Bayesian Analyses of the Two-Stage CARR-Return Models with Applications to COVID-19 Impact on the Cryptocurrency Market

3.1 CARR Model

Engle and Russell (1998) studied the irregular time intervals between transactions, particularly financial market transactions such as trading stocks. They found that there was clustering of transactions, and hence, a high autocorrelation in the time intervals between trades. The autoregressive conditional duration (ACD) model was then proposed to model the conditional density of the durations of such transactions. Chou (2005) extended this idea by using range-based volatility measures in the ACD model, proposing the Conditional Autoregressive Range (CARR) model. Let V _t be the random variable for the volatility process at time t which can be measured by (3) to (6). The CARR ( p , q ) model can be written as

where λ _t is the conditional mean of the volatility measure and ɛ _t is the error term which has a mean of one and is assumed to follow some distribution defined on the positive real domain. For this process to be stationary, the following conditions must hold

Parameters β_1,i are the autoregressive terms which capture the short-term volatility while β_2,i measures the persistence by giving weight to the previous volatility estimates which slow the rate of change for the estimates. The long term unconditional mean is captured in the term β₀.

Chou (2005) showed that this model can be extended to capture other exogenous variables denoted by X_i,t. The conditional mean in the CARRX model can be written as

Some exogenous variables that could be used include lagged returns, trading volume, or variables which capture seasonal affects.

With volatility, there can be asymmetry in future volatility based on whether the previous days return was positive or negative. This is known as the leverage effect. Tan et al. (2019) considered three asymmetric mean functions denoted as LCARR ( p , q , w ) and can be written as

where w = 1 , 1 λ t − 1 , 1 V t − 1 are denoted by a , b , and c respectively to capture the leverage effect and 1 _E is an indicator function of event E such that it is 1 when E holds and 0 otherwise.

Storti and Vitale (2003) proposed a GARCH model which included a bi-linear effect, capturing the interaction between past observed and estimated volatilities. This interaction effect will be incorporated into the CARR model, denoted by BCARR ( p , q , s ) and will be written as

The parameter β₅ is used here to avoid confusion between the parameters which measure the leverage effect, β₃ and β₄.

In the CARR model, the error term (ɛ _t ) is assumed to follow some positive distribution. Previous works on the CARR models typically used the Weibull (Wei) distribution. However, the GB2 distribution is more general as it nests many other distributions as special cases such as the Wei and GG distribution. See Table 1 for a list of distributions used in this paper and Figure 1 of Chan et al. (2018) for a list of nested distributions.

3.2 Return Model

In order to get VaR estimates of returns using the CARR model, Shao et al. (2009) denoted a return process to include a conditional return with volatility inputs λ _t from the CARR models. Let R _t be the random variable of return at time t measured by (2). This model is referred to as stage two of the CARR-return model and can be written as

where E ( z t ) = 0 , Var(z _t ) = 1, and z _t can follow any distribution defined on the real domain such as Normal, Student-t and VG distributions. Read Fung and Seneta (2007) for details of the symmetric VG distribution including the pdf and the fact that it contains the symmetric Laplace distribution as a special case when the shape parameter ν = 1. The parameter λ _t is the fitted volatility using the CARR models in (8), (10) and (11) and ρ is a scaling parameter to ensure that the volatility estimate σ t 2 = ρ λ t is unbiased for the true volatility in (1) which is further assumed to change over time. The mean μ _t can follow any model such as a constant mean or autoregressive model. An AR(1) model

is used in this analysis as the autocorrelation of returns (see Figure 7) at the first lag is sometimes significant, however, typically there is not much autocorrelation in returns making it difficult to forecast. This model provides an alternative way to model returns in comparison to the GARCH model. The volatility is modelled separately in the two-stage model and is assumed to follow some distribution such as GB2. This is beneficial as the stochastic nature of volatility allows one to estimate VoaR and CVoaR which cannot be done with the GARCH model. In addition, any volatility measure such as the RPK volatility measure in (5) can be used for the volatility model and they capture more information about volatility throughout a day. The GARCH model is restricted to using the return squared although realised GARCH model allows more flexibility. Because the two-stage model takes in two data sources, it is often more efficient than the GARCH model.

3.3 Model Evaluation

3.4 Bayesian Inference

4.1 Volatility Data

The data used for the analysis comes from the cryptocurrency exchange Binance and is extracted using the Python package binance.client. Data is collected in the time intervals 1 day, 1 h, 5 min, and 1 min for the two cryptocurrencies BTC and ETH. All timestamps are in Coordinated Universal Time (UTC) time zone. The time period that will be considered in the analysis is 2019-01-01 to 2022-12-31.

