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University of Toronto St. George
Statistical Sciences
Zhou Zhou

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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