# ARIMA

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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.