Load Required Libraries
Load and Prepare Data
data_path <- "financial_Data.csv"
import <- read.csv(data_path)
h <- import$High
log.h <- na.omit(diff(h))
log.h_ts <- ts(log.h)
Exploratory Data Analysis
par(mfrow = c(2, 2))
plot(h, type = "l", main = "Raw High Prices", ylab = "Price")
plot(log.h, type = "l", main = "Log Returns", ylab = "Return")
acf(log.h, main = "ACF of Log Returns")
pacf(log.h, main = "PACF of Log Returns")
par(mfrow = c(1,1))
GARCH(1,1), GARCH(1,2), GARCH(2,1), GARCH(2,2) using
fGarch
garch11_model <- garchFit(~ garch(1, 1), data = log.h, trace = FALSE)
## Warning in sqrt(diag(fit$cvar)): NaNs produced
garch12_model <- garchFit(~ garch(1, 2), data = log.h, trace = FALSE)
## Warning in sqrt(diag(fit$cvar)): NaNs produced
garch21_model <- garchFit(~ garch(2, 1), data = log.h, trace = FALSE)
## Warning in sqrt(diag(fit$cvar)): NaNs produced
garch22_model <- garchFit(~ garch(2, 2), data = log.h, trace = FALSE)
## Warning in sqrt(diag(fit$cvar)): NaNs produced
summary(garch11_model)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = log.h, trace = FALSE)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x0000023546c8e7b0>
## [data = log.h]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## -0.10302076 0.41140792 0.00000001 0.99999999
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -1.030e-01 9.894e-01 -0.104 0.917
## omega 4.114e-01 1.110e+00 0.371 0.711
## alpha1 1.000e-08 NaN NaN NaN
## beta1 1.000e+00 4.215e-03 237.257 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -1030.33 normalized: -4.171378
##
## Description:
## Mon May 5 20:28:20 2025 by user: victo
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 1.812496e+05 0.0000000
## Shapiro-Wilk Test R W 2.702137e-01 0.0000000
## Ljung-Box Test R Q(10) 1.407700e+00 0.9991952
## Ljung-Box Test R Q(15) 2.281588e+00 0.9999295
## Ljung-Box Test R Q(20) 2.822732e+00 0.9999976
## Ljung-Box Test R^2 Q(10) 1.188498e-01 1.0000000
## Ljung-Box Test R^2 Q(15) 1.799509e-01 1.0000000
## Ljung-Box Test R^2 Q(20) 2.529911e-01 1.0000000
## LM Arch Test R TR^2 1.515726e-01 1.0000000
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 8.375146 8.431978 8.374632 8.398027
summary(garch12_model)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 2), data = log.h, trace = FALSE)
##
## Mean and Variance Equation:
## data ~ garch(1, 2)
## <environment: 0x00000235482614d8>
## [data = log.h]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1 beta2
## -0.09197634 0.00025330 0.00000001 0.99999999 0.00153416
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -9.198e-02 9.881e-01 -0.093 0.925840
## omega 2.533e-04 1.510e+00 0.000 0.999866
## alpha1 1.000e-08 NaN NaN NaN
## beta1 1.000e+00 2.611e-01 3.830 0.000128 ***
## beta2 1.534e-03 2.657e-01 0.006 0.995393
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -1030.135 normalized: -4.170587
##
## Description:
## Mon May 5 20:28:20 2025 by user: victo
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 1.783573e+05 0.0000000
## Shapiro-Wilk Test R W 2.715645e-01 0.0000000
## Ljung-Box Test R Q(10) 1.420902e+00 0.9991613
## Ljung-Box Test R Q(15) 2.296993e+00 0.9999264
## Ljung-Box Test R Q(20) 2.842354e+00 0.9999974
## Ljung-Box Test R^2 Q(10) 1.204329e-01 1.0000000
## Ljung-Box Test R^2 Q(15) 1.824890e-01 1.0000000
## Ljung-Box Test R^2 Q(20) 2.566449e-01 1.0000000
## LM Arch Test R TR^2 1.537039e-01 1.0000000
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 8.381660 8.452700 8.380862 8.410261
summary(garch21_model)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(2, 1), data = log.h, trace = FALSE)
##
## Mean and Variance Equation:
## data ~ garch(2, 1)
## <environment: 0x000002354a2a08c8>
## [data = log.h]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 alpha2 beta1
## -0.10189597 0.41736275 0.00000001 0.00000001 0.99999999
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -1.019e-01 9.891e-01 -0.103 0.918
