# Nonlinear Autoregressive Distributed Lag Approach: An Application on the Connectedness between Bitcoin Returns and the Other Ten Most Relevant Cryptocurrency Returns

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Data

#### 3.2. Methodology

_{jt}= α

_{0}+ α

^{+}·BR

_{t}

^{+}+ α

^{−}·BR

_{t}

^{−}+ ɛ

_{jt}

_{jt}and BR

_{t}are scalar I(1) variables. In detail, R

_{jt}is the returns from the j-th alternative cryptocurrency returns corresponding to period t, for j = 1,…9, BR

_{t}is the Bitcoin returns for period t which is decomposed as BR

_{t}= BR

_{0}+ BR

_{t}

^{+}+ BR

_{t}

^{−}, where BR

_{t}

^{+}and BR

_{t}

^{−}are partial sums of positive (appreciations) and negative (depreciations) changes in Bitcoin returns, ε

_{jt}and v

_{t}are random disturbances and α = (α

_{0}, α

^{+}, α

^{−}) is a vector of long-run parameters to be estimated.

^{+}and α

^{−}, in Equation (1), capture the long-run relationship between each of the top alternative cryptocurrency returns and increases (α

^{+}) or decreases (α

^{−}), respectively, in the Bitcoin returns. Finally, we study whether the long-run relationship reflects asymmetric long-run Bitcoin returns passthrough to each of the alternative cryptocurrency returns.

_{jt}and BR

_{t}is modelled as piecewise asymmetric linear function subject to the decomposition of BR

_{t}because if we suppose that |α

^{+}|<|α

^{−}| in Equation (1), the long-run effect of a unit negative change in BR

_{t}will increase BR

_{t}by a greater amount than a unit positive change would reduce it. Therefore, reference [40] confirms that the NARDL model includes a regime-switching cointegrating relationship in which regime transitions are governed by the sign of ∆BR

_{t}.

_{t}is a k × 1 vector of multiple regressors defined such that BR

_{t}= BR

_{0}+ BR

_{t}

^{+}+ BR

_{t}

^{−}, ϕ

_{i}is the autoregressive parameter, p is the number of lagged dependent variables and q is the number of lags for regressors, γ

_{i}

^{+}and γ

_{i}

^{−}are the asymmetric distributed lag parameters, and, finally, ε

_{jt}is an iid process with zero mean and constant variance σ

_{ε}

^{2}.

^{+}= −β

_{2}/β

_{1}, α

^{−}= −β

_{3}/β

_{1}, are the coefficients of long-run impacts of Bitcoin return increases and decreases respectively on each of the nine alternative cryptocurrency returns. On the other hand, $\sum}_{i=0}^{q}{\gamma}_{i}^{+$ and $\sum}_{i=0}^{q}{\gamma}_{i}^{-$ measure the short-run influences of increases and decreases respectively of Bitcoin returns on each of the top nine alternative cryptocurrency returns. Thus, not only are the asymmetric long-run relationship considered, but the asymmetric short-run influences of Bitcoin returns changes on the top ten cryptocurrency returns are also captured in order to identify differences in the response of economic agents to positive and negative shocks.

_{0}: PCorr = 0); second, the presence of cointegration by the Wald F test for the joint null hypothesis that coefficients on the level variables are jointly equal to zero (H

_{0}: β

_{1}= β

_{2}= β

_{3}= 0); third, the cointegration equation (long-run elasticities) between variables; fourth, the long-run symmetry by means of the Wald test, with symmetry implying H

_{0}: −β

_{2}/β

_{1}= −β

_{3}/β

_{1}; fifth, the short-run symmetry in the short-run model by the Wald test for the null of short-run symmetry defined by γ

_{i}

^{+}= γ

_{i}

^{−}and sixth, the effect of the cumulative sum of positive and negative changes (respectively) in Bitcoin returns for 1 to 4 lags on the rest of cryptocurrencies’ returns.

## 4. Results and Discussion

#### 4.1. Results of the NARDL Models: Daily Frequency

^{2}of each cryptocurrency.

