4时间序列参数估计
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时间序列模型参数估计1理论基础
1.1矩估计
1.1.1AR模型
矩估计法参数估计的思路:
即从样本中依次求中r k然后求其对应的参数Φk值
方差:
1.1.2MA模型
对于MA模型采用矩估计是比较不精确的,所以这里不予讨论
1.1.3ARMA(1,1)矩估计法参数估计的思路:
方差:
1.2最小二乘估计
1.2.1AR模型
最小二乘参数估计的思路:
对于AR(P)而言也可以得到类似矩估计得到的方程,即最小二乘与矩估计得到的估计量相同。
1.2.2MA模型
最小二乘参数估计的思路:
1.2.3ARMA模型
最小二乘参数估计的思路:
1.3极大似然估计与无条件最小二乘估计
2R中如何实现时间序列参数估计2.1对于AR模型
ar(x, aic = TRUE, order.max = NULL,
method=c("yule-walker", "burg", "ols", "mle", "yw"),
na.action, series, ...)
> ar(ar1.s,order.max=1,AIC=F,method='yw')#即矩估计
Call:
ar(x = ar1.s, order.max = 1, method = "yw", AIC = F)
Coefficients:
1
0.8314
Order selected 1 sigma^2 estimated as 1.382
> ar(ar1.s,order.max=1,AIC=F,method='ols')#最小二乘估计
Call:
ar(x = ar1.s, order.max = 1, method = "ols", AIC = F)
Coefficients:
1
0.857
Intercept: 0.02499 (0.1308)
Order selected 1 sigma^2 estimated as 1.008
> ar(ar1.s,order.max=1,AIC=F,method='mle')#极大似然估计
Call:
ar(x = ar1.s, order.max = 1, method = "mle", AIC = F)
Coefficients:
1
0.8924
Order selected 1 sigma^2 estimated as 1.041
采用自编函数总结三个不同的估计值
> Myar(ar2.s,order.max=3)
最小二乘估计矩估计极大似然估计
1 1.5137146 1.4694476 1.5061369
2 -0.8049905 -0.7646034 -0.7964453
2.2对于ARMA模型
arima(x, order = c(0, 0, 0), seasonal = list(order = c(0, 0, 0), period = NA), xreg = NULL, include.mean = TRUE, transform.pars = TRUE, fixed = NULL,
init = NULL, method = c("CSS-ML", "ML", "CSS"), n.cond, optim.control = list(), kappa = 1e+06, io = NULL, xtransf, transfer = NULL)
order的三个参数分别代表AR,差分MA的阶数
> arima(arma11.s,order=c(1,0,1),method='CSS')
Call:
arima(x = arma11.s, order = c(1, 0, 1), method = "CSS")
Coefficients:
ar1 ma1 intercept
0.5586 0.3669 0.3928
s.e. 0.1219 0.1564 0.3380
sigma^2 estimated as 1.199: part log likelihood = -150.98
> arima(arma11.s,order=c(1,0,1),method='ML')
Call:
arima(x = arma11.s, order = c(1, 0, 1), method = "ML")
Coefficients:
ar1 ma1 intercept
0.5647 0.3557 0.3216
s.e. 0.1205 0.1585 0.3358
sigma^2 estimated as 1.197: log likelihood = -151.33, aic = 308.65 采用自编函数总结三个不同的估计值
> Myarima(arma11.s,order=c(1,0,1))
$coef
条件SS估计极大似然估计条件似然估计
ar1 0.5585828 0.5647477 0.5647498
ma1 0.3668814 0.3556965 0.3556973
intercept 0.3927654 0.3216166 0.3216152
$log
条件SS估计极大似然估计条件似然估计
[1,] -150.984 -151.3268 -151.3268
$sigma2
条件SS估计极大似然估计条件似然估计
[1,] 1.199378 1.196984 1.196984
$aic
条件SS估计极大似然估计条件似然估计