#Loading Libraries
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':
##
## sigma
library(timeSeries)
## Loading required package: timeDate
##
## Attaching package: 'timeSeries'
## The following object is masked from 'package:rugarch':
##
## quantile
## The following objects are masked from 'package:graphics':
##
## lines, points
library(forecast)
## Warning: package 'forecast' was built under R version 4.4.3
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
#Setting Working Directory and Importing Dataset
setwd("C:\\Users\\derek\\OneDrive\\Desktop\\R Class")
data= read.csv("bills.csv");data
## Month BONDS BILLS
## 1 Jul 9.77600 16.91
## 2 Mar 9.88250 17.73
## 3 Mar 8.71850 17.87
## 4 Feb 8.38350 17.84
## 5 Mar 23.73200 20.34
## 6 Jun 8.26625 16.06
## 7 Feb 10.63040 17.91
## 8 Jun 7.39000 18.18
## 9 Jul 10.56575 15.75
## 10 Jan 8.58650 15.93
## 11 Apr 8.91927 18.04
## 12 Feb 8.38350 17.84
## 13 Jun 7.39000 18.18
## 14 Jan 9.25900 17.03
## 15 Feb 9.16000 17.06
## 16 Mar 9.00700 16.91
## 17 May 10.44000 20.12
## 18 Mar 8.71850 17.87
## 19 Mar 9.00700 16.91
## 20 Apr 8.80300 16.70
## 21 Sep 7.77200 19.73
## 22 May 8.82100 16.97
## 23 Nov 8.64300 15.94
## 24 Aug 10.92900 20.13
## 25 Feb 9.16000 17.06
## 26 Jul 11.95420 20.15
## 27 Jun 8.26625 16.06
## 28 Sep 7.77200 19.73
## 29 Jul 5.91720 17.02
## 30 Dec 8.57800 15.99
## 31 Feb 8.38350 17.84
## 32 May 9.46200 17.45
## 33 Jul 10.56575 15.75
## 34 Mar 10.17980 15.46
## 35 May 8.05750 18.22
## 36 Dec 8.24760 18.15
## 37 Mar 9.88250 17.73
## 38 Aug 8.41060 16.26
## 39 Jun 8.26625 16.06
## 40 Sep 14.61320 16.82
## 41 Jul 11.95420 20.15
## 42 Sep 7.77200 19.73
## 43 May 8.25650 15.26
## 44 Oct 9.19600 19.04
## 45 Nov 9.94500 16.89
## 46 Mar 9.00700 16.91
## 47 May 8.05750 18.22
## 48 Jul 11.95420 20.15
## 49 May 8.05750 18.22
## 50 Mar 9.00700 16.91
## 51 Aug 8.41060 16.26
## 52 Feb 8.59000 15.47
## 53 Jan 9.25900 17.03
## 54 Oct 8.68400 16.00
## 55 May 8.82100 16.97
## 56 May 10.44000 20.12
## 57 Aug 10.92900 20.13
## 58 Dec 8.24760 18.15
## 59 Jan 8.07880 18.13
## 60 Jun 6.20600 16.97
## 61 Sep 8.47900 16.04
## 62 Mar 9.00700 16.91
## 63 Jan 9.25900 17.03
## 64 Feb 8.59000 15.47
## 65 Sep 9.68650 16.86
## 66 Mar 9.88250 17.73
## 67 May 8.25650 15.26
## 68 Nov 12.33340 17.16
## 69 Feb 9.16000 17.06
## 70 Jul 9.77600 16.91
## 71 Apr 10.37900 17.87
## 72 Apr 8.91927 18.04
## 73 Jun 9.80960 16.36
## 74 Jun 10.18300 20.30
## 75 Jul 10.56575 15.75
## 76 Aug 10.92900 20.13
## 77 Oct 9.71700 17.00
## 78 Oct 9.71700 17.00
## 79 Mar 10.17980 15.46
## 80 Jul 9.77600 16.91
## 81 Dec 9.90890 18.30
## 82 May 9.46200 17.45
## 83 Jun 7.39000 18.18
## 84 Feb 10.63040 17.91
## 85 Jan 11.35820 18.00
## 86 Oct 8.68400 16.00
## 87 Apr 20.01700 20.22
## 88 Jan 8.07880 18.13
## 89 May 10.44000 20.12
## 90 Jun 7.39000 18.18
## 91 Mar 8.71850 17.87
## 92 Jan 9.25900 17.03
## 93 Apr 8.91927 18.04
## 94 Oct 9.71700 17.00
## 95 Jan 9.25900 17.03
## 96 Aug 8.41060 16.26
## 97 Oct 21.65400 16.58
## 98 Jan 9.25900 17.03
## 99 Nov 9.94500 16.89
## 100 Nov 8.64300 15.94
## 101 Apr 8.42250 15.40
## 102 Dec 9.90890 18.30
## 103 Jun 9.80960 16.36
## 104 Dec 8.24760 18.15
## 105 Jul 9.77600 16.91
## 106 Oct 21.65400 16.58
## 107 Jun 9.80960 16.36
## 108 Nov 8.64300 15.94
## 109 Feb 8.38350 17.84
## 110 May 8.82100 16.97
#Selecting the Treasury Bills Column
D1=data$BILLS;D1
## [1] 16.91 17.73 17.87 17.84 20.34 16.06 17.91 18.18 15.75 15.93 18.04 17.84
## [13] 18.18 17.03 17.06 16.91 20.12 17.87 16.91 16.70 19.73 16.97 15.94 20.13
## [25] 17.06 20.15 16.06 19.73 17.02 15.99 17.84 17.45 15.75 15.46 18.22 18.15
## [37] 17.73 16.26 16.06 16.82 20.15 19.73 15.26 19.04 16.89 16.91 18.22 20.15
## [49] 18.22 16.91 16.26 15.47 17.03 16.00 16.97 20.12 20.13 18.15 18.13 16.97
## [61] 16.04 16.91 17.03 15.47 16.86 17.73 15.26 17.16 17.06 16.91 17.87 18.04
## [73] 16.36 20.30 15.75 20.13 17.00 17.00 15.46 16.91 18.30 17.45 18.18 17.91
## [85] 18.00 16.00 20.22 18.13 20.12 18.18 17.87 17.03 18.04 17.00 17.03 16.26
## [97] 16.58 17.03 16.89 15.94 15.40 18.30 16.36 18.15 16.91 16.58 16.36 15.94
## [109] 17.84 16.97
#(I) GARCH MODEL
#Time Series Plot
ts.plot(D1,
main = "Time Series of Treasury Bills Rate of Return",
ylab = "Amount (KSH)",
xlab = "Observation Index",
col = "darkblue",
lwd = 2,
lty = 1, # Solid line
ylim = c(min(D1) - 1, max(D1) + 1)) # Adjust y-axis range