_{0}: PCorr = 0) is rejected by all the top ten cryptocurrencies. More specifically, a high positive correlation is observed between Bitcoin returns and all the rest of the top ten cryptocurrency returns. All of them exhibit statistical significance at the 1% level, showing Pearson’s correlation coefficients between 43.3% and 82.2%, except for Tether that shows statistical significance at the 5% level and the lowest Pearson’s correlation coefficient of 10.7%.

_{0}: β

_{1}= β

_{2}= β

_{3}= 0) is rejected by five cryptocurrencies (XRP, Bitcoin_cash, Tether, EOS, and Binance coin). Thus, the F statistics show long-run relationships, i.e., cointegration, between changes in Bitcoin returns and XRP, Bitcoin_cash, Tether, EOS and Binance_coin returns for daily frequency. Additionally, the long-run coefficients of changes in Bitcoin returns are positive and statistically significant at 1% level for these five cryptocurrencies, where the highest values are for XRP and Theter.

_{jt−i}= e

^{+}·BR

^{+}

_{t−i}+ e

^{−}·BR

^{−}

_{t−i}(long-run elasticities) between Bitcoin returns (BR) and the rest of the top ten cryptocurrencies’ returns (R

_{jt−i}). Thus, regarding the long-run elasticities for the cumulative sum of positive changes in Bitcoin returns) BR

^{+}

_{t−i}and the cumulative sum of negative changes in Bitcoin returns BR

^{−}

_{t-i}, all cryptocurrency returns respond in the same way to positive and negative changes in Bitcoin returns. Additionally, the coefficients are quite similar and are of modest size for all cryptocurrencies. The largest coefficients correspond to Bitcoin_sv returns that respond more to positive and negative changes in Bitcoin returns (4.5% versus 5.7%, respectively). Moreover, the long-run elasticities for the cumulative sum of positive and negative changes in Bitcoin returns are statistically significant just for four cryptocurrencies, EOS, XRP, Tether and Binance_coin. Moreover, the coefficients are negative for XRP and EOS, meaning they move in the opposite direction to the changes in Bitcoin returns, but are positive for Tether and Binance_coin, meaning they fluctuate in line with Bitcoin returns.

_{0}: −β

_{2}/β

_{1}= −β

_{3}

**/**β

**), is rejected only by two cryptocurrencies: XRP and Binance_coin. Thus, the Wald test indicates that there could be asymmetry in the long-run impact of Bitcoin returns on XRP and Binance_coin returns for daily data, corroborating previous results obtained with long-run elasticities.**

_{1}_{0}: γ

_{i}

^{+}= γ

_{i}

^{−}), is rejected by all the cryptocurrencies as all cryptocurrencies show positive and statistically significant coefficients at the 1% significance level. Therefore, there is strong evidence of asymmetric short-run responses of all cryptocurrency returns to changes in Bitcoin returns for daily frequency. Thus, nonlinear asymmetries are important in the short-run relationship between Bitcoin returns and the remaining top ten cryptocurrencies’ returns for daily data.

#### 4.2. Results of the NARDL Models: Weekly Frequency

^{2}to be a bit higher for weekly than for daily frequencies.

#### 4.3. Results of the NARDL Models: Monthly Frequency

^{2}of 26.8% for the Tether returns to a maximum of 77.3% for EOS returns. It is noticeable that the two most recently issued cryptocurrencies with the smallest sample size have the highest adjusted R

^{2}; 96.6% for Bitcoin_sv and 80.1% for Tezos. In any event, there is a clear tendency for the explanatory power of the NARDL model to rise as the sampling frequency decreases. For example, for EOS the explanatory power steadily increases as we move from daily, weekly, and monthly frequency, achieving R

^{2}of 40.4%, 50% and 77.3% respectively.

^{2}of all, showing that there is no correlation between bitcoin returns and the returns of these more recent cryptocurrencies.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of the Bitcoin and the rest of relevant cryptocurrencies daily prices (Bitcoin prices in the right-axis and the rest of cryptocurrencies prices in the left-axis).

