1. ARMA models 2012 International Finance CYCU
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-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 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

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 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

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
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. ARMA=AR+MA, i.e. general form

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