#Differencing and Plotting The Differenced Series
log.D1=diff(D1);log.D1
## [1] 0.82 0.14 -0.03 2.50 -4.28 1.85 0.27 -2.43 0.18 2.11 -0.20 0.34
## [13] -1.15 0.03 -0.15 3.21 -2.25 -0.96 -0.21 3.03 -2.76 -1.03 4.19 -3.07
## [25] 3.09 -4.09 3.67 -2.71 -1.03 1.85 -0.39 -1.70 -0.29 2.76 -0.07 -0.42
## [37] -1.47 -0.20 0.76 3.33 -0.42 -4.47 3.78 -2.15 0.02 1.31 1.93 -1.93
## [49] -1.31 -0.65 -0.79 1.56 -1.03 0.97 3.15 0.01 -1.98 -0.02 -1.16 -0.93
## [61] 0.87 0.12 -1.56 1.39 0.87 -2.47 1.90 -0.10 -0.15 0.96 0.17 -1.68
## [73] 3.94 -4.55 4.38 -3.13 0.00 -1.54 1.45 1.39 -0.85 0.73 -0.27 0.09
## [85] -2.00 4.22 -2.09 1.99 -1.94 -0.31 -0.84 1.01 -1.04 0.03 -0.77 0.32
## [97] 0.45 -0.14 -0.95 -0.54 2.90 -1.94 1.79 -1.24 -0.33 -0.22 -0.42 1.90
## [109] -0.87
plot(log.D1,
type = "l", # Line plot
col = "darkred",
lwd = 2,
main = "First Differences of Treasury Bills Rate Of Return",
xlab = "Time",
ylab = "Difference in Rate of Return (%)")
#Autocorrelation Analysis (ACF/PACF)
par(mfrow = c(1, 2))
acf(log.D1, main = "ACF of Differenced Series", col = "blue")
pacf(log.D1, main = "PACF of Differenced Series", col = "darkgreen")
#Removing Negatives
S=log.D1^2;S
## [1] 0.6724 0.0196 0.0009 6.2500 18.3184 3.4225 0.0729 5.9049 0.0324
## [10] 4.4521 0.0400 0.1156 1.3225 0.0009 0.0225 10.3041 5.0625 0.9216
## [19] 0.0441 9.1809 7.6176 1.0609 17.5561 9.4249 9.5481 16.7281 13.4689
## [28] 7.3441 1.0609 3.4225 0.1521 2.8900 0.0841 7.6176 0.0049 0.1764
## [37] 2.1609 0.0400 0.5776 11.0889 0.1764 19.9809 14.2884 4.6225 0.0004
## [46] 1.7161 3.7249 3.7249 1.7161 0.4225 0.6241 2.4336 1.0609 0.9409
## [55] 9.9225 0.0001 3.9204 0.0004 1.3456 0.8649 0.7569 0.0144 2.4336
## [64] 1.9321 0.7569 6.1009 3.6100 0.0100 0.0225 0.9216 0.0289 2.8224
## [73] 15.5236 20.7025 19.1844 9.7969 0.0000 2.3716 2.1025 1.9321 0.7225
## [82] 0.5329 0.0729 0.0081 4.0000 17.8084 4.3681 3.9601 3.7636 0.0961
## [91] 0.7056 1.0201 1.0816 0.0009 0.5929 0.1024 0.2025 0.0196 0.9025
## [100] 0.2916 8.4100 3.7636 3.2041 1.5376 0.1089 0.0484 0.1764 3.6100
## [109] 0.7569
summary(S)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.1156 1.0816 3.6932 4.3681 20.7025
shapiro.test(D1)
##
## Shapiro-Wilk normality test
##
## data: D1
## W = 0.92318, p-value = 8.65e-06
M1=garchFit(~garch(1,1),data = log.D1,trace = F);M1
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = log.D1, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x00000277a2923eb0>
## [data = log.D1]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 0.0055046 1.7887270 0.2972967 0.2046660
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.005505 0.147082 0.037 0.9701
## omega 1.788727 0.768708 2.327 0.0200 *
## alpha1 0.297297 0.133754 2.223 0.0262 *
## beta1 0.204666 0.219461 0.933 0.3510
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -220.414 normalized: -2.022147
##
## Description:
## Sun May 18 15:18:36 2025 by user: derek
M2=garchFit(~garch(1,2),data = log.D1,trace = F);M2
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = log.D1, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000277a4309808>
## [data = log.D1]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1 beta2
## 0.00550459 1.79245128 0.28948183 0.21003982 0.00000001
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 5.505e-03 1.520e-01 0.036 0.9711
## omega 1.792e+00 8.412e-01 2.131 0.0331 *
## alpha1 2.895e-01 1.339e-01 2.162 0.0306 *
## beta1 2.100e-01 6.394e-01 0.329 0.7425
## beta2 1.000e-08 5.383e-01 0.000 1.0000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -220.5755 normalized: -2.023628
##
## Description:
## Sun May 18 15:18:36 2025 by user: derek
M3=garchFit(~garch(2,1),data = log.D1,trace = F);M3
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(2, 1), data = log.D1, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(2, 1)
## <environment: 0x000002779304cbd8>
## [data = log.D1]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 alpha2 beta1
## 0.00550459 2.21807069 0.27678708 0.10274595 0.00000001
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 5.505e-03 1.467e-01 0.038 0.9701
## omega 2.218e+00 1.422e+00 1.559 0.1189
## alpha1 2.768e-01 1.322e-01 2.093 0.0363 *
## alpha2 1.027e-01 2.150e-01 0.478 0.6328
## beta1 1.000e-08 5.621e-01 0.000 1.0000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -220.4115 normalized: -2.022124
##
## Description:
## Sun May 18 15:18:36 2025 by user: derek
#Checking And Comparing AIC/BIC
