What does correlation have to do with causation?

JDN 2457345

I’ve been thinking of expanding the topics of this blog into some basic statistics and econometrics. It has been said that there are “Lies, damn lies, and statistics”; but in fact it’s almost the opposite—there are truths, whole truths, and statistics. Almost everything in the world that we know—not merely guess, or suppose, or intuit, or believe, but actually know, with a quantifiable level of certainty—is done by means of statistics. All sciences are based on them, from physics (when they say the Higgs discovery is a “5-sigma event”, that’s a statistic) to psychology, ecology to economics. Far from being something we cannot trust, they are in a sense the only thing we can trust.

The reason it sometimes feels like we cannot trust statistics is that most people do not understand statistics very well; this creates opportunities for both accidental confusion and willful distortion. My hope is therefore to provide you with some of the basic statistical knowledge you need to combat the worst distortions and correct the worst confusions.

I wasn’t quite sure where to start on this quest, but I suppose I have to start somewhere. I figured I may as well start with an adage about statistics that I hear commonly abused: “Correlation does not imply causation.”

Taken at its original meaning, this is definitely true. Unfortunately, it can be easily abused or misunderstood.

In its original meaning, the formal sense of the word “imply” meaning logical implication, to “imply” something is an extremely strong statement. It means that you logically entail that result, that if the antecedent is true, the consequent must be true, on pain of logical contradiction. Logical implication is for most practical purposes synonymous with mathematical proof. (Unfortunately, it’s not quite synonymous, because of things like Gödel’s incompleteness theorems and Löb’s theorem.)

And indeed, correlation does not logically entail causation; it’s quite possible to have correlations without any causal connection whatsoever, simply by chance. One of my former professors liked to brag that from 1990 to 2010 whether or not she ate breakfast had a statistically significant positive correlation with that day’s closing price for the Dow Jones Industrial Average.

How is this possible? Did my professor actually somehow influence the stock market by eating breakfast? Of course not; if she could do that, she’d be a billionaire by now. And obviously the Dow’s price at 17:00 couldn’t influence whether she ate breakfast at 09:00. Could there be some common cause driving both of them, like the weather? I guess it’s possible; maybe in good weather she gets up earlier and people are in better moods so they buy more stocks. But the most likely reason for this correlation is much simpler than that: She tried a whole bunch of different combinations until she found two things that correlated. At the usual significance level of 0.05, on average you need to try about 20 combinations of totally unrelated things before two of them will show up as correlated. (My guess is she used a number of different stock indexes and varied the starting and ending year. That’s a way to generate a surprisingly large number of degrees of freedom without it seeming like you’re doing anything particularly nefarious.)

But how do we know they aren’t actually causally related? Well, I suppose we don’t. Especially if the universe is ultimately deterministic and nonlocal (as I’ve become increasingly convinced by the results of recent quantum experiments), any two data sets could be causally related somehow. But the point is they don’t have to be; you can pick any randomly-generated datasets, pair them up in 20 different ways, and odds are, one of those ways will show a statistically significant correlation.

All of that is true, and important to understand. Finding a correlation between eating grapefruit and getting breast cancer, or between liking bitter foods and being a psychopath, does not necessarily mean that there is any real causal link between the two. If we can replicate these results in a bunch of other studies, that would suggest that the link is real; but typically, such findings cannot be replicated. There is something deeply wrong with the way science journalists operate; they like to publish the new and exciting findings, which 9 times out of 10 turn out to be completely wrong. They never want to talk about the really important and fascinating things that we know are true because we’ve been confirming them over hundreds of different experiments, because that’s “old news”. The journalistic desire to be new and first fundamentally contradicts the scientific requirement of being replicated and confirmed.

So, yes, it’s quite possible to have a correlation that tells you absolutely nothing about causation.

But this is exceptional. In most cases, correlation actually tells you quite a bit about causation.

And this is why I don’t like the adage; “imply” has a very different meaning in common speech, meaning merely to suggest or evoke. Almost everything you say implies all sorts of things in this broader sense, some more strongly than others, even though it may logically entail none of them.

