Recently, the precision matrix has attracted increasing interest in the financial context, given its various and useful applications, from portfolio optimization, in which the closed-form solution to the global minimum variance portfolio involves the precision matrix (Torri et al. 2019), to systemic risk analysis, in which the partial correlations are more informative than independent correlations when analyzing the financial system as a whole. Many measures based on the precision matrix have been proposed to assess both systemic risk and portfolio strategy performances; see, for instance, Senneret et al. (2016) and Torri et al. (2019).

In this real data application, we show how the estimation of CCs’ precision matrix (together with other interesting financial quantities) changes according to the choice of the estimator, that is, when considering a robust approach against a non-robust one or assuming a time-varying approach instead of a time-invariant one. The existence and extent of these variations must necessarily be considered by those decision makers who rely on financial quantity estimates associated with different time horizons.

Data

For this study, we consider the daily close value, (p_{t}), of 12 CCs, namely, Bitcoin (BTC), Ethereum (ETH), XRP (XRP), Litecoin (LTC), Stellar (XLM), Monero (XMR), Dash (DASH), Ethereum Classic (ETC), NEM (XEM), Zcash (ZEC), Dogecoin (DOGE), and Waves (WAVES), from 01/02/2017 to 26/07/2021, downloaded from CoinMarketCap (https://coinmarketcap.com/it/). The dynamics of CCs’ log returns and histograms (Figs. 3, 4) make it evident that their distribution cannot be considered Gaussian because of the presence of extreme events. Such a feature is confirmed by the sample statistics in Table 1, in which we report robust statistics, namely median, interquartile range, measures of skewness, and excess kurtosis based on quantiles [see Kim and White (2004)], minimum, and maximum. We also performed the Jarque-Bera test for non-normality but did not include the p-values, all of which were zero. Some CCs, such as XRP, XEM, and DOGE, exhibit very high kurtosis, confirming the presence of extreme events.

The dynamics of (vert r_{t}-mu vert) are reported in Fig. 2, where (mu) is the mean of (r_{t}), and show that the variances of the CCs’ log return series cannot be considered constant over time, whereas (investigated but not shown) the auto-correlations and cross-correlations for different lags in the CCs’ log return series can be considered—except for lag 0—zero with good approximation.

Fig. 2
figure 2

Dynamics of (vert r_{t}-mu vert), where (r_{t}) are the 12 CCs’ log return considered and (mu) is the corresponding mean

Table 1 Sample statistics of the 12 CCs’ log return considered

We consider the model discussed in Eq. (1) to describe the evolution of the selected 12 CCs. First, we estimate the precision matrix for several values of the robustness parameter, namely, for (gamma =0,0.01, 0.02, 0.05 text{ and } 0.1), and several bandwidth values, parameter, namely for (h=20, 50, 100 text{ and } infty). We attempt different values of (gamma), and choose those that provide significantly different estimates. Therefore, we do not consider values (gamma > 0.1) because the estimates do not change significantly.

We stress that the estimation windows around the reference time are larger than the bandwidth; however, significant weights are given only to observations inside the bandwidth.

Results

Precision matrices are used to construct Gaussian graphical models, which are statistical models describing relationships among variables in the form of graphs. Therefore, the estimated precision matrices are presented in “Conditional correlation graphs” section in the form of a graph. In particular, we show the five local estimates obtained considering a neighbor of the following dates: June 30, 2017, January 16, 2018, September 8, 2019, May 15, 2020, and April 30, 2021. The first date corresponds to a period in which investors started to observe the CCs market with more interest, the second date is immediately after the huge increase in Bitcoin—that is when the interest toward CCs was very high, and the third date corresponds to a period in which the interest toward this market was still high but not characterized by dramatic events. The last two dates correspond to the outbreak of the pandemic and to the second increase in the value of Bitcoin in April 2021, when its value reached 53,000 dollars. Figures 5, 6, 7, 8 and 9 shows that, in general, the smaller the value of the bandwidth of the kernel, the higher the interconnection between cryptocurrencies. This phenomenon results from the significantly not constant dependencies among the cryptocurrencies that, not being regulated by any institution and having no fundamentals from which to extract a price (Baek and Elbeck 2015; Huang et al. 2019) have wild variations that depend only on the whims of the market. With a large bandwidth, the time-varying dependency is mediated, resulting in a smaller interconnection. This evidence is the first of the importance of using a time-varying approach. Moreover, especially in Figs. 6 and 9, it is possible to observe high interconnection between cryptocurrency when the (gamma) parameter is high. These two figures are both associated with periods in which CCs are characterized by high growth and high volatility, and in the underlying process, CCs turn out to be conditionally dependent; however, the dependencies are hidden by extreme noisy events. Instead, Figs. 7 and 8 show how extreme events can distort the amplitude of the dependence between cryptocurrencies.

