# Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Data and Methods

#### 3.1. Data

#### 3.2. Volatility Rescaled Returns

#### 3.3. Methods

#### 3.3.1. Analysis on the Normality of Returns

#### Kolmogorov–Smirnov Normality Test

#### Dvoretzky–Kiefer–Wolfowitz Bounds

#### 3.3.2. Comparison with Other Distributions

- Generalized hyperbolic (GH) distribution,$$f(x;\lambda ,\alpha ,\beta ,\delta ,\mu )=\frac{{\gamma}^{\lambda}}{\sqrt{2\pi}\phantom{\rule{0.166667em}{0ex}}{\delta}^{\lambda +1}\gamma {K}_{\lambda}}\xb7\frac{{\alpha}^{\lambda +1/2}\sqrt{{\delta}^{2}+{(x-\mu )}^{2}}\phantom{\rule{0.166667em}{0ex}}{K}_{\lambda -1/2}}{{({\delta}^{2}+{(x-\mu )}^{2})}^{1/4-\lambda /2}}\phantom{\rule{2.em}{0ex}}(x\in \mathbb{R}),$$
- Generalized Pareto (GP) distribution,$$\frac{{(1+\xi \left(\frac{x-\mu}{\sigma}\right))}^{-(1/\xi +1)}}{\sigma}\phantom{\rule{2.em}{0ex}}(x>\mu ),$$
- Exponential distribution, obtained by the GP distribution (6) when $\xi =\mu =0$.

#### 3.3.3. Analysis on the Autocorrelation

#### Ljung-Box Q-Test

#### ARCH Test

#### 3.3.4. Analysis on the Stationarity

#### KPSS Test

#### Hassani Test

## 4. Empirical Results

#### 4.1. Statistical Properties and Analysis on Original Log Returns

#### USD Basis Swap 1Mv3M (USBAAC) Index: Original Data

#### 4.2. Statistical Properties and Analysis on Averaged Log Returns

#### USD Basis Swap 1Mv3M (USBAAC) Index: Averaged Log Returns

#### 4.3. Statistical Properties and Analysis on Volatility Rescaled Log Returns

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Analysis on S&P 500 Index

**Figure A4.**Empirical CDF versus standard normal CDF for S&P 500 returns. The dotted black lines represent the DKW upper and lower bounds.

**Table A1.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 1.6502 × 10${}^{-19}$ | 74.92% | 0 | 0.0482 | 0 | 0.7256 | 1 | 0 | 1 | 0 |

Monthly | 1 | 8.3615 × 10${}^{-9}$ | 58.32% | 0 | 0.1985 | 0 | 0.9714 | 1 | 0 | 1 | 0 |

Yearly | 1 | 2.7038 × 10${}^{-30}$ | 68.55% | 1 | 8.6098 × 10${}^{-7}$ | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 2.7038 × 10${}^{-30}$ | 68.55% | 1 | 8.6098 × 10${}^{-7}$ | 1 | 0 | 1 | 0 | 1 | 0 |

**Table A2.**Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 1.9720 × 10${}^{-5}$ | 1 | 0 | 1 | 0 |

Monthly | 1 | 0.0031 | 1 | 0 | 1 | 0 |

Yearly | 0 | 0.7234 | 1 | 0 | 0 | 0.2308 |

Yearly Shuffled | 1 | 0 | 0 | 0.9306 | 1 | 0 |

#### Appendix A.2. Analysis on Bovespa Index

**Figure A9.**Empirical CDF versus standard normal CDF for Bovespa returns. The dotted black lines represent the DKW upper and lower bounds.

