### Abstract

We consider the following problem. There is a structural equation of interest that contains an explanatory variable that theory predicts is endogenous. There are one or more instrumental variables that credibly are exogenous with regard to this structural equation, but which have limited explanatory power for the endogenous variable. Further, there is one or more potentially 'strong' instruments, which has much more explanatory power but which may not be exogenous. Hausman (1978) provided a test for the exogeneity of the second instrument when none of the instruments are weak. Here, we focus on how the standard Hausman test does in the presence of weak instruments using the StaigerStock asymptotics. It is natural to conjecture that the standard version of the Hausman test would be invalid in the weak instrument case, which we confirm. However, we provide a version of the Hausman test that is valid even in the presence of weak IV and illustrate how to implement the test in the presence of heteroskedasticity. We show that the situation we analyze occurs in several important economic examples. Our Monte Carlo experiments show that our procedure works relatively well in finite samples. We should note that our test is not consistent, although we believe that it is impossible to construct a consistent test with weak instruments.

Original language | English |
---|---|

Pages (from-to) | 289-299 |

Number of pages | 11 |

Journal | Journal of Econometrics |

Volume | 160 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 Feb 1 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Economics and Econometrics
- Applied Mathematics
- History and Philosophy of Science

### Cite this

*Journal of Econometrics*,

*160*(2), 289-299. https://doi.org/10.1016/j.jeconom.2010.09.009

}

*Journal of Econometrics*, vol. 160, no. 2, pp. 289-299. https://doi.org/10.1016/j.jeconom.2010.09.009

**The Hausman test and weak instruments.** / Hahn, Jinyong; Ham, John C.; Moon, Hyungsik Roger.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Hausman test and weak instruments

AU - Hahn, Jinyong

AU - Ham, John C.

AU - Moon, Hyungsik Roger

PY - 2011/2/1

Y1 - 2011/2/1

N2 - We consider the following problem. There is a structural equation of interest that contains an explanatory variable that theory predicts is endogenous. There are one or more instrumental variables that credibly are exogenous with regard to this structural equation, but which have limited explanatory power for the endogenous variable. Further, there is one or more potentially 'strong' instruments, which has much more explanatory power but which may not be exogenous. Hausman (1978) provided a test for the exogeneity of the second instrument when none of the instruments are weak. Here, we focus on how the standard Hausman test does in the presence of weak instruments using the StaigerStock asymptotics. It is natural to conjecture that the standard version of the Hausman test would be invalid in the weak instrument case, which we confirm. However, we provide a version of the Hausman test that is valid even in the presence of weak IV and illustrate how to implement the test in the presence of heteroskedasticity. We show that the situation we analyze occurs in several important economic examples. Our Monte Carlo experiments show that our procedure works relatively well in finite samples. We should note that our test is not consistent, although we believe that it is impossible to construct a consistent test with weak instruments.

AB - We consider the following problem. There is a structural equation of interest that contains an explanatory variable that theory predicts is endogenous. There are one or more instrumental variables that credibly are exogenous with regard to this structural equation, but which have limited explanatory power for the endogenous variable. Further, there is one or more potentially 'strong' instruments, which has much more explanatory power but which may not be exogenous. Hausman (1978) provided a test for the exogeneity of the second instrument when none of the instruments are weak. Here, we focus on how the standard Hausman test does in the presence of weak instruments using the StaigerStock asymptotics. It is natural to conjecture that the standard version of the Hausman test would be invalid in the weak instrument case, which we confirm. However, we provide a version of the Hausman test that is valid even in the presence of weak IV and illustrate how to implement the test in the presence of heteroskedasticity. We show that the situation we analyze occurs in several important economic examples. Our Monte Carlo experiments show that our procedure works relatively well in finite samples. We should note that our test is not consistent, although we believe that it is impossible to construct a consistent test with weak instruments.

UR - http://www.scopus.com/inward/record.url?scp=78650519831&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650519831&partnerID=8YFLogxK

U2 - 10.1016/j.jeconom.2010.09.009

DO - 10.1016/j.jeconom.2010.09.009

M3 - Article

AN - SCOPUS:78650519831

VL - 160

SP - 289

EP - 299

JO - Journal of Econometrics

JF - Journal of Econometrics

SN - 0304-4076

IS - 2

ER -