All simulators
Over-Identification & the J Test
Give one instrument a direct effect and see the estimates drift and the J test catch it only sometimes.
Day 66: Over-Identification & Instrument Validitymean IV, instrument 1
0.501
mean IV, instrument 2
0.662
theory 0.5 + δ/π₂ = 0.667
mean 2SLS
0.588
J test rejection rate
12.0%
5% if both valid
IV with instrument 1IV with instrument 22SLS with both
Two instruments for one endogenous regressor (true effect 0.5); 400 simulated studies of 500 observations. Give instrument 2 a direct effect on the outcome and watch its estimate drift, 2SLS follow it, and the J test reject only some of the time.