#QUESTION 1
library(rugarch)
## Loading required package: parallel
##
## Attaching package: 'rugarch'
## The following object is masked from 'package:stats':
##
## sigma
library(timeSeries)
## Loading required package: timeDate
##
## Attaching package: 'timeSeries'
## The following object is masked from 'package:rugarch':
##
## quantile
## The following objects are masked from 'package:graphics':
##
## lines, points
library(fGarch)
## NOTE: Packages 'fBasics', 'timeDate', and 'timeSeries' are no longer
## attached to the search() path when 'fGarch' is attached.
##
## If needed attach them yourself in your R script by e.g.,
## require("timeSeries")
library(PerformanceAnalytics)
## Loading required package: xts
## Loading required package: zoo
##
## Attaching package: 'zoo'
## The following object is masked from 'package:timeSeries':
##
## time<-
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
##
## Attaching package: 'PerformanceAnalytics'
## The following objects are masked from 'package:fGarch':
##
## ES, VaR
## The following objects are masked from 'package:timeDate':
##
## kurtosis, skewness
## The following object is masked from 'package:graphics':
##
## legend
data= read.csv("DATA25.csv");data
## Year Month STOCK GDP CPI PENSION INFLATION CLAIMS INTEREST
## 1 2025 April 3.74 4.11 6.12 13.50 5.58 4.59 4.12
## 2 2025 March 3.81 3.62 6.33 13.59 5.53 4.53 3.97
## 3 2025 February 3.98 3.45 6.47 13.49 5.70 4.53 4.07
## 4 2025 January 4.21 3.28 6.48 13.47 4.04 4.63 4.20
## 5 2024 December 4.50 2.99 6.51 13.70 4.35 4.95 4.44
## 6 2024 November 4.81 2.75 6.56 13.73 4.28 5.20 4.62
## 7 2024 October 5.14 2.72 6.57 13.71 3.95 5.61 4.45
## 8 2024 September 5.50 3.56 6.64 13.83 3.73 6.24 4.38
## 9 2024 August 5.77 4.36 6.66 13.85 4.18 6.89 4.42
## 10 2024 July 5.97 4.31 6.68 13.76 4.46 7.40 4.40
## 11 2024 June 4.64 6.22 6.68 13.66 4.83 7.79 4.37
## 12 2024 May 6.49 5.10 6.81 13.65 4.50 7.98 4.50
## 13 2024 April 6.73 5.00 7.06 13.61 4.73 8.15 4.21
## 14 2024 March 6.97 5.70 7.16 13.61 5.72 8.33 4.22
## 15 2024 February 7.26 6.31 7.19 13.81 7.06 8.40 4.30
## 16 2024 January 7.50 6.85 7.19 14.23 8.04 8.36 4.59
## 17 2023 December 7.67 6.63 7.20 14.29 7.47 8.21 4.73
## 18 2023 November 7.87 6.80 7.25 14.39 9.21 8.13 4.96
## 19 2023 October 8.10 6.92 7.23 14.24 11.70 7.84 4.78
## 20 2023 September 8.32 6.78 7.25 14.25 11.48 7.20 4.56
## 21 2023 August 8.52 6.73 7.27 14.45 10.28 6.76 4.56
## 22 2023 July 8.68 7.28 7.33 15.32 9.04 6.43 4.55
## 23 2023 June 8.77 7.88 7.36 15.49 6.99 6.26 4.57
## 24 2023 May 8.78 8.03 7.46 17.28 6.35 6.30 4.57
## 25 2023 April 8.71 7.90 7.62 18.47 6.68 6.43 5.70
## 