To compare how the features of volatility and returns have changed throughout the pandemic, three time periods:

1. Period 1: 2019-01-01 to 2019-12-31 (pre-pandemic; T* = 365)

2. Period 2: 2020-01-01 to 2020-12-31 (early pandemic; T* = 366)

3. Period 3: 2021-01-01 to 2022-07-01 (late pandemic; T* = 547)

Table 3 reports the summary statistics for the volatility measures PK, RPK and QPK defined in (3), (5) and (6), respectively, for both BTC and ETH as well as for each time period. It can be seen that the RPK measures usually have the lowest variance for time period 1 and QPK measures for time period 3 demonstrating their relative efficiencies under the pre and post pandemic periods respectively. Moreover, QPK measures have lower mean by excluding outliers. Across periods, period 2 has the highest variance, skewness and kurtosis and next comes period 3. These are due to more frequent market shocks which cause short periods of heighten measures during the pandemic particularly the early period.

The test statistic LB Q₁₀ tests whether the group of autocorrelation’s up to lag 10 are different from 0. A higher test statistic indicates that autocorrelation’s are further from 0. All test statistic are significant at the 5 % level except the QPK(5, 95) measure in Period 1 for ETH which is highlighted in boldface. Hence, it is important to model the short and long-term memory in volatility as it is clearly present. The RPK, 1 m (1 min) measure always has the highest LB Q₁₀ test statistic followed by the RPK, 1 h (1 h) measure which could mean that realised volatility measures have higher short-term memory, however, this will be explored later in the analysis.

Figure 1 visualises the data characteristics. The times series plot (column 1) for period 2 shows that there is a period of extremely high volatility in early 2020, particularly on the 12th and 13th of March. This was just after the World Health Organisation (WHO) declared COVID-19 a pandemic on the 11th of March which highlights the major impact it had on the cryptocurrency market. The x-axis of the density plots (column 2) is capped at 0.02 to exclude these extreme outliers as it would be hard to see the shape of the distribution otherwise. The exclusion of outliers can also be seen from the robustness of the QPK measure: in period 2, the max of the QPK(5, 95) measure is 0.068 (see Table 3) which is less than the max of 0.126 for the PK measure by excluding these outliers. From the ACF plots (column 3) in Figure 1, there appears to be more autocorrelation for the RPK measures as seen from the higher spikes in the ACF plots and the higher Ljung-Box test statistics in Table 3. Moreover, the ACF plots do not display periodic persistence. Summary plots for ETH can be found in the Appendix A. The behaviour of ETH is relatively similar to BTC.

Figure 1:

BTC time series, density, and ACF plots for selected volatility measures.

Figure 1:

(continued)

4.2 Comparison of Pandemic Periods, Error Distributions and Volatility Measures

Different error distributions, Wei, GG, and GB2, are compared using the CARR(1, 1) model in (7) and (8) with order p = q = 1 where the PK, RPK and QPK volatility measures are used. For each period, they are compared using both BTC and ETH. DIC is used to evaluate the model-fit and determine the best choice of error distribution where a lower DIC is preferred. Table 4 reports regression parameter estimates using each combination of volatility measures, error distributions, periods and both BTC and ETH. We do not report shape parameters (a, p, q) as they do not provide direct interpretation of the distribution properties such as moments. Instead we focus on evaluating persistence effects. In terms of computational speed, the models took roughtly 0.5, 1.5, 5 min to run using Wei, GG and GB2 distributions respectively. This time is about 50 % longer when running the models on period 3 reflecting model complexity and data size. The log density is manually specified for the GG and GB2 distributions as no inbuilt functions are available in RStan.

Comparing error distributions, the models using GB2 always have the lowest DIC followed by the GG distribution as expected. Comparing persistence parameter estimates across more generalised distributions, the short-term memory β₁ drops, the persistence β₂ increases and their sum (β₁ + β₂ or total persistence) also increases. For example, β₂ for RPK, 1 m measure using BTC in period 1 rises from 0.052 for Wei to 0.541 for GB2 distribution. The averaged persistence and total persistence across coins, periods and models for the three distributions in Table 4 are (0.31, 0.48, 0.63) and (0.76, 0.87, 0.95) showing very clear increases. This is because a more generalised distribution can capture the outliers better with a heavier tail so that the persistence becomes more clear. Across periods, the persistence and total persistence averaged across models for BTC ( 0.45 , 0.52 , 0.58 ) and ( 0.88 , 0.87 , 0.90 ) and ETH ( 0.38 , 0.40 , 0.52 ) and ( 0.79 , 0.84 , 0.91 ) also increases in general revealing increasing persistence across pandemic. Reversed trends are also displayed for short-term memory β₁.

