22. Heavy-Tailed Distributions#

22.1.1. Introduction: light tails#

Most commonly used probability distributions in classical statistics and the natural sciences have “light tails.”

To explain this concept, let’s look first at examples.

We can see this when we plot the density and show a histogram of observations, as with the following code (which assumes $\mu =0$ and $\sigma =1$).

fig, ax = plt.subplots()
X = norm.rvs(size=1_000_000)
ax.hist(X, bins=40, alpha=0.4, label='histogram', density=True)
x_grid = np.linspace(-4, 4, 400)
ax.plot(x_grid, norm.pdf(x_grid), label='density')
ax.legend()
plt.show()

Notice how

• the density’s tails converge quickly to zero in both directions and

• even with 1,000,000 draws, we get no very large or very small observations.

We can see the last point more clearly by executing

X.min(), X.max()

(-4.887962116860806, 5.048642271206914)

Here’s another view of draws from the same distribution:

n = 2000
fig, ax = plt.subplots()
data = norm.rvs(size=n)
ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)
ax.vlines(list(range(n)), 0, data, lw=0.2)
ax.set_ylim(-15, 15)
ax.set_xlabel('$i$')
ax.set_ylabel('$X_i$', rotation=0)
plt.show()

We have plotted each individual draw $X_i$ against $i$.

None are very large or very small.

In other words, extreme observations are rare and draws tend not to deviate too much from the mean.

Putting this another way, light-tailed distributions are those that rarely generate extreme values.

(A more formal definition is given below.)

Many statisticians and econometricians use rules of thumb such as “outcomes more than four or five standard deviations from the mean can safely be ignored.”

But this is only true when distributions have light tails.

22.1.2. When are light tails valid?#

In probability theory and in the real world, many distributions are light-tailed.

For example, human height is light-tailed.

Yes, it’s true that we see some very tall people.

• For example, basketballer Sun Mingming is 2.32 meters tall

But have you ever heard of someone who is 20 meters tall? Or 200? Or 2000?

Have you ever wondered why not?

After all, there are 8 billion people in the world!

In essence, the reason we don’t see such draws is that the distribution of human height has very light tails.

In fact the distribution of human height obeys a bell-shaped curve similar to the normal distribution.

22.1.3. Returns on assets#

But what about economic data?

Let’s look at some financial data first.

Our aim is to plot the daily change in the price of Amazon (AMZN) stock for the period from 1st January 2015 to 1st July 2022.

This equates to daily returns if we set dividends aside.

The code below produces the desired plot using Yahoo financial data via the yfinance library.

data = yf.download('AMZN', '2015-1-1', '2022-7-1')

[*********************100%***********************]  1 of 1 completed

YF.download() has changed argument auto_adjust default to True

s = data['Close']
r = s.pct_change()

fig, ax = plt.subplots()

ax.plot(r, linestyle='', marker='o', alpha=0.5, ms=4)
ax.vlines(r.index, 0, r.values, lw=0.2)
ax.set_ylabel('returns', fontsize=12)
ax.set_xlabel('date', fontsize=12)

plt.show()

This data looks different to the draws from the normal distribution we saw above.

Several of observations are quite extreme.

We get a similar picture if we look at other assets, such as Bitcoin

data = yf.download('BTC-USD', '2015-1-1', '2022-7-1')

[*********************100%***********************]  1 of 1 completed

s = data['Close']
r = s.pct_change()

fig, ax = plt.subplots()

ax.plot(r, linestyle='', marker='o', alpha=0.5, ms=4)
ax.vlines(r.index, 0, r.values, lw=0.2)
ax.set_ylabel('returns', fontsize=12)
ax.set_xlabel('date', fontsize=12)

plt.show()

The histogram also looks different to the histogram of the normal distribution:

r = np.random.standard_t(df=5, size=1000)

fig, ax = plt.subplots()
ax.hist(r, bins=60, alpha=0.4, label='bitcoin returns', density=True)

xmin, xmax = plt.xlim()
x = np.linspace(xmin, xmax, 100)
p = norm.pdf(x, np.mean(r), np.std(r))
ax.plot(x, p, linewidth=2, label='normal distribution')

ax.set_xlabel('returns', fontsize=12)
ax.legend()

plt.show()

If we look at higher frequency returns data (e.g., tick-by-tick), we often see even more extreme observations.

See, for example, [Mandelbrot, 1963] or [Rachev, 2003].

22.1.4. Other data#

The data we have just seen is said to be “heavy-tailed”.

With heavy-tailed distributions, extreme outcomes occur relatively frequently.

Later in this lecture, we examine heavy tails in these distributions.

22.1.5. Why should we care?#

Heavy tails are common in economic data but does that mean they are important?