**Figure 2.**Time evolution of the Bitcoin returns and the rest of relevant cryptocurrency returns. Compiled by the authors, based on the information provided by the Coinmarketcap website.

**Table 1.**Top 10 Cryptocurrencies by Market Capitalization (Date: 7 March 2020/22 March 2020) (Total market capitalization: $251.5 billion/$167.1 billion).

^{1}

Name | Market Cap | Price | Volume (24 h) | Circulating Supply | Change (24 h) | Starting Date |
---|---|---|---|---|---|---|

Bitcoin | $166,743,993,933 $106,591,196,069 | $8887.80 $5830.25 | $47,868,579,352 $40,099,664,740 | 18,238,800 BTC 18,282,425 BTC | −0.21% −5.73% | 01/26/2015 |

Ethereum | $26,966,016,878 $13,590,860,527 | $237.32 $123.32 | $25,206,666,119 $12,497,707,224 | 109,863,231 ETH 110,207,055 ETH | 2.07% −7.05% | 03/10/2016 |

XRP | $10,688,702,708 $6,585,765,149 | $0.23624 $0.150214 | $3,252,412,868 $1,864,979,798 | 43,749,413,421 XRP 43,842,625,397 XRP | −0.88% −5.02% | 01/26/2015 |

Bitcoin_Cash | $6,364,459,307 $3,736,418,941 (5) | $330.77 $203.67 | $6,617,099,625 $4,015,953,536 | 18,300,000 BCH 18,345,250 BCH | −0.25% −7.47% | 08/03/2017 |

Tether | $4,641,437,047 $4,637,871,717 (4) | $1.0047 $0.99903 | $66,519,050,406 $49,036,623,749 | 4,642,367,414 USDT 4,642,367,414 USDT | 0.16% −0.21% | 04/15/2017 |

Bitcoin_SV | $4,439,960,724 $2,894,145,363 | $233.95 $157.78 | $3,344,789,290 $3,365,019,330 | 18,297,290 BSV 18,342,440 BSV | −1.66% −6.35% | 11/19/2018 |

Litecoin | $4,072,866,599 $2,292,391,578 | $60.45 $35.63 | $6,342,837,357 $3,148,219,029 | 64,168,987 LTC 64,342,318 LTC | −0.77% −7.34% | 08/24/2016 |

EOS | $3,526,893,934 $1,965,191,547 | $3.64 $2.13 | $6,064,573,978 $2,921,411,201 | 920,452,308 EOS 921,045,767 EOS | −0.47% −6.45% | 07/02/2017 |

Binance Coin | $3.292,877,236 $1,735,514,181 | $20.24 $11.16 | $427,799,971 $308,670,064 | 155,536,713 BNB 155,536,713 BNB | −1.68% −7.48% | 11/09/2017 |

Tezos | $2,250,710,445 $1,038,511,561 | $2.98 $1.47 | $317,321,520 $113,589,399 | 702,028,555 XTZ 704,565,511 XTZ | −0.04% −11.11% | 02/02/2018 |

^{1}Compiled by the authors, based on the information provided by the Coinmarketcap website.

**Table 2.**Descriptive statistics of Bitcoin returns and returns of the rest of the top ten cryptocurrency returns.

^{1}

Panel A: Daily Frequency. | |||||||||||

Name | Mean | Median | Max. | Min. | Std. Dev. | Skewness | Kurtosis | JB Stat. | ADF Stat. | PP Stat. | KPSS Stat. |

Bitcoin returns | 0.0019 | 0.0019 | 0.2276 | −0.1869 | 0.0376 | −0.1471 | 7.3114 | 1453 *** | −43.873 *** | −43.881 *** | 0.1581 |

Ethereum returns | 0.0021 | −0.0001 | 0.2586 | −0.3134 | 0.0574 | −0.0418 | 6.4015 | 703.3 *** | −38.679 *** | −38.816 *** | 0.3182 |

