IntermediateEconometrics · Chapter 1About 9 min

A memo says AI tools raise wages, with a very confident interval,and you check what it assumes

A confident memo

The Ministry of Finance of Halcyon. A two-paragraph memo lands on your desk.

1. A
   The adviserWorkers who use AI tools all day earn more per hour. Here’s the slope.
2. A
   The adviserAnd the 95% interval is tiny. Very precise.
3. You
   YouWho’s going to use it?
4. A
   The adviserThe Minister. To decide how much to spend on AI training.
5. H
   Your head of unitBefore it goes up: check what it assumes.

Predict first

Wages vary far more among heavy AI users — some gain a lot, others nothing. What does that do to the memo’s interval?

What OLS assumes

OLS picks the line that minimises the sum of squared residuals. What that line means depends on the error u in:

y_i = \beta_0 + \beta_1 x_i + u_i

For an unbiased slope, four assumptions are enough:

1. linear in the parameters,
2. a random sample,
3. no regressor is an exact mix of the others,
4. the error has mean zero whatever x is: \mathbb{E}[u_i \mid x_i] = 0.

The usual standard error needs one more — homoskedasticity, the same error variance for everyone:

\widehat{\operatorname{se}}(\hat\beta_1) = \sqrt{\frac{\hat\sigma^2}{\sum_i (x_i - \bar x)^2}}

With all five, OLS is the best linear unbiased estimator (Gauss–Markov).

AI data break the fifth one. Step through it.

1. The memo’s line through the survey data.
2. Little AI use: points stay close to the line.
3. Heavy AI use: points spread far apart.
4. The usual formula assumes the same spread everywhere.
5. The robust formula lets the spread grow. That changes the interval, not the line.

The bench

Simulated survey data: y = 1 + x + u. x is the share of working time with AI tools (0 to 1). The error’s spread is e^{h(x - 0.5)}.

• h = 0: same spread for everyone.
• h = 3: heavy users’ errors about 20 times as spread out as non-users’.

Model bench — two standard errors

Challenge. Find the setting where the memo’s usual standard error is exactly right.

Simulated data, the same for every reader. Lines: the exact spread of the estimate and the expected values of the two standard errors for this design; coverage in the readout comes from simulated samples.

As h grows, the robust standard error tracks the real spread. The usual one falls behind. At 200 workers and h = 3: 0.45 against 0.57 — about a fifth too small. Its 95% interval covers the true slope less than 95% of the time.

An AI’s answer

Spot the error

Question to an AI assistant My regression of wages on AI use shows heteroskedasticity. What should I do?

Answer (written for this exercise)

¹ Heteroskedasticity means the variance of the error differs across observations. ² It does not bias the OLS estimate of the slope. ³ It does make the usual standard errors unreliable. ⁴ Robust standard errors fix this by correcting the estimated slope.

Which sentence is wrong?

Sentence 1 Sentence 2 Sentence 3 Sentence 4

What goes back

1. You
   YouSame slope. New standard error — robust. The interval is about a quarter wider.
2. A
   The adviserWhy wider?
3. You
   YouWages vary more among heavy AI users. The old formula assumed they didn’t.
4. H
   Your head of unitGood. Send it up.

Your prediction

You did not make a prediction at the start of the chapter. Here is what it asked:

Wages vary far more among heavy AI users — some gain a lot, others nothing. What does that do to the memo’s interval?

The slope is fine. The usual standard error, which assumes equal spread, is not — on the bench it is about a fifth too small.

Check yourself

The errors are heteroskedastic, but \mathbb{E}[u \mid x] = 0 holds. The OLS slope is:

Biased Unbiased, but the usual standard error is wrong Unbiased, and the usual standard error is right

Which assumption makes OLS unbiased?

\mathbb{E}[u \mid x] = 0 \operatorname{Var}(u \mid x) = \sigma^2 Normally distributed errors

The robust standard error of a slope is 0.56. What is the half-width of the 95% interval, using 1.96? (Two decimals.)

Your answer

Take it with you

This chapter on two pages

The ideas, the formulas, a worked example, four traps — and six questions to try on your own, with answers.

Download the sheet (PDF) 2 pages · 140 KB

Take it to your AI

Want to go further? Paste this into the AI you use. It asks the AI to quiz you, not to hand you answers.

I am learning about heteroskedasticity in regression. A report regresses workers' hourly wages on the share of their working time spent with AI tools, and wages vary much more among heavy AI users than among non-users. The report uses the usual OLS standard errors.
Please act as my tutor. Do not give me the answers. Ask me one question at a time to help me work out:
1. whether the OLS slope is still unbiased,
2. what goes wrong with the usual standard errors, and
3. what heteroskedasticity-robust standard errors change and what they leave unchanged.
Wait for my answer after each question. At the end, tell me what I got right and correct anything I got wrong.

Key points

• OLS is unbiased if: the model is linear, the sample is random, no regressor is an exact mix of others, and \mathbb{E}[u \mid x] = 0.
• The usual standard error also needs homoskedasticity: the same error spread for every x.
• With heteroskedasticity, the slope is still unbiased. The usual standard errors are wrong.
• Residual plots and the Breusch–Pagan or White tests look for it.
• Robust (White) standard errors change the standard error, not the estimate.

The Minister signs off the robust interval. The next file has twenty years of quarterly data. To be continued — Chapter 2: heteroskedasticity and serial correlation.