\(d=0\) Arima(x, order=c(1,0,0), include.drift=FALSE, include.mean=TRUE)

\(m\) is estimated and called intercept.

\[(x_t-m) = \phi_1 (x_{t-1}-m) + w_t\] Arima(x, order=c(1,0,0), include.drift=TRUE, include.mean=FALSE)

\(\mu\) is estimated and called drift.

\[x_t = \mu + \phi_1 x_{t-1} + w_t\]

Arima(x, order=c(1,0,0), include.drift=TRUE, include.mean=TRUE)

\(\mu\) and \(m\) are estimated and called drift and intercept.

\[(x_t-m) = \mu + \phi_1 (x_{t-1}-m) + w_t\]

If \(d=1\), then include.mean is ignored in Arima() and include.drift estimates an intercept like include.mean in the \(d=0\) case, but it is called drift in the output. \(y_t = x_t-x_{t-1}\).

• Arima(x, order=c(1,1,0), include.drift=TRUE)

\(m\) is estimated and called drift.

\[(y_t-m) = \phi_1 (y_{t-1}-m) + w_t\]

• Arima(x, order=c(1,1,0), include.drift=FALSE)

\[y_t = \phi_1 y_{t-1} + w_t\]

• Arima(x, order=c(0,1,0), include.drift=TRUE)

This is a random walk with drift.

\[(y_t-m) = w_t\] which is

\[x_t = m + x_{t-1} + w_t\]

If \(d\ge2\), then both include.mean and include.drift are ignored. \(z_t = y_t-y_{t-1} = (x_t - x_{t-1}) - (x_{t-1}-x_{t-2})\).

• Arima(x, order=c(1,2,0))

\[z_t = \phi_1 z_{t-1} + w_t\]

If \(d=0\),

• arima(x, order=c(1,0,0), include.mean=TRUE)

\(m\) is estimated and called intercept.

\[(x_t-m) = \phi_1 (x_{t-1}-m) + w_t\]

If \(d=1\), then include.mean is ignored and no intercept can be estimated.

• arima(x, order=c(1,1,0), include.mean=TRUE)

\[y_t = \phi_1 y_{t-1}+ w_t\]

• arima(x, order=c(0,1,0))

Because an intercept cannot be estimated, this means that a random walk with drift cannot be estimated by arima().

\[y_t = w_t\] only can be estimated which is random walk without drift.

\[x_t = x_{t-1} + w_t\]