The answer to this question is affirmative!

When distributions are heavy-tailed, we need to think carefully about issues like

• diversification and risk

• forecasting

• taxation (across a heavy-tailed income distribution), etc.

We return to these points below.

22.2.1. Simulations#

The figure below shows a simulation.

The top two subfigures each show 120 independent draws from the normal distribution, which is light-tailed.

The bottom subfigure shows 120 independent draws from the Cauchy distribution, which is heavy-tailed.

n = 120
np.random.seed(11)

fig, axes = plt.subplots(3, 1, figsize=(6, 12))

for ax in axes:
    ax.set_ylim((-120, 120))

s_vals = 2, 12

for ax, s in zip(axes[:2], s_vals):
    data = np.random.randn(n) * s
    ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)
    ax.vlines(list(range(n)), 0, data, lw=0.2)
    ax.set_title(fr"draws from $N(0, \sigma^2)$ with $\sigma = {s}$", fontsize=11)

ax = axes[2]
distribution = cauchy()
data = distribution.rvs(n)
ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)
ax.vlines(list(range(n)), 0, data, lw=0.2)
ax.set_title(f"draws from the Cauchy distribution", fontsize=11)

plt.subplots_adjust(hspace=0.25)

plt.show()

In the top subfigure, the standard deviation of the normal distribution is 2, and the draws are clustered around the mean.

In the middle subfigure, the standard deviation is increased to 12 and, as expected, the amount of dispersion rises.

The bottom subfigure, with the Cauchy draws, shows a different pattern: tight clustering around the mean for the great majority of observations, combined with a few sudden large deviations from the mean.

This is typical of a heavy-tailed distribution.

22.2.2. Nonnegative distributions#

Let’s compare some distributions that only take nonnegative values.

One is the exponential distribution, which we discussed in our lecture on probability and distributions.

The exponential distribution is a light-tailed distribution.

Here are some draws from the exponential distribution.

n = 120
np.random.seed(11)

fig, ax = plt.subplots()
ax.set_ylim((0, 50))

data = np.random.exponential(size=n)
ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)
ax.vlines(list(range(n)), 0, data, lw=0.2)

plt.show()

Another nonnegative distribution is the Pareto distribution.

If $X$ has the Pareto distribution, then there are positive constants $\overline{x}$ and $\alpha$ such that

The parameter $\alpha$ is called the tail index and $\overline{x}$ is called the minimum.

The Pareto distribution is a heavy-tailed distribution.

One way that the Pareto distribution arises is as the exponential of an exponential random variable.

In particular, if $X$ is exponentially distributed with rate parameter $\alpha$, then

is Pareto-distributed with minimum $\overline{x}$ and tail index $\alpha$.

Here are some draws from the Pareto distribution with tail index $1$ and minimum $1$.

n = 120
np.random.seed(11)

fig, ax = plt.subplots()
ax.set_ylim((0, 80))
exponential_data = np.random.exponential(size=n)
pareto_data = np.exp(exponential_data)
ax.plot(list(range(n)), pareto_data, linestyle='', marker='o', alpha=0.5, ms=4)
ax.vlines(list(range(n)), 0, pareto_data, lw=0.2)

plt.show()

Notice how extreme outcomes are more common.

22.2.3. Counter CDFs#

For nonnegative random variables, one way to visualize the difference between light and heavy tails is to look at the counter CDF (CCDF).

For a random variable $X$ with CDF $F$, the CCDF is the function

(Some authors call $G$ the “survival” function.)

The CCDF shows how fast the upper tail goes to zero as $x\to \mathrm{\infty }$.

If $X$ is exponentially distributed with rate parameter $\alpha$, then the CCDF is

This function goes to zero relatively quickly as $x$ gets large.

The standard Pareto distribution, where $\overline{x}=1$, has CCDF

This function goes to zero as $x\to \mathrm{\infty }$, but much slower than $G_E$.

Here’s a plot that illustrates how $G_E$ goes to zero faster than $G_P$.

x = np.linspace(1.5, 100, 1000)
fig, ax = plt.subplots()
alpha = 1.0
ax.plot(x, np.exp(- alpha * x), label='exponential', alpha=0.8)
ax.plot(x, x**(- alpha), label='Pareto', alpha=0.8)
ax.set_xlabel('X value')
ax.set_ylabel('CCDF')
ax.legend()
plt.show()

Here’s a log-log plot of the same functions, which makes visual comparison easier.

fig, ax = plt.subplots()
alpha = 1.0
ax.loglog(x, np.exp(- alpha * x), label='exponential', alpha=0.8)
ax.loglog(x, x**(- alpha), label='Pareto', alpha=0.8)
ax.set_xlabel('log value')
ax.set_ylabel('log prob')
ax.legend()
plt.show()

In the log-log plot, the Pareto CCDF is linear, while the exponential one is concave.

