modification

Author: shlff

lectures/heavy_tails.md

@@ -125,7 +125,7 @@ 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...
+But this is only true when distributions have light tails.
 
 
 ### When are light tails valid?
@@ -400,7 +400,7 @@ Notice how extreme outcomes are more common.
 
 ### Counter CDFs
 
-For nonnegative random varialbes, one way to visualize the difference between
+For nonnegative random variables, one way to visualize the difference between
 light and heavy tails is to look at the 
 **counter CDF** (CCDF).
 
@@ -424,7 +424,7 @@ $$ G_P(x) = x^{- \alpha} $$
 
 This function goes to zero as $x \to \infty$, but much slower than $G_E$.
 
-Here's a plot that illustrates how $G_E$ goes to zero faster that $G_P$.
+Here's a plot that illustrates how $G_E$ goes to zero faster than $G_P$.
 
 ```{code-cell} ipython3
 x = np.linspace(1.5, 100, 1000)
@@ -452,12 +452,12 @@ 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
-visualations --- we return to this point below.
+visualisations --- we return to this point below.
 
 
 ### Empirical CCDFs
 
-The sample countpart of the CCDF function is the **empirical CCDF**.
+The sample counterpart of the CCDF function is the **empirical CCDF**.
 
 Given a sample $x_1, \ldots, x_n$, the empirical CCDF is given by
 
@@ -529,7 +529,7 @@ 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** $\alpha$ if
+has a **Pareto tail** with **tail index** $\alpha$.
 
 Notice that every Pareto distribution with tail index $\alpha$ 
 has a **Pareto tail** with **tail index** $\alpha$.
@@ -548,7 +548,7 @@ As mentioned above, heavy tails are pervasive in economic data.
 
 In fact power laws seem to be very common as well.
 
-We now illustrate this by showing the empirical CCDF of 
+We now illustrate this by showing the empirical CCDF of heavy tails.
 
 All plots are in log-log, so that a power law shows up as a linear log-log
 plot, at least in the upper tail.
@@ -642,7 +642,7 @@ def extract_wb(varlist=['NY.GDP.MKTP.CD'],
 
 ### Firm size
 
-Here is a plot of the firm size distribution taken from Forbes Global 2000.
+Here is a plot of the firm size distribution for the largest 500 firms in 2020 taken from Forbes Global 2000.
 
 ```{code-cell} ipython3
 :tags: [hide-input]
@@ -652,46 +652,39 @@ 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'])[0:500], ax, label=label, add_reg_line=True)
+empirical_ccdf(np.asarray(d['Market Value'])[:top], ax, label=label, add_reg_line=True)
 
 plt.show()
 ```
 
 ### City size
 
-Here is a plot of the city size distribution for the US, where size is
-measured by population.
+Here are plots of the city size distribution for the US and brazil in 2023 from world population review.
+
+The size is measured by population.
 
 ```{code-cell} ipython3
 :tags: [hide-input]
 
-df_cs_us = pd.read_csv('https://raw.githubusercontent.com/QuantEcon/high_dim_data/update_csdata/cross_section/cities_us.txt', delimiter="\t", header=None)
-df_cs_us = df_cs_us[[0, 3]]
-df_cs_us.columns = 'rank', 'pop'
-x = np.asarray(df_cs_us['pop'])
-citysize = []
-for i in x:
-    i = i.replace(",", "")
-    citysize.append(int(i))
-df_cs_us['pop'] = citysize
-df_cs_br = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/update_csdata/cross_section/cities_brazil.csv', delimiter=",", header=None)
-df_cs_br.columns = df_cs_br.iloc[0]
-df_cs_br = df_cs_br[1:401]
-df_cs_br = df_cs_br.astype({"pop2023": float})
+# 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/update_csdata/cross_section/cities_us.csv')
+df_cs_br = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/update_csdata/cross_section/cities_brazil.csv')
 
 fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))
-empirical_ccdf(np.asarray(df_cs_us['pop']), axes[0], label="US", add_reg_line=True)
+
+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()
 ```
 
 ### Wealth
 
-Here is a plot of the upper tail of the wealth distribution.
+Here is a plot of the upper tail (top 500) of the wealth distribution.
 
-The data is from the Forbes billionaires list.
+The data is from the Forbes Billionaires list in 2020.
 
 ```{code-cell} ipython3
 :tags: [hide-input]
@@ -710,10 +703,11 @@ 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))
-    if len(z) <= 500:    # cut-off number: top 500
-        z = z[0:500]
+    top = 500           # cut-off number: top 500
+    if len(z) <= top:    
+        z = z[:top]
 
-    empirical_ccdf(z[0:500], axs[i], label=c, xlabel='log wealth', add_reg_line=True)
+    empirical_ccdf(z[:top], axs[i], label=c, xlabel='log wealth', add_reg_line=True)
     
 fig.tight_layout()
 
@@ -742,6 +736,8 @@ df_gdp1 = extract_wb(varlist=variable_code,
 ```
 
 ```{code-cell} ipython3
+:tags: [hide-input]
+
 fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))
 
 for name, ax in zip(variable_names, axes):
@@ -856,7 +852,7 @@ You will find that
 
 Diversification reduces risk, as expected.
 
-But there is a hidden assumption here: the variance is of returns is finite.
+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.
@@ -1071,9 +1067,6 @@ Present discounted value of tax revenue will be estimated by
 1. multiplying by the tax rate, and
 1. summing the results with discounting to obtain present value.
 
-If $X$ has the Pareto distribution, then there are positive constants $\bar x$
-and $\alpha$ such that
-
 The Pareto distribution is assumed to take the form {eq}`pareto` with $\bar x = 1$ and $\alpha = 1.05$.
 
 (The value the tail index $\alpha$ is plausible given the data {cite}`gabaix2016power`.)