26 2023 March 8.59 9.19 7.72 22.51 6.47 6.48 5.75
## 27 2023 February 8.30 9.23 7.75 23.03 6.34 6.50 5.52
## 28 2023 January 7.95 8.98 7.77 23.04 6.26 6.47 5.27
## 29 2022 December 7.66 9.06 7.78 23.43 6.40 6.44 5.53
## 30 2022 November 7.38 9.48 7.86 22.36 5.80 6.46 5.79
## 31 2022 October 7.48 9.59 7.87 23.27 5.00 6.59 5.04
## 32 2022 September 6.81 9.18 7.88 20.84 5.27 6.72 4.87
## 33 2022 August 6.61 8.53 7.95 19.98 6.45 6.88 4.71
## 34 2022 July 6.45 8.32 8.02 18.61 6.84 6.87 4.76
## 35 2022 June 6.29 7.91 8.06 17.76 7.78 6.77 4.91
## 36 2022 May 6.16 7.08 8.21 17.42 8.01 6.58 4.94
## 37 2022 April 6.05 6.47 8.41 16.69 7.32 6.42 4.83
## 38 2022 March 6.29 5.56 8.43 16.67 6.72 6.31 4.59
## 39 2022 February 6.23 5.08 8.60 16.76 5.97 6.29 4.53
## 40 2022 January 6.08 5.39 8.66 15.11 5.84 6.34 5.20
## 41 2021 December 5.62 5.73 8.69 14.05 6.62 6.54 4.74
## 42 2021 November 6.10 5.80 8.83 13.05 7.03 6.63 5.50
## 43 2021 October 6.07 6.45 8.84 12.20 6.87 6.65 5.04
## 44 2021 September 5.35 6.91 8.96 12.08 7.08 6.69 5.68
## 45 2021 August 5.71 6.57 9.13 11.30 6.31 6.63 6.57
## 46 2021 July 5.53 6.55 9.16 10.83 5.61 6.63 6.92
## 47 2021 June 5.35 6.32 9.19 9.81 5.53 6.74 9.25
## 48 2021 May 5.20 5.87 9.28 10.13 6.02 6.88 9.06
## 49 2021 April 4.66 5.76 9.57 10.32 6.09 6.97 9.15
## 50 2021 March 5.17 5.90 9.60 11.90 6.43 7.08 9.29
## 51 2021 February 5.16 5.78 9.76 11.80 6.60 7.19 10.22
## 52 2021 January 5.74 5.69 9.85 11.97 8.36 7.33 10.47
## 53 2020 December 5.41 5.62 9.92 12.46 7.67 7.19 11.03
## 54 2020 November 5.53 5.33 10.06 12.77 7.39 7.05 11.70
## 55 2020 October 5.67 4.84 10.16 13.33 7.30 6.85 10.85
## 56 2020 September 5.79 4.20 10.11 13.62 6.41 6.58 8.63
## 57 2020 August 5.87 4.36 10.23 13.89 6.27 6.39 7.64
## 58 2020 July 6.01 4.36 10.28 14.23 6.86 6.21 8.15
## 59 2020 June 6.16 4.59 10.34 14.19 7.21 6.01 9.32
## 60 2020 May 6.18 5.33 10.54 14.17 7.15 5.72 8.48
## 61 2020 April 6.03 6.01 10.92 14.05 7.36 5.39 7.29
## 62 2020 March 5.84 5.84 10.96 13.90 7.76 5.05 7.01
## 63 2020 February 5.72 7.17 11.17 14.28 8.29 4.75 6.84
## 64 2020 January 5.29 5.78 11.32 14.19 6.67 4.50 6.82
## 65 2019 December 5.20 5.82 11.28 14.29 6.03 4.44 7.23
## 66 2019 November 5.19 5.56 11.43 14.65 4.91 4.56 6.33
## 67 2019 October 5.19 4.95 11.46 14.54 4.05 4.96 5.98
## 68 2019 September 5.24 3.83 11.49 14.47 4.14 5.61 5.86
## 69 2019 August 5.40 5.00 11.77 14.33 4.11 6.33 5.00
## 70 2019 July 5.32 6.27 12.18 14.24 4.45 7.24 4.66
## 71 2019 June 5.16 5.70 12.47 14.95 3.67 8.20 4.43
## 72 2019 May 5.04 4.49 12.79 13.89 3.20 9.38 4.45
## 73 2019 April 4.91 6.58 12.73 14.11 3.25 10.67 4.06
## 74 2019 