#  Predicting Value at Risk for Cryptocurrencies With Generalized Random
Forests

Konstantin Görgen ,3 Department of Economics (ECON), Karlsruhe Institute of
Technology, Blücherstr.17, 76185 Karlsruhe, Germany Jonas Meirer Eidgenössiche
Technische Hochschule Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Melanie Schienle 11footnotemark: 1

###### Abstract

We study the prediction of Value at Risk (VaR) for cryptocurrencies. In
contrast to classic assets, returns of cryptocurrencies are often highly
volatile and characterized by large fluctuations around single events.
Analyzing a comprehensive set of 105 major cryptocurrencies, we show that
Generalized Random Forests (GRF) (Athey et al., 2019) adapted to quantile
prediction have superior performance over other established methods such as
quantile regression, GARCH-type and CAViaR models. This advantage is
especially pronounced in unstable times and for classes of highly-volatile
cryptocurrencies. Furthermore, we identify important predictors during such
times and show their influence on forecasting over time. Moreover, a
comprehensive simulation study also indicates that the GRF methodology is at
least on par with existing methods in VaR predictions for standard types of
financial returns and clearly superior in the cryptocurrency setup.

Keywords: Generalized Random Forests, Value at Risk, Quantile Prediction,
Backtesting, Cryptocurrencies, Conditional Predictive Ability.

JEL Classification: C58,G17,C22

33footnotetext: Corresponding author: Konstantin Görgen; email:
<EMAIL_ADDRESS>phone: +49/721/608/43793

##  1 Introduction

Cryptocurrencies are an important and rising part of today’s digital economy.
Currently, the market capitalization of the top 10 cryptocurrencies in the
world is close to $2 trillion and growing111See e.g.
https://coinmarketcap.com/charts/; accessed at 22nd March 2022.. The use of
cryptocurrencies in terms of daily volume exploded from 2016 to
2018††footnotemark: , which not only attracts individuals but also business
users such as hedge funds or merchants as well as long-term investors such as
crypto-focused and traditional investment funds (Vigliotti and Jones, 2020).
The crypto asset market, however, remains highly volatile where e.g. an
investment in Bitcoin in 2013 would have seen a return of roughly $20,000\%$
in 2017, but an investment in 2017 would have led to a performance of -75% in
2019††footnotemark: . Consequently, there is a need to predict and monitor the
risks associated with cryptocurrencies. To address this, we find that classic
approaches such as historical simulation, GARCH-type, or CaViaR methods are
too restrictive. More general non-linear methods provide more flexibility to
account for such non-standard time series behaviour that might be attributed
to a large extend to speculators.

In this paper, we propose a novel flexible way for out-of-sample prediction of
the Value at Risk for cryptocurrencies. We use a quantile version of
Generalized Random Forests (GRF, see Athey et al., 2019), which builds upon
mean random forests (Breiman, 2001) now tailored to quantiles. This framework
shows to be especially promising when dealing with more volatile classes of
cryptocurrencies due to the non-linear structure of their returns. In a
comprehensive out-of-sample scenario using more than 100 of the largest
cryptocurrencies, GRF outperforms other established methods such as CAViaR
(Engle and Manganelli, 2004), quantile regression (Koenker and Hallock, 2001)
or GARCH-models (Bollerslev, 1986; Glosten et al., 1993) over a rolling
window, particularly in unstable times. This can be attributed to the non-
parametric approach of random forests that is flexible and adaptable
considering important factors and non-linearity. We further analyze
performance in different important subperiods, consider different classes of
cryptocurrencies, and employ different sets of covariates with the forest-
based methods and the benchmark procedures.

Previous studies have confirmed that there exist speculative bubbles (Cheah
and Fry, 2015; Hafner, 2020), and we find that our approach assesses risks
especially well during such times. Moreover, we account for a large number of
covariates that describe volatility, liquidity, and supply (Liu and Tsyvinski,
2020). It can be seen that variable importance differs substantially depending
on time, where long-term measures of standard deviation, that are an important
predictor in stable times, are not relevant predictors for VaR in unstable,
volatile times. Furthermore, only few of the additional covariates beside
lagged standard deviations and lagged returns are relevant. We find that for
other, less volatile classes of cryptocurrencies such as stablecoins,
especially GJR-GARCH models and quantile regression can compete with GRF. A
simulation study also highlights that the proposed GRF methodology is at least
on par in prediction of VaR also for standard-type financial returns with
clear advantages in the crypto-currency type case.

Our paper contributes to the growing literature on cryptocurrencies. Analyses
performed in the past include GARCH models (Chu et al., 2017) as well as ARMA-
GARCH models (Platanakis and Urquhart, 2019), approaches using RiskMetrics
(Pafka and Kondor, 2001) and GAS-models (Liu et al., 2020), application of
extreme value theory (Gkillas and Katsiampa, 2018), vine copula-based
approaches (Trucios et al., 2020), Markov-Switching GARCH models (Maciel,
2020), non-causal autoregressive models (Hencic and Gouriéroux, 2015) and also
some machine learning-based approaches (see e.g. Takeda and Sugiyama, 2008).
Additionally, cryptocurrencies can be used for diversification in investment
strategies with other, traditional assets (see e.g. Trimborn et al., 2020;
Petukhina et al., 2021), as the correlation between them and more established
assets tends to be low (Elendner et al., 2017; Platanakis and Urquhart, 2019).
This again poses the question of assessing the risks of cryptocurrencies,
where new methods of addressing the above mentioned challenges need to be
explored.

The paper is structured as follows. Section 2 presents the underlying data and
cryptocurrencies with descriptive statistics and standard Box-Jenkins time-
series checks. In Section 3, we introduce the main random forest-type
techniques for conditional quantiles and present the employed evaluation tests
and framework. The comprehensive simulation study in Section 4 demonstrates
the performance of the different methods under various data generating
processes. The empirical results are contained in Section 5, where we present
aggregate prediction results for all currencies in coverage performance
(Section 5.1.1) as well as in pairwise comparison tests (Section 5.1.2). In
Section 5.2, we analyze representative important currencies in detail and
study the importance of specific factors. Finally, we conclude in Section 6.

##  2 Data

We use daily log-returns of 105 of the largest cryptocurrencies222The data was
obtained on 23rd March 2022 from https://docs.coinmetrics.io/ using the
community data set, which can be downloaded from a public Github repository at
https://github.com/coinmetrics/data/. from coinmetrics by market
capitalization333All currencies have a maximum market capitalization of more
than 15 million USD each. in US-Dollar (USD) at time of retrieval, in the
period from 07/2010 to 03/2022. The coinmetrics data includes spot-market
information from 30 different exchanges, such as Binance, ZB.COM, FTX, OKX,
Coinbase, KuCoin, or Kraken444See https://docs.coinmetrics.io/exchanges/all-
exchanges for an overview of all exchanges included.. Depending on the
currency (i.e. the first trade date on the exchanges), the number of available
observations varies between 261 and 4264. The data is summarized in Table 1.
We see that there are some very negative and positive returns in the data, as
well as high excess-kurtosis confirming the observations in the depicted
quantiles. Furthermore, there are some assets with a high skewness, both
positive and negative, indicating asymmetry in the distribution of returns.

We do not detect any stochastic non-stationarities in the data, which is
supported by Augmented Dickey-Fuller (ADF) tests and Kwiatkowski-Phillips-
Schmidt-Shin (KPSS) tests (Kwiatkowski et al., 1992). With Alpha Finance Lab
(alpha), Polymath (poly), and Synthetix (snx), KPSS tests against level
stationarity seem slightly significant, while trend KPSS tests and ADF tests
suggest stationarity. With Algorand (algo), Binance Coin (bnb), Curve DAO
Token (crv), FTX Token (ftt), Internet Computer (icp), Aave (lend), OMG
Network (omg), SushiSwap (sushi), and Monero (xmr), KPSS tests against trend
stationarity are slightly significant, while ADF tests and level KPSS tests
again suggest stationarity. All results of the stationarity tests can be found
in Table 10 in the appendix.

Table 1: Descriptive Statistics of Log-Returns of Cryptocurrencies

| Min | 1% | 5% | Median | 95% | 99% | Max | Skewness | Excess-Kurtosis | Standard Deviation | Observations
---|---|---|---|---|---|---|---|---|---|---|---
Min% | -1.264 | -0.308 | -0.168 | -0.006 | 0.001 | 0.002 | 0.006 | -3.556 | 1.671 | 0.001 | 261.000
1% | -1.120 | -0.268 | -0.158 | -0.006 | 0.001 | 0.002 | 0.006 | -2.691 | 1.965 | 0.001 | 314.960
5% | -0.830 | -0.243 | -0.139 | -0.004 | 0.002 | 0.006 | 0.016 | -1.195 | 3.043 | 0.002 | 526.200
25% | -0.576 | -0.199 | -0.114 | -0.001 | 0.089 | 0.170 | 0.342 | -0.190 | 6.482 | 0.060 | 717.000
50% | -0.492 | -0.181 | -0.104 | 0.000 | 0.109 | 0.209 | 0.461 | 0.298 | 10.708 | 0.073 | 1354.000
75% | -0.362 | -0.160 | -0.086 | 0.001 | 0.120 | 0.239 | 0.704 | 1.030 | 23.164 | 0.080 | 1669.000
95% | -0.022 | -0.006 | -0.002 | 0.002 | 0.153 | 0.318 | 1.258 | 2.241 | 75.697 | 0.101 | 2851.800
99% | -0.006 | -0.002 | -0.001 | 0.003 | 0.197 | 0.412 | 1.431 | 3.472 | 160.129 | 0.117 | 3264.120
Max% | -0.005 | -0.002 | -0.001 | 0.003 | 0.201 | 0.539 | 1.462 | 3.753 | 216.567 | 0.136 | 4264.000

Note: The rows show the quantiles of the sample-measures in the columns that
are for log-returns of all cryptocurrencies combined. For example, over all
cryptocurrencies, there is a median of 1354 observations per cryptocurrency.
Figure 1: Pointwise median returns over all currencies with $5\%$ and $95\%$
sample quantiles in blue over each date. The number of currencies that have a
return at the time is indicated in red. The orange, green, and purple shaded
areas correspond to the three periods we analyze closer in Section 5.

For the cryptocurrencies, Figure 1 illustrates the median returns (black) over
all cryptocurrencies by date. We can see that in the beginning of the time
period, only one currency, namely Bitcoin, was present in the data set. From
2014, we see an incremental increase (red line), while there is a jump up in
2017 and an consecutively faster increase in available cryptocurrencies. We
also see that the returns are very different between currencies from 2014 to
2018 (blue), which marks a period of hype leading to a crash in the beginning
of 2018. Later, there are large negative spikes corresponding to the many
waves of the Covid-19 pandemic. Based on these observations, we divide our
data into three periods. The first period ranges from August 2015 (2015-08-22)
to the end of 2017 (2017-12-21), while the second period subsequently lasts
until November 2020 (2020-11-05). The last period then covers the rest of our
data (until 2022-03-20)555The specific dates account for the training periods
of currencies and creation of new assets to make sure that we capture a
maximum number of cryptocurrencies in each time period..

Since we are in a time series setup, we include classic covariates based on
lagged return in our analysis. Additionally, we employ information specific to
each cryptocurrency in 7 external covariates. The five time-series based
covariates consist of the one-day lagged return and the lagged 3,7,30, and 60
day return standard deviation. The external covariates are the number of
unique active daily addresses (Active_Users), the number of unique addresses
that hold any amount of native units of that currency or at least 10 or 100
USD equivalent (Total_Users, Total_Users_USD10,Total_Users_USD100), the supply
equality ratio (SER), i.e. the ratio of supply held by addresses with less
than $1/10^{7}$ of the current supply to the top 1% of addresses with the
highest current supply, the number of initiated transactions (Transactions),
and the velocity of supply in the current year (Velocity), which describes the
the ratio of current supply to the sum of the value transferred in the last
year. See also Table 12 in the appendix for details on the covariates.

Table 2: Summary of Covariates for Different Time Periods

Quantile | Ret | Active_Users | Total_Users | Total_Users_USD100 | Total_Users_USD10 | SER | Transactions | Velocity | sd_3 | sd_7 | sd_30 | sd_60 | CapMrktCurMUSD
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Period 1: 5 Currencies
5% | -0.094 | 7691 | 233361 | 7768 | 32688 | 0.001 | 3304 | 9.321 | 0.007 | 0.014 | 0.022 | 0.026 | 17
Median | -0.001 | 127679 | 2579591 | 297029 | 615602 | 0.013 | 49170 | 33.708 | 0.036 | 0.044 | 0.051 | 0.055 | 2611
95% | 0.104 | 324662 | 9697275 | 3562101 | 5930594 | 0.031 | 110698 | 109.281 | 0.137 | 0.134 | 0.123 | 0.124 | 185407
Period 2: 15 Currencies
5% | -0.085 | 3592 | 95320 | 5271 | 16960 | 0.001 | 16618 | 4.668 | 0.008 | 0.015 | 0.024 | 0.027 | 70
Median | -0.001 | 66921 | 2629309 | 233835 | 540993 | 0.008 | 128481 | 16.244 | 0.034 | 0.041 | 0.047 | 0.049 | 5008
95% | 0.094 | 179852 | 8601014 | 2135275 | 4469914 | 0.018 | 655221 | 67.849 | 0.122 | 0.119 | 0.111 | 0.109 | 99979
Period 3: 77 Currencies
5% | -0.082 | 2138 | 533215 | 15647 | 41396 | 0.002 | 25250 | 5.255 | 0.008 | 0.017 | 0.026 | 0.030 | 233
Median | -0.000 | 19371 | 1218761 | 80236 | 215034 | 0.007 | 137082 | 15.325 | 0.035 | 0.041 | 0.047 | 0.049 | 1794
95% | 0.086 | 76081 | 2793940 | 565048 | 1195791 | 0.012 | 424953 | 53.743 | 0.111 | 0.105 | 0.101 | 0.096 | 26105
Full Data: 105 Currencies
5% | -0.089 | 1857 | 559862 | 16964 | 44887 | 0.002 | 19135 | 5.626 | 0.009 | 0.019 | 0.028 | 0.033 | 461
Median | -0.000 | 15406 | 1095198 | 72651 | 186751 | 0.006 | 106286 | 14.371 | 0.038 | 0.045 | 0.051 | 0.054 | 2367
95% | 0.095 | 62448 | 2389786 | 434200 | 917576 | 0.010 | 339294 | 44.627 | 0.120 | 0.113 | 0.108 | 0.102 | 21085

Note: values are means of quantiles over all assets contained in the specific
time period. CapMrktCurMUSD describe the market cap in Mio. USD and is not
included as a covariate due to multicollinearity reasons.

