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# H7 - H7 Given a set of observations cfw X 1 X n from a...

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H7 Given a set of observations 1 { ,..., } n X X from a stationary time series, the ACVF is estimated by the sample autocovariance function defined as 1 1 ˆ () ( ) ( ) γ + = =− nh th n t n t hX X X n X , where 1 1 n nn t X X n = = . This also leads to estimating the ACF by the sample autocorrelation function ˆ ˆˆ / ( 0 ) ρ γγ = hh . Using the divisor n (instead of ) in the formula for ˆ h ensures that ˆ h (and therefore ˆ h ) is a nonnegative-definite function, which is the necessary and sufficient condition for a function to be an ACVF of a stationary process: ___________________________________________________________________________________________ It may be shown that under certain general conditions, ( ˆ ˆ ˆ ~ N , / , (1), . .., ( ) Wn k ρρ ) = ± ρ , where W is the covariance matrix (whose elements are given by the so-called Bartlett’s formula ). In particular, 1) Purely random process If { } 2 ~ IID(0, ) t X σ , then ( ) 1 ˆ()~AN 0 , , 0 in i . Then a 95% CI for is 11 1.96 1.96 ⎛⎞ −< < + ⎜⎟ ⎝⎠ , hence we are 95% confident that ˆ 1.96 1.96 < + , i.e., if we plot ˆ j ,
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