April 4.91 6.58 13.07 14.06 4.14 12.04 3.77
## 75 2019 March 4.67 4.35 13.31 14.53 5.32 13.29 4.22
## 76 2019 February 4.65 4.14 13.32 15.21 6.09 14.33 3.52
## 77 2019 January 4.68 4.70 13.67 15.88 7.74 15.27 3.85
## 78 2018 December 4.69 5.71 13.79 16.02 10.05 15.97 4.07
## 79 2018 November 4.59 5.58 13.96 15.94 12.22 16.40 3.88
## 80 2018 October 4.53 5.53 14.01 15.88 13.06 16.50 4.19
## 81 2018 September 4.53 5.70 14.38 15.89 15.61 16.45 3.88
## 82 2018 August 4.63 4.04 14.40 18.37 16.69 15.93 3.50
## 83 2018 July 4.95 4.35 14.48 17.85 18.31 15.10 3.53
## 84 2018 June 5.20 4.28 14.56 16.87 18.93 14.02 3.47
## 85 2018 May 5.61 3.95 14.83 16.67 19.72 12.82 3.31
## 86 2018 April 6.24 3.73 14.97 16.35 18.91 11.49 3.10
## 87 2018 March 6.89 4.18 16.27 15.96 17.32 10.18 3.01
## 88 2018 February 7.40 4.46 16.40 15.39 16.67 9.00 2.82
## 89 2018 January 7.79 4.83 16.94 14.67 15.53 7.88 2.59
## 90 2017 December 7.98 4.50 18.68 12.99 14.48 6.88 2.36
## 91 2017 November 8.15 4.73 18.46 11.25 12.95 5.96 2.21
## 92 2017 October 8.33 5.72 18.54 9.66 12.05 5.20 1.78
## 93 2017 September 8.40 7.06 18.85 8.93 9.19 4.49 1.65
## 94 2017 August 8.36 8.04 18.62 8.18 6.54 4.05 1.71
## 95 2017 July 8.21 7.47 18.92 7.55 5.42 3.93 1.38
## 96 2017 June 8.13 9.21 19.46 7.83 4.51 3.96 1.25
## 97 2017 May 7.84 11.70 19.59 7.65 3.84 4.02 1.16
## 98 2017 April 7.20 11.48 21.16 7.79 3.18 4.12 0.86
## 99 2017 March 6.76 10.28 21.83 8.44 3.21 4.40 0.75
## 100 2017 February 6.43 9.04 23.31 9.10 3.22 4.69 0.93
## 101 2017 January 6.26 6.99 24.16 9.48 3.57 5.03 1.26
## 102 2016 December 6.30 6.35 25.99 9.74 3.49 5.43 1.26
## 103 2016 November 6.43 6.68 26.47 10.38 3.88 5.85 1.18
## 104 2016 October 6.48 6.47 27.38 9.17 3.66 6.32 1.46
## 105 2016 September 6.50 6.34 29.02 8.01 3.97 7.03 1.24
## 106 2016 August 6.47 6.26 29.32 7.61 5.18 7.88 1.11
## 107 2016 July 6.44 6.40 29.63 7.21 5.95 8.64 1.00
## 108 2016 June 6.46 5.80 30.11 7.01 5.32 9.24 1.15
## 109 2016 May 6.59 5.00 31.64 6.67 5.00 10.24 0.96
## 110 2016 April 6.72 5.27 32.65 6.26 6.62 11.42 0.88
## 111 2016 March 6.88 6.45 33.54 6.22 6.74 12.41 0.73
## 112 2016 February 6.87 6.84 34.44 6.22 7.36 13.42 0.96
## 113 2016 January 6.77 7.78 34.09 6.20 8.44 14.35 0.76
## 114 2015 December 6.58 8.01 33.46 6.22 8.60 15.11 0.52
## 115 2015 November 6.42 7.32 33.70 6.54 9.61 15.93 0.43
## 116 2015 October 6.31 6.72 33.26 6.55 12.42 16.72 0.43
## 117 2015 September 6.29 5.97 32.75 6.92 14.60 17.07 0.32
## 118 2015 August 6.34 5.84 32.92 6.58 14.69 16.87 0.64
## 119 2015 July 6.54 6.62 32.12 6.44 13.22 16.56 0.88
D1=data$STOCK;D1
