Tendencia aleatoria o determinística: una nueva prueba basada en la teoría tradicional

Abstract

Several procedures to test the null hypothesis on the random or deterministic origin of the trend in a time series are found in the specialized literature. Most of these tests are based on the analysis of the unit roots of the autoregressive or moving average operators. The procedures are based on the nonstandard theory associated to a Wiener process. In this paper it is proposed a test that uses the autocorrelation function (ACF) of the residuals considering the null hypothesis H0 : Z_{t} = β_{0} + Z_{t-1} + a_{t}, and the alternative hypothesis H1 : Z_{t} = β_{0} + β_{1}t+a_{t}. The distribution of the test statistics for finite sample sizes and the asymptotic approximation are obtained using the usual theory, assuming that at is a gaussian white noise. The procedure is generalized for the case where at is a correlated white noise. The results obtained using simulation show that the proposed test has in general high power and specially when it is compared the well known Dicker-Fuller Augmented test (ADF), in the case when the roots of the autoregressive or moving average operators are close to one. The proposed procedure has also better approximation to the nominal test size when it is also compared with the ADF.

Several procedures to test the null hypothesis on the random or deterministic origin of the trend in a time series are found in the specialized literature. Most of these tests are based on the analysis of the unit roots of the autoregressive or moving average operators. The procedures are based on the nonstandard theory associated to a Wiener process. In this paper it is proposed a test that uses the autocorrelation function (ACF) of the residuals considering the null hypothesis H0 : Z_{t} = β_{0} + Z_{t-1} + a_{t}, and the alternative hypothesis H1 : Z_{t} = β_{0} + β_{1}t+a_{t}. The distribution of the test statistics for finite sample sizes and the asymptotic approximation are obtained using the usual theory, assuming that at is a gaussian white noise. The procedure is generalized for the case where at is a correlated white noise. The results obtained using simulation show that the proposed test has in general high power and specially when it is compared the well known Dicker-Fuller Augmented test (ADF), in the case when the roots of the autoregressive or moving average operators are close to one. The proposed procedure has also better approximation to the nominal test size when it is also compared with the ADF.