Stationarity against integration in the autoregressive process with polynomial trend

Frédéric Proïa (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

We tackle the stationarity issue of an autoregressive path with a polynomial trend, and we 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 $r$. 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 $-1$. We also observe it on simulated data. Finally, we describe some useful stochastic processes that appear in our limiting distributions.

Mots clés: Donsker's invariance principle; ARIMA process; Unit root; Stationarity testing procedure; LMC test; Continuous mapping theorem; Random walk; Polynomial trend; Stochastic nonstationarity; Integrated process; KPSS test

Domaines: Statistiques [math.ST]

Frédéric Proïa. Stationarity against integration in the autoregressive process with polynomial trend. Probability and Mathematical Statistics, 2018, 38 (1), pp.1-26. ⟨10.19195/0208-4147.38.1.1⟩. ⟨hal-01171154⟩