ARIMA模型與非正常的錯誤

Question

我適合ARIMA(0,0,1)模型在R與一個外生變量。

性擬合後，我測試的誤差項，它的高度非正常（這就像t-distributed錯誤）：

我的問題是：是否有任何包R可以容納ARIMA模型t-distributed錯誤？或者有沒有其他補救措施來解決這個問題？

數據已經過對數轉換的數據，所以我想我不能執行另一個數據轉換。

非常感謝您的幫助！

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下面是數據：

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) 

而且我的代碼：

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

這個QQ圖是表示heavy - tailed distribution的。你可以參考this question瞭解各種類型的q-q圖。 要回答你的問題，有些軟件包可以更好地處理非正態分佈。嘗試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 

這裏auto.arima自動選擇基於該AIC值比ARIMA(0,0,1)與AIC = 219.96更好的最佳ARIMA(4,0,2)模型。 擬合也較好如圖此分位圖 -

Answer 2

還有一個包中的R稱爲Autobox。它可以從autobox.com（我隸屬於它）。

標準化曲線示出了X被設置爲Y.

模型差分，x變量和3個離羣值有關。注意.257係數要低得多。

通過測試方差變化，並使用加權最小二乘（GLM）我們已經確定的方差在期間44開始改變，見紙here。

殘值