Overview

1. Take the time series data given and model it to a Johnson SU distribution.
2. After modeling create a rolling z-score to model the deviation the series is on with a look back of 50 or so using the johnson SU distribution

Setup

loading the packages into R work space.

library(SuppDists)
library(readr)
library(ExtDist)
library(rugarch)
library(tseries)

Importing the data set and converting to a data frame

Data <- c(0.0275, 0.0295, 0.028, 0.0291, 0.0323, 0.0315, 0.0365, 0.0335, 0.0338, 0.0347, 0.0325, 0.034, 0.0323, 0.0308,0.0295, 0.0284, 0.0299, 0.0283, 0.0289, 0.0264, 0.026, 0.0263, 0.0276, 0.0239, 0.026, 0.0262, 0.0261, 0.0233, 0.0286, 0.0304,0.0323, 0.0327, 0.0359, 0.0346, 0.0347, 0.0346, 0.0341, 0.0337, 0.0366, 0.0339, 0.0357, 0.039, 0.0367, 0.0358, 0.0347, 0.0363, 0.0394, 0.0384, 0.0408, 0.0405, 0.0397, 0.0394, 0.0401, 0.0388, 0.0385, 0.0383, 0.037, 0.0355, 0.0366, 0.0361, 0.0347, 0.0339,0.0355, 0.0332, 0.0339, 0.0354, 0.0339, 0.0342, 0.035, 0.0346, 0.0314, 0.033, 0.0317, 0.0306, 0.0288, 0.0289, 0.0307, 0.0319, 0.0306, 0.0295, 0.0296, 0.0285, 0.0267, 0.0257, 0.0262, 0.0236, 0.0252, 0.0245, 0.0231, 0.0223, 0.0231, 0.0228, 0.0231, 0.0235,0.0243, 0.0236, 0.023, 0.022, 0.0208, 0.0197, 0.0227, 0.0238, 0.0254, 0.026, 0.0263, 0.0245, 0.0246, 0.025, 0.0258, 0.0263, 0.0258,0.0265, 0.0295, 0.032, 0.0299, 0.0315, 0.0294, 0.032, 0.0304, 0.0296, 0.0295, 0.0303, 0.0301, 0.0291, 0.0298, 0.0312, 0.0328, 0.0351, 0.0355, 0.0359, 0.0333, 0.0322, 0.0345, 0.0313, 0.0308, 0.032, 0.0288, 0.0245, 0.0286, 0.0283, 0.027, 0.0252, 0.0247, 0.0245, 0.0253, 0.0261, 0.0272, 0.0274, 0.0258, 0.0286, 0.0288, 0.0294, 0.0265, 0.0269, 0.0267, 0.0326, 0.0317, 0.0288, 0.0249, 0.0273, 0.0262, 0.03,0.0288, 0.0304, 0.0297, 0.0299, 0.0281, 0.0304, 0.0317, 0.0293, 0.0299, 0.0274, 0.0265, 0.0259, 0.0251, 0.0217, 0.0181, 0.0145, 0.0197,0.0211, 0.0223, 0.0216, 0.0223, 0.0211, 0.0203, 0.0202, 0.019, 0.0194, 0.0186, 0.0205, 0.0194, 0.0197, 0.019, 0.0187, 0.0198,0.0198, 0.019, 0.0201, 0.0166, 0.0177, 0.0201, 0.0212, 0.0186, 0.0201, 0.0208, 0.0206, 0.0214, 0.021, 0.0201, 0.0217, 0.0196,0.0214, 0.0221, 0.0264, 0.0232, 0.0229, 0.0246, 0.028, 0.0272, 0.0266, 0.0272, 0.0265, 0.0277, 0.0277, 0.0277, 0.0268, 0.0263, 0.0265, 0.0261, 0.0273, 0.0278, 0.0286, 0.0276, 0.028, 0.0291, 0.0278, 0.0285, 0.0293, 0.0244, 0.0239, 0.0221, 0.0214, 0.0215,0.0224, 0.0247, 0.0241, 0.0227, 0.0222, 0.0202, 0.0198, 0.0205, 0.0214, 0.0225, 0.0238, 0.0226, 0.0227, 0.0222, 0.021, 0.022, 0.0211,0.0242, 0.0225, 0.0251, 0.025, 0.0264, 0.0277, 0.0264, 0.0277, 0.0247, 0.0253, 0.0251, 0.025, 0.0255, 0.0252, 0.0249, 0.0243, 0.0258, 0.0236, 0.0242, 0.0249, 0.0266, 0.0282, 0.0304, 0.0301, 0.0295, 0.0302, 0.0278, 0.0255, 0.0251, 0.024, 0.0225, 0.0236, 0.0263, 0.0273, 0.0257, 0.0281,0.0282, 