1. 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

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

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

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

5. 4 Realizations from the AR(4) Model
stationary

6. 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

7. (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

8. 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)

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

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

11. 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

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

13. 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

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

15. Findley-Quinn Theorem (Theorem 5.1)
Note:

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

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

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

20. 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:

21. 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 i
1 - B + B2
.167 i
1+B2
.25
+ i
1 + B + B2
.333 i
.417 i
1 + B .5 -1

22. 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))

23. More General Seasonal Models

24. Airline Data (log)
Why?

25. More General Seasonal Models

26. Other Nonstationary Models
Random Walk
where
is a white noise sequence

27. Random Walk with Drift
where
is a white noise sequence

28. Random Walk
Random Walk with Drift
Same noise sequence

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