## omega 4.174e-01 1.116e+00 0.374 0.709
## alpha1 1.000e-08 NaN NaN NaN
## alpha2 1.000e-08 NaN NaN NaN
## beta1 1.000e+00 4.233e-03 236.257 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -1030.29 normalized: -4.171213
##
## Description:
## Mon May 5 20:28:20 2025 by user: victo
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 1.809666e+05 0.0000000
## Shapiro-Wilk Test R W 2.703369e-01 0.0000000
## Ljung-Box Test R Q(10) 1.408933e+00 0.9991920
## Ljung-Box Test R Q(15) 2.282837e+00 0.9999293
## Ljung-Box Test R Q(20) 2.824183e+00 0.9999976
## Ljung-Box Test R^2 Q(10) 1.190041e-01 1.0000000
## Ljung-Box Test R^2 Q(15) 1.802002e-01 1.0000000
## Ljung-Box Test R^2 Q(20) 2.533500e-01 1.0000000
## LM Arch Test R TR^2 1.517803e-01 1.0000000
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 8.382912 8.453952 8.382114 8.411513
summary(garch22_model)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(2, 2), data = log.h, trace = FALSE)
##
## Mean and Variance Equation:
## data ~ garch(2, 2)
## <environment: 0x000002354b655628>
## [data = log.h]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 alpha2 beta1 beta2
## -0.09198733 0.00025330 0.00000001 0.00000001 0.99999999 0.00153417
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -9.199e-02 9.881e-01 -0.093 0.925831
## omega 2.533e-04 1.502e+00 0.000 0.999865
## alpha1 1.000e-08 NaN NaN NaN
## alpha2 1.000e-08 NaN NaN NaN
## beta1 1.000e+00 2.608e-01 3.834 0.000126 ***
## beta2 1.534e-03 2.655e-01 0.006 0.995389
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -1030.135 normalized: -4.170587
##
## Description:
## Mon May 5 20:28:20 2025 by user: victo
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 1.783571e+05 0.0000000
## Shapiro-Wilk Test R W 2.715645e-01 0.0000000
## Ljung-Box Test R Q(10) 1.420903e+00 0.9991612
## Ljung-Box Test R Q(15) 2.296994e+00 0.9999264
## Ljung-Box Test R Q(20) 2.842354e+00 0.9999974
## Ljung-Box Test R^2 Q(10) 1.204331e-01 1.0000000
## Ljung-Box Test R^2 Q(15) 1.824893e-01 1.0000000
## Ljung-Box Test R^2 Q(20) 2.566453e-01 1.0000000
## LM Arch Test R TR^2 1.537041e-01 1.0000000
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 8.389758 8.475006 8.388614 8.424079
predict(garch11_model, n.ahead = 120, plot = TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 -0.1030208 18.82311 18.82311 -36.99564 36.78959
## 2 -0.1030208 18.83403 18.83403 -37.01705 36.81101
## 3 -0.1030208 18.84495 18.84495 -37.03845 36.83241
## 4 -0.1030208 18.85586 18.85586 -37.05984 36.85380
## 5 -0.1030208 18.86677 18.86677 -37.08121 36.87517
## 6 -0.1030208 18.87767 18.87767 -37.10258 36.89653
## 7 -0.1030208 18.88856 18.88856 -37.12393 36.91789
## 8 -0.1030208 18.89945 18.89945 -37.14527 36.93922
## 9 -0.1030208 18.91033 18.91033 -37.16659 36.96055
## 10 -0.1030208 18.92121 18.92121 -37.18791 36.98186
## 11 -0.1030208 18.93208 18.93208 -37.20921 37.00317
## 12 -0.1030208 18.94294 18.94294 -37.23050 37.02446
## 13 -0.1030208 18.95379 18.95379 -37.25177 37.04573
## 14 -0.1030208 18.96464 18.96464 -37.27304 37.06700
## 15 -0.1030208 18.97549 18.97549 -37.29429 37.08825
## 16 -0.1030208 18.98632 18.98632 -37.31553 37.10949
## 17 -0.1030208 18.99716 18.99716 -37.33676 37.13072
## 18 -0.1030208 19.00798 19.00798 -37.35798 37.15194
## 19 -0.1030208 19.01880 19.01880 -37.37918 37.17314
## 20 -0.1030208 19.02961 19.02961 -37.40038 37.19434
## 21 -0.1030208 19.04042 19.04042 -37.42156 37.21552
## 22 -0.1030208 19.05122 19.05122 -37.44273 37.23668
## 23 -0.1030208 19.06201 19.06201 -37.46388 37.25784
## 24 -0.1030208 19.07280 19.07280 -37.48503 37.27899
## 25 -0.1030208 19.08358 19.08358 -37.50616 37.30012
## 26 -0.1030208 19.09436 19.09436 -37.52728 37.32124
## 27 -0.1030208 19.10513 19.10513 -37.54839 37.34235
## 28 -0.1030208 19.11589 19.11589 -37.56949 37.36344
## 29 -0.1030208 19.12665 19.12665 -37.59057 37.38453
## 30 -0.1030208 19.13740 19.13740 -37.61164 37.40560
## 31 -0.1030208 19.14815 19.14815 -37.63271 37.42666
## 32 -0.1030208 19.15889 19.15889 -37.65375 37.44771
## 33 -0.1030208 19.16962 19.16962 -37.67479 37.46875
## 34 -0.1030208 19.18035 19.18035 -37.69582 37.48978
## 35 -0.1030208 19.19107 19.19107 -37.71683 37.51079
## 36 -0.1030208 19.20179 19.20179 -37.73784 