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Lecture

# January 28.docx

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University of Toronto St. George

Statistical Sciences

STA457H1

Zhou Zhou

Winter

Description

January 28, 2014
White Noise
1. White noise process may not be independent
Eg. If Y , 1, Y are ind N(0,1)
Let X = 1 , X 1 Y Y2, …, X1= 2 Y …Y n 1 2 n
It is easy to see that {X} is notiindependent process
We show that X , X are not independent
1 2
Suppose that X , X were independent
1 2
E[X X1] =2E[X ]E[X ] 12 22
RHS = 1
LHS = E[Y Y ] =1E[Y2]E[Y ] = 3 14 22
On the other hand, we will show that {X} is a white noist.
E[X] i E[Y ]E[Y1]…E[Y ]2= 0 n
2 2 2
V[X] i E[Y ]E[Y1]…E[Y ] 2 1 n
2 2 2
For any i > j, Cov[X, X] = EiXX]j= E[(Y Y iY)jY Y …Y)] =1E[2 Y …Y i …Y1 2 j 1 2 i i+1 j 2 2 2
= E[Y ]E1Y ]…E[Y 2E[Y ]…E[Y] i 1*1*…*i+1*…*0 = 0 j
Removal of seasonal trends
1. Estimate and subtract the seasonal trend
Suppose X = S + Yi wheri Y isistationary, S is a seasonal trend witi S = S i i+d for all i, d is
known
Si
We can estimate S by i = 1/#A ∑jϵi X, wheri Aj= {k | kϵZ, 1≤i≤n, k-i can be divided by d}
n
A i {1, 1+d, 1+2d, 1+3d, …, 1+( ⌈ ⌉ -1)d}
d
S i
It is easy to show, under some mild conditions that -p> S as i -> ∞ for any i
̂
Then X – i S i will be approximately stationary
2. Remove seasonal trend by differencing
d
Define W = X –IX = (1iB )X i-d i
W is weakly stationary
i
Proof: W = X i X = Si+ Y –i-d – Y i = (S i S ) +i-d – Y i-d Y – Y -i statii-dry i i-d i i-d
Goodness of fit tests
1. Make sure that our time series model has modelled the dependence structure of the
sequence adequately
2. Tool: Residual analysis
3. Standard: If the residuals are approximately white noise then the model passes the test
no trend trend
non-constant variance
no changing variance
4. Sample ACF plot helps us check no-correlation

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