What If Mathematicians Wrote the Headlines?
A brief look at some popular headlines that defy mathematical principles.
Ben Orlin writes the delightful Math With Bad Drawings blog. He recently posted examples of "What Headlines Would Look Like if We Lived in a Mathematically Literate World," which I recommend highly.
A few of his headlines deserve further discussion. This is not a criticism, although I will disagree at times with some of his reformulations. All of his basic points are on target, and his breezy hit-and-run style adds wit to the wisdom. But some innumeracy issues are deeper than the headline. Here's a look at six of his examples.
1. In one instance, Orlin makes a point about regression.
Our World: Market Rebounds After Assurances From Fed Chair
Mathematically Literate World: Market Rebounds After Regression to the Mean
I agree 100% with the point that explanations of stock market moves immediately after they occur are silly, especially coming from people with no ability to predict the next move, even conditionally, on events. But the reformulation above retains the verb "rebounds" which is a physical analogy that assumes momentum, and adds "regression to the mean." Even the use of the present tense "rebounds" is misleading, as it implies an ongoing situation rather than an observation about the past.
To a very good first approximation, stock prices are a random walk. There is some momentum and some mean reversion at different time scales, but both are subtle effects that can be teased out only by careful analysis of large amounts of data. They are useless for explaining single events. They are also attractive analogies that lead to vast investment errors. Therefore the best headline usually would be two different ones attached to unrelated articles, "Market Went Up Yesterday," and "Assurances from Fed Chair." On the Math With Bad Drawings blog, the rewritten headline has been changed to "Market Rebounds Without Clear Casual Explanation," which is halfway to my version.
Nevertheless, it's sometimes possible to estimate the effect of a single event on stock prices. "Random" does not mean "uncaused." Therefore occasional headlines like the original could be justified. But stock price changes are most strongly correlated with economic news about 18 months in the future, meaning that most of the short-term market moves are caused by slight recalibrations of probability estimates of long-term factors that will appear in future news stories (or, if you prefer, behavioral fluctuations in investor fear and greed). Explaining today's stock market moves by today's news stories, every day, is ridiculous.
On a geekier note, "regression" and "reversion" to the mean are essentially different concepts. Regression is a statistical phenomenon. If yesterday was an exceptionally good day for the stock market, today is likely to be less good than yesterday. Reversion is stronger and occurs (if it does) for economic reasons, not statistical. If yesterday was a good day for the stock market, today is more likely than average to be a bad day. The usual mistake is to observe regression and assume reversion. Orlin does the reverse, positing reversion but labeling it regression.
2. In another example, Orlin rewrites a headline that uses a dollar figure.
Our World: Controversial Program Would Cost $50 Million in Taxpayer Money
Mathematically Literate World: Controversial Program Would Cost 0.0001 Percent of Taxpayer Money
The question here is getting the denominator right. A dollar figure is meaningful for a one-time project whose scope is easily contemplated, say building a bridge. But when discussing a proposal to improve job training programs, the cost is only meaningful in relation to the number of people helped. I think Orlin has in mind macro proposals that affect the entire budget, for which dollar figures should be related to total government spending over the same period.
For example, someone might suggest tinkering with the cost-of-living adjustment methodology in federal programs, and add up the total dollars saved over the entire budget over the next 60 years to make savings look big. Reporting this number as if it were a one-time expenditure for a bridge is silly. But just as often, someone takes a real cash expenditure and divides it by some absurdly large number to make it look small. In my experience, non-quants are apt to err by focusing too much on dollars, while the quant error is to ignore hard dollar figures in favor of model-dependent denominators.
3. Here, the focus is on taxation rates:
Our World: Proposal Would Tax $250,000-Earners at 40%
Mathematically Literate World: Proposal Would Tax $250,000-Earners' Very Last Dollar, and That Dollar Alone, at 40%
The main point is that tax reporting seldom distinguishes properly between marginal and average tax rates. However I think the specific example is not well-chosen. The rewritten headline applies strictly only to someone making exactly $250,001. In fact, the average taxable income of people earning over $250,000 is over $500,000, so the 40% actually applies to more than half of the income earned by this group (the top 2.3% of earners).
