Using ivreg2 but f statistic too large

[Jenny Nguyen]

I investigate a household data with IVREG2. I have only 1 endogenous variable and 1 instrument. This mean just identification. However, Kleibergen-Paap Wald rk F statistic is too large if I cluster standard errors at district level (F statistic = 32035), and still very large if I cluster at provincial level (F statistic = 1784.64 as below). If I create a variable combining province with type of work organization and cluster standard errors by that variable then the F statistic is equal to 151 (still large compare to Stock-Yogo indicator).

I want to know if I can use the combined variable to cluster or not. Is F statistic equaling to 151 acceptable compared to Stock-Yogo below? Is there any other way to reduce F statistic? By the way, is Anderson Rubin test important for my case of just identification?

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Weak identification test
Ho: equation is weakly identified
Cragg-Donald Wald F statistic 78695.51
Kleibergen-Paap Wald rk F statistic 1784.64
Stock-Yogo weak ID test critical values for K1=1 and L1=1:
10% maximal IV size 16.38
15% maximal IV size 8.96
20% maximal IV size 6.66
25% maximal IV size 5.53
Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and orthogonality conditions are valid
Anderson-Rubin Wald test F(1,62)= 23.34 P-val=0.0000
Anderson-Rubin Wald test Chi-sq(1)= 23.72 P-val=0.0000
Stock-Wright LM S statistic Chi-sq(1)= 31.95 P-val=0.0000

[George Ford]

what is the null of the Kleibergen-Paap Wald test?