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By Jean Meeus

Acknowledged in book," nearly each prior instruction manual on celestial calcuations was once compelled to depend on formaulae for solar Moon and planets that have been constructed within the final century or sooner than 1920.

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XQ) . ,A) i f and onl y if A ! an d A 2 ar e variation free; tha t is , i f an d onl y if so tha t th e paramete r spac e A i s th e direct produc t o f A j an d A 2. I n other words , fo r an y value s o f A j an d A 2, admissibl e value s o f th e parameters A of th e join t distributio n ca n b e recovered . Th e essentia l element o f weak exogeneit y is that th e margina l distribution contain s n o information relevan t to A ! (for an exposition , se e Ericsson 1992) . Weak exogeneity: z t i s weakl y exogenou s fo r a se t o f parameter s o f interest ij> i f an d onl y i f ther e exist s a partitio n (A j , A2) o f A such that (i ) t/> i s a functio n o f A j alone , an d (ii ) [ ( y t z t ' , ^ i ) , (z t\ A^) ] operate s a sequential cut .

Phillip s (1987a ) provide s a goo d account o f this issue, an d a discussion is also containe d i n Ch . 3 . 5. Ste p representation o f a random walk over 100 0 points empirical researc h involvin g 1(1) tim e serie s hav e differen t distribution s from thos e tha t aris e wit h 1(0 ) data . I n particular , man y statistics i n 1(1) processes d o no t converge t o constants , a s i n th e 1(0 ) case , bu t instea d converge t o rando m variables . Thus , differen t critica l value s ma y b e required fo r tests , dependin g o n th e degre e o f integratio n o f th e tim e series.

Super exogeneity: z t i s supe r exogenou s fo r t y i f an d onl y i f z t i s weakly exogenous fo r \l> and A \ i s invariant t o intervention s affecting A^ . Weak exogeneit y ensure s tha t ther e i s n o los s o f informatio n abou t parameters o f interest fro m analysin g only the conditiona l distribution ; a variable z t i s weakl y exogenou s fo r a se t o f parameter s t/ > i f inferenc e concerning t/ ; can b e mad e conditiona l o n z t wit h no los s o f information relative t o tha t whic h could be obtaine d usin g the joint density o f y t an d Introduction and Overvie w 1 9 Zf Stron g exogeneit y i s necessar y fo r multi-ste p forecastin g whic h proceeds b y forecasting future z s an d the n forecastin g ys conditiona l on those zs .

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