# How to Analyze the Volatility of stock returns with the (G)ARCH model part II

Empirical results of Logarithmic Returns/Skewness and Kurtosis

March 15, 2023

10

min

Empirical results of Logarithmic Returns/Skewness and Kurtosis

In order to get a first impression of the data, it makes sense to get an overview of the data.As already explained in the second chapter (2. Data), logarithmic returns were used for the analysis. Figure 1.1 shows the log returns of the Nasdaq100 and the price of gold. Figure 1.2shows the squared log returns of the Nasdaq100 and the price of gold. The Nasdaq100 is represented by the black line and the price of gold is represented by the blue line. This is constant for all further mappings. It is clear that there are differences in log returns between the two asset classes. Figure 1.1 shows that gold fluctuates less, so the time series is concentrated closer to zero. It can also be seen that the Nasdaq100 forms more extreme maxima than gold.

A certain anti-cyclical behavior of the two asset classes can also be seen, for example theNasdaq100 shows the largest fluctuation in March 2020 (corona crisis), while gold shows a small fluctuation in this period. This contrasts with the fact that gold made its maximum in2013 (strong economic data from the USA and an associated increase in the US dollar), while the Nasdaq100 fluctuated very little during this period. Furthermore, both graphs show atypical volatility structure of capital markets (Schmelzer 2009: 1). After extreme shocks, the fluctuation around this area is very high. The time series needs time to develop to its average level (Schmelzer 2009: 4). Between the extreme points it can be clearly seen that there are long phases in which the fluctuation is very small (Schmelzer 2009: 4). The extreme volatility phases have a major impact on later models (Schmelzer 2009: 4). Also, standard errors tend to be lower, since the number of observations is very large. A tendency towards higher kurtosis is also typical of the return distributions of capital market data (Schmelzer2009: 15).

An analysis of the distribution of both log returns is now required. With the help of the skewness, statements can be made about the symmetry of the empirical return distributions(Schmelzer 2009: 15). The skewness of a symmetrical distribution, such as the normal distribution, has a value of zero (Schmelzer 2009: 15). The values of the skewness are both negative. This suggests that a left-skewed distribution occurs here (Schmelzer 2009: 15).Another measure to characterize the shape of a distribution is the kurtosis of the distribution (Schmelzer 2009: 15). With a normal distribution, the kurtosis has a value of three (Schmelzer 2009: 15). As can be seen, the kurtosis values for the Nasdaq100 and gold are well above three. Therefore, the data form significantly higher peaks and broader ends than in the case of a normal distribution (Schmelzer 2009: 15). Thus, a leptokurtic distribution of the observed data can be assumed (Schmelzer 2009: 15).

Marcus Schmelzer, (2009), The volatility of financial market data, theoretical foundations andempirical analyzes of hourly return time series and risk measures, Universität Köln

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