comparison <- data.frame(
Model = c("GARCH(1,1)", "GARCH(1,2)", "GARCH(2,1)"),
AIC = c(M1@fit$ics["AIC"], M2@fit$ics["AIC"], M3@fit$ics["AIC"]),
BIC = c(M1@fit$ics["BIC"], M2@fit$ics["BIC"], M3@fit$ics["BIC"])
)
print(comparison)
## Model AIC BIC
## 1 GARCH(1,1) 4.117688 4.216453
## 2 GARCH(1,2) 4.139000 4.262456
## 3 GARCH(2,1) 4.135991 4.259448
#Significance of Coefficients
summary(M1)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = log.D1, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x00000277a2923eb0>
## [data = log.D1]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 0.0055046 1.7887270 0.2972967 0.2046660
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.005505 0.147082 0.037 0.9701
## omega 1.788727 0.768708 2.327 0.0200 *
## alpha1 0.297297 0.133754 2.223 0.0262 *
## beta1 0.204666 0.219461 0.933 0.3510
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -220.414 normalized: -2.022147
##
## Description:
## Sun May 18 15:18:36 2025 by user: derek
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 1.9705271 0.3733408172
## Shapiro-Wilk Test R W 0.9779825 0.0677367869
## Ljung-Box Test R Q(10) 33.3740420 0.0002356307
## Ljung-Box Test R Q(15) 43.5270458 0.0001303603
## Ljung-Box Test R Q(20) 44.2494630 0.0013933457
## Ljung-Box Test R^2 Q(10) 5.7710271 0.8341196225
## Ljung-Box Test R^2 Q(15) 9.3529708 0.8583505082
## Ljung-Box Test R^2 Q(20) 11.1563062 0.9420678282
## LM Arch Test R TR^2 10.2125898 0.5973163147
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 4.117688 4.216453 4.115120 4.157741
summary(M2)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = log.D1, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000277a4309808>
## [data = log.D1]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1 beta2
## 0.00550459 1.79245128 0.28948183 0.21003982 0.00000001
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 5.505e-03 1.520e-01 0.036 0.9711
## omega 1.792e+00 8.412e-01 2.131 0.0331 *
## alpha1 2.895e-01 1.339e-01 2.162 0.0306 *
## beta1 2.100e-01 6.394e-01 0.329 0.7425
## beta2 1.000e-08 5.383e-01 0.000 1.0000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -220.5755 normalized: -2.023628
##
## Description:
## Sun May 18 15:18:36 2025 by user: derek
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 1.9405147 0.3789854922
## Shapiro-Wilk Test R W 0.9782288 0.0710801820
## Ljung-Box Test R Q(10) 33.5889540 0.0002167586
## Ljung-Box Test R Q(15) 43.8519796 0.0001159868
## Ljung-Box Test R Q(20) 44.5687624 0.0012620585
## Ljung-Box Test R^2 Q(10) 5.8807540 0.8251802569
## Ljung-Box Test R^2 Q(15) 9.4543594 0.8525884717
## Ljung-Box Test R^2 Q(20) 11.2154316 0.9404431795
## LM Arch Test R TR^2 10.2404177 0.5948779555
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 4.139000 4.262456 4.135033 4.189066
summary(M3)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(2, 1), data = log.D1, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(2, 1)
## <environment: 0x000002779304cbd8>
## [data = log.D1]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 alpha2 beta1
## 0.00550459 2.21807069 0.27678708 0.10274595 0.00000001
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 5.505e-03 1.467e-01 0.038 0.9701
## omega 2.218e+00 1.422e+00 1.559 0.1189
## alpha1 2.768e-01 1.322e-01 2.093 0.0363 *
## alpha2 1.027e-01 2.150e-01 0.478 0.6328
## beta1 1.000e-08 5.621e-01 0.000 1.0000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -220.4115 normalized: -2.022124
##
## Description:
## Sun May 18 15:18:36 2025 by user: derek
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 2.3235348 0.3129326182
## Shapiro-Wilk Test R W 0.9764039 0.0497583829
## Ljung-Box Test R Q(10) 33.5772536 0.0002177468
## Ljung-Box Test R Q(15) 43.4813182 0.0001325178
## Ljung-Box Test R Q(20) 44.2115063 0.0014097968
## Ljung-Box Test R^2 Q(10) 6.0591424 0.8102701112
## Ljung-Box Test R^2 Q(15) 9.7042261 0.8379209427
## Ljung-Box Test R^2 Q(20) 11.0814459 0.9440829710
## LM Arch Test R TR^2 10.3761132 0.5830010032
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 4.135991 4.259448 4.132024 4.186057
#--------------------------------------------------------------------------------
#--------------------------------------------------------------------------------
#(II) Exponential GARCH (EGARCH) MODEL
