A Volatility Analysis of a Four-Cryptocurrency Portfolio
Mateusz Kowalski & Aleksandra Ciesińska
2024-07-04
We will be using the following libraries:
library(dplyr)
library(xts)
library(fBasics)
library(tseries)
library(car)
library(FinTS)
library(fGarch)
library(rugarch)
library(ggplot2)
library(tidyr)
library(scales)1. Data
We will be analyzing a portfolio that consists of 4 cryptocurrencies: Avalanche, Cardano, Komodo and Monero. Each day, the percentage allocation of each cryptocurrency in the portfolio is determined by its market capitalization. The plot below illustrates the daily allocation of each cryptocurrency within the portfolio (please note: Komodo’s allocation is approx. 0.1%).
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.2525899 -0.0206787 0.0011353 -0.0007867 0.0204785 0.1559860
2. GARCH models
In this section we will be considering 5 chosen types of GARCH models: GARCH, EGARCH, GARCH-t, TGARCH and GARCH-In-Mean. If the model turns out successful, there are going to be additional plots presented and only 1 final model shown. Otherwise, we will present the process of finding the optimal model, indicating signs of misspecification within these models.
2.1. GARCH(1,1)
##
## *---------------------------------*
## * 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 0.000131 0.000035 3.7184 0.000201
## alpha1 0.168103 0.033779 4.9766 0.000001
## beta1 0.764141 0.041760 18.2982 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000131 0.000059 2.2213 0.026328
## alpha1 0.168103 0.048569 3.4611 0.000538
## beta1 0.764141 0.059514 12.8396 0.000000
##
## LogLikelihood : 2014.659
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.6776
## Bayes -3.6639
## Shibata -3.6776
## Hannan-Quinn -3.6724
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01262 0.9105
## Lag[2*(p+q)+(p+q)-1][2] 0.05876 0.9496
## Lag[4*(p+q)+(p+q)-1][5] 3.30202 0.3547
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4155 0.5192
## Lag[2*(p+q)+(p+q)-1][5] 1.1754 0.8189
## Lag[4*(p+q)+(p+q)-1][9] 2.4711 0.8420
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.4745 0.500 2.000 0.4909
## ARCH Lag[5] 1.5460 1.440 1.667 0.5803
## ARCH Lag[7] 2.4837 2.315 1.543 0.6158
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.6361
## Individual Statistics:
## omega 0.4032
## alpha1 0.4343
## beta1 0.5229
##
## 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 0.2067 0.8363
## Negative Sign Bias 0.1548 0.8770
## Positive Sign Bias 1.4243 0.1546
## Joint Effect 4.1789 0.2428
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 78.94 2.827e-09
## 2 30 85.05 2.063e-07
## 3 40 96.64 8.509e-07
## 4 50 109.93 1.424e-06
##
##
## Elapsed time : 0.151243##
## please wait...calculating quantiles...
Parameters
- omega (cons.): insignificant in the robust variant
- alpha1, beta1 > 0: variance greater than 0
- alpha1 + beta1 < 1: GARCH(1,1) process is stationary
Tests & Plots
- (+/-) Autocorrelation of st. residuals: no
autocorrelation up to the last lag taken into account
- but! ACF plot: 3rd and 9th lags are problematic
- (+) Autocorrelation of sq. st. residuals: no
autocorrelation up to the last lag taken into account
- ACF: results are consistent
- (+) Nyblom Stability Test: not rejecting H0 postulating parameters’ stability
- (+) LM Arch Test: no ARCH effects up to the last lag taken into account
- (+) Sign Test: not rejecting H0 postulating no presence of leverage effects in the standardized residuals
- (-) Pearson Goodness-of-Fit: rejecting H0 postulating that the empirical distribution is equal to the assumed distribution (irrespective of the choice of bins)
Leverage Effects
Leverage effects are understood as the impact of positive and negative return shocks on volatility not predicted by the model under consideration (their presence indicates misspecification).
More complicated versions of this GARCH type model produced worse results. Thus, there is no point in making the model too complex if the simpler alternative provides better fit.
