Chapter 5 Nonstationary Time series Models NOTE: Some slides have blank sections. They are based on a teaching style in which the corresponding blank (derivations, theorem proofs, examples, …) are worked out in class on the board or overhead projector. 1

Deterministic Signal-plus-Noise Models Example Signals: C constant

Sometimes it’s not easy to tell whether a deterministic signal is present in the data Is there a deterministic signal?

Realizations - is there a deterministic signal? No Yes Sometimes it’s not easy to tell whether a deterministic signal is present in the data.

4 Realizations from the AR(4) Model stationary

Definition 5.1 (a) Nonstationary ARIMA(p,d,q) Model The autoregressive integrated moving average process of orders p, d, and q (denoted ARIMA(p,d,q)) is a process, Xt , whose differences (1- B)d Xt satisfy a (stationary) ARMA(p,q) model, where d is a non-negative integer We use the notation You can generate realizations from the ARIMA(p,d,q) model using gen.arima.wge

(b) Nonstationary ARUMA(p,d,q) Model The autoregressive unit root moving average process of orders p, d, and q (denoted ARUMA(p,d,q)) is a process, Xt , for which the process l(B) Xt satisfies a (stationary) ARMA(p,q) model, where l(B) = 1 - l1B -  - ld Bd is an operator whose characteristic equation has all its roots on the unit circle. We use the notation: Note: An ARIMA(p,d,q) process is an ARUMA(p,d,q) process with l(B) = (1- B)d You can generate realizations from the ARUMA(p,d,q) model using gen.aruma.wge

tswge demo gen.aruma.wge(n,phi,theta,d,s,lambda,sn) gen.aruma.wge(n=200,phi=.7,d=1,s=0,lambda=0) gen.aruma.wge(n=120,phi=-.4,d=0,s=12,lambda=0) gen.aruma.wge(n=200,phi=.7,d=1,s=12,lambda=c(1.6,-1),theta=-.8)

Note: 1-a1 B factor dominates behavior (of rk) becomes “first order”

Note: 1-a1B- a2 B2 factor dominates behavior (of rk) becomes “second order”

General Property As some roots of the AR characteristic equation approach the unit circle, r k seems to “nearly satisfy” a lower order difference equation

Question: What is r k for an ARUMA(p, d, q) process?

Clearly we need a new definition of autocorrelation for the case (1-B) Xt = at a similar situation arises for all ARUMA(p,d,q) models extended autocorrelation function

Extended Autocorrelation Function Autocorrelation functions are defined as a limit when some roots are on the unit circle: - or the “extended autocorrelation function”

Findley-Quinn Theorem (Theorem 5.1) Note:

Examples: (1) l(B) = 1 - B (2) l(B) = 1 - B 2

Types of ARUMA Models Non-cyclic ARUMA Models - these are the ARIMA models given by Box and Jenkins - all unit circle roots are +1

Cyclic ARUMA Models - at least one of the unit roots is not +1

Seasonal Models - a special case of cyclic models containing factors such as (1 - B s) - Monthly Data (1 - B 12), Quarterly Data (1 - B 4), … Example:

Factor Tables Factor Abs Recip f Root(s) 1-B 1 1+B2 .25  i 1+B .5 -1 1+B2 .25  i 1+B .5 -1 Factor Abs Recip f Root(s) 1 – B 1 .083 .866 + .5i 1 - B + B2 .167 .5 + .866i 1+B2 .25 + i 1 + B + B2 .333 -.5 + .866i .417 -.866 + .5i 1 + B .5 -1

tswge demo To obtain factor tables on the previous slide us factor.wge(phi=c(0,0,0,1)) factor.wge(phi=c(0,0,0,0,0,0,0,0,0,0,0,1))

More General Seasonal Models

Airline Data (log) Why?

More General Seasonal Models

Other Nonstationary Models Random Walk where is a white noise sequence

Random Walk with Drift where is a white noise sequence

Random Walk Random Walk with Drift Same noise sequence

TVF Signals - these are signals with time varying frequencies (TVF) Nonstationary Chapters 12 and 13