XRP returns | 0.0015 | −0.0013 | 1.0280 | −0.9965 | 0.0994 | 0.8984 | 30.2463 | 58000 *** | −32.003 *** | −59.811 *** | 0.1527 |

Bitcoin_cash returns | 0.0001 | −0.0038 | 0.4355 | −0.4792 | 0.0780 | 0.6110 | 10.6729 | 2382 *** | −28.553 *** | −28.566 *** | 0.1053 |

Theter returns | 0.0000 | 0.0000 | 0.0453 | −0.0575 | 0.0063 | 0.0252 | 19.1176 | 11441 *** | −22.254 *** | −47.324 *** | 0.0110 |

Bitcoin_sv returns | 0.0026 | −0.0014 | 0.8979 | −0.3259 | 0.0860 | 3.6652 | 34.7653 | 20990 *** | −23.548 *** | −23.516 *** | 0.0578 |

Litecoin returns | 0.0022 | −0.0024 | 0.6070 | −0.3080 | 0.0619 | 1.7426 | 16.6638 | 10696 *** | −36.409 *** | −36.453 *** | 0.3425 |

EOS returns | 0.0003 | −0.0015 | 0.3559 | −0.3567 | 0.0757 | 0.4055 | 7.6595 | 912.4 *** | −32.951 *** | −32.980 *** | 0.0918 |

Binance_coin returns | 0.0028 | 0.0007 | 0.4874 | −0.4023 | 0.0626 | 0.9070 | 13.6192 | 4105.6 *** | −27.227 *** | −27.191 *** | 0.2255 |

Tezos returns | 0.0000 | −0.0042 | 0.2525 | −0.4094 | 0.0667 | −0.1728 | 6.4442 | 381.4 *** | −26.555 *** | −26.563 *** | 0.3154 |

Panel B: Weekly Frequency. | |||||||||||

Name | Mean | Median | Max. | Min. | Std. Dev. | Skewness | Kurtosis | JB Stat. | ADF Stat. | PP Stat. | KPSS Stat. |

Bitcoin returns | 0.0136 | 0.0093 | 0.3446 | −0.3686 | 0.1007 | −0.0770 | 4.9667 | 43.128 *** | −15.549 *** | −15.547 *** | 0.1537 |

Ethereum returns | 0.0138 | 0.0083 | 0.7457 | −0.3951 | 0.1592 | 0.9938 | 6.4246 | 135.227 *** | −12.899 *** | −13.087 *** | 0.2326 |

XRP returns | 0.0103 | −0.0124 | 1.2546 | −0.9822 | 0.2240 | 1.7314 | 12.631 | 1161.02 *** | −16.056 *** | −16.074 *** | 0.1336 |

Bitcoin_cash returns | 0.0005 | −0.0087 | 0.8526 | −0.7188 | 0.2199 | 0.7793 | 6.1413 | 68.656 *** | −10.451 *** | −10.422 *** | 0.1020 |

Theter returns | 0.0004 | 0.0001 | 0.0439 | −0.0444 | 0.0105 | −0.4256 | 8.2501 | 176.799 *** | −8.8943 *** | −14.437 *** | 0.1301 |

Bitcoin_sv returns | 0.0216 | −0.0036 | 0.9894 | −0.4649 | 0.2205 | 1.6966 | 8.6941 | 122.655 *** | −7.6877 *** | −7.6881 *** | 0.0484 |

Litecoin returns | 0.0150 | −0.0033 | 1.1406 | −0.3031 | 0.1828 | 2.6024 | 16.126 | 1528.52 *** | −13.285 *** | −13.310 *** | 0.2772 |

EOS returns | 0.0017 | −0.0064 | 0.7216 | −0.4452 | 0.1966 | 0.5641 | 3.8327 | 11.387 *** | −9.8301 *** | −9.8971 *** | 0.0679 |

Binance coin returns | 0.0213 | 0.0102 | 0.6706 | −0.3331 | 0.1645 | 1.3036 | 6.8411 | 107.756 *** | −10.142 *** | −10.433 *** | 0.2077 |