Table 2 gives an overview of the employed covariates and their values in the
different time periods. We can see that in Period 1, we have the most extreme
returns on average as well as the most extreme lagged standard deviations.
This is not surprising looking at the first period (853 days), which arguably
marks the most volatile period, with many new currencies being created, as
well as the second longest period. In the following period (1050 days), the
average median market cap reaches a high as well as the number of users
invested in the currencies, indicating that the market is growing while
stabilizing more. This is followed by a sharp drop in the market cap for the
last, shortest period (500 days), which starts at the beginning of the
Covid-19 pandemic. There, the number of active users as well as the SER
decreases, indicating that more smaller addresses are pushed off the market,
while it is the period with the most currencies.

Given this hugely dynamic cryptocurrency market evolution we distinguish
between the three systematically different periods in terms of actively traded
currencies and market capitalization. The first one is characterized by a few,
rapidly changing currencies and extreme returns and volatilities, while the
second period is less extreme and more characterized by a strong increase in
median market caps with a few dozens of market components. The third period,
in the end, is very short but contains more than five times the number of
currencies in comparison to the second period.

##  3 Methodology

For the prediction of cryptocurrencies, we advocate the use of non-linear
machine learning based techniques. In this way, we intend to accommodate the
documented large share of speculation (Ghysels and Nguyen, 2019; Baur et al.,
2018; Selmi et al., 2018; Glaser et al., 2014) and resulting frequent changes
in unconditional volatility which make predictions in this market peculiar. In
particular, we focus on generalized random forest methods that are tailored
for conditional quantiles of returns and thus allow to forecast the VaR. The
flexible but interpretable non-linearity of the approach allows for a direct
comparison to standard linear and (G)ARCH-type models. We also argue that the
difference in forecasting performance can moreover be employed to detect
periods of bubbles and extensive speculation.

Recall that for daily log returns $r_{t}$ the $VaR_{t}$ at level
$\alpha\in(0,1)$ conditional on some covariates $x_{t-1}$ is defined as

$VaR^{\alpha}_{t}(x_{t-1})=\sup_{r_{t}}\left(F(r_{t}|x_{t-1})<\alpha\right)\
,$ (1)

where $F$ marks the distribution of $r_{t}$ conditional on $x_{t-1}$.
Generally, the conditioning variables could consist of past lagged returns,
standard deviations, but also of external (market) information or other
assets. We employ these as covariates that are explained in Section 2.

We propose the use of two different types of random forest-based techniques
which directly model the conditional VaR in (1). Both build on the classic
random forest (Breiman, 2001) which is an ensemble of (decorrelated) decision
trees (see e.g. Hastie et al., 2009) for the mean of $r_{t}$. In a decision
tree, each outcome $r_{t}$ is sorted into leafs of the tree by binary splits.
These splits are performed based on different $x_{t-1}$ components falling
above or below specific adaptive threshold values that need to be calculated,
for example by the Gini Impurity or MSE-splitting in Classification and
Regression Trees (CART) (Breiman et al., 1984), or using other criteria.
Finally, the prediction for a new $r_{t}$ is a weighted version of each tree
prediction.

In the proposed method that we employ, the generalized random forest (GRF)
from on Athey et al. (2019), the random forest split criterion is adapted to
mimic the task of quantile regression rather than minimizing a standard mean
squared loss criterion for mean regression tasks. Intuitively, the splits in
each leaf are conducted by minimizing the Gini-loss, which separates the
returns $r_{t}$ as best as possible at different quantiles. To transform the
minimizing problem in the splits into a classification task, the response
variable $r_{t}$ is transformed in each split to obtain pseudo-outcomes
$\rho_{t}=\sum_{k=1}^{K}1\\{r_{t}>\theta_{k}\\}$, where
$\Theta=(\theta_{q_{1}},\dots,\theta_{q_{K}})$ describe a set of $K$ pilot-
quantiles of $r_{t}$ in the parent node. These quantiles with levels
$\tau=q_{1},\dots,q_{K}$ are then used to calibrate the split666We use the
tuning parameters $K=1$ and $\tau=\alpha$, i.e. the level of VaR we are
analyzing, in this classification pre-step.. In Athey et al. (2019), this is
formally motivated by moment conditions and gradient approximations, but
practically, $r_{t}$ is relabeled to a nominal scale depending on the largest
quantile it does not exceed. In a final step, the optimal split on a variable
component $p$ of $x_{t-1}$ and $j=1,\dots,J$ observations in the parent node
is then based on minimizing the above-mentioned Gini impurity criterion for
classification. For a separation into two possible leaf sets $v=l,r$, the Gini
impurity for one leaf $v$ is $G_{p}^{v}=1-\sum_{k=1}^{K}p_{k,v}^{2}$, where
$p_{k,v}=\sum_{j=1}^{J}1\\{\rho_{j}=k\textit{ and }\rho_{j}\in v\\}/|v|$ is
the proportion of $\rho_{j}$ in group $v$ with value $k=1,\dots,K$. The full
loss is then an average weighted by leaf size, yielding

$\displaystyle G_{p}=(|l|G_{p}^{l}+|r|G_{p}^{r})/(|l|+|r|)\ .$ (2)

We choose the Gini-loss since it is fast and, for certain configurations,
produces purer nodes than for example using entropy as a splitting criterion
(see e.g Breiman, 1996). This can be particularly helpful when dealing with
changing variance (and thus time-varying quantiles) of returns, where we would
like to detect single extreme events. For our specific case of $\alpha=0.05$,
this implies that values larger than $\theta_{0.05}$ in the parent node are
given the value $1$, while others are $0$. Algorithm 1 briefly summarizes the
tree building algorithm from Athey et al. (2019) for the quantile version of
GRF, where the main differences with regard to the splitting regime in
comparison to a classic CART occur in every step the tree is grown.

In addition to that, the outcome that is predicted is not the mean but the
$\alpha$-quantile (i.e. $VaR^{\alpha}$), which is done in a way that you do
not calculate a weighted average of $r_{t}$ but a weighted average of the
empirical CDF $\hat{F}(r_{t}\rvert x_{t-1})=E[1_{\\{r_{t}\\}|x_{t-1}}]$.
Intuitively, log returns $r_{t}$ that have similar $x_{t-1}$ in comparison to
a new observation $x_{v}$ receive higher weight in the empirical CDF.
Similarity weights $w_{t}(x_{v})$ are measured as the relative frequency on
how often $x_{v}$ falls in the same terminal leaf as $x_{t-1}$, for
$t=1,\dots,T$, and averaged over all trees for each $x_{t-1}$. This last step
was originally introduced by Meinshausen (2006) for random forests.

As a benchmark, we employ the quantile regression forest (QRF) based on
Meinshausen (2006). This random forest, however, uses the same splitting
regime as the original CART random forest and therefore does not account
explicitly for situations where the variance and therefore the quantile
changes, as splits are conducted based on a mean-squared error criterion.
Since such volatility changes are to be expected for cryptocurrencies in our
data, we expect GRF to perform better than QRF, but still include both in the
analysis to see potential differences in predictions. Furthermore, GRF uses
so-called “honest” trees, meaning that different data (usually the subsampled
data for each tree is split again in half) is used for building and “filling”
each of the trees with values.

Algorithm 1 Generalized Random Forest - Tree Building

1:Set of “honest”, subsampled observations $X_{T}$ and $R_{T}$; minimum node
size $n_{m}$; quantile probabilities $\tau_{K}=(\tau_{1},\dots,\tau_{K})$

2:Growing the tree Create root node $P_{0}$

3:Initialize queue Q with $P_{0}$

4:while Queue is not empty do

5: Take the oldest element from Q (Parent Node $P$) and remove it from Q

6: Take a random subsample of $p$ variables index by set
$P_{sub}=\\{\tilde{1},\dots,\tilde{p}\\}$ from $X_{T}$ on which to potentially
split and take observations
$x^{(P_{sub})}_{i}=(x^{(\tilde{1})}_{i},\dots,x^{(\tilde{P})}_{i})$ from $P$.

7: Set $loss=\infty$

8: for h in 1 to p do

9: Compute quantiles $\theta_{k}$ of $r_{t}$ from parent node $P$ at
$\tau_{1},\dots,\tau_{K}$ and compute pseudo outcomes
$\rho_{t}=\sum_{k=1}^{K}1\\{r_{t}>\theta_{k}\\}$ for each $r_{t}\in P$.

10: For each possible split point in $x^{(h)}$, compute the criterion from
Equation (2)

11: Save loss $s_{h}$ that minimizes this splitting criterion

12: Save optimal split point $split_{h}$

13: if $s_{h}<loss$ then

14: $loss\leftarrow s_{h}$

15: $ind_{h}\leftarrow h$

16: end if

17: end for

18: if Split on variable $h$ with $split_{ind_{h}}$ succeeded (based on
hyperparameters) then

19: Determine children $C_{1},C_{2}$ according to optimal split

20: Add both children $C_{1},C_{2}$ to a new daughter node each with
corresponding observations left and add these to Q

21: end if

22:end while

23:One tree of the forest

As benchmarks we further include two types of standard time series methods. We
use the CAViaR (CAV) methodology by Engle and Manganelli (2004) and standard
quantile regression (Koenker and Hallock, 2001). Both make use of quantile
regression (QR) techniques (Koenker and Bassett, 1978) that do not minimize
the squared error as in ordinary regression, but use the check function
$\rho_{\alpha}(u)=u\left(\alpha-1\\{u\leq 0\\}\right)$ to minimize
$L_{\alpha}(f_{\alpha}(\cdot),x_{t})=\sum_{t=1}^{T}\rho_{\alpha}\left(r_{t}-f_{\alpha}(x_{t})\right)$.
For CAV, we use a symmetric absolute value (SAV) component for
$f_{\alpha}(\cdot)$, i.e
$f_{\alpha}(x_{t},r_{t})=\beta_{1}+\beta_{2}f_{\alpha}(x_{t-1},r_{t-1})+\beta_{3}|r_{t-1}|$,
and $f_{\alpha}(r_{t-1},x_{t})=\beta_{4}^{\prime}x_{t}+\beta_{5}r_{t-1}$ for
the quantile regression. In contrast to the former methods, they can only
capture parametric (non)-linear effects which limits their flexibility.

For comparison, we use a GJR-GARCH(1,1) model (Glosten et al., 1993), a
GARCH(1,1) (Bollerslev, 1986) model, a simple historical simulation (Hist),
meaning that we predict $VaR^{\alpha}_{t+1}$ at level $\alpha$ as the sample
$\alpha$-quantile of the preceding returns in a window of length $K$, i.e.
$(r_{t-K+1},\dots,r_{t})$, and one that fits a normal distribution to the
sample data and uses the theoretical fitted $\alpha$-quantile as the
prediction for $VaR^{\alpha}_{t+1}$ (NormFit). We do not expect the latter to
perform well as we have high skewness and excess-kurtosis in the data (see
Table 1 in Section 2).

For the proposed random forest-type and all benchmark procedures that allow
for additional covariates (GRF, QRF, QR) we include lagged standard deviations
(SD) in addition to the lagged level $r_{t-1}$ in the model. We expect these
to capture the strongly varying levels of unconditional volatility in
particular for the cryptocurrencies in the non-linear structure. Additionally,
we also employ the above methods and the GARCH(1,1) model using both the
latter covariates as well as the 7 external covariates described in Section 2.
These methods are GRF-X, QRF-X, QR-X, and GARCH-X. To establish a fair common
ground in used model complexity for the QR, QRF, GARCH-X, and GRF, we select a
common set of different SD lags as covariates from an additional Monte-Carlo
study. In this, we use the simple SAV-model from Section 4 as a baseline of
which the respective specification is tailored to regime changes in the
unconditional volatilities as observed form the cryptocurrencies. The model is
essentially linear autoregressive of order 1 in the VaR and thus directly
yields the VaR as outputs. With this, it is possible to use a MSE-minimizing
criterion and select the MSE-minimizing variables for the subsequent
simulations and data analysis. Table 3 summarizes the results of this short
simulation study, which is why we use 3, 7, 30, and 60 day lagged SD as
covariates.

