APPENDIX
Lag 1 Correlation in a MA(1) Series
The variance of ytcan be calculated using the preceding formulas as
Therefore, unlike the white noise series, this series has a nontrivial correla-
tion structure.
Lag 1 Correlation in a AR(1) Series
The variance at each time instant is given as
Ifa> 1, the series explodes and the variance becomes infinity. However,
whena> 1, the variance can be calculated as the sum of an infinite geo-
metric series and written as
The covariance is given as
cov()yytt, −−−− 1111 =Eyy[]tt=+E y[]()αε αt tty = var()yt
var
var
()yt = t
()
−
ε
1 α^2
var()yEytt= ()^2246 =++++...() 1 ααα var()εt
y
t
t
var
var
=
()
()
=
1 +^2
βε β
β
corryy
Eyy
yy
E
yy
tt
tt
tt
t
tt
,
var var var var
−
−
−
−
−
()=
[]
() ( )
=
()
() ( )
1 =
1
1
1
2
1
βε
=+() 1 βε^2 var()t
var()yttt=+var()εβε−−− 1 =var()ε β ε β εεt+^2 var()t 11 + 2 cov()tt, =
34 BACKGROUND MATERIAL