# Import libraries
library(rugarch)
library(forecast)
# Set working directory and import dataset
setwd("C:/Users/derek/OneDrive/Desktop/R Class")
data <- read.csv("bills.csv")
# Select the Treasury Bills column
d2 <- data$BILLS
# Define the eGARCH model specification
egm2 <- ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE), # <- change here
distribution.model = "norm"
)
# Fit the eGARCH model
egm2_fit <- ugarchfit(spec = egm2, data = d2)
egm2_fit
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 17.420734 0.164916 105.63409 0.00000
## omega 0.068730 0.083592 0.82220 0.41096
## alpha1 -0.065874 0.103935 -0.63379 0.52621
## beta1 0.889915 0.138093 6.44434 0.00000
## gamma1 -0.070927 0.118264 -0.59974 0.54868
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 17.420734 0.425703 40.92230 0.00000
## omega 0.068730 0.067817 1.01347 0.31084
## alpha1 -0.065874 0.283103 -0.23268 0.81601
## beta1 0.889915 0.093713 9.49620 0.00000
## gamma1 -0.070927 0.136008 -0.52149 0.60202
##
## LogLikelihood : -187.6314
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.5024
## Bayes 3.6251
## Shibata 3.4985
## Hannan-Quinn 3.5522
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04092 0.8397
## Lag[2*(p+q)+(p+q)-1][2] 0.48513 0.7006
## Lag[4*(p+q)+(p+q)-1][5] 1.26482 0.7974
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1831 0.6687
## Lag[2*(p+q)+(p+q)-1][5] 1.2283 0.8062
## Lag[4*(p+q)+(p+q)-1][9] 2.3875 0.8541
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.6457 0.500 2.000 0.4217
## ARCH Lag[5] 0.7079 1.440 1.667 0.8211
## ARCH Lag[7] 1.7794 2.315 1.543 0.7639
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.7115
## Individual Statistics:
## mu 0.17004
## omega 0.35284
## alpha1 0.07106
## beta1 0.37865
## gamma1 0.07738
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.90461 0.3677
## Negative Sign Bias 0.07789 0.9381
## Positive Sign Bias 0.85743 0.3932
## Joint Effect 1.51995 0.6777
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 111.5 4.297e-15
## 2 30 128.9 1.476e-14
## 3 40 134.4 2.095e-12
## 4 50 139.1 1.460e-10
##
##
## Elapsed time : 0.3751609
# Display the fit summary
egm2_fit
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 17.420734 0.164916 105.63409 0.00000
## omega 0.068730 0.083592 0.82220 0.41096
## alpha1 -0.065874 0.103935 -0.63379 0.52621
## beta1 0.889915 0.138093 6.44434 0.00000
## gamma1 -0.070927 0.118264 -0.59974 0.54868
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 17.420734 0.425703 40.92230 0.00000
## omega 0.068730 0.067817 1.01347 0.31084
## alpha1 -0.065874 0.283103 -0.23268 0.81601
## beta1 0.889915 0.093713 9.49620 0.00000
## gamma1 -0.070927 0.136008 -0.52149 0.60202
##
## LogLikelihood : -187.6314
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.5024
## Bayes 3.6251
## Shibata 3.4985
## Hannan-Quinn 3.5522
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04092 0.8397
## Lag[2*(p+q)+(p+q)-1][2] 0.48513 0.7006
## Lag[4*(p+q)+(p+q)-1][5] 1.26482 0.7974
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1831 0.6687
## Lag[2*(p+q)+(p+q)-1][5] 1.2283 0.8062
## Lag[4*(p+q)+(p+q)-1][9] 2.3875 0.8541
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.6457 0.500 2.000 0.4217
## ARCH Lag[5] 0.7079 1.440 1.667 0.8211
## ARCH Lag[7] 1.7794 2.315 1.543 0.7639
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.7115
## Individual Statistics:
## mu 0.17004
## omega 0.35284
## alpha1 0.07106
## beta1 0.37865
## gamma1 0.07738
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.90461 0.3677
## Negative Sign Bias 0.07789 0.9381
## Positive Sign Bias 0.85743 0.3932
## Joint Effect 1.51995 0.6777
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 111.5 4.297e-15
## 2 30 128.9 1.476e-14
## 3 40 134.4 2.095e-12
## 4 50 139.1 1.460e-10
##
##
## Elapsed time : 0.3751609
#PLotting The Fitted Model
par(mfrow = c(3, 4))
for(i in 1:12){
plot(egm2_fit, which = i)
}
##
## please wait...calculating quantiles...
#Forecasting
forc=ugarchforecast(egm2_fit);forc
##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: eGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=1970-04-21]:
## Series Sigma
## T+1 17.42 1.589
## T+2 17.42 1.562
## T+3 17.42 1.540
## T+4 17.42 1.519
## T+5 17.42 1.502
## T+6 17.42 1.486
## T+7 17.42 1.473
## T+8 17.42 1.460
## T+9 17.42 1.450
## T+10 17.42 1.440
#Plotting The Forecasting results
plot(forc, which = 1)