**Table A3.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 4.0876 × 10${}^{-19}$ | 69.62% | 0 | 0.7870 | 0 | 0.9870 | 1 | 0 | 1 | 0 |

Monthly | 1 | 8.4152 × 10${}^{-7}$ | 31.55% | 0 | 0.6679 | 0 | 0.9977 | 1 | 0 | 1 | 0 |

Yearly | 1 | 6.7479 × 10${}^{-103}$ | 81.43% | 1 | 3.7439 × 10${}^{-12}$ | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 6.7479 × 10${}^{-103}$ | 81.43% | 1 | 3.7439 × 10${}^{-12}$ | 1 | 0 | 1 | 0 | 1 | 0 |

**Table A4.**Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 0 | 1 | 0 | 1 | 0 |

Monthly | 1 | 0 | 1 | 0 | 1 | 1.3679 × 10${}^{-4}$ |

Yearly | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 0 | 0.8933 | 0 | 0.5147 | 0 | 0.7809 |

#### Appendix A.3. Analysis on US Corporate Index

**Figure A14.**Empirical CDF versus standard normal CDF for US Corp. returns. The dotted black lines represent the DKW upper and lower bounds.

**Table A5.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 0.0063 | 0.73% | 0 | 0.3201 | 0 | 0.9748 | 1 | 0 | 1 | 0 |

Monthly | 0 | 0.1877 | 0% | 0 | 0.9965 | 0 | 0.9900 | 1 | 0 | 1 | 0 |

Yearly | 0 | 0.0712 | 0% | 0 | 0.0698 | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 0 | 0.0712 | 0% | 0 | 0.0698 | 1 | 0 | 1 | 0 | 1 | 0 |

**Table A6.**Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 2.4980 × 10${}^{-14}$ | 1 | 0 | 1 | 6.0678 × 10${}^{-4}$ |

Monthly | 0 | 0.7075 | 0 | 0.2615 | 1 | 4.8873 × 10${}^{-5}$ |

Yearly | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 0 | 0.2325 | 0 | 0.5581 | 0 | 0.7833 |

#### Appendix A.4. Analysis on Emerging Markets Corporate Plus Index

**Figure A19.**Empirical CDF versus standard normal CDF for EmMkt Corp. returns. The dotted black lines represent the DKW upper and lower bounds.

**Table A7.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 8.3270 × 10${}^{-8}$ | 55.68% | 0 | 0.8814 | 0 | 0.997 | 1 | 0 | 1 | 0 |

Monthly | 1 | 0.0012 | 11.68% | 0 | 0.9927 | 0 | 0.9999 | 1 | 0 | 1 | 0 |

Yearly | 1 | 5.8695 × 10${}^{-4}$ | 8.54% | 0 | 0.2340 | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 0 | 0.0138 | 0% | 0 | 0.3056 | 1 | 0 | 1 | 0 | 1 | 0 |

**Table A8.**Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 0 | 1 | 0 | 1 | 0 |

Monthly | 0 | 0.7652 | 0 | 0.9976 | 1 | 0.0018 |

Yearly | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 0 | 0.6017 | 1 | 0 | 0 | 0.9606 |

#### Appendix A.5. Analysis on 3-Month Treasury Constant Maturity Rate

**Figure A24.**Empirical CDF versus standard normal CDF for DGS3M returns. The dotted black lines represent the DKW upper and lower bounds.

**Table A9.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 1.6567 × 10${}^{-20}$ | 47.48% | 1 | 1.7558 × 10${}^{-20}$ | 1 | 1.46 × 10${}^{-20}$ | 1 | 0 | 1 | 0 |

Monthly | 1 | 1.0664 × 10${}^{-71}$ | 75.69% | 1 | 3.6783 × 10${}^{-7}$ | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly | 1 | 2.0644 × 10${}^{-286}$ | 79.83% | 1 | 2.2533 × 10${}^{-26}$ | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 3.0634 × 10${}^{-297}$ | 80.24% | 1 | 1.3943 × 10${}^{-28}$ | 1 | 0 | 1 | 0 | 1 | 0 |

**Table A10.**Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 0 | 1 | 0 | 1 | 0 |

Monthly | 1 | 2.4962 × 10${}^{-4}$ | 1 | 0 | 1 | 0.0283 |

Yearly | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 3.0097 × 10${}^{-5}$ | 0 | 0.6966 | 1 | 0 |

#### Appendix A.6. Analysis on 10-Year Treasury Constant Maturity Rate

**Figure A29.**Empirical CDF versus standard normal CDF for DGS10Y returns. The dotted black lines represent the DKW upper and lower bounds.