Table 3: MSE Prediction Error for Different Covariate Combinations

Lagged SD (in days) | 3 | 7 | 30 | 3 and 7 | 3 and 30 | 7 and 30 | 3, 7 and 30 | 3, 7, 30, and 60
---|---|---|---|---|---|---|---|---
GRF-MSE | 0.124 | 0.095 | 0.089 | 0.075 | 0.070 | 0.069 | 0.059 | 0.057

MSE prediction error for a simulated SAV-model as in Section 4 for GRF (QRF).
The minimum MSE is marked in bold.

To compare the performance of the above methods, we use two types of
evaluation approaches. First, we test how well each model predicts the
conditional $\alpha$-VaR over the entire out-of-sample horizon using three
different sets of evaluation techniques. The simplest way of checking whether
a model predicts $VaR^{\alpha}$ correctly over a time horizon is to look at
its coverage meaning the number of times $r_{t}$ is smaller than the predicted
$VaR^{\alpha}_{t}$. Ideally, this should be exactly $\alpha T$ times. This
measure is called Actual over Expected Exceedances (AoE) and is computed as
$AoE_{\alpha}=(\alpha T)^{-1}\sum_{t=1}^{T}1\\{r_{t}<VaR^{\alpha}_{t}\\}$. To
test this intuition formally, we employ three tests, the DQ-test777We use the
implementation from the GAS-package in R (Ardia et al., 2019) including 4
lagged Hit-values, a constant, the VaR-forecast, and the squared lagged
(log-)return in the regression model. from Engle and Manganelli (2004), the
Christoffersen-test (Christoffersen, 1998) and the Kupiec-test (Kupiec, 1995).
All three tests assume that the forecasts have correct coverage under the null
hypothesis. The Kupiec-test is simply the formalization of the above
intuition, the Christoffersen-test is robust against serial correlation by
assuming that $g_{t}=1\\{r_{t}<VaR^{\alpha}_{t}\\}\sim Bern(\alpha)$, and the
DQ-test additionally accounts for problems with conditional coverage due to
clustering of the hits exceedance sequences $g_{t}$ with a regression-based
approach. In the empirical analysis, we only report the values of the DQ-test,
which is the strictest of the tests, for reasons of clarity. Results for the
other tests do not differ substantially and are available upon request from
the authors.

Secondly, for comparing the forecast performance of two models 1 and 2
directly, we implement the one-step ahead test for conditional predictive
ability (CPA) from Giacomini and White (2006), Theorem 1, that assumes under
the null hypothesis that forecasts of model 1 and model 2 have on average
equal predictive ability conditional on previous information. As suggested by
Giacomini and Komunjer (2005), we use the quantile loss function $L_{\alpha}$
for the test. This tests assumes under the null hypothesis that $H_{0}:\
E\left[\Delta
L_{t}|\mathcal{F}_{t-1}\right]=E\left[L_{\alpha}(f_{\alpha}^{(1)},r_{t})-L_{\alpha}(f_{\alpha}^{(2)},r_{t})|\mathcal{F}_{t-1}\right]\equiv
E\left[h_{t-1}\Delta L_{t}\right]=0$ and that this loss difference is a
martingale difference sequence, where $\mathcal{F}_{t-1}$ contains all
information up to time $t-1$ and $f_{\alpha}^{(1)}$ and $f_{\alpha}^{(2)}$ are
two competing forecasts. The test statistic is computed using a Wald-type test
with a set of factors $h_{t-1}$ that can possibly predict the loss difference
$\Delta L_{t}$ and is $\chi^{2}_{q}$-distributed under $H_{0}$. More
specifically, we choose $h_{t-1}=(1,\Delta L_{t-1})$ (i.e. $q=2$), i.e. using
the lagged loss difference and an intercept as predictors in a linear
regression with parameter $\beta_{0}$ for the simulation and application.

##  4 Simulation

In this section, we study the finite sample forecast performance in quantiles
for true DGPs of standard stock-type dynamics as well as for set-ups of
cryptocurrency type. Overall, we find that, as expected, GRF performs best in
all settings, in particular in settings which are designed to mimic the
cryptocurrency behaviour over time but also in those similar to stock index
behaviour. For QRF, the performance is entirely different, at least for the
chosen parsimonious model specification and the relatively short estimation
intervals.

We study two different types of DGPs. The first one is a standard GARCH(1,1)
process, i.e

$\displaystyle r_{t}$ $\displaystyle=z_{t}\sigma_{t}$ (3)
$\displaystyle\sigma^{2}_{t}$
$\displaystyle=\omega+\beta_{0}\varepsilon^{2}_{t-1}+\beta_{1}\sigma^{2}_{t-1}\
,$ (4)

where the parameters are estimated on the full Bitcoin data to mimic the
behavior of cryptocurrencies (with $z_{t}\sim N(0,1)$). We denote this setting
as sim GARCH Bitcoin fit. Moreover, we also consider the specification with
$\beta_{0}=0.1$, $\beta_{1}=0.8$, $\omega=10^{-4}$ and $z_{t}\sim N(0,1)$ (sim
GARCH) and with $z_{t}$ as $t_{5}$-distributed (sim GARCH t), which
corresponds to standard stock index data. Secondly, for the Sim SAV-Model
setting, we fit a symmetric absolute value (SAV) model to normal returns, i.e.

$\displaystyle\textit{VaR}_{t+1}=\gamma_{0}+\gamma_{1}\textit{VaR}_{t}+\gamma_{2}|r_{t}^{(\textit{init})}-\gamma_{3}|\
,$ (5)

with $r_{t}^{(\textit{init})}\sim N(0,\sigma_{t}^{2})$ and
$\dfrac{\sigma_{t}}{65}\sim\chi^{2}_{2}$, where new draws of $\sigma_{t}$ are
only taken every 100 observations, keeping $\sigma_{t}$ constant meanwhile. We
then generate the final return as $r_{t}\sim
N\left(0,\dfrac{\widehat{VaR}_{t}}{\Phi(\alpha)^{-1}}\right)$ from the fitted
SAV-model, where $\Phi(\alpha)^{-1}$ is the quantile function of a standard
normal variable. We do this to obtain returns that have exactly the VaR that
we obtained from the SAV model before.

Table 4: Simulation: $5\%$ VaR

Rolling Window | $l=500$ | $l=1000$ |
---|---|---|---
| DQ | Kupiec | Christoffersen | AoE |  | DQ | Kupiec | Christoffersen | AoE
Sim GARCH Normal |  |  |  |  |  |
QRF |  $0.940$ $(0.008)$ |  $0.325$ $(0.200)$ |  $0.200$ $(0.275)$ | 1.185 |  |  $0.725$ $(0.063)$ |  $0.160$ $(0.377)$ |  $0.090$ $(0.393)$ | 1.148
GRF |  $0.495$ $(0.163)$ |  $0.000$ $(0.583)$ |  $0.015$ $(0.549)$ | 1.040 |  |  $0.225$ $(0.317)$ |  $0.020$ $(0.547)$ |  $0.030$ $(0.525)$ | 1.030
QR |  $0.760$ $(0.048)$ |  $0.085$ $(0.441)$ |  $0.090$ $(0.438)$ | 1.095 |  |  $0.305$ $(0.270)$ |  $0.040$ $(0.537)$ |  $0.040$ $(0.496)$ | 1.042
Hist |  $0.835$ $(0.045)$ |  $0.015$ $(0.554)$ |  $0.195$ $(0.365)$ | 1.047 |  |  $0.630$ $(0.110)$ |  $0.060$ $(0.490)$ |  $0.195$ $(0.368)$ | 1.030
NormFit |  $0.740$ $(0.071)$ |  $0.040$ $(0.551)$ |  $0.205$ $(0.329)$ | 1.006 |  |  $0.555$ $(0.130)$ |  $0.065$ $(0.475)$ |  $0.195$ $(0.345)$ | 1.001
CAV |  $0.785$ $(0.051)$ |  $0.025$ $(0.514)$ |  $0.010$ $(0.545)$ | 1.073 |  |  $0.275$ $(0.286)$ |  $0.055$ $(0.513)$ |  $0.025$ $(0.511)$ | 1.032
GARCH(1,1) |  $0.445$ $(0.209)$ |  $0.030$ $(0.511)$ |  $0.155$ $(0.394)$ | 1.046 |  |  $0.205$ $(0.327)$ |  $0.055$ $(0.531)$ |  $0.095$ $(0.428)$ | 1.028
Sim GARCH t |  |  |  |  |  |
QRF |  $0.920$ $(0.018)$ |  $0.315$ $(0.203)$ |  $0.210$ $(0.277)$ | 1.188 |  |  $0.655$ $(0.085)$ |  $0.175$ $(0.348)$ |  $0.100$ $(0.399)$ | 1.162
GRF |  $0.435$ $(0.188)$ |  $0.010$ $(0.579)$ |  $0.025$ $(0.537)$ | 1.029 |  |  $0.260$ $(0.341)$ |  $0.030$ $(0.522)$ |  $0.020$ $(0.518)$ | 1.022
QR |  $0.770$ $(0.060)$ |  $0.130$ $(0.407)$ |  $0.105$ $(0.407)$ | 1.109 |  |  $0.275$ $(0.303)$ |  $0.040$ $(0.507)$ |  $0.035$ $(0.468)$ | 1.049
Hist |  $0.725$ $(0.077)$ |  $0.020$ $(0.573)$ |  $0.200$ $(0.360)$ | 1.036 |  |  $0.580$ $(0.121)$ |  $0.115$ $(0.440)$ |  $0.185$ $(0.334)$ | 1.016
NormFit |  $0.695$ $(0.090)$ |  $0.310$ $(0.271)$ |  $0.370$ $(0.220)$ | 0.839 |  |  $0.550$ $(0.155)$ |  $0.340$ $(0.275)$ |  $0.370$ $(0.239)$ | 0.817
CAV |  $0.525$ $(0.135)$ |  $0.010$ $(0.523)$ |  $0.010$ $(0.542)$ | 1.059 |  |  $0.175$ $(0.331)$ |  $0.055$ $(0.499)$ |  $0.040$ $(0.502)$ | 1.032
GARCH(1,1) |  $0.405$ $(0.232)$ |  $0.135$ $(0.381)$ |  $0.210$ $(0.316)$ | 0.893 |  |  $0.270$ $(0.308)$ |  $0.200$ $(0.368)$ |  $0.220$ $(0.344)$ | 0.862
Sim SAV-Model |  |  |  |  |  |
QRF |  $0.960$ $(0.010)$ |  $0.315$ $(0.188)$ |  $0.180$ $(0.265)$ | 1.187 |  |  $0.735$ $(0.071)$ |  $0.150$ $(0.352)$ |  $0.115$ $(0.413)$ | 1.158
GRF |  $0.495$ $(0.148)$ |  $0.005$ $(0.571)$ |  $0.025$ $(0.570)$ | 1.047 |  |  $0.210$ $(0.317)$ |  $0.040$ $(0.518)$ |  $0.035$ $(0.548)$ | 1.048
QR |  $0.895$ $(0.024)$ |  $0.050$ $(0.435)$ |  $0.045$ $(0.452)$ | 1.102 |  |  $0.310$ $(0.234)$ |  $0.045$ $(0.490)$ |  $0.045$ $(0.530)$ | 1.064
Hist |  $0.290$ $(0.220)$ |  $0.035$ $(0.595)$ |  $0.070$ $(0.523)$ | 1.047 |  |  $0.165$ $(0.353)$ |  $0.070$ $(0.525)$ |  $0.080$ $(0.496)$ | 1.048
NormFit |  $0.215$ $(0.360)$ |  $0.040$ $(0.539)$ |  $0.110$ $(0.502)$ | 0.994 |  |  $0.145$ $(0.418)$ |  $0.080$ $(0.501)$ |  $0.070$ $(0.474)$ | 0.999
CAV |  $0.775$ $(0.053)$ |  $0.030$ $(0.525)$ |  $0.030$ $(0.563)$ | 1.067 |  |  $0.235$ $(0.289)$ |  $0.045$ $(0.527)$ |  $0.045$ $(0.563)$ | 1.038
GARCH(1,1) |  $0.135$ $(0.338)$ |  $0.030$ $(0.567)$ |  $0.065$ $(0.528)$ | 1.021 |  |  $0.065$ $(0.469)$ |  $0.045$ $(0.546)$ |  $0.040$ $(0.524)$ | 1.018
Sim GARCH Bitcoin fit |  |  |  |  |  |
QRF |  $0.875$ $(0.018)$ |  $0.325$ $(0.200)$ |  $0.200$ $(0.275)$ | 1.185 |  |  $0.585$ $(0.110)$ |  $0.160$ $(0.377)$ |  $0.090$ $(0.393)$ | 1.148
GRF |  $0.255$ $(0.303)$ |  $0.000$ $(0.583)$ |  $0.015$ $(0.549)$ | 1.040 |  |  $0.160$ $(0.454)$ |  $0.020$ $(0.547)$ |  $0.030$ $(0.525)$ | 1.030
QR |  $0.635$ $(0.095)$ |  $0.085$ $(0.441)$ |  $0.090$ $(0.438)$ | 1.095 |  |  $0.170$ $(0.380)$ |  $0.040$ $(0.537)$ |  $0.040$ $(0.496)$ | 1.042
Hist |  $0.600$ $(0.102)$ |  $0.015$ $(0.554)$ |  $0.195$ $(0.365)$ | 1.047 |  |  $0.385$ $(0.269)$ |  $0.060$ $(0.490)$ |  $0.195$ $(0.368)$ | 1.030
NormFit |  $0.540$ $(0.159)$ |  $0.040$ $(0.551)$ |  $0.205$ $(0.329)$ | 1.006 |  |  $0.330$ $(0.299)$ |  $0.065$ $(0.475)$ |  $0.195$ $(0.345)$ | 1.001
CAV |  $0.970$ $(0.007)$ |  $0.375$ $(0.178)$ |  $0.280$ $(0.224)$ | 0.814 |  |  $0.655$ $(0.055)$ |  $0.305$ $(0.236)$ |  $0.270$ $(0.273)$ | 0.791
GARCH(1,1) |  $0.230$ $(0.319)$ |  $0.030$ $(0.536)$ |  $0.180$ $(0.399)$ | 1.054 |  |  $0.100$ $(0.458)$ |  $0.040$ $(0.545)$ |  $0.080$ $(0.438)$ | 1.032

The table displays rejection rates of t-tests of empirical quantile levels
against the nominal level of $5\%$ for DQ-, Kupiec- and Christoffersen-tests
and mean p-values in parentheses. Thus higher p-values and lower rejection
rates indicate better model performance. The GARCH case uses predictions from
the QMLE-fit of a GARCH(1,1) specification with normally distributed errors
and can therefore be seen as an oracle for the GARCH-simulated specifications.