#-------------------------------------------------------------------------------
#-------------------------------------------------------------------------------
#(III) GJR-GARCH MODEL
#Importing Libraries
library(rugarch)
library(forecast)
#Setting Working Directory and Importing Dataset
setwd("C:\\Users\\derek\\OneDrive\\Desktop\\R Class")
data= read.csv("bills.csv");data
## Month BONDS BILLS
## 1 Jul 9.77600 16.91
## 2 Mar 9.88250 17.73
## 3 Mar 8.71850 17.87
## 4 Feb 8.38350 17.84
## 5 Mar 23.73200 20.34
## 6 Jun 8.26625 16.06
## 7 Feb 10.63040 17.91
## 8 Jun 7.39000 18.18
## 9 Jul 10.56575 15.75
## 10 Jan 8.58650 15.93
## 11 Apr 8.91927 18.04
## 12 Feb 8.38350 17.84
## 13 Jun 7.39000 18.18
## 14 Jan 9.25900 17.03
## 15 Feb 9.16000 17.06
## 16 Mar 9.00700 16.91
## 17 May 10.44000 20.12
## 18 Mar 8.71850 17.87
## 19 Mar 9.00700 16.91
## 20 Apr 8.80300 16.70
## 21 Sep 7.77200 19.73
## 22 May 8.82100 16.97
## 23 Nov 8.64300 15.94
## 24 Aug 10.92900 20.13
## 25 Feb 9.16000 17.06
## 26 Jul 11.95420 20.15
## 27 Jun 8.26625 16.06
## 28 Sep 7.77200 19.73
## 29 Jul 5.91720 17.02
## 30 Dec 8.57800 15.99
## 31 Feb 8.38350 17.84
## 32 May 9.46200 17.45
## 33 Jul 10.56575 15.75
## 34 Mar 10.17980 15.46
## 35 May 8.05750 18.22
## 36 Dec 8.24760 18.15
## 37 Mar 9.88250 17.73
## 38 Aug 8.41060 16.26
## 39 Jun 8.26625 16.06
## 40 Sep 14.61320 16.82
## 41 Jul 11.95420 20.15
## 42 Sep 7.77200 19.73
## 43 May 8.25650 15.26
## 44 Oct 9.19600 19.04
## 45 Nov 9.94500 16.89
## 46 Mar 9.00700 16.91
## 47 May 8.05750 18.22
## 48 Jul 11.95420 20.15
## 49 May 8.05750 18.22
## 50 Mar 9.00700 16.91
## 51 Aug 8.41060 16.26
## 52 Feb 8.59000 15.47
## 53 Jan 9.25900 17.03
## 54 Oct 8.68400 16.00
## 55 May 8.82100 16.97
## 56 May 10.44000 20.12
## 57 Aug 10.92900 20.13
## 58 Dec 8.24760 18.15
## 59 Jan 8.07880 18.13
## 60 Jun 6.20600 16.97
## 61 Sep 8.47900 16.04
## 62 Mar 9.00700 16.91
## 63 Jan 9.25900 17.03
## 64 Feb 8.59000 15.47
## 65 Sep 9.68650 16.86
## 66 Mar 9.88250 17.73
## 67 May 8.25650 15.26
## 68 Nov 12.33340 17.16
## 69 Feb 9.16000 17.06
## 70 Jul 9.77600 16.91
## 71 Apr 10.37900 17.87
## 72 Apr 8.91927 18.04
## 73 Jun 9.80960 16.36
## 74 Jun 10.18300 20.30
## 75 Jul 10.56575 15.75
## 76 Aug 10.92900 20.13
## 77 Oct 9.71700 17.00
## 78 Oct 9.71700 17.00
## 79 Mar 10.17980 15.46
## 80 Jul 9.77600 16.91
## 81 Dec 9.90890 18.30
## 82 May 9.46200 17.45
## 83 Jun 7.39000 18.18
## 84 Feb 10.63040 17.91
## 85 Jan 11.35820 18.00
## 86 Oct 8.68400 16.00
## 87 Apr 20.01700 20.22
## 88 Jan 8.07880 18.13
## 89 May 10.44000 20.12
## 90 Jun 7.39000 18.18
## 91 Mar 8.71850 17.87
## 92 Jan 9.25900 17.03
## 93 Apr 8.91927 18.04
## 94 Oct 9.71700 17.00
## 95 Jan 9.25900 17.03
## 96 Aug 8.41060 16.26
## 97 Oct 21.65400 16.58
## 98 Jan 9.25900 17.03
## 99 Nov 9.94500 16.89
## 100 Nov 8.64300 15.94
## 101 Apr 8.42250 15.40
## 102 Dec 9.90890 18.30
## 103 Jun 9.80960 16.36
## 104 Dec 8.24760 18.15
## 105 Jul 9.77600 16.91
## 106 Oct 21.65400 16.58
## 107 Jun 9.80960 16.36
## 108 Nov 8.64300 15.94
## 109 Feb 8.38350 17.84
## 110 May 8.82100 16.97
#Selecting the Treasury Bills Column
D3=data$BILLS;D3
## [1] 16.91 17.73 17.87 17.84 20.34 16.06 17.91 18.18 15.75 15.93 18.04 17.84
## [13] 18.18 17.03 17.06 16.91 20.12 17.87 16.91 16.70 19.73 16.97 15.94 20.13
## [25] 17.06 20.15 16.06 19.73 17.02 15.99 17.84 17.45 15.75 15.46 18.22 18.15
## [37] 17.73 16.26 16.06 16.82 20.15 19.73 15.26 19.04 16.89 16.91 18.22 20.15
## [49] 18.22 16.91 16.26 15.47 17.03 16.00 16.97 20.12 20.13 18.15 18.13 16.97
## [61] 16.04 16.91 17.03 15.47 16.86 17.73 15.26 17.16 17.06 16.91 17.87 18.04
## [73] 16.36 20.30 15.75 20.13 17.00 17.00 15.46 16.91 18.30 17.45 18.18 17.91
## [85] 18.00 16.00 20.22 18.13 20.12 18.18 17.87 17.03 18.04 17.00 17.03 16.26
## [97] 16.58 17.03 16.89 15.94 15.40 18.30 16.36 18.15 16.91 16.58 16.36 15.94
## [109] 17.84 16.97
#GJR-GARCH(1,1) Model With Standard Normal Distribution
N1=ugarchspec(variance.model=list(model="gjrGARCH",garchOrder=c(1,1)),
mean.model=list(armaOrder=c(0,0),include.mean=F),
distribution.model="norm");N1
##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : gjrGARCH(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 : FALSE
#GJR-GARCH(1,1) Model With Student-T Distribution
S1=ugarchspec(variance.model=list(model="gjrGARCH",garchOrder=c(1,1)),
mean.model=list(armaOrder=c(0,0),include.mean=F),