Correlation does in fact suggest causation. Like any suggestion, it can be overridden. If we know that 20 different combinations were tried until one finally yielded a correlation, we have reason to distrust that correlation. If we find a correlation between A and B but there is no logical way they can be connected, we infer that it is simply an odd coincidence.

But when we encounter any given correlation, there are three other scenarios which are far more likely than mere coincidence: A causes B, B causes A, or some other factor C causes A and B. These are also not mutually exclusive; they can all be true to some extent, and in many cases are.

A great deal of work in science, and particularly in economics, is based upon using correlation to infer causation, and has to be—because there is simply no alternative means of approaching the problem.

Yes, sometimes you can do randomized controlled experiments, and some really important new findings in behavioral economics and development economics have been made this way. Indeed, much of the work that I hope to do over the course of my career is based on randomized controlled experiments, because they truly are the foundation of scientific knowledge. But sometimes, that’s just not an option.

Let’s consider an example: In my master’s thesis I found a strong correlation between the level of corruption in a country (as estimated by the World Bank) and the proportion of that country’s income which goes to the top 0.01% of the population. Countries that have higher levels of corruption also tend to have a larger proportion of income that accrues to the top 0.01%. That correlation is a fact; it’s there. There’s no denying it. But where does it come from? That’s the real question.

Could it be pure coincidence? Well, maybe; but when it keeps showing up in several different models with different variables included, that becomes unlikely. A single p < 0.05 will happen about 1 in 20 times by chance; but five in a row should happen less than 1 in 1 million times (assuming they’re independent, which, to be fair, they usually aren’t).

Could it be some artifact of the measurement methods? It’s possible. In particular, I was concerned about the possibility of Halo Effect, in which people tend to assume that something which is better (or worse) in one way is automatically better (or worse) in other ways as well. People might think of their country as more corrupt simply because it has higher inequality, even if there is no real connection. But it would have taken a very large halo bias to explain this effect.

So, does corruption cause income inequality? It’s not hard to see how that might happen: More corrupt individuals could bribe leaders or exploit loopholes to make themselves extremely rich, and thereby increase inequality.

Does inequality cause corruption? This also makes some sense, since it’s a lot easier to bribe leaders and manipulate regulations when you have a lot of money to work with in the first place.

Does something else cause both corruption and inequality? Also quite plausible. Maybe some general cultural factors are involved, or certain economic policies lead to both corruption and inequality. I did try to control for such things, but I obviously couldn’t include all possible variables.

So, which way does the causation run? Unfortunately, I don’t know. I tried some clever statistical techniques to try to figure this out; in particular, I looked at which tends to come first—the corruption or the inequality—and whether they could be used to predict each other, a method called Granger causality. Those results were inconclusive, however. I could neither verify nor exclude a causal connection in either direction. But is there a causal connection? I think so. It’s too robust to just be coincidence. I simply don’t know whether A causes B, B causes A, or C causes A and B.

Imagine trying to do this same study as a randomized controlled experiment. Are we supposed to create two societies and flip a coin to decide which one we make more corrupt? Or which one we give more income inequality? Perhaps you could do some sort of experiment with a proxy for corruption (cheating on a test or something like that), and then have unequal payoffs in the experiment—but that is very far removed from how corruption actually works in the real world, and worse, it’s prohibitively expensive to make really life-altering income inequality within an experimental context. Sure, we can give one participant $1 and the other $1,000; but we can’t give one participant $10,000 and the other $10 million, and it’s the latter that we’re really talking about when we deal with real-world income inequality. I’m not opposed to doing such an experiment, but it can only tell us so much. At some point you need to actually test the validity of your theory in the real world, and for that we need to use statistical correlations.

Or think about macroeconomics; how exactly are you supposed to test a theory of the business cycle experimentally? I guess theoretically you could subject an entire country to a new monetary policy selected at random, but the consequences of being put into the wrong experimental group would be disastrous. Moreover, nobody is going to accept a random monetary policy democratically, so you’d have to introduce it against the will of the population, by some sort of tyranny or at least technocracy. Even if this is theoretically possible, it’s mind-bogglingly unethical.

