Lesson preview · Working with evidence
Testing causal claims and reading a regression
~8 min · Free to read
Imagine a minister in Country X with five years of figures. In quarters (three-month periods) when shoppers feel confident, the economy grows faster. Growth here means real GDP growth: the rise in gross domestic product (GDP), the value of everything the country produces, once price rises are taken out. “So let’s cheer people up,” she says, “and growth will follow.” Two things that move together show a correlation. She claims more: that confidence causes growth. Is she right?
Key Term
Three ways a pattern can fool you
Before accepting a cause, rule out three other stories.
- Reverse causation. The cause runs the other way.
- A third variable. Something else drives both.
- Chance. In a small sample (only a few cases), patterns can turn up by luck. More data usually make chance patterns fade; real ones stay.
All three patterns were strong: R², a score from 0 to 1 for how closely the dots follow a straight line, was 0.80, 0.83 and 0.86. Yet none showed a cause. Most likely, crime drove police numbers, age drove shoe size and reading, and the ice-cream link was luck: over 30 seasons, R² fell to 0.01. A pattern shows that things move together, not why.
To test a cause cleanly, researchers toss a coin to split people into two groups. They change one thing for one group and compare. When that’s impossible, economists look for a natural experiment: a chance event or a change in the law that splits similar people almost as a coin toss would.
In April 1992, the US state of New Jersey raised its minimum wage, the lowest hourly pay allowed by law, from 4.25 to 5.05 dollars. Simple supply and demand predicts that a minimum wage above the market wage costs jobs. Next door, Pennsylvania kept its minimum wage the same, and fast-food restaurants on each side of the border were much alike. So Pennsylvania showed what would probably have happened in New Jersey without the rise.
Economists David Card and Alan Krueger surveyed 410 of these restaurants before and after the rise. They compared the change in jobs on each side and found no sign that the rise cut jobs. A 1995 study of separately collected payroll records found a small fall compared with Pennsylvania. Card and Krueger’s reply, using official employment records, found none. The debate ran for years. In 2021, Card shared the Nobel memorial prize in economics, partly for this work.
Key Term
Reading a fitted line
Plot 20 students as dots (invented data): the more hours they revise, the higher they score. A fitted line (a regression line, or line of best fit) is the straight line closest to all the dots.
- The dependent variable (Y), here the test score, is the one you’re explaining. On a scatter plot it goes on the vertical axis. (Supply-and-demand diagrams break this rule by putting price there.)
- The independent variable (X), here hours of revision, is the one that explains it. It goes on the horizontal axis.
- The slope says: for each extra unit of X, Y changes by about this much. Here, each extra hour goes with about 4.8 more points.
- r, the correlation coefficient, runs from −1 to 1. Its sign gives the direction: above 0, Y tends to rise as X rises; below 0, it tends to fall. Its size gives the strength: near 1 or −1, the dots hug a straight line; near 0, there’s no straight-line link.
- R² is r squared, from 0 to 1: the share of the variation in Y (its ups and downs) that the line explains.
R² fell from 0.90 to 0.31 to 0.01 as the dots spread out: it measures how well the line fits, nothing more. Even 0.90 doesn’t prove that revision raises scores. Keen students may revise more and also work harder in class.
The minister’s figures back her only in part. R² = 0.49: the line explains about half of the variation in growth, so other things drive growth too. Nor can the line say which way the cause runs: people may feel confident because the economy is already growing.
Trap · Common Misconception
"R² = 0.49 means a 49% chance"
R² is not a probability, and it isn’t proof of a cause. An R² of 0.49 doesn’t mean a 49% chance that confidence raises growth. Shoe size and reading had an R² of 0.83, yet bigger feet don’t help anyone read.
Tip
Writing about a chart
Quote the value, name the dependent variable, and state one limit. For the shops: “R² = 0.31, so the line explains about a third of the variation in sales, the dependent variable. But busy shops may just spend more on adverts: the cause could run backwards.”
Check yourself · no marks
A scatter plot has r = −0.7. What is R², and what does the minus sign tell you?
Commit to one first. Guessing and being wrong beats reading the answer cold.
Practice · 1 / 6
Imagine figures for 40 cafés in one city: cafés with more staff serve more customers a day. The owner of a quiet café decides to hire two more staff to bring in customers. What is the main problem with her reasoning?
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