SWB
Wafula
2025-05-18
#eGarch
#import data
D1=read.csv("Inflation rates.csv");D1## Year Month Annual Monthly
## 1 2025 April 3.74 4.11
## 2 2025 March 3.81 3.62
## 3 2025 February 3.98 3.45
## 4 2025 January 4.21 3.28
## 5 2024 December 4.50 2.99
## 6 2024 November 4.81 2.75
## 7 2024 October 5.14 2.72
## 8 2024 September 5.50 3.56
## 9 2024 August 5.77 4.36
## 10 2024 July 5.97 4.31
## 11 2024 June 4.64 6.22
## 12 2024 May 6.49 5.10
## 13 2024 April 6.73 5.00
## 14 2024 March 6.97 5.70
## 15 2024 February 7.26 6.31
## 16 2024 January 7.50 6.85
## 17 2023 December 7.67 6.63
## 18 2023 November 7.87 6.80
## 19 2023 October 8.10 6.92
## 20 2023 September 8.32 6.78
## 21 2023 August 8.52 6.73
## 22 2023 July 8.68 7.28
## 23 2023 June 8.77 7.88
## 24 2023 May 8.78 8.03
## 25 2023 April 8.71 7.90
## 26 2023 March 8.59 9.19
## 27 2023 February 8.30 9.23
## 28 2023 January 7.95 8.98
## 29 2022 December 7.66 9.06
## 30 2022 November 7.38 9.48
## 31 2022 October 7.48 9.59
## 32 2022 September 6.81 9.18
## 33 2022 August 6.61 8.53
## 34 2022 July 6.45 8.32
## 35 2022 June 6.29 7.91
## 36 2022 May 6.16 7.08
## 37 2022 April 6.05 6.47
## 38 2022 March 6.29 5.56
## 39 2022 February 6.23 5.08
## 40 2022 January 6.08 5.39
## 41 2021 December 5.62 5.73
## 42 2021 November 6.10 5.80
## 43 2021 October 6.07 6.45
## 44 2021 September 5.35 6.91
## 45 2021 August 5.71 6.57
## 46 2021 July 5.53 6.55
## 47 2021 June 5.35 6.32
## 48 2021 May 5.20 5.87
## 49 2021 April 4.66 5.76
## 50 2021 March 5.17 5.90
## 51 2021 February 5.16 5.78
## 52 2021 January 5.74 5.69
## 53 2020 December 5.41 5.62
## 54 2020 November 5.53 5.33
## 55 2020 October 5.67 4.84
## 56 2020 September 5.79 4.20
## 57 2020 August 5.87 4.36
## 58 2020 July 6.01 4.36
## 59 2020 June 6.16 4.59
## 60 2020 May 6.18 5.33
## 61 2020 April 6.03 6.01
## 62 2020 March 5.84 5.84
## 63 2020 February 5.72 7.17
## 64 2020 January 5.29 5.78
## 65 2019 December 5.20 5.82
## 66 2019 November 5.19 5.56
## 67 2019 October 5.19 4.95
## 68 2019 September 5.24 3.83
## 69 2019 August 5.40 5.00
## 70 2019 July 5.32 6.27
## 71 2019 June 5.16 5.70
## 72 2019 May 5.04 4.49
## 73 2019 April 4.91 6.58
## 74 2019 April 4.91 6.58
## 75 2019 March 4.67 4.35
## 76 2019 February 4.65 4.14
## 77 2019 January 4.68 4.70
## 78 2018 December 4.69 5.71
## 79 2018 November 4.59 5.58
## 80 2018 October 4.53 5.53
## 81 2018 September 4.53 5.70
## 82 2018 August 4.63 4.04
## 83 2018 July 4.95 4.35
## 84 2018 June 5.20 4.28
## 85 2018 May 5.61 3.95
## 86 2018 April 6.24 3.73
## 87 2018 March 6.89 4.18
## 88 2018 February 7.40 4.46
## 89 2018 January 7.79 4.83
## 90 2017 December 7.98 4.50
## 91 2017 November 8.15 4.73
## 92 2017 October 8.33 5.72
## 93 2017 September 8.40 7.06
## 94 2017 August 8.36 8.04
## 95 2017 July 8.21 7.47
## 96 2017 June 8.13 9.21
## 97 2017 May 7.84 11.70
## 98 2017 April 7.20 11.48
## 99 2017 March 6.76 10.28
## 100 2017 February 