For all settings, we generate 2000 return observations and forecast the one-
step ahead VaR over the different rolling window lengths $l=500,1000$. We
repeat this generation process 200 times for $\alpha=0.01$ and $\alpha=0.05$.
For comparison of the different methods described in Section 3, we use the DQ-
test, the Kupiec test, the Christoffersen-test, and the AoE. Note that for all
tests, we present aggregate results from two-sided t-tests of the empirical
versus the nominal coverage. The results are therefore rejection rates of
t-tests against the nominal level of 5%. Therefore, a lower rejection rate and
higher mean p-values (in parentheses) indicate better performance. For GRF,
QRF, and QR, we use a common set of lagged covariates as described at the end
of Section 3.

Table 4 summarizes the results of the simulation for the 5% $VaR$. According
to the more advanced $DQ$\- and Christoffersen-tests for evaluation, GRF
consistently outperforms the other methods in almost all cases, indicating
superior predictive quality. This is very much in contrast to the QRF which is
consistently dominated by the other models. For the Sim GARCH and Sim GARCH t
cases which mimic standard stock indices, the performances of GRF, CAV and QR
appear in a similar range with mostly advantages for GRF in particular for
smaller sample sizes. This holds generally for normally distributed as well as
heavy tailed innovations which lead to similar results also in magnitude of
the rejection rates. As expected, forecasting performance increases throughout
all models with larger estimation windows, though with CAV often profiting the
most from the larger sample sizes. For these settings with GARCH as the true
DGP, the performance of the GARCH model serves as an oracle reference. In the
t-innovation case, it has coverage problems and GRF is even able to outperform
it in Christoffersen-tests.

For the Sim SAV Model and the Sim GARCH Bitcoin fit the situation, however,
differs substantially. In these cryptocurrency-like cases, the GRF clearly
dominates the QR and CAV particularly strongly in the small 500 observations
setting. Considering our application where only a relatively small time span
is available, this seems crucial. Moreover, CAV runs into coverage problems
according to the AoE results which even deteriorate for larger sample sizes
for Sim GARCH Bitcoin fit. In the latter case, the GRF rejection rates are
close to the GARCH benchmark while the strong conditional dependence structure
in the tails of the Sim SAV Model setting shows that for such extreme cases,
the unconditional coverage of GRF is still excellent, but the conditional
coverage measured by the DQ-test is only average among all models for the
larger estimation samples and even below for the smaller ones. These relative
findings generally prevail for the 1% VaR forecasts, but absolute performance
is generally worse for all methods, especially with a smaller rolling window
of 500 (see Table 16 in the appendix.). Intuitively, this finding is
reasonable, since relevant observations for the 1% level should in theory only
occur in 5 of the 500 observations, making it harder for the data-driven
methods to predict such unlikely events.

Additionally, we conduct direct pairwise comparison tests between the superior
random forest type method GRF against the best performing non-oracle other
parametric methods via CPA-tests for each scenario. The respective results are
reported in Table 5. In the majority of cases, GRF outperforms its competitors
on average, however, mean p-values are mostly not significant. For example,
GRF has a smaller loss (i.e. quantile loss) than QR in about 90% of the
forecasts (aggregated over all runs) but a mean p-value of 0.298, which would
not qualify as a significant out-performance (on average). This of course does
not mean that the test never rejects but is likely be caused by high variances
in the p-values over different simulation runs. Furthermore, the competing
methods are also slightly improving with window length, which further implies
that GRF can deal better with a smaller training time frame.

Table 5: CPA-tests on Predictions of $5\%$ VaR for Different Window Lengths
and Different Models

Rolling Window | $l=500$ |  | $l=1000$
---|---|---|---
GRF vs: | QR | Hist | CAV |  | QR | Hist | CAV
Sim GARCH Normal |  |  |  |  |
Mean P-Value | ${0.429}$ | ${0.351}$ | ${0.334}$ |  | ${0.491}$ | ${0.388}$ | ${0.463}$
No. P-Values $<0.1$ | ${34}$ | ${55}$ | ${50}$ |  | ${24}$ | ${44}$ | ${27}$
GRF-Performance | ${0.780}$ | ${0.755}$ | ${0.847}$ |  | ${0.501}$ | ${0.744}$ | ${0.659}$
Sim GARCH t |  |  |  |  |
Mean P-Value | ${0.392}$ | ${0.382}$ | ${0.396}$ |  | ${0.431}$ | ${0.397}$ | ${0.446}$
No. P-Values $<0.1$ | ${43}$ | ${40}$ | ${34}$ |  | ${30}$ | ${37}$ | ${15}$
GRF-Performance | ${0.732}$ | ${0.707}$ | ${0.692}$ |  | ${0.439}$ | ${0.717}$ | ${0.529}$
Sim SAV-Model |  |  |  |  |
Mean P-Value | ${0.298}$ | ${0.311}$ | ${0.374}$ |  | ${0.471}$ | ${0.381}$ | ${0.448}$
No. P-Values $<0.1$ | ${53}$ | ${63}$ | ${42}$ |  | ${26}$ | ${50}$ | ${22}$
GRF-Performance | ${0.903}$ | ${0.279}$ | ${0.775}$ |  | ${0.657}$ | ${0.321}$ | ${0.658}$
Sim GARCH Bitcoin fit |  |  |  |  |
Mean P-Value | ${0.429}$ | ${0.351}$ | ${0.001}$ |  | ${0.491}$ | ${0.388}$ | ${0.003}$
No. P-Values $<0.1$ | ${34}$ | ${55}$ | ${200}$ |  | ${24}$ | ${44}$ | ${199}$
GRF-Performance | ${0.780}$ | ${0.755}$ | ${0.995}$ |  | ${0.501}$ | ${0.745}$ | ${0.994}$

Note: the table displays CPA-tests for $5\%$ one-day ahead VaR-forecasts of
the best performing random forest type techniques GRF in Table 4 versus the
best parametric time series models. We report mean p-values, the number of
significant p-values over 200 iterations and the rate at which GRF outperforms
the competing method (i.e. a value of 0.8 means that GRF has a smaller error
loss than the competing method in 80% of the rolling window forecasts over all
runs). Low p-values paired with performance rates larger than 0.5 indicate
that GRF outperforms the competing methods.

##  5 Results

In this section, we highlight the advantages from using non-linear machine
learning-based methods for forecasting the VaR of cryptocurrencies. In
particular, we show for a large cross-section of more than 100
cryptocurrencies that the proposed random forest method GRF yields superior
performance across a wide range of different types of cryptocurrencies and
different time periods. Investigating the underlying drivers, we illustrate
that the non-linear model predictions excel especially for assets that are
frequently traded by a large amount of different users, and for more volatile
assets and times.

More specifically, we predict the $5\%-\textit{VaR}$ as a key quantity in risk
management for our comprehensive set of cryptocurrencies. In an extensive out-
of-sample forecasting study, we compare the random forest-based machine
learning methods to standard linear time series and GARCH-type models
including approaches with exogenous asset information in covariates. The
prediction performance is assessed with the DQ-test to obtain an overall
aggregate picture on the realized coverages as well as pairwise CPA-tests
across different time periods and types of cryptocurrencies. Based on these
findings, we focus our analysis on three important selected currencies
comprising Bitcoin (btc) as the largest currency by far regarding market cap,
Tether (usdt) as a stablecoin with lower volatility, and Cardano (ada) as a
currency specifically allowing for smart contracts. For these we also consider
predicted loss series by CPA-tests and variable importance measures to uncover
important drivers. We furthermore identify single events that majorly affect
the predictive ability of the procedures.

###  5.1 Aggregated Forecasting Performance

In this section, we provide results on aggregate forecast performance of the
different modeling approaches over all cryptocurrencies.

####  5.1.1 Backtesting

Table 6: Medians of P-Values for DQ-Tests in Different Time Periods

Period | GRF | QRF | QR | CAV | GJR-GARCH | Hist | GRF-X | QRF-X | QR-X | GARCH-X
---|---|---|---|---|---|---|---|---|---|---
1 | 0.16 | 0.29 | 0.08 | 0.06 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
2 | 0.27 | 0.09 | 0.06 | 0.27 | 0.17 | 0.08 | 0.13 | 0.06 | 0.00 | 0.00
3 | 0.18 | 0.05 | 0.07 | 0.03 | 0.30 | 0.01 | 0.20 | 0.10 | 0.00 | 0.03

Note: Higher median p-values indicate correct conditional coverage according
to the DQ-test. Periods correspond to the times indicated in Figure 1. Table
7: Medians of P-Values for DQ-Tests Sorted by SER in Different Time Periods

Group | GRF | QRF | QR | CAV | GJR-GARCH | Hist | GRF-X | QRF-X | QR-X | GARCH-X
---|---|---|---|---|---|---|---|---|---|---
Period 1
High | 0.16 | 0.92 | 0.00 | 0.00 | 0.12 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00
Low | 0.13 | 0.28 | 0.10 | 0.10 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Period 2
High | 0.16 | 0.08 | 0.20 | 0.19 | 0.11 | 0.08 | 0.13 | 0.07 | 0.00 | 0.00
Low | 0.29 | 0.10 | 0.06 | 0.31 | 0.25 | 0.10 | 0.32 | 0.06 | 0.00 | 0.09
Period 3
High | 0.47 | 0.05 | 0.22 | 0.11 | 0.21 | 0.01 | 0.51 | 0.12 | 0.00 | 0.00
Low | 0.07 | 0.04 | 0.03 | 0.02 | 0.32 | 0.00 | 0.18 | 0.07 | 0.00 | 0.06

Note: The currencies are divided into two groups based on specific covariate
values for each time period. The groups are constructed to best separate the
sorted covariate values of the currencies. They are therefore separated around
their steepest decay to obtain two groups with homogeneous SER values.
Detailed plots for the group separation can be found in Figure 10, column 1.
The remaining details correspond to Table 6.

We analyze the three fundamentally distinct periods 08/2015–12/2017,
01/2018–11/2020, and 12/2020–03/2022 of the cryptocurrency market which differ
according to the number of actively traded currencies, the market
capitalization and the overall market situation (see Section 2). We thus
analyze each period separately, with a focus on the last period 3 in the
aggregate, where the number of available currencies is relatively high. Given
the largely different market situations, we refrain from reporting results for
the full time period. We assess the performance of VaR forecasts by their
conditional coverage and the dependence structure of exceedances, which we
test with the DQ-test (see Section 3 for details). This test is the most
strict of all backtests, conditioning on past information, and therefore most
accurately captures the coverage abilities of the procedures888Detailed
results for the other tests are omitted here for reasons of clarity, do not
differ qualitatively, and are available upon request from the authors.. The
results over the different time periods for the 5% VaR-predictions are shown
in Table 6. We can see that depending on the time period, the median p-values
vary strongly, which is not surprising giving the different characteristics of
each period and the increasing number of cryptocurrencies in the later
periods. In general, GRF is the only method with median values consistently
over the 10% level, indicating that it is the most consistently calibrated
forecasting method. The QRF performs extremely well in the first time period,
but rejects the test often in the third period. Adding external covariates as
listed in Table 2 does not improve performance and in general leads to
substantially lower p-values especially for the QR. Only in the final third
period, adding external covariates increases the p-values slightly for the GRF
and substantially for the QRF (GRF-X, QRF-X). This indicates that those extra
covariates are not necessarily predictive for extreme returns, or rather that
the existing measures such as lagged returns and SD comprise the information
already quite well. Compared to its non-forest counterparts, only CAV and GJR-
GARCH can partly keep up with GRF. GJR-GARCH has higher median p-values than
GRF in the third period, which is the shortest and which contains the most
currencies. It is also marked by less extreme returns and a large reduction in
active users (see Table 2), which could indicate that the forest methods excel
in particular in highly volatile periods, when large shifts in the market are
present. On the other hand, CAV has a similar, slightly worse performance in
the second period while performing much worse than GRF in the other periods.
This highlights the inability of the parametric methods to adapt to rapidly
changing situations such as in Period 1 and 3. QR, Hist, and GARCH-X are all
fully dominated by GRF throughout the three time periods as expected, as QR
can only incorporate changes linearly, and GARCH-X and Hist serve as simple
baselines. Additionally, when checking the unconditional (median) coverage
represented as the ratio of exceedances relative to the expected ones, random
forest-type methods also show superior performance over standard GARCH-type
methods, which have quite low coverage in all (GARCH-X) or the first period
(GJR-GARCH)999Detailed results are available on request..

