Moving from univariate to bivariate jointly dependent long memory time seriesintroduces a phase parameter (?), at the frequency of principal interest, zero; for shortmemory series ? = 0 automatically. The latter case has also been stressed under longmemory, along with the "fractional differencing" case ( ) / 2; 2 1 ? = d - d p where 1 2 d , dare the memory parameters of the two series. We develop time domain conditionsunder which these are and are not relevant, and relate the consequent properties ofcross-autocovariances to ones of the (possibly bilateral) moving averagerepresentation which, with martingale difference innovations of arbitrary dimension,is used in asymptotic theory for local Whittle parameter estimates depending on asingle smoothing number. Incorporating also a regression parameter (ß) which, whennon-zero, indicates cointegration, the consistency proof of these implicitly-definedestimates is nonstandard due to the ß estimate converging faster than the others. Wealso establish joint asymptotic normality of the estimates, and indicate how thisoutcome can apply in statistical inference on several questions of interest. Issues ofimplementation are discussed, along with implications of knowing ß and of correct orincorrect specification of ? , and possible extensions to higher-dimensional systemsand nonstationary series.
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Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number
/2007/525.
References listed on IDEAS Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
Clifford M. Hurvich & Eric Moulines & Philippe Soulier, 2005.
"Estimating Long Memory in Volatility,"
Econometrica,
Econometric Society, vol. 73(4), pages 1283-1328, 07.
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