2.2. GARCH-t(1,1)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00008 0.000033 2.3918 0.016767
## alpha1 0.17848 0.045236 3.9455 0.000080
## beta1 0.79929 0.045819 17.4444 0.000000
## shape 4.51930 0.626864 7.2094 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00008 0.000042 1.9212 0.054712
## alpha1 0.17848 0.048602 3.6722 0.000240
## beta1 0.79929 0.055004 14.5316 0.000000
## shape 4.51930 0.566515 7.9774 0.000000
##
## LogLikelihood : 2062.36
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.7630
## Bayes -3.7447
## Shibata -3.7630
## Hannan-Quinn -3.7561
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04558 0.8309
## Lag[2*(p+q)+(p+q)-1][2] 0.15941 0.8798
## Lag[4*(p+q)+(p+q)-1][5] 3.41980 0.3357
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4378 0.5082
## Lag[2*(p+q)+(p+q)-1][5] 1.1534 0.8241
## Lag[4*(p+q)+(p+q)-1][9] 2.4370 0.8470
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.479 0.500 2.000 0.4889
## ARCH Lag[5] 1.456 1.440 1.667 0.6040
## ARCH Lag[7] 2.357 2.315 1.543 0.6421
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.0066
## Individual Statistics:
## omega 0.6794
## alpha1 0.5139
## beta1 0.7364
## shape 0.5475
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.3396 0.7342
## Negative Sign Bias 0.1630 0.8705
## Positive Sign Bias 1.5162 0.1298
## Joint Effect 4.5255 0.2100
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 28.12 0.08113
## 2 30 36.84 0.15034
## 3 40 49.62 0.11862
## 4 50 58.65 0.16263
##
##
## Elapsed time : 0.2038231##
## please wait...calculating quantiles...
Parameters
- omega (cons.): insignificant in the robust variant
- alpha1, beta1 > 0: variance greater than 0
- alpha1 + beta1 < 1: GARCH(1,1) process is stationary
Tests & Plots
- (+/-) Autocorrelation of st. residuals: no
autocorrelation up to the last lag taken into account
- but! ACF plot: 3rd and 9th lags are problematic
- (+) Autocorrelation of sq. st. residuals: no
autocorrelation up to the last lag taken into account
- ACF: results are consistent
- (+) Nyblom Stability Test: not rejecting H0 postulating parameters’ stability
- (+) LM Arch Test: no ARCH effects up to the last lag taken into account
- (+) Sign Test: not rejecting H0 postulating no presence of leverage effects in the standardized residuals
- (-/0) Pearson Goodness-of-Fit: rejecting H0 postulating that the empirical distribution is equal to the assumed distribution (irrespective of the choice of bins), however, choosing 20 bins we would be able not to reject it with alpha = 10%.
Also, the QQPlot looks way better - theoretical quantiles fit empirical ones better than while considering GARCH(1,1).
To sum up, GARCH-t(1,1) seems to be better than GARCH(1,1). From now on, we will always check variants with t-distributed residuals.
