PRACTICAL ACTUARIAL MODELLING - EXAM
KARIUKI GILBERT GIKWA = BUS-244-020/2021
2025-05-21
#QUESTION ONE
#install and load necessary packages
library(fGarch)## Warning: package 'fGarch' was built under R version 4.3.3## 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 4.3.3## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:stats':
##
## sigmalibrary(fitdistrplus)## Warning: package 'fitdistrplus' was built under R version 4.3.3## Loading required package: MASS## Loading required package: survival## Warning: package 'survival' was built under R version 4.3.3##
## Attaching package: 'fitdistrplus'## The following object is masked from 'package:rugarch':
##
## fitdistlibrary(timeSeries)## Warning: package 'timeSeries' was built under R version 4.3.3## Loading required package: timeDate## Warning: package 'timeDate' was built under R version 4.3.3##
## Attaching package: 'timeSeries'## The following object is masked from 'package:rugarch':
##
## quantile## The following objects are masked from 'package:graphics':
##
## lines, points#load the data
Data1= read.csv("DATA25.csv");Data1## 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 0.88Data2= Data1$STOCK;Data2## [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#Model Fitting
M1=ugarchspec(variance.model = list(model = "eGARCH", garchOrder =c(1,2)), mean.model = list(armaOrder = c(1,2), include.mean = FALSE), distribution.model = "norm");M1##
## *---------------------------------*
## * 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 : norm
## Includes Skew : FALSE
## Includes Shape : FALSE
## Includes Lambda : FALSE#Summary
summary(M1)## Length Class Mode
## 1 uGARCHspec S4#Log Returns
log_return = diff(Data2);log_return## [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 log returns
plot(log_return)
ts.plot(log_return)
#Remove the negatives
d= log_return^2;d## [1] 0.0049 0.0289 0.0529 0.0841 0.0961 0.1089 0.1296 0.0729 0.0400 1.7689
## [11] 3.4225 0.0576 0.0576 0.0841 0.0576 0.0289 0.0400 0.0529 0.0484 0.0400
## [21] 0.0256 0.0081 0.0001 0.0049 0.0144 0.0841 0.1225 0.0841 0.0784 0.0100
## [31] 0.4489 0.0400 0.0256 0.0256 0.0169 0.0121 0.0576 0.0036 0.0225 0.2116
## [41] 0.2304 0.0009 0.5184 0.1296 0.0324 0.0324 0.0225 0.2916 0.2601 0.0001
## [51] 0.3364 0.1089 0.0144 0.0196 0.0144 0.0064 0.0196 0.0225 0.0004 0.0225
## [61] 0.0361 0.0144 0.1849 0.0081 0.0001 0.0000 0.0025 0.0256 0.0064 0.0256
## [71] 0.0144 0.0169 0.0000 0.0576 0.0004 0.0009 0.0001 0.0100 0.0036 0.0000
## [81] 0.0100 0.1024 0.0625 0.1681 0.3969 0.4225 0.2601 0.1521 0.0361 0.0289
## [91] 0.0324 0.0049 0.0016 0.0225 0.0064 0.0841 0.4096 0.1936 0.1089 0.0289
## [101] 0.0016 0.0169 0.0025 0.0004 0.0009 0.0009 0.0004 0.0169 0.0169 0.0256
## [111] 0.0001 0.0100 0.0361 0.0256 0.0121 0.0004 0.0025 0.0400plot(d)
ts.plot(d)
#Model fitting
M1fit= ugarchfit(data = log_return, spec = M1);M1fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,2)
## Mean Model : ARFIMA(1,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## ar1 0.311434 0.042572 7.3154 0.000000
## ma1 0.010609 0.018537 0.5723 0.567121
## ma2 0.073935 0.012835 5.7607 0.000000
## omega -0.527684 0.313081 -1.6855 0.091901
## alpha1 0.299353 0.160045 1.8704 0.061424
## beta1 0.425412 0.122959 3.4598 0.000541
## beta2 0.338834 0.125646 2.6967 0.007002
## gamma1 1.196180 0.261571 4.5731 0.000005
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## ar1 0.311434 0.044898 6.93644 0.000000
## ma1 0.010609 0.001744 6.08321 0.000000
## ma2 0.073935 0.001853 39.89242 0.000000
## omega -0.527684 0.561063 -0.94051 0.346957
## alpha1 0.299353 0.231491 1.29315 0.195959
## beta1 0.425412 0.116389 3.65509 0.000257
## beta2 0.338834 0.135398 2.50250 0.012332
## gamma1 1.196180 0.301857 3.96273 0.000074
##
## LogLikelihood : -0.1537826
##
## Information Criteria
## ------------------------------------
##
## Akaike 0.13820
## Bayes 0.32604
## Shibata 0.12976
## Hannan-Quinn 0.21447
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2713 0.6025