distribution.model="std");S1
##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : gjrGARCH(1,1)
## 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 : FALSE
#GJR-GARCH(1,1) Model With General Error Distribution (GED)
G1=ugarchspec(variance.model=list(model="gjrGARCH",garchOrder=c(1,1)),
mean.model=list(armaOrder=c(0,0),include.mean=F),
distribution.model="ged");G1
##
## *---------------------------------*
## * GARCH Model Spec *
## *---------------------------------*
##
## Conditional Variance Dynamics
## ------------------------------------
## GARCH Model : gjrGARCH(1,1)
## Variance Targeting : FALSE
##
## Conditional Mean Dynamics
## ------------------------------------
## Mean Model : ARFIMA(0,0,0)
## Include Mean : FALSE
## GARCH-in-Mean : FALSE
##
## Conditional Distribution
## ------------------------------------
## Distribution : ged
## Includes Skew : FALSE
## Includes Shape : TRUE
## Includes Lambda : FALSE
#Model Fitting
S1fit <- ugarchfit(spec = S1, data = D3, solver = "nlminb")
#Plotting the Results
S1fit
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Convergence Problem:
## Solver Message: false convergence (8)
# Print forecast summary
print(forc)
##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: eGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=1970-04-21]:
## Series Sigma
## T+1 17.42 1.589
## T+2 17.42 1.562
## T+3 17.42 1.540
## T+4 17.42 1.519
## T+5 17.42 1.502
## T+6 17.42 1.486
## T+7 17.42 1.473
## T+8 17.42 1.460
## T+9 17.42 1.450
## T+10 17.42 1.440
print(forc, which = 1)
## <S4 Type Object>
## attr(,"forecast")
## attr(,"forecast")$n.ahead
## [1] 10
##
## attr(,"forecast")$N
## [1] 110
##
## attr(,"forecast")$n.start
## [1] 0
##
## attr(,"forecast")$n.roll
## [1] 0
##
## attr(,"forecast")$sigmaFor
## 1970-04-21
## T+1 1.588618
## T+2 1.562481
## T+3 1.539583
## T+4 1.519489
## T+5 1.501827
## T+6 1.486282
## T+7 1.472584
## T+8 1.460500
## T+9 1.449829
## T+10 1.440399
##
## attr(,"forecast")$seriesFor
## 1970-04-21
## T+1 17.42073
## T+2 17.42073
## T+3 17.42073
## T+4 17.42073
## T+5 17.42073
## T+6 17.42073
## T+7 17.42073
## T+8 17.42073
## T+9 17.42073
## T+10 17.42073
##
## attr(,"model")
## attr(,"model")$modelinc
## mu ar ma arfima archm mxreg omega alpha
## 1 0 0 0 0 0 1 1
## beta gamma eta1 eta2 delta lambda vxreg skew
## 1 1 0 0 0 0 0 0
## shape ghlambda xi aux aux aux
## 0 0 0 0 1 NA
##
## attr(,"model")$modeldesc
## attr(,"model")$modeldesc$distribution
## [1] "norm"
##
## attr(,"model")$modeldesc$distno
## [1] 1
##
## attr(,"model")$modeldesc$vmodel
## [1] "eGARCH"
##
##
## attr(,"model")$modeldata
## attr(,"model")$modeldata$T
## [1] 110
##
## attr(,"model")$modeldata$data
## [1] 16.91 17.73 17.87 17.84 20.34 16.06 17.91 18.18 15.75 15.93 18.04 17.84
## [13] 18.18 17.03 17.06 16.91 20.12 17.87 16.91 16.70 19.73 16.97 15.94 20.13
## [25] 17.06 20.15 16.06 19.73 17.02 15.99 17.84 17.45 15.75 15.46 18.22 18.15
## [37] 17.73 16.26 16.06 16.82 20.15 19.73 15.26 19.04 16.89 16.91 18.22 20.15
## [49] 18.22 16.91 16.26 15.47 17.03 16.00 16.97 20.12 20.13 18.15 18.13 16.97
## [61] 16.04 16.91 17.03 15.47 16.86 17.73 15.26 17.16 17.06 16.91 17.87 18.04
## [73] 16.36 20.30 15.75 20.13 17.00 17.00 15.46 16.91 18.30 17.45 18.18 17.91
## [85] 18.00 16.00 20.22 18.13 20.12 18.18 17.87 17.03 18.04 17.00 17.03 16.26
## [97] 16.58 17.03 16.89 15.94 15.40 18.30 16.36 18.15 16.91 16.58 16.36 15.94
## [109] 17.84 16.97
##
## attr(,"model")$modeldata$index
## [1] "1970-01-02 UTC" "1970-01-03 UTC" "1970-01-04 UTC" "1970-01-05 UTC"
## [5] "1970-01-06 UTC" "1970-01-07 UTC" "1970-01-08 UTC" "1970-01-09 UTC"
## [9] "1970-01-10 UTC" "1970-01-11 UTC" "1970-01-12 UTC" "1970-01-13 UTC"
## [13] "1970-01-14 UTC" "1970-01-15 UTC" "1970-01-16 UTC" "1970-01-17 UTC"
## [17] "1970-01-18 UTC" "1970-01-19 UTC" "1970-01-20 UTC" "1970-01-21 UTC"
## [21] "1970-01-22 UTC" "1970-01-23 UTC" "1970-01-24 UTC" "1970-01-25 UTC"
## [25] "1970-01-26 UTC" "1970-01-27 UTC" "1970-01-28 UTC" "1970-01-29 UTC"
## [29] "1970-01-30 UTC" "1970-01-31 UTC" "1970-02-01 UTC" "1970-02-02 UTC"
## [33] "1970-02-03 UTC" "1970-02-04 UTC" "1970-02-05 UTC" "1970-02-06 UTC"
## [37] "1970-02-07 UTC" "1970-02-08 UTC" "1970-02-09 UTC" "1970-02-10 UTC"
## [41] "1970-02-11 UTC" "1970-02-12 UTC" "1970-02-13 UTC" "1970-02-14 UTC"
## [45] "1970-02-15 UTC" "1970-02-16 UTC" "1970-02-17 UTC" "1970-02-18 