Inference of the Trend in a Partially Linear Model with Locally Stationary Regressors

Abstract

In this article, we construct the uniform confidence band (UCB) of nonparametric trend in a partially linear model with locally stationary regressors. A two-stage semiparametric regression is employed to estimate the trend function. Based on this estimate, we develop an invariance principle to construct the UCB of the trend function. The proposed methodology is used to estimate the Non-Accelerating Inflation Rate of Unemployment (NAIRU) in the Phillips Curve and to perform inference of the parameter based on its UCB. The empirical results strongly suggest that the U.S. NAIRU is time-varying.

Keywords:

• Invariane principle
• Local-stationarity
• Nonparametric trend
• Partially linear model
• Phillips curve
• Semiparametric regression
• Time-varying NAIRU
• Uniform confidence band

JEL Classification:

• C12
• C13
• C14

ACKNOWLEDGMENT

I thank the suggestions from the editor, associate editor, and the two anonymous referees. Discussions on this paper with Hira Koul, Chae Young Lim, Wei Biao Wu, and Zhibiao Zhao led to significant improvements to the earlier draft. However, I am responsible for all errors.

Notes

¹We use index i for time. The traditional time index t is reserved for continuous time in [0, 1].

²From Figure 1, the U.S. unemployment rate appears to be a non-stationary (and possibly locally stationary) process. Thus, our assumption of locally stationary regressors for model (29) seems appealling.

³We use an Epanechnikov kernel. The GCV chooses b_n = 0.11 for (30).

⁴The width of 95% UCB of time-varying NAIRU is 2.38% while that of 99% UCB is 2.73%.

⁵The width of band is constant because the band is uniform and the model error is stationary.