0.0299, 0.0299, 0.028, 0.0294, 0.0288, 0.0288, 0.0294, 0.0286, 0.0282, 0.0295, 0.0267, 0.0281, 0.0281, 0.0295, 0.031, 0.0273, 0.0276, 0.0291,0.0293, 0.0291, 0.0284, 0.027, 0.0248, 0.0251, 0.0248, 0.0243, 0.0241, 0.0256, 0.0253, 0.0267, 0.0274, 0.0245, 0.0242, 0.0199, 0.0207, 0.0229, 0.0229, 0.0241, 0.0253, 0.0256, 0.0309, 0.0301, 0.0294, 0.0268, 0.0256, 0.0253, 0.0226, 0.0262, 0.0237, 0.0238, 0.0229, 0.0235, 0.0232, 0.0241, 0.0238, 0.023, 0.0233,0.0243, 0.0231, 0.0244, 0.0281, 0.0248, 0.0262, 0.0253, 0.0267, 0.0251, 0.0276, 0.027, 0.0279, 0.0266, 0.0242, 0.0245, 0.0249, 0.0276, 0.0256, 0.0256, 0.0267,0.0269, 0.0271, 0.0293, 0.0319, 0.0303, 0.0298, 0.0283, 0.0313, 0.028, 0.0276, 0.028, 0.0276, 0.0278, 0.0286, 0.0313, 0.0299, 0.028, 0.0276, 0.0281, 0.0305,0.0299, 0.0313, 0.0295, 0.0354, 0.0255, 0.0269, 0.0268, 0.0304, 0.0311, 0.0327, 0.0321, 0.0321, 0.0302, 0.0308, 0.0282, 0.0268, 0.0251, 0.0254, 0.026, 0.0268,0.0269, 0.0267, 0.0253, 0.0255, 0.0273, 0.0278, 0.0268, 0.029, 0.0282, 0.0292, 0.0314, 0.0308, 0.0293, 0.0288, 0.0304, 0.0276, 0.0308, 0.0318, 0.0315, 0.0323,0.0326, 0.0336, 0.0313, 0.0312, 0.0294, 0.0313, 0.0305, 0.0298, 0.0286, 0.0273, 0.0295, 0.0268, 0.0266, 0.0258, 0.0251, 0.022, 0.0224, 0.024, 0.0219, 0.0215, 0.0292, 0.0278, 0.0279, 0.0272, 0.0283, 0.0266, 0.0313, 0.0322, 0.0314, 0.0337, 0.0313, 0.0332, 0.0346, 0.0356, 0.0351, 0.034, 0.0333, 0.0338, 0.0317, 0.0318,0.033, 0.0327, 0.0347, 0.0329, 0.0386, 0.0371, 0.0383, 0.0362, 0.0352, 0.034, 0.0335, 0.0362, 0.0365, 0.0333, 0.0322, 0.0312, 0.0278, 0.0305, 0.0295, 0.0308,0.0306, 0.0284, 0.0316, 0.0299, 0.03, 0.0293, 0.0299, 0.0289, 0.031, 0.0313, 0.0294, 0.0326, 0.0307, 0.0322, 0.0243, 0.0242, 0.0255, 0.0269, 0.0266, 0.0266,0.029, 0.0289, 0.03, 0.0287, 0.0287, 0.0282, 0.0327, 0.0332, 0.0345, 0.0319, 0.0338, 0.0311, 0.0307, 0.0311, 0.031, 0.0299, 0.0299, 0.0316, 0.0289, 0.0297, 0.0314, 0.0301, 0.0287, 0.0281, 0.0312, 0.0331, 0.0322, 0.0298, 0.0279, 0.0286, 0.0307, 0.0295, 0.0279, 0.0297, 0.0294, 0.029, 0.0288, 0.0286, 0.0284, 0.0288, 0.0296, 0.0278, 0.0272, 0.024, 0.0225, 0.0227, 0.0246, 0.0232, 0.026, 0.0261, 0.0259, 0.0275, 0.0276, 0.0275, 0.0241, 0.0225, 0.024, 0.0248, 0.0228, 0.0231, 0.0251, 0.025, 0.0257, 0.0261, 0.024, 0.0244, 0.0209, 0.0187, 0.019, 0.0242, 0.0255, 0.026, 0.0233, 0.0239, 0.0199, 0.0224, 0.0214, 0.0216, 0.0214, 0.0248,0.0255, 0.0228, 0.0231, 0.0228, 0.0238, 0.0234, 0.0239, 0.023, 0.024, 0.0241, 0.0235, 0.0235, 0.0215, 0.0208, 0.0211, 0.0185, 0.0188, 0.0174, 0.02, 0.0185,0.0181, 0.0188, 0.0193, 0.019, 0.0214, 0.0211, 0.0221, 0.0213, 0.0213, 0.0211, 0.0217, 0.0196, 0.0205, 0.0207, 0.0215, 0.0236, 0.0234, 0.0232, 0.0225, 0.0222, 0.0279, 0.027, 0.0276, 0.0287, 0.0291, 0.032, 0.0329, 0.0304, 0.0304, 0.0276, 0.0275, 0.028, 0.0322, 0.0305, 