Now, you might be thinking: But we do change real-world policies, right? Couldn’t we use those changes as a sort of “experiment”? Yes, absolutely; that’s called a quasi-experiment or a natural experiment. They are tremendously useful. But since they are not truly randomized, they aren’t quite experiments. Ultimately, everything you get out of a quasi-experiment is based on statistical correlations.

Thus, abuse of the adage “Correlation does not imply causation” can lead to ignoring whole subfields of science, because there is no realistic way of running experiments in those subfields. Sometimes, statistics are all we have to work with.

This is why I like to say it a little differently:

Correlation does not prove causation. But correlation definitely can suggest causation.

Why the Republican candidates like flat income tax—and we really, really don’t

JDN 2456160 EDT 13:55.

The Republican Party is scrambling to find viable Presidential candidates for next year’s election. The Democrats only have two major contenders: Hillary Clinton looks like the front-runner (and will obviously have the most funding), but Bernie Sanders is doing surprisingly well, and is particularly refreshing because he is running purely on his principles and ideas. He has no significant connections, no family dynasty (unlike Jeb Bush and, again, Hillary Clinton) and not a huge amount of wealth (Bernie’s net wealth is about $500,000, making him comfortably upper-middle class; compare to Hillary’s $21.5 million and her husband’s $80 million); but he has ideas that resonate with people. Bernie Sanders is what politics is supposed to be. Clinton’s campaign will certainly raise more than his; but he has already raised over $4 million, and if he makes it to about $10 million studies suggest that additional spending above that point is largely negligible. He actually has a decent chance of winning, and if he did it would be a very good sign for the future of America.

But the Republican field is a good deal more contentious, and the 19 candidates currently running have been scrambling to prove that they are the most right-wing in order to impress far-right primary voters. (When the general election comes around, whoever wins will of course pivot back toward the center, changing from, say, outright fascism to something more like reactionism or neo-feudalism. If you were hoping they’d pivot so far back as to actually be sensible center-right capitalists, think again; Hillary Clinton is the only one who will take that role, and they’ll go out of their way to disagree with her in every way they possibly can, much as they’ve done with Obama.) One of the ways that Republicans are hoping to prove their right-wing credentials is by proposing a flat income tax and eliminating the IRS.

Unlike most of their proposals, I can see why many people think this actually sounds like a good idea. It would certainly dramatically reduce bureaucracy, and that’s obviously worthwhile since excess bureaucracy is pure deadweight loss. (A surprising number of economists seem to forget that government does other things besides create excess bureaucracy, but I must admit it does in fact create excess bureaucracy.)

Though if they actually made the flat tax rate 20% or even—I can’t believe this is seriously being proposed—10%, there is no way the federal government would have enough revenue. The only options would be (1) massive increases in national debt (2) total collapse of government services—including their beloved military, mind you, or (3) directly linking the Federal Reserve quantitative easing program to fiscal policy and funding the deficit with printed money. Of these, 3 might not actually be that bad (it would probably trigger some inflation, but actually we could use that right now), but it’s extremely unlikely to happen, particularly under Republicans. In reality, after getting a taste of 2, we’d clearly end up with 1. And then they’d be complaining about the debt and clamor for more spending cuts, more spending cuts, ever more spending cuts, but there would simply be no way to run a functioning government on 10% of GDP in anything like our current system. Maybe you could do it on 20%—maybe—but we currently spend more like 35%, and that’s already a very low amount of spending for a First World country. The UK is more typical at 47%, while Germany is a bit low at 44%; Sweden spends 52% and France spends a whopping 57%. Anyone who suggests we cut government spending from 35% to 20% needs to explain which 3/7 of government services are going to immediately disappear—not to mention which 3/7 of government employees are going to be immediately laid off.

