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Prueba de hipótesis sobre la existencia de una raíz fraccional en una serie de tiempo no estacionaria Resumen: En este trabajo se propone una modificación de la prueba de hipótesis propuesta por Castaño, Gómez y Gallón (2008) para determinar la existencia de memoria larga en un proceso ARFIMA(p,d,q) estacionario e invertible. En el caso puntual de los procesos ARFIMA(p,d,q), esta modificación permite determinar la existencia de una raíz fraccional en una serie de tiempo no estacionaria cuyo componente ARMA de corto plazo es indeterminado o desconocido. Vía simulaciones de Monte Carlo, se validan los resultados analíticos obtenidos en el trabajo y se demuestra el buen comportamiento de la prueba propuesta, en términos de potencia y tamaño, en comparación con otras metodologías disponibles en la literatura. Palabras clave: Series de tiempo de memoria larga, parámetro de diferenciación fraccional, aproximación autorregresiva, proceso ARFIMA no estacionario. Clasificación JEL: C15, C22, C52. A test for the existence of a fractional root in a non-stationary time series Abstract: In this work, we present a modification of the hypothesis testing procedure for the existence of long memory in the stationary and invertible ARFIMA(p,d,q) process proposed by Castaño, Gómez and Gallón (2008). This modification allows assessing the existence of a fractional root in a non-stationary time series when the short-term ARMA component is undetermined or unknown, especially in ARFIMA(p,d,q) processes. We validate, via Monte Carlo simulations, the analytical results and demonstrate the good performance of the proposed test in terms of both power and size, in comparison to other well-known tests in the literature. Keywords: Long memory time series, fractional differencing parameter, autoregressiveapproximation, non-stationary ARFIMA process. JEL classification: C15, C22, C52. Un test d’hypothèses sur l'existence d'une racine fractionnaire sur une série chronologique non stationnaire Résumé: Cet article présente une modification du test d'hypothèse proposée par Castaño, Gomez et Gallon (2008), laquelle détermine l'existence d'une mémoire longue dans un processus du type ARFIMA (p, d, q) stationnaire et décomposable. Dans le cas spécifique du processus ARFIMA (p, d, q), cette modification permet de déterminer l'existence d'une racine fractionnaire sur une série chronologique non stationnaire, dont la composante ARMA à court terme est considérée indéterminée ou inconnue. Tout en faisant recours à la méthode Monte Carlo, les résultats obtenus sont validés et ils démontrent la bonne performance du test, aussi bien en termes de puissance et que de taille par rapport aux autres méthodes disponibles dans la littérature. Mots-clés: séries chronologiques à longue mémoire, paramètre de différenciation fractionnaire, approchement autorégressif, processus ARFIMA non-stationnaire. Classification JEL: C15, C22, C52.

In this paper we propose tests for the null hypothesis that a time series process displays a constant level against the alternative that it displays (possibly) multiple changes in level. Our proposed tests are based on functions of appropriately standardized sequences of the differences between sub-sample mean estimates from the series under investigation. The tests we propose differ notably from extant tests for level breaks in the literature in that they are designed to be robust as to whether the process admits an autoregressive unit root (the data are I ( 1 ) ) or stable autoregressive roots (the data are I ( 0 ) ). We derive the asymptotic null distributions of our proposed tests, along with representations for their asymptotic local power functions against Pitman drift alternatives under both I ( 0 ) and I ( 1 ) environments. Associated estimators of the level break fractions are also discussed. We initially outline our procedure through the case of non-trending series, but our analysis is subsequently extended to allow for series which display an underlying linear trend, in addition to possible level breaks. Monte Carlo simulation results are presented which suggest that the proposed tests perform well in small samples, showing good size control under the null, regardless of the order of integration of the data, and displaying very decent power when level breaks occur.

En este trabajo se propone una modificación de la prueba de hipótesis propuesta por Castaño, Gómez y Gallón (2008) para determinar la existencia de memoria larga en un proceso ARFIMA(p,d,q) estacionario e invertible. En el caso puntual de los procesos ARFIMA(p,d,q), esta modificación permite determinar la existencia de una raíz fraccional en una serie de tiempo no estacionaria cuyo componente ARMA de corto plazo es indeterminado o desconocido. Vía simulaciones de Monte Carlo, se validan los resultados analíticos obtenidos en el trabajo y se demuestra el buen comportamiento de la prueba propuesta, en términos de potencia y tamaño, en comparación con otras metodologías disponibles en la literatura. In this work, we present a modification of the hypothesis testing procedure for the existence of long memory in the stationary and invertible ARFIMA(p,d,q) process proposed by Castaño, Gómez and Gallón (2008). This modification allows assessing the existence of a fractional root in a non-stationary time series when the short-term ARMA component is undetermined or unknown, especially in ARFIMA(p,d,q) processes. We validate, via Monte Carlo simulations, the analytical results and demonstrate the good performance of the proposed test in terms of both power and size, in comparison to other well-known tests in the literature. Cet article présente une modification du test d'hypothèse proposée par Castaño, Gomez et Gallon (2008), laquelle détermine l'existence d'une mémoire longue dans un processus du type ARFIMA (p, d, q) stationnaire et décomposable. Dans le cas spécifique du processus ARFIMA (p, d, q), cette modification permet de déterminer l'existence d'une racine fractionnaire sur une série chronologique non stationnaire, dont la composante ARMA à court terme est considérée indéterminée ou inconnue. Tout en faisant recours à la méthode Monte Carlo, les résultats obtenus sont validés et ils démontrent la bonne performance du test, aussi bien en termes de puissance et que de taille par rapport aux autres méthodes disponibles dans la littérature.

