Parameter Estimation for Periodically Stationary Time Series

Abstract

The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule-Walker estimates.

Covariance estimation for multivariate conditionally Gaussian dynamic linear models

“…For comparison purposes, (2) and this demonstrates the accuracy of the estimator S t . We observe that under Σ 3 , the MSSE (1) has values significantly smaller than 1 as compared to the MSSE (2) using the true value of Σ 3 .…”

“…It appears that the estimator S t for all models converges to the true values of Σ, but the rate of convergence depends on the underlying state space model (here LL performs faster convergence) and on the prior S 0 . Table 2 shows the averaged (over all 1000 simulated time series) mean vector of squared standardized one-step forecast errors (MSSE (1) ), for each of the three models (LL, LT, SE) and for each of Σ (Σ 1 , Σ 2 , Σ 3 ). For comparison purposes, (2) and this demonstrates the accuracy of the estimator S t .…”

“…The following procedure applies to any conditionally Gaussian dynamic linear model in the form of equation (1). For the simulation studies considered in this paper, given data y N = (y 1 , y 2 , .…”

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

“…Anderson and Meerschaert [4] develop the asymptotics necessary to determine which of these estimates are statistically different from zero, under the classical assumption that the noise sequence has finite fourth moment. In this paper, we extend those results to the case where the noise sequence has finite second moment but infinite fourth moment.…”

“…An important class of stochastic models for describing periodically stationary time series are the periodic ARMA models, in which the model parameters are allowed to vary with the season. Periodic ARMA models are developed by many authors including [1,2,[4][5][6][7]20,[22][23][24]26,28,30,31,[33][34][35][36][37][38][39][40][41].…”