## Lag[2*(p+q)+(p+q)-1][8] 4.5994 0.4182
## Lag[4*(p+q)+(p+q)-1][14] 8.5612 0.2816
## d.o.f=3
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.004073 0.9491
## Lag[2*(p+q)+(p+q)-1][8] 0.457946 0.9971
## Lag[4*(p+q)+(p+q)-1][14] 0.884533 0.9998
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.1196 0.500 2.000 0.7295
## ARCH Lag[6] 0.2442 1.461 1.711 0.9587
## ARCH Lag[8] 0.4800 2.368 1.583 0.9834
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.06
## Individual Statistics:
## ar1 0.3068
## ma1 0.4813
## ma2 0.1811
## omega 0.2880
## alpha1 0.1142
## beta1 0.3859
## beta2 0.3969
## gamma1 0.2344
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.1323865 0.8949
## Negative Sign Bias 0.0001613 0.9999
## Positive Sign Bias 0.5647460 0.5734
## Joint Effect 0.3264686 0.9550
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 32.17 0.02991
## 2 30 41.15 0.06678
## 3 40 55.56 0.04147
## 4 50 68.62 0.03351
##
##
## Elapsed time : 0.374718#Plot for M1
plot(M1fit, which=7)
plot(M1fit, which=9)
plot(M1fit, which=12)
#Summary for M1
summary(M1fit)## Length Class Mode
## 1 uGARCHfit S4#QUESTION TWO
library(survival)
# Load the AML dataset
data(aml)## Warning in data(aml): data set 'aml' not found# View the first few rows of the dataset
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# Fit the Kaplan-Meier survival model
km_fit <- survfit(Surv(time, status) ~ x, data = aml)
# Print summary of Kaplan-Meier model
summary(km_fit)## Call: survfit(formula = Surv(time, status) ~ 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 Kaplan-Meier survival curves
plot(km_fit, col = c("blue", "red"), lwd = 2, xlab = "Time", ylab = "Survival Probability",
main = "Kaplan-Meier Survival Curves for AML Patients")
legend("topright", legend = c("Maintained", "Non-maintained"), col = c("blue", "red"), lwd = 2)
# Log-Rank Test for statistical comparison
survdiff(Surv(time, status) ~ x, data = aml)## Call:
## survdiff(formula = Surv(time, status) ~ 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 THREE
library(fGarch)
library(rugarch)
library(fitdistrplus)
library(timeSeries)
#load the data
E1= read.csv("DATA25.csv");E1## 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 0.88E2= E1$PENSION;E2## [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 6.44E3= diff(E2);E3## [1] 0.09 -0.10 -0.02 0.23 0.03 -0.02 0.12 0.02 -0.09 -0.10 -0.01 -0.04
## [13] 0.00 0.20 0.42 0.06 0.10 -0.15 0.01 0.20 0.87 0.17 1.79 1.19
## [25] 4.04 0.52 0.01 0.39 -1.07 0.91 -2.43 -0.86 -1.37 -0.85 -0.34 -0.73
## [37] -0.02 0.09 -1.65 -1.06 -1.00 -0.85 -0.12 -0.78 -0.47 -1.02 0.32 0.19
## [49] 1.58 -0.10 0.17 0.49 0.31 0.56 0.29 0.27 0.34 -0.04 -0.02 -0.12
## [61] -0.15 0.38 -0.09 0.10 0.36 -0.11 -0.07 -0.14 -0.09 0.71 -1.06 0.22
## [73] -0.05 0.47 0.68 0.67 0.14 -0.08 -0.06 0.01 2.48 -0.52 -0.98 -0.20
## [85] -0.32 -0.39 -0.57 -0.72 -1.68 -1.74 -1.59 -0.73 -0.75 -0.63 0.28 -0.18
## [97] 0.14 0.65 0.66 0.38 0.26 0.64 -1.21 -1.16 -0.40 -0.40 -0.20 -0.34
## [109] -0.41 -0.04 0.00 -0.02 0.02 0.32 0.01 0.37 -0.34 -0.14#Maximum likelihood estimator
E3_positive = E3[E3 > 0]
fit_ln= fitdist(data = E3_positive, distr = "lnorm", method = c("mle"))
summary(fit_ln)## Fitting of the distribution ' lnorm ' by maximum likelihood
## Parameters :
## estimate Std. Error
## meanlog -1.447937 0.1898052
## sdlog 1.381802 0.1342122
## Loglikelihood: -15.60268 AIC: 35.20536 BIC: 39.14594
## Correlation matrix:
## meanlog sdlog
## meanlog 1.000000e+00 -4.684208e-11
## sdlog -4.684208e-11 1.000000e+00plot(fit_ln)
E3_positive= E3[E3 > 0]
