4、平稳时间序列的的拟合 （1）参数估计：（矩估计）
1AR(p)模型参数的矩估计 ○
对于：xt??1xt?1??2xt?2????pxt?p?at
~?~2x??x??(1??)x??(1??)xt?1?ttt?1???1??xt(1)?(xt(1)?xt?xt?1???xt?n?2)n有：? ???????1?xt(l)?(xt(l?1)?xt(l?2)???xt(1)?xt?xt?n?l)n?于是可得如下的Yule-Walk方程：
??1??1?0??2?1????p?p?1???2??1?1??2?0????p?p?2 ????????????????1p?12p?2p0?p?k 用 ?代替 ?k ，并解上述方程 组，就可得：
???1??1??????1???2???????????p?1???????p??
?1?1??p?2??2??1??p?2???p?3????1????p?3???p?1????1???p?2????2????? ???1????????p??????1
根据自协方差公式有：
?0?E(xt2)?E(xt(?1xt?1??2xt?2????pxt?p?at))??1?1??2?2????p?p??2a2a
于是可得到?的矩估计
2?????0(1??????a???11??2?2????p?p)
MA(q)模型参数的矩估计
第三章已经推导出MA(q)的自协方差结果，将代替，代替(i=1,2…q) , 得如下方程组：
2 ○
2?a??k,?2?k,?a??i?i?2?2?2?2?k?0a?(1??1??2????q)?? ?k??2??????a(??k??1?k?1????q?k?q)k?1,2?,q???上式是含有q+1个参数的非线性方程
组，解此方程组，即 可以求出各参数：??,??,???,??
方程组可以直接求解，也可以用迭代法求解。
2a12q
5、平稳时间序列的预测： （1）简单移动平均
???xt(1)?????xt(1)???????x(l)?t??1(xt?xt?1???xt?n?1)n1?(xt(1)?xt?xt?1???xt?n?2)n??1?(xt(l?1)?xt(l?2)???xt(1)?xt?xt?n?l)n
（2）指数平滑法（单指数）
xt(1)??xt??(1??)xt?1??(1??)2xt?1?
~或xt??xt??(1??)xt?1
（3）ARMA预测法：（P225）
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