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Lecture 2

Lecture 2. ARMA models 2012 International Finance CYCU. White noise?. About color? About sounds? Remember the statistical definition!. White noise. Def: {  t } is a white-noise process if each value in the series: zero mean constant variance no autocorrelation In statistical sense:

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Lecture 2

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  1. Lecture 2 ARMA models 2012 International Finance CYCU

  2. White noise? • About color? • About sounds? • Remember the statistical definition!

  3. White noise • Def: {t} is a white-noise process if each value in the series: • zero mean • constant variance • no autocorrelation • In statistical sense: • E(t) = 0, for all t • var(t) = 2 , for all t • cov(tt-k ) = cov(t-jt-k-j ) = 0, for all j, k, jk

  4. White noise • w.n. ~ iid (0, 2 )iid: independently identical distribution • white noise is a statistical “random” variable in time series

  5. The AR(1) model (with w.n.) • yt = a0 + a1 yt-1 + t • Solution by iterations • yt = a0 + a1 yt-1 + t • yt-1 = a0 + a1 yt-2 + t-1 • yt-2 = a0 + a1 yt-3 + t-2 •  • y1 = a0 + a1 y0 + 1

  6. General form of AR(1) • Taking E(.) for both sides of the eq.

  7. Compare AR(1) models • Math. AR(1) • “true” AR(1) in time series

  8. Infinite population {yt} • If yt is an infinite DGP, E(yt) implies • Why? If |a1| < 1

  9. Stationarity in TS • In strong form • f(y|t) is a distribution function in time t • f(.) is strongly stationary if f(y|t) = f(y|t-j) for all j • In weak form • constant mean • constant variance • constant autocorrelation

  10. Weakly Stationarity in TS Also called “Covariance-stationarity” • Three key features • constant mean • constant variance • constant autocorrelation • In statistical sense: if {yt} is weakly stationary, • E(yt) = a constant, for all t • var(yt) = 2 (a constant), for all t • cov(yt yt-k ) = cov(yt-j yt-k-j ) =a constant, for all j, k, jk

  11. AR(p) models • where t ~ w. n. • For example: AR(2) • yt = a0 + a1 yt-1 + a2 yt-2 + t • EX. please write down the AR(5) model

  12. The AR(5) model • yt=a0 +a1 yt-1+a2 yt-2+a3 yt-3+a4 yt-4+a5 yt-5+ t

  13. Stationarity Restrictions for ARMA(p,q) • Enders, p.60-61. • Sufficient condition • Necessary condition

  14. MA(q) models • MA: moving average • the general form • where t ~ w. n.

  15. MA(q) models • MA(1) • Ex. Write down the MA(2) model...

  16. The MA(2) model • Make sure you can write down MA(2) as... • Ex. Write down the MA(5) model...

  17. The MA(5) model • yt=a0+a1yt-1+a2yt-2+a3yt-3+a4 yt-4 + a5 yt-5 + t

  18. ARMA(p,q) models • ARMA=AR+MA, i.e. • general form • ARMA=AR+MA, i.e. • ARMA(1,1) = AR(1) + MA(1)

  19. Ex. ARMA(1,2) & ARMA(1,2) • ARMA(1,2) • Please write donw: ARMA(1,2) !

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