Examination
MOREEN MWANGI-BUS-244-025/2021
2025-05-25
library(fGarch)## Warning: package 'fGarch' was built under R version 4.4.3## 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(rugarch)## Warning: package 'rugarch' was built under R version 4.4.3## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:stats':
##
## sigmadata = read.csv("data.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 0.88d1 = 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 6.54d2 = 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 0.20ts.plot(d2)
ts.plot(d2, main = "Monthly Log Returns of STOCK (2016-2025)", ylab = "Log Returns")
FA1 = garchFit(formula = ~garch(1,2), data = d2, cond.dist = "snorm"); FA1##
## 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 0.01708317 secs
##
## --- END OF TRACE ---## Warning in sqrt(diag(fit$cvar)): NaNs produced##
## Time to Estimate Parameters:
## Time difference of 0.169044 secs##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = d2, cond.dist = "snorm")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000208ef00d1c8>
## [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:
## Mon May 26 22:37:50 2025 by user: usersummary(FA1)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = d2, cond.dist = "snorm")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000208ef00d1c8>
## [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:
## Mon May 26 22:37:50 2025 by user: user
##
##
## 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 0.2039940FA2 = garchFit(formula = ~garch(1,2), data = d2, cond.dist = "sged"); FA1##
## 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: sged
## 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 TRUE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1 beta2 skew shape
## 1 2 3 5 6 8 9
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 213.68376: 0.0719082 0.100000 0.100000 0.400000 0.400000 1.00000 4.00000
## 1: 200.06244: 0.0718654 0.134540 0.140391 0.430148 0.430664 0.992563 3.99534
## 2: 192.33629: 0.0717702 0.144817 0.173268 0.390317 0.389600 0.978545 3.98567
## 3: 190.03555: 0.0714745 0.169166 0.218462 0.395394 0.391824 0.937305 3.96651
## 4: 184.76687: 0.0710772 0.152310 0.226224 0.369224 0.360484 0.889237 3.94655
## 5: 182.29342: 0.0707325 0.173144 0.274999 0.368365 0.356799 0.850328 3.92688
## 6: 178.70717: 0.0701994 0.139821 0.286318 0.346441 0.329068 0.808050 3.90638
## 7: 176.30057: 0.0695120 0.142211 0.323833 0.356896 0.333574 0.757956 3.88173
## 8: 173.47083: 0.0687405 0.103473 0.341936 0.340971 0.308941 0.719735 3.85968
## 9: 171.20786: 0.0676017 0.0958126 0.386422 0.357560 0.312140 0.682194 3.83197
## 10: 169.59078: 0.0662598 0.0566461 0.413236 0.350153 0.284416 0.655960 3.80671
## 11: 167.39389: 0.0640990 0.0990788 0.416403 0.363961 0.260410 0.641755 3.77402
## 12: 165.54554: 0.0622314 0.0940374 0.433919 0.355701 0.211635 0.625766 3.74175
## 13: 163.45953: 0.0604648 0.0968479 0.473400 0.387876 0.195692 0.611484 3.70878
## 14: 162.02637: 0.0591630 0.0820251 0.495863 0.381985 0.146771 0.588441 3.68157
## 15: 159.91425: 0.0575610 0.111081 0.505231 0.416475 0.114658 0.577436 3.65026
## 16: 158.56292: 0.0568854 0.102066 0.537251 0.396412 0.0710994 0.549347 3.62864
## 17: 156.52930: 0.0555880 0.120865 0.559769 0.439950 0.0449740 0.542997 3.59787
## 18: 155.53283: 0.0546011 0.103352 0.578667 0.443496 1.00000e-08 0.524725 3.57142
## 19: 153.86862: 0.0525623 0.115664 0.598057 0.468855 1.00000e-08 0.519438 3.51899
## 20: 