**Table A11.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 1.2613 × 10${}^{-19}$ | 74.57% | 0 | 0.7075 | 0 | 0.9404 | 1 | 0 | 1 | 0 |

Monthly | 1 | 2.3517 × 10${}^{-6}$ | 39.47% | 0 | 0.93367 | 0 | 0.9775 | 1 | 0 | 1 | 0 |

Yearly | 1 | 3.2079 × 10${}^{-7}$ | 41.77% | 1 | 0.0020 | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 3.2079 × 10${}^{-7}$ | 41.77% | 1 | 0.0020 | 1 | 0 | 1 | 0 | 1 | 0 |

**Table A12.**Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 0 | 0 | 0.9879 | 1 | 0 |

Monthly | 1 | 0.0248 | 0 | 0.9987 | 0 | 0.0721 |

Yearly | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 0 | 0.2294 | 1 | 0 | 0 | 0.5526 |

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**Figure 1.**USBAAC log-returns histograms. Weekly, monthly and yearly sampling generates different distributions. In terms of distributions, there is no difference between shuffled and not shuffled yearly log returns.

**Figure 2.**USBAAC weekly and monthly log-returns Q-Q plots. This is the graphic representation of distribution quantiles comparing the CDF of the observed time series, which is unknown, a priori, with that of a specified distribution, chosen as benchmark. If the observed variable follows the theoretical distribution chosen, the Q-Q plot thickens across the line that connects the first and third quantiles of the data.

**Figure 3.**USBAAC yearly and yearly randomly shuffled log-returns Q-Q plots. In both cases, distributions do not look Gaussian.

**Figure 4.**Empirical CDF versus standard normal CDF for USBAAC returns. The dotted black lines represent the DKW upper and lower bounds.

Index | Code | Description | Asset Class | Market | Time Frame |
---|---|---|---|---|---|

a | USBAAC | USD Basis Swap 1Mv3M | Swap | Developed | 12 February 2007–30 March 2020 |

b | SPX | S&P 500 | Equity | Developed | 30 December 1927–29 May 2020 |

c | IBOV | Bovespa | Equity | Emerging | 5 January 1927–29 May 2020 |

d | BAMLCC0A2AATRIV | AA US Corp.TR | Bond Corporate | Developed | 23 December 1988–29 May 2020 |

e | BAMLEM1BRRAAA2ACRPITRIV | AAA-A Em. Mkt Corp TR | Bond Corporate | Emerging | 8 January 1999–29 May 2020 |

f | DGS3MO | 3-M Treasury Const. Mty | Bond Government | Developed | 11 January 1982–25 May 2020 |

g | DGS10 | 10-Y Treasury Const. Mty | Bond Government | Developed | 8 January 1962–25 May 2020 |

Statistical Characteristics of Weekly Returns | |||||||
---|---|---|---|---|---|---|---|

Mom.\Des. | USD Swap 1Mv3M | S&P 500 | Bovespa | AA US Corp.TR | Em Mk | 3m Tbill | 10Y Tbond |

St. Dev. | 0.0480 | 0.0250 | 0.0616 | 0.0054 | 0.0051 | 0.0001 | 0.0262 |

Mean | −7.0055 × 10${}^{-5}$ | 0.0011 | 0.0092 | 0.0012 | 0.0012 | 0.0001 | −0.0006 |

Kurtosis | 27.4840 | 9.6489 | 19.7571 | 19.4851 | 23.2641 | 2.7099 | 26.0274 |

Skew | 0.3850 | −0.6135 | 1.5259 | −1.3492 | −2.2882 | 0.5302 | −1.3782 |

Statistical Characteristics of Monthly Returns | |||||||

Mom.\Des. | USD Swap 1Mv3M | S&P 500 | Bovespa | AA US Corp.TR | Em Mk | 3m Tbill | 10Y Tbond |