Figure 2: Boxplots of p-values of DQ-tests over all cryptocurrencies contained
in Period 3 separated into groups with high vs low SER for every method. For
details on the groups, see Table 7. The dashed horizontal line depicts a level
of 0.1.

We investigate Period 3, which contains 77 cryptocurrencies, more in detail,
especially with regard to how conditional coverage is characterized by
external information. For this, we split the currencies into two
groups101010Split points are plotted in Figure 10 in the appendix and
correspond to obtaining two groups with mean SER values as homogeneous as
possible. They are therefore separated around their steepest decay. depending
on the value of the SER, which characterizes the concentration of supply to
users. A low SER consequently implies that most supply is concentrated on a
few, large users, often indicating a more stable currency. Figure 2 depicts
boxplots over p-values of DQ-tests for each group and procedure in Period 3.
GRF(-X) performs particularly well for the groups with a high SER, which
arguably are more prone to speculations during hype/bubble periods due to the
large amount of supply held by many small addresses. For the (arguably) more
stable addresses with a smaller SER, GJR-GARCH has higher p-values, followed
by GRF-X, which has the most consistent performance in this period. This is in
contrast to CAV, QR, or QRF, which are always outperformed by GRF(-X), as well
as the simple baselines Hist and GARCH-X. For details on all time periods and
groups according to the different observable factors active users and lagged
standard deviation, see Table 13 in the appendix.

####  5.1.2 Direct Pairwise Forecast Comparisons

In addition to assessing the coverage performance, we conduct pairwise CPA-
tests for all considered methods in the three different specified time periods
over all cryptocurrencies. The results of the tests are contained in Table 8
and Figure 4. Note that the CPA-tests require the rolling window length to be
smaller than the out-of-sample forecast window to produce valid results.
Therefore the results from the shortest period Period 3, where both window
sizes are equal to 500 might have less power. Thus by construction, for the
only recently introduced cryptocurrencies in period 3, the out-of-sample size
is too low for the CPA-tests to have high power. Therefore, we additionally
look at the direct comparisons of predicted losses (as suggested by Giacomini
and White (2006), Section 4)111111We use the lagged loss difference and an
intercept for loss prediction in an autoregressive setup since these are the
main drivers of the test statistic in the CPA test., where we compare in the
loss series how often GRF is better, i.e. has a smaller loss, than its
competitors (GRF-Performance). Note that a value of one thus indicates that
GRF has a smaller predicted loss over the full loss series.

In general, GRF performs better than its competitors for a majority of
crypotocurrencies over all time periods. Table 8 summarizes the results. We
can see that QRF is almost always outperformed, and for around 50% of
cryptocurrencies, losses are even significantly smaller. This is not
surprising, as QRF has a similar structure to GRF while not being tuned to
predict the quantiles directly. Thus, we expect it to be less sensitive to
changes that only affect the quantile of the return distribution, for example
large shock events. The same holds for the GARCH-X and Hist, which are clearly
outperformed by the GRF, as well as the QR-X and QRF-X. Adding exogenous
information in covariates as part of the non-linear GRF (i.e. GRF-X) is better
especially in later periods (see Figure 9 in the appendix). This is
interesting, since the other methods cannot benefit as much as GRF from
additional covariates. Generally, for cryptocurrencies, the non-parametric
form of the GRF helps to extract information from exogenous covariates in
contrast to standard parametric methods such as QR and GARCH. As GRF accounts
specifically for the quantiles in the random forest splitting function, this
helps to also favorably integrate additional covariates $X$ in contrast to
QRF. Overall, however, both GRF-procedures seem to perform very similarly,
especially in pairwise comparisons. For full results of the GRF-X, see Table
14 and Figure 9 in the appendix. While CAV, QR, and GJR-GARCH are outperformed
over the majority of cryptocurrencies, only 20% to 50% of these out-
performances reach significance. In subsection 5.2, we will focus on specific
cryptocurrencies for a more in-depth understanding.

Table 8: Performance and Significance of CPA-tests Over Different Time Periods
for GRF Without Additional Covariates

GRF vs.: | QRF | QR | CAV | GJR-GARCH | Hist | GRF-X | QRF-X | QR-X | GARCH-X
---|---|---|---|---|---|---|---|---|---
Share of GRF With Better Performance
Period 1 | 1.00 | 0.80 | 0.80 | 1.00 | 1.00 | 0.60 | 1.00 | 1.00 | 1.00
Period 2 | 1.00 | 0.73 | 0.87 | 0.80 | 0.93 | 0.40 | 1.00 | 1.00 | 1.00
Period 3 | 0.92 | 0.68 | 0.71 | 0.64 | 0.90 | 0.44 | 0.92 | 0.96 | 0.70
Full Data | 0.91 | 0.74 | 0.75 | 0.71 | 0.82 | 0.44 | 0.90 | 0.97 | 0.72
Share of GRF With Significantly Better Performance
Period 1 | 0.40 | 0.40 | 0.20 | 0.20 | 1.00 | 0.00 | 1.00 | 1.00 | 0.60
Period 2 | 0.53 | 0.33 | 0.27 | 0.33 | 0.73 | 0.00 | 1.00 | 0.87 | 0.67
Period 3 | 0.43 | 0.21 | 0.23 | 0.19 | 0.51 | 0.09 | 0.74 | 0.78 | 0.53
Full Data | 0.41 | 0.31 | 0.24 | 0.18 | 0.41 | 0.12 | 0.69 | 0.83 | 0.54

The top part shows summary values that are shares over all cryptocurrencies in
the respective time period. It describes the number of times that GRF had a
better performance (i.e. more than 50% of predicted losses by the CPA test
were smaller for the GRF) relative to all crytpocurrencies (in that period),
and the number of times that GRF was significantly better (at least at a 10%
level) as judged by the CPA test over all crytpocurrencies (in that period).
The bottom part shows the detailed results of CPA-tests with the color of each
box indicating the performance of GRF. Blue signifies a performance of 1,
meaning that GRF has a smaller predicted loss in 100% of cases. *, **, ***
shows significance on a level of 10%, 5%, and 1%. The values are ordered by 30
day lagged standard deviation from highest to lowest (top/left to
bottom/right).

Figure 4: Overview of results for CPA-tests of GRF vs. all other methods for
cryptocurrencies in the third period ordered by 30 day lagged standard
deviation from highest to lowest, left to right. The color of each box
indicates the performance of GRF, with 1 (blue) indicating that GRF has a
smaller predicted loss in 100% of cases. *, **, *** indicate significance on a
level of 10%, 5%, and 1%.

When considering the single time periods, it is notable how for Period 1 and 2
(bottom part of Table 8), GRF(-X) is constantly outperforming the other
methods for most cryptocurrencies, and only has somewhat worse performance for
doge and the stablecoins, although these are insignificant (see below). For
the first two periods, QR is maybe the most competitive of the other methods,
while there is a general tendency for the classic methods to perform worse
with higher volatility of returns, which can be seen in Table 8 for Period 2
where the currencies are ordered from highest 30 day lagged SD on the left to
the lowest on the right. This becomes more apparent in Period 3, where we deal
with much more cryptocurrencies (77) and a much shorter time horizon (500 out-
of-sample observations). Here, methods such as GJR-GARCH and QR are on par
with GRF or even better when looking at low-volatility (partly regulated)
stablecoins such as pax, gusd, tusd, dai, and usdt with its derivatives (e.g.
usdt_eth,usdt_trx). CAV, on the other hand, is only rarely better here (as
indicated by CPA tests), while being significantly outperformed for important
and large assets such as btc, ada, xlm, and most stablecoins where the QR and
GJR-GARCH performed well.

To highlight the specific properties of currencies where GRF outperforms the
other methods, we split the assets into two groups. The first group
(Group_low) contains assets where GRF performance is low in comparison to the
three other methods that were able to compete in some cases with GRF, namely
QR, CAV, and GJR-GARCH. We add an asset into that group when at least two of
the methods outperform GRF (in terms of loss difference) for that asset,
separately for each time period. All other assets are sorted into the second
group (Group_high), indicating high performance of GRF. Table 9 summarizes the
results over these groups for each time period and covariate. In each group,
we take the mean over all cryptocurrencies of median values for each
covariate. We then divide Group_low by Group_high. For example, An SER of 0.15
in Period 3 indicates that cryptocurrencies in Group_low have, on average, a
median SER that is only 15% to that of Group_high, or in other words, the
median SER for Group_high is around $6.7=\dfrac{1}{0.15}$ times higher than
that of Group_low on average. We see that covariates of cryptocurrencies for
which GRF performs better have much higher volatility (especially for the
second and third period), a much higher SER121212Apart from the first period
where the only currency belonging to Group_low is doge., indicating a larger
concentration of supply at a lot of small addresses, a higher market
capitalization, a lower rate of turnover (Velocity), and more active and total
users. To summarize, this confirms the observation that GRF performs better
for assets with highly varying returns that are traded by a large amount of
users, which could thus also be prone to speculation. On the other side,
methods such as QR or GJR-GARCH are better with more stable currencies that
are used more as a hedging device (e.g. stablecoins). This confirms our
findings from the backtests in Section 5.1.1, where GRF excels in particular
for currencies with high SER values, high volatility, and a large number of
active users.

Table 9: Difference Between Covariates of Cryptos Where GRF is Better vs.
Worse

| Period 1 | Period 2 | Period 3 | Full Data
---|---|---|---|---
Ret | 1.20 | 0.16 | 1.05 | 0.92
Active_Users | 0.34 | 0.15 | 0.26 | 0.34
Total_Users | 0.79 | 0.18 | 0.13 | 0.15
Total_Users_USD100 | 0.22 | 0.17 | 0.27 | 0.35
Total_Users_USD10 | 0.41 | 0.17 | 0.19 | 0.25
CapMrktCurUSD | 0.08 | 0.11 | 0.37 | 0.33
SER | 1.07 | 0.50 | 0.15 | 0.21
Transactions | 0.42 | 0.04 | 1.20 | 1.53
VelCur1yr | 4.68 | 3.11 | 1.48 | 1.43
sd_3 | 0.89 | 0.52 | 0.59 | 0.58
sd_7 | 0.91 | 0.54 | 0.59 | 0.57
sd_30 | 0.90 | 0.52 | 0.59 | 0.58
sd_60 | 0.91 | 0.54 | 0.58 | 0.57

Note: Values are shares of groups of cryptocurrencies where at least two of
CAV, QR, and GJR-GARCH have better CPA-performance than GRF divided by the
remaining rest. Raw values before division are mean values over all
cryptocurrencies for the median of each covariate in the respective time
period. A value smaller than 1 indicates that currencies where GRF performs
better have a higher average median values of the respective covariate than
the rest of the currencies.

###  5.2 Extension: In-Depth Analysis of Specific Classes of Assets

To identify which specific events drive the performance of these methods, we
analyze the predicted loss series of CPA tests over the full horizon of
availability for the three cryptocurrencies Bitcoin, Cardano, and Tether
separately. We furthermore show which covariates are important over the full
data and specific time periods using variable importance measures of GRF-X.

We choose Bitcoin since it is the largest currency by market cap, with the
longest data availability, Tether as the largest stablecoin by daily volume
and market cap, and Cardano as a fairly new (i.e. fewer observations), however
large currency (again by market cap), which can be used for smart contracts,
identity verification, or supply chain tracking131313See e.g.
https://cardano.org/enterprise/, accessed 19/05/2022.. Since we deal with VaR-
predictions, the initial loss function is the quantile loss with a quantile
$\alpha=0.05$.

Figure 5: Rolling 180-day mean of predicted loss difference series (red) of
$h_{t}\hat{\beta}_{0}$ of CPA tests on Bitcoin (btc) predicted 5% VaR with
$l=500$ for GRF vs. CAV (left), GJR-GARCH (right), and QR (bottom center) with
rolling mean 10-day log-returns in grey. A positive predicted loss difference
indicates that the prediction error of GRF is smaller than of the compared
method.

First, we look at Bitcoin (btc), the largest and most popular currency, where
GRF largely outperforms QR, CAV, and GJR-GARCH in the CPA-tests. Figure 5
shows the predicted loss difference for each of the different methods. GRF
outperforms the other methods consistently for most time frames. This is most
likely due to the specific tailoring of the methodology to quantiles, as it
outperforms QRF (not plotted) consistently here. For the parametric methods
GJR-GARCH and QR, there are two short time periods where they have a smaller
loss. For GJR-GARCH, this happens in the very beginning of the out-of-sample
periods in April 2013, where btc crashed with negative log-returns of up to
$-0.66$. Since this was the first drop of that magnitude for btc, GRF, as a
forest based-method, had never seen such an extreme event, therefore could
have technically not predicted it. GJR-GARCH, on the other hand, as a
parametric method, has no range restriction in that regard. In the following
crashes, GRF correctly predicts these extreme events better than GJR-GARCH,
which is visible from the loss series. For QR, the short time frame is only
caused by the predictions of the CPA test itself, while actual losses are
smaller for GRF141414It might indicate, however, that the methods perform
quite equally during that specific time, since there is no predictable
difference..

