By Terence C. Mills, Raphael N. Markellos
Evidently patched jointly from subject matters written over a time period, this ebook isn't cohesive nor comprehensible. turbines does not spend any phrases constructing his issues nor explaning the improvement. Spend your assets on Hamilton's vintage and nice definative bible, Time sequence research in its place.
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Extra resources for The Econometric Modelling of Financial Time Series
15 Univariate linear stochastic models: basic concepts We can now deduce the ACF of an AR(1) process. 3), at xtÀk ¼ 1 i¼0 at atÀkÀi : As at is white noise, any term in atatÀkÀi has zero expectation if k þ i > 0. 4) simplifies to k ¼
kÀ1 ; for all k>0 and, consequently, k ¼ k 0. An AR(1) process therefore has an ACF given by k ¼ k. Thus, if > 0, the ACF decays exponentially to zero, while, if > 0, the ACF decays in an oscillatory pattern, both decays being slow if is close to the non-stationary boundaries of þ1 and À1.
One possibility is that the mean evolves as a polynomial of order d in time. This will arise if xt can be decomposed into a trend component, given by the polynomial, and a stochastic, stationary, but possibly autocorrelated, zero mean error component. This is always possible given Cramer’s (1961) extension of Wold’s decomposition theorem to non-stationary processes. Thus, we may have xt ¼ t þ "t ¼ d X j t j þ ðB Þat ð2:12Þ j¼0 Since E ð"t Þ ¼ ðBÞE ðat Þ ¼ 0 we have E ðxt Þ ¼ E ðt Þ ¼ d X j t j j¼0 and, as the j coefficients remain constant through time, such a trend in the mean is said to be deterministic.
The general autoregressive model of order p, AR(p), can be written as xt À 1 xtÀ1 À 2 xtÀ2 À . . À p xtÀp ¼ at or À Á 1 À 1 B À 2 B 2 À . . À p Bp xt ¼ ðB Þxt ¼ at The linear filter representation xt ¼ (B)at can be obtained by equating coefficients in (B) (B) ¼ 1 (see Mills, 1990, chap. 5, for examples of how to do this). The stationarity conditions required for convergence of the -weights are that the roots of the characteristic equation À Á ðB Þ ¼ ð1 À g1 B Þð1 À g2 BÞ. . 8 70 80 90 100 (continued) are such that jgi j<1 for i ¼ 1; 2; .
The Econometric Modelling of Financial Time Series by Terence C. Mills, Raphael N. Markellos