Lecture 2 ARMA models 2012 International Finance CYCU 1 White noise? • About color? • About sounds? • Remember the statistical definition! 2 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, jk 3 White noise • w.n. ~ iid (0, 2 ) iid: independently identical distribution • white noise is a statistical “random” variable in time series 4 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 5 General form of AR(1) t 1 t 1 i 0 i 0 y t a 0 a1i a1t y0 a1i t i • Taking E(.) for both sides of the eq. t 1 t 1 i t i E( y t ) E a 0 a1 E a1 y0 E a1 t i i 0 i 0 t 1 E( y t ) a 0 a a y 0 i 0 i 1 t 1 6 Compare AR(1) models • Math. AR(1) 1 a1t t yt a 0 (a 1 ) y 0 1 a1 • “true” AR(1) in time series 1 a1t (a1 ) t y 0 E(y t ) a 0 1 a1 7 Infinite population {yt} • If yt is an infinite DGP, E(yt) implies a0 lim y t (note: a constant) t (1 a1 ) • Why? If |a1| < 1 1 a t yt a 0 (a 1 ) y 0 1 a1 t 1 8 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 9 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, jk 10 AR(p) models p y t a 0 a i y t i t i 1 – 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 11 The AR(5) model • yt=a0 +a1 yt-1+a2 yt-2+a3 yt-3+a4 yt-4+a5 yt-5+ t 12 Stationarity Restrictions for ARMA(p,q) • Enders, p.60-61. • Sufficient condition p | a i 1 i | 1 • Necessary condition p a i 1 i 1 13 MA(q) models • MA: moving average – the general form q y t a 0 t b i t i i 1 – where t ~ w. n. 14 MA(q) models • MA(1) yt a 0 t b1 t 1 • Ex. Write down the MA(2) model... 15 The MA(2) model • Make sure you can write down MA(2) as... yt a 0 ε t b1ε t 1 b2ε t 2 • Ex. Write down the MA(5) model... 16 The MA(5) model • yt=a0+a1yt-1+a2yt-2+a3yt-3+a4 yt-4 + a5 yt-5 + t 17 ARMA(p,q) models • ARMA=AR+MA, i.e. – general form p q i 1 i 1 y t a 0 a i y t i ε t b i ε t i • ARMA=AR+MA, i.e. – ARMA(1,1) = AR(1) + MA(1) yt a 0 a i yt i ε t bi ε t i 18 Ex. ARMA(1,2) & ARMA(1,2) • ARMA(1,2) yt a 0 a1 yt 1 ε t b1 ε t 1 b2ε t 2 • Please write donw: ARMA(1,2) ! 19