37.53179
## 37 -0.1030208 19.21250 19.21250 -37.75883 37.55278
## 38 -0.1030208 19.22320 19.22320 -37.77981 37.57376
## 39 -0.1030208 19.23390 19.23390 -37.80077 37.59473
## 40 -0.1030208 19.24459 19.24459 -37.82173 37.61569
## 41 -0.1030208 19.25528 19.25528 -37.84267 37.63663
## 42 -0.1030208 19.26596 19.26596 -37.86360 37.65756
## 43 -0.1030208 19.27663 19.27663 -37.88453 37.67848
## 44 -0.1030208 19.28730 19.28730 -37.90543 37.69939
## 45 -0.1030208 19.29796 19.29796 -37.92633 37.72029
## 46 -0.1030208 19.30862 19.30862 -37.94722 37.74118
## 47 -0.1030208 19.31927 19.31927 -37.96809 37.76205
## 48 -0.1030208 19.32991 19.32991 -37.98896 37.78292
## 49 -0.1030208 19.34055 19.34055 -38.00981 37.80377
## 50 -0.1030208 19.35119 19.35119 -38.03065 37.82461
## 51 -0.1030208 19.36181 19.36181 -38.05148 37.84544
## 52 -0.1030208 19.37243 19.37243 -38.07229 37.86625
## 53 -0.1030208 19.38305 19.38305 -38.09310 37.88706
## 54 -0.1030208 19.39366 19.39366 -38.11390 37.90785
## 55 -0.1030208 19.40426 19.40426 -38.13468 37.92864
## 56 -0.1030208 19.41486 19.41486 -38.15545 37.94941
## 57 -0.1030208 19.42545 19.42545 -38.17621 37.97017
## 58 -0.1030208 19.43604 19.43604 -38.19696 37.99092
## 59 -0.1030208 19.44662 19.44662 -38.21770 38.01166
## 60 -0.1030208 19.45720 19.45720 -38.23842 38.03238
## 61 -0.1030208 19.46777 19.46777 -38.25914 38.05310
## 62 -0.1030208 19.47833 19.47833 -38.27984 38.07380
## 63 -0.1030208 19.48889 19.48889 -38.30054 38.09450
## 64 -0.1030208 19.49944 19.49944 -38.32122 38.11518
## 65 -0.1030208 19.50999 19.50999 -38.34189 38.13585
## 66 -0.1030208 19.52053 19.52053 -38.36255 38.15651
## 67 -0.1030208 19.53106 19.53106 -38.38320 38.17716
## 68 -0.1030208 19.54159 19.54159 -38.40383 38.19779
## 69 -0.1030208 19.55211 19.55211 -38.42446 38.21842
## 70 -0.1030208 19.56263 19.56263 -38.44507 38.23903
## 71 -0.1030208 19.57314 19.57314 -38.46568 38.25964
## 72 -0.1030208 19.58365 19.58365 -38.48627 38.28023
## 73 -0.1030208 19.59415 19.59415 -38.50685 38.30081
## 74 -0.1030208 19.60465 19.60465 -38.52742 38.32138
## 75 -0.1030208 19.61514 19.61514 -38.54798 38.34194
## 76 -0.1030208 19.62562 19.62562 -38.56853 38.36249
## 77 -0.1030208 19.63610 19.63610 -38.58907 38.38303
## 78 -0.1030208 19.64657 19.64657 -38.60960 38.40355
## 79 -0.1030208 19.65704 19.65704 -38.63011 38.42407
## 80 -0.1030208 19.66750 19.66750 -38.65062 38.44458
## 81 -0.1030208 19.67796 19.67796 -38.67111 38.46507
## 82 -0.1030208 19.68841 19.68841 -38.69159 38.48555
## 83 -0.1030208 19.69885 19.69885 -38.71207 38.50602
## 84 -0.1030208 19.70929 19.70929 -38.73253 38.52649
## 85 -0.1030208 19.71973 19.71973 -38.75298 38.54694
## 86 -0.1030208 19.73016 19.73016 -38.77342 38.56738
## 87 -0.1030208 19.74058 19.74058 -38.79385 38.58781
## 88 -0.1030208 19.75100 19.75100 -38.81426 38.60822
## 89 -0.1030208 19.76141 19.76141 -38.83467 38.62863
## 90 -0.1030208 19.77182 19.77182 -38.85507 38.64903
## 91 -0.1030208 19.78222 19.78222 -38.87545 38.66941
## 92 -0.1030208 19.79261 19.79261 -38.89583 38.68979
## 93 -0.1030208 19.80300 19.80300 -38.91619 38.71015
## 94 -0.1030208 19.81339 19.81339 -38.93655 38.73051
## 95 -0.1030208 19.82377 19.82377 -38.95689 38.75085
## 96 -0.1030208 19.83414 19.83414 -38.97722 38.77118
## 97 -0.1030208 19.84451 19.84451 -38.99755 38.79150
## 98 -0.1030208 19.85487 19.85487 -39.01786 38.81182
## 99 -0.1030208 19.86523 19.86523 -39.03816 38.83212
## 100 -0.1030208 19.87558 19.87558 -39.05845 38.85241
## 101 -0.1030208 19.88593 19.88593 -39.07873 38.87269
## 102 -0.1030208 19.89627 19.89627 -39.09900 38.89295
## 103 -0.1030208 19.90661 19.90661 -39.11925 38.91321
## 104 -0.1030208 19.91694 19.91694 -39.13950 38.93346
## 105 -0.1030208 19.92726 19.92726 -39.15974 38.95370
## 106 -0.1030208 19.93758 19.93758 -39.17997 38.97393
## 107 -0.1030208 19.94790 19.94790 -39.20018 38.99414
## 108 -0.1030208 19.95821 19.95821 -39.22039 39.01435
## 109 -0.1030208 19.96851 19.96851 -39.24059 39.03454
## 110 -0.1030208 19.97881 19.97881 -39.26077 39.05473
## 111 -0.1030208 19.98910 19.98910 -39.28095 39.07490
## 112 -0.1030208 19.99939 19.99939 -39.30111 39.09507
## 113 -0.1030208 20.00968 20.00968 -39.32126 39.11522
## 114 -0.1030208 20.01995 20.01995 -39.34141 39.13537
## 115 -0.1030208 20.03023 20.03023 -39.36154 39.15550
## 116 -0.1030208 20.04049 20.04049 -39.38166 39.17562
## 117 -0.1030208 20.05075 20.05075 -39.40178 39.19573
## 118 -0.1030208 20.06101 20.06101 -39.42188 39.21584
## 119 -0.1030208 20.07126 20.07126 -39.44197 39.23593
## 120 -0.1030208 20.08151 20.08151 -39.46205 39.25601
predict(garch12_model, n.ahead = 120, plot = TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 -0.09197634 19.19048 19.19048 -37.70463 37.52067
## 2 -0.09197634 19.20518 19.20518 -37.73343 37.54948
## 3 -0.09197634 19.21989 19.21989 -37.76227 37.57831
## 4 -0.09197634 19.23461 19.23461 -37.79112 37.60717
## 5 -0.09197634 19.24934 19.24934 -37.82000 37.63604
## 6 -0.09197634 19.26409 19.26409 -37.84889 37.66494
## 7 -0.09197634 19.27884 19.27884 -37.87781 37.69386
## 8 -0.09197634 19.29361 19.29361 -37.90676 37.72280
## 9 -0.09197634 19.30839 19.30839 -37.93572 37.75177
## 10 -0.09197634 19.32318 19.32318 -37.96471 37.78075
## 11 -0.09197634 19.33798 19.33798 -37.99372 37.80976
## 12 -0.09197634 19.35279 19.35279 -38.02275 37.83879
## 13 -0.09197634 19.36761 19.36761 -38.05180 37.86785
## 14 -0.09197634 19.38245 19.38245 -38.08088 37.89692
## 15 -0.09197634 19.39729 19.39729 -38.10997 37.92602
## 16 -0.09197634 19.41215 19.41215 -38.13909 37.95514
## 17 -0.09197634 19.42702 19.42702 -38.16824 37.98428
## 18 -0.09197634 19.44190 19.44190 -38.19740 38.01345
## 19 -0.09197634 19.45679 19.45679 -38.22659 38.04263
## 20 -0.09197634 19.47169 19.47169 -38.25580 38.07184
## 21 -0.09197634 19.48661 19.48661 -38.28503 38.10107
## 22 -0.09197634 19.50153 19.50153 -38.31428 38.13033
## 23 -0.09197634 19.51647 19.51647 -38.34356 38.15960
## 24 -0.09197634 19.53142 19.53142 -38.37286 38.18890
## 25 -0.09197634 19.54638 19.54638 -38.40218 38.21822
## 26 -0.09197634 19.56135 19.56135 -38.43152 38.24757
## 27 -0.09197634 19.57633 19.57633 -38.46089 38.27693
## 28 -0.09197634 19.59133 19.59133 -38.49027 38.30632
## 29 -0.09197634 19.60633 19.60633 -38.51969 38.33573
## 30 -0.09197634 19.62135 19.62135 -38.54912 38.36517
## 31 -0.09197634 19.63638 19.63638 -38.57857 38.39462
## 32 -0.09197634 19.65142 19.65142 -38.60805 38.42410
## 33 -0.09197634 19.66647 19.66647 -38.63755 38.45360
## 34 -0.09197634 19.68154 19.68154 -38.66708 38.48313
## 35 -0.09197634 19.69661 19.69661 -38.69662 38.51267
## 36 -0.09197634 19.71170 19.71170 -38.72619 38.54224
## 37 -0.09197634 19.72680 19.72680 -38.75578 38.57183
## 38 -0.09197634 19.74190 19.74190 -38.78540 38.60145
## 39 -0.09197634 19.75703 19.75703 -38.81504 38.63108
## 40 -0.09197634 19.77216 19.77216 -38.84469 38.66074
## 41 -0.09197634 19.78730 19.78730 -38.87438 38.69042
## 42 -0.09197634 19.80246 19.80246 -38.90408 38.72013
## 43 -0.09197634 19.81763 19.81763 -38.93381 38.74986
## 44 -0.09197634 19.83281 19.83281 -38.96356 38.77961
## 45 -0.09197634 19.84800 19.84800 -38.99333 38.80938
## 46 -0.09197634 19.86320 19.86320 -39.02313 38.83918
## 47 -0.09197634 19.87841 19.87841 -39.05295 38.86899
## 48 -0.09197634 19.89364 19.89364 -39.08279 38.89884
## 49 -0.09197634 19.90887 19.90887 -39.11265 38.92870
## 50 -0.09197634 19.92412 19.92412 -39.14254 38.95859
## 51 -0.09197634 19.93938 19.93938 -39.17245 38.98850
## 52 -0.09197634 19.95466 19.95466 -39.20238 39.01843
## 53 -0.09197634 19.96994 19.96994 -39.23234 39.04839
## 54 -0.09197634 19.98524 19.98524 -39.26232 39.07837
## 55 -0.09197634 20.00054 20.00054 -39.29232 39.10837
## 56 -0.09197634 20.01586 20.01586 -39.32234 39.13839
## 57 -0.09197634 20.03119 20.03119 -39.35239 39.16844
## 58 -0.09197634 20.04654 20.04654 -39.38246 39.19851
## 59 -0.09197634 20.06189 20.06189 -39.41256 39.22860
## 60 -0.09197634 20.07726 20.07726 -39.44267 39.25872
## 61 -0.09197634 20.09263 20.09263 -39.47281 39.28886
## 62 -0.09197634 20.10802 20.10802 -39.50298 39.31902
## 63 -0.09197634 20.12342 20.12342 -39.53316 39.34921
## 64 -0.09197634 20.13884 20.13884 -39.56337 39.37942
## 65 -0.09197634 20.15426 20.15426 -39.59360 39.40965
## 66 -0.09197634 20.16970 20.16970 -39.62386 39.43991
## 67 -0.09197634 20.18515 20.18515 -39.65414 39.47018
## 68 -0.09197634 20.20061 20.20061 -39.68444 39.50049
## 69 -0.09197634 20.21608 20.21608 -39.71476 39.53081
## 70 -0.09197634 20.23156 20.23156 -39.74511 39.56116
## 71 -0.09197634 20.24706 20.24706 -39.77548 39.59153
## 72 -0.09197634 20.26257 20.26257 -39.80588 39.62192
## 73 -0.09197634 20.27809 20.27809 -39.83630 39.65234
## 74 -0.09197634 20.29362 20.29362 -39.86674 39.68278
## 75 -0.09197634 20.30916 20.30916 -39.89720 39.71325
## 76 -0.09197634 20.32472 20.32472 -39.92769 39.74374
## 77 -0.09197634 20.34028 20.34028 -39.95820 39.77425
## 78 -0.09197634 20.35586 20.35586 -39.98873 39.80478
## 79 -0.09197634 20.37145 20.37145 -40.01929 39.83534
## 80 -0.09197634 20.38706 20.38706 -40.04987 39.86592
## 81 -0.09197634 20.40267 20.40267 -40.08048 39.89652
## 82 -0.09197634 20.41830 20.41830 -40.11110 39.92715
## 83 -0.09197634 20.43394 20.43394 -40.14176 39.95780
## 84 -0.09197634 20.44959 20.44959 -40.17243 39.98848
## 85 -0.09197634 20.46525 20.46525 -40.20313 40.01918
## 86 -0.09197634 20.48092 20.48092 -40.23385 40.04990
## 87 -0.09197634 20.49661 20.49661 -40.26460 40.08064
## 88 -0.09197634 20.51231 20.51231 -40.29536 40.11141
## 89 -0.09197634 20.52802 20.52802 -40.32616 40.14220
## 90 -0.09197634 20.54374 20.54374 -40.35697 40.17302
## 91 -0.09197634 20.55948 20.55948 -40.38781 40.20386
## 92 -0.09197634 20.57522 20.57522 -40.41868 40.23472
## 93 -0.09197634 20.59098 20.59098 -40.44956 40.26561
## 94 -0.09197634 20.60675 20.60675 -40.48047 40.29652
## 95 -0.09197634 20.62254 20.62254 -40.51141 40.32745
## 96 -0.09197634 20.63833 20.63833 -40.54236 40.35841
## 97 -0.09197634 20.65414 20.65414 -40.57335 40.38939
## 98 -0.09197634 20.66996 20.66996 -40.60435 40.42040
## 99 -0.09197634 20.68579 20.68579 -40.63538 40.45143
## 100 -0.09197634 20.70163 20.70163 -40.66643 40.48248
## 101 -0.09197634 20.71749 20.71749 -40.69751 40.51356
## 102 -0.09197634 20.73336 20.73336 -40.72861 40.54466
## 103 -0.09197634 20.74924 20.74924 -40.75973 40.57578
## 104 -0.09197634 20.76513 20.76513 -40.79088 40.60693
## 105 -0.09197634 20.78103 20.78103 -40.82205 40.63810
## 106 -0.09197634 20.79695 20.79695 -40.85325 40.66929
## 107 -0.09197634 20.81288 20.81288 -40.88447 40.70051
## 108 -0.09197634 20.82882 20.82882 -40.91571 40.73176
## 109 -0.09197634 20.84477 20.84477 -40.94698 40.76302
## 110 -0.09197634 20.86074 20.86074 -40.97827 40.79432
## 111 -0.09197634 20.87671 20.87671 -41.00958 40.82563
## 112 -0.09197634 20.89270 20.89270 -41.04092 40.85697
## 113 -0.09197634 20.90870 20.90870 -41.07228 40.88833
## 114 -0.09197634 20.92472 20.92472 -41.10367 40.91972
## 115 -0.09197634 20.94075 20.94075 -41.13508 40.95113
## 116 -0.09197634 20.95678 20.95678 -41.16652 40.98257
## 117 -0.09197634 20.97283 20.97283 -41.19798 41.01402
## 118 -0.09197634 20.98890 20.98890 -41.22946 41.04551
## 119 -0.09197634 21.00497 21.00497 -41.26097 41.07701
## 120 -0.09197634 21.02106 21.02106 -41.29250 41.10855
predict(garch22_model, n.ahead = 120, plot = TRUE)
## meanForecast meanError standardDeviation lowerInterval upperInterval
## 1 -0.09198733 19.19051 19.19051 -37.70470 37.52073
## 2 -0.09198733 19.20521 19.20521 -37.73351 37.54954
## 3 -0.09198733 19.21992 19.21992 -37.76234 37.57837
## 4 -0.09198733 19.23464 19.23464 -37.79120 37.60722
## 5 -0.09198733 19.24938 19.24938 -37.82007 37.63610
## 6 -0.09198733 19.26412 19.26412 -37.84897 37.66500
## 7 -0.09198733 19.27888 19.27888 -37.87789 37.69392
## 8 -0.09198733 19.29364 19.29364 -37.90683 37.72286
## 9 -0.09198733 19.30842 19.30842 -37.93580 37.75182
## 10 -0.09198733 19.32321 19.32321 -37.96479 37.78081
## 11 -0.09198733 19.33801 19.33801 -37.99380 37.80982
## 12 -0.09198733 19.35282 19.35282 -38.02283 37.83885
## 13 -0.09198733 19.36765 19.36765 -38.05188 37.86791
## 14 -0.09198733 19.38248 19.38248 -38.08096 37.89698
## 15 -0.09198733 19.39733 19.39733 -38.11005 37.92608
## 16 -0.09198733 