## [1] 3.74 3.81 3.98 4.21 4.50 4.81 5.14 5.50 5.77 5.97 4.64 6.49 6.73 6.97 7.26
## [16] 7.50 7.67 7.87 8.10 8.32 8.52 8.68 8.77 8.78 8.71 8.59 8.30 7.95 7.66 7.38
## [31] 7.48 6.81 6.61 6.45 6.29 6.16 6.05 6.29 6.23 6.08 5.62 6.10 6.07 5.35 5.71
## [46] 5.53 5.35 5.20 4.66 5.17 5.16 5.74 5.41 5.53 5.67 5.79 5.87 6.01 6.16 6.18
## [61] 6.03 5.84 5.72 5.29 5.20 5.19 5.19 5.24 5.40 5.32 5.16 5.04 4.91 4.91 4.67
## [76] 4.65 4.68 4.69 4.59 4.53 4.53 4.63 4.95 5.20 5.61 6.24 6.89 7.40 7.79 7.98
## [91] 8.15 8.33 8.40 8.36 8.21 8.13 7.84 7.20 6.76 6.43 6.26 6.30 6.43 6.48 6.50
## [106] 6.47 6.44 6.46 6.59 6.72 6.88 6.87 6.77 6.58 6.42 6.31 6.29 6.34 6.54
D2=diff(D1);D2
## [1] 0.07 0.17 0.23 0.29 0.31 0.33 0.36 0.27 0.20 -1.33 1.85 0.24
## [13] 0.24 0.29 0.24 0.17 0.20 0.23 0.22 0.20 0.16 0.09 0.01 -0.07
## [25] -0.12 -0.29 -0.35 -0.29 -0.28 0.10 -0.67 -0.20 -0.16 -0.16 -0.13 -0.11
## [37] 0.24 -0.06 -0.15 -0.46 0.48 -0.03 -0.72 0.36 -0.18 -0.18 -0.15 -0.54
## [49] 0.51 -0.01 0.58 -0.33 0.12 0.14 0.12 0.08 0.14 0.15 0.02 -0.15
## [61] -0.19 -0.12 -0.43 -0.09 -0.01 0.00 0.05 0.16 -0.08 -0.16 -0.12 -0.13
## [73] 0.00 -0.24 -0.02 0.03 0.01 -0.10 -0.06 0.00 0.10 0.32 0.25 0.41
## [85] 0.63 0.65 0.51 0.39 0.19 0.17 0.18 0.07 -0.04 -0.15 -0.08 -0.29
## [97] -0.64 -0.44 -0.33 -0.17 0.04 0.13 0.05 0.02 -0.03 -0.03 0.02 0.13
## [109] 0.13 0.16 -0.01 -0.10 -0.19 -0.16 -0.11 -0.02 0.05 0.20
plot(D2)
ts.plot(D2)
ts.plot(D2)
ts.plot(D2, main = "stock Log Return", ylab = "Log Returns")
M1 = garchFit(formula = ~garch(1,2), data = D2, cond.dist = "snorm"); M1
##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 2)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 2
## Max GARCH Order: 2
## Maximum Order: 2
## Conditional Dist: snorm
## h.start: 3
## llh.start: 1
## Length of Series: 118
## Recursion Init: mci
## Series Scale: 0.3299878
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.71908150 0.7190815 0.07190815 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.10000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.40000000 TRUE
## beta2 0.00000001 1.0000000 0.40000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 TRUE
## shape 1.00000000 10.0000000 4.00000000 FALSE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1 beta2 skew
## 1 2 3 5 6 8
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 158.22240: 0.0719082 0.100000 0.100000 0.400000 0.400000 1.00000