What we have just said can be further detailed by observing Fig. 10 in “Conditional correlations dynamics” section, where we reported some conditional correlation dynamics, that is, the evolution of some elements of the precision matrix. Indeed, it turns out that, for a small value of h (first column), conditional correlations show a significant differences at varying (gamma), meaning that extreme events hide the underlying conditional correlation dynamics. Worth noting is that discrepancies among different estimates can be significant—more than one hundred percent. For high values of h (second column), the large bandwidth averaged the genuine time-dependent dynamics of the process, similar to what was observed from the synthetic stochastic model in “Time-varying estimation” section; however, differences in the estimation of the conditional correlation at varying (gamma) are still evident. Although the analysis is carried out on the 12 CCs, here we report only three examples of conditional correlations between CC pairs as an example of the type of analysis and the results that can be achieved using the proposed methodology.

In Fig. 11 in “Quantitative measures” section, we consider a few indicators to quantify the discrepancies among the estimated precision matrices. The first indicator is the normalized Frobenius norm of the differences between the estimated precision matrix obtained for (gamma =0.1), for which we expect more accurate estimates and the one obtained for (gamma =0,0.01, 0.02, 0.05), that is

$$begin{aligned} varDelta Fleft( t,gamma right) = frac{sqrt{sum _{i,j}{left( varvec{hat{varTheta }}left( t,0.1right) -varvec{hat{varTheta }}left( t,gamma right) right) ^{2}_{i,j}}}}{sqrt{sum _{i,j}{left( varvec{hat{varTheta }}left( t,0.1right) right) ^{2}_{i,j}}}}. end{aligned}$$

The computation of (varDelta Fleft( t,gamma right)) is reported in the first row of Fig. 11, which shows that, even if this measure is an average quantity, the relative differences associated with the different values of (gamma) are remarkable for both small and large bandwidth values, touching one hundred percent for the smallest bandwidth.

Of course, the discrepancies among different estimates affect the computation of financial quantities. To measure such an impact, we also consider the following generally used quantities [see Billio et al. (2012) for more details]:

  • the sum of the elements of the precision matrix, that is

    $$begin{aligned} sleft( t, gamma right) =sum _{i,j}left| {varvec{hat{varTheta }}left( t, gamma right) _{i,j}}right| , end{aligned}$$

    gives a measure of interconnectedness of the entire system;

  • let (lambda _{k}left( t,gamma right)) for (k=1,dots ,d) be the eigenvalues of the correlation matrix (hat{varvec{varSigma }}left( t,gamma right)) and (nu in left{ 1,dots , dright}), then

    $$begin{aligned} wleft( t, gamma right) =frac{sum _{k=1}^{nu }{lambda _{k}left( t,gamma right) }}{sum _{k=1}^{d}{lambda _{k}left( t,gamma right) }} end{aligned}$$

    is a measure of interconnectedness. Indeed, the numerator is the risk associated with the first (nu) principal components, whereas the denominator is the total risk of the system.

As previously done, we considered the relative differences of these measures using as reference the estimated precision matrix obtained for (gamma =0.1), for which we expect more accurate estimates, that is (varDelta Sleft( t,gamma right) = frac{sleft( t, 0.1right) -sleft( t, gamma right) }{sleft( t, 0.1right) }) and (varDelta Wleft( t, gamma right) = frac{wleft( t, 0.1right) – wleft( t,gamma right) }{wleft( t, 0.1right) }) for (gamma =0,0.01, 0.02, 0.05), to better highlight the effect of different estimations on the financial quantities of interest. We report the results in the second and third rows of Fig. 11, which show that both measures present significant differences by varying parameter (gamma). Such differences can also be observed when varying the bandwidth parameter, even if, as previously noted, at a small bandwidth ((h=20)), the differences are greater than those for the large bandwidth ((h=100)) because of the usual average effect that emerges in the last case.

Considering the three quantities reported in Fig. 11, the absolute value gradually decreases from (varDelta F) (first row), (varDelta S) (second row), and (varDelta W) (third row). This result is expected. In fact, (varDelta F), measuring the sum of the differences between the elements of the precision matrices, is an accurate measure of the discrepancies between the estimates obtained at varying (gamma), whereas (varDelta S), measuring the differences between the overall connections in the CC market (obtained at varying (gamma)) can be considered a difference between composite variables that usually fluctuate less than the individual variables of which it is composed. Lastly, (varDelta W), measuring the relative differences in the percentages of variance explained by the first (nu) eigenvalues of the correlation matrix, is an extensive quantity least affected by different estimates obtained at varying (gamma).

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