Suppose I have hierarchical data such as students clustered into classrooms. I want to use a two stage least squares regression with an instrument that affects students at the classroom level to test my hypothesis. What is the appropriate method to test if my instrument is weak? The Cragg-Donald F-statistic with Stock and Yogo critical values seems appropriate for the general case of identifying weak instruments but it doesn't build in any adjustment for clustering.
The appropriate weak instrument test for testing for weak instruments in panel data or more generally data that is non-i.i.d is the Kleibergen-Paap Wald rk F statistic. From the STATA documentation of the ivreg2 command:
When the i.i.d. assumption is dropped and ivreg2 is invoked with the robust, bw or cluster options, the Cragg-Donald-based weak instruments test is no longer valid. ivreg2 instead reports a correspondingly-robust Kleibergen-Paap Wald rk F statistic. The degrees of freedom adjustment for the rk statistic is (N-L)/L1, as with the Cragg-Donald F statistic, except in the cluster-robust case, when the adjustment is N/(N-1) * (N_clust-1)/N_clust, following the standard Stata small-sample adjustment for cluster-robust. In the case of two-way clustering, N_clust is the minimum of N_clust1 and N_clust2. The critical values reported by ivreg2 for the Kleibergen-Paap statistic are the Stock-Yogo critical values for the Cragg-Donald i.i.d. case. The critical values reported with 2-step GMM are the Stock-Yogo IV critical values, and the critical values reported with CUE are the LIML critical values.