H7 - H7 Given a set of observations cfw_ X 1 ,., X n from a...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ,
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.

Ask a homework question - tutors are online