Secondly, as summarized on the left in Figure 6, we look at Cardano (ada), a
large and fairly new currency offering e.g. smart contracts or supply chain
tracking. There, GRF is significantly outperforming GJR-GARCH and CAV, while
being slightly better than QR, although not reaching a significant level. This
lack of power for QR is likely due to the small out-of-sample size (roughly
1000) for the fairly new asset compared to the training window of $l=500$.
Again, we can see that around the most extreme spike in March 2020, GJR-GARCH
is slightly better, as GRF has not yet seen such an extreme event, therefore
is not able to correctly predict the size of the loss. One normal solution
would be to increase the training length, which is not possible in this case
with a fairly new currency. Interestingly, in the later extreme events in May
2022, GRF increases in performance compared to GJR-GARCH, confirming the
challenge with lacking training data.

Finally, we also look more closely at Tether (usdt) as the largest stablecoin
that is roughly bound to the USD151515As it is backed by USD cash reserves,
see https://tether.to/en/.. The right part of Figure 6 shows the 30-day
rolling means of the predicted loss differences. Notably, the rolling loss
difference and rolling mean-return are already around 10 times smaller than
those of btc or ada, indicating that usdt substantially differs from the other
two currencies. While the losses of QR are very similar to those of GRF, CAV
is significantly worse. Only GJR-GARCH consistently has lower predicted losses
than GRF, although they are deemed not significant by the CPA tests (see
Figure 13 in the appendix for detailed results for the full time frame). We
can see, however, that for the relatively rare tail events in 2017, which are
still not too extreme, there is some variation in the predicted loss
difference. For the first event that is quite extreme, GJR-GARCH reacts too
late, and only the second event is correctly detected by GJR-GARCH, as there
is some limited information of anticipation. In general, GRF tends too
overshoot less in these situations, which is also reflected in the DQ-tests,
where GRF tends to be better for assets that are more prone to speculative
bubbles.

Figure 6: Rolling 30-day mean of predicted loss difference series of
$h_{t}\hat{\beta}_{0}$ of CPA tests on Cardano/Tether (ada/usdt) predicted 5%
VaR with $l=500$ for GRF vs. CAV (orange), GJR-GARCH (green), and QR (purple)
with rolling mean 10-day log-returns in grey. A positive predicted loss
difference indicates that the prediction error of GRF is smaller than of the
compared method.

To further understand the drivers of the GRF performance, we obtain variable
importance measures that depict the frequency of inclusion in splits of the
forest. The variable importance of a covariate $x_{p}$ is measured as the
proportion of splits on $x_{p}$ relative to all splits in a respective layer
$l$ (over all trees in a trained forest), weighted by layer $l$161616We use a
maximum depth of $d_{max}=5$ corresponding to the number of covariates and a
weight decay of 2, meaning a split further down in each tree receives less
weight $w_{l}$ in the final frequency as it is less important for the three
specific currencies analyzed in the previous section. Specifically, for layer
$l=1,\dots,5$, $w_{l}=\dfrac{l^{-2}}{\sum_{l=1}^{5}l^{-2}}$.. In Figure 7, the
importance difference of certain covariates over time for the three currencies
is clearly visible.

Overall, the lagged return is very important for predicting VaR when returns
are quite extreme relative to all returns in a specific asset, in the case of
btc in times of hypes and crashes. Intuitively, this finding seems reasonable
as in times of bubbles, when the volatility is driven by some short, bubble-
like events and returns are highly variable, volatility lagged over a longer
time horizon is less predictive for VaR and predictions are driven by events
happening shortly before the prediction. In rather unstable times, but not in
extreme cases, the lagged SD-measures gain importance, while the extra
covariates only play a role for assets with relatively small volumes, e.g.
when new currencies are created. This also explains why GRF-X performs much
better for new, low-market-cap assets in Period 3 (see e.g. Figure 12 in the
appendix).

Figure 7: Rolling 30-day mean of the GRF variable importance in the out-of-
sample period for the full data of ada, btc, and usdt, for predicted one-day
ahead 5% VaR with $l=500$. Plotted are the 5 most important variables for each
cryptocurrency. Variable importance of covariate $x_{p}$ is measured as the
proportion of splits on $x_{p}$ relative to all splits in a respective layer
$l$ (over all trees in a trained forest), weighted by layer $l$. Variable
description can be seen in Section 2.

Starting with ada, we see that it is the only asset of the three where the
number of active addresses plays an important role, where for the other two
assets, the 60-day lagged standard deviation is more important. The importance
of variables can be split into two periods. The period until the beginning of
2021 is largely dominated by measures that somehow account for trading
activity (Active_Users,Total_Users/_USD100/_USD10, Transactions). For reasons
of clarity, we only plot the most important one of these measures,
Active_Users. The spike of the latter in importance at the beginning of 2021
is likely caused by the massive increase in price and market cap during that
time, representing a period of hype with many actively trading users171717See
also Figure 14 in the appendix for an overview of the log-returns of the three
currencies.. For the rest of the time period, lagged SD, mostly 3-day lagged
SD, followed by 30-day and 60-day SD, is dominating the predictions of GRF.
This change of importance seems reasonable as the structure of the asset
fundamentally changes with the price increasing tenfold and the volume
increasing strongly at the same time.

For btc, we have much more data covering 10 years, which is why the important
variables change frequently in different periods. Lagged return is naturally
important in phases of extreme hype and crashes that are characterized by
large positive and negative returns, e.g. in the very beginning (where the
price was still quite low), at the end of 2013 (the first time btc had a price
of USD 1000), at the end of 2017 (with a price over USD 19,000), and from mid
2020 to the mid 2021, where there were multiple hypes and crashes during the
Covid-19 pandemic. Between these hype periods, the lagged SDs are most
important. In 2014-2015 and from the end of 2019 to mid 2021 , 30-day SD is
contributing most to the GRF-predictions, followed by 7-day SD in 2016-2018
and 60-day SD 20 from 2018 to the end of 2019. This changing scheme is
interesting, as 30-day SD seems to be a good predictor especially in very
unstable times (return-wise), while 7-day and 60-day SD are more important in
relatively stable times.

Finally, the stablecoin usdt is an exception, being largely dominated by the
lagged return. Given that by construction usdt is essentially bound to the US
dollar, its dynamic properties also largely correspond to those of standard
currencies. From mid 2019, the prices and returns are rather stable and the
volume increases strongly, and the influence lagged 7-day SD increases
slightly, while still being less important than lagged return. This is not
surprising, as usdt is quite stable in comparison to btc and ada.

##  6 Conclusion

In this paper, we show that random forests can significantly improve the
forecasting performance for VaR-predictions when tailored to conditional
quantiles. In both simulations and analyzing return data of 105 of the largest
cryptocurrencies, the proposed GRF-type random forest shows superior
prediction performance. In particular, the adaptive non-linear form of GRF
appears to capture time-variations of volatility and spike-behavior in
cryptocurrency returns especially well in contrast to more conventional
financial econometric methods. We further show that the GRF is better in
assessing the tail risk of cryptocurrencies in times where volatility in
returns is high which often coincides with increased speculation in the
market. In such periods, standard time-series based procedures, even when
augmented with external factors, are substantially inferior in VaR predictions
suggesting that a comparison of predicted losses could serve as an easy,
empirical pre-screening device to detect such speculative bubbles.

Our findings are highly relevant for the risk assessment of cryptocurrencies
where we show that classic methods might lead to a miss-quantification of
associated risks. Such results directly impact investment and hedging
decisions for respective assets. Specifically, we detect superior performance
of GRF especially for highly volatile cryptocurrencies that have a high number
of active users and could thus be potentially prone to speculation and hypes.
On the other hand, for the class of stablecoins that are usually tied to some
large, classic currency such as the USD, and which are usually dominated by a
smaller number of large accounts, other classic methods such as GJR-GARCH or
quantile regression are more on par with GRF. This corresponds also to the
simulation results mimicking the behavior of conventional assets such as as
stock returns where the advantage of GRF decreases. Major gains of GRF,
however, occur the more cryptocurrency-like the set-up becomes.

The proposed random forest methodology allows identifying important external
factors despite the non-linearity of the methodology. We show that such
factors are time-varying and changing particularly in unstable times. For
future research, this set of covariates could even be further augmented with
other potentially driving real-time factors, such as for example properly
extracted and filtered social media information. The relevance of such factors
might also provide additional guidance for relevant exogenous information to
be included in standard parametric models such as CAViaR.

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##  Appendix A Appendix

Table 10: Overview of Non-Stationarity Tests

| KPSS_level | KPSS_trend | ADF
---|---|---|---
algo | 0.09 | 0.02 | 0.01
alpha | 0.02 | 0.10 | 0.01
bnb | 0.06 | 0.05 | 0.01
bnb_eth | 0.06 | 0.05 | 0.01
btc | 0.05 | 0.10 | 0.01
crv | 0.10 | 0.03 | 0.01
dash | 0.07 | 0.10 | 0.01
dcr | 0.10 | 0.08 | 0.01
dot | 0.10 | 0.10 | 0.01
eth | 0.10 | 0.08 | 0.01
ftt | 0.10 | 0.04 | 0.01
gno | 0.10 | 0.10 | 0.01
gnt | 0.10 | 0.04 | 0.01
icp | 0.10 | 0.02 | 0.01
lend | 0.10 | 0.02 | 0.01
loom | 0.05 | 0.10 | 0.01
neo | 0.10 | 0.08 | 0.01
omg | 0.10 | 0.02 | 0.01
poly | 0.04 | 0.10 | 0.01
ppt | 0.10 | 0.07 | 0.01
snx | 0.02 | 0.10 | 0.01
sushi | 0.10 | 0.01 | 0.01
uni | 0.10 | 0.08 | 0.01
weth | 0.05 | 0.10 | 0.01
wtc | 0.10 | 0.09 | 0.01
xem | 0.07 | 0.10 | 0.01
xmr | 0.10 | 0.04 | 0.01
xtz | 0.10 | 0.09 | 0.01
yfi | 0.08 | 0.10 | 0.01

Note: Values in columns show p-values for KPSS-test and ADF-tests as described
in Section 2. We only show values for assets that have values smaller than
$0.1$ for the KPSS-tests. No asset had p-values larger than $0.01$ for the
ADF-tests.

Table 11: Overview of All Employed Crypto Assets

id | Asset | Start-Date | End-Date | Obs. | X | Time Periods | id | Asset | Start-Date | End-Date | Obs. | X | Time Periods
---|---|---|---|---|---|---|---|---|---|---|---|---|---
1inch | 1inch | 2020-12-26 | 2022-03-21 | 451 | 7 |  | eos | EOS | 2018-06-09 | 2022-03-21 | 1382 | 2 | 3
aave | Aave | 2020-10-10 | 2022-03-21 | 528 | 7 |  | eos_eth | EOS ETH | 2017-06-29 | 2018-06-02 | 339 | 7 |
ada | Cardano | 2017-12-01 | 2022-03-21 | 1572 | 7 | 3 | etc | Ethereum Classic | 2016-07-25 | 2022-03-21 | 2066 | 9 | 3
algo | Algorand | 2019-06-22 | 2022-03-21 | 1004 | 7 | 3 | eth | Ethereum | 2015-08-08 | 2022-03-21 | 2418 | 9 | 2,3
alpha | Alpha Finance Lab | 2020-10-11 | 2022-03-21 | 527 | 7 |  | ftt | FTX Token | 2019-08-20 | 2022-03-21 | 945 | 7 | 3
ant | Aragon | 2017-08-29 | 2022-03-21 | 1666 | 7 | 3 | fun | FunFair | 2017-09-02 | 2022-03-21 | 1662 | 7 | 3
bal | Balancer | 2020-06-25 | 2022-03-21 | 635 | 7 |  | fxc | Flexacoin | 2019-07-17 | 2021-01-25 | 559 | 6 |
bat | Basic Attention Token | 2017-10-06 | 2022-03-21 | 1628 | 7 | 3 | gas | Gas | 2017-08-08 | 2022-03-21 | 1687 | 7 | 3
bch | Bitcoin Cash | 2017-08-01 | 2022-03-21 | 1694 | 8 | 3 | gno | Gnosis | 2017-05-02 | 2022-03-21 | 1785 | 7 | 3
bnb | Binance Coin | 2017-07-15 | 2019-04-22 | 647 | 7 |  | gnt | Golem (gnt) | 2017-02-19 | 2022-03-21 | 1857 | 7 | 3
bnb_eth | bnb_eth | 2017-07-15 | 2019-04-22 | 647 | 7 |  | grin | Grin | 2019-01-29 | 2022-03-21 | 1148 | 2 | 3
bsv | Bitcoin SV | 2018-11-15 | 2022-03-21 | 1220 | 8 |  | gusd | Gemini Dollar | 2018-09-16 | 2022-03-21 | 1283 | 7 | 3
btc | Bitcoin | 2010-07-18 | 2022-03-21 | 4265 | 9 | 1,2,3 | hbtc | Huobi Bitcoin | 2019-12-09 | 2022-03-21 | 834 | 5 | 3
btg | Bitcoin Gold | 2017-10-25 | 2022-03-21 | 1609 | 7 | 3 | hedg | HedgeTrade | 2019-11-02 | 2022-01-27 | 818 | 7 |
busd | Binance USD | 2019-09-20 | 2022-03-21 | 914 | 7 | 3 | ht | Huobi Token | 2019-03-06 | 2022-03-21 | 1112 | 7 | 3
comp | Compound | 2020-06-18 | 2022-03-21 | 642 | 7 |  | husd | HUSD | 2019-07-20 | 2022-03-21 | 976 | 7 | 3
cro | Crypto.com Coin | 2019-03-20 | 2022-03-21 | 1098 | 7 | 3 | icp | Internet Computer | 2021-05-11 | 2022-03-21 | 315 | 7 |
crv | Curve DAO Token | 2020-08-15 | 2022-03-21 | 584 | 7 |  | kcs | KuCoin Token | 2020-04-04 | 2022-03-22 | 718 | 0 |
cvc | Civic | 2017-09-11 | 2022-03-21 | 1653 | 7 | 3 | knc | Kyber Network | 2017-09-27 | 2022-03-21 | 1637 | 7 | 3
dai | Dai | 2019-11-20 | 2022-03-21 | 853 | 7 | 3 | lend | Aave (lend) | 2017-12-09 | 2022-03-21 | 1564 | 7 | 3
dash | Dash | 2014-02-08 | 2022-03-21 | 2964 | 8 | 1,2,3 | leo_eos | UNUS SED LEO EOS | 2019-05-21 | 2022-03-21 | 1036 | 3 | 3
dcr | Decred | 2016-05-17 | 2022-03-21 | 2133 | 8 | 2 | leo_eth | UNUS SED LEO ETH | 2019-05-21 | 2022-03-21 | 1036 | 7 | 3
dgb | DigiByte | 2015-02-10 | 2022-03-21 | 2597 | 8 | 2,3 | link | Chainlink | 2017-09-29 | 2022-03-21 | 1635 | 7 | 3
doge | Dogecoin | 2014-01-23 | 2022-03-21 | 2980 | 8 | 1,2,3 | loom | Loom Network | 2018-05-03 | 2022-03-21 | 1419 | 7 | 3
dot | Polkadot | 2020-08-20 | 2022-03-21 | 579 | 7 |  | lpt | Livepeer | 2018-12-20 | 2022-03-21 | 1182 | 7 | 3
drgn | Dragonchain | 2018-01-03 | 2022-03-21 | 1539 | 7 | 3 | ltc | Litecoin | 2013-04-01 | 2022-03-21 | 3277 | 8 | 1,2,3
elf | aelf | 2017-12-22 | 2022-03-21 | 1551 | 7 | 3 | maid | MaidSafeCoin | 2014-07-10 | 2022-03-21 | 2812 | 7 | 2,3

