4时间序列参数估计

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