6.43 9.04
## 101 2017 January 6.26 6.99
## 102 2016 December 6.30 6.35
## 103 2016 November 6.43 6.68
## 104 2016 October 6.48 6.47
## 105 2016 September 6.50 6.34
## 106 2016 August 6.47 6.26
## 107 2016 July 6.44 6.40
## 108 2016 June 6.46 5.80
## 109 2016 May 6.59 5.00
## 110 2016 April 6.72 5.27
## 111 2016 March 6.88 6.45
## 112 2016 February 6.87 6.84
## 113 2016 January 6.77 7.78
## 114 2015 December 6.58 8.01
## 115 2015 November 6.42 7.32
## 116 2015 October 6.31 6.72
## 117 2015 September 6.29 5.97
## 118 2015 August 6.34 5.84
## 119 2015 July 6.54 6.62D2=D1$Annual;D2## [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 6.54D3=diff(D2);D3## [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 0.20library(rugarch)## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:stats':
##
## sigmam1=ugarchspec(variance.model = list("eGarch",garchOrder=c(1,1)),mean.model = list(armaOrder=c(0,0),include.mean=F),distribution.model = "norm");m1## Warning: unidentified option(s) in variance.model:
## ##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : sGARCH(1,1)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(0,0,0)
## Include Mean : FALSE
## GARCH-in-Mean : FALSE
##
## Conditional Distribution
## ------------------------------------
## Distribution : norm
## Includes Skew : FALSE
## Includes Shape : FALSE
## Includes Lambda : FALSEsummary(m1)## Length Class Mode
## 1 uGARCHspec S4m1fit=ugarchfit(data=D3,spec=m1);m1fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.012176 0.007073 1.7214 0.085185
## alpha1 0.645848 0.200403 3.2228 0.001270
## beta1 0.353152 0.103362 3.4167 0.000634
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.012176 0.009043 1.3464 0.178161
## alpha1 0.645848 0.216870 2.9780 0.002901
## beta1 0.353152 0.113637 3.1077 0.001885
##
## LogLikelihood : -12.42652
##
## Information Criteria
## ------------------------------------
##
## Akaike 0.26147
## Bayes 0.33191
## Shibata 0.26022
## Hannan-Quinn 0.29007
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.624 0.0100597
## Lag[2*(p+q)+(p+q)-1][2] 10.653 0.0012849
## Lag[4*(p+q)+(p+q)-1][5] 15.333 0.0003617
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1804 0.6710
## Lag[2*(p+q)+(p+q)-1][5] 0.7792 0.9073
## Lag[4*(p+q)+(p+q)-1][9] 1.1479 0.9793
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.01557 0.500 2.000 0.9007
## ARCH Lag[5] 0.29971 1.440 1.667 0.9403
## ARCH Lag[7] 0.56999 2.315 1.543 0.9716
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.8506
## Individual Statistics:
## omega 0.5299
## alpha1 0.4818
## beta1 0.4534
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.2820 0.7785
## Negative Sign Bias 0.1199 0.9048
## Positive Sign Bias 0.4589 0.6472
## Joint Effect 0.2257 0.9733
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 22.00 0.2843
## 2 30 35.56 0.1868
## 3 40 46.07 0.2030
## 4 50 50.83 0.4015
##
##
## Elapsed time : 0.4813418plot(m1fit, which=9)
plot(m1fit, which=11)