St. Dev. | 0.0737 | 0.0540 | 0.1302 | 0.0125 | 0.0144 | 3.4106 × 10^{−4} | 0.0591 |

Mean | 3.3382 × 10${}^{-4}$ | 0.0042 | 0.0355 | 0.0049 | 0.0047 | 4.2310 × 10^{−4} | −0.0023 |

Kurtosis | 8.3043 | 12.4736 | 8.7592 | 5.7354 | 39.0341 | 2.6788 | 25.9171 |

Skew | −0.6069 | −0.4424 | 1.0511 | −0.3696 | −3.7233 | 0.52482 | −2.1663 |

Statistical Characteristics of Yearly Returns | |||||||

Mom.\Des. | USD Swap 1Mv3M | S&P 500 | Bovespa | AA US Corp.TR | Em Mk | 3m Tbill | 10Y Tbond |

St. Dev. | 0.2386 | 0.1954 | 0.9815 | 0.0504 | 0.0505 | 0.0040 | 0.2029 |

Mean | −0.0046 | 0.0526 | 0.4455 | 0.0602 | 0.0588 | 0.0052 | −0.0170 |

Kurtosis | 6.3841 | 6.9968 | 8.3539 | 2.5708 | 5.2510 | 2.0359 | 7.7485 |

Skew | 0.8520 | −1.1589 | 2.4618 | −0.0347 | 0.1058 | 0.3025 | −1.0258 |

**Table 3.**KPSS test to assess if the time series are trend stationary against the alternative of a unit root. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

USBAAC | S&P 500 | Bovespa | US Corp. | EmMkt Corp. | DGS3M | DGS10Y | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 0.0048 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

Monthly | 1 | 0.0046 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0.0405 | 1 | 0 | 1 | 0 |

Yearly | 0 | 0.6661 | 0 | 0.7321 | 0 | 0.0767 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

**Table 4.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-Skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 1.8548 × 10${}^{-25}$ | 75.19% | 0 | 0.7614 | 0 | 0.7115 | 1 | 0 | 1 | 0 |

Monthly | 1 | 6.0526 × 10${}^{-5}$ | 30.01% | 0 | 0.8750 | 0 | 0.8683 | 1 | 0 | 1 | 0 |

Yearly | 1 | 7.1011 × 10${}^{-11}$ | 45.09% | 0 | 0.0376 | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 7.1011 × 10${}^{-11}$ | 45.09% | 0 | 0.0376 | 1 | 0 | 1 | 0 | 1 | 0 |

**Table 5.**Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 1.3188 × 10${}^{-8}$ | 0 | 0.9282 | 1 | 1.5504 × 10${}^{-12}$ |

Monthly | 1 | 0.0127 | 0 | 0.9693 | 1 | 5.0823 × 10${}^{-6}$ |

Yearly | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 0.0149 | 0 | 0.9900 | 0 | 0.6897 |

**Table 6.**K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Weekly | 1 | 1.6502 × 10${}^{-19}$ | 74.92% | 0 | 0.0482 | 0 | 0.7256 | 1 | 0 | 1 | 0 |

Monthly | 1 | 8.3615 × 10${}^{-9}$ | 58.32% | 0 | 0.1985 | 0 | 0.9714 | 1 | 0 | 1 | 0 |

Yearly | 1 | 2.7038 × 10${}^{-30}$ | 68.55% | 1 | 8.6098 × 10${}^{-7}$ | 1 | 0 | 1 | 0 | 1 | 0 |

Yearly Shuffled | 1 | 2.7038 × 10${}^{-30}$ | 68.55% | 1 | 8.6099 × 10${}^{-7}$ | 1 | 0 | 1 | 0 | 1 | 0 |

**Table 7.**Ljung-Box Q-test and ARCH test to detect autocorrelation for the average series. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Ljung-Box Q-Test | ARCH Test | |||||
---|---|---|---|---|---|---|

$\mathit{m}=ln\left(\mathit{n}\right)$ | $\mathit{m}=(\mathit{n}-\mathbf{1})$ | |||||

resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Monthly | 1 | 1.3188 × 10${}^{-8}$ | 0 | 0.9654 | 0 | 0.4820 |

Yearly | 1 | 0.0071 | 0 | 0.7184 | 0 | 0.6229 |

**Table 8.**K-S test to detect the original distribution for the average series. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case.

