### Abstract

Varying-coefficient linear models arise from multivariate nonparametric regression, non-linear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an index. To enlarge the modelling capacity substantially, this paper explores a class of varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varying-coefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a one-step scheme and estimating coefficient functions through one-dimensional local linear smoothing. The locally significant variables are selected in terms of a combined use of the t-statistic and the Akaike information criterion. We further extend the algorithm for models with two indices. Simulation shows that the methodology proposed has appreciable flexibility to model complex multivariate non-linear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink-muskrat data in 1925-1994 and the pound-dollar exchange rates in 1974-1983.

Original language | English (US) |
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Pages (from-to) | 57-80 |

Number of pages | 24 |

Journal | Journal of the Royal Statistical Society. Series B: Statistical Methodology |

Volume | 65 |

Issue number | 1 |

DOIs | |

State | Published - 2003 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Akaike information criterion
- Backfilling algorithm
- Generalized cross-validation
- Local linear regression
- Local significant variable selection
- One-step estimation
- Smoothing index

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## Cite this

*Journal of the Royal Statistical Society. Series B: Statistical Methodology*,

*65*(1), 57-80. https://doi.org/10.1111/1467-9868.00372