Advances in Box‐Jenkins modeling: 1. Model construction

Abstract

Box‐Jenkins modeling of time series data can be improved and simplified by adhering to contemporary modeling procedures. This paper gives the theory and techniques of the application of many recent advances that have been made at the identification, estimation, and diagnostic check stages of model development. The inverse autocorrelation function and the inverse partial autocorrelation function are demonstrated to be useful identification tools for both nonseasonal and seasonal models. Parameters can be estimated more efficiently by employing the modified sum of squares technique. At the estimation stage it is also possible to obtain a maximum likelihood estimate for a Box‐Cox power transformation. The Akaike information criterion is introduced to formalize mathematically the concept of model parsimony. When checking for model adequacy, knowing the distribution of the residual autocorrelation allows for a sensitive test for residual whiteness. Diagnostic checks are given for verifying the assumption of homoscedasticity of the model residuals. In practice, heteroscedasticity and nonnormality of the residuals can often be removed by a Box‐Cox transformation.

Revue bibliographique des méthodes de prévision des débits

“…Cependant, chaque ajout accroît le nombre de paramètres et la complexité du modèle. Cependant, HiPEL et al (1977) et LJUNG (1987 ont proposé des méthodes pratiques d'estimation et d'identification du modèle.…”

Stochastic modeling of surface stream flow at different time scales: Sangsoorakh karst basin, Iran

“…building: identification, estimation, and diagnostic checking, which is discussed in detail by Hipel et al (1977). The 95% confidence limits are calculated using…”

“…The third stage of time-series analysis consists of diagnostic checking of the estimated model residuals and is used to evaluate the ARIMA and DARMA models performance. To determine whether the residual are white noise and normally independently distributed, residual autocorrelation function (RACF) tests (Hipel et al, 1977) are employed. The Akaike information criteria (AIC) (Akaike, 1974) and Schwars' approximation of the Bayes Information Criterion (BIC) (Schwars, 1978) were utilized to select the most appropriate model from the candidate set of calibrated models…”

“…Finally, the notation DARMA (p,q) is employed to represent this type of deseasonalized model where p and q stand for the orders of the nonseasonal and seasonal AR and MA operators presented in Equations (2) and (1), respectively. Calibration of time-series models is conducted based on the three stages of model building: identification, estimation, and diagnostic checking (Box and Jenkins, 1976;Hipel et al, 1977;Hipel and McLeod, 1994). The purpose of the identification stage is to determine the differencing required to produce stationarity and the persistence structure in the series using interpretation of the autocorrelation function (ACF) and the partial autocorrelation function (PACF).…”

Forecasting seasonal hydrologic response in major river basins.

“…After Box and Jenkins (1968) and Box and Jenkins (1970), numerous studies have been published in literature including Carlson et al (1970) Hipel et al (1977);McLeod et al (1977); Bras and Rodríguez-Iturbe (1994). Brief descriptions of the processes are summarized later.…”