Periodic Autoregressive Model Research Articles

Testing for periodic integration

A periodic autoregressive time-series model assumes that the autoregressive parameters vary with the season. This model can also be represented by a multivariate model for the annual vector containing the seasonal observations. When this multivariate model contains one unit root, a time-series is said to be periodically integrated of order 1. In this paper we propose tests for such a single unit root. These tests for periodic integration are applied to a periodic model for the quarterly German consumption series.

The effects of seasonally adjusting a periodic autoregressive process

Traditional methods for the analysis of seasonal and nonstationary time series assume that seasonality and a stochastic trend can be separated in some way. However, several macroeconomic time series display patterns which indicate that separation may not be valid. Such patterns occur if seasonal movements change slowly over time and the timing of changes depends on exogenous shocks from e.g., a business cycle. Periodic autoregressive processes with unit roots are suitable for modeling and forecasting such series. Using Monte-Carlo simulations the Census X-11 adjustment and the Box-Jenkins analysis are compared considering the case that the data are generated by periodic processes. It appears, for example, that the intrinsic periodicity is removed only partially, that a test for a unit root is robust, and that the most sensible practical strategy seems to be to start with a general periodic autoregressive model. If necessary, one can then switch to a usual seasonal adjustment procedure if the hypothesis of nonperiodicity cannot be rejected. The quartely real German GNP series is used to illustrate that a periodic model can yield superior modeling and forecasting.

PERIODIC CORRELATION IN STRATOSPHERIC OZONE DATA

Abstract. A 50‐year time series of monthly stratospheric ozone readings from Arosa, Switzerland, is analyzed. The time series exhibits the properties of a periodically correlated (PC) random sequence with annual periodicities. Spectral properties of PC random sequences are reviewed and a test to detect periodic correlation is presented. An autoregressive moving‐average (ARMA) model with periodically varying coefficients (PARMA) is fitted to the data in two stages. First, a periodic autoregressive model is fitted to the data. This fit yields residuals that are stationary but non‐white. Next, a stationary ARMA model is fitted to the residuals and the two models are combined to produce a larger model for the data. The combined model is shown to be a PARMA model and yields residuals that have the correlation properties of white noise.

DIAGNOSTIC CHECKING OF PERIODIC AUTOREGRESSION MODELS WITH APPLICATION

Abstract. An overview of model building with periodic autoregression (PAR) models is given emphasizing the three stages of model development:identification, estimation and diagnostic checking. New results on the distribution of residual autocorrelations and suitable diagnostic checks are derived. The validity of these checks is demonstrated by simulation. The methodology discussed is illustrated with an application. It is pointed out that the PAR approach to model development offers some important advantages over the more general approach using periodic autoregressive moving‐average models.

Comparison of non-Gaussian multicomponent and periodic autoregressive models for river flow

The Non-Gaussian Multicomponent model for river flow (NGM) of Vandewiele and Dom is modified in order to facilitate maximum likelihood estimation. It is also generalized so that a wider variety of river flows at a diversity of time steps can be modeled. This model is applied to two basins in Belgium and France with very different areas, both at monthly and weekly time scale. Results on the quality of forecasting and simulation (especially simulation of low and high flow volumes) are compared with those of classical Periodic Autoregressive models (PAR). Results with NGM are always better, in most cases considerably better. This is due to the fact that NGM models explicitly take into consideration the presence of so called flow components, like baseflow and direct flow recession, which are phenomena well known to hydrologists.

Generation of multivariate autoregressive sequences with emphasis on initial values

Certain aspects of data generation are studied through multivariate autoregressive (AR) models. The main emphasis is on the preservation of certain desired moments and the effect of initial values on these moments. The problem of preservation of moments is approached in a nontraditional way by starting with the initial values. For this purpose, general AR processes with a random start and with time-varying parameters are introduced to lay a foundation for the analysis of all types of AR processes, including the periodic cases. It is shown that an AR process with a random start and with parameters obtained from the moment equations is capable of generating jointly multivariate normal vectors with any specified means and covariance matrices, and with any specified autocovariance matrices up to a given lag. With a random start, there is no transition period involved for achieving these moments. A simple solution is proposed for matrix equations of the form BB T = A which appear in the moment equations. The aggregation properties of general AR process are also studied. A more detailed analysis is given for the two-period first-order periodic autoregressive model, PAR 2(1). For the PAR 2(1) process with a random start and with parameters obtained from the moment equations, it is shown that the autocovariance function depends only on the period and the lag, and therefore the process is periodic (covariance) stationary. The PAR 2(1) process with a fixed start is also studied. It is shown that the moments of this process depend on the absolute time, in addition to the period and the lag, and therefore the process is not periodic stationary. This dependence diminishes with time, and periodic stationarity is realized if the AR parameters satisfy certain conditions. In that case, the PAR 2(1) process with a fixed start converges to that with a random start, but only after a certain transition period. This proves the superiority of a random start over a fixed start.

