21 Time Series
Mean function
Z 1
xt = E [xt] = |
xft(x) dx |
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Autocovariance function
x(s; t) = E [(xs s)(xt t)] = E [xsxt] s t
x(t; t) = E (xt t)2 = V [xt]
Autocorrelation function (ACF)
(s; t) = |
Cov [xs; xt] |
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(s; t) |
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p |
V [xs] V [xt] |
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p |
(s; s) (t; t) |
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Cross-covariance function (CCV) |
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xy(s; t) = E [(xs xs )(yt yt )]
Cross-correlation function (CCF)
xy(s; t) = p
xy(s; t)
x(s; s) y(t; t)
Backshift operator
Bk(xt) = xt k
Di erence operator
rd = (1 B)d
White noise
wt wn(0; w2 )
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Gaussian: wt |
iid |
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0; 2 |
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E [wt] = 0 t |
TN |
w |
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2 |
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V [wt] = 2 t 2 T
w(s; t) = 0 s 6= t ^ s; t 2 T
Random walk |
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Drift |
t |
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Et[xt] = tPj=1 |
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x = t + |
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Symmetric moving average
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k |
k |
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X |
X |
mt = |
ajxt j where aj = a j 0 and |
aj = 1 |
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j= k |
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21.1Stationary Time Series
Strictly stationary
P [xt1 c1; : : : ; xtk ck] = P [xt1+h c1; : : : ; xtk+h ck]
8k 2 N; tk; ck; h 2 Z
Weakly stationary
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xt2 |
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t 2 Z |
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E |
xt2 |
< 1 8t 2 Z |
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x(s; |
t) = x(s +8r; t + r) |
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r; s; t |
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Z |
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Autocovariance function
(h) = E [(xt+h )(xt )] |
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8h 2 Z |
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(0) |
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(0) |
j (h)j |
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(h) = ( h) |
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Autocorrelation function (ACF) |
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x(h) = |
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Cov [xt+h; xt] |
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(t + h; t) |
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p |
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(0) |
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V [xt+h] V [xt] |
(t + h; t + h) (t; t) |
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Jointly stationary time series
xy(h) = E [(xt+h x)(yt y)]
xy(h) = p
xy(h)
x(0) y(h)
Linear process
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1 |
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X |
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j jj < 1 |
xt = + |
jwt j |
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j=1 |
j=1 |
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1 |
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(h) = w2 |
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j+h j |
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j=1 |
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