For the volatility measures fitted to the CARR models, four sampling frequency, 1 h, 15 min, 5 min, and 1 min of the RPK measures are originally considered but only results for 1 h and 1 min are reported in Table 4 because the trends of effect are roughly in order of sampling frequency. As the sampling frequency increases, information for the short-term memory will also increase. For example, a RPK, 1 m measure has 1,440 intervals during the day in comparison to 24 intervals when using the RPK, 1 h measure. Hence, the RPK, 1 m is capturing more information about the volatility allowing for better predictions of the short-term memory in volatility. This can also be seen from the ACF plots in Figure 1 that the RPK, 1 m measure has higher autocorrelation for all periods. Andersen and Bollerslev (1998) noted how as the number of intervals in a realised volatility measure goes to infinity, it approaches the true volatility. Hence, it is evident that as the sampling frequency increases, the short-term memory parameter β₁ become higher and so explains why β₁ from RPK, 1 m is higher relative to RPK, 1 h.

Similarly, the symmetric quantile ranges (1, 99), (2, 98), (3, 97), (4, 96) and (5, 95) are also applied to the CARR models and only results from (1, 99) and (5, 95) are reported. Symmetric ranges are more efficient as suggested by Tan et al. (2019). Between (1, 99) and (5, 95), the persistence (short-term memory) of using (5, 95) seems to be slightly higher (lower) as more outliers are dropped. Comparing across the three types of volatility measures PK, RPK and QPK, the persistence and total persistence averaged over periods, distributions and models (1 h, 1 m for RPK; (1, 99), (5, 95) for RPK) are (0.60, 0.35, 0.64) and (0.87, 0.89, 0.87) for BTC and (0.51, 0.29, 0.53) and (0.82, 0.89, 0.81) for ETH in Table 4 showing a clear drop of persistence using RPK measures for both coins as expected and a clear increase of total persistence using RPK measures for ETH. Table 14 in Appendix C reports the 95 % credible intervals for all the parameters providing some ideas of their estimates’ precision.

Figure 2 visualises the impact of changing the sampling interval length of the RPK measure and the error distribution used on short-term memory and persistence. This graph is created using the data from period 1. As can be seen on the left hand side of the graph, when the interval size is 1 min, β₁ is highest which can be seen from the dotted lines and it declines as the interval size increases. Simultaneously, the persistence term β₂ increases which is expected as there is a trade-off between long-term persistence and short-term memory. However, after an interval size of about 120 min (2 h) there is not much difference in short-term memory. So in practice, sampling frequency less than 2 h would be needed to capture the extra information in a realised volatility measure. Also, it is clear that when using a more generalised distribution, β₁ is always lower and β₂ is higher as explained before.

Figure 2:

RPK interval length and distribution impact on short-term memory and persistence for BTC in period 1.

Figure 3a and b compares the observed and theoretical error densities from the CARR(1, 1) model for each error distribution and period using the RPK, 1 m volatility for BTC and ETH respectively. It is clear that the GB2 distribution (column 3) captures observed error density better. The QQ-plots for each model using the different error distributions for BTC and ETH across periods can be seen in Figure 4. While the observed quantiles from 0 to 2 are relatively close to the theoretical quantiles in each graph, the right tails are usually far off the theoretical quantiles when using the Wei and GG error distributions. On the other hand, models using the GB2 distribution are able to model these heavy tails better, however, not perfectly which highlights the difficulty in modelling the tails of volatility. There is one extreme observation for BTC in period 2 which can be seen in the middle row that none of the distributions could capture. This observation was on March 12th, just after COVID-19 was declared a pandemic on March 11th. Similar observations are seen for ETH.

Figure 3:

Theoretical versus actual density for the CARR(1, 1) model errors using RPK, 1 m volatility measure.

Figure 4:

QQ plots for the CARR(1, 1) model error using RPK, 1 m volatility measure.

In summary, results highlight the importance of exploring different error distributions when modelling volatility and that a more generalised distribution such as GB2 provides better model fit in terms of DIC in Table 4, density plots in Figure 3 and QQ plots in Figure 4. This is because its flexible tail can downweigh the effect of outliers on the model-fit and so is more able to indicate a true lower short-term memory and capture stronger persistence. On the other hand, a potential reason for higher short-term memory when using Weibull is because the error terms will take higher values with lower probability due to the thinner tails with the distribution.

