【正文】 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 to determine if that operation results in a stationary series. ? The process is continued until a stationary time series is found. ? This then determines the value of d. Identification Determination of the values of p and q. To determine the value of p and q we use the graphical properties of the autocorrelation function and the partial autocorrelation function. Again recall the following: A ut o c or re l a t i on f un c t i onP a r t i a l A ut oc or r e l a t i on f un c t i onC u t s o f fC u t s o f fI n f i n i t e . T a i l s o f f .D a m p e d E x p o n e n t i a l s a n d / o r C o s i n e w a v e sI n f i n i t e . T a i l s o f f .I n f i n i t e . T a i l s o f f .I n f i n i t e . T a i l s o f f .D o m i n a t e d b y d a m p e d E x p o n e n t i a l s amp。 ?1, ?2, ... ,?q 。 ?, ?, d) (The jacobian of the transformation is 1) Then the joint density of x given x* and u* is given by: ? ?2,*,*, sdαβuxxf? ??????? ???????? ??Nttnu122 ,*,*,21e x p21 dss?αβux? ??????? ???????? dss?,*21e x p212 αβSn? ? ? ????NttuS12 ,*,*,* w h e r e dd αβuxαβLet: ? ?2**, , sdαβuxxL? ??????? ???????? ??Nttnu122 ,*,*,21e x p21 dss?αβux? ??????? ???????? dss?,*21e x p212 αβSn? ? ? ????NttuS12 ,*,*,*a g a i n dd αβuxαβ = “conditional likelihood function” ? ? ? ?2**,2**, ,ln, sdsd αβαβ uxxuxx Ll ?? ? ? ???????Nttunn1222 ,*,*,21ln22 dss αβux? ? ? ? ? ?dss? ,*2 12ln22ln2 22 αβSnn ????“conditional log likelihood function” = ? ? ? ?2**,2**, , a n d , sdsd αβαβ uxxuxx Ll? ? ? ????NttuS12 ,*,*,* dd αβuxαβThe values that maximize are the values that minimize d?,?,? αβ? ? ? ?dds ?,?,?*1?,?,?*,*,1?122 αβαβux SnunNtt ?? ??with ? ? ? ????NttuS12 ,*,*,* dd αβuxαβComment: Requires a iterative numerical minimization procedure to find: The minimization of: d?,?,? αβ? Steepest descent ? Simulated annealing ? etc ? ? ? ????NttuS12 ,*,*,* dd αβuxαβComment: for specific values of The putation of: can be achieved by using the forecast equations d,αβ? ?1? 1??? ttt xxu? ? ? ????NttuS12 ,*,*,* dd αβuxαβComment: assumes we know the value of starting values of the time series {xt| t ? T} and {ut| t ? T} The minimization of : Namely x* and u*. * of c o m p o n e n ts f o r th e 0* of c o m p o n e n ts f o r th e uxxApproaches: 1. Use estimated values: 2. Use forecasting and backcasting equations to estimate the values: Backcasting: If the time series {xt|t ? T} satisfies the equation: 2211 qtqttt uuuu ??? ????? ??? ? 2211 d??? ????? ??? ptpttt xxxx ?It can also be shown to satisfy the equation: 2211 qtqttt uuuu ??? ????? ??? ? 2211 d??? ????? ??? ptpttt xxxx ?Both equations result in a time series with the same mean, variance and autocorrelation function: In the same way that the first equation can be used to forecast into the future the second equation can be used to backcast into the past: * of c o m p o n e n ts f o r th e 0* of c o m p o n e n ts f o r th e uxxApproaches to handling starting values of the series {xt|t ? T} and {ut|t ? T} 1. Initially start with the values: 2. Estimate the parameters of the model using Maximum Likelihood estimation and the conditional Likelihood function. 3. Use the estimated parameters to backcast the ponents of x*. The backcasted ponents of u* will still be zero. 4. Repeat steps 2 and 3 until the estimates stablize. This algorithm is an application of the EM algorithm This general algorithm is frequently used when there are missing values. The E stands for Expectation (using a model to estimate the missing values) The M stands for Maximum Likelihood Estimation, the process used to estimate the parameters of the model. Some Examples using: ? Minitab ? Statistica ? SPlus ? SAS