19.41219 19.41219 -38.13917 37.95520
## 17 -0.09198733 19.42706 19.42706 -38.16832 37.98434
## 18 -0.09198733 19.44194 19.44194 -38.19748 38.01351
## 19 -0.09198733 19.45683 19.45683 -38.22667 38.04269
## 20 -0.09198733 19.47173 19.47173 -38.25588 38.07190
## 21 -0.09198733 19.48664 19.48664 -38.28511 38.10113
## 22 -0.09198733 19.50157 19.50157 -38.31436 38.13039
## 23 -0.09198733 19.51651 19.51651 -38.34364 38.15967
## 24 -0.09198733 19.53146 19.53146 -38.37294 38.18896
## 25 -0.09198733 19.54642 19.54642 -38.40226 38.21829
## 26 -0.09198733 19.56139 19.56139 -38.43160 38.24763
## 27 -0.09198733 19.57637 19.57637 -38.46097 38.27700
## 28 -0.09198733 19.59137 19.59137 -38.49036 38.30639
## 29 -0.09198733 19.60637 19.60637 -38.51977 38.33580
## 30 -0.09198733 19.62139 19.62139 -38.54920 38.36523
## 31 -0.09198733 19.63642 19.63642 -38.57866 38.39469
## 32 -0.09198733 19.65146 19.65146 -38.60814 38.42417
## 33 -0.09198733 19.66651 19.66651 -38.63764 38.45367
## 34 -0.09198733 19.68157 19.68157 -38.66717 38.48319
## 35 -0.09198733 19.69665 19.69665 -38.69671 38.51274
## 36 -0.09198733 19.71174 19.71174 -38.72628 38.54231
## 37 -0.09198733 19.72683 19.72683 -38.75587 38.57190
## 38 -0.09198733 19.74194 19.74194 -38.78549 38.60151
## 39 -0.09198733 19.75707 19.75707 -38.81512 38.63115
## 40 -0.09198733 19.77220 19.77220 -38.84478 38.66081
## 41 -0.09198733 19.78734 19.78734 -38.87447 38.69049
## 42 -0.09198733 19.80250 19.80250 -38.90417 38.72020
## 43 -0.09198733 19.81767 19.81767 -38.93390 38.74993
## 44 -0.09198733 19.83285 19.83285 -38.96365 38.77968
## 45 -0.09198733 19.84804 19.84804 -38.99342 38.80945
## 46 -0.09198733 19.86324 19.86324 -39.02322 38.83925
## 47 -0.09198733 19.87845 19.87845 -39.05304 38.86907
## 48 -0.09198733 19.89368 19.89368 -39.08288 38.89891
## 49 -0.09198733 19.90892 19.90892 -39.11275 38.92877
## 50 -0.09198733 19.92417 19.92417 -39.14263 38.95866
## 51 -0.09198733 19.93943 19.93943 -39.17254 38.98857
## 52 -0.09198733 19.95470 19.95470 -39.20248 39.01850
## 53 -0.09198733 19.96998 19.96998 -39.23243 39.04846
## 54 -0.09198733 19.98528 19.98528 -39.26241 39.07844
## 55 -0.09198733 20.00059 20.00059 -39.29242 39.10844
## 56 -0.09198733 20.01591 20.01591 -39.32244 39.13847
## 57 -0.09198733 20.03124 20.03124 -39.35249 39.16851
## 58 -0.09198733 20.04658 20.04658 -39.38256 39.19858
## 59 -0.09198733 20.06193 20.06193 -39.41265 39.22868
## 60 -0.09198733 20.07730 20.07730 -39.44277 39.25880
## 61 -0.09198733 20.09268 20.09268 -39.47291 39.28894
## 62 -0.09198733 20.10807 20.10807 -39.50307 39.31910
## 63 -0.09198733 20.12347 20.12347 -39.53326 39.34929
## 64 -0.09198733 20.13888 20.13888 -39.56347 39.37950
## 65 -0.09198733 20.15431 20.15431 -39.59370 39.40973
## 66 -0.09198733 20.16974 20.16974 -39.62396 39.43998
## 67 -0.09198733 20.18519 20.18519 -39.65424 39.47026
## 68 -0.09198733 20.20065 20.20065 -39.68454 39.50056
## 69 -0.09198733 20.21612 20.21612 -39.71486 39.53089
## 70 -0.09198733 20.23161 20.23161 -39.74521 39.56124
## 71 -0.09198733 20.24711 20.24711 -39.77558 39.59161
## 72 -0.09198733 20.26261 20.26261 -39.80598 39.62200
## 73 -0.09198733 20.27813 20.27813 -39.83640 39.65242
## 74 -0.09198733 20.29366 20.29366 -39.86684 39.68286
## 75 -0.09198733 20.30921 20.30921 -39.89730 39.71333
## 76 -0.09198733 20.32476 20.32476 -39.92779 39.74382
## 77 -0.09198733 20.34033 20.34033 -39.95830 39.77433
## 78 -0.09198733 20.35591 20.35591 -39.98884 39.80486
## 79 -0.09198733 20.37150 20.37150 -40.01940 39.83542
## 80 -0.09198733 20.38710 20.38710 -40.04998 39.86600
## 81 -0.09198733 20.40272 20.40272 -40.08058 39.89661
## 82 -0.09198733 20.41835 20.41835 -40.11121 39.92724
## 83 -0.09198733 20.43398 20.43398 -40.14186 39.95789
## 84 -0.09198733 20.44964 20.44964 -40.17254 39.98856
## 85 -0.09198733 20.46530 20.46530 -40.20324 40.01926
## 86 -0.09198733 20.48097 20.48097 -40.23396 40.04998
## 87 -0.09198733 20.49666 20.49666 -40.26470 40.08073
## 88 -0.09198733 20.51236 20.51236 -40.29547 40.11150
## 89 -0.09198733 20.52807 20.52807 -40.32627 40.14229
## 90 -0.09198733 20.54379 