Moving forward, only the GB2 error distribution will be used for the CARR models as GB2 distribution models the error term the best allowing for more accurate parameter estimates. Also, the RPK, 1 m measure will be used for the two-stage model in Section 5 as this measure captures more information about the true volatility.

4.3 Comparison of Mean Functions and Time Periods

Using the GB2 error distribution and the RPK, 1 m measure, the CARR model with different mean functions are compared using different orders, as well as the leverage and bilinear effects. The models compared are the CARR ( p , q ) , LCARR ( p , q , a ) and BCARR ( p , q , s ) as defined in (8), (10) and (11), respectively. We also consider leverage effect choices a , b , c and the choice a in which w = 1 often gives better DIC. Hence, only the LCARR ( p , q , a ) model is reported in Table 5 and considered in subsequent analyses. Moreover, the BCARR(1, 1, 1) model gives very similar results to CARR(1, 1) model. Hence, it is reported in Table 16 in Appendix D together with LCARR ( 1,1 , b ) and LCARR ( 1,1 , c ) models for both coins. In Table 5, we drop parameter a from their model names without causing confusion and report parameter estimates as well as DICs. Again, 95 % credible intervals showing precision of all parameter estimates are reported in Table 15.

As shown in Table 5 and 16, the LCARR(1, 2, a ) model has the best performance due to the lowest DIC for each period. This model has second order persistence and the leverage affect. This LCARR(1, 2, a ) model shows an improvement in DIC relative to CARR(1, 1) from −4,588 to −4,602. Clearly, the improvement is less than −4,350 to −4,588 when the errors distribution changes from Wei to GB2 on the CARR(1, 1) model.

Comparing between models, CARR(2, 1) models have higher DIC (as β₁₂ are often tiny) than CARR(1, 2) showing that the higher order terms should be applied to long-term persistence as expected. On the other hand, the sum of β₂₁ + β₂₂ in CARR(1, 2) is close β₂ in CARR(2, 1) or CARR(1, 1) showing no obvious overall increases in persistence for higher order models. Comparing CARR(1, 2) with LCARR(1, 2, a ) models, the latter with leverage effect display lower short-term memory and stronger persistence, hence demonstrating better model-fit. Another observation is on parameters β₃ and β₄ for the LCARR(1, 2, a ) model which are always positive indicating that a larger previous day return in either direction would result in higher volatility.

Comparing effects across periods, β₄ (0.00818) is much higher than β₃ (0.00260) for period 2 or early pandemic, meaning that a positive return would have a higher impact on volatility in comparison to a negative return of the same magnitude. This trend is different in period 3 as β₃ is higher than β₄ meaning that negative returns had a higher impact on volatility as expected. This indicates that the stress of early pandemic has reversed the usual leverage effect in the market. Moreover, there is lower short-term memory and higher persistence during the early pandemic period (period 2). For example, when considering the LCARR(1, 2, a ) model, β₁ is 0.310 in comparison to 0.361 and 0.397 in the other two periods. This lower short-term memory could be due to more uncertainty in the market with sudden shocks from the two days of extremely high volatility in period 2 which had an impact on the parameter estimates as discussed in Section 4.4.

Similar observations are seen for ETH. For example, the LCARR(1, 2, a ) is the best model in each period and as well, the usual leverage effect is reversed in period 2. Similar results are also observed when fitting RPK, 5 m measures to the CARR models in Table 5. Again, the best model is LCARR(1, 2, a ) for all periods and both coins according to DIC. As the trends are similar, this table is omitted.

4.4 Model Robustness – Removing Outliers in Period 2

In period 2, there are two days of extremely high volatility and large negative and positive returns on March 12th and 13th, 2020. The RPK, 1 m measure on these two days is 0.06 and 0.09 which is approximately 10 and 15 standard deviations away from the mean volatility. The returns on these two days also have a high magnitude being 12 and 3.5 standard deviations from the mean return on the 12th and 13th of March, respectively. This was just after the WHO declared COVID-19 a pandemic. Outliers can have an impact on the parameter estimates so it is important to check the sensitivity of parameter estimates when the outliers are removed. In order to remove the outliers, the volatility on the 12th and 13th of March was set to be equal to the volatility on the 14th of March. Also, the returns on the 12th and 13th were set to be equal to 0.

