Testing overidentifying restrictions with many instruments and heteroscedasticity using regularised jackknife IV

This paper proposes a new overidentifying restrictions test in a linear model when the number of instruments (possibly weak) may be smaller or larger than the sample size n or even infinite in a heteroscedastic framework. The proposed J test combines two techniques: the jackknife method and the regularisation technique which consists in stabilising the projection matrix. We theoretically show that our new test achieves the asymptotically correct size in the presence of many instruments. The simulation results demonstrate that our modified J statistic test has better empirical properties in small samples than existing J tests. We also propose a regularised F-test to assess the strength of the instruments, which is robust to heteroscedasticity and many instruments.