ARIMA Model

The autoregressive integrated moving average (ARIMA) process generates nonstationary series that are integrated of order D, denoted I(D). A nonstationary I(D) process is one that can be made stationary by taking D differences. Such processes are often called difference-stationary or unit root processes.

A series that you can model as a stationary ARMA(p,q) process after being differenced D times is denoted by ARIMA(p,D,q). The form of the ARIMA(p,D,q) model in Econometrics Toolbox™ is

Δ^{D} y_{t} = c + ϕ_{1} Δ^{D} y_{t - 1} + \dots + ϕ_{p} Δ^{D} y_{t - p} + ε_{t} + θ_{1} ε_{t - 1} + \dots + θ_{q} ε_{t - q},

(1)

where

Δ^{D} y_{t}

denotes a Dth differenced time series, and

ε_{t}

is an uncorrelated innovation process with mean zero.

In lag operator notation, $L^{i} y_{t} = y_{t - i}$ . You can write the ARIMA(p,D,q) model as

ϕ^{*} (L) y_{t} = ϕ (L) {(1 - L)}^{D} y_{t} = c + θ (L) ε_{t} .

(2)

Here,

ϕ^{*} (L)

is an unstable AR operator polynomial with exactly D unit roots. You can factor this polynomial as

ϕ (L) {(1 - L)}^{D},

where

ϕ (L) = (1 - ϕ_{1} L - \dots - ϕ_{p} L^{p})

is a stable degree p AR lag operator polynomial (with all roots lying outside the unit circle). Similarly,

θ (L) = (1 + θ_{1} L + \dots + θ_{q} L^{q})

is an invertible degree q MA lag operator polynomial (with all roots lying outside the unit circle).

The signs of the coefficients in the AR lag operator polynomial, $ϕ (L)$ , are opposite to the right side of Equation 1. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 1.

Note

In the original Box-Jenkins methodology, you difference an integrated series until it is stationary before modeling. Then, you model the differenced series as a stationary ARMA(p,q) process [1]. Econometrics Toolbox fits and forecasts ARIMA(p,D,q) processes directly, so you do not need to difference data before modeling (or backtransform forecasts).

References

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

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ARIMA Model

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