H14 - H14 The second-order properties of a time series are...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
H14 The second-order properties of a time series are completely described by its ACVF γ h , or equivalently, under mild conditions (a sufficient condition is || =−∞ < ∞ h h ), by its Fourier transform, which is called the spectral density function or the spectrum : 0 1 11 () 2 c o s 22 ih h hh h f e ω ωγ ππ ∞∞ =−∞ = ⎛⎞ == + ⎝⎠ ∑∑ h . (1) The spectrum of a time series can be obtained as long as we know the model satisfied by the time series. Often in practice we only have a finite set of time series data and we would like to estimate from it the spectrum of the process. Given a finite realization 1 { ,..., } N X X of a stationary time series, the ACVF is estimated by the sample autocovariance function 1 1 ˆ ( ) ( ) Nh Nt h N t t hX X XXX N + = N . ( H7 ) It would seem that the natural way to estimate f would be to replace h in (1) by its sample estimate: 1 (1 ) 1 ˆ ˆ c o s 2 N h hN f h π =− = . (2) Note that the sum now is restricted to < , since for a time series of length N ,
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

H14 - H14 The second-order properties of a time series are...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online