2.3. EGARCH(1,1)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.33897 0.133239 -2.54408 0.010956
## alpha1 0.01236 0.028572 0.43258 0.665317
## beta1 0.94802 0.020386 46.50456 0.000000
## gamma1 0.29957 0.062693 4.77842 0.000002
## shape 4.60801 0.646964 7.12251 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.33897 0.148015 -2.29012 0.022015
## alpha1 0.01236 0.029344 0.42121 0.673598
## beta1 0.94802 0.022645 41.86429 0.000000
## gamma1 0.29957 0.071151 4.21044 0.000025
## shape 4.60801 0.583926 7.89142 0.000000
##
## LogLikelihood : 2067.08
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.7698
## Bayes -3.7470
## Shibata -3.7698
## Hannan-Quinn -3.7612
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01937 0.8893
## Lag[2*(p+q)+(p+q)-1][2] 0.13158 0.8980
## Lag[4*(p+q)+(p+q)-1][5] 3.41791 0.3360
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.001189 0.9725
## Lag[2*(p+q)+(p+q)-1][5] 0.850823 0.8924
## Lag[4*(p+q)+(p+q)-1][9] 2.247564 0.8735
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.4555 0.500 2.000 0.4997
## ARCH Lag[5] 1.4314 1.440 1.667 0.6105
## ARCH Lag[7] 2.3497 2.315 1.543 0.6436
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.1014
## Individual Statistics:
## omega 0.66120
## alpha1 0.28489
## beta1 0.63525
## gamma1 0.04854
## shape 0.27686
##
## 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 0.1694 0.8655
## Negative Sign Bias 0.3846 0.7006
## Positive Sign Bias 1.4740 0.1408
## Joint Effect 4.9123 0.1783
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 28.45 0.07515
## 2 30 37.06 0.14469
## 3 40 43.11 0.29977
## 4 50 49.05 0.47100
##
##
## Elapsed time : 0.2431729##
## please wait...calculating quantiles...
Parameters
- alpha1: insignificant
- Variance: is always greater than 0 in this GARCH type
- gamma > 0: variance reaction to shocks is symmetrical (EGARCH allows negative gamma meaning that the reaction is not symmetrical)
Tests & Plots
- (+/-) Autocorrelation of st. residuals: no
autocorrelation up to the last lag taken into account
- but! ACF plot: 3rd and 9th lags are problematic
- (+) Autocorrelation of sq. st. residuals: no
autocorrelation up to the last lag taken into account
- ACF: results are consistent
- (+) Nyblom Stability Test: not rejecting H0 postulating parameters’ stability
- (+) LM Arch Test: no ARCH effects up to the last lag taken into account
- (+) Sign Test: not rejecting H0 postulating no presence of leverage effects in the standardized residuals
- (-) Pearson Goodness-of-Fit: rejecting H0 postulating that the empirical distribution is equal to the assumed distribution (irrespective of the choice of bins)
More complicated versions of this GARCH model type produced worse results.
2.4. Threshold GARCH
Version I: TGARCH(1, 1)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(1,1)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001630 0.000827 1.97222 0.048584
## alpha1 0.156684 0.039685 3.94821 0.000079
## beta1 0.846300 0.044727 18.92159 0.000000
## eta11 -0.066003 0.109022 -0.60542 0.544903
## shape 4.631263 0.645590 7.17369 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001630 0.001105 1.47590 0.139971
## alpha1 0.156684 0.050909 3.07775 0.002086
## beta1 0.846300 0.061119 13.84677 0.000000
## eta11 -0.066003 0.125133 -0.52746 0.597871
## shape 4.631263 0.583568 7.93611 0.000000
##
## LogLikelihood : 2067.062
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.7698
## Bayes -3.7469
## Shibata -3.7698
## Hannan-Quinn -3.7611
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.002624 0.9591
## Lag[2*(p+q)+(p+q)-1][2] 0.125971 0.9018
## Lag[4*(p+q)+(p+q)-1][5] 3.436014 0.3332
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1333 0.7150
## Lag[2*(p+q)+(p+q)-1][5] 1.0194 0.8553
## Lag[4*(p+q)+(p+q)-1][9] 2.4705 0.8421
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.3776 0.500 2.000 0.5389
## ARCH Lag[5] 1.4176 1.440 1.667 0.6142
## ARCH Lag[7] 2.3352 2.315 1.543 0.6466
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.0844
## Individual Statistics:
## omega 0.5694
## alpha1 0.5021
## beta1 0.5919
## eta11 0.3485
## shape 0.4554
##
## 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 0.1387 0.8897
## Negative Sign Bias 0.5371 0.5913
## Positive Sign Bias 1.4815 0.1388
## Joint Effect 5.3174 0.1500
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 29.07 0.06487
## 2 30 34.76 0.21264
## 3 40 54.23 0.05332
## 4 50 59.38 0.14710
##
##
## Elapsed time : 0.3373291The most important observation is that eta11 is insignificant which makes this model tantamount to ordinary GARCH.
