3 mo treasury yield borrowing costs dow industrials ny times 18 sept 2008 front page
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3 mo treasury yield
borrowing costs
Dow industrials
NY Times
18 Sept 2008
front page
Stat 153 - 16 Sept 2008 D. R. Brillinger
Chapter 3
mean function
)]([)( tXEt
variance function
)]([)(2 tXVart
autocovariance
)(),((),(2121
tXtXCovtt
Strictly stationary
Normal/gaussian - all joint distributions jointly normal
Wide sense stationary vs. second-order
)(t22 )( t
)](),([)( tXtXCov
lag:),0(/)()(
Properties of autocovariance
)()(
1|)(|
Useful models
Purely random
Building block
0 ,0
0 ,1)(
k
kk
,...2/,1/ ,0
0 ,),()( 2
k
kZZCovk Zktt
Random walk
not stationary
0, 01 XZXX ttt
tXE t )(
2)( Zt tXVar
randompurelyZXXX tttt ,1
t
iit ZX
1
*
)()( )( YbEXaEbYaXE
)(),(2)( )( 22 YVarbYXabCovXVarabYaXVar
),(),(),(),(
),(
VYbdCovUYbcCovVXadCovUXacCov
dVcUbYaXCov
Moving average, MA(q)
qtqtttt ZZZX ...110
otherwise
k
kkMA
,0
1/ ),1/(
0 ,1)( ).1(2
11
From *
0)( ,0)( tt XEZEIf
stationary
)(
,...,1,0 ,
,0
)(
0
2
k
qk
qk
k
kq
ikiiZ
Backward shift operator
Linear process. )(MA
jtt
j XXB
0iitit ZX
Need convergence condition
q
q
t
q
q
tt
BBB
ZBB
ZBX
qMA
...)(
)...(
)(
)(
10
10
autoregressive process, AR(p)
first-order, AR(1) Markov
Linear process
For convergence/stationarity
1||
tt
tptptt
ZXB
ZXXX
)(
...11
ttt ZXX 1
...
)(
2
2
1
21
ttt
tttt
ZZZ
XZZX
*
a.c.f. From *
||
22||
)(
,...2/,1/,0 ),1/()(k
Z
k
k
kk
p.a.c.f.
In general case,
Very useful for prediction
tptptt ZXXX ...11
tystationarifor 1||in 0(z) of roots need
)(
z
ZXB tt
ARMA(p,q)
)()( tt ZXB
qtqtttptptt ZZZXXX ...... 11011
ARIMA(p,d,q).
stationary is t
d X
0)ARIMA(0,1,
)1(
walkRandom
1
tt
ttt
ZX
XBXX
Yule-Walker equations for AR(p).
Correlate, with Xt-k , each side of
tptptt ZXXX ...11
0 ),(...)1()( 1 kpkkk p
For AR(1)
0 ),0()( kk k