Table 6 shows the effect of removing the outliers for the LCARR(1, 2, a ) model in period 2 using both BTC and ETH. It is clear that the short-term memory β₁ is now higher for both coins. Parameter β₁ is likely to be lower as the outliers will mask the autocorrelation pattern (short-term memory) in the data. Now its clear that after removing outliers, there is an increase in short-term memory and a drop in persistence for period 2. Between periods, the β₁ (β₂₁ + β₂₂) value of 0.393 (0.533) for BTC after removing the outliers is closer to the β₁ (β₂₁ + β₂₂) values of 0.361 (0.520) and 0.397 (0.498) for the other two periods using the LCARR(1, 2, a ) model (see Table 5). Hence, it appears that COVID-19 has a mild gradual increase on short-term memory and no obvious change on persistence if there are no shocks to the market on the 12th and 13th of March, 2020. In terms of the leverage effect, both β₃ and β₄ become smaller but not by a large amount. Parameter β₄ also remains larger than β₃ meaning that a positive return has a larger impact on volatility than a negative return of the same magnitude. Lastly, DICs for both coins improve after removing the outliers.

In summary, these modelling results show that capturing different effects in volatility such as the leverage effect and second order persistence does improve the in-sample model fit as indicated by the lower DIC’s in Table 5. However, the choice of error distribution has the biggest impact on the DIC value. Hence, it is more important to ensure that an accurate error distribution is being used when modelling the volatility as capturing the fatter tail of the distribution is more effective than fitting a more complicated conditional volatility mean function.

Comparing the COVID-19 impact on volatility across periods, it did appear that there was lower short-term memory due to the lower β₁ estimates in Table 5 for period 2. However, once the two outliers on the 12th and 13th of March, 2020 were manipulated to have a much lower volatility by setting them to the volatility on the 14th of March, β₁ became higher as seen in Table 6 and is closer to the β₁ values in the other two periods. Hence, it appears that COVID-19 did not have a major impact on short-term memory and persistence after discounting the large shocks. The outliers in period 2 may violate the thinner tail assumption such that the effect may contaminate the short-term memory and persistence effects in the mean function. This could be managed by either adopting more general distributions with flexible tails or manipulating the mean function to capture a possible jump.

Comparing the leverage affect across periods, it did appear to change across periods. During the early pandemic period (period 2), β₄ (0.00818) is the highest and larger than β₃ indicating that a positive return would have a larger impact on the next day volatility. But for the late pandemic (period 3), we do observe the usual phenomenon that β₃ is much larger than β₄ indicating higher impacts on volatility from negative returns and so, the market is starting to behave more like the usual financial asset markets.

While these analyses compare the features of volatility across the three periods, it is important to note that differences between periods may not be due to the COVID-19 pandemic alone. For example, the changing popularity of investing in cryptocurrencies as well as regulation may also be strong drivers of the differences between periods.

4.5 Out-Of-Sample Volatility Forecast Performance

So far, the models have only been compared based on their in-sample performance. Forecast performance comparison is important to evaluate models’ ability to generalise to unseen data. Section 3.4.4 details the one-step-ahead risk measure forecasts based on the 100 moving window W _s = (V,R)_(s+1):(s+T), s = 0, …, 99 with T = T* − 100 days of window size for model training. Taking period 1 with T* = 365 days as an example, the window size is 265 and windows (1, 265), (2, 251), …, (100, 364) are used to train 100 models and forecast day 266, 267, …, 365 (2019-09-23 to 2019-12-31) respectively. For period 2 and 3 with sample sizes T* = 366, 547, the forecast periods are 2020-09-23 to 2020-12-31 and 2022-03-24 to 2022-07-01 with 266 and 447 window sizes (days for model training) respectively. These forecast periods can be seen in Figure 5.

Figure 5:

One-day-ahead forecasts of VoaR and CVoaR using the LCARR(1, 2, a ) model for the RPK, 1 m measure across periods for BTC and ETH.