Version II: TGARCH(2, 1)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(2,1)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001632 0.000826 1.975852 0.048172
## alpha1 0.156577 0.039609 3.953025 0.000077
## alpha2 0.000000 0.000018 0.000027 0.999978
## beta1 0.846266 0.044689 18.936649 0.000000
## eta11 -0.066999 0.109084 -0.614196 0.539086
## eta12 0.164905 0.373495 0.441518 0.658838
## shape 4.643109 0.646818 7.178388 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001632 0.001105 1.476890 0.139705
## alpha1 0.156577 0.050846 3.079439 0.002074
## alpha2 0.000000 0.000000 0.040246 0.967897
## beta1 0.846266 0.061125 13.844912 0.000000
## eta11 -0.066999 0.125466 -0.534001 0.593341
## eta12 0.164905 0.502722 0.328025 0.742893
## shape 4.643109 0.585341 7.932320 0.000000
##
## LogLikelihood : 2067.01
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.7660
## Bayes -3.7340
## Shibata -3.7661
## Hannan-Quinn -3.7539
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.002307 0.9617
## Lag[2*(p+q)+(p+q)-1][2] 0.126495 0.9015
## Lag[4*(p+q)+(p+q)-1][5] 3.430414 0.3340
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1339 0.7144
## Lag[2*(p+q)+(p+q)-1][8] 2.1406 0.8367
## Lag[4*(p+q)+(p+q)-1][14] 4.6656 0.8096
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.4209 0.500 2.000 0.5165
## ARCH Lag[6] 1.9891 1.461 1.711 0.4932
## ARCH Lag[8] 2.8966 2.368 1.583 0.5630
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.1145
## Individual Statistics:
## omega 0.5754
## alpha1 0.5061
## alpha2 0.5814
## beta1 0.5969
## eta11 0.3425
## eta12 0.6022
## shape 0.4655
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.1412 0.8877
## Negative Sign Bias 0.5398 0.5894
## Positive Sign Bias 1.4825 0.1385
## Joint Effect 5.3453 0.1482
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 29.44 0.05942
## 2 30 34.87 0.20896
## 3 40 55.03 0.04587
## 4 50 58.47 0.16669
##
##
## Elapsed time : 0.916533It is important to note that alpha2 is insignificant, so using TGARCH(2,1), in other words adding alpha2 to the previous version of the model, was pointless.
Version III: TGARCH(1, 2)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : fGARCH(1,2)
## fGARCH Sub-Model : TGARCH
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001638 0.000832 1.96874 0.048983
## alpha1 0.160365 0.043875 3.65502 0.000257
## beta1 0.797406 0.243369 3.27653 0.001051
## beta2 0.045967 0.226386 0.20305 0.839097
## eta11 -0.067596 0.109361 -0.61811 0.536506
## shape 4.638206 0.645601 7.18432 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.001638 0.001098 1.49271 0.135513
## alpha1 0.160365 0.047012 3.41114 0.000647
## beta1 0.797406 0.177597 4.48998 0.000007
## beta2 0.045967 0.186854 0.24601 0.805676
## eta11 -0.067596 0.126117 -0.53598 0.591972
## shape 4.638206 0.586842 7.90367 0.000000
##
## LogLikelihood : 2067.031
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.7679
## Bayes -3.7405
## Shibata -3.7679
## Hannan-Quinn -3.7575
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.00161 0.9680
## Lag[2*(p+q)+(p+q)-1][2] 0.13184 0.8979
## Lag[4*(p+q)+(p+q)-1][5] 3.45830 0.3297
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.09413 0.7590
## Lag[2*(p+q)+(p+q)-1][8] 2.15505 0.8344
## Lag[4*(p+q)+(p+q)-1][14] 4.72349 0.8029
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.4087 0.500 2.000 0.5226
## ARCH Lag[6] 2.0143 1.461 1.711 0.4876
## ARCH Lag[8] 2.9599 2.368 1.583 0.5510
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.3226
## Individual Statistics:
## omega 0.5658
## alpha1 0.4970
## beta1 0.5859
## beta2 0.5844
## eta11 0.3362
## shape 0.4570
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.1209 0.9038
## Negative Sign Bias 0.5013 0.6163
## Positive Sign Bias 1.5196 0.1289
## Joint Effect 5.3144 0.1502
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 28.27 0.07842
## 2 30 35.74 0.18114
## 3 40 49.03 0.13024
## 4 50 54.54 0.27213
##
##
## Elapsed time : 0.455431Now beta2 is insigificant, so using TGARCH(1,2), i.e. adding beta2, turned out counterproductive.
In spite of seemingly promising test results for all tested versions, TGARCH models do not seem to be a good option for this analysis due to insignificant parameters. Each time the crucial parameter, either eta which is typical to TGARCH modelling or parameters vital to make the model a bit more complicated, is not significant.
We also tried to manipulate with the mean equation trying: ARFIMA(0,0,0), ARFIMA(1,0,0) and ARFIMA(0,0,1), however the results did not differ much. Thus, we are going to skip TGARCH models with different mean specifications.