plot(m1fit, which=7)
#forecasting
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zooforc=ugarchforecast(m1fit,n.ahead = 30);forc##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: sGARCH
## Horizon: 30
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=1970-04-29]:
## Series Sigma
## T+1 0 0.2145
## T+2 0 0.2411
## T+3 0 0.2651
## T+4 0 0.2870
## T+5 0 0.3073
## T+6 0 0.3264
## T+7 0 0.3444
## T+8 0 0.3615
## T+9 0 0.3778
## T+10 0 0.3934
## T+11 0 0.4084
## T+12 0 0.4228
## T+13 0 0.4368
## T+14 0 0.4503
## T+15 0 0.4634
## T+16 0 0.4761
## T+17 0 0.4885
## T+18 0 0.5006
## T+19 0 0.5123
## T+20 0 0.5238
## T+21 0 0.5351
## T+22 0 0.5461
## T+23 0 0.5569
## T+24 0 0.5674
## T+25 0 0.5778
## T+26 0 0.5879
## T+27 0 0.5979
## T+28 0 0.6077
## T+29 0 0.6173
## T+30 0 0.6268plot(forc, which=1)
plot(forc, which=3)
#fGarch
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(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, pointsimport=read.csv("Inflation rates.csv");import## Year Month Annual Monthly
## 1 2025 April 3.74 4.11
## 2 2025 March 3.81 3.62
## 3 2025 February 3.98 3.45
## 4 2025 January 4.21 3.28
## 5 2024 December 4.50 2.99
## 6 2024 November 4.81 2.75
## 7 2024 October 5.14 2.72
## 8 2024 September 5.50 3.56
## 9 2024 August 5.77 4.36
## 10 2024 July 5.97 4.31
## 11 2024 June 4.64 6.22
## 12 2024 May 6.49 5.10
## 13 2024 April 6.73 5.00
## 14 2024 March 6.97 5.70
## 15 2024 February 7.26 6.31
## 16 2024 January 7.50 6.85
## 17 2023 December 7.67 6.63
## 18 2023 November 7.87 6.80
## 19 2023 October 8.10 6.92
## 20 2023 September 8.32 6.78
## 21 2023 August 8.52 6.73
## 22 2023 July 8.68 7.28
## 23 2023 June 8.77 7.88
## 24 2023 May 8.78 8.03
## 25 2023 April 8.71 7.90
## 26 2023 March 8.59 9.19
## 27 2023 February 8.30 9.23
## 28 2023 January 7.95 8.98
## 29 2022 December 7.66 9.06
## 30 2022 November 7.38 9.48
## 31 2022 October 7.48 9.59
## 32 2022 September 6.81 9.18
## 33 2022 August 6.61 8.53
## 34 2022 July 6.45 8.32
## 35 2022 June 6.29 7.91
## 36 2022 May 6.16 7.08
## 37 2022 April 6.05 6.47
## 38 2022 March 6.29 5.56
## 39 2022 February 6.23 5.08
## 40 2022 January 6.08 5.39
## 41 2021 December 5.62 5.73
## 42 2021 November 6.10 5.80
## 43 2021 October 6.07 6.45
## 44 2021 September 5.35 6.91
## 45 2021 August 5.71 6.57
## 46 2021 July 5.53 6.55
## 47 2021 June 5.35 6.32
## 48 2021 May 5.20 5.87
## 49 2021 April 4.66 5.76
## 50 2021 March 5.17 5.90
## 51 2021 February 5.16 5.78
## 52 2021 January 5.74 5.69
## 53 2020 December 5.41 5.62
## 54 2020 November 5.53 5.33
## 55 2020 October 5.67 4.84
## 56 2020 September 5.79 4.20
## 57 2020 August 5.87 4.36
## 58 2020 July 6.01 4.36
## 59 2020 June 6.16 4.59
## 60 2020 May 6.18 5.33
## 61 2020 April 6.03 6.01
## 62 2020 March 5.84 5.84
## 63 2020 February 5.72 7.17
## 64 2020 January 5.29 5.78
## 65 2019 December 5.20 5.82
## 