20.54379 -40.35708 40.17311
## 91 -0.09198733 20.55953 20.55953 -40.38792 40.20395
## 92 -0.09198733 20.57527 20.57527 -40.41879 40.23481
## 93 -0.09198733 20.59103 20.59103 -40.44967 40.26570
## 94 -0.09198733 20.60680 20.60680 -40.48058 40.29661
## 95 -0.09198733 20.62259 20.62259 -40.51152 40.32754
## 96 -0.09198733 20.63838 20.63838 -40.54248 40.35850
## 97 -0.09198733 20.65419 20.65419 -40.57346 40.38948
## 98 -0.09198733 20.67001 20.67001 -40.60446 40.42049
## 99 -0.09198733 20.68584 20.68584 -40.63549 40.45152
## 100 -0.09198733 20.70169 20.70169 -40.66654 40.48257
## 101 -0.09198733 20.71754 20.71754 -40.69762 40.51365
## 102 -0.09198733 20.73341 20.73341 -40.72872 40.54475
## 103 -0.09198733 20.74929 20.74929 -40.75985 40.57587
## 104 -0.09198733 20.76518 20.76518 -40.79099 40.60702
## 105 -0.09198733 20.78109 20.78109 -40.82217 40.63819
## 106 -0.09198733 20.79700 20.79700 -40.85336 40.66939
## 107 -0.09198733 20.81293 20.81293 -40.88458 40.70061
## 108 -0.09198733 20.82887 20.82887 -40.91583 40.73185
## 109 -0.09198733 20.84482 20.84482 -40.94709 40.76312
## 110 -0.09198733 20.86079 20.86079 -40.97838 40.79441
## 111 -0.09198733 20.87677 20.87677 -41.00970 40.82573
## 112 -0.09198733 20.89276 20.89276 -41.04104 40.85706
## 113 -0.09198733 20.90876 20.90876 -41.07240 40.88843
## 114 -0.09198733 20.92477 20.92477 -41.10379 40.91982
## 115 -0.09198733 20.94080 20.94080 -41.13520 40.95123
## 116 -0.09198733 20.95684 20.95684 -41.16664 40.98266
## 117 -0.09198733 20.97289 20.97289 -41.19810 41.01412
## 118 -0.09198733 20.98895 20.98895 -41.22958 41.04561
## 119 -0.09198733 21.00503 21.00503 -41.26109 41.07711
## 120 -0.09198733 21.02112 21.02112 -41.29262 41.10864
eGARCH Models using rugarch
# eGARCH Models using `rugarch`
# Define eGARCH model specifications
egarch_models <- list(
egarch11 = ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE),
distribution.model = "norm"
),
egarch12 = ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE),
distribution.model = "norm"
),
egarch21 = ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE),
distribution.model = "norm"
),
egarch22 = ugarchspec(
variance.model = list(model = "eGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE),
distribution.model = "norm"
)
)
# Fit models to the data (ensure `log.h` is defined and numeric)
fit_egarch <- lapply(egarch_models, function(spec) {
ugarchfit(spec = spec, data = log.h, solver = "hybrid")
})
# Extract model coefficients
egarch_coefs <- lapply(fit_egarch, coef)
print(egarch_coefs)
## $egarch11
## mu omega alpha1 beta1 gamma1
## 0.279731070 3.549580822 1.003625132 -0.005222232 1.055261970
##
## $egarch12
## mu omega alpha1 beta1 beta2 gamma1
## -1.4500122 6.4319537 2.4091016 0.3962988 -0.3797081 2.9621385
##
## $egarch21
## mu omega alpha1 alpha2 beta1 gamma1
## 0.08387409 -0.08023885 0.11971505 0.04468266 0.99999999 -0.54928852
## gamma2
## 0.42233103
##
## $egarch22
## mu omega alpha1 alpha2 beta1 beta2 gamma1
## 0.1654386 -0.2142866 0.9316198 -0.2810897 0.6377920 0.4064154 0.5958231
## gamma2
## -0.4636316
# Optional: Forecast using the fitted models
egarch_forecasts <- lapply(fit_egarch, function(fit) ugarchforecast(fit, n.ahead = 10))
# You can inspect a specific forecast like this:
print(egarch_forecasts$egarch11)
##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: eGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=1970-09-05]:
## Series Sigma
## T+1 0.2797 3.848
## T+2 0.2797 5.858
## T+3 0.2797 5.845
## T+4 0.2797 5.845
## T+5 0.2797 5.845
## T+6 0.2797 5.845
## T+7 0.2797 5.845
## T+8 0.2797 5.845
## T+9 0.2797 5.845
## T+10 0.2797 5.845
# Optional: Plot diagnostics
#lapply(fit_egarch, plot)
GJR-GARCH Models using rugarch
gjr_models <- list(
gjr11 = ugarchspec(variance.model = list(model="gjrGARCH", garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0)), distribution.model="norm"),
gjr12 = ugarchspec(variance.model = list(model="gjrGARCH", garchOrder=c(1,2)), mean.model=list(armaOrder=c(0,0)), distribution.model="norm"),
