Covariate-Adjusted Regression for Time Series

Abstract

The main purpose of this article is to consider the covariate-adjusted regression (CAR) model for time series. The CAR model was initially proposed by Sentürk and Müller (2005) for such situations where predictor and response variables are not directly observed, but are distorted by some common observable covariate. Despite CAR being originally designed for independent cross-sectional data, multiple works have extended this method to dependent data setting. In this article, the authors extend CAR to the distorted time series setting. This extension is meaningful in many fields such as econometrics, mathematical finance, and signal processing. The estimates of regression parameters are proposed by establishing connection with functional-coefficient time series model. The consistency and asymptotic normality of the proposed estimates are investigated under the α-mixing conditions. Real data and simulated examples are provided for illustration.

Keywords:

• α-mixing
• Asymptotic normality
• Covariate-adjusted regression
• Functional-coefficient regression model
• Goodness-of-fit test
• Time series

Mathematics Subject Classification:

• Primary 62M10
• Secondary 62F12

Acknowledgments

This research was supported by the National Natural Science Foundation of China grant 11071146 and the National Basic Research Program of China (973 Program, No. 2007CB814901).

Notes

Notes: The MSE of estimates , , and . The simulation is repeated 300 times for each of sample sizes of 50, 100, and 200.