具有非正常错误的ARIMA模型

Question

我在ARIMA(0,0,1)中使用一个外生变量拟合R模型。

在拟合之后，我测试了错误术语，它非常不正常（就像t-distributed错误）：

我的问题是：R中是否有任何可以使ARIMA模型符合t-distributed错误的包？或者还有其他任何解决这个问题的方法吗？

数据已经是对数转换后的数据，所以我想我无法执行其他数据转换。

提前感谢您的帮助！

以下是数据：

dput(x)
c(1.098612289, 0, 1.791759469, 1.386294361, 0, 2.079441542, 2.772588722, 
2.564949357, 3.737669618, 3.761200116, 3.891820298, 3.555348061, 
2.944438979, 2.772588722, 1.791759469, 2.772588722, 2.564949357, 
3.258096538, 3.295836866, 2.890371758, 2.772588722, 2.197224577, 
4.077537444, 4.828313737, 5.855071922, 6.620073207, 7.561641746, 
7.887208586, 7.557472902, 6.747586527, 5.583496309, 4.465908119, 
3.526360525, 2.890371758, 2.564949357, 2.397895273, 2.302585093, 
0.693147181, 1.386294361, 0.693147181, 0.693147181, 0, 0, 1.098612289, 
0.693147181, 0, 0, 0, 0, 0, 0, 0, 0.693147181, 0.693147181, 0, 
0, 0.693147181, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0.693147181, 0, 0.693147181, 0.693147181, 1.386294361, 
0.693147181, 1.098612289, 2.564949357, 3.555348061, 4.744932128, 
4.615120517, 4.934473933, 4.779123493, 5.308267697, 5.303304908, 
5.416100402, 5.379897354, 5.153291594, 5.081404365, 4.927253685, 
4.86753445, 4.356708827, 4.060443011, 3.891820298, 3.091042453, 
3.091042453, 2.995732274, 2.302585093, 2.079441542, 1.609437912, 
0.693147181, 0, 0)

dput(y)
c(-2.760818612, -0.969058209, -1.374522756, -2.760817117, -0.681374268, 
0.011775716, -0.195861406, 0.976866516, 1.000404862, 1.131034014, 
0.794568131, 0.183662413, 0.011814959, -0.96901336, 0.011818696, 
-0.195818426, 0.497333426, 0.535078613, 0.129616682, 0.01183645, 
-0.5635262, 1.316797505, 2.067596972, 3.094420195, 3.859561475, 
4.801489346, 5.127554079, 4.798176537, 3.988449441, 2.824408827, 
1.706836735, 0.767295318, 0.131309734, -0.19411042, -0.361162633, 
-0.456471128, -2.065908853, -1.372761111, -2.065908104, -2.065907917, 
-2.759055098, -2.759055098, -1.660442435, -2.065907356, -2.759054536, 
-2.759054536, -2.759054536, -2.759054536, -2.759054536, -2.759054536, 
-2.759054536, -2.065907168, -2.065906981, -2.759054162, -2.759054162, 
-2.065906794, -2.759053975, -2.759053975, -2.759053975, -2.759053975, 
-2.759053975, -2.759053975, -2.759053975, -2.759053975, -2.759053975, 
-2.759053975, -2.759053975, -2.759053975, -2.759053975, -2.759053975, 
-2.759053975, -2.759053975, -2.759053975, -2.759053975, -2.065906607, 
-2.759053787, -2.06590642, -2.065906232, -1.37275849, -2.065905484, 
-1.660440001, -0.194100686, 0.796304383, 1.985909791, 1.856116899, 
2.17549615, 2.020167801, 2.549349637, 2.544424292, 2.657261726, 
2.621099122, 2.394525569, 2.3226683, 2.168543275, 2.108848197, 
1.598036993, 1.301781851, 1.133168127, 0.332394215, 0.332398148, 
0.237091526, -0.456053969, -0.679196209, -1.149199089, -2.065489634, 
-2.758636814, -2.758636814, -2.758636814)

我的代码：

y1 = y
x_data1 = matrix(c(x), ncol = 1)
ts_mod1 = arima(y1, order = c(0,0,1), xreg = x_data1)
ts_res1 = ts_mod1$residuals

qqnorm(ts_res1, main = "", cex.axis = 1.2, cex.lab = 1.45)
qqline(ts_res1, col = "red")

Answer 1

此q-q图表示heavy - tailed distribution。您可以参考this question来了解各种类型的q-q图。 要回答你的问题，有些软件包可以更好地处理非正态分发。试试forecast包 -

require('forecast')
ts_mod1 <- auto.arima(y1,xreg = x_data1)
ts_mod1

# Series: y1 
# ARIMA(4,0,2) with non-zero mean 
# 
# Coefficients:
#     ar1      ar2     ar3      ar4      ma1     ma2  intercept  x_data1
# 0.7269  -0.3027  0.2060  -0.0391  -0.6260  0.4672    -2.4920   0.8695
# s.e.  0.4409   0.4004  0.1771   0.1796   0.4577  0.3664     0.2536   0.1102
# 
# sigma^2 estimated as 0.3996:  log likelihood=-99.8
# AIC=217.6   AICc=219.44   BIC=241.74

此处auto.arima会根据ARIMA(4,0,2)值自动选择最佳AIC模型，该值优于ARIMA(0,0,1) AIC = 219.96。 拟合也更好，如q-q图所示 -

Answer 2

R中有另一个名为Autobox的软件包。它可以从autobox.com获得（我附属于它）。

标准化图表显示X与Y相关。

带差分的模型，x变量和3个异常值。注意.257系数要低得多。

通过测试方差变化并使用加权最小二乘法（GLM），我们确定了从第44期开始的方差变化。参见文章here。

残差