153.51581: 0.0514952 0.0967008 0.630800 0.437155 1.00000e-08 0.488976 3.48484
## 21: 152.09572: 0.0496647 0.123835 0.638436 0.426913 1.00000e-08 0.505643 3.43075
## 22: 151.30806: 0.0483636 0.112071 0.675247 0.414353 1.00000e-08 0.474803 3.38808
## 23: 150.47667: 0.0465411 0.126729 0.685926 0.404100 1.00000e-08 0.492471 3.33009
## 24: 147.87837: 0.0369779 0.125736 0.715727 0.297225 1.00000e-08 0.572771 3.01120
## 25: 139.97933: -0.000405331 0.0530910 0.738071 0.232390 1.00000e-08 0.849237 1.64980
## 26: 138.75942: -0.0665448 0.397418 0.501703 0.276136 1.00000e-08 0.646511 1.00000
## 27: 135.80787: -0.0677727 0.330670 0.590754 0.262543 1.00000e-08 0.574103 1.17880
## 28: 134.13980: -0.0680699 0.273506 0.598810 0.213618 1.00000e-08 0.597494 1.17552
## 29: 132.76728: -0.0692124 0.133271 0.787867 0.233650 1.00000e-08 0.517698 1.26387
## 30: 132.64053: -0.0693464 0.222659 0.818711 0.257270 1.00000e-08 0.579494 1.27844
## 31: 131.30126: -0.0696963 0.177293 0.820826 0.221640 1.00000e-08 0.584440 1.28028
## 32: 131.03384: -0.0707889 0.142906 0.851483 0.207439 1.00000e-08 0.590756 1.30836
## 33: 130.88656: -0.0729791 0.145092 0.885807 0.220199 1.00000e-08 0.603306 1.33923
## 34: 130.74579: -0.0769934 0.138208 0.889760 0.206781 1.00000e-08 0.600432 1.34320
## 35: 130.64066: -0.0808478 0.156036 0.897656 0.196886 1.00000e-08 0.597173 1.33855
## 36: 129.93968: -0.127139 0.144771 0.967635 0.201344 1.00000e-08 0.592556 1.40322
## 37: 129.44821: -0.173683 0.162905 0.955718 0.150997 1.00000e-08 0.567915 1.37021
## 38: 128.69774: -0.219913 0.166358 1.00000 0.153993 1.00000e-08 0.574711 1.30825
## 39: 128.35108: -0.320717 0.223340 1.00000 0.171702 1.00000e-08 0.520779 1.04351
## 40: 128.09259: -0.285665 0.198140 1.00000 0.160919 1.00000e-08 0.544229 1.16004
## 41: 128.05463: -0.267954 0.199448 1.00000 0.167311 1.00000e-08 0.559199 1.12191
## 42: 127.99689: -0.267565 0.200995 1.00000 0.173057 0.00511300 0.550515 1.12106
## 43: 127.99385: -0.268043 0.201528 1.00000 0.169279 1.00000e-08 0.551970 1.12141
## 44: 127.98047: -0.268039 0.203556 1.00000 0.172089 0.00165295 0.549360 1.12188
## 45: 127.96723: -0.268376 0.206708 1.00000 0.177959 1.00000e-08 0.547865 1.10337
## 46: 127.96430: -0.270345 0.208439 1.00000 0.179477 1.00000e-08 0.546598 1.10194
## 47: 127.96425: -0.270345 0.208460 1.00000 0.179510 1.00000e-08 0.546553 1.10195
## 48: 127.96424: -0.270345 0.208464 1.00000 0.179516 1.00000e-08 0.546547 1.10195
## 49: 127.96424: -0.270345 0.208467 1.00000 0.179520 1.00000e-08 0.546544 1.10195
## 50: 127.96424: -0.270344 0.208467 1.00000 0.179521 1.00000e-08 0.546548 1.10195
## 51: 127.96405: -0.269992 0.208686 1.00000 0.180610 1.00000e-08 0.546720 1.10065
## 52: 127.96404: -0.270342 0.208920 1.00000 0.180689 1.00000e-08 0.546686 1.10041
## 53: 127.96404: -0.270226 0.208898 1.00000 0.180597 1.00000e-08 0.546703 1.10060
## 54: 127.96404: -0.270211 0.208916 1.00000 0.180590 1.00000e-08 0.546699 1.10056
## 55: 127.96404: -0.270214 0.208935 1.00000 0.180582 1.00000e-08 0.546695 1.10053
## 56: 127.96404: -0.270216 0.208941 1.00000 0.180580 1.00000e-08 0.546693 1.10052
## 57: 127.96404: -0.270216 0.208936 1.00000 0.180581 1.00000e-08 0.546695 1.10053
##
## Final Estimate of the Negative LLH:
## LLH: -2.862507 norm LLH: -0.02425853
## mu omega alpha1 beta1 beta2 skew
## -0.08916807 0.02275146 0.99999999 0.18058116 0.00000001 0.54669455
## shape
## 1.10053358
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1 beta2 skew
