Integration Versus Trend Stationary in Time Series

1. INTRODUCTION
A WELL-KNOWN APPROACH to modeling macroeconomic time series is to assume that the
natural logarithm of the series can be represented by the sum of a deterministic time
trend and a stochastic term. The trend need not literally be part of the data generation
process, but may be viewed as a substitute for a complicated and unknown function of
population, capital accumulation, technical progress, etc. Within this approach there are
two competing models; in the trendatationary specification the stochastic term follows a
stationary process, while in the integrated specification the stochastic term follows a
random walk. The essential difference between the models is the nature of the process
driving the stochastic component, not whether the series is trended. The conclusion of
this study is that it is difficult to discriminate between the two models using classical
testing methods. This is the consequence of low power: the powers of integration tests
against plausible trend-stationary alternatives can be quite low, as can the powers of
trend-stationarity tests against integrated alternatives. Our analysis thus suggests that it
is premature to accept the integration hypothesis as a stylized fact of macroeconomic
time series.
The leading case we examine is a model with linear trend and iid normal innovations,
which we use to study the power of integration and trend-stationarity tests. This strategy
is motivated by the idea that the study should begin with the case which is most favorable
to high power; the presumption is that the finite sample powers of tests designed for this
case are superior to the powers of tests designed for models with more general
innovation sequences.
2. LEADING CASE
Let the time series {y,} be the stochastic process generated by the linear model
and the first-order autoregressive (AR) process
It is assumed that the innovation sequence {u,} is iid N(0, a'), and xo is an unknown
constant. Thus model (2.1)-(2.2) can be interpreted as a random walk about a linear
trend when /3 = 1 and an asymptotically stationary AR(1) process about a linear trend
when I < 1. In either case, the standardized initial displacement plays an important
role below, and will be denoted by x;j:= xo/a = (yo -ao)/u. This parameter measures
the distance (in units of innovation standard deviations) between the initial value yo and
the trend line.