Autoregressive Integrated Moving Average (ARIMA) model

ARIMA

For nonstationary time series, applying difference operators repeatedly to the data \(\{X_t\}\) until the differenced observations resemble a realization of some stationary process \(\{W_t\}\)

ARIMA(p,q,d) \(\{X_t\}\) is said to follow an ARIMA(p,q,d) model if \(W_t = (1-B)^d X_t\) is stationary ARMA model. i.e.

\[(1-B)^d \Phi(B)X_t = \Theta(B)a_t, a_t \sim NID(0,\sigma^2)\]

A series follows a stationary ARMA model after differencing \(d\) times, i.e. \((1-B)^d X_t\), such process is called an \(I(d)\) process.

Example

whether \((1-B)^2 X_y\) is stationary

\((1-B)^2 X_t = X_t-2B X_t + B^2 X_t = X_t - 2X_{t-1}+X_{t-2}\)

Dickey Fuller unit root test

The DF test is used to test \(I(1)\) processes.
Consider \(X_t = \phi X_{t-1} + a_t. a_t \sim NID(0,\sigma^2)\),
then \(\Delta X_t = (\phi - 1) X_{t-1} + a_t = \pi X_{t-1} + a_t\)
\(H_0: \pi = 0, i.e. X_t\sim I(1)\)

The general DF test may contain an intercept and a deterministic time trend as \(\Delta X_t = a+\tau^T DR_t + \pi X_{t-1} + a_t\)

Augmented Dickey-Fuller test

Problems with basic DF

The basic DF test considers only a single unit root
Correct model specification
- correct specification of time trend and intercept
- The DGP may contain both AR and MA terms
- There might be structural breaks in the data

ADF test equation

\[\Delta X_t = \tau^T DR_t + \pi X_{t-1}+\sum_{j=1}^k \gamma_j\Delta X_{t-j} + a_t\]

where \(k=p-1\), the equation use the autoregression to take into account the presence of serial correlated errors.

Box-cov transformation

If the variance is not stationary, we can try stablize it with a Box Cox transformation, i.e. include \(\lambda\) as one of the parameters \(\Phi(B)(X_t^\lambda - \mu) = \Theta(B)a_t\), then choose \(\lambda, \Phi, \Theta\) based on Minimized RMSE.