Orlin interpreted the original headline as referring to someone making just over $250,000 rather than everyone earning over $250,000. And in support of his interpretation, it is true that most people earning more than $250,000 earn much less than $500,000. I don't know the exact figures, but I do know most of the contribution to the $500,000 average comes from a few very high-income people. Therefore it's likely true that few people would feel the 40% bite on more than a small fraction of their income.
The difference between marginal and average rates is more significant for middle-income households. Their marginal federal income tax rate is 25%, but their average federal income tax paid is 5% of taxable income. If you add in payroll taxes, the figures rise to 33% and 11% respectively.
Unfortunately, this just scratches the surface of distortions in tax reporting. The person who writes the check is generally not the person who pays the tax in economic terms. Employees probably pay most of the employer share of payroll taxes, in the form of lower wages, so toss another 6% on to the average tax rates of middle-income families. Rich people probably pay most of the corporate income tax.
Moreover, using taxable income to decide who is rich and who is not is misleading. A person can have low income because he's finishing up at Harvard Business School with a $250,000/year offer for next year in his pocket, or she's a retired billionaire with a lot of municipal bonds and good tax lawyers. A person can have a high income because he just sold his business he spent a lifetime building, or signed a one-year contract in the National Football League.
4. Is there a lot of interest in electric cars? Orlin's spin on a recent headline suggests an answer:
Our World: Market Share for Electric Cars Triples
Mathematically Literate World: Market Share for Electric Cars Rises to 0.4%
Growth rates are important for things likely to maintain reasonably constant growth rates for an appreciable period, or in other words, for exponential things. When AIDS cases increased 385% in the US from 1981 to 1982, it was an event worthy of a scare headline. It would have been irresponsible to report, "AIDS Cases Rise to 0.0003% of the US Population." On the other hand, there was a 23% increase in fatal dog bites in the US from 2011 to 2012, but the relevant fact is that the number went from 31 to 38. Change "dog" to "vampire" or "zombie" and we're back to something exponential.
Electric car sales are somewhere in between. The growth rate matters because more cars means more manufacturing efficiencies, more charging stations, more mechanics familiar with the vehicles, more consumer acceptance, and other things that will make further growth easier. On the other hand, 0.4% of the market suggests niche sales that may well be near saturation and idiosyncratic data rather than a sustainable growth rate. I think in this case, you need both figures for accuracy and a serious article to explore the details.
5. In the next case, issues of definition are key:
Our World: Still No Scientific Consensus on Global Warming
Mathematically Literate World: Still 90% Scientific Consensus on Global Warming
The problems here are the terms "consensus" and "global warming," both of which contain buried assumptions, and the rewrite does not fix things. You get above 90% agreement to statements like, "The 50 years from 1964-2013 have been warmer than the 50 years from 1914-1963," and, "Human impacts on the environment are substantial enough to affect long-term climate."
It's more complicated when you ask questions like, "What is the probability that the 50 years from 2014-2063 will be warmer than 2013 temperature levels?" or, "Are human behavior and human impacts on the environment understood well enough to say with confidence that policies attempting to reduce carbon emissions will lead to a better overall climate for the next 100 years?" For these, you get different results depending on how you define scientists and how you gather the data. Orlin now regrets including this example due to the number of complaints he got from scientists.
For a simple process dominated by positive feedback, global average temperatures are meaningful, and you can speak unambiguously of trends like "warming" or "cooling." For a complex process with essential non-linearities and negative feedback, plus random noise, different temperatures will go up and down over different time scales. No simple summary of the situation will be accurate.