Tezos returns | 0.0016 | 0.0051 | 0.4392 | −0.6843 | 0.1690 | −0.4786 | 5.1781 | 25.471 *** | −8.8875 *** | −8.9152 *** | 0.2496 |

Panel C: Monthly Frequency. | |||||||||||

Name | Mean | Median | Max. | Min. | Std. Dev. | Skewness | Kurtosis | JB Stat. | ADF Stat. | PP Stat. | KPSS Stat. |

Bitcoin returns | 0.0625 | 0.0437 | 0.8826 | −0.5717 | 0.2452 | 0.7046 | 4.8384 | 13.414 *** | −7.6711 *** | −7.6713 *** | 0.1204 |

Ethereum returns | 0.0640 | 0.0000 | 1.2973 | −0.7859 | 0.4150 | 0.5850 | 3.63045 | 3.4593 | −6.2936 *** | −6.3522 *** | 0.1704 |

XRP returns | 0.0541 | −0.0258 | 2.0518 | −0.5347 | 0.4546 | 2.5123 | 10.569 | 206.345 *** | −6.2751 *** | −5.1123 *** | 0.1214 |

Bitcoin_cash returns | 0.0130 | −0.0169 | 1.3271 | −1.5992 | 0.5085 | −0.3969 | 5.6314 | 9.4425 *** | −5.3384 *** | −5.3394 *** | 0.1039 |

Theter returns | 0.0003 | 0.0003 | 0.0302 | −0.0441 | 0.0124 | −0.8521 | 6.8862 | 25.510 *** | −5.9941 *** | −14.375 *** | 0.5000 ** |

Bitcoin_sv returns | 0.1087 | 0.0293 | 1.1937 | −0.4832 | 0.4831 | 1.2566 | 3.6701 | 3.9463 | −4.7496 *** | −4.7496 *** | 0.1259 |

Litecoin returns | 0.0732 | 0.0373 | 1.5685 | −0.6346 | 0.3906 | 1.5518 | 7.1185 | 45.431 *** | −5.2324 *** | −5.2614 *** | 0.2251 |

EOS returns | 0.0417 | 0.1166 | 1.5578 | −0.9160 | 0.5107 | 0.6028 | 4.3839 | 4.3512 | −3.9072 *** | −4.5276 *** | 0.3110 |

Binance_coin returns | 0.1019 | 0.0534 | 1.5514 | −0.6107 | 0.4498 | 1.2385 | 5.5057 | 13.966 *** | −4.3508 *** | −4.7401 *** | 0.1590 |

Tezos returns | 0.0073 | −0.0174 | 0.8747 | −1.0750 | 0.4401 | −0.1028 | 3.4300 | 0.2271 | −3.6817 ** | −3.6335 ** | 0.2478 |

^{1}This table presents the descriptive statistics of daily (Panel A), weekly (Panel B) and monthly (Panel C) Bitcoin returns and returns of the rest of relevant cryptocurrencies over the period from January 2015 to March 2020. They include mean, median, minimum (Min.) and maximum (Max.) values, standard deviation (Std. Dev.) and Skewness and Kurtosis measures. JB denotes the statistic of the Jarque–Bera test for normality. The results of the Augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) unit root tests and the Kwiatkowski et al. (KPSS) stationarity test are also reported in the last three columns. As usual, *, **, *** indicate statistical significance at the 10%, 5% and 1% levels, respectively.

**Table 3.**Regression results of nonlinear ARDL models: asymmetry and cointegration tests between Bitcoin returns and the rest of relevant cryptocurrencies’ returns: daily frequency.

