heavy_tails.md identifies power laws only visually — add a goodness-of-fit test

heavy_tails.md defines power laws and Pareto tails formally, then identifies them by eye: "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." The lecture builds empirical CCDFs and Q-Q plots for firm size and city size, and stops there.

It has no goodness-of-fit test, no tail-index estimator, and no note that log-log linearity is a weak diagnostic. Searching the lecture for "goodness", "test for", "Hill estimator", "KS test", "Kolmogorov" or "Clauset" returns nothing.

Why this matters

Exercise ht_ex4 asks the reader to compare a Pareto distribution against a mean-and-median-matched lognormal, for the present discounted value of corporate tax revenue, and to observe the difference. The lecture therefore poses the Pareto-versus-lognormal question and gives the reader no way to settle it from data. The same comparison appears, also unresolved, as Exercise 2.2.10 of Economic Networks.

Eyeballing a log-log plot for straightness is precisely the practice the goodness-of-fit literature exists to caution against, so teaching only the visual method leaves readers with a diagnostic that looks more reliable than it is.

Suggested scope

A short section, not a new lecture:

• Estimating the tail index — the Hill estimator, and its sensitivity to where the tail is deemed to start
• Testing the hypothesis — the Clauset–Shalizi–Newman KS-based procedure is the standard reference and has a widely used implementation
• A sentence of honesty in the existing visual material, noting that log-log linearity is suggestive rather than conclusive

Enough that a reader can answer "is this actually a power law?" instead of "does this look straight?". Wiring it back to ht_ex4 would close the loop on an exercise that currently ends in an observation rather than an answer.

Spun out of QuantEcon/meta#141, which collected it as the one concrete deliverable inside a broader proposal.

Start with lectures/heavy_tails.md, especially the empirical CCDF, Q-Q plot, and ht_ex4 material. Research the Hill estimator and the Clauset–Shalizi–Newman KS-based procedure, then add a short section, qualify the existing visual diagnostic, and connect the method back to ht_ex4 so readers can assess Pareto versus lognormal data.