Table 11: (Continued)

id | Asset | Start-Date | End-Date | Obs. | X | Time Periods | id | Asset | Start-Date | End-Date | Obs. | X | Time Periods
---|---|---|---|---|---|---|---|---|---|---|---|---|---
mana | Decentraland | 2017-08-25 | 2022-03-21 | 1670 | 7 | 3 | tusd | TrueUSD | 2018-07-06 | 2022-03-21 | 1355 | 0 | 3
matic_eth | matic_eth | 2019-04-27 | 2022-03-21 | 1060 | 7 | 3 | uma | UMA | 2020-09-08 | 2022-03-21 | 560 | 7 |
mkr | Maker | 2017-12-26 | 2022-03-21 | 1547 | 7 | 3 | uni | Uniswap | 2020-09-18 | 2022-03-21 | 550 | 7 |
neo | Neo | 2017-07-15 | 2022-03-21 | 1711 | 7 | 3 | usdc | USD Coin | 2018-09-28 | 2022-03-21 | 1271 | 7 | 3
nxm | Nexus Mutual | 2020-08-26 | 2022-03-21 | 573 | 7 |  | usdk | USDK | 2020-06-13 | 2022-03-21 | 647 | 7 |
omg | OMG Network | 2017-07-15 | 2022-03-21 | 1711 | 7 | 3 | usdt | Tether | 2014-10-06 | 2022-03-21 | 2724 | 7 | 2,3
pax | Paxos Standard | 2018-11-30 | 2022-03-21 | 1208 | 7 | 3 | usdt_eth | TetherETH | 2017-11-28 | 2022-03-21 | 1575 | 7 | 3
paxg | PAX Gold | 2020-02-15 | 2022-03-21 | 766 | 7 | 3 | usdt_omni | usdt_omni | 2014-10-06 | 2022-03-21 | 2724 | 7 | 2,3
pay | TenX | 2017-10-03 | 2022-03-21 | 1631 | 7 | 3 | usdt_trx | TetherTRON | 2019-04-16 | 2022-03-21 | 1071 | 7 | 3
perp | Perpetual Protocol | 2021-02-04 | 2022-03-21 | 411 | 7 |  | vtc | Vertcoin | 2014-01-29 | 2022-03-21 | 2974 | 8 | 1,2,3
poly | Polymath | 2018-06-15 | 2022-03-21 | 1376 | 7 | 3 | wbtc | Wrapped Bitcoin | 2018-11-27 | 2022-03-21 | 1211 | 7 | 3
powr | Power Ledger | 2017-11-02 | 2022-03-21 | 1601 | 7 | 3 | weth | Wrapped Ether | 2017-12-18 | 2022-03-21 | 1555 | 7 | 3
ppt | Populous | 2017-09-20 | 2022-03-21 | 1644 | 7 | 3 | wnxm | Wrapped NXM | 2020-08-26 | 2022-03-21 | 573 | 7 |
qash | QASH | 2017-11-06 | 2022-03-21 | 1597 | 7 | 3 | wtc | Waltonchain | 2017-08-28 | 2022-03-21 | 1667 | 7 | 3
qnt | Quant | 2019-03-16 | 2022-03-21 | 1102 | 7 | 3 | xaut | Tether Gold | 2020-02-24 | 2022-03-21 | 757 | 7 | 3
ren | Ren | 2018-12-07 | 2022-03-21 | 1201 | 7 | 3 | xem | NEM | 2015-04-01 | 2022-03-21 | 2547 | 2 | 2,3
renbtc | renBTC | 2020-05-13 | 2022-03-21 | 678 | 7 | 3 | xlm | Stellar | 2015-09-30 | 2022-03-21 | 2365 | 7 | 2,3
rep | Augur | 2016-10-04 | 2022-03-21 | 1995 | 7 | 3 | xmr | Monero | 2014-05-20 | 2022-03-21 | 2863 | 3 | 2,3
rev_eth | rev_eth | 2020-03-26 | 2022-03-21 | 726 | 7 | 3 | xrp | XRP | 2014-08-15 | 2022-03-21 | 2776 | 7 | 2,3
sai | Sai | 2017-12-23 | 2019-11-30 | 708 | 7 |  | xtz | Tezos | 2018-06-30 | 2022-03-21 | 1361 | 7 | 3
snt | Status | 2017-06-20 | 2022-03-21 | 1736 | 7 | 3 | xvg | Verge | 2017-09-30 | 2022-03-21 | 1634 | 8 | 3
snx | Synthetix | 2020-04-09 | 2022-03-21 | 712 | 7 | 3 | yfi | yearn.finance | 2020-07-25 | 2022-03-21 | 605 | 7 |
srm | Serum | 2020-08-11 | 2022-03-21 | 588 | 7 |  | zec | Zcash | 2016-10-29 | 2022-03-21 | 1970 | 8 | 3
sushi | SushiSwap | 2020-09-01 | 2022-03-21 | 567 | 7 |  | zrx | 0x | 2017-08-11 | 2022-03-21 | 1684 | 7 | 3
swrv | Swerve | 2020-09-22 | 2022-03-21 | 546 | 7 |  |  |  |  |  |  |  |
trx | TRON | 2018-06-25 | 2022-03-21 | 1366 | 2 | 3 |  |  |  |  |  |  |
trx_eth | TronETH | 2017-10-07 | 2018-06-25 | 262 | 7 |  |  |  |  |  |  |  |

Note: Obs. is the number of observations in total, i.e. including also the
time points used for training. X depicts the number of additional covariates
contained for each cryptocurrency, and Time Periods describes the sub-periods
that the specific currency is in. Blank spaces indicate that a currency is not
contained in any sub-period, which can occur when the currency is not
available for the full out-of-sample sub-period.

Table 12: External Covariates and Descriptions Variable Name | Coding Coinmetrics |  Description
---|---|---
Active_Users | AdrActCnt |  The number of unique active daily addresses
Total_Users | AdrBalCnt |  The number of unique addresses that hold any amount of native units of that currency
Total_Users_USD100 | AdrBalUSD100Cnt |  The number of unique addresses that hold at least 100 USD of native units of that currency
Total_Users_USD10 | AdrBalUSD10Cnt |  The number of unique addresses that hold at least 10 USD of native units of that currency
SER | SER |  The supply equality ratio, i.e. the ratio of supply held by addresses with less than 1 over 10 millionth of the current supply to the top one percent of addresses with the highest current supply
Transactions | TxCnt |  The number of daily initiated transactions
Velocity | VelCur1yr |  The velocity of supply in the current year, which describes the the ratio of current supply to the sum of the value transferred in the last year
Note: variable coding corresponds to https://docs.coinmetrics.io/. Detailed
variable descriptions are available on
https://docs.coinmetrics.io/info/metrics. Table 13: Medians of P-Values for
DQ-Tests Sorted by Covariate Values in Different Time Periods

Period | Sorting | GRF | QRF | QR | CAV | GJR-GARCH | Hist | GRF-X | QRF-X | QR-X | GARCH-X
---|---|---|---|---|---|---|---|---|---|---|---
High Group
SER
1 | 1 Highest | 0.16 | 0.92 | 0.00 | 0.00 | 0.12 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00
2 | 5 Highest | 0.16 | 0.08 | 0.20 | 0.19 | 0.11 | 0.08 | 0.13 | 0.07 | 0.00 | 0.00
3 | 15 Highest | 0.47 | 0.05 | 0.22 | 0.11 | 0.21 | 0.01 | 0.51 | 0.12 | 0.00 | 0.00
Number of Active Users
1 | 1 Highest | 0.16 | 0.92 | 0.00 | 0.00 | 0.12 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00
2 | 5 Highest | 0.16 | 0.09 | 0.40 | 0.19 | 0.11 | 0.08 | 0.13 | 0.07 | 0.00 | 0.00
3 | 15 Highest | 0.38 | 0.06 | 0.05 | 0.01 | 0.21 | 0.00 | 0.17 | 0.18 | 0.00 | 0.00
7-Day Lagged Standard Deviation
1 | 1 Highest | 0.16 | 0.02 | 0.03 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02
2 | 12 Highest | 0.29 | 0.06 | 0.23 | 0.21 | 0.20 | 0.12 | 0.16 | 0.05 | 0.00 | 0.00
3 | 60 Highest | 0.19 | 0.02 | 0.03 | 0.01 | 0.23 | 0.01 | 0.21 | 0.05 | 0.00 | 0.04
Low Group
SER
1 | 4 Lowest | 0.13 | 0.28 | 0.10 | 0.10 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
2 | 10 Lowest | 0.29 | 0.10 | 0.06 | 0.31 | 0.25 | 0.10 | 0.32 | 0.06 | 0.00 | 0.09
3 | 62 Lowest | 0.07 | 0.04 | 0.03 | 0.02 | 0.32 | 0.00 | 0.18 | 0.07 | 0.00 | 0.06
Number of Active Users
1 | 4 Lowest | 0.13 | 0.28 | 0.10 | 0.10 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
2 | 10 Lowest | 0.42 | 0.09 | 0.06 | 0.31 | 0.20 | 0.10 | 0.26 | 0.06 | 0.00 | 0.09
3 | 62 Lowest | 0.13 | 0.04 | 0.07 | 0.03 | 0.32 | 0.01 | 0.21 | 0.07 | 0.00 | 0.07
7-Day Lagged Standard Deviation
1 | 4 Lowest | 0.13 | 0.42 | 0.10 | 0.10 | 0.07 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00
2 | 3 Lowest | 0.00 | 0.11 | 0.00 | 0.32 | 0.00 | 0.00 | 0.00 | 0.06 | 0.00 | 0.00
3 | 17 Lowest | 0.06 | 0.47 | 0.24 | 0.30 | 0.60 | 0.00 | 0.15 | 0.45 | 0.00 | 0.00

Note: The currencies are divided into two groups based on specific covariate
values for each time period. The groups are constructed to best separate the
sorted covariate values of the currencies. They are therefore separated at
their steepest decay to obtain two groups with homogeneous covariate values.
Detailed plots for the cutoff can be found in Figure 10. The remaining details
correspond to Table 6.

