### Abstract

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coefficient functions can easily be estimated via a simple local regression. This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental data set.

Original language | English (US) |
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Pages (from-to) | 1491-1518 |

Number of pages | 28 |

Journal | Annals of Statistics |

Volume | 27 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1999 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Local linear fit
- Mean-squared errors
- Optimal rate of convergence
- Varying coefficient models

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## Cite this

*Annals of Statistics*,

*27*(5), 1491-1518. https://doi.org/10.1214/aos/1017939139