^{1}

Cryptocurrencies | PCorr | Coint | Eq | LAsym | SAsym | Lags^{+} | Lags^{−} | Adj. R^{2} |
---|---|---|---|---|---|---|---|---|

Ethereum returns | 0.8242 *** | 0.6334 | e^{+}: 0.0370e ^{−}: 0.0500 | 0.3384 | 17.776 *** | (2): 0.0935 * (4): 0.1477 *** | (3): −0.1319 ** | 0.3254 |

XRP returns | 0.7266 *** | 60.617 *** | e^{+}: −0.0226 **e ^{−}: −0.0272 ** | 3.3268 * | 8.1825 *** | (1): 0.2196 ** (3): 0.1807 ** | - | 0.1619 |

Bitcoin_cash returns | 0.6778 *** | 15.534 *** | e^{+}: 0.0203e ^{−}: 0.0230 | 0.8904 | 13.737 *** | (1): −0.1787 ^{**} | (1): −0.3240 *** | 0.3091 |

Theter returns | 0.1069 ** | 54.861 *** | e^{+}: 0.0019 **e ^{−}: 0.0020 * | 0.2310 | - | - | (1): −0.0124 * (2): −0.0224 *** | 0.1449 |

Bitcoin_sv returns | 0.4328 *** | 0.3960 | e^{+}: 0.4491e ^{−}: 0.5710 | 0.2313 | 6.7191 **^{*} | (2): 0.3620 ** | - | 0.1824 |

Litecoin returns | 0.7694 *** | 0.6729 | e^{+}: −0.0390e ^{−}: −0.0550 | 0.4228 | 18.475 *** | (1): 0.1033 * (2): 0.1408 ** | - | 0.3601 |

EOS returns | 0.7609 *** | 5.7063 *** | e^{+}: −0.4973 ***e ^{−}: −0.5148 *** | 0.9959 | 18.881 *** | - | (4): −0.2319 *** | 0.4045 |

Binance_coin returns | 0.6222 *** | 10.605 *** | e^{+}: 0.0561 *e ^{−}: 0.0668 ** | 3.9280 ** | 17.722 *** | - | (1): −0.3004 *** | 0.4023 |

Tezos returns | 0.5006 *** | 1.0487 | e^{+}: 0.1403e ^{−}: 0.1275 | 0.3006 | 10.531 *** | - | - | 0.1936 |

^{1}This table reports the coefficient estimates of the NARDL model between Bitcoin returns and the rest of relevant cryptocurrencies’ returns. PCorr refers to the Pearson’s correlation coefficients defined by the null of PCorr = 0. Coint refers to the Wald test for the presence of cointegration defined by β

_{1}= β

_{2}= β

_{3}= 0. Eq shows the cointegration equation (long-run elasticities) between Bitcoin returns (BR) and the rest of relevant cryptocurrencies’ returns R

_{jt}

_{-i}= e

^{+}·BR

^{+}

_{t}

_{-i}+ e

^{−}·BR

^{−}

_{t}

_{-i}. LAsym refers to the Wald test for the null of long-run symmetry defined by −β

_{2}/β

_{1}= −β

_{3}/β

_{1}. SAsym refers to the Wald test for the null of short-run symmetry defined by γ

_{i}

^{+}= γ

_{i}

^{−}. Lags

^{+}and Lags

^{−}show the effect of the cumulative sum of positive and negative changes (respectively) in Bitcoin returns for ()-lags on the rest of relevant cryptocurrency returns. As usual, *, **, *** indicate statistical significance at the 10%, 5% and 1% levels, respectively. The critical values are available in [43], in case of small sample size.

**Table 4.**Regression results of nonlinear ARDL models: asymmetry and cointegration tests between Bitcoin returns and the rest of relevant cryptocurrencies’ returns: weekly frequency.

