### Abstract

This paper investigates the asymptotic size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t-statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS), depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a positive lower bound on the strength of the instruments. The asymptotic size equals 1 for empirically relevant choices of the parameter space. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS-based t-statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t-statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).

Original language | English (US) |
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Pages (from-to) | 369-382 |

Number of pages | 14 |

Journal | Econometric Theory |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Social Sciences (miscellaneous)
- Economics and Econometrics

### Cite this

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*Econometric Theory*, vol. 26, no. 2, pp. 369-382. https://doi.org/10.1017/S0266466609100026

**The impact of a Hausman pretest on the asymptotic size of a hypothesis test.** / Guggenberger, Patrik.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The impact of a Hausman pretest on the asymptotic size of a hypothesis test

AU - Guggenberger, Patrik

PY - 2010/4/1

Y1 - 2010/4/1

N2 - This paper investigates the asymptotic size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t-statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS), depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a positive lower bound on the strength of the instruments. The asymptotic size equals 1 for empirically relevant choices of the parameter space. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS-based t-statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t-statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).

AB - This paper investigates the asymptotic size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t-statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS), depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a positive lower bound on the strength of the instruments. The asymptotic size equals 1 for empirically relevant choices of the parameter space. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS-based t-statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t-statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).

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UR - http://www.scopus.com/inward/citedby.url?scp=65949089210&partnerID=8YFLogxK

U2 - 10.1017/S0266466609100026

DO - 10.1017/S0266466609100026

M3 - Article

AN - SCOPUS:65949089210

VL - 26

SP - 369

EP - 382

JO - Econometric Theory

JF - Econometric Theory

SN - 0266-4666

IS - 2

ER -