UTC"
## [49] "1970-02-19 UTC" "1970-02-20 UTC" "1970-02-21 UTC" "1970-02-22 UTC"
## [53] "1970-02-23 UTC" "1970-02-24 UTC" "1970-02-25 UTC" "1970-02-26 UTC"
## [57] "1970-02-27 UTC" "1970-02-28 UTC" "1970-03-01 UTC" "1970-03-02 UTC"
## [61] "1970-03-03 UTC" "1970-03-04 UTC" "1970-03-05 UTC" "1970-03-06 UTC"
## [65] "1970-03-07 UTC" "1970-03-08 UTC" "1970-03-09 UTC" "1970-03-10 UTC"
## [69] "1970-03-11 UTC" "1970-03-12 UTC" "1970-03-13 UTC" "1970-03-14 UTC"
## [73] "1970-03-15 UTC" "1970-03-16 UTC" "1970-03-17 UTC" "1970-03-18 UTC"
## [77] "1970-03-19 UTC" "1970-03-20 UTC" "1970-03-21 UTC" "1970-03-22 UTC"
## [81] "1970-03-23 UTC" "1970-03-24 UTC" "1970-03-25 UTC" "1970-03-26 UTC"
## [85] "1970-03-27 UTC" "1970-03-28 UTC" "1970-03-29 UTC" "1970-03-30 UTC"
## [89] "1970-03-31 UTC" "1970-04-01 UTC" "1970-04-02 UTC" "1970-04-03 UTC"
## [93] "1970-04-04 UTC" "1970-04-05 UTC" "1970-04-06 UTC" "1970-04-07 UTC"
## [97] "1970-04-08 UTC" "1970-04-09 UTC" "1970-04-10 UTC" "1970-04-11 UTC"
## [101] "1970-04-12 UTC" "1970-04-13 UTC" "1970-04-14 UTC" "1970-04-15 UTC"
## [105] "1970-04-16 UTC" "1970-04-17 UTC" "1970-04-18 UTC" "1970-04-19 UTC"
## [109] "1970-04-20 UTC" "1970-04-21 UTC"
##
## attr(,"model")$modeldata$period
## Time difference of 1 days
##
## attr(,"model")$modeldata$sigma
## [1] 1.337851 1.378118 1.394762 1.400353 1.407532 1.252326 1.297123 1.307832
## [9] 1.299238 1.339601 1.377176 1.372601 1.382109 1.367619 1.405714 1.440614
## [17] 1.472035 1.324873 1.336180 1.376584 1.413063 1.295084 1.338958 1.376612
## [25] 1.236746 1.285331 1.151197 1.203190 1.100753 1.158548 1.209863 1.231820
## [33] 1.279637 1.321535 1.359361 1.344014 1.334637 1.354853 1.392022 1.425520
## [41] 1.458055 1.310247 1.200305 1.246811 1.185463 1.237303 1.285453 1.275832
## [49] 1.142387 1.142559 1.197374 1.246715 1.290390 1.334791 1.372942 1.410431
## [57] 1.268449 1.136760 1.142080 1.148444 1.203027 1.251388 1.298482 1.342244
## [65] 1.378407 1.415145 1.428636 1.456869 1.487467 1.514993 1.539549 1.532177
## [73] 1.513966 1.537207 1.373247 1.407546 1.265185 1.311465 1.354112 1.389267
## [81] 1.425204 1.399020 1.433385 1.414581 1.415744 1.410660 1.442385 1.291609
## [89] 1.287651 1.155187 1.157294 1.180704 1.233245 1.239718 1.287924 1.332519
## [97] 1.371533 1.408129 1.442739 1.473918 1.499815 1.521926 1.487188 1.512939
## [105] 1.489088 1.516077 1.539683 1.560469 1.578166 1.569136
##
## attr(,"model")$modeldata$residuals
## [1] -0.51073393 0.30926607 0.44926607 0.41926607 2.91926607 -1.36073393
## [7] 0.48926607 0.75926607 -1.67073393 -1.49073393 0.61926607 0.41926607
## [13] 0.75926607 -0.39073393 -0.36073393 -0.51073393 2.69926607 0.44926607
## [19] -0.51073393 -0.72073393 2.30926607 -0.45073393 -1.48073393 2.70926607
## [25] -0.36073393 2.72926607 -1.36073393 2.30926607 -0.40073393 -1.43073393
## [31] 0.41926607 0.02926607 -1.67073393 -1.96073393 0.79926607 0.72926607
## [37] 0.30926607 -1.16073393 -1.36073393 -0.60073393 2.72926607 2.30926607
## [43] -2.16073393 1.61926607 -0.53073393 -0.51073393 0.79926607 2.72926607
## [49] 0.79926607 -0.51073393 -1.16073393 -1.95073393 -0.39073393 -1.42073393
## [55] -0.45073393 2.69926607 2.70926607 0.72926607 0.70926607 -0.45073393
## [61] -1.38073393 -0.51073393 -0.39073393 -1.95073393 -0.56073393 0.30926607
## [67] -2.16073393 -0.26073393 -0.36073393 -0.51073393 0.44926607 0.61926607
## [73] -1.06073393 2.87926607 -1.67073393 2.70926607 -0.42073393 -0.42073393
## [79] -1.96073393 -0.51073393 0.87926607 0.02926607 0.75926607 0.48926607
## [85] 0.57926607 -1.42073393 2.79926607 0.70926607 2.69926607 0.75926607
## [91] 0.44926607 -0.39073393 0.61926607 -0.42073393 -0.39073393 -1.16073393
## [97] -0.84073393 -0.39073393 -0.53073393 -1.48073393 -2.02073393 0.87926607
## [103] -1.06073393 0.72926607 -0.51073393 -0.84073393 -1.06073393 -1.48073393
## [109] 0.41926607 -0.45073393
##
##
## attr(,"model")$pars
## Level Fixed Include Estimate LB UB
## mu 17.42073393 0 1 1 -1744.209 1744.209
## ar 0.00000000 0 0 0 NA NA
## ma 0.00000000 0 0 0 NA NA
## arfima 0.00000000 0 0 0 NA NA
## archm 0.00000000 0 0 0 NA NA
## mxreg 0.00000000 0 0 0 NA NA
## omega 0.06872993 0 1 1 -10.000 10.000
## alpha1 -0.06587363 0 1 1 -10.000 10.000
## beta1 0.88991519 0 1 1 -1.000 1.000
## gamma1 -0.07092741 0 1 1 -10.000 10.000
## eta1 0.00000000 0 0 0 NA NA
## eta2 0.00000000 0 0 0 NA NA
## delta 