With Bayesian inference, posterior predictive distributions V s + T + 1 ( 1 : K ) (size K = 10,000) for the volatility forecasts after burn-in (see Section 3.4.4) can be simulated from the model trained using data window W _s , s = 0, …, 99. Taking the best LCARR(1, 2, a ) model with GB2 distribution as an example, the one-step ahead forecast of V_s+T+1 from window W _s in the MCMC iteration k ∈ 1: K is given by

(see Table 1) where λ s + T + 1 ( k ) is given by (10) using the data window W _s , p = 1 , q = 2 , and parameter β b , i = β b , i ( s , k ) , b = 0,1,2 , … at iteration k. In this way, we could obtain the posterior predictive sample V s + T + 1 ( k ) : k = 1 , … , K for the forecast day T + s + 1, s = 0, …, 99, that is, the last 100 days of each pandemic period. For the VoaR and CVoaR forecasts, it is clear that only the upper quantile levels α₂ are of interest and they can be estimated based on the quantiles and conditional mean of the posterior sample, that is,

where V ̂ s + T + 1 ( k α 2 ) denotes the kα₂-th posterior volatility forecast sample arranged in ascending order from V ̂ s + T + 1 ( 1 : K ) defined in (25). As a demonstration, Figure 13d present the posterior predictive distribution of V ̂ T + 1 ( 1 : K ) ( s = 0 ) in (25) for the first one-step-ahead forecast, fitting V_1:T for BTC to the first model of Table 4, that is CARR(1, 1) with Weibull distribution. The window size T = 265 in period 1 and the posterior sample size K = 10,000. Hence, the estimate V ̂ 266 indicated by the red dash line is the median V ̂ 266 ( 5,000 ) of the distribution V ̂ 266 ( 1 : 10,000 ) in ascending order whereas V o a R 266 ( 0.9 ) = V ̂ 266 ( 9,000 ) takes the 9,000-th value and CVoaR₂₆₆(0.9) takes the average of the 9,000-th to 10,000-th values.

4.6 Extension of Lagged Predictors to Varying Interval Length to Identify Pandemic Impacts

In the current time series models, variables such as return or volatility are regressed on their lagged measures which capture information from the whole previous days. For the LCARR(1, 2, a ) model with the best in-sample and out-of-sample performance, the volatility is regressed on the previous day volatility, and the leverage effect uses the previous day return. This model has an implicit assumption that the volatility from the previous day is the best predictor of next days volatility for the autoregressive component, and that this previous day volatility is essentially uniform in a day. Now consider two different days within a time series where their volatility measures are almost equal. On one of these days, the volatility is increasing, and the other day it is decreasing. Clearly, we would expect the day with increasing volatility to have a higher volatility the next day on average. However, current models would treat these two days the same as they only measure volatility on a daily base.

This section aims to cover some potential improvements to the CARR model, however, these improvements can be applied to other models such as the GARCH model. We focus only on the LCARR(1, 2, a ) model in this analysis as it is the best performing volatility model in Table 5.

4.6.1 Varying Period Length of Lagged Volatility for Short Memory

Looking at the LCARR(1, 2, a ) model, the previous days volatility V_t−1 in the conditional mean function λ _t could be replaced by a measure which only captures volatility from the second half of the previous day to predict the next days whole volatility. This section will explore the effect of regressing on previous volatility, captured from different lengths, e.g. the last 12, 16, and 20 h from the previous day.

To explore the effect, attempts are made to regress on previous volatilities, captured from time intervals of different lengths. The idea can also be extended to regress on multiple ( m ) non-overlapping previous volatilities. For example, treating the volatilities from the first and second half of the previous day as two separate variables ( m = 2 ) in the model. In theory, this model would perform just as well as the model which only uses the previous day volatility as this is a special case of the extended model when the parameter estimates are equal.

To define this model, take the conditional mean of the LCARR ( p , q , a ) model given by (10) with w = 1 and replace the term ∑ i = 1 p β 1 , i V t − i by ∑ i = 1 m α i v t − 1 , i where m is the specified number of variables to regress on and v_t−1 are the set of non-overlapping volatilities captured from previous time intervals t − 1 of different lengths. Note, it is not required that p = m . Then the new model denoted by LCARR * ( m , q , a ) can be written as

(27) λ t = β 0 + ∑ i = 1 m α i v t − 1 , i + ∑ i = 1 q β 2 , i λ t − i + β 3 | R t − i | 1 R t − 1 < 0 + β 4 R t − i 1 R t − 1 > 0 .

The stationary conditions in (9) become more complicated when we regress on varying time lengths. For example, if v t − 1 , i * comes from a time interval smaller than a day, then α _i can take a value larger than 1. Since E v t − 1 , i * = p E ( V t − 1 ) where p is the length of the time interval as a proportion of 1 day, the volatility measures v_t−1,i will be multiplied by 1/p, that is v t − 1 , i = v t − 1 , i * / p to ensure that E ( v t − 1 , i ) = E ( V t − 1 ) . Then the same stationary conditions

(28) β 0 , α i , β 2 , j > 0   and   ∑ i = 1 m α i + ∑ i = 1 q β 2 , i < 1

apply.

