【正文】 ?, ?, d) ... uN = uN (x, x*, u*。,ss?ss u?It is difficult to determine the exact density function of x1,x2, ... , xN from this information however if we assume that p starting values on the xprocess x* = (x1p,x2p, ... , xo) and q starting values on the uprocess u* = (u1q,u2q, ... , uo) have been observed then the conditional distribution of x = (x1,x2, ... , xN) given x* = (x1p,x2p, ... , xo) and u* = (u1q,u2q, ... , uo) can easily be determined. The system of equations : x1 = ?1x0 + ?2x1 +... +?px1p + d + u1 +?1u0 + ?2u1 +... + ?qu1q x2 = ?1x1 + ?2x0 +... +?px2p + d + u2 +?1u1 + ?2u0 +... +?qu2q ... xN= ?1xN1 + ?2xN2 +... +?pxNp + d + uN +?1uN1 + ?2uN2 +... + ?quNq can be solved for: u1 = u1 (x, x*, u*。 s2) = g(u 。 d , s2 by the method of Maximum Likelihood estimation we need to find the joint density function of x1, x2, ...,xN f(x1, x2, ..., xN |?1, ?2, ... ,?p 。q1,q2, ... , qk) is the joint density function of the observations x1,x2, ... , xN. L(q1,q2, ... , qk) is called the Likelihood function. It is important to note that: finding the values q1,q2, ... , qk to maximize L(q1,q2, ... , qk) is equivalent to finding the values to maximize l(q1,q2, ... , qk) = ln L(q1,q2, ... , qk). l(q1,q2, ... , qk) is called the logLikelihood function. Again let {ut : t ?T} be identically distributed and uncorrelated with mean zero. In addition assume that each is normally distributed . Consider the time series {xt : t ?T} defined by the equation: (*) xt = ?1xt1 + ?2xt2 +... +?pxtp + d + ut +?1ut1 + ?2ut2 +... +?qutq Assume that x1, x2, ...,xN are observations on the time series up to time t = N. To estimate the p + q + 2 parameters ?1, ?2, ... ,?p 。 ?1, ?2, ... , ?q by replacing rh by rh. Estimation of parameters of an ARMA(p,q) series ? ?? ?112112111111 211?rr???????r??????Example: The ARMA(1,1) process The expression for r1 and r2 in terms of ?1 and ?1 are: Further ? ? ? ?021 11121212xtuV ar s????s?????? ?? ?112112111111???2?1????1????????rrr??????Thus the expression for the estimates of ?1, ?1, and s2 are : and ? ?0??2?1?1?1121212xC????s????? ? ? ?? ?111111211121????1??2?1a n d ???????????????rrrHence or ???????? ????????? ?????????? ??121121121211 ??1?2?1 rrrrrrr ????This is a quadratic equation which can be solved 0?12?12112122221121 ????????? ?????????? ??????????? ?rrrrrrrrr ??Example (ChemicalConcentration Data) the time series was identified as either an ARIMA(1,0,1) time series or an ARIMA(0,1,1) series. If we use the first identification then series xt is an ARMA(1,1) series. Identifying the series xt is an ARMA(1,1) series. The autocorrelation at lag 1 is r1 = and the autocorrelation at lag 2 is r2 = . Thus the estimate of ?1 is Also the quadratic equation bees 0?12?12112122221121 ????????? ?????????? ??????????? ?rrrrrrrrr ??02 9 8 6 4 9 8 121 ??? ??which has the two solutions and . Again we select as our estimate of ?1 to be the solution , resulting in an invertible estimated series. Since d = m(1 ?1) the estimate of d can be puted as follows: Thus the identified model in this case is xt = xt1 + ut ut1 + ? ? )( 1 ????? ?d xIf we use the second identification then series ?xt = xt – xt1 is an MA(1) series. Thus the estimate of ?1 is: 14121?2111 ??? rr?The value of r1 = . Thus the estimate of ?1 is: ? ? ? ? ??????????? 11?21?The estimate of ?1 = , corresponds to an invertible time series. This is the solution that we will choose The estimate of the parameter m is the sample mean. Thus the identified model in this case is: ?xt = ut ut1 + or xt = xt1 + ut ut1 + This pares with the other identification: xt = xt1 + ut ut1 + (An ARIMA(1,0,1) model) (An ARIMA(0,1,1) model) Preliminary Estimation of the Parameters of an AR(p) Process ? ?pp r?r?ss?????11210111 1 ???? pp r??r ?2112 ???? pp r?r?r ??and 111 ppp ?r?r ??? ? ?The regression coefficients ?1, ?2, …., ?p and the auto correlation function rh satisfy the YuleWalker equations: ? ? ? ?ppx rrC ??s ??10? 112 ????? ?111 ?1? ???? pp rr ?? ?2112 ?? ???? pp rrr ?? ??and 1?? 11 ppp rr ?? ??? ? ?The YuleWalker equations can be used to estimate the regression coeff