We develop a method of testing linearity using power transforms of regressors, allowing for stationary processes and time trends. The linear model is a simplifying hypothesis that derives from the power transform model in three different ways, each producing its own identification problem. We call this modeling difficulty the trifold identification problem and show that it may be overcome using a test based on the quasi-likelihood ratio (QLR) statistic. More specifically, the QLR statistic may be approximated under each identification problem and the separate null approximations may be combined to produce a composite approximation that embodies the linear model hypothesis. The limit theory for the QLR test statistic depends on a Gaussian stochastic process. In the important special case of a linear time trend regressor and martingale difference errors asymptotic critical values of the test are provided. Test power is analyzed and an empirical application to crop-yield distributions is provided. The paper also considers generalizations of the Box-Cox transformation, which are associated with the QLR test statistic.
Bibliographical noteFunding Information:
The co-editor, Jianqing Fan, Associated editor, and two anonymous referees provided helpful comments for which we are grateful. We have also benefited from discussions with Shun-ichiro Bessho, Seonghoon Cho, Stan Hurn, Isao Ishida, Tae-Hwan Kim, Shandre Thangavelu, Byungsam Yoo, Valentin Zelenyuk, and other participants at the NZESG meeting (Auckland, 2013) and the 8th Joint Economics Symposium of Four Leading East Asian Universities (Shanghai, 2014). Baek and Cho acknowledge research support from the second stage BK21 project and research grant ( NRF-2010-332-B00025 ) of the National Research Foundation of Korea , respectively, and Phillips acknowledges support from a Kelly Fellowship and the National Science Foundation of USA under Grant No. SES 12-58258 .
© 2015 Elsevier B.V.
All Science Journal Classification (ASJC) codes
- Economics and Econometrics