^{1}

Cryptocurrencies | PCorr | Coint | Eq | LAsym | SAsym | Lags^{+} | Lags^{−} | Adj. R^{2} |
---|---|---|---|---|---|---|---|---|

Ethereum returns | 0.8123 *** | 2.3692 * | e^{+}: 0.0529e ^{−}: 0.0821 | 0.3332 | 6.9406 *** | - | - | 0.3861 |

XRP returns | 0.7392 *** | 0.8958 | e^{+}: −1.1248 *e ^{−}: −1.7386 * | 0.2152 | 3.5334 *** | - | - | 0.0666 |

Bitcoin_cash returns | 0.7315 *** | 0.5972 | e^{+}: −0.9784e ^{−}: −1.0266 | 0.0613 | 6.8692 *** | (2): 0.3845 ** (4): 0.3768 * | (1): 0.5360 ** (3): 0.7716 *** | 0.5155 |

Theter returns | −0.4073 *** | 2.8918 ** | e^{+}: 0.0388 ***e ^{−}: 0.0429 *** | 0.6522 | - | (1): 0.0440 *** (3): 0.0196 * | - | 0.1409 |

Bitcoin_sv returns | 0.4208 *** | 1.0911 | e^{+}: −0.7533e ^{−}: −1.4758 | 0.6861 | 2.6063 *** | (1): 0.8402 ** | (1): −1.0168 ** | 0.2719 |

Litecoin returns | 0.6745 *** | 0.2642 | e^{+}: 0.0899e ^{−}: −0.0127 | 0.1199 | 5.3563 *** | - | - | 0.3196 |

EOS returns | 0.6991 *** | 3.1813 ** | e^{+}: 0.6927 **e ^{−}: 0.8068 ** | 0.7554 | 7.7183 *** | (3): −0.5188 *** | (1): −0.4054 *** | 0.5000 |

Binance_coin returns | 0.5308 *** | 1.9915 * | e^{+}: 0.1923e ^{−}: 1.1908 | 0.0867 | 6.2489 *** | (2): 0.4735 *** | - | 0.3054 |

Tezos returns | 0.5138 *** | 0.9228 | e^{+}: 0.5929e ^{−}: 0.4970 | 0.2075 | 6.2904 *** | - | - | 0.2798 |

^{1}This table reports the coefficient estimates of the NARDL model between Bitcoin returns and the rest of relevant cryptocurrencies’ returns. PCorr refers to the Pearson’s correlation coefficients defined by the null of PCorr = 0. Coint refers to the Wald test for the presence of cointegration defined by β

_{1}= β

_{2}= β

_{3}= 0. Eq shows the cointegration equation (long-run elasticities) between Bitcoin returns (BR) and the rest of relevant cryptocurrencies’ returns R

_{j}

_{−i}= e

^{+}·BR

^{+}

_{t}

_{−i}+ e

^{−}·BR

^{−}

_{t}

_{−i}. LAsym refers to the Wald test for the null of long-run symmetry defined by −β

_{2}/β

_{1}= −β

_{3}/β

_{1}. SAsym refers to the Wald test for the null of short-run symmetry defined by γ

_{i}

^{+}= γ

_{i}

^{−}. Lags

^{+}and Lags

^{−}show the effect of the cumulative sum of positive and negative changes (respectively) in Bitcoin returns for ()-lags on the rest of relevant cryptocurrency returns. As usual, *, **, *** indicate statistical significance at the 10%, 5% and 1% levels, respectively. The critical values are available in [43], in case of small sample size.

**Table 5.**Regression results of nonlinear ARDL models: asymmetry and cointegration tests between Bitcoin returns and the rest of relevant cryptocurrencies’ returns: monthly frequency.

^{1}

Cryptocurrencies | PCorr | Coint | Eq | LAsym | SAsym | Lags^{+} | Lags^{−} | Adj. R^{2} |
---|---|---|---|---|---|---|---|---|

Ethereum returns | 0.6352 *** | 0.1902 | e^{+}: −0.8061e ^{−}: −1.0821 | 0.0205 | 3.9753 *** | - | - | 0.4302 |

XRP returns | 0.4454 * | 4.4249 *** | e^{+}: 0.1575e ^{−}: 0.4109 | 0.9089 | 2.7308 *** | - | - | 0.2721 |

Bitcoin_cash returns | 0.5927 ** | 0.4673 | e^{+}: 0.7670e ^{−}: 0.4763 | 0.1481 | 4.8457 *** | (1): 1.1441 *** | - | 0.5652 |