## mu -18337.894 -25039.292 -1415.6779 -2258.6687 -2519.8732 4403.9370
## omega -25039.292 -40094.459 -2253.3084 -3186.2919 -3620.3201 4332.7413
## alpha1 -1415.678 -2253.308 -354.7735 -405.4328 -448.6927 499.6435
## beta1 -2258.669 -3186.292 -405.4328 -680.1415 -857.0798 720.9675
## beta2 -2519.873 -3620.320 -448.6927 -857.0798 -1457.9312 847.4049
## skew 4403.937 4332.741 499.6435 720.9675 847.4049 -2549.9059
## shape -1242.802 -2179.034 -255.2144 -302.9894 -393.5820 431.4117
## shape
## mu -1242.8022
## omega -2179.0337
## alpha1 -255.2144
## beta1 -302.9894
## beta2 -393.5820
## skew 431.4117
## shape -266.9376
## attr(,"time")
## Time difference of 0.0459621 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.1860218 secs##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = d2, cond.dist = "snorm")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000208ef00d1c8>
## [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:
## Mon May 26 22:37:50 2025 by user: usersummary(FA2)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = d2, cond.dist = "sged")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000208f0670a88>
## [data = d2]
##
## Conditional Distribution:
## sged
##
## Coefficient(s):
## mu omega alpha1 beta1 beta2 skew
## -0.08916807 0.02275146 0.99999999 0.18058116 0.00000001 0.54669455
## shape
## 1.10053358
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -8.917e-02 4.017e-02 -2.220 0.0264 *
## omega 2.275e-02 2.559e-02 0.889 0.3740
## alpha1 1.000e+00 1.335e-01 7.491 6.82e-14 ***
## beta1 1.806e-01 1.238e-01 1.459 0.1447
## beta2 1.000e-08 5.836e-02 0.000 1.0000
## skew 5.467e-01 4.541e-02 12.040 < 2e-16 ***
## shape 1.101e+00 1.581e-01 6.962 3.37e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 2.862507 normalized: 0.02425853
##
## Description:
## Mon May 26 22:37:50 2025 by user: user
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 65.8660178 4.996004e-15
## Shapiro-Wilk Test R W 0.9049246 4.257435e-07
## Ljung-Box Test R Q(10) 19.5518912 3.378755e-02
## Ljung-Box Test R Q(15) 26.7102125 3.118807e-02
## Ljung-Box Test R Q(20) 33.9609046 2.639037e-02
## Ljung-Box Test R^2 Q(10) 4.4190899 9.264674e-01
## Ljung-Box Test R^2 Q(15) 6.9958760 9.577641e-01
## Ljung-Box Test R^2 Q(20) 11.5740517 9.299572e-01
## LM Arch Test R TR^2 7.2803815 8.385422e-01
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 0.07012700 0.23448965 0.06360023 0.13686308FA3 = garchFit(formula = ~garch(1,2), data = d2, cond.dist = "std"); FA1##
## 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: std
## 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 FALSE
## shape 1.00000000 10.0000000 4.00000000 TRUE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1 beta2 shape
## 1 2 3 5 6 9
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 147.45016: 0.0719082 0.100000 0.100000 0.400000 0.400000 4.00000
## 1: 146.71427: 0.0718894 0.0893871 0.0948900 0.387497 0.387457 3.99967
## 2: 139.42908: 0.0458939 0.233096 1.00000 1.00000e-08 1.00000e-08 3.97586
## 3: 137.87767: 0.0541041 0.283005 1.00000 0.0306658 0.0680911 3.46048
## 4: 137.62666: 0.0455829 0.240842 1.00000 0.0636565 0.0672508 3.97991
## 5: 137.13516: 0.0307210 0.103671 1.00000 0.223152 0.000331189 4.41614
## 6: 137.01132: 0.0226453 0.182205 1.00000 0.160134 1.00000e-08 4.23722
## 7: 136.92987: 0.0228678 0.155016 1.00000 0.229540 1.00000e-08 4.15202
## 8: 136.89040: 0.0226057 0.156048 0.995538 0.201667 1.00000e-08 4.19149
## 9: 136.88596: 0.0226284 0.151838 0.995681 