0.00000000 0 0 0 NA NA
## lambda 0.00000000 0 0 0 NA NA
## vxreg 0.00000000 0 0 0 NA NA
## skew 0.00000000 0 0 0 NA NA
## shape 0.00000000 0 0 0 NA NA
## ghlambda 0.00000000 0 0 0 NA NA
## xi 0.00000000 0 0 0 NA NA
##
## attr(,"model")$start.pars
## NULL
##
## attr(,"model")$fixed.pars
## NULL
##
## attr(,"model")$maxOrder
## [1] 1
##
## attr(,"model")$pos.matrix
## start stop include
## mu 1 1 1
## ar 0 0 0
## ma 0 0 0
## arfima 0 0 0
## archm 0 0 0
## mxreg 0 0 0
## omega 2 2 1
## alpha 3 3 1
## beta 4 4 1
## gamma 5 5 1
## eta1 0 0 0
## eta2 0 0 0
## delta 0 0 0
## lambda 0 0 0
## vxreg 0 0 0
## skew 0 0 0
## shape 0 0 0
## ghlambda 0 0 0
## xi 0 0 0
## aux 0 0 0
## aux 6 6 1
##
## attr(,"model")$fmodel
## NULL
##
## attr(,"model")$pidx
## begin end
## mu 1 1
## ar 2 2
## ma 3 3
## arfima 4 4
## archm 5 5
## mxreg 6 6
## omega 7 7
## alpha 8 8
## beta 9 9
## gamma 10 10
## eta1 11 11
## eta2 12 12
## delta 13 13
## lambda 14 14
## vxreg 15 15
## skew 16 16
## shape 17 17
## ghlambda 18 18
## xi 19 19
##
## attr(,"model")$n.start
## [1] 0
##
## attr(,"class")
## [1] "uGARCHforecast"
## attr(,"class")attr(,"package")
## [1] "rugarch"
#-------------------------------------------------------------------------------
#-------------------------------------------------------------------------------
# (IV) Kaplan-Meir Model and Cummulative Hazard Function Analysis
#Importing Libraries
library(survival)
library(survminer)
## Warning: package 'survminer' was built under R version 4.4.3
## Loading required package: ggplot2
## Loading required package: ggpubr
## Warning: package 'ggpubr' was built under R version 4.4.3
##
## Attaching package: 'ggpubr'
## The following object is masked from 'package:forecast':
##
## gghistogram
##
## Attaching package: 'survminer'
## The following object is masked from 'package:survival':
##
## myeloma
#Dataframe
D1=data.frame(time=c(1,2,2,3,4,5,6,8,9,10),event=c(0,1,0,1,0,1,0,1,0,1))
kmc=with(D1,Surv(time,event));kmc
## [1] 1+ 2 2+ 3 4+ 5 6+ 8 9+ 10
#Fitting the Kaplan-Meir Model
kmc2 = surv_fit(Surv(time, event) ~ 1, data = D1); kmc2
## Call: survfit(formula = Surv(time, event) ~ 1, data = structure(list(
## time = c(1, 2, 2, 3, 4, 5, 6, 8, 9, 10), event = c(0, 1,
## 0, 1, 0, 1, 0, 1, 0, 1)), class = "data.frame", row.names = c(NA,
## -10L)))
##
## n events median 0.95LCL 0.95UCL
## [1,] 10 5 8 5 NA
summary(kmc2)
## Call: survfit(formula = Surv(time, event) ~ 1, data = structure(list(
## time = c(1, 2, 2, 3, 4, 5, 6, 8, 9, 10), event = c(0, 1,
## 0, 1, 0, 1, 0, 1, 0, 1)), class = "data.frame", row.names = c(NA,
## -10L)))
##
## time n.risk n.event survival std.err lower 95% CI upper 95% CI
## 2 9 1 0.889 0.105 0.706 1
## 3 7 1 0.762 0.148 0.521 1
## 5 5 1 0.610 0.181 0.341 1
## 8 3 1 0.406 0.205 0.151 1
## 10 1 1 0.000 NaN NA NA
#Plotting Kaplan-Meir curve
plot(kmc,
col = "blue",
lwd = 2,
xlab = "Time",
ylab = "Survival Probability",
main = "Kaplan-Meier Curve")
# Plotting cumulative hazard
grid()
plot(
kmc2,
fun = "cumhaz",
col = "purple",
lwd = 2,
xlab = "Time (Days)",
ylab = "Cumulative Hazard",
main = "Cumulative Hazard Function"
)
#-------------------------------------------------------------------------------
#-------------------------------------------------------------------------------
#Data Analysis and Plotting in RStudio; First Class
setwd("C:\\Users\\derek\\OneDrive\\Documents\\DDMA")
colortaste_data<-read.csv('colortaste.csv')
head(colortaste_data)
## X color taste weight size eaten
## 1 1 red sweet 1.5 5.7 FALSE
## 2 2 blue sour 1.6 6.8 TRUE
## 3 3 green sour 1.7 6.2 FALSE
## 4 4 orange salty 1.5 5.6 FALSE
## 5 5 purple salty 1.7 5.9 FALSE
## 6 6 orange salty 1.6 5.4 TRUE
# Scatterplot of Size vs Weight
plot(colortaste_data$size,colortaste_data$weight,
xlab='size',ylab='weight',
main='scatteplot of size against weight')
# Aggregate the mean size by color
mean_size <- tapply(colortaste_data$size, colortaste_data$color, mean)
#Bar Plot
barplot(mean_size,
col = 'blue',
xlab = "Color",
ylab = "Average Size",
main = "Bar Plot of Average Size by Color")
#histogram
hist(colortaste_data$weight,
col = 'green',
xlab = 'weight',ylab='frequency',
main = 'histogram of weiight frequency')
#-------------------------------------------------------------------------------
#THE END
#-------------------------------------------------------------------------------
