Modeling and Multivariate Methods - SAS

366 Performing Time Series Analysis Chapter 14
ARIMA Model
B is the backshift operator defined as
By t
= y t – 1
w t
= ( 1 – B) d y t
is the response series after differencing
μ is the intercept or mean term.
φ( B) and θ( B)
, respectively, the autoregressive operator and the moving average operator and are
written
φ( B) = 1 – φ 1 B – φ 2 B 2 – … – φ p B p andθ( B) = 1 – θ 1 B – θ 2 B 2 – … – θ q B q
a t
are the sequence of random shocks.
The a t
are assumed to be independent and normally distributed with mean zero and constant variance.
The model can be rewritten as
φ( B)w t
= δ + θ( B)a t where the constant estimate δ is given by the relation
δ = φ( B)μ = μ – φ 1
μ – φ 2
μ – … – φ p
μ .
The ARIMA command displays the Specify ARIMA Model dialog, which allows you to specify the ARIMA
model you want to fit. The results appear when you click Estimate.
Use the Specify ARIMA Model dialog for the following three orders that can be specified for an ARIMA
model:
1. The Autoregressive Order is the order (p) of the polynomial ϕ( B)
operator.
2. The Differencing Order is the order (d) of the differencing operator.
3. The Moving Average Order is the order (q) of the differencing operator θ( B)
.
4. An ARIMA model is commonly denoted ARIMA(p,d,q). If any of p,d, or q are zero, the corresponding
letters are often dropped. For example, if p and d are zero, then model would be denoted MA(q).