To show how the DIC of the LCARR * ( 1,2 , a ) model changes with increasing length of the interval that capture previous volatility, Figure 6a plots DIC against the last n hours of the day (n = 4, 6, 8, …, 22) that measures v_t−1 and replaces the lagged volatility term V_t−1 (n = 24). This was run for each time period using the RPK, 1 m measure and BTC. In the figure, the red line shows the DIC from the LCARR(1, 2, a ) model using V_t−1 (n = 24) as the benchmark while the black line shows the DIC of the LCARR * ( 1,2 , a ) model for different interval lengths of v_t−1. For period 1, regressing on the previous 18 h results in the lowest DIC of −4,606 which is lower than the DIC of −4,602 for the LCARR(1, 2, a ) model using 24 h. However, a longer period length has a lower DIC for period 2 and 3, which indicates that regressing on volatility measures captured from shorter periods generally has no obvious improvement. Hence, results show that using the whole previous days volatility measure is perhaps a good choice.

Figure 6:

DIC against interval length in the previous day volatility for the autoregressive effect (subfigures (a) and (b)) and in the previous day returns for the leverage effect (subfigure (c)) across the three periods.

Another LCARR * ( 2,2 , a ) model is run where the previous day volatility V_t−1 is replaced by two volatility measures, v_t−1,1 and v_t−1,2, computed using non-overlapping intervals from the previous day. For example, one volatility measure is computed from the last 8 h of the previous day denoted as v_t−1,1, and the other volatility measure is computed from the first 16 h denoted as v_t−1,2. This model can be written as (27) with m = 2 . This setting aims to see whether each of the 2 intervals from the previous day has different correlation with the next day volatility. Hence, they should be given different weights or coefficients α₁ and α₂.

Figure 6b plots the DIC against the length n in hours for the volatility measure v_t−1,1 computed from the last n hours of the previous day. Note that the other volatility measure v_t−1,2 is computed using volatility from the first 24 − n hours of the previous day. In the figure, we can see that splitting up the volatility measure from the previous day into two measures results in an improvement on the LCARR * ( 2,2 , a ) model for periods 1 and 3. This can be seen as the black line which indicates the DIC’s for the new model does go lower than the benchmark red line which is the DIC for the LCARR(1, 2, a ) model. In period 3, when v_t−1,1 is computed using the last 4 h of volatility from the previous day, and v_t−1,2 from the first 20 h, the DIC is about −6,327 in comparison to −6,309 for the LCARR(1, 2, a ) model. This is quite a large improvement and shows that in period 3, the most recent hours from the previous day would have higher predictive power. Hence, v_t−1,1 should be treated as a separate variable. Moreover, for period 1, it is evident that regressing on the previous volatility computed from a smaller interval from the end of the previous day improves the DIC. However, the new model performed worse for period 2 as seen by the higher DIC’s. Overall, these results indicate that in some cases, it may be appropriate to split up the previous day volatility into two volatility measures from two non-overlapping intervals and regress on them as two separate measures.

5.1 Return Data

The daily close-to-close log return is considered as input to the return models. Table 9 summarises and compares this return data across the three periods for both BTC and ETH. From the table, the mean return is clear highest for period 2 for both coins. This is because the price of BTC went from 7,201 to 28,924 USD in 2020 and ETH went from 131 to 736 USD. The variance of returns is also the highest in period 2 which shows how the uncertainty in the market due to COVID-19 interacted with the increasing popularity of cryptocurrencies may impact the volatility. For periods 1 and 3, the skewness and kurtosis are relatively low in comparison to period 2. The LB Q₁₀ test for autocorrelations up to lag 10 are significant with the test statistics highlighted in boldface except periods 1 and 3 for BTC. The LB test statistics are also higher for ETH across all periods.

The bottom row of Table 9 is the p-values from the Kolmogorov-Smirnov (KS) test for normality. Most of the p-values are below 0.05 indicating that the returns do not come from a normal distribution highlighting the need to use heavy tail distributions hinted from the high kurtoses. There is one exception for ETH in period 3. Moreover, the p-value for BTC in period 3 is just marginally significant. This closer to normality observation for both coins in period 3 is also evidenced from the fact that period 3 is, in fact, longer than the other periods, and so is more powerful in testing against normality. On the other hand (Cont 2001), proposed the idea of aggregational normality: as one increases the time scale Δt (decrease the sampling frequency) over which returns are calculated, their distribution looks more and more like a normal distribution.

