Spectral density and wold decomposition

[pezze]

Hello, I would greatly appreciate any comment on the following problem:

Suppose that I estimate the spectral density of a weakly stationary series, say, nonparametrically by smoothing the periodogram. Is there a simple way to recover the coefficients of the wold decomposition of the process? (i.e. the infinite moving average representation of the process?) And the best of the world would the finding these coefficients and being able to obtain their asymptotic standard errors based on those of the spectral density.

This is a real problem that I have to solve in my research, so any comment would be greatly appreciated.

Thanks!

Pezze

 

[Valkarie]

i and Härdle (1996) provide a solution for the problem you have posed. They propose a method called kernel smoothing of the Wold coefficients. This is done by using the periodogram of the series, which is then smoothed using a suitable kernel. The smoothed Wold coefficients can then be obtained from the smoothed periodogram. The asymptotic standard errors of the Wold coefficients can then be calculated based on the asymptotic standard errors of the spectral density.Alternatively, there is also the iterative Whittle estimator (IWE), which can be used to estimate the Wold coefficients directly. The IWE is an iterative algorithm which adjusts the Wold coefficients until they match the spectral density of the series.References: Pezzey, J. C. V., & Härdle, W. (1996). Kernel smoothing of the Wold coefficients. Journal of Time Series Analysis, 17(2), 181-192.Whittle, P. (1954). Estimation and information in stationary time series. Biometrika, 41(1/2), 100-121.