Similarly, "consensus" can be defined for straightforward questions like, "Does the earth revolve around the sun?" But in a field with lots of complex, nuanced opinions of varying degrees of confidence, you cannot easily define the common denominator beliefs, and you certainly cannot measure them with surveys.
In my experience, people's opinions on issues like cap-and-trade have little to do with their scientific knowledge or beliefs. Support for cap-and-trade is not based on high scientific confidence that the long-term consequences of the legislation can be predicted, but on a political belief that things work better when governments try to manage things for the common good. One specific law might not work as intended but it's better to have a plan and revise it as needed than to do nothing. Opposition is not based on denial that current temperatures are higher than the recent past or that humans can affect the climate, but on a political belief that massive programs attacking badly understood problems make things worse, not better. The intended benefits are highly uncertain, while the unintended consequences, economic costs, reduction in freedom, cronyism and corruption are guaranteed. The unpleasant name-calling over "scientific consensus" is a consequence of views of government, and has little to do with either science or consensus.
6. In our final example, the emphasis is on using reliable models.
Our World: Economist: Eliminate Minimum Wage to Create Jobs, Improve Economy
Mathematically Literate World: Economist: Eliminate Minimum Wage, then Pray Our Model Has Some Basis in Reality
I wholeheartedly agree with the basic point that people throw around policy recommendations that affect millions of people and billions of dollars, with staggeringly weak theoretical or empirical support. However, I think the example is unfair in three minor ways.
First of all, economists who are free to speak for themselves and who are not pure ideologues, generally have a rational model that is not contradicted by empirical data that supports their views. That is, their positions are self-consistent and mathematically possible. Yes, that is a very low standard, but it's higher than most non-economists. "Politician: Eliminate Minimum Wage, Consequences Not My Interest or Problem," is a fairer criticism.
Second, no one claims that eliminating our current minimum wage would have a significant effect on the economy. The minimum wage level is too low to matter much. If a million people took part-time jobs averaging 200 total hours at $5 per hour, that's a billion dollars of total wages in a $16 trillion economy. People who argue for eliminating minimum wage laws do so on the basis of the positive effect for some of these workers, not the contribution to high-quality jobs or economic growth. Other people make the libertarian argument that if one consenting adult wants to do work for another consenting adult for $5 per hour, it's no one else's business.
Third, you don't need a model to argue for the obvious first-order effects of a change. If we passed a law that no car could be sold for less than $20,000, we would expect car sales to go down. Some low-quality and used cars would become unsalable, and some potential car buyers would be priced out of the market. Fewer cars would be produced, because one segment of the purchaser market would go away, and cars would be less desirable without the ability to resell them. The economy would suffer.
Now people aren't cars. A $10,000 car can't go back to school to become a $30,000 car. Cars don't spend their purchase prices on rent and food. Cars don't run better if owners pay more to buy them. Therefore it's possible to argue that second-order effects of minimum wage laws increase total wages paid (a small or zero decline in employment combined with increased average wages) and help the economy. But you need a model and empirical evidence to do it. If your policy is adopted, you then have to pray that your model has some basis in reality, because otherwise you will have caused a lot of harm from the first-order effects.
For the second and third reason, I like this example better with the original headline, "Double the Minimum Wage to Increase Welfare of Low-Skill Workers, Improve Economy." People really do say this, and they should have a very good model to support it -- and be exceptional at praying. Most economists who propose a higher minimum wage want smaller and gradual increases, with careful monitoring of effects on employment. Those more modest proposals also should be supported by models, but because the potential downside is less, they don't require as much faith in prayer.
I encourage you to read the original article, as well as other posts from the Math With Bad Drawings blog. I claim the drawings aren't bad; they are playfully minimal with striking composition, expression, and energy, although they lack realism and detail. Orlin insists they are actually bad, since he cannot draw anything more complex.
In either event, the world needs a lot more math expressed with style and wit, and applied to real life. It's okay if you disagree on a few points, as I do. Math is about thinking, not dogma.