gjr21 = ugarchspec(variance.model = list(model="gjrGARCH", garchOrder=c(2,1)), mean.model=list(armaOrder=c(0,0)), distribution.model="norm"),
gjr22 = ugarchspec(variance.model = list(model="gjrGARCH", garchOrder=c(2,2)), mean.model=list(armaOrder=c(0,0)), distribution.model="norm")
)
fit_gjr_models <- lapply(gjr_models, function(spec) ugarchfit(data=log.h, spec=spec, solver="hybrid"))
lapply(fit_gjr_models, coef)
## $gjr11
## mu omega alpha1 beta1 gamma1
## -0.93841343 0.41903772 0.01262167 0.99557055 -0.01928445
##
## $gjr12
## mu omega alpha1 beta1 beta2
## 7.382630e-02 1.069272e-01 1.036676e-20 3.928563e-01 6.144344e-01
## gamma1
## -1.985695e-02
##
## $gjr21
## mu omega alpha1 alpha2 beta1
## 1.811668e-01 1.664226e-06 1.065664e-10 3.628408e-18 9.885479e-01
## gamma1 gamma2
## -9.252576e-01 9.139672e-01
##
## $gjr22
## mu omega alpha1 alpha2 beta1
## 0.3009698435 0.0003810766 0.0062263714 0.1428815489 0.0476039400
## beta2 gamma1 gamma2
## 0.8234459067 -0.0067442899 -0.1424269253
lapply(fit_gjr_models, summary)
## $gjr11
## Length Class Mode
## 1 uGARCHfit S4
##
## $gjr12
## Length Class Mode
## 1 uGARCHfit S4
##
## $gjr21
## Length Class Mode
## 1 uGARCHfit S4
##
## $gjr22
## Length Class Mode
## 1 uGARCHfit S4
# lapply(fit_gjr_models, plot)
Compare GJR-GARCH Models
model_names <- names(fit_gjr_models)
gjr_ic <- sapply(fit_gjr_models, function(fit) infocriteria(fit))
colnames(gjr_ic) <- model_names
gjr_llh <- sapply(fit_gjr_models, function(fit) likelihood(fit))
names(gjr_llh) <- model_names
print("Information Criteria (lower is better):")
## [1] "Information Criteria (lower is better):"
print(round(gjr_ic, 4))
## gjr11 gjr12 gjr21 gjr22
## [1,] 7.7157 7.4058 6.9727 6.9024
## [2,] 7.7867 7.4910 7.0722 7.0161
## [3,] 7.7149 7.4046 6.9712 6.9004
## [4,] 7.7443 7.4401 7.0128 6.9482
print("Log-Likelihoods (higher is better):")
## [1] "Log-Likelihoods (higher is better):"
print(round(gjr_llh, 4))
## gjr11 gjr12 gjr21 gjr22
## -947.8897 -908.6129 -854.1328 -844.4491
GJR-GARCH(2,2) with Different Distributions
fit_gjr_by_dist <- function(distribution) {
spec <- ugarchspec(
variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(0, 0)),
distribution.model = distribution
)
ugarchfit(spec = spec, data = log.h, solver = "hybrid")
}
model_norm <- fit_gjr_by_dist("norm")
model_std <- fit_gjr_by_dist("std")
model_ged <- fit_gjr_by_dist("ged")
ic_table <- cbind(
norm = infocriteria(model_norm),
std = infocriteria(model_std),
ged = infocriteria(model_ged)
)
colnames(ic_table) <- c("Normal", "Student-t", "GED")
print(round(ic_table, 4))
## Normal Student-t GED
## Akaike 6.8026 5.7560 15.4882
## Bayes 6.9163 5.8839 15.6161
## Shibata 6.8006 5.7535 15.4857
## Hannan-Quinn 6.8484 5.8075 15.5397
llh <- c(
Normal = likelihood(model_norm),
Student_t = likelihood(model_std),
GED = likelihood(model_ged)
)
print(round(llh, 4))
## Normal Student_t GED
## -832.1244 -701.8682 -1903.7953
Extra: Built-in Data Extraction and Correlation/Regression
Analysis
# Extract and preview data
data2 <- read.csv("financial_Data.csv")
data3 <- data2$Adj.Close
# Visualization
hist(data3)
barplot(data3)
hist(data3, col = "blue")
# Regression and Correlation
d <- data2$Close
h <- data2$High
g <- data2$Low
df <- data.frame(d, h, g)
cor(d, h)
## [1] 0.9996436
cor(d, g)
## [1] 0.9996625
cor(h, g)
## [1] 0.999654
lm_model <- lm(d ~ g + h)
aov(lm_model)
## Call:
## aov(formula = lm_model)
##
## Terms:
## g h Residuals
## Sum of Squares 2064833.6 319.3 1075.1
## Deg. of Freedom 1 1 245
##
## Residual standard error: 2.094752
## Estimated effects may be unbalanced
summary(lm_model)
##
## Call:
## lm(formula = d ~ g + h)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.0668 -1.0546 -0.0468 1.0636 7.2652
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.17232 0.34677 0.497 0.62
## g 0.53490 0.05621 9.517 < 2e-16 ***
## h 0.46548 0.05456 8.531 1.53e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.095 on 245 degrees of freedom
## Multiple R-squared: 0.9995, Adjusted R-squared: 0.9995
## F-statistic: 2.353e+05 on 2 and 245 DF, p-value: < 2.2e-16
# Plots
hist(d)
hist(h)
hist(g)
plot(lm_model)