## 1: 157.00115: 0.0718866 0.0848552 0.0963416 0.382913 0.382824 0.997854
## 2: 156.11519: 0.0718169 0.0908528 0.123383 0.381533 0.381947 0.989882
## 3: 155.35927: 0.0717291 0.0795271 0.134394 0.365653 0.366012 0.981090
## 4: 154.52650: 0.0715898 0.0901513 0.157160 0.363245 0.363594 0.967384
## 5: 153.85902: 0.0714515 0.0827135 0.166454 0.347021 0.346636 0.955627
## 6: 153.09593: 0.0712784 0.0971498 0.186338 0.344822 0.343462 0.941139
## 7: 152.49917: 0.0711504 0.0882191 0.194524 0.328256 0.325769 0.931290
## 8: 151.80588: 0.0709914 0.104495 0.214564 0.328031 0.324132 0.918630
## 9: 151.14289: 0.0708532 0.0958646 0.222030 0.311531 0.305882 0.908875
## 10: 150.48882: 0.0706986 0.110922 0.243120 0.310850 0.303484 0.896545
## 11: 149.77795: 0.0705455 0.103160 0.251082 0.294777 0.284927 0.886372
## 12: 148.70988: 0.0701797 0.130087 0.293028 0.288816 0.273359 0.859006
## 13: 147.14418: 0.0697308 0.122400 0.300881 0.258228 0.230864 0.836039
## 14: 146.14907: 0.0693365 0.148798 0.344514 0.257389 0.221559 0.809382
## 15: 145.03265: 0.0688743 0.128836 0.355393 0.230236 0.179659 0.789948
## 16: 144.07209: 0.0682625 0.158863 0.393506 0.239960 0.169948 0.761431
## 17: 143.18237: 0.0676038 0.135146 0.411172 0.222054 0.128009 0.741666
## 18: 142.26338: 0.0666659 0.154691 0.450737 0.246569 0.115836 0.717959
## 19: 141.46902: 0.0657696 0.128013 0.475605 0.245733 0.0750194 0.701332
## 20: 140.58825: 0.0647495 0.140135 0.512082 0.280969 0.0573670 0.687308
## 21: 139.78260: 0.0638370 0.118387 0.536955 0.292982 0.0143258 0.673924
## 22: 139.29099: 0.0632218 0.144400 0.565803 0.326805 1.00000e-08 0.663623
## 23: 138.71445: 0.0618675 0.0988726 0.588715 0.340646 1.00000e-08 0.646750
## 24: 138.45571: 0.0586914 0.110209 0.624425 0.348787 1.00000e-08 0.649521
## 25: 138.09292: 0.0552808 0.105936 0.636322 0.316986 1.00000e-08 0.652348
## 26: 137.89714: 0.0528465 0.116320 0.667950 0.320506 1.00000e-08 0.618259
## 27: 137.65249: 0.0491862 0.0917482 0.681645 0.314804 1.00000e-08 0.621970
## 28: 137.40234: 0.0458058 0.105439 0.702605 0.305152 1.00000e-08 0.644302
## 29: 137.21752: 0.0418394 0.104567 0.700496 0.290545 1.00000e-08 0.631357
## 30: 137.02326: 0.0378728 0.106251 0.714713 0.302440 1.00000e-08 0.625087
## 31: 136.89262: 0.0339824 0.0922285 0.731010 0.301017 1.00000e-08 0.631189
## 32: 136.68177: 0.0298741 0.103516 0.733704 0.295725 1.00000e-08 0.632549
## 33: 136.53865: 0.0258629 0.101379 0.733617 0.282447 1.00000e-08 0.620881
## 34: 135.46147: -0.0148118 0.0869752 0.775421 0.287650 1.00000e-08 0.641636
## 35: 134.43178: -0.0554155 0.121756 0.741130 0.283905 1.00000e-08 0.608036
## 36: 134.26206: -0.0953977 0.145425 0.788345 0.200213 1.00000e-08 0.666517
## 37: 133.59275: -0.135793 0.130915 0.822745 0.218306 1.00000e-08 0.595349
## 38: 133.52026: -0.140075 0.132369 0.830699 0.235119 1.00000e-08 0.621230
## 39: 133.48864: -0.149418 0.137116 0.829051 0.219430 1.00000e-08 0.619217
