ARIMA

Posted on Jul 11, 2017 0 Comments

Tags: Time Series

Any time series without a constant mean over time is nonstationary.

General Model

\[Y_t=\mu_t + X_t\]

where $\mu_t$ is a nonconstant mean function and $X_t$ is a zero-mean, stationary series.

Stationary Through Differencing

the first difference of $Y_t$

\[\nabla Y_t = Y_t - Y_{t-1}\]

ARIMA Models

A time series $\{Y_t\}$ is said to follow an integrated autoregressive moving average model if the $d$-th difference $W_t=\nabla^dY_t$ is a stationary ARMA process.

If $\{W_t\}$ follows an ARMA(p, q) model, we say $\{Y_t\}$ is an ARIMA(p, d, q) process.

IMA(1,1) Model

\[\nabla Y_t = e_t - \theta e_{t-1}\]

IMA(2,2) Model

\[\nabla^2 Y_t = e_t-\theta_1 e_{t-1} -\theta_2 e_{t-2}\]

ARI(1,1) Model

\[\nabla Y_t = \nabla Y_{t-1} + e_t\]

References

Time Series Analysis With Applications in R Second Edition