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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

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