66 2019 November 5.19 5.56
## 67 2019 October 5.19 4.95
## 68 2019 September 5.24 3.83
## 69 2019 August 5.40 5.00
## 70 2019 July 5.32 6.27
## 71 2019 June 5.16 5.70
## 72 2019 May 5.04 4.49
## 73 2019 April 4.91 6.58
## 74 2019 April 4.91 6.58
## 75 2019 March 4.67 4.35
## 76 2019 February 4.65 4.14
## 77 2019 January 4.68 4.70
## 78 2018 December 4.69 5.71
## 79 2018 November 4.59 5.58
## 80 2018 October 4.53 5.53
## 81 2018 September 4.53 5.70
## 82 2018 August 4.63 4.04
## 83 2018 July 4.95 4.35
## 84 2018 June 5.20 4.28
## 85 2018 May 5.61 3.95
## 86 2018 April 6.24 3.73
## 87 2018 March 6.89 4.18
## 88 2018 February 7.40 4.46
## 89 2018 January 7.79 4.83
## 90 2017 December 7.98 4.50
## 91 2017 November 8.15 4.73
## 92 2017 October 8.33 5.72
## 93 2017 September 8.40 7.06
## 94 2017 August 8.36 8.04
## 95 2017 July 8.21 7.47
## 96 2017 June 8.13 9.21
## 97 2017 May 7.84 11.70
## 98 2017 April 7.20 11.48
## 99 2017 March 6.76 10.28
## 100 2017 February 6.43 9.04
## 101 2017 January 6.26 6.99
## 102 2016 December 6.30 6.35
## 103 2016 November 6.43 6.68
## 104 2016 October 6.48 6.47
## 105 2016 September 6.50 6.34
## 106 2016 August 6.47 6.26
## 107 2016 July 6.44 6.40
## 108 2016 June 6.46 5.80
## 109 2016 May 6.59 5.00
## 110 2016 April 6.72 5.27
## 111 2016 March 6.88 6.45
## 112 2016 February 6.87 6.84
## 113 2016 January 6.77 7.78
## 114 2015 December 6.58 8.01
## 115 2015 November 6.42 7.32
## 116 2015 October 6.31 6.72
## 117 2015 September 6.29 5.97
## 118 2015 August 6.34 5.84
## 119 2015 July 6.54 6.62l=import$Monthly;l## [1] 4.11 3.62 3.45 3.28 2.99 2.75 2.72 3.56 4.36 4.31 6.22 5.10
## [13] 5.00 5.70 6.31 6.85 6.63 6.80 6.92 6.78 6.73 7.28 7.88 8.03
## [25] 7.90 9.19 9.23 8.98 9.06 9.48 9.59 9.18 8.53 8.32 7.91 7.08
## [37] 6.47 5.56 5.08 5.39 5.73 5.80 6.45 6.91 6.57 6.55 6.32 5.87
## [49] 5.76 5.90 5.78 5.69 5.62 5.33 4.84 4.20 4.36 4.36 4.59 5.33
## [61] 6.01 5.84 7.17 5.78 5.82 5.56 4.95 3.83 5.00 6.27 5.70 4.49
## [73] 6.58 6.58 4.35 4.14 4.70 5.71 5.58 5.53 5.70 4.04 4.35 4.28
## [85] 3.95 3.73 4.18 4.46 4.83 4.50 4.73 5.72 7.06 8.04 7.47 9.21
## [97] 11.70 11.48 10.28 9.04 6.99 6.35 6.68 6.47 6.34 6.26 6.40 5.80
## [109] 5.00 5.27 6.45 6.84 7.78 8.01 7.32 6.72 5.97 5.84 6.62plot(l)
ts.plot(l)
log.l=diff(l);log.l## [1] -0.49 -0.17 -0.17 -0.29 -0.24 -0.03 0.84 0.80 -0.05 1.91 -1.12 -0.10
## [13] 0.70 0.61 0.54 -0.22 0.17 0.12 -0.14 -0.05 0.55 0.60 0.15 -0.13
## [25] 1.29 0.04 -0.25 0.08 0.42 0.11 -0.41 -0.65 -0.21 -0.41 -0.83 -0.61
## [37] -0.91 -0.48 0.31 0.34 0.07 0.65 0.46 -0.34 -0.02 -0.23 -0.45 -0.11
## [49] 0.14 -0.12 -0.09 -0.07 -0.29 -0.49 -0.64 0.16 0.00 0.23 0.74 0.68
## [61] -0.17 1.33 -1.39 0.04 -0.26 -0.61 -1.12 1.17 1.27 -0.57 -1.21 2.09
## [73] 0.00 -2.23 -0.21 0.56 1.01 -0.13 -0.05 0.17 -1.66 0.31 -0.07 -0.33
## [85] -0.22 0.45 0.28 0.37 -0.33 0.23 0.99 1.34 0.98 -0.57 1.74 2.49
## [97] -0.22 -1.20 -1.24 -2.05 -0.64 0.33 -0.21 -0.13 -0.08 0.14 -0.60 -0.80
## [109] 0.27 1.18 0.39 0.94 0.23 -0.69 -0.60 -0.75 -0.13 0.78plot(log.l)
ts.plot(log.l)