## 40: 133.48742: -0.148312 0.136969 0.831257 0.220073 1.00000e-08 0.615241
## 41: 133.48730: -0.148787 0.137222 0.827499 0.221000 1.00000e-08 0.615725
## 42: 133.48726: -0.148882 0.137433 0.828886 0.220425 1.00000e-08 0.615890
## 43: 133.48725: -0.148890 0.137445 0.828808 0.220453 1.00000e-08 0.615758
## 44: 133.48725: -0.148898 0.137457 0.828774 0.220459 1.00000e-08 0.615751
## 45: 133.48725: -0.148899 0.137459 0.828774 0.220458 1.00000e-08 0.615751
##
## Final Estimate of the Negative LLH:
## LLH: 2.660707 norm LLH: 0.02254837
## mu omega alpha1 beta1 beta2 skew
## -0.04913478 0.01496817 0.82877385 0.22045798 0.00000001 0.61575116
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1 beta2
## mu -1849.870609 -3513.099 -7.438658 58.47853 -18.663082
## omega -3513.099186 -71069.441 -1088.691371 -3132.40646 -3512.182244
## alpha1 -7.438658 -1088.691 -53.973799 -97.56155 -103.545884
## beta1 58.478534 -3132.406 -97.561550 -380.13691 -313.240476
## beta2 -18.663082 -3512.182 -103.545884 -313.24048 -214.546412
## skew -96.063547 -1257.281 -21.331750 -61.43075 -1.605818
## skew
## mu -96.063547
## omega -1257.280880
## alpha1 -21.331750
## beta1 -61.430754
## beta2 -1.605818
## skew -234.599533
## attr(,"time")
## Time difference of 0.04183602 secs
##
## --- END OF TRACE ---
## Warning in sqrt(diag(fit$cvar)): NaNs produced
##
## Time to Estimate Parameters:
## Time difference of 0.214797 secs
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = D2, cond.dist = "snorm")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000173dd5b6238>
## [data = D2]
##
## Conditional Distribution:
## snorm
##
## Coefficient(s):
## mu omega alpha1 beta1 beta2 skew
## -0.04913478 0.01496817 0.82877385 0.22045798 0.00000001 0.61575116
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -4.913e-02 2.599e-02 -1.891 0.05866 .
## omega 1.497e-02 4.978e-03 3.007 0.00264 **
## alpha1 8.288e-01 1.800e-01 4.605 4.12e-06 ***
## beta1 2.205e-01 5.654e-02 3.899 9.66e-05 ***
## beta2 1.000e-08 NaN NaN NaN
## skew 6.158e-01 6.112e-02 10.075 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -2.660707 normalized: -0.02254837
##
## Description:
## Thu May 29 12:26:35 2025 by user: SILVENUS
summary(M1)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = D2, cond.dist = "snorm")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000173dd5b6238>
## [data = D2]
##
## Conditional Distribution:
## snorm
##
## Coefficient(s):
## mu omega alpha1 beta1 beta2 skew
## -0.04913478 0.01496817 0.82877385 0.22045798 0.00000001 0.61575116
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -4.913e-02 2.599e-02 -1.891 0.05866 .