2.5. GARCH-In-Mean
Version I
##
## *---------------------------------*
## * 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|)
## mu -0.003325 0.004357 -0.76296 0.445486
## archm 0.072339 0.120330 0.60117 0.547727
## omega 0.000132 0.000036 3.69078 0.000224
## alpha1 0.170371 0.033989 5.01254 0.000001
## beta1 0.761675 0.042151 18.07013 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.003325 0.004302 -0.77273 0.439682
## archm 0.072339 0.112830 0.64113 0.521437
## omega 0.000132 0.000060 2.18472 0.028909
## alpha1 0.170371 0.048489 3.51361 0.000442
## beta1 0.761675 0.059862 12.72393 0.000000
##
## LogLikelihood : 2015.112
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.6748
## Bayes -3.6520
## Shibata -3.6748
## Hannan-Quinn -3.6661
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01933 0.8894
## Lag[2*(p+q)+(p+q)-1][2] 0.07527 0.9373
## Lag[4*(p+q)+(p+q)-1][5] 3.40630 0.3378
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4837 0.4867
## Lag[2*(p+q)+(p+q)-1][5] 1.2221 0.8077
## Lag[4*(p+q)+(p+q)-1][9] 2.5697 0.8273
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.4314 0.500 2.000 0.5113
## ARCH Lag[5] 1.5656 1.440 1.667 0.5752
## ARCH Lag[7] 2.5365 2.315 1.543 0.6049
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.935
## Individual Statistics:
## mu 0.09314
## archm 0.07781
## omega 0.37996
## alpha1 0.43244
## beta1 0.50651
##
## 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 0.6855 0.4932
## Negative Sign Bias 0.1604 0.8726
## Positive Sign Bias 1.2678 0.2052
## Joint Effect 4.8460 0.1834
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 80.26 1.677e-09
## 2 30 89.16 4.895e-08
## 3 40 97.08 7.411e-07
## 4 50 100.42 2.106e-05
##
##
## Elapsed time : 0.3935249The parameter archm is insignificant which makes the model ordinary GARCH, as in the Section 2.4.
Version II
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(2,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002909 0.003230 -0.90044 0.367886
## archm 0.075249 0.090244 0.83384 0.404371
## omega 0.000092 0.000043 2.12703 0.033418
## alpha1 0.169811 0.059688 2.84500 0.004441
## alpha2 0.029912 0.077961 0.38368 0.701216
## beta1 0.775046 0.065716 11.79378 0.000000
## shape 4.529187 0.634414 7.13917 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002909 0.002942 -0.98870 0.322808
