📂Statistical Analysis

Heteroskedasticity and Volatility Clustering in Time Series Analysis

Definition ¹

Given a time series data $\left\{ p_{t} \right\}$.

1. When the variance of $\left\{ p_{t} \right\}$ depends on $t$, $\left\{ p_{t} \right\}$ is said to have Heteroscedasticity.
2. The phenomenon of the variance of $\left\{ p_{t} \right\}$, which has Heteroscedasticity, increasing and decreasing repeatedly is referred to as Volatility Clustering.
3. The following defined $r_{t}$ is referred to as (Log) Return in $t$. $$ r_{t} := \nabla \log p_{t} = \log {{ p_{t} } \over { p_{t-1} }} $$

Explanation

Heteroscedasticity is read as [heteroscedasticity] and Volatility Clustering as [volatility clustering]. There are few documents in Korean that explain these two terms, probably because those who study this see no need to simplify the terminology. While writing and reading about Heteroscedasticity and Volatility Clustering does not feel particularly unnatural, when speaking, the long words are often pronounced as they are.

Heteroscedasticity

ARIMA models are interested in the mean of the data, as is the case with most statistical analyses. What matters is how the actual values move. But not only the values themselves depend on time; the variation itself can also change over time. That’s why the word Heteroscedasticity was coined. If the variance of time series data is always constant regardless of time, the term would be unnecessary.

The interest in the variance of time series is mainly in the field of economics and finance. Changes in data signify changes in value, and can be directly used when discussing the risk of an asset. Since risk is just another form of profit, it certainly is a matter of interest. Time series data is commonly written as $\left\{ y_{t} \right\}$, but when focusing on Heteroscedasticity, writing it as $\left\{ p_{t} \right\}$ is also for that reason. $p_{t}$’s $p$ comes from Price.

Volatility Clustering

The phenomenon where the variance increases and decreases is called Volatility Clustering, and the reason for having such a term is simple. There are no instances where the volatility in the market only increases or decreases, and even if there were, it would not be complex enough to require statistical analysis. This could be described more elegantly in statistical terms, but for now, let’s stick to this explanation.

Return

Return is a useful expression for identifying and analyzing Heteroscedasticity. By taking the log, if $p_{t}$ is greater than $p_{t-1}$ (if the price goes up), it becomes a positive number, and if it’s smaller (if the price goes down), a negative number. Multiplying by 100 makes it easier to view as a percentage. Since it uses the rate of change rather than just how big or small the difference is, another advantage is that it is less affected by the unique characteristics of the data.

Practice

The graph above is drawn by extracting only DAX from the built-in data EuStockMarkets, showing the German DAX index from 1991 to 1999.

Return is drawn as shown above, and if the image with the highlight seems to show that the volatility is alternating between high and low, then it can be considered that Volatility Clustering has been understood.

Even if it doesn’t seem so, that’s fine. It is the role of statistics to solidify such ‘arguments based on sensation’, and no statistical techniques have been applied yet.

Code

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