This paper considers speciﬁcation testing for instrumental variables estimation in the presence of many instruments. The test proposed is a modiﬁed version of the Sargan (1958, Econometrica 26(3): 393-415) test of overidentifying restrictions. The test statistic asymptotically follows the standard normal distribution under the null hypothesis of correct speciﬁcation when the number of instruments increases with the sample size. We ﬁnd that the new test statistic is numerically equivalent up to a sign to the test statistic proposed by Hahn and Hausman (2002, Econometrica 70(1): 163-189). We also assess the size and power properties of the test.
Lee, Yoonseok and Okui, Ryo, "A Specification Test for Instrumental Variables Regression with Many Instruments" (2009). Cowles Foundation Discussion Papers. 2065.