ARMA models 2012 International Finance CYCU 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: 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

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

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

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

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

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

Stationarity in TS In strong form In weak 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

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

AR(p) models For example: AR(2) EX. please write down the AR(5) model 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

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

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

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

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

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

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

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

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