Table 14: Performance and Significance of CPA-tests with GRF-X Over Different
Time Periods

GRF-X vs.: | GRF | QRF | QR | CAV | GJR-GARCH | Hist | QRF-X | QR-X | GARCH-X
---|---|---|---|---|---|---|---|---|---
Share of GRF With Better Performance
Period 1 | 0.40 | 1.00 | 0.80 | 0.60 | 0.80 | 1.00 | 1.00 | 1.00 | 1.00
Period 2 | 0.60 | 1.00 | 0.67 | 0.87 | 0.80 | 1.00 | 1.00 | 1.00 | 1.00
Period 3 | 0.55 | 0.90 | 0.66 | 0.75 | 0.66 | 0.90 | 0.94 | 0.96 | 0.73
Full Data | 0.54 | 0.89 | 0.72 | 0.76 | 0.73 | 0.83 | 0.92 | 0.97 | 0.76
Share of GRF With Significantly Better Performance
Period 1 | 0.00 | 0.20 | 0.20 | 0.00 | 0.20 | 1.00 | 1.00 | 1.00 | 0.60
Period 2 | 0.20 | 0.40 | 0.27 | 0.20 | 0.40 | 0.80 | 0.93 | 0.93 | 0.67
Period 3 | 0.12 | 0.40 | 0.19 | 0.23 | 0.19 | 0.58 | 0.73 | 0.81 | 0.51
Full Data | 0.13 | 0.39 | 0.30 | 0.24 | 0.19 | 0.47 | 0.69 | 0.86 | 0.52

The top part shows summary values that are shares over all cryptocurrencies in
the respective time period. It describes the number of times that GRF-X had a
better performance (i.e. more than 50% of predicted losses by the CPA test
were smaller for the GRF-X) relative to all crytpocurrencies (in that period),
and the number of times that GRF-X was significantly better (at least at a 10%
level) as judged by the CPA test over all crytpocurrencies (in that period).
The bottom part shows the detailed results of CPA-tests with the color of each
box indicating the performance of GRF-X. Blue signifies a performance of 1,
meaning that GRF-X has a smaller predicted loss in 100% of cases. *, **, ***
shows significance on a level of 10%, 5%, and 1%. The values are ordered by 30
day lagged standard deviation from highest to lowest.

Figure 9: Overview of results for CPA-tests of GRF-X vs. all other methods for
cryptocurrencies in the third period ordered by 30 day lagged standard
deviation from highest to lowest, left to right. The color of each box
indicates the performance of GRF-X, with 1 indicating that GRF-X has a smaller
predicted loss in 100% of cases. *, **, *** indicate significance on a level
of 10%, 5%, and 1%. Table 15: Difference Between Covariates of Cryptos Where
GRF-X is Better vs. Worse

| Period 1 | Period 2 | Period 3 | Full Data
---|---|---|---|---
Ret | 1.20 | 0.63 | -0.40 | -0.08
Active_Users | 0.34 | 0.22 | 0.32 | 0.39
Total_Users | 0.79 | 0.26 | 0.31 | 0.34
Total_Users_USD100 | 0.22 | 0.23 | 0.22 | 0.27
Total_Users_USD10 | 0.41 | 0.23 | 0.16 | 0.20
CapMrktCurUSD | 0.08 | 0.16 | 0.45 | 0.38
SER | 1.07 | 0.72 | 0.17 | 0.22
Transactions | 0.42 | 0.06 | 3.26 | 3.74
Velocity | 4.68 | 4.44 | 1.31 | 1.23
sd_3 | 0.89 | 0.31 | 0.65 | 0.65
sd_7 | 0.91 | 0.32 | 0.64 | 0.64
sd_30 | 0.90 | 0.33 | 0.63 | 0.64
sd_60 | 0.91 | 0.34 | 0.63 | 0.65

Note: values are shares of groups of cryptocurrencies where at least two of
CAV, QR, and GJR-GARCH have better CPA-performance than GRF-X divided by the
remaining rest. Raw values before division are mean values over all
cryptocurrencies for the median of each covariate in the respective time
period.

Figure 10: Sorted variable means and cutoff values for groups that separate
the cryptocurrencies based on variables for each period. These groups are used
in backtests in Table 13. The cutoff value is depicted in red dotted lines fro
each variable and time period.

Figure 11: Overview of results for CPA-tests of GRF vs. all other methods for
cryptocurrencies in the first and second period ordered by market cap from
highest to lowest. The color of each box indicates the performance of GRF,
with 1 indicating that GRF has a smaller predicted loss in 100% of cases. *,
**, *** indicate significance on a level of 10%, 5%, and 1%.

Figure 12: Overview of results for CPA-tests of GRF vs. all other methods for
cryptocurrencies in the third period ordered by market cap from highest to
lowest, left to right. The color of each box indicates the performance of GRF,
with 1 indicating that GRF has a smaller predicted loss in 100% of cases. *,
**, *** indicate significance on a level of 10%, 5%, and 1%.

Figure 13: Overview of results for CPA-tests of GRF vs. all other methods for
cryptocurrencies for the full data set ordered alphabetically. The color of
each box indicates the performance of GRF, with 1 indicating that GRF has a
smaller predicted loss in 100% of cases. *, **, *** indicate significance on a
level of 10%, 5%, and 1%. Figure 14: Log-returns of the specific
cryptocurrencies analyzed in Substection 5.2. Table 16: Simulation: $1\%$ VaR

Rolling Window | $l=500$ | $l=1000$ |
---|---|---|---
| DQ | Kupiec | Christoffersen | AoE |  | DQ | Kupiec | Christoffersen | AoE
Sim GARCH Normal |  |  |  |  |  |
QRF |  $0.995$ $(0.002)$ |  $0.910$ $(0.017)$ |  $0.855$ $(0.029)$ | 1.894 |  |  $0.830$ $(0.053)$ |  $0.580$ $(0.128)$ |  $0.395$ $(0.180)$ | 1.751
GRF |  $0.635$ $(0.121)$ |  $0.120$ $(0.396)$ |  $0.115$ $(0.388)$ | 1.265 |  |  $0.410$ $(0.318)$ |  $0.100$ $(0.459)$ |  $0.050$ $(0.518)$ | 1.193
QR |  $0.945$ $(0.018)$ |  $0.500$ $(0.122)$ |  $0.485$ $(0.155)$ | 1.575 |  |  $0.520$ $(0.235)$ |  $0.140$ $(0.433)$ |  $0.090$ $(0.469)$ | 1.273
Hist |  $0.830$ $(0.046)$ |  $0.045$ $(0.421)$ |  $0.195$ $(0.339)$ | 1.231 |  |  $0.675$ $(0.171)$ |  $0.120$ $(0.477)$ |  $0.140$ $(0.456)$ | 1.122
NormFit |  $0.700$ $(0.116)$ |  $0.055$ $(0.470)$ |  $0.200$ $(0.392)$ | 1.165 |  |  $0.625$ $(0.209)$ |  $0.130$ $(0.441)$ |  $0.145$ $(0.450)$ | 1.152
CAV |  $0.890$ $(0.038)$ |  $0.365$ $(0.181)$ |  $0.340$ $(0.215)$ | 1.486 |  |  $0.450$ $(0.304)$ |  $0.130$ $(0.435)$ |  $0.080$ $(0.508)$ | 1.245
GARCH(1,1) |  $0.440$ $(0.268)$ |  $0.060$ $(0.468)$ |  $0.150$ $(0.412)$ | 1.128 |  |  $0.310$ $(0.408)$ |  $0.060$ $(0.532)$ |  $0.075$ $(0.525)$ | 1.072
Sim GARCH t |  |  |  |  |  |
QRF |  $0.990$ $(0.002)$ |  $0.890$ $(0.016)$ |  $0.845$ $(0.029)$ | 1.942 |  |  $0.815$ $(0.060)$ |  $0.665$ $(0.108)$ |  $0.500$ $(0.152)$ | 1.801
GRF |  $0.625$ $(0.127)$ |  $0.135$ $(0.372)$ |  $0.155$ $(0.396)$ | 1.286 |  |  $0.410$ $(0.297)$ |  $0.100$ $(0.455)$ |  $0.055$ $(0.523)$ | 1.222
QR |  $0.925$ $(0.014)$ |  $0.570$ $(0.106)$ |  $0.575$ $(0.134)$ | 1.676 |  |  $0.565$ $(0.227)$ |  $0.165$ $(0.393)$ |  $0.130$ $(0.431)$ | 1.290
Hist |  $0.755$ $(0.077)$ |  $0.080$ $(0.412)$ |  $0.215$ $(0.385)$ | 1.232 |  |  $0.665$ $(0.160)$ |  $0.125$ $(0.454)$ |  $0.175$ $(0.434)$ | 1.133
NormFit |  $0.890$ $(0.028)$ |  $0.655$ $(0.092)$ |  $0.670$ $(0.096)$ | 1.693 |  |  $0.760$ $(0.083)$ |  $0.510$ $(0.184)$ |  $0.450$ $(0.209)$ | 1.669
CAV |  $0.815$ $(0.042)$ |  $0.320$ $(0.190)$ |  $0.310$ $(0.230)$ | 1.466 |  |  $0.425$ $(0.298)$ |  $0.155$ $(0.410)$ |  $0.090$ $(0.472)$ | 1.248
GARCH(1,1) |  $0.855$ $(0.045)$ |  $0.750$ $(0.057)$ |  $0.745$ $(0.068)$ | 1.769 |  |  $0.595$ $(0.174)$ |  $0.500$ $(0.176)$ |  $0.430$ $(0.196)$ | 1.680
Sim SAV-Model |  |  |  |  |  |
QRF |  $0.990$ $(0.001)$ |  $0.940$ $(0.015)$ |  $0.835$ $(0.028)$ | 1.891 |  |  $0.810$ $(0.063)$ |  $0.575$ $(0.133)$ |  $0.405$ $(0.187)$ | 1.749
GRF |  $0.585$ $(0.140)$ |  $0.065$ $(0.417)$ |  $0.045$ $(0.462)$ | 1.251 |  |  $0.300$ $(0.352)$ |  $0.080$ $(0.460)$ |  $0.035$ $(0.550)$ | 1.180
QR |  $0.940$ $(0.015)$ |  $0.445$ $(0.129)$ |  $0.375$ $(0.179)$ | 1.553 |  |  $0.560$ $(0.200)$ |  $0.190$ $(0.406)$ |  $0.090$ $(0.473)$ | 1.289
Hist |  $0.360$ $(0.239)$ |  $0.060$ $(0.515)$ |  $0.075$ $(0.534)$ | 1.195 |  |  $0.275$ $(0.415)$ |  $0.060$ $(0.513)$ |  $0.060$ $(0.597)$ | 1.097
NormFit |  $0.215$ $(0.449)$ |  $0.105$ $(0.452)$ |  $0.110$ $(0.527)$ | 1.166 |  |  $0.240$ $(0.520)$ |  $0.135$ $(0.448)$ |  $0.090$ $(0.553)$ | 1.147
CAV |  $0.855$ $(0.035)$ |  $0.255$ $(0.207)$ |  $0.250$ $(0.239)$ | 1.444 |  |  $0.485$ $(0.273)$ |  $0.075$ $(0.439)$ |  $0.050$ $(0.518)$ | 1.221
GARCH(1,1) |  $0.215$ $(0.407)$ |  $0.090$ $(0.502)$ |  $0.090$ $(0.532)$ | 1.155 |  |  $0.210$ $(0.513)$ |  $0.105$ $(0.471)$ |  $0.055$ $(0.574)$ | 1.122
Sim GARCH Bitcoin fit |  |  |  |  |  |
QRF |  $0.975$ $(0.004)$ |  $0.910$ $(0.017)$ |  $0.855$ $(0.029)$ | 1.894 |  |  $0.750$ $(0.090)$ |  $0.580$ $(0.128)$ |  $0.395$ $(0.180)$ | 1.751
GRF |  $0.470$ $(0.207)$ |  $0.120$ $(0.396)$ |  $0.115$ $(0.388)$ | 1.265 |  |  $0.275$ $(0.424)$ |  $0.100$ $(0.459)$ |  $0.050$ $(0.518)$ | 1.193
QR |  $0.880$ $(0.040)$ |  $0.500$ $(0.122)$ |  $0.485$ $(0.155)$ | 1.575 |  |  $0.370$ $(0.328)$ |  $0.140$ $(0.433)$ |  $0.090$ $(0.469)$ | 1.273
Hist |  $0.675$ $(0.090)$ |  $0.045$ $(0.421)$ |  $0.195$ $(0.339)$ | 1.231 |  |  $0.530$ $(0.259)$ |  $0.120$ $(0.477)$ |  $0.140$ $(0.456)$ | 1.122
NormFit |  $0.540$ $(0.203)$ |  $0.055$ $(0.470)$ |  $0.200$ $(0.392)$ | 1.165 |  |  $0.500$ $(0.296)$ |  $0.130$ $(0.441)$ |  $0.145$ $(0.450)$ | 1.152
CAV |  $0.745$ $(0.091)$ |  $0.105$ $(0.405)$ |  $0.155$ $(0.400)$ | 1.243 |  |  $0.435$ $(0.345)$ |  $0.045$ $(0.486)$ |  $0.030$ $(0.555)$ | 1.046
GARCH(1,1) |  $0.345$ $(0.328)$ |  $0.065$ $(0.459)$ |  $0.190$ $(0.385)$ | 1.157 |  |  $0.215$ $(0.510)$ |  $0.050$ $(0.537)$ |  $0.070$ $(0.512)$ | 1.082

Results for $1\%$ VaR show rejection rates of t-tests of empirical levels
against the nominal level of $1\%$ for DQ-, Kupiec- and Christoffersen-tests
and mean p-values in parentheses. Higher p-values and lower rejection rates
indicate better model performance. GARCH depicts an oracle GARCH-model that
fits an GARCH(1,1) process with normally distributed errors.