Figure 7 displays the time series, density, and ACF plots for the returns of BTC. Those plots for ETH are similar and so are omitted. The time series plot for period 2 shows the large outlier on the 12th of March after COVID-19 was declared a pandemic. On that date, the return was about −0.5 as compared to the mean of 0.0038. From the ACF plot for period 2, there is a significant negative autocorrelation at the first lag which is not surprising given the high LB Q₁₀ test statistic in Table 9 and explains why an AR(1) in (13) will be used for the conditional mean of the return model.

Figure 7:

BTC time series, density, and ACF plots for returns.

5.2 Return Models

In this section, the return models in (12) and (13) for the stage two of the CARR-return models are analysed and compared across periods. The volatility input from the stage one CARR model to the state two return model are volatility estimates from the LCARR(1, 2, a ) model using the RPK, 1 m measure as this volatility model has the lowest DIC and better out-of-sample forecast accuracy. Table 10 reports the parameter estimates and DIC of the return model for each period and for BTC and ETH. The parameter μ₀ in (13) is the unconditional return, ϕ₁ is the autoregressive term for the AR(1) process of the return, and ρ in (12) is the scaling parameter to ensure that the volatility estimate ρλ _t where λ _t is given in (10), is unbiased for the true return variance. Unlike the shape parameters a, p, q for volatility distributions, we report ν in Table 10 as it is directly related to kurtosis of the return distributions; hence revealing the effects of outliers. The lowest DIC in each period for each coin and the parameters significant at the 5 % level are highlighted in boldface.

5.2.1 Comparison of Error Distributions

For the error distribution, three distributions namely Normal (N), Student-t (ST), and Variance-Gamma (VG) distributions are considered. While the Normal distribution is a special case of the ST and VG distributions when ν goes to infinity, the ST and VG distributions are not special cases of each other and have different shapes. Relative to ST distribution, VG distribution has lower density for the mid-regions defined as the regions between the peak and the tails on each side (Fung and Seneta 2007). Hence, it is important to compare these two distributions. For computational time, the return models take less time to run likely because of less parameters and complexity in the model. The running times are around 15 s, 30 s, and 1 min for the Normal, ST, and VG distributions respectively.

As shown by the DICs, the ST distribution is the best for BTC in periods 1 and 2 and ETH is periods 2 and 3 while the VG distribution modelled returns the best for the remaining cases. While the VG distribution modelled returns better than ST distribution in some cases, the improvement is only marginal. For example, in period 3 for BTC, the difference in DIC for ST and VG is only 2.26 while the difference in DIC is much larger when the ST distribution performs better. Hence, the ST distribution is better at modelling this return data overall.

The residual density plots in Figure 8 visualises the comparison across distributions. The density plot for period 2 is cropped on the left tail to exclude the extreme negative return on March 12th because it would be hard to see the shape of the density plots if that observation is included. Overall, it is clear that the ST and VG distributions model the error term well. But we can see how the shape of these two distributions are different. The VG distribution is more peaked around 0 so it can be good for modelling returns when there is infrequent trading resulting in more around 0 returns. This is likely for high frequency data which often come with high kurtosis. Between ST and VG distribution, the daily returns for BTC and ETH is usually better modelled by the ST distribution and in the cases where the VG distribution fit the data better, it is only marginally better than the ST distribution. Also, for period 3, it is clear that the Normal distribution models the error term better in comparison to the other two periods supporting the observation that returns are closer to a Normal distribution in period 3 relative to periods 1 and 2.

Figure 8:

Theoretical versus actual density for the return errors across distributions and periods.

Furthermore, Figure 9 graphs the QQ plots for the error terms from the return model across distributions and periods for both coins. The QQ plots highlight how the ST and VG distributions are better at modelling the tails of the return distribution in comparison to the Normal distribution. However, there are still some outliers on the tails which cannot be captured by the distributions, highlighting the difficulty in modelling the extreme ends of the tails. For period 1, the VG error distribution appears to model the error terms near 0 better in comparison to the ST distribution as the observed quantiles are closer to the theoretical quantiles in the middle, showing how the peak of VG distributions is better at capturing the central portion of the data with high kurtosis.

Figure 9:

QQ plots for return error across distributions and periods.

5.3 Out-Of-Sample Forecast Performance

Again, return models are also evaluated in terms of out-of-sample forecast to test the generalisation capacity to unseen data.