The subject of this paper is autoregressive (AR) modeling of a stationary, Gaussian discrete time process, based on a finite sequence of observations. The process is assumed to admit an AR(∞) representation with exponentially decaying coefficients. We adopt the nonparametric minimax framework and study how well the process can be approximated by a finiteorder ARmodel. A lower bound on the accuracy of ARapproximations is derived, and a nonasymptotic upper bound on the accuracy of the regularized least squares estimator is established. It is shown that with a “proper” choice of the model order, this estimator is minimax optimal in order. These considerations lead also to a nonasymptotic upper bound on the mean squared error of the associated one-step predictor. A numerical study compares the common model selection procedures to the minimax optimal order choice. 1. Introduction. The standard methods for estimating parameters of time series are based on the assumption that the observations come from an autoregressive (AR), moving average (MA), or mixed (ARMA) model of known orders. This assumption can rarely be justified in practice, and the less stringent assumption is that the time series data are observations from a linear stationary process. A common approach to modeling linear stationary processes is based on an ARapproximation. In this framework a finite order ARmodel is fitted to the observations. The order of the ARmodel should provide an “optimal” finite ARapproximation to the process, and it is usually chosen by selection procedures based on the data. This nonparametric ARapproach to modeling linear stationary processes has been investigated by Shibata (1980), Bhansali (1981, 1986), An, Chen and Hannan (1982) and Hannan and Kavalieris (1986). Shibata (1980) considered the problem of predicting a Gaussian infiniteorder ARprocess by fitting a finite ARmodel. The notion of optimality for the model selection procedure proposed by Shibata (1980) is based on an asymptotic lower bound on the mean squared prediction error. Specifically, the procedure is asymptotically efficient if it attains the lower bound asymptotically. Shibata (1980) also established that the final prediction error (FPE) [Akaike (1970)] and the AIC [Akaike (1974)] criteria are asymptotically efficient in the above sense, provided that the linear process does not degenerate to a finite order autoregression. A similar result has been obtained

We study a bootstrap method for stationary real-valued time series, which is based on the sieve of autoregressive processes. Given a sample [math] from a linear process [math] , we approximate the underlying process by an autoregressive model with order [math] , where [math] as the sample size [math] . Based on such a model, a bootstrap process [math] is constructed from which one can draw samples of any size. ¶ We show that, with high probability, such a sieve bootstrap process { X t *} t ∈ℤ satisfies a new type of mixing condition. This implies that many results for stationary mixing sequences carry over to the sieve bootstrap process. As an example we derive a functional central limit theorem under a bracketing condition.

We study a bootstrap method which is based on the method of sieves. A linear process is approximated by a sequence of autoregressive processes of order p=p(n), where p(n)→∞, p(n)=o(n) as the sample size n→∞. For given data, we then estimate such an AR(p(n)) model and generate a bootstrap sample by resampling from the residuals. This sieve bootstrap enjoys a nice nonparametric property, being model-free within a class of linear processes.

Tracking of an unknown frequency embedded in noise is widely applied in a variety of applications. Unknown frequencies can be obtained by approximating generalized spectral density of a periodic process by an autoregressive (AR) model. The advantage is that an AR model has a simple structure and its parameters can be easily estimated iteratively, which is crucial for online (real-time) applications. Typically, the order of the AR approximation is chosen by information criteria. However, with an increase of a sample size, model order may change, which leads to re-estimation of all model parameters. We propose a new iterative procedure for frequency detection based on a regularization of an empirical information matrix. The suggested method enables to avoid the repeated model selection as well as parameter estimation steps and therefore optimize computational costs. The asymptotic properties of the proposed regularized AR (RAR) frequency estimates are derived and performance of RAR is evaluated by numerical examples.

Theorems are proved relating to the rate of almost sure convergence of autocovariances, and hence autocorrelations, to their true values. These rates are uniform in the lag up to some order $P(T)$, increasing with $T$. The key assumption is that the process is stationary and the best linear predictor is the best predictor. In particular for an ARMA process and $P(T) = O\{(\ln T)^a\}, a < \infty$, the rate is $O\{(\ln \ln T/T)^{1/2}\}$. These results are used to discuss autoregressions and the use of autoregressions to approximate the structure of a more general process by increasing the order of the autoregression with $T$.