## omega 1.497e-02 4.978e-03 3.007 0.00264 **
## alpha1 8.288e-01 1.800e-01 4.605 4.12e-06 ***
## beta1 2.205e-01 5.654e-02 3.899 9.66e-05 ***
## beta2 1.000e-08 NaN NaN NaN
## skew 6.158e-01 6.112e-02 10.075 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -2.660707 normalized: -0.02254837
##
## Description:
## Thu May 29 12:26:35 2025 by user: SILVENUS
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 63.6187925 1.532108e-14
## Shapiro-Wilk Test R W 0.9162099 1.739095e-06
## Ljung-Box Test R Q(10) 21.6429491 1.703055e-02
## Ljung-Box Test R Q(15) 29.4520794 1.405805e-02
## Ljung-Box Test R Q(20) 36.4413440 1.364209e-02
## Ljung-Box Test R^2 Q(10) 3.4703174 9.680966e-01
## Ljung-Box Test R^2 Q(15) 5.2805158 9.894731e-01
## Ljung-Box Test R^2 Q(20) 8.6683565 9.864399e-01
## LM Arch Test R TR^2 6.5679093 8.847989e-01
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 0.1467916 0.2876739 0.1419466 0.2039940
return_stats = table.Stats(D2)
## Warning in log(1 + x): NaNs produced
skewness = PerformanceAnalytics::skewness(D2)
kurtosis = PerformanceAnalytics::kurtosis(D2)
View(return_stats)
# Forecast using best model
spec = ugarchspec(variance.model = list(model = "eGARCH", garchOrder = c(1,2)),
mean.model = list(armaOrder = c(1,2), include.mean = FALSE),
distribution.model = "std"); spec
##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : eGARCH(1,2)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(1,0,2)
## Include Mean : FALSE
## GARCH-in-Mean : FALSE
##
## Conditional Distribution
## ------------------------------------
## Distribution : std
## Includes Skew : FALSE
## Includes Shape : TRUE
## Includes Lambda : FALSE
fit = ugarchfit(spec = spec, data = D2)
forecast = ugarchforecast(fit, n.ahead = 60, conf.int = 0.99)
plot(forecast, which=1)
plot(forecast,which=3)
#QUESTION 2
#Kaplan Meier
library(survival)
data(package = "survival")
str(aml)
## 'data.frame': 23 obs. of 3 variables:
## $ time : num 9 13 13 18 23 28 31 34 45 48 ...
## $ status: num 1 1 0 1 1 0 1 1 0 1 ...
## $ x : Factor w/ 2 levels "Maintained","Nonmaintained": 1 1 1 1 1 1 1 1 1 1 ...
head(aml)
## time status x
## 1 9 1 Maintained
## 2 13 1 Maintained
## 3 13 0 Maintained
## 4 18 1 Maintained
## 5 23 1 Maintained
## 6 28 0 Maintained
surv_object = Surv(time = aml$time, event = aml$status)
km_fit = survfit(surv_object ~ x, data = aml)
print(km_fit)
## Call: survfit(formula = surv_object ~ x, data = aml)
##
## n events median 0.95LCL 0.95UCL
## x=Maintained 11 7 31 18 NA
## x=Nonmaintained 12 11 23 8 NA
summary(km_fit)
## Call: survfit(formula = surv_object ~ x, data = aml)
##
## x=Maintained
## time n.risk n.event survival std.err lower 95% CI upper 95% CI
## 9 11 1 0.909 0.0867 0.7541 1.000
## 13 10 1 0.818 0.1163 0.6192 1.000
## 18 8 1 0.716 0.1397 0.4884 1.000
## 23 7 1 0.614 0.1526 0.3769 0.999
## 31 5 1 0.491 0.1642 0.2549 0.946
## 34 4 1 0.368 0.1627 0.1549 0.875
## 48 2 1 0.184 0.1535 0.0359 0.944
##
## x=Nonmaintained
## time n.risk n.event survival std.err lower 95% CI upper 95% CI
## 5 12 2 0.8333 0.1076 0.6470 1.000
## 8 10 2 0.6667 0.1361 0.4468 0.995
## 12 8 1 0.5833 0.1423 0.3616 0.941
## 23 6 1 0.4861 0.1481 0.2675 0.883
## 27 5 1 0.3889 0.1470 0.1854 0.816
## 30 4 1 0.2917 0.1387 0.1148 0.741
## 33 3 1 0.1944 0.1219 0.0569 0.664
## 43 2 1 0.0972 0.0919 0.0153 0.620
## 45 1 1 0.0000 NaN NA NA
plot(km_fit,
