Chapter 4 Other Stationary Time Series Models

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Presentation transcript:

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.

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

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:

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)

Autocorrelations and Spectrum of Harmonic Signal + Noise Model

Harmonic Component Models ARMA Approximation to Harmonic Component Models

ARMA Approximation to Harmonic Component Models

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

ARMA Approximation to Harmonic Component Models

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

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

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

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.

ARCH(1) Realizations with various values of a1

ARCH and GARCH Model Realizations

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)

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)