【正文】 rhh10 20 30 40xt 1 . 00 . 01 . 0xtFkk10 20 30 40k 1 . 00 . 01 . 0? xt10 20 30 40hrh 1 . 00 . 01 . 0? x t10 20 30 40Fkk 1 . 00 . 01 . 0? x t2rh10 20 30 40h 1 . 00 . 01 . 010 20 30 40Fkkk? xt2Possible Identification 1. d = 0, p = 2, q= 0 I B M s t oc k d at a: 3 0 02 0 01 0 003 0 04 0 05 0 06 0 07 0 0D a i l y I B M C o m m o n S t o c k C l o s i n g P r i c e sM a y 1 7 1 9 6 1 N o v e m b e r 2 1 9 6 2D a yP r i c e ( $ ) S u m m a r y S t at is t ics ACF and PACF for xt ,?xt and ?2xt (IBM Stock Price Data) 1 . 00 . 01 . 0xtrh10 20 30 40h 1 . 00 . 01 . 010 20 30 40kFkkxt 1 . 00 . 01 . 0rh? xt10 20 30 40h 1 . 00 . 01 . 0? xtFkk10 20 30 40k 1 . 00 . 01 . 0rh2? xt10 20 30 40h 1 . 00 . 01 . 02? xtFkk10 20 30 40kPossible Identification 1. d = 1, p =0, q= 0 Estimation of ARIMA parameters Preliminary Estimation Using the Method of moments Equate sample statistics to population paramaters Estimation of parameters of an MA(q) series The theoretical autocorrelation function in terms the parameters of an MA(q) process is given by. ??????????????????qhqhqqhqhhh011 2222111????????r ??To estimate ?1, ?2, … , ?q we solve the system of equations: qhrqqhqhhh ?????????? ?? 1???1?????2222111??????????This set of equations is nonlinear and generally very difficult to solve For q = 1 the equation bees: Thus 2111 ?1?????r? ? 0??1 1121 ??? ?? ror 0?? 11211 ??? rr ??This equation has the two solutions 14121?2111 ??? rr?One solution will result in the MA(1) time series being invertible For q = 2 the equations bee: 22212111 ??1????????????r222122 ??1???????rEstimation of parameters of an ARMA(p,q) series We use a similar technique. Namely: Obtain an expression for rh in terms ?1, ?2 , ... , ?p 。Model Building For ARIMA time series Consists of three steps 1. Identification 2. Estimation 3. Diagnostic checking ARIMA Model building Identification Determination of p, d and q To identify an ARIMA(p,d,q) we use extensively the autocorrelation function {rh : ? h ?} and the partial autocorrelation function, {Fkk: 0 ? k ?}. The definition of the sample covariance function {Cx(h) : ? h ?} and the sample autocorrelation function {rh: ? h ?} are given below: ? ? ? ?? ????? ???hTthttx xxxxThC11? ?? ?0 a n d xxh ChCr ?The divisor is T, some statisticians use T – h (If T is large, both give approximately the same results.) It can be shown that: ? ? ??????? ?tkttkhh TrrC o v rr1,Thus ? ??????? ??? ???????qtttth rTTrV a r122 2111 rAssuming rk = 0 for k q ????qttr rTs h12211L e t The sample partial autocorrelation function is defined by: 11111?212111212111?????????????????????FkkkkkkkkkrrrrrrrrrrrrrIt can be shown that: ? ? TV a r kk 1? ?FTs kk1L e t ? ?FIdentification of an Arima process Determining the values of p,d,q ? Recall that if a process is stationary one of the roots of the autoregressive operator is equal to one. ? This will cause the limiting value of the autocorrelation function to be nonzero. ? Thus a nonstationary process is identified by an autocorrelation function that does not tail away to zero quickly or cutoff after a finite number of steps. To determine the value of d Note: the autocorrelation function for a stationary ARMA time series satisfies the following difference equation 1 1 2 2h h h p h pr ? r ? r ? r? ? ?? ? ? ?The solution to this equation has general form 12121 1 1hph h hpc c cr r rr ? ? ? ?where r1, r2, r1, … rp, are the roots of the polynomial ? ? 2121 ppx x x x? ? ? ?? ? ? ? ?For a stationary ARMA time series Therefore 12121 1 1 0 a s hp h h hpc c c hr r rr ? ? ? ? ? ? ?The roots r1, r2, r1, … rp, have absolute value greater than 1. If the ARMA time series is nonstationary some of the roots r1, r2, r1, … rp, have absolute value equal to 1, and 12121 1 1 0 a s hp h h hpc c c a hr r rr ? ? ? ? ? ? ? ?010 3 6 912 15 18 21 24 27 30stationary 010 3 6 912 15 18 21 24 27 30nonstationary ? If the process is nonstationary then first differences of the series are puted t