## archm 0.075249 0.078378 0.96009 0.337012
## omega 0.000092 0.000057 1.61765 0.105738
## alpha1 0.169811 0.055188 3.07698 0.002091
## alpha2 0.029912 0.085873 0.34833 0.727592
## beta1 0.775046 0.084748 9.14528 0.000000
## shape 4.529187 0.582620 7.77382 0.000000
##
## LogLikelihood : 2062.796
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.7583
## Bayes -3.7263
## Shibata -3.7584
## Hannan-Quinn -3.7462
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.07285 0.7872
## Lag[2*(p+q)+(p+q)-1][2] 0.19006 0.8603
## Lag[4*(p+q)+(p+q)-1][5] 3.44509 0.3318
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.3471 0.5558
## Lag[2*(p+q)+(p+q)-1][8] 2.1993 0.8271
## Lag[4*(p+q)+(p+q)-1][14] 4.1976 0.8608
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.7975 0.500 2.000 0.3718
## ARCH Lag[6] 2.1231 1.461 1.711 0.4642
## ARCH Lag[8] 2.5862 2.368 1.583 0.6234
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.2002
## Individual Statistics:
## mu 0.06714
## archm 0.06052
## omega 0.70263
## alpha1 0.50846
## alpha2 0.54844
## beta1 0.75740
## shape 0.55695
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.0286 0.3039
## Negative Sign Bias 0.5212 0.6023
## Positive Sign Bias 1.1807 0.2380
## Joint Effect 5.4740 0.1402
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 35.84 0.01106
## 2 30 40.68 0.07342
## 3 40 55.18 0.04461
## 4 50 59.38 0.14710
##
##
## Elapsed time : 0.570493Now most of the parameters are insignificant and the model is really bad.
Version III
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002765 0.003241 -0.853124 0.393590
## archm 0.071548 0.090455 0.790974 0.428959
## omega 0.000084 0.000036 2.348957 0.018826
## alpha1 0.185268 0.052675 3.517200 0.000436
## beta1 0.792156 0.254048 3.118132 0.001820
## beta2 0.000001 0.223230 0.000003 0.999997
## shape 4.508110 0.627300 7.186533 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.002765 0.002965 -0.932634 0.351009
## archm 0.071548 0.078431 0.912238 0.361643
## omega 0.000084 0.000043 1.966920 0.049192
## alpha1 0.185268 0.045363 4.084117 0.000044
## beta1 0.792156 0.180683 4.384240 0.000012
## beta2 0.000001 0.174942 0.000004 0.999996
## shape 4.508110 0.568647 7.927782 0.000000
##
## LogLikelihood : 2062.724
##
## Information Criteria
## ------------------------------------
##
## Akaike -3.7582
## Bayes -3.7262
## Shibata -3.7583
## Hannan-Quinn -3.7461
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06273 0.8022
## Lag[2*(p+q)+(p+q)-1][2] 0.19087 0.8598
## Lag[4*(p+q)+(p+q)-1][5] 3.52160 0.3200
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.5571 0.4554
## Lag[2*(p+q)+(p+q)-1][8] 2.2747 0.8146
## Lag[4*(p+q)+(p+q)-1][14] 4.3356 0.8464
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.7531 0.500 2.000 0.3855
## ARCH Lag[6] 2.1344 1.461 1.711 0.4619
## ARCH Lag[8] 2.6369 2.368 1.583 0.6133
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.2581
## Individual Statistics:
## mu 0.06731
## archm 0.05950
## omega 0.68024
## alpha1 0.51285
## beta1 0.74181
## beta2 0.74640
## shape 0.54689
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.9535 0.3406
## Negative Sign Bias 0.6136 0.5396
## Positive Sign Bias 1.2884 0.1979
## Joint Effect 5.4807 0.1398
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 32.22 0.02955
## 2 30 36.35 0.16366
## 3 40 54.23 0.05332
## 4 50 61.30 0.11166
##
##
## Elapsed time : 0.3855731Again, most of the parameters are insignificant.
GARCH-In-Mean models do not seem to be a good option for this analysis due to insignificant parameters. Each time the crucial parameter archm is insignificant, which suggests that a typical GARCH model could be better.
Attempts to successfully extend either variance or mean equations turn out counterproductive - more complex models are even worse.
Conclusions
Important conclusion to be drawn is that assuming residuals to be t-distributed had a neutral or positive impact on the quality of the model (comparing it with a normal distribution). This is an expected outcome as financial data are often leptokurtic - t or GED distributions are the common choice.
From all the aforementioned models the best ones were EGARCH and GARCH-t based on AIC & BIC, with almost the same values. The second best model was ordinary GARCH(1,1).
3. Conditional variance in-sample forecast
In this section we will use our best GARCH models for predicting the level of unconditional variance. After that we will present on the plot of the convergence of the conditional variance to the unconditional variance. The forecast is going to be calculated for 500 days ahead.
We can see that in each case the conditional variance converges to the unconditional variance predicted by our GARCH models.
4. 99% Value-at-Risk out-of-sample forecast
In this section we are going to calculate 99% Value-at-Risk the period from April to June 2024. The exceedance percentages are going to be computed at the end of this section, providing us with final conclusions for this analysis.
Percentages of observations exceeding VaR:
## Percentages of observations exceeding VaR for GARCH(1,1): 0.01111111## Percentages of observations exceeding VaR for EGARCH(1,1): 0.02222222## Percentages of observations exceeding VaR for GARCH-t(1,1): 0.01111111Conclusions
The results indicate that all three models generally perform similarly in estimating risk, as indicated by their close exceedance rates for the out-of-sample period. However, the fact that the lowest exceedance rate was obtained for the GARCH-t(1,1) model may imply that it provides a slightly more conservative estimate of risk compared to GARCH(1,1) and EGARCH(1,1). This may be connected with the model’s ability to capture fat-tailed distributions or higher volatility periods more effectively, which are common characteristics in financial markets.