## Abstract

Suppose Y has a linear regression on X_{1}, X_{2}, but observations are only available on (Y, X_{1}). If large scale data on (X_{1}, X_{2}) are available, which do not include Y, and if the regression of X_{2}, given X_{1}, is nonlinear, then one may estimate the regression coefficients of Y by using the proxy g(X_{1}) {colon equals} E(X_{2}|X_{1}) for X_{2}, or an instrument φ(X_{1}) which is uncorrelated with X_{2}. Both methods provide estimators which are asymptotically normal around the true parameter values under appropriate assumptions. A computation of the optimal instrument is provided, and the asymptotic relative efficienties of the two types of estimators compared.

Original language | English (US) |
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Pages (from-to) | 123-138 |

Number of pages | 16 |

Journal | Journal of Multivariate Analysis |

Volume | 47 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1993 |

Externally published | Yes |

## Keywords

- Efficiency
- Linear regression
- Nonlinear regression
- Optimal instrument
- Proxy

## ASJC Scopus subject areas

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty