By Terence C. Mills, Raphael N. Markellos
Evidently patched jointly from themes written over a time period, this ebook isn't cohesive nor comprehensible. generators does not spend any phrases constructing his subject matters 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
The behaviour of the ACF is determined by the difference equation ðB Þk ¼ 0; k>0 ð2:6Þ which has the solution k ¼ A1 g1k þ A2 g2k þ . . þ Ap gpk Since jgi j<1, the ACF is thus described by a mixture of damped exponentials (for real roots) and damped sine waves (for complex roots). As an example, consider the AR(2) process À Á 1 À 1 B À 2 B2 xt ¼ at with characteristic equation ðB Þ ¼ ð1 À g1 B Þð1 À g2 BÞ ¼ 0 The roots g1 and g2 are given by À Á1=2 1 1 Æ 21 þ 42 g1 ; g2 ¼ 2 and can both be real, or they can be a pair of complex numbers.
22) can be inverted to give x t ¼ S d wt ð2:23Þ where S is the infinite summation, or integral, operator defined by À S ¼ 1 þ B þ B 2 þ . 20) can be obtained by summing, or ‘integrating’, the stationary process d times: hence the term ‘integrated process’. Box and Jenkins (1976, chap. 4) refer to this type of non-stationary behaviour as homogeneous non-stationarity, and it is important to discuss why this form of non-stationarity is felt to be useful in describing the behaviour of many financial time series.
15) is explosive if > 1 but stationary if < 1. The case ¼ 1 provides a process that is neatly balanced between the two. If xt is generated by the model xt ¼ xtÀ1 þ at ð2:17Þ then xt is said to follow a random walk. If we allow a constant, 0, to be included, so that xt ¼ xtÀ1 þ 0 þ at ð2:18Þ then xt will follow a random walk with drift. 0 43 Univariate linear stochastic models: basic concepts Thus, the correlation between xt and xt Àk is given by rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t Àk t Àk k;t ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ t t ðt À k Þ If t is large compared to k, all k,t will be approximately unity.