xlab = "Time (Weeks)",
ylab = "Survival Probability",
col = c("blue", "red"),
lty = c(1, 2),
main = "Kaplan-Meier Survival Curves for AML Patients by Treatment",
mark.time = TRUE)
legend("topright",
legend = levels(aml$x),
col = c("blue", "red"),
lty = c(1, 2))
# Optional: Confidence intervals
plot(km_fit,
xlab = "Time (Weeks)",
ylab = "Survival Probability",
col = c("blue", "red"),
lty = c(1, 2),
main = "Kaplan-Meier Survival Curves with Confidence Intervals",
mark.time = TRUE,
conf.int = TRUE)
legend("topright",
legend = levels(aml$x),
col = c("blue", "red"),
lty = c(1, 2))
surv_diff = survdiff(surv_object ~ x, data = aml)
print(surv_diff)
## Call:
## survdiff(formula = surv_object ~ x, data = aml)
##
## N Observed Expected (O-E)^2/E (O-E)^2/V
## x=Maintained 11 7 10.69 1.27 3.4
## x=Nonmaintained 12 11 7.31 1.86 3.4
##
## Chisq= 3.4 on 1 degrees of freedom, p= 0.07
#QUESTION FOUR
library(fitdistrplus)
## Loading required package: MASS
##
## Attaching package: 'fitdistrplus'
## The following object is masked from 'package:rugarch':
##
## fitdist
library(goftest)
data = read.csv("DATA25.csv")
claims = data$CLAIMS
D1 = diff(claims); D1
## [1] -0.06 0.00 0.10 0.32 0.25 0.41 0.63 0.65 0.51 0.39 0.19 0.17
## [13] 0.18 0.07 -0.04 -0.15 -0.08 -0.29 -0.64 -0.44 -0.33 -0.17 0.04 0.13
## [25] 0.05 0.02 -0.03 -0.03 0.02 0.13 0.13 0.16 -0.01 -0.10 -0.19 -0.16
## [37] -0.11 -0.02 0.05 0.20 0.09 0.02 0.04 -0.06 0.00 0.11 0.14 0.09
## [49] 0.11 0.11 0.14 -0.14 -0.14 -0.20 -0.27 -0.19 -0.18 -0.20 -0.29 -0.33
## [61] -0.34 -0.30 -0.25 -0.06 0.12 0.40 0.65 0.72 0.91 0.96 1.18 1.29
## [73] 1.37 1.25 1.04 0.94 0.70 0.43 0.10 -0.05 -0.52 -0.83 -1.08 -1.20
## [85] -1.33 -1.31 -1.18 -1.12 -1.00 -0.92 -0.76 -0.71 -0.44 -0.12 0.03 0.06
## [97] 0.10 0.28 0.29 0.34 0.40 0.42 0.47 0.71 0.85 0.76 0.60 1.00
## [109] 1.18 0.99 1.01 0.93 0.76 0.82 0.79 0.35 -0.20 -0.31
D1_positive = D1[D1 > 0]
fitwMGEAD = fitdist(data = D1_positive, distr = "weibull", method = "mge", gof = "AD" )
summary(fitwMGEAD)
## Fitting of the distribution ' weibull ' by maximum goodness-of-fit
## Parameters :
## estimate
## shape 0.9491740
## scale 0.4775768
## Loglikelihood: -14.89649 AIC: 33.79297 BIC: 38.23199
fitwMGEKS = fitdist(data = D1_positive, distr = "weibull", method = "mge", gof = "KS" )
summary(fitwMGEKS)
## Fitting of the distribution ' weibull ' by maximum goodness-of-fit
## Parameters :
## estimate
## shape 0.9092325
## scale 0.4803812
## Loglikelihood: -15.56588 AIC: 35.13177 BIC: 39.57078
fitwMGECvM = fitdist(data = D1_positive, distr = "weibull", method = "mge", gof = "CvM" )
summary(fitwMGECvM)
## Fitting of the distribution ' weibull ' by maximum goodness-of-fit
## Parameters :
## estimate
## shape 0.8976101
## scale 0.4866929
## Loglikelihood: -15.85954 AIC: 35.71907 BIC: 40.15809
set.seed(123)
weibull_data = rweibull(100, shape = 2, scale = 3)
fit_weibull = fitdist(weibull_data, "weibull")
# Histogram with density
hist(weibull_data, probability = TRUE, main = "Histogram of Weibull Data with Fit",
xlab = "Value", col = "blue", border = "white")
# Add Weibull fit curve
curve(dweibull(x, shape = fit_weibull$estimate["shape"], scale = fit_weibull$estimate["scale"]),
col = "red", lwd = 2, add = TRUE)
# P–P plot
ppcomp(list(fit_weibull), main = "P–P Plot for Weibull Fit")