Normal | t-skew | Gen. Hyperbolic | Gen. Pareto | Exp. Pareto | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

resp. | p-Value | DKW Exceeds | resp. | p-Value | resp. | p-Value | resp. | p-Value | resp. | p-Value | |

Av. Monthly | 1 | 0.0003 | 75.19% | 0 | 0.8706 | 0 | 0.0414 | 1 | 0 | 1 | 0 |

Av. Yearly | 0 | 0.5382 | 24.43% | 0 | 0.9961 | 0 | 0.0752 | 1 | 0 | 1 | 0 |

Normal | ||||
---|---|---|---|---|

Index | Sampling | resp. | p-Value | DKW Exceeds |

USBAAC | Weekly | 1 | 5.0443 × 10${}^{-15}$ | 67.16% |

Monthly | 1 | 5.3363 × 10${}^{-4}$ | 25.74% | |

Yearly | 1 | 1.0622 × 10${}^{-7}$ | 28.05% | |

Yearly Shuffled | 1 | 2.2856 × 10${}^{-11}$ | 54.27% | |

S&P 500 | Weekly | 1 | 2.8341 × 10${}^{-9}$ | 57.24% |

Monthly | 1 | 1.3090 × 10${}^{-5}$ | 46.72% | |

Yearly | 1 | 3.3137 × 10${}^{-40}$ | 70.12% | |

Yearly Shuffled | 1 | 5.0050 × 10${}^{-30}$ | 69.52% | |

Bovespa | Weekly | 1 | 5.7367 × 10${}^{-4}$ | 16.51% |

Monthly | 0 | 0.0229 | 0% | |

Yearly | 1 | 5.2297 × 10${}^{-7}$ | 33.15% | |

Yearly Shuffled | 1 | 7.6801 × 10${}^{-104}$ | 80.69% | |

US Corp. | Weekly | 0 | 0.0199 | 0% |

Monthly | 0 | 0.1236 | 0% | |

Yearly | 1 | 8.4622 × 10${}^{-7}$ | 34.55% | |

Yearly Shuffled | 0 | 0.1165 | 0% | |

EmMkt Corp. | Weekly | 1 | 1.2991 × 10${}^{-5}$ | 42.61% |

Monthly | 1 | 8.7935 × 10${}^{-4}$ | 8.27% | |

Yearly | 1 | 3.2051 × 10${}^{-7}$ | 32.73% | |

Yearly Shuffled | 0 | 0.0888 | 0% | |

DGS3M | Weekly | 1 | 5.5608 × 10${}^{-10}$ | 51.97% |

Monthly | 1 | 1.0300 × 10${}^{-119}$ | 86.22% | |

Yearly | 1 | 1.1038 × 10${}^{-62}$ | 75.85% | |

Yearly Shuffled | 1 | 8.6683 × 10${}^{-122}$ | 89.42% | |

DGS10Y | Weekly | 0 | 0.0101 | 0% |

Monthly | 1 | 0.0038 | 1.19% | |

Yearly | 1 | 1.0023 × 10${}^{-6}$ | 44.23% | |

Yearly Shuffled | 1 | 2.0652 × 10${}^{-7}$ | 53.54% |

**Table 10.**K-S test to detect the normality of the rescaled returns with $({K}^{*},{\beta}^{*})$ (see Equation (4)).

Normal | ||||||
---|---|---|---|---|---|---|

Index | Sampling | resp. | p-Value | DKW Exceeds | ${\mathit{K}}^{*}$ | ${\mathit{\beta}}^{*}$ |

