7.4  Unconditional Leptokurtosis and
Conditional Heteroskedasticity

While leptokurtosis and heteroskedasticity are different notions, both arise in financial time series analysis, and one can manifest itself as the other.

Exhibit 7.8 indicates a histogram of daily log returns for the Toronto Stock Exchange TSE-100 Total Return Index during the 5-year period 1995 through 1999. A normal distribution has been fit to the data based upon the data’s sample mean and sample standard deviation.

Exhibit 7.8: Histogram of daily log returns for the Toronto Stock Exchange TSE-100 Total Return Index during the 5-year period 1995 through 1999. A normal distribution has been fit to the data based upon its sample mean and sample standard deviation. Of 1,259 data points, 8 in the left tail and 3 in the right tail are not displayed because they fall outside the range [–.035, .035].

Comparing the normal curve and the histogram in Exhibit 7.8, we see that the histogram is  leptokurtic. Exhibit 7.9 compares the mean, standard deviation, skewness, and kurtosis of the normal curve and histogram.

Exhibit 7.0: Parameters of the normal distribution of Exhibit 7.8 are compared with sample parameters of the histogram of returns from the same exhibit. The histogram of returns is highly leptokurtic.

This leptokurtosis is not an isolated result. Histograms of log returns for financial assets often exhibit leptokurtosis. For commodities or energy products, such leptokurtosis is often extreme. It has prompted some authors to propose that log returns be modeled with leptokurtic distributions. For example, Wilson (1993) proposes that key factors for value-at-risk measures be modeled with Student t distributions.

Replacing normal distributions with leptokurtic distributions complicates value-at-risk measures. Furthermore, both theory and empirical evidence suggest that it is not necessary to do so. For measuring value-at-risk, we are interested in modeling the conditional distributions of key factors, but histograms such as Exhibit 7.8 are more descriptive of unconditional distributions.3 Is it possible for a stochastic process to have leptokurtic unconditional distributions but non-leptokurtic conditional distributions? Certainly. This is a common phenomenon with conditionally heteroskedastic processes.

Soon, we shall describe how conditional heteroskedasticity can manifest itself as unconditional leptokurtosis. Before we do so, let’s take a different look at the TSE-100 returns of Exhibit 7.8. In Exhibit 7.10, we graph them chronologically as a time series. They exhibit conditional heteroskedasticity.

Exhibit 7.10: The same returns as in Exhibit 7.8 are presented chronologically as a time series. The returns exhibit conditional heteroskedasticity.

7.4.1 Experiment With Conditional Heteroskedasticity

Let’s conduct an experiment with conditional heteroskedasticity. I will do it here, but I would like you to reproduce the same work in your own spreadsheet. Generate a time series of 1,500 pseudorandom variates. Let the first 500 be N(0,25). Let the next 500 be N(0,9). Let the last 500 be N(0,1). Graph your time series. Exhibit 7.11 shows my graph.

Exhibit 7.11: A time series constructed from 1,500 pseudorandom variates. The first 500 are N(0,25). The next 500 are N(0,9). The last 500 are N(0,1).

Now calculate the sample mean, standard deviation, skewness, and kurtosis of the first 500 values of your time series. Do so for the second 500 values and the third 500 values. Finally, calculate the same parameters for the entire time series. My results are indicated in Exhibit 7.12.

Exhibit 7.12: Sample parameters for sections of the time series as well as for the entire time series. Sample parameters for the sections are consistent with normal distributions. Sample parameters for the entire time series are not. In particular, the entire time series exhibits kurtosis that is almost twice that of a normal distribution.

A normal distribution has a kurtosis of 3. The first, second, and third sections of my time series each have sample kurtoses consistent with a normal distribution. The overall time series does not. This is illustrated in Exhibit 7.13 with a histogram of all the time series’ 1,500 values.

Exhibit 7.13: Histogram of time series values from the experiment. A normal distribution has been fit to the data based upon its sample mean and sample standard deviation. Of 1,500 data points, 5 in the left tail and 5 in the right tail are not displayed because they fall outside the range [–12, 12].

Each value in the time series was generated from a normal distribution, but the histogram of all values is distinctly non-normal. The sample leptokurtosis of the overall time series arises much as it can with a mixed-normal distribution, as discussed in Section 3.11.2.

This experiment is a warning. Conditional heteroskedasticity can manifest itself as unconditional leptokurtosis. Just because a histogram of time series values exhibits leptokurtosis does not mean those values were drawn from leptokurtic conditional distributions. In the context of value-at-risk, if a histogram of historical data for a key factor is leptokurtic, that does not mean the key factor needs to be modeled with a conditionally leptokurtic distribution. The unconditional leptokurtosis may reflect conditional heteroskedasticity, as in our experiment.

7.4.2 Modeling Unconditional Leptokurtosis

It is important to distinguish between markets and models for those markets. Markets generate data. There are various models we might fit to that data. Consider again Exhibit 7.8, which indicates a leptokurtic histogram of total returns for the TSE-100. Of the many models we might fit to that data, let’s explore two competing white noise models:

1. Returns are conditionally homoskedastic and are drawn from a fixed conditional distribution that is leptokurtic. A Student t distribution might be used for this purpose.
2. Returns are conditionally heteroskedastic and are drawn from conditional distributions that are normal. A GARCH model might be used for this purpose.

Both models could produce the leptokurtic histogram of Exhibit 7.8. The first would do so directly with a conditional distribution that is leptokurtic. The second would do so indirectly with conditional heteroskedasticity. Both approaches are studied in the financial literature.

There is a characteristic of financial time series that makes us favor the second model over the first. This is volatility clustering. For the most part, extreme market moves do not occur in isolation. They cluster. Markets experience periods of turmoil and periods of tranquility. Our first model cannot reproduce this behavior. It will produce consecutive extreme market moves only by coincidence. The second model easily reproduces volatility clustering. Indeed, GARCH models are designed specifically for this purpose.

We have many options in how we model markets. We don’t have to choose between a conditionally leptokurtic model and one that is conditionally heteroskedastic. Both of these properties might be combined in a single model. A regime-switching model might be implemented with conditional heteroskedasticity and leptokurtic mixed-normal conditional distributions.

7.4.3 Implications for Value-at-Risk Measures

In Chapter 4, we discussed multivariate GARCH and multivariate regime-switching processes that can be used to model conditional heteroskedasticity. These are actively being researched, but cannot yet be recommended for production value-at-risk measures. Some versions of these, such as orthogonal GARCH, have known shortcomings. Others hold promise, but require further study before they can be recommended.

Despite their shortcomings, the UWMA and EWMA methods are simple and reliable. They can be trusted to produce covariance matrices that, although imperfect, are reasonable. These are the standard methods used today.

This conclusion is necessarily tentative. We have described in detail the challenges of modeling financial markets. Yet, available solutions fall short of the needs we have identified. Still, our discussion is useful. If we find ourselves using a value-at-risk measure that employs UWMA or EWMA, it is important to know the limitations of those methods. Also, our discussion may spur further research into alternative modeling techniques.