0.209764 1.00000e-08 4.18694
## 10: 136.88426: 0.0225073 0.151525 0.993537 0.213050 1.00000e-08 4.19620
## 11: 136.88352: 0.0223897 0.148215 0.991522 0.214552 1.00000e-08 4.19795
## 12: 136.87694: 0.0206467 0.150141 0.982458 0.209955 1.00000e-08 4.23833
## 13: 136.85339: 0.0128715 0.146758 0.927438 0.222137 1.00000e-08 4.48504
## 14: 136.84368: 0.0165704 0.141028 0.931850 0.217525 1.00000e-08 4.64811
## 15: 136.84215: 0.0188297 0.141285 0.916402 0.221270 1.00000e-08 4.67465
## 16: 136.84211: 0.0191467 0.141099 0.919151 0.221736 1.00000e-08 4.67473
## 17: 136.84209: 0.0190217 0.141023 0.918139 0.221726 1.00000e-08 4.67718
## 18: 136.84209: 0.0190140 0.141038 0.918108 0.221713 1.00000e-08 4.67739
## 19: 136.84209: 0.0190109 0.141037 0.918099 0.221716 1.00000e-08 4.67750
##
## Final Estimate of the Negative LLH:
## LLH: 6.015549 norm LLH: 0.05097923
## mu omega alpha1 beta1 beta2 shape
## 0.006273365 0.015357775 0.918099153 0.221715770 0.000000010 4.677500167
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1 beta2
## mu -1531.764781 -235.29583 -1.772935 -22.652570 -121.418104
## omega -235.295830 -32108.91276 -478.579950 -1572.650273 -1700.738776
## alpha1 -1.772935 -478.57995 -18.619276 -38.718023 -52.675496
## beta1 -22.652570 -1572.65027 -38.718023 -171.630280 -205.687277
## beta2 -121.418104 -1700.73878 -52.675496 -205.687277 -236.772636
## shape -3.013536 -86.22731 -2.224710 -4.526777 -6.763704
## shape
## mu -3.0135360
## omega -86.2273133
## alpha1 -2.2247104
## beta1 -4.5267765
## beta2 -6.7637038
## shape -0.6219538
## attr(,"time")
## Time difference of 0.02115512 secs
##
## --- END OF TRACE ---## Warning in sqrt(diag(fit$cvar)): NaNs produced##
## Time to Estimate Parameters:
## Time difference of 0.06096601 secs##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = d2, cond.dist = "snorm")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000208ef00d1c8>
## [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:
## Mon May 26 22:37:50 2025 by user: usersummary(FA3)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = d2, cond.dist = "std")
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000208f349af60>
## [data = d2]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu omega alpha1 beta1 beta2 shape
## 0.00627336 0.01535777 0.91809915 0.22171577 0.00000001 4.67750017
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 6.273e-03 2.318e-02 0.271 0.78668
## omega 1.536e-02 7.430e-03 2.067 0.03874 *
## alpha1 9.181e-01 3.638e-01 2.524 0.01161 *
## beta1 2.217e-01 NaN NaN NaN
## beta2 1.000e-08 NaN NaN NaN
## shape 4.678e+00 1.743e+00 2.683 0.00729 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -6.015549 normalized: -0.05097923
##
## Description:
## Mon May 26 22:37:51 2025 by user: user
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 110.4985376 0.000000e+00
## Shapiro-Wilk Test R W 0.9186264 2.382849e-06
## Ljung-Box Test R Q(10) 21.3196537 1.897163e-02
## Ljung-Box Test R Q(15) 29.2166590 1.508164e-02
## Ljung-Box Test R Q(20) 35.2636248 1.874434e-02
## Ljung-Box Test R^2 Q(10) 1.7138937 9.980999e-01
## Ljung-Box Test R^2 Q(15) 2.7336826 9.997753e-01
## Ljung-Box Test R^2 Q(20) 4.1088667 9.999421e-01
## LM Arch Test R TR^2 5.3489252 9.452982e-01
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 0.2036534 0.3445356 0.1988083 0.2608557return_stats =(d2)
skewness = PerformanceAnalytics::skewness(d2)
kurtosis = PerformanceAnalytics::kurtosis(d2)
View(return_stats)Forecast using best model
