ARIMA(p,d,q) Model

Definition

Let $W_t$ : ARMA(p,q) that is weakly stationary

An Autoregressive, Integrated, Moving Average model $Z_t$, often referred to ARIMA(p,d,q), is defined as

$$W_t={\nabla }^d{Z}_t$$

Formulations

ARIMA(p,1,q)

Let $W_t$ be a weakly stationary ARMA(p,q) model, defined as

$W_t={\varphi }_1{W}_{t-1}+\cdots +{\varphi }_p{W}_{t-p}+a_t-{\theta }_1{a}_{t-1}-\cdots -{\theta }_q{a}_{t-q}$

The model ARIMA(p,1,q) is defined as:

$$\begin{array}{rl}{W}_t& ={\nabla }^1{Z}_t\\ {\varphi }_1{W}_{t-1}+\cdots +{\varphi }_p{W}_{t-p}+a_t-{\theta }_1{a}_{t-1}-\cdots -{\theta }_q{a}_{t-q}& =(Z_t-Z_{t-1})\\ Z_{t-1}+{\varphi }_1{W}_{t-1}+\cdots +{\varphi }_p{W}_{t-p}+a_t-{\theta }_1{a}_{t-1}-\cdots -{\theta }_q{a}_{t-q}& =Z_t\\ Z_{t-1}+{\varphi }_1(Z_{t-1}-Z_{t-2})+\cdots +{\varphi }_p(Z_{t-p}-Z_{t-p-1})+a_t-\dots & =Z_t\end{array}$$

It resembles an ARMA(p+1, q) model but with one root of the AR polynomial exactly equal to 1.

Property: Characteristic Polynomial of ARIMA(p,1,q)

The AR operator using the backshift operator $B$ is: $\varphi (B)(1-B)Z_t=\theta (B)a_t$ The characteristic polynomial is: $\varphi (x)(1-x)=0$ This polynomial has one root equal to 1 (the $(1-x)$ factor), which confirms the non-stationarity of the original series. The remaining roots must lie outside the unit circle for the differenced series to be stationary.

Constant Term

If the differenced series $W_t={\nabla }^d{Z}_t$ has a non-zero mean ${\mu }_W$, a constant term ${\theta }_0$ is added: $\varphi (B)W_t={\theta }_0+\theta (B)a_t$ where ${\theta }_0={\mu }_W(1-{\varphi }_1-\cdots -{\varphi }_p)$.

• If $d=1$, a constant ${\theta }_0\ne 0$ implies a deterministic linear trend in the original series $Z_t$.
• If $d=2$, it implies a deterministic quadratic trend.

Sub-models

Integrated Moving Average (IMA)

An ARIMA(p,d,q) model where the autoregressive order $p=0$ and is integrated. Denoted as IMA(d,q).

IMA(1,1)

$\nabla Z_t=a_t-\theta a_{t-1}\text{or}{Z}_t=Z_{t-1}+a_t-\theta a_{t-1}$ Non-stationary due to the unit root in the AR part; variance increases linearly with time. Common for series with a stochastic level.

IMA(2,2)

${\nabla }^2{Z}_t=a_t-{\theta }_1{a}_{t-1}-{\theta }_2{a}_{t-2}\text{or}{Z}_t=2Z_{t-1}-Z_{t-2}+a_t-{\theta }_1{a}_{t-1}-{\theta }_2{a}_{t-2}$ Useful for modeling series with a stochastic trend.

Autoregressive Integrated (ARI)

An ARIMA(p,d,q) model where the moving average order $q=0$. Denoted as ARI(p,d).

ARI(1,1)

$\nabla Z_t=\varphi \nabla Z_{t-1}+a_t\text{or}{Z}_t=(1+\varphi )Z_{t-1}-\varphi Z_{t-2}+a_t$ where $|\varphi |<1$ for the differenced series to be stationary.

Procedure: Determining Weights for ARI(1,1)

Representing an ARI(1,1) process as a general linear process $Z_t=\sum {\psi }_j{a}_{t-j}$:

1. Use the relationship $(1-\varphi B)(1-B)\psi (B)=1$.
2. Equate coefficients of $B^k$ on both sides.
3. The recursive solution for ${\psi }_k$ is ${\psi }_k=(1+\varphi ){\psi }_{k-1}-\varphi {\psi }_{k-2}$ with ${\psi }_0=1$ and ${\psi }_1=1+\varphi$.
4. The explicit solution is ${\psi }_k=\frac{1-{\varphi }^{k+1}}{1-\varphi }$.