To study the extreme values in the return – target of forecasting – we need to analyze the data using two major econometrics: Excess Kurtosis and Skewness.

These statistical tools better represent the extremes of the data set rather than focusing solely on the average.

Positive extreme returns are critical to holding a short position. It is important as a short seller to know when the positive returns are extreme, so, when the forecast shows a regression to the mean this would allow the short seller to invest in the right time by being able to identify the peak and knowing in advance – through the return forecasting – that a reversion to the mean is expected. Whereas the negative extreme returns are important in risk management since it is important to know when the negative returns are extreme and there will again be a reversion to the mean and to positive returns.

#### Excess Kurtosis

The excess kurtosis of a normal random variable is zero. A distribution with positive excess kurtosis is said to have heavy tails – which is the case with the price distribution of the two cryptocurrencies – implying that the distribution puts more mass on the tails of its support than a normal distribution does.

In practice, this means that a random sample from such a distribution tends to contain more extreme values. Such a distribution is said to be leptokurtic

Sample kurtosis can be calculated through the following equation

\(\hat{K}(x) = \frac{1}{(T-1)\hat{\sigma}^4_x}\sum\limits_{t=1}^T(x_t-\hat{\mu}_x)^4\cdot\)

T is the number of observations

Bitcoin and Ethereum return Kurtosis respectively

Bitcoin Return Kurtosis |
Ethereum Return Kurtosis |

65.83791065718604 |
75.85892119668719 |

From the output, we could see that the excess kurtosis is quite high – leptokurtic – for both cryptocurrencies compared to the kurtosis of a normal distribution, which means that we have a lot of extreme values

#### Skewness in Return

Skewness in the return distribution can be visualized through the asymmetry that deviates from the symmetrical bell curve where the data piles to the left of the curve (positive) skewness or to the right (negative)

From the visualization, the return looks almost symmetrical; but to get more precise numbers, and to find if the data is skewed to the right or the left, we calculated the skewness for both Bitcoin and Ethereum returns

Sample skewness can be calculated through the following equation

\(\hat{S}(x) = \frac{1}{(T-1)\hat{\sigma}_x^3}\sum\limits_{t=1}^T(x_t - \hat{\mu}_x)^3\)

T is the number of observations, û is the mean of the distribution, and sigma is the standard deviation

Bitcoin and Ethereum Skewness respectively

Bitcoin Return Kurtosis |
Ethereum Return Kurtosis |

1.4296588537329864 |
0.6970627924698324 |