This idea is often used to separate light- and heavy-tailed distributions in visualisations — we return to this point below.

22.2.4. Empirical CCDFs#

The sample counterpart of the CCDF function is the empirical CCDF.

Given a sample $x_1,\dots ,x_n$, the empirical CCDF is given by

Thus, $\hat{G}(x)$ shows the fraction of the sample that exceeds $x$.

def eccdf(x, data):
    "Simple empirical CCDF function."
    return np.mean(data > x)

Here’s a figure containing some empirical CCDFs from simulated data.

# Parameters and grid
x_grid = np.linspace(1, 1000, 1000)
sample_size = 1000
np.random.seed(13)
z = np.random.randn(sample_size)

# Draws
data_exp = np.random.exponential(size=sample_size)
data_logn = np.exp(z)
data_pareto = np.exp(np.random.exponential(size=sample_size))

data_list = [data_exp, data_logn, data_pareto]

# Build figure
fig, axes = plt.subplots(3, 1, figsize=(6, 8))
axes = axes.flatten()
labels = ['exponential', 'lognormal', 'Pareto']

for data, label, ax in zip(data_list, labels, axes):

    ax.loglog(x_grid, [eccdf(x, data) for x in x_grid], 
        'o', markersize=3.0, alpha=0.5, label=label)
    ax.set_xlabel("log value")
    ax.set_ylabel("log prob")
    
    ax.legend()
    
    
fig.subplots_adjust(hspace=0.4)

plt.show()

As with the CCDF, the empirical CCDF from the Pareto distributions is approximately linear in a log-log plot.

We will use this idea below when we look at real data.

22.2.4.1. Q-Q Plots#

We can also use a qq plot to do a visual comparison between two probability distributions.

The statsmodels package provides a convenient qqplot function that, by default, compares sample data to the quintiles of the normal distribution.

If the data is drawn from a normal distribution, the plot would look like:

data_normal = np.random.normal(size=sample_size)
sm.qqplot(data_normal, line='45')
plt.show()

We can now compare this with the exponential, log-normal, and Pareto distributions

# Build figure
fig, axes = plt.subplots(1, 3, figsize=(12, 4))
axes = axes.flatten()
labels = ['exponential', 'lognormal', 'Pareto']
for data, label, ax in zip(data_list, labels, axes):
    sm.qqplot(data, line='45', ax=ax, )
    ax.set_title(label)
plt.tight_layout()
plt.show()

22.2.5. Power laws#

One specific class of heavy-tailed distributions has been found repeatedly in economic and social phenomena: the class of so-called power laws.

A random variable $X$ is said to have a power law if, for some $\alpha >0$,

We can write this more mathematically as

It is also common to say that a random variable $X$ with this property has a Pareto tail with tail index $\alpha$.

Notice that every Pareto distribution with tail index $\alpha$ has a Pareto tail with tail index $\alpha$.

We can think of power laws as a generalization of Pareto distributions.

They are distributions that resemble Pareto distributions in their upper right tail.

Another way to think of power laws is a set of distributions with a specific kind of (very) heavy tail.

22.3.1. Firm size#

Here is a plot of the firm size distribution for the largest 500 firms in 2020 taken from Forbes Global 2000.

Show source Hide code cell source

df_fs = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-global2000.csv')
df_fs = df_fs[['Country', 'Sales', 'Profits', 'Assets', 'Market Value']]
fig, ax = plt.subplots(figsize=(6.4, 3.5))

label="firm size (market value)"
top = 500 # set the cutting for top
d = df_fs.sort_values('Market Value', ascending=False)
empirical_ccdf(np.asarray(d['Market Value'])[:top], ax, label=label, add_reg_line=True)

plt.show()

22.3.2. City size#

Here are plots of the city size distribution for the US and Brazil in 2023 from the World Population Review.

The size is measured by population.

Show source Hide code cell source

# import population data of cities in 2023 United States and 2023 Brazil from world population review
df_cs_us = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_us.csv')
df_cs_br = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_brazil.csv')

fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))

empirical_ccdf(np.asarray(df_cs_us["pop2023"]), axes[0], label="US", add_reg_line=True)
empirical_ccdf(np.asarray(df_cs_br['pop2023']), axes[1], label="Brazil", add_reg_line=True)

plt.show()

22.3.3. Wealth#

Here is a plot of the upper tail (top 500) of the wealth distribution.

The data is from the Forbes Billionaires list in 2020.