And then they want to add investment deductions; in general investment deductions are a good thing, as long as you tie them to actual investments in genuinely useful things like factories and computer servers. (Or better yet, schools, research labs, or maglev lines, but private companies almost never invest in that sort of thing, so the deduction wouldn’t apply.) The kernel of truth in the otherwise ridiculous argument that we should never tax capital is that taxing real investment would definitely be harmful in the long run. As I discussed with Miles Kimball (a cognitive economist at Michigan and fellow econ-blogger I hope to work with at some point), we could minimize the distortionary effects of corporate taxes by establishing a strong deduction for real investment, and this would allow us to redistribute some of this enormous wealth inequality without dramatically harming economic growth.

But if you deduct things that aren’t actually investments—like stock speculation and derivatives arbitrage—then you reduce your revenue dramatically and don’t actually incentivize genuinely useful investments. This is the problem with our current system, in which GE can pay no corporate income tax on $108 billion in annual profit—and you know they weren’t using all that for genuinely productive investment activities. But then, if you create a strong enforcement system for ensuring it is real investment, you need bureaucracy—which is exactly what the flat tax was claimed to remove. At the very least, the idea of eliminating the IRS remains ridiculous if you have any significant deductions.

Thus, the benefits of a flat income tax are minimal if not outright illusory; and the costs, oh, the costs are horrible. In order to have remotely reasonable amounts of revenue, you’d need to dramatically raise taxes on the majority of people, while significantly lowering them on the rich. You would create a direct transfer of wealth from the poor to the rich, increasing our already enormous income inequality and driving millions of people into poverty.

Thus, it would be difficult to more clearly demonstrate that you care only about the interests of the top 1% than to propose a flat income tax. I guess Mitt Romney’s 47% rant actually takes the cake on that one though (Yes, all those freeloading… soldiers… and children… and old people?).

Many Republicans are insisting that a flat tax would create a surge of economic growth, but that’s simply not how macroeconomics works. If you steeply raise taxes on the majority of people while cutting them on the rich, you’ll see consumer spending plummet and the entire economy will be driven into recession. Rich people simply don’t spend their money in the same way as the rest of us, and the functioning of the economy depends upon a continuous flow of spending. There is a standard neoclassical economic argument about how reducing spending and increasing saving would lead to increased investment and greater prosperity—but that model basically assumes that we have a fixed amount of stuff we’re either using up or making more stuff with, which is simply not how money works; as James Kroeger cogently explains on his blog “Nontrivial Pursuits”, money is created as it is needed; investment isn’t determined by people saving what they don’t spend. Indeed, increased consumption generally leads to increased investment, because our economy is currently limited by demand, not supply. We could build a lot more stuff, if only people could afford to buy it.

And that’s not even considering the labor incentives; as I already talked about in my previous post on progressive taxation, there are two incentives involved when you increase someone’s hourly wage. On the one hand, they get paid more for each hour, which is a reason to work; that’s the substitution effect. But on the other hand, they have more money in general, which is a reason they don’t need to work; that’s the income effect. Broadly speaking, the substitution effect dominates at low incomes (about $20,000 or less), the income effect dominates at high incomes (about $100,000 or more), and the two effects cancel out at moderate incomes. Since a tax on your income hits you in much the same way as a reduction in your wage, this means that raising taxes on the poor makes them work less, while raising taxes on the rich makes them work more. But if you go from our currently slightly-progressive system to a flat system, you raise taxes on the poor and cut them on the rich, which would mean that the poor would work less, and the rich would also work less! This would reduce economic output even further. If you want to maximize the incentive to work, you want progressive taxes, not flat taxes.

Flat taxes sound appealing because they are so simple; even the basic formula for our current tax rates is complicated, and we combine it with hundreds of pages of deductions and credits—not to mention tens of thousands of pages of case law!—making it a huge morass of bureaucracy that barely anyone really understands and corporate lawyers can easily exploit. I’m all in favor of getting rid of that; but you don’t need a flat tax to do that. You can fit the formula for a progressive tax on a single page—indeed, on a single line: r = 1 – I^-p

That’s it. It’s simple enough to be plugged into any calculator that is capable of exponents, not to mention efficiently implemented in Microsoft Excel (more efficiently than our current system in fact).

