and the residuals are heteroscedastic and/or serially correlated.
The test is based on what it calls the “effective F statistic” which corrects for heteroscedasticity and/or serial correlation as specified in the options of the estimating equation. It provides critical values for two estimation methods, TSLS and LIML

We now apply the test to the demand for cigarettes example allowing for heteroscedasticity in the residuals. We first run the model with one instrument, then two. After each estimation we test for weak identification.
⚠ The estimation command for the model has to be run with the robust option (or any other option specifying heteroscedasticity or serially correlation). Otherwise the program will hang.

. quietly ivreg2 lpackpc (lravgprs = rtaxso), r
. weakivtest
(obs=48)
Montiel-Pflueger robust weak instrument test
--------------------------------------------
Effective F statistic:       40.385
Confidence level alpha:          5%
--------------------------------------------
--------------------------------------------
Critical Values             TSLS      LIML
--------------------------------------------
% of Worst Case Bias
tau=5%                    37.418    37.418
tau=10%                   23.109    23.109
tau=20%                   15.062    15.062
tau=30%                   12.039    12.039
--------------------------------------------
 
. quietly ivreg2 lpackpc (lravgprs = rtaxso rtax),  r
. weakivtest
(obs=48)
Montiel-Pflueger robust weak instrument test
--------------------------------------------
Effective F statistic:      227.507
Confidence level alpha:          5%
--------------------------------------------
--------------------------------------------
Critical Values             TSLS      LIML
--------------------------------------------
% of Worst Case Bias
tau=5%          16.392    25.517
tau=10%     10.771    15.896
tau=20%     7.534    10.489
tau=30%     6.297     8.460
--------------------------------------------

The interpretation of the output is similar to that of the Stock-Yogo maximal bias test. It tests the null hypothesis that the estimator’s approximate asymptotic bias exceeds a certain fraction, τ (tau), of a “worst-case” benchmark.
In both cases of one instrument and two instruments, the effective F statistic is larger than the largest critical value: the null hypothesis of (jointly) weak instruments is rejected.

Google for Stock, Weak Instruments: Beyond the i.i.d. Homoskedastic Case
On Edit; the data for the demand for cigarettes example comes from Stock and Watson's textbook