0.0323, 0.0314, 0.0304, 0.0275, 0.027, 0.0268, 0.0223, 0.0231, 0.0241, 0.0249, 0.0224, 0.0231, 0.0245, 0.0249, 0.0267, 0.0276, 0.0285, 0.0305, 0.0313, 0.0318, 0.0314, 0.0308, 0.0261, 0.0259, 0.025, 0.0273, 0.0283, 0.0286, 0.0273, 0.0279, 0.031, 0.0325, 0.0328, 0.0329, 0.0323, 0.033, 0.0334, 0.0354, 0.0348, 0.0355, 0.0343, 0.0339, 0.0339, 0.0326, 0.0314, 0.0311, 0.0328, 0.0285, 0.03, 0.029, 0.0279, 0.0299, 0.0355, 0.0367, 0.0346, 0.0368, 0.0366, 0.0366, 0.0353, 0.0338, 0.0339, 0.0352, 0.0352, 0.0354, 0.0354, 0.0351, 0.0363, 0.0337, 0.0339, 0.0317, 0.0295, 0.0319, 0.0309, 0.0301, 0.0304, 0.0326, 0.0304, 0.0284, 0.03, 0.0306, 0.0282, 0.029, 0.0307, 0.0332, 0.0346, 0.0335, 0.0326, 0.0353, 0.0321, 0.0335, 0.0318, 0.0328,0.0334, 0.0363, 0.0331, 0.0353, 0.0355, 0.0353, 0.0336, 0.0357, 0.0362, 0.0378, 0.0384, 0.035, 0.0366, 0.0382, 0.0375, 0.0362, 0.0374, 0.0372, 0.037, 0.0352, 0.0375, 0.0392, 0.0412, 0.0405, 0.041, 0.043, 0.0419, 0.0431, 0.0414, 0.0413, 0.0403, 0.042, 0.0404, 0.0396, 0.0385, 0.0377, 0.0369, 0.036, 0.0373, 0.0369, 0.0377, 0.0394, 0.0373, 0.0378, 0.0398, 0.0376, 0.0347, 0.0373, 0.0364, 0.0356, 0.0332, 0.0334, 0.033, 0.0327, 0.0336, 0.0312, 0.0391, 0.0385, 0.0365, 0.0379, 0.0366, 0.0397, 0.0372, 0.0365, 0.039, 0.0389, 0.0407, 0.0408, 0.0392, 0.0384, 0.0376, 0.0389, 0.0375, 0.0381, 0.0368, 0.0358, 0.0364, 0.0373, 0.035, 0.0372, 0.0396, 0.0385, 0.0388, 0.0378,0.0364, 0.037, 0.0367, 0.0373, 0.0364, 0.0367, 0.0351, 0.0344, 0.0359, 0.0364, 0.0355, 0.0344, 0.0332, 0.0329, 0.0311, 0.032, 0.0322, 0.0308, 0.0299, 0.0299, 0.0293,0.03, 0.0313, 0.0283, 0.0269, 0.0254, 0.0242, 0.0241, 0.025, 0.0222, 0.0226, 0.022, 0.0233, 0.025, 0.0259, 0.0254, 0.0248, 0.025, 0.0226, 0.0239, 0.0235, 0.0217,0.0219, 0.0217, 0.0201, 0.0231, 0.0213, 0.0208, 0.0205, 0.0207, 0.0229, 0.022, 0.0219, 0.0174, 0.017, 0.0193, 0.0158, 0.0146, 0.015, 0.0115, 0.0074, 0.007, 0.003,0.0081, 0.0078, 0.0059, -0.0031, 0.0019, 0.0067, 0.0023, 0.004, 0.0078, 0.0123, 0.0124, 0.009, 0.0089, 0.0108, 0.0128, 0.0144, 0.0146, 0.0147, 0.0179, 0.0223,0.0177, 0.0225, 0.0226, 0.0233, 0.0264, 0.0295, 0.027, 0.0261, 0.025, 0.0193, 0.0199, 0.0197, 0.0187, 0.0191, 0.0153, 0.0199, 0.0207, 0.0215, 0.0189, 0.0178, 0.0181, 0.0189, 0.0195, 0.0188, 0.0212, 0.0257, 0.0249, 0.0192, 0.0214, 0.0211, 0.0216, 0.0176, 0.0162, 0.0179, 0.0214, 0.0202, 0.0203, 0.0203, 0.021, 0.0135,0.0147, 0.0144, 0.0149, 0.0176, 0.0181, 0.0199, 0.0189, 0.0202, 0.0197, 0.02, 0.0229, 0.0241, 0.0257, 0.0216, 0.0225, 0.0241, 0.0259, 0.0209, 0.0226, 0.0217,0.0194, 0.0207, 0.0198, 0.0225, 0.0218, 0.0231, 0.0236, 0.0207, 0.021, 0.0218, 0.0224, 0.0192, 0.0213, 0.0208, 0.0205, 0.019, 0.0189, 0.019, 0.0201, 0.0192, 0.0196, 0.0181, 0.0184, 0.0199, 0.0179, 0.0195, 0.019, 0.0171, 0.0174, 0.0177, 0.0179, 0.0177, 0.0167)

