STATIONARITY AGAINST INTEGRATION IN THE AUTOREGRESSIVE
PROCESS WITH POLYNOMIAL TREND
Abstract: We tackle the stationarity issue of an autoregressive path with a polynomial trend, and
generalize some aspects of the LMC test, the testing procedure of Leybourne and McCabe.
First, we show that it is possible to get the asymptotic distribution of the test statistic
under the null hypothesis of trend-stationarity as well as under the alternative of
nonstationarity for any polynomial trend of order . Then, we explain the reason why the
LMC test, and by extension the KPSS test, does not reject the null hypothesis of
trend-stationarity, mistakenly, when the random walk is generated by a unit root located at
. We also observe it on simulated data and correct the procedure. Finally, we
describe some useful stochastic processes that appear in our limiting distributions.
2010 AMS Mathematics Subject Classification: Primary: 62M10, 60G10, 62M02;
Secondary: 91G70, 91B84.
Keywords and phrases: LMC test, KPSS test, unit root, stationarity testing
procedure, polynomial trend, stochastic nonstationarity, random walk, integrated
process, ARIMA process, Donsker’s invariance principle, continuous mapping