Combined with that simple formula, you could list all of the sensible deductions on a couple of additional pages (business investments and educational expenses, mostly—poverty should be addressed by a basic income, not by tax deductions on things like heating and housing, which are actually indirect corporate subsidies), along with a land tax (one line: $3000 per hectare), a basic income (one more line: $8,000 per adult and $4,000 per child), and some additional excise taxes on goods with negative externalities (like alcohol, tobacco, oil, coal, and lead), with a line for each; then you can provide a supplementary manual of maybe 50 pages explaining the detailed rules for applying each of those deductions in unusual cases. The entire tax code should be readable by an ordinary person in a single sitting no longer than a few hours. That means no more than 100 pages and no more than a 7th-grade reading level.

Why do I say this? Isn’t that a ridiculous standard? No, it is a Constitutional imperative. It is a fundamental violation of your liberty to tax you according to rules you cannot reasonably understand—indeed, bordering on Kafkaesque. While this isn’t taxation without representation—we do vote for representatives, after all—it is something very much like it; what good is the ability to change rules if you don’t even understand the rules in the first place? Nor would it be all that difficult: You first deduct these things from your income, then plug the result into this formula.

So yes, I absolutely agree with the basic principle of tax reform. The tax code should be scrapped and recreated from scratch, and the final product should be a primary form of only a few pages combined with a supplementary manual of no more than 100 pages. But you don’t need a flat tax to do that, and indeed for many other reasons a flat tax is a terrible idea, particularly if the suggested rate is 10% or 15%, less than half what we actually spend. The real question is why so many Republican candidates think that this will appeal to their voter base—and why they could actually be right about that.

Part of it is the entirely justified outrage at the complexity of our current tax system, and the appealing simplicity of a flat tax. Part of it is the long history of American hatred of taxes; we were founded upon resisting taxes, and we’ve been resisting taxes ever since. In some ways this is healthy; taxes per se are not a good thing, they are a bad thing, a necessary evil.

But those two things alone cannot explain why anyone would advocate raising taxes on the poorest half of the population while dramatically cutting them on the top 1%. If you are opposed to taxes in general, you’d cut them on everyone; and if you recognize the necessity of taxation, you’d be trying to find ways to minimize the harm while ensuring sufficient tax revenue, which in general means progressive taxation.

To understand why they would be pushing so hard for flat taxes, I think we need to say that many Republicans, particularly those in positions of power, honestly do think that rich people are better than poor people and we should always give more to the rich and less to the poor. (Maybe it’s partly halo effect, in which good begets good and bad begets bad? Or maybe just world theory, the ingrained belief that the world is as it ought to be?)

Romney’s 47% rant wasn’t an exception; it was what he honestly believes, what he says when he doesn’t know he’s on camera. He thinks that he earned every penny of his $250 million net wealth; yes, even the part he got from marrying his wife and the part he got from abusing tax laws, arbitraging assets and liquidating companies. He thinks that people who live on $4,000 or even $400 a year are simply lazy freeloaders, who could easily work harder, perhaps do some arbitrage and liquidation of their own (check out these alleged “rags to riches” stories including the line “tried his hand at mortgage brokering”), but choose not to, and as a result deserve what they get. (It’s important to realize just how bizarre this moral attitude truly is; even if I thought you were the laziest person on Earth, I wouldn’t let you starve to death.) He thinks that the social welfare programs which have reduced poverty but never managed to eliminate it are too generous—if he even thinks they should exist at all. And in thinking these things, he is not some bizarre aberration; he is representing an entire class of people, nearly all of whom vote Republican.

The good news is, these people are still in the minority. They hold significant sway over the Republican primary, but will not have nearly as much impact in the general election. And right now, the Republican candidates are so numerous and so awful that I have trouble seeing how the Democrats could possibly lose. (But please, don’t take that as a challenge, you guys.)