Transforming a non-stationary series into a stationary series is a critical step for applying ARMA models for forecasting

ARMA Models

Autoregressive and moving average models can be combined together to form ARMA models. •

Definition – A time series {xt ;t = 0, ±1, ±2, . . . } is ARMA(p, q) if it is stationary and

Xt = wt + X p i=1 φiXt−i + X q j=1 θjwt−j ,

φ(B)Xt = θ(B)wt

“Stationarize” Nonstationary Time Series

One limitation of ARMA models is the stationarity condition. •

In many situations, time series can be thought of as being composed of two components, a non-stationary trend series and a zero-mean stationary series, i.e. Xt = µt + Yt . •

Strategies

Detrending: Subtracting with an estimate for trend and deal with residuals. Yˆ t = Xt − µˆt

Differencing: Recall that random walk with drift is capable of representing trend, thus we can model trend as a stochastic component as well.

µt = δ + µt−1 + wt

∇Xt = Xt − Xt−1 = δ + wt + (Yt − Yt−1) = δ + wt + ∇Yt

∇ is defined as the first difference and it can be extended to higher orders.

One advantage of differencing over detrending for trend removal is that no parameter estimation is required. •

In fact, differencing operation can be repeated.

The first difference eliminates a linear trend.

A second difference, i.e. the difference of first difference, can eliminate a quadratic trend.