acf(log.l)
pacf(log.l)
s=log.l^2;s## [1] 0.2401 0.0289 0.0289 0.0841 0.0576 0.0009 0.7056 0.6400 0.0025 3.6481
## [11] 1.2544 0.0100 0.4900 0.3721 0.2916 0.0484 0.0289 0.0144 0.0196 0.0025
## [21] 0.3025 0.3600 0.0225 0.0169 1.6641 0.0016 0.0625 0.0064 0.1764 0.0121
## [31] 0.1681 0.4225 0.0441 0.1681 0.6889 0.3721 0.8281 0.2304 0.0961 0.1156
## [41] 0.0049 0.4225 0.2116 0.1156 0.0004 0.0529 0.2025 0.0121 0.0196 0.0144
## [51] 0.0081 0.0049 0.0841 0.2401 0.4096 0.0256 0.0000 0.0529 0.5476 0.4624
## [61] 0.0289 1.7689 1.9321 0.0016 0.0676 0.3721 1.2544 1.3689 1.6129 0.3249
## [71] 1.4641 4.3681 0.0000 4.9729 0.0441 0.3136 1.0201 0.0169 0.0025 0.0289
## [81] 2.7556 0.0961 0.0049 0.1089 0.0484 0.2025 0.0784 0.1369 0.1089 0.0529
## [91] 0.9801 1.7956 0.9604 0.3249 3.0276 6.2001 0.0484 1.4400 1.5376 4.2025
## [101] 0.4096 0.1089 0.0441 0.0169 0.0064 0.0196 0.3600 0.6400 0.0729 1.3924
## [111] 0.1521 0.8836 0.0529 0.4761 0.3600 0.5625 0.0169 0.6084m2=garchFit(~garch(1,1),data = s,trace = F);m2##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = s, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x000001be4cc53dd8>
## [data = s]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 0.37718 0.16609 1.00000 0.26924
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.37718 0.05859 6.438 1.21e-10 ***
## omega 0.16609 0.08168 2.033 0.0420 *
## alpha1 1.00000 0.41289 2.422 0.0154 *
## beta1 0.26924 0.12652 2.128 0.0333 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -146.7789 normalized: -1.243889
##
## Description:
## Sun May 18 10:18:57 2025 by user: petersummary(m2)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = s, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x000001be4cc53dd8>
## [data = s]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 0.37718 0.16609 1.00000 0.26924
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.37718 0.05859 6.438 1.21e-10 ***
## omega 0.16609 0.08168 2.033 0.0420 *
## alpha1 1.00000 0.41289 2.422 0.0154 *
## beta1 0.26924 0.12652 2.128 0.0333 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -146.7789 normalized: -1.243889
##
## Description:
## Sun May 18 10:18:57 2025 by user: peter
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 344.5777330 0.000000e+00
## Shapiro-Wilk Test R W 0.7075887 5.111457e-14
## Ljung-Box Test R Q(10) 12.9991126 2.237214e-01
## Ljung-Box Test R Q(15) 18.5794522 2.334273e-01
## Ljung-Box Test R Q(20) 24.1434939 2.361802e-01
## Ljung-Box Test R^2 Q(10) 2.0081647 9.962772e-01
## Ljung-Box Test R^2 Q(15) 6.5529652 9.688898e-01
## Ljung-Box Test R^2 Q(20) 8.2853815 9.898050e-01
## LM Arch Test R TR^2 3.5410980 9.903686e-01
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 2.555575 2.649496 2.553376 2.593710plot(m2, which=10)
plot(m2, which=9)