Show source Hide code cell source

df_w = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-billionaires.csv')
df_w = df_w[['country', 'realTimeWorth', 'realTimeRank']].dropna()
df_w = df_w.astype({'realTimeRank': int})
df_w = df_w.sort_values('realTimeRank', ascending=True).copy()
countries = ['United States', 'Japan', 'India', 'Italy']  
N = len(countries)

fig, axs = plt.subplots(2, 2, figsize=(8, 6))
axs = axs.flatten()

for i, c in enumerate(countries):
    df_w_c = df_w[df_w['country'] == c].reset_index()
    z = np.asarray(df_w_c['realTimeWorth'])
    # print('number of the global richest 2000 from '+ c, len(z))
    top = 500           # cut-off number: top 500
    if len(z) <= top:    
        z = z[:top]

    empirical_ccdf(z[:top], axs[i], label=c, xlabel='log wealth', add_reg_line=True)
    
fig.tight_layout()

plt.show()

22.3.4. GDP#

Of course, not all cross-sectional distributions are heavy-tailed.

Here we show cross-country per capita GDP.

# get gdp and gdp per capita for all regions and countries in 2021

variable_code = ['NY.GDP.MKTP.CD', 'NY.GDP.PCAP.CD']
variable_names = ['GDP', 'GDP per capita']

df_gdp1 = extract_wb(varlist=variable_code, 
                     c="all", 
                     s=2021, 
                     e=2021, 
                     varnames=variable_names)
df_gdp1.dropna(inplace=True)

Show source Hide code cell source

fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))

for name, ax in zip(variable_names, axes):
    empirical_ccdf(np.asarray(df_gdp1[name]).astype("float64"), ax, add_reg_line=False, label=name)

plt.show()

The plot is concave rather than linear, so the distribution has light tails.

One reason is that this is data on an aggregate variable, which involves some averaging in its definition.

Averaging tends to eliminate extreme outcomes.

22.5.1. Diversification#

One of the most important ideas in investing is using diversification to reduce risk.

This is a very old idea — consider, for example, the expression “don’t put all your eggs in one basket”.

To illustrate, consider an investor with one dollar of wealth and a choice over $n$ assets with payoffs $X_1,\dots ,X_n$.

Suppose that returns on distinct assets are independent and each return has mean $\mu$ and variance ${\sigma }^2$.

If the investor puts all wealth in one asset, say, then the expected payoff of the portfolio is $\mu$ and the variance is ${\sigma }^2$.

If instead the investor puts share $1/n$ of her wealth in each asset, then the portfolio payoff is

Try computing the mean and variance.

You will find that

• The mean is unchanged at $\mu$, while

• the variance of the portfolio has fallen to ${\sigma }^2/n$.

Diversification reduces risk, as expected.

But there is a hidden assumption here: the variance of returns is finite.

If the distribution is heavy-tailed and the variance is infinite, then this logic is incorrect.

For example, we saw above that if every $X_i$ is Cauchy, then so is $Y_n$.

This means that diversification doesn’t help at all!

22.5.2. Fiscal policy#

The heaviness of the tail in the wealth distribution matters for taxation and redistribution policies.

The same is true for the income distribution.

For example, the heaviness of the tail of the income distribution helps determine how much revenue a given tax policy will raise.

22.6.1. Light and heavy tails#

A distribution $F$ with density $f$ on ${\mathbb{R}}_{+}$ is called heavy-tailed if

We say that a nonnegative random variable $X$ is heavy-tailed if its density is heavy-tailed.

This is equivalent to stating that its moment generating function $m(t):=\mathbb{E}\exp (tX)$ is infinite for all $t>0$.

For example, the log-normal distribution is heavy-tailed because its moment generating function is infinite everywhere on $(0,\mathrm{\infty })$.

The Pareto distribution is also heavy-tailed.

Less formally, a heavy-tailed distribution is one that is not exponentially bounded (i.e. the tails are heavier than the exponential distribution).

A distribution $F$ on ${\mathbb{R}}_{+}$ is called light-tailed if it is not heavy-tailed.

A nonnegative random variable $X$ is light-tailed if its distribution $F$ is light-tailed.

For example, every random variable with bounded support is light-tailed. (Why?)

As another example, if $X$ has the exponential distribution, with cdf $F(x)=1-\exp (-\lambda x)$ for some $\lambda >0$, then its moment generating function is

In particular, $m(t)$ is finite whenever $t<\lambda$, so $X$ is light-tailed.

One can show that if $X$ is light-tailed, then all of its moments are finite.

Conversely, if some moment is infinite, then $X$ is heavy-tailed.

The latter condition is not necessary, however.

For example, the lognormal distribution is heavy-tailed but every moment is finite.