Johnson SU distribution model

parms<-JohnsonFit(Data)
parms

## $gamma
## [1] 0.2070377
## 
## $delta
## [1] 1.613749
## 
## $xi
## [1] 0.007967222
## 
## $lambda
## [1] 0.04188204
## 
## $type
## [1] "SB"

pJohnsonSU <- function(q, gamma = 0.2070377
    , delta = 1.613749 , xi = 0.007967222, lambda = 0.04188204  , params = list(gamma = 2.21600011, delta = 1.613749, xi =  0.007967222, lambda = 0.04188204),...){
  if(!missing(params)){
    gamma <- params$gamma
    delta <- params$delta
    xi <- params$xi
    lambda <- params$lambda
  }
  
  u = (q - xi)/lambda
  out = pnorm(delta*asinh(u)+gamma)
  return(out)
}
z.score = function(Data) {
  return((Data - mean(Data)) / sd(Data))
}
specgarchsk = ugarchspec(variance.model = list(garchOrder=c(1,1)), distribution.model ="ged")
roll = ugarchroll(specgarchsk, data = z.score(Data), n.ahead = 1,refit.every = 50, refit.window = "rolling")
roll

## 
## *-------------------------------------*
## *              GARCH Roll             *
## *-------------------------------------*
## No.Refits        : 10
## Refit Horizon    : 50
## No.Forecasts : 500
## GARCH Model      : sGARCH(1,1)
## Distribution : ged 
## 
## Forecast Density:
##                         Mu  Sigma Skew  Shape Shape(GIG) Realized
## 1971-05-17 03:00:00 0.2829 0.2560    0 1.4247          0   0.3767
## 1971-05-18 03:00:00 0.3426 0.2563    0 1.4247          0   0.2217
## 1971-05-19 03:00:00 0.2295 0.2575    0 1.4247          0   0.5472
## 1971-05-20 03:00:00 0.4690 0.2754    0 1.4247          0   0.5937
## 1971-05-21 03:00:00 0.5434 0.2577    0 1.4247          0   0.2992
## 1971-05-22 03:00:00 0.3223 0.2670    0 1.4247          0   0.7951
## 
## ..........................
##                          Mu  Sigma Skew  Shape Shape(GIG) Realized
## 1972-09-22 03:00:00 -1.2940 0.2654    0 1.2063          0  -1.6070
## 1972-09-23 03:00:00 -1.5473 0.2687    0 1.2063          0  -1.5605
## 1972-09-24 03:00:00 -1.5437 0.2652    0 1.2063          0  -1.5140
## 1972-09-25 03:00:00 -1.5041 0.2619    0 1.2063          0  -1.4830
## 1972-09-26 03:00:00 -1.4725 0.2587    0 1.2063          0  -1.5140
## 1972-09-27 03:00:00 -1.4941 0.2559    0 1.2063          0  -1.6690
## 
## Elapsed: 11.49303 secs