fit_gm= fitdist(data = E3_positive, distr = "gamma", method = c("mle"))
summary(fit_gm)## Fitting of the distribution ' gamma ' by maximum likelihood
## Parameters :
## estimate Std. Error
## shape 0.8084506 0.1356321
## rate 1.6523797 0.3749279
## Loglikelihood: -14.25895 AIC: 32.51789 BIC: 36.45847
## Correlation matrix:
## shape rate
## shape 1.0000000 0.7393854
## rate 0.7393854 1.0000000plot(fit_gm)
#moment matching estimation
fitln_mme = fitdist(data = E3_positive, distr = "lnorm", method = c("mme"))
summary(fitln_mme)## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate Std. Error
## meanlog -1.248413 0.4414005
## sdlog 1.032978 0.4616989
## Loglikelihood: -22.09074 AIC: 48.18148 BIC: 52.12206
## Correlation matrix:
## meanlog sdlog
## meanlog 1.0000000 -0.9175477
## sdlog -0.9175477 1.0000000plot(fitln_mme)
E3_positive= E3[E3 > 0]
fitgm_mme = fitdist(data = E3_positive, distr = "gamma", method = c("mme"))
summary(fitgm_mme)## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate Std. Error
## shape 0.5244463 0.1736935
## rate 1.0719496 0.4091231
## Loglikelihood: -17.2375 AIC: 38.47499 BIC: 42.41558
## Correlation matrix:
## shape rate
## shape 1.0000000 0.8677666
## rate 0.8677666 1.0000000plot(fitgm_mme)
#QUESTION FOUR
library(goftest)
library(fGarch)
library(fitdistrplus)
Data_F= read.csv("DATA25.csv");Data_F## 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 0.88f1= Data_F$CPI;f1## [1] 6.12 6.33 6.47 6.48 6.51 6.56 6.57 6.64 6.66 6.68 6.68 6.81
## [13] 7.06 7.16 7.19 7.19 7.20 7.25 7.23 7.25 7.27 7.33 7.36 7.46
## [25] 7.62 7.72 7.75 7.77 7.78 7.86 7.87 7.88 7.95 8.02 8.06 8.21
## [37] 8.41 8.43 8.60 8.66 8.69 8.83 8.84 8.96 9.13 9.16 9.19 9.28
## [49] 9.57 9.60 9.76 9.85 9.92 10.06 10.16 10.11 10.23 10.28 10.34 10.54
## [61] 10.92 10.96 11.17 11.32 11.28 11.43 11.46 11.49 11.77 12.18 12.47 12.79
## [73] 12.73 13.07 13.31 13.32 13.67 13.79 13.96 14.01 14.38 14.40 14.48 14.56
## [85] 14.83 14.97 16.27 16.40 16.94 18.68 18.46 18.54 18.85 18.62 18.92 19.46
## [97] 19.59 21.16 21.83 23.31 24.16 25.99 26.47 27.38 29.02 29.32 29.63 30.11
## [109] 31.64 32.65 33.54 34.44 34.09 33.46 33.70 33.26 32.75 32.92 32.12f2= diff(f1);f2## [1] 0.21 0.14 0.01 0.03 0.05 0.01 0.07 0.02 0.02 0.00 0.13 0.25
## [13] 0.10 0.03 0.00 0.01 0.05 -0.02 0.02 0.02 0.06 0.03 0.10 0.16
## [25] 0.10 0.03 0.02 0.01 0.08 0.01 0.01 0.07 0.07 0.04 0.15 0.20
## [37] 0.02 0.17 0.06 0.03 0.14 0.01 0.12 0.17 0.03 0.03 0.09 0.29
## [49] 0.03 0.16 0.09 0.07 0.14 0.10 -0.05 0.12 0.05 0.06 0.20 0.38
## [61] 0.04 0.21 0.15 -0.04 0.15 0.03 0.03 0.28 0.41 0.29 0.32 -0.06
## [73] 0.34 0.24 0.01 0.35 0.12 0.17 0.05 0.37 0.02 0.08 0.08 0.27
## [85] 0.14 1.30 0.13 0.54 1.74 -0.22 0.08 0.31 -0.23 0.30 0.54 0.13
## [97] 1.57 0.67 1.48 0.85 1.83 0.48 0.91 1.64 0.30 0.31 0.48 1.53
## [109] 1.01 0.89 0.90 -0.35 -0.63 0.24 -0.44 -0.51 0.17 -0.80#Fit the distribution
f2_positive= f2[f2 > 0]
fitlnMGEKS = fitdist(data = f2_positive, distr = "weibull", method = "mge", gof = "KS")
summary(fitlnMGEKS)## Fitting of the distribution ' weibull ' by maximum goodness-of-fit
## Parameters :
## estimate
## shape 0.8401815
## scale 0.2120730
## Loglikelihood: 34.87007 AIC: -65.74015 BIC: -60.43222fitlnMGCvM = fitdist(data = f2_positive, distr = "weibull", method = "mge", gof = "CvM")
summary(fitlnMGCvM)## Fitting of the distribution ' weibull ' by maximum goodness-of-fit
## Parameters :
## estimate
## shape 0.8263981
## scale 0.1991286
## Loglikelihood: 34.45488 AIC: -64.90977 BIC: -59.60185fitlnMGEAD = fitdist(data = f2_positive, distr = "weibull", method = "mge", gof = "AD")
summary(fitlnMGEAD)## Fitting of the distribution ' weibull ' by maximum goodness-of-fit
## Parameters :
## estimate
## shape 0.7634249
## scale 0.2174130
## Loglikelihood: 36.49133 AIC: -68.98267 BIC: -63.67475