Discussion Paper Series 2006-E-1

Estimating Deterministic Trends with an Integrated or Stationary Noise Component

Pierre Perron and Tomoyoshi Yabu

We propose a test for the slope of a trend function when it is a priori unknown whether the series is trend-stationary or contains an autoregressive unit root. Let be the sum of the autoregressive coefficients in the autoregressive representation of the series. The procedure is based on a Feasible Quasi Generalized Least Squares method from an AR(1) specification with parameter . The estimate of is the OLS estimate obtained from an autoregression applied to detrended data and is truncated to take a value 1 whenever the estimate is in a neighborhood of 1. This makes the estimate "super-efficient" when =1 and implies that inference on the slope parameter can be performed using the standard Normal distribution whether =1 or ||<1. Theoretical arguments and simulation evidence show that =1/2 is the appropriate choice. Simulations show that our procedure has good size properties and greater power than the tests proposed by Vogelsang (1998). Applications to inference about the growth rates of GNP for many countries show the usefulness of the method.

Key words: Linear Trend, Unit Root, Median-Unbiased Estimates, GLS Procedure, Super Efficient Estimates


Views expressed in Discussion Paper Series are those of the authors and do not necessarily reflect those of the Bank of Japan or Institute for Monetary and Economic Studies.

Copyright 2006 Bank of Japan All Rights Reserved.