Theter returns | −0.1473 | 3.8636 ** | e^{+}: 0.0203 **e ^{−}: 0.0289 ** | 1.8779 | −2.5775 ** | (1): 0.0210 ** | (1): −0.0292 * | 0.2680 |

Bitcoin_sv returns | 0.2854 | 34.743 *** | e^{+}: 0.7260e ^{−}: 6.0939 * | 46.084 *** | −3.2676 *** | (1): 2.8139 * (2): 2.4948 * | - | 0.9657 |

Litecoin returns | 0.4924 * | 2.7840 ** | e^{+}: 3.0736 ***e ^{−}: 4.2521 ** | 0.1822 | 3.4526 *** | (1): 0.7763 ** (4): 0.8604 *** | (3): −0.6674 * | 0.4907 |

EOS returns | 0.4932 * | 2.7137 * | e^{+}: 1.4434e ^{−}: 2.6779 ** | 0.3991 | 3.2146 *** | (1): 0.8562 ^{***} | (1): −1.0826 ** (3): −0.7961 *** | 0.7731 |

Binance_coin returns | 0.5057 * | 1.8156 | e^{+}: 0.2134e ^{−}: 0.2746 | 0.0705 | 2.4323 *** | (1): 1.3610 *** (4): 0.3091 * | (2): −0.6275 ** (3): −1.1079 *** (4): −0.6770 ** | 0.7481 |

Tezos returns | 0.2630 | 14.1765 *** | e^{+}: 1.5210 ***e ^{−}: 3.2410 *** | 20.439 *** | 3.0335 *** | (2): −2.1163 *** | (1): 3.5387 *** (2): 2.1296 *** (3): 1.9299 *** | 0.8079 |

^{1}This table reports the coefficient estimates of the NARDL model between Bitcoin returns and the rest of relevant cryptocurrencies’ returns. PCorr refers to the Pearson’s correlation coefficients defined by the null of PCorr = 0. Coint refers to the Wald test for the presence of cointegration defined by β

_{1}= β

_{2}= β

_{3}= 0. Eq shows the cointegration equation (long-run elasticities) between Bitcoin returns (BR) and the rest of relevant cryptocurrencies’ returns R

_{jt}

_{−i}= e

^{+}·BR

^{+}

_{t}

_{−i}+ e

^{−}·BR

^{−}

_{t}

_{−i}. LAsym refers to the Wald test for the null of long-run symmetry defined by −β

_{2}/β

_{1}= −β

_{3}/β

_{1}. SAsym refers to the Wald test for the null of short-run symmetry defined by γ

_{i}

^{+}= γ

_{i}

^{−}. Lags

^{+}and Lags

^{−}show the effect of the cumulative sum of positive and negative changes (respectively) in Bitcoin returns for ()-lags on the rest of relevant cryptocurrency returns. As usual, *, **, *** indicate statistical significance at the 10%, 5% and 1% levels, respectively. The critical values are available in [43], in case of small sample size.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

González, M.d.l.O.; Jareño, F.; Skinner, F.S.
Nonlinear Autoregressive Distributed Lag Approach: An Application on the Connectedness between Bitcoin Returns and the Other Ten Most Relevant Cryptocurrency Returns. *Mathematics* **2020**, *8*, 810.
https://doi.org/10.3390/math8050810

**AMA Style**

González MdlO, Jareño F, Skinner FS.
Nonlinear Autoregressive Distributed Lag Approach: An Application on the Connectedness between Bitcoin Returns and the Other Ten Most Relevant Cryptocurrency Returns. *Mathematics*. 2020; 8(5):810.
https://doi.org/10.3390/math8050810

**Chicago/Turabian Style**

González, María de la O, Francisco Jareño, and Frank S. Skinner.
2020. "Nonlinear Autoregressive Distributed Lag Approach: An Application on the Connectedness between Bitcoin Returns and the Other Ten Most Relevant Cryptocurrency Returns" *Mathematics* 8, no. 5: 810.
https://doi.org/10.3390/math8050810