spec = ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1,2)),
mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
distribution.model = "std");spec##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : gjrGARCH(1,2)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(0,0,0)
## Include Mean : FALSE
## GARCH-in-Mean : FALSE
##
## Conditional Distribution
## ------------------------------------
## Distribution : std
## Includes Skew : FALSE
## Includes Shape : TRUE
## Includes Lambda : FALSEfit=ugarchfit(spec = spec, data=d2)
forecast = ugarchforecast(fit, n.ahead = 60, conf.int = 0.99)
plot(forecast, which=1)
#QUESTION FOUR
library(survival)
d1=read.csv("data.csv");d1## 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 0.88library(fitdistrplus)## Warning: package 'fitdistrplus' was built under R version 4.4.3## Loading required package: MASS##
## Attaching package: 'fitdistrplus'## The following object is masked from 'package:rugarch':
##
## fitdistclaims=d1$CLAIMS;claims## [1] 4.59 4.53 4.53 4.63 4.95 5.20 5.61 6.24 6.89 7.40 7.79 7.98
## [13] 8.15 8.33 8.40 8.36 8.21 8.13 7.84 7.20 6.76 6.43 6.26 6.30
## [25] 6.43 6.48 6.50 6.47 6.44 6.46 6.59 6.72 6.88 6.87 6.77 6.58
## [37] 6.42 6.31 6.29 6.34 6.54 6.63 6.65 6.69 6.63 6.63 6.74 6.88
## [49] 6.97 7.08 7.19 7.33 7.19 7.05 6.85 6.58 6.39 6.21 6.01 5.72
## [61] 5.39 5.05 4.75 4.50 4.44 4.56 4.96 5.61 6.33 7.24 8.20 9.38
## [73] 10.67 12.04 13.29 14.33 15.27 15.97 16.40 16.50 16.45 15.93 15.10 14.02
## [85] 12.82 11.49 10.18 9.00 7.88 6.88 5.96 5.20 4.49 4.05 3.93 3.96
## [97] 4.02 4.12 4.40 4.69 5.03 5.43 5.85 6.32 7.03 7.88 8.64 9.24
## [109] 10.24 11.42 12.41 13.42 14.35 15.11 15.93 16.72 17.07 16.87 16.56fit <- fitdist(claims, "weibull")
summary(fit)## Fitting of the distribution ' weibull ' by maximum likelihood
## Parameters :
## estimate Std. Error
## shape 2.390864 0.1583517
## scale 9.177913 0.3745349
## Loglikelihood: -313.3678 AIC: 630.7356 BIC: 636.2939
## Correlation matrix:
## shape scale
## shape 1.0000000 0.3426565
## scale 0.3426565 1.0000000plot(fit)
#QUESTION THREE
library(survival)
d1=read.csv("data.csv");d1## 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 0.88install.packages("fitdistrplus")## Warning: package 'fitdistrplus' is in use and will not be installedlibrary(fitdistrplus)
pension=d1$PENSION;pension## [1] 13.50 13.59 13.49 13.47 13.70 13.73 13.71 13.83 13.85 13.76 13.66 13.65
## [13] 13.61 13.61 13.81 14.23 14.29 14.39 14.24 14.25 14.45 15.32 15.49 17.28
## [25] 18.47 22.51 23.03 23.04 23.43 22.36 23.27 20.84 19.98 18.61 17.76 17.42
## [37] 16.69 16.67 16.76 15.11 14.05 13.05 12.20 12.08 11.30 10.83 9.81 10.13
## [49] 10.32 11.90 11.80 11.97 12.46 12.77 13.33 13.62 13.89 14.23 14.19 14.17
## [61] 14.05 13.90 14.28 14.19 14.29 14.65 14.54 14.47 14.33 14.24 14.95 13.89
## [73] 14.11 14.06 14.53 15.21 15.88 16.02 15.94 15.88 15.89 18.37 17.85 16.87
## [85] 16.67 16.35 15.96 15.39 14.67 12.99 11.25 9.66 8.93 8.18 7.55 7.83
## [97] 7.65 7.79 8.44 9.10 9.48 9.74 10.38 9.17 8.01 7.61 7.21 7.01
## [109] 6.67 6.26 6.22 6.22 6.20 6.22 6.54 6.55 6.92 6.58 6.44fit <- fitdist(pension, "gamma")
summary(fit)## Fitting of the distribution ' gamma ' by maximum likelihood
## Parameters :
## estimate Std. Error
## shape 9.8271656 1.25294192
## rate 0.7386947 0.09662708
## Loglikelihood: -336.7268 AIC: 677.4537 BIC: 683.0119
## Correlation matrix:
## shape rate
## shape 1.0000000 0.9746921
## rate 0.9746921 1.0000000plot(fit)