The Performance of Periodic Autoregressive Models in Forecasting Seasonal U.K. Consumption

The parameters of a periodic model are allowed to vary according to the time at which observations are made. Periodic autoregressive models are fitted to the quarterly values of seasonally unadjusted real nondurable consumers' expenditure for the United Kingdom and its components. The periodic model offers no improvement over conventional specifications if the aggregate is modeled directly. On the other hand, periodic models generally perform well for the components, which contain additional seasonal information. The choice between a periodic or nonperiodic specification is also shown to have an important influence on the resulting dynamic properties.

The Performance of Periodic Autoregressive Models in Forecasting Seasonal U.K. Consumption

The Performance of Periodic Autoregressive Models in Forecasting Seasonal U.K. Consumption

Combining Hydrologic Forecasts

Forecasts of river flows are useful in optimizing the operation of multipurpose reservoir systems. Using two case studies, the usefulness of combination techniques for improving forecasts is examined. In the first study, a transfer function‐noise model, a periodic autoregressive model, and a conceptual model are employed to forecast quarter‐monthly river flows. These models all approach the modeling and forecasting problem from three different perspectives, and each has its own particular strengths and weaknesses. The forecasts generated by the individual models are combined in an effort to exploit the strengths of each model. The results of this case study indicate that significantly better forecasts can be obtained when forecasts from different types of models are combined. In the second study, periodic autoregressive models and seasonal autoregressive integrated moving average models are used to forecast monthly river flows. Combining the individual forecasts from these two statistical time series models does not result in significantly better forecasts.

Recent trends in random data analysis.

Recent trends in random data analysis with parametric and nonparametric approaches are reviewed. First, for nonparametric spectrum analysis, the effectiveness of using data windows is emphasized with an example. Secondly, based on a recursive algorithm by Levinson and Durbin, various autoregressive (AR) spectrum estimation techniques by Yule-Walker, Burg and others are explained together with their specific properties. Also, selection criteria for the order of AR model fitting are briefly discussed. As for autoregressive moving average (ARMA) spectrum estimation, the high order Yule-Walker method and its variants are mentioned. Finally, a periodic AR model is introduced for treating random data with periodic structures. An example of applying this model to real temperature data is also given.

FORECASTING QUARTER‐MONTHLY RIVERFLOW1

ABSTRACTRecent developments with respect to transfer function‐noise models are reviewed and used to model and forecast quarter‐monthly (i.e., near‐weekly) natural inflows to the Lac St‐Jean reservoir in the Province of Quebec, Canada. The covariate series are rainfall and snowmelt, the latter being a novel derivation from daily rainfall, snowfall and temperature series. It is clearly demonstrated using the residual variance and the Akaike information criterion that modeling is improved as one starts with a deseasonalized ARMA model of the inflow series and successively adds transfer functions for the rainfall and snowmelt series. It is further demonstrated that the transfer function‐noise model is better than a periodic autoregressive model of the inflow series. A split‐sample experiment is used to compare one‐step‐ahead forecasts from this transfer function‐noise model with forecasts from other stochastic models as well as with forecasts from a so‐called conceptual hydrological model (i.e., a model which attempts to mathematically simulate the physical processes involved in the hydrological cycle). It is concluded that the transfer function‐noise model is the preferred model for forecasting the quarter‐monthly Lac St‐Jean inflow series.

Forecasting monthly riverflow time series

Mean monthly flows from thirty rivers in North and South America are used to test the short-term forecasting ability of seasonal ARIMA, deseasonalized ARMA, and periodic autoregressive models. The series were split into two sections and models were calibrated to the first portion of the data. The models were then used to generate one-step-ahead forecasts for the second portion of the data. The forecast performance is compared using various measures of accuracy. The results suggest that a periodic autoregressive model, identified by using the partial autocorrelation function, provided the most accurate forecasts