Box–Jenkins Time Series Models

Introduction

We are going to examine the Autoregressive Moving Average (ARMA) process for identifying the serial correlation attributes of a stationary time series (see Boland 2008; Box and Jenkins 1970). Another name for the processes that we will undertake is the Box–Jenkins (BJ) Methodology, which describes an iterative process for identifying a model and then using that model for forecasting. The Box–Jenkins methodology comprises four steps:

• Identification of process

• Estimation of parameters

• Verification of model

• Forecasting

Identification of Process

Assume we have a (at least weakly) stationary time series, i.e., no trend, seasonality, and it is homoscedastic (constant variance). Stationarity will be discussed further in section Stationarity. The general form of an ARMA model is

$${X}_{t} - {\phi }_{1}{X}_{t-1} -\cdots - {\phi }_{p}{X}_{t-p} = {Z}_{t} + {\theta }_{1}{Z}_{t-1} + \cdots + {\theta }_{q}{Z}_{t-q},$$

(1)

where {X _t } are identically distributed random variables ∼ (0, σ...