1. Chapter 4 Other Stationary 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.

2. Harmonic Component Models
Recall:
- is a stationary process
- spectrum has an infinite spike at f

3. Discrete Harmonic Component Model
where
• H t is referred to as the “harmonic component” or “signal”
• H t and N t are uncorrelated processes
• this is an example of a “signal + noise” model
More General:

4. tswge demo
gen.sigplusnoise.wge(n,b0,b1,coef,freq,psi,phi,vara,sn)
Generates a realization of length n from the model x(t)=b0+b1*t+coef[1]*cos(2*pi*freq[1]*t+psi[1]) coef[2]*cos(2*pi*freq[2]*t+psi[2])+z(t)
gen.sigplusnoise.wge(n=50,b0=0,b1=0,coef=c(4,2), freq=c(.05,.15),psi=c(1.1,2.7),phi=.7,vara=1)
gen.sigplusnoise.wge(n=50,b0=10,b1=.2)

5. Autocorrelations and Spectrum of Harmonic Signal + Noise Model

6. Harmonic Component Models
ARMA Approximation to
Harmonic Component Models

7. ARMA Approximation to Harmonic Component Models

8. ARMA Approximation to Harmonic Component Models
Factor Table for ARMA(2,2) Model
Factor f0
AR Part
1-1.72B +.99B 2
.995
.08
MA Part
1-1.37B +.72B 2
.85
.10

9. ARMA Approximation to Harmonic Component Models

10. ARCH and GARCH Processes
DOW daily rate of return
Stock Market Crash
October 29, 1929
Higher variability during Depression
Black Monday
October 19, 1987
Notes:
(3) When the conditional variance depends on t, the process is said to be volatile

11. Question: Note: Example: ARCH Model
The answer is “No” if the at are normally distributed.
Why?
The answer is “Yes” if at are not normally distributed
Example: ARCH Model
The ARCH(1) process at is white noise with a fat tail (leptokurtic distribution)
ARCH processes are “strange” types of white noise
See Section 4.2

12. Gaussian White Noise
Sample Autocorrs.
ARCH (White) Noise
Sample Autocorrs.
Squares of Above Data
Sample Autocorrs.
Squares of Above Data
Sample Autocorrs.

13. ARCH(q0) Model
GARCH(p0, q0) Model
These models have been developed by Econometricians to describe the types of volatility behavior seen in economic data.

14. ARCH(1) Realizations with various values of a1

15. ARCH and GARCH Model Realizations

16. tswge demo gen.arch.wge(n,alpha0,alpha, plot='TRUE',sn)
gen.arch.wge(n=500,alpha0=.1,alpha=c(.36,.27,.18,.09))
gen.garch.wge(n,alpha0,alpha,beta plot='TRUE',sn)
gen.garch.wge(n=500,alpha0=.1,alpha=.45,beta=.45)

17. Other topics: AR processes with ARCH or GARCH noise
(a) White Noise
(b) AR(2) with noise in (a)
(c) ARCH(1) noise
(d) AR(2) with noise in (c)
(e) GARCH(1,1) noise
(f) AR(2) with noise in (e)