Advanced Econometrics · Chapter 1

One number

The Governor’s note is three lines long.

The GovernorSince firms adopted AI tools, productivity has jumped.
The GovernorThe models need to know how persistent productivity shocks are. I need a number — and how sure we are.
Ilse VargaHere’s the series. Two hundred quarters, fifty years. AI arrived in the last twenty.
YouUse all 200, or only the 20 since AI?
Ilse VargaThat’s your first decision. Estimate $\rho$ .

Predict first

Estimate $\rho$ from all 200 quarters, or only the 20 since AI. How much wider is the 95% interval with 20?

The macro team’s process:

z_t = \rho \, z_{t-1} + \varepsilon_t, \qquad \varepsilon_t \sim N(0, \sigma^2)

Asking the data

Maximum likelihood turns the model around. For given $\rho$ and $\sigma^2$ , the model says how likely the data are. Read the other way: which $\rho$ fits the data best?

Given the first observation, the log-likelihood is:

\ell(\rho, \sigma^2) = -\frac{T}{2} \ln (2\pi\sigma^2) - \frac{1}{2\sigma^2} \sum_{t=1}^{T} (z_t - \rho z_{t-1})^2

Its maximum is the least-squares regression of today on yesterday:

\hat\rho = \frac{\sum_t z_t z_{t-1}}{\sum_t z_{t-1}^2}, \qquad \widehat{\operatorname{se}}(\hat\rho) = \sqrt{\hat\sigma^2 \Big/ \textstyle\sum_t z_{t-1}^2}

How sure comes from the shape of the curve. Step through it.

The likelihood of each $\rho$ , using 200 quarters.
Its peak is the estimate: 0.913.
Values within 1.92 of the peak: the 95% interval, 0.86–0.97.
Now only 20 quarters. The curve is much flatter.
Interval 0.70–1.10. It includes $\rho = 1$ — a boost that never fades.

Why 1.92? Twice the drop from the peak is the likelihood-ratio statistic, about $\chi^2_1$ . Its 5% cut-off is 3.84, and $3.84 / 2 = 1.92$ .

The bench

Model bench — the likelihood curve

Challenge. Find a sample length at which the data rule out ρ = 1 — a boost that never fades.

A simulated series for the fictional island, the same for every reader. The first T quarters are used; the likelihood is conditional on the first observation.

20 quarters: interval 0.70–1.10. 200 quarters: estimate 0.913, standard error 0.029, interval 0.86–0.97.

An AI’s answer

Spot the error

Question to an AI assistant I estimated the persistence of productivity by maximum likelihood on 200 quarters and got 0.913. What does that tell me?

Answer (written for this exercise)

¹ Maximum likelihood picks the parameter value that makes the observed data most probable. ² For a Gaussian AR(1), the MLE of the persistence equals the least-squares estimate. ³ Its standard error shrinks roughly with the square root of the sample size. ⁴ So with 200 quarters the estimate is exact, and the true persistence must be 0.913.

Which sentence is wrong? Sentence 1 Sentence 2 Sentence 3 Sentence 4

Friday

YouFifty years of data: about 0.9. The interval stays below one.
YouOnly the five years since AI: we can’t rule out that the boost is permanent.
The GovernorSo it depends on whether fifty years ago still counts.
YouYes. The likelihood says how sure the data are. It can’t say whether AI changed the process.

Your prediction

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

Estimate $\rho$ from all 200 quarters, or only the 20 since AI. How much wider is the 95% interval with 20?

Ten times fewer quarters: an interval about 3.5 times wider — from 0.86–0.97 to 0.70–1.10.

Check yourself

For a Gaussian AR(1), given the first observation, the MLE of

\rho

equals: The sample autocorrelation at lag 2 The least-squares coefficient from regressing

z_t

z_{t-1}

The sample mean of

z_t

The sample size quadruples. The standard error of

\hat\rho

roughly: Halves Falls to a quarter Stays the same

The log-likelihood is 2.5 points higher at the estimate than at

\rho_0 = 1

. What is the LR statistic?

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 maximum likelihood estimation. I estimate the persistence rho of a Gaussian AR(1) productivity series by maximum likelihood: with 200 quarters the estimate is 0.913 (standard error 0.029); with only the last 20 quarters, the 95% likelihood-ratio interval runs from 0.70 to 1.10.
Please act as my tutor. Do not give me the answers. Ask me one question at a time to help me work out:
1. why the maximum likelihood estimate equals least squares here,
2. why the interval is so much wider with 20 quarters, and
3. what assumption I make when I use all 200 quarters.
Wait for my answer after each question. At the end, tell me what I got right and correct anything I got wrong.

Key points

The likelihood is the probability of the data, read as a function of the parameters. The MLE is where it peaks.
For a Gaussian AR(1), given the first observation, the MLE of $\rho$ is the least-squares estimate.
A sharper peak means a smaller standard error. It shrinks with $1/\sqrt{T}$ .
LR $= 2[\ell(\hat\theta) - \ell(\theta_0)] \approx \chi^2_1$ . Values within 1.92 of the peak form a 95% interval.
A long sample gives a narrow interval — only if the process stayed the same.

Ilse reads your note twice. “Everything here rests on the shocks being normal. What if all you trust is a few averages?” To be continued — Chapter 2: the generalised method of moments.