H5 - H5 The Yule-Walker equations: AR(1): AR(2): (h) = (h -...

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H5 The Yule-Walker equations: AR(1) : () ( 1 ) , 0 ρ φρ =− > hhh , with initial condition (0) 1 = . AR(2) : 12 ) ( 2 ) , 1 +− hh h h > , with initial conditions 1 2 (1) 1 φ = , 2 1 2 2 (2) 1 = + . Example 1 Suppose for a stationary AR(2) model, 112 2 , .5, (2) .1 tt t t XX XZ −− = ++ = = , Determine 2 . In general, solutions or the Yule-Walker equations : 1 ( ) ( 1) ... ( ) 0 φ ρ = −++ − = p h p , have the form || 1 1 p p hA A ππ =+ + . It turns put that the condition, ||1 i π < for all i , is equivalent to that of causality, h tends to zero as increases. h In contrast to the correlation function the Partial Correlation Function (PACF), h α , starts from lag 1, = . ACF h for MA processes cuts off at ( ) q hq = , i.e., () 0 , q = > , PACF h for processes cuts off at AR( ) q hp = , i.e.,
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This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.

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