USBAAC | Weekly | 1 | 1.5841 × 10${}^{-5}$ | 33.76% | 2.0166 | 0.3653 |

Monthly | 1 | 0.0028 | 6.62% | 12.9821 | 0.3374 | |

Yearly | 1 | 0.0063 | 1.22% | 6.7090 | 0.1032 | |

Yearly Shuffled | 1 | 2.1026 × 10${}^{-10}$ | 40.24% | 7.6792 | 0.0053 | |

S&P 500 | Weekly | 1 | 1.0061 × 10${}^{-7}$ | 44.27% | 8.3221 | 0.1905 |

Monthly | 1 | 2.0385 × 10${}^{-5}$ | 32.34% | 9.5246 | 0.1070 | |

Yearly | 1 | 1.2624 × 10${}^{-24}$ | 57.09% | 5.5153 | 0.0050 | |

Yearly Shuffled | 1 | 1.0678 × 10${}^{-27}$ | 73.56% | 8.5980 | 0.0393 | |

Bovespa | Weekly | 1 | 0.0029 | 8.89% | 6.7990 | 0.1540 |

Monthly | 0 | 0.0260 | 0% | 1.1396 | 0.1014 | |

Yearly | 0 | 0.5326 | 0% | 5.2612 | 0.0116 | |

Yearly Shuffled | 1 | 1.3433 × 10${}^{-98}$ | 80.01% | 3.9341 | 0.0293 | |

US Corp. | Weekly | 0 | 0.0626 | 0% | 9.6300 | 0.1515 |

Monthly | 0 | 0.2102 | 0% | 7.8913 | 0.0015 | |

Yearly | 0 | 0.4253 | 0% | 5.5781 | 0.0040 | |

Yearly Shuffled | 0 | 0.0886 | 0% | 2.3842 | 0.0948 | |

EmMkt Corp. | Weekly | 1 | 0.0016 | 5.95% | 11.7189 | 0.2936 |

Monthly | 0 | 0.0109 | 0% | 4.6068 | 0.3794 | |

Yearly | 1 | 1.3835 × 10${}^{-4}$ | 22.16% | 5.9900 | 0.0044 | |

Yearly Shuffled | 0 | 0.0234 | 0% | 4.5304 | 0.0279 | |

DGS3M | Weekly | 1 | 2.3686 × 10${}^{-10}$ | 49.58% | 4.2862 | 0.0244 |

Monthly | 1 | 1.1081 × 10${}^{-29}$ | 64.83% | 5.0489 | 0.0161 | |

Yearly | 1 | 1.4193 × 10${}^{-67}$ | 70.98% | 33.7053 | 0.0185 | |

Yearly Shuffled | 1 | 2.9770 × 10${}^{-48}$ | 73.56% | 62.4370 | 0.1358 | |

DGS10Y | Weekly | 0 | 0.2669 | 0% | 7.4724 | 0.1314 |

Monthly | 1 | 0.0060 | 0.26% | 7.6785 | 0.1516 | |

Yearly | 1 | 0.0024 | 2.54% | 5.3205 | 0.0047 | |

Yearly Shuffled | 1 | 4.2387 × 10${}^{-7}$ | 40.33% | 10.0382 | 0.0009 |

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## Share and Cite

**MDPI and ACS Style**

Orlando, G.; Bufalo, M.
Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions. *Risks* **2021**, *9*, 88.
https://doi.org/10.3390/risks9050088

**AMA Style**

Orlando G, Bufalo M.
Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions. *Risks*. 2021; 9(5):88.
https://doi.org/10.3390/risks9050088

**Chicago/Turabian Style**

Orlando, Giuseppe, and Michele Bufalo.
2021. "Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions" *Risks* 9, no. 5: 88.
https://doi.org/10.3390/risks9050088