[phillips] Judgment calls left open after the review of #1105 #1106

A review of #1105 found the corrections in the PR sound, and the clear-cut fixes (two execution failures, the PDF break, and a handful of factual and typo corrections) have been pushed to the PR branch. The items below are judgment calls about the economics or the framing, so they are collected here rather than changed on the branch. @thomassargent30, could you look through them and say which you would like changed? Items 1 to 3 are the ones most worth your time; the rest are smaller.

Line links point to the PR branch at 4d8ee89.

1. "Flat" and "steep" are used in opposite senses, and the Bernanke contrast in phillips_lost_conquest may depend on it

The same event, the estimated response of unemployment to inflation shrinking toward zero during an escape toward Ramsey, is called steepening in two places and flattening in two others:

credible_policies:1263 defines the term the textbook way: in the $(U, y)$ plane the curve has slope $-1/\theta$, so a larger $\theta$ is a flatter curve. By that definition, the curve described at lost_conquest:134 ("a flatter curve means that surprise inflation does less to real activity") is a steeper one.

This matters for lost_conquest:132, which says a flat Phillips curve "points policy in opposite directions here and in the Conquest". With the textbook convention used throughout, the two cases seem to line up rather than oppose each other. In Sargent (1999), Nash inflation $\theta U^*$ rises as the curve flattens (larger $\theta$). In the 2020s model a flatter curve raises the sacrifice ratio and counsels patience. In both, a flatter curve means tolerating more inflation. Recommendation: choose one convention for the suite (the textbook one already stated in phillips_credible_policies), and then decide whether the "opposite directions" paragraph at lines 132 to 140 should stay, be reworded, or go.

2. Pivetta and Reis (2007) at lost_conquest:122

The lecture says they "cautioned that part of the apparent decline may reflect smaller shocks rather than a structural break". My recollection, held with medium confidence and not checked against the paper during the review, is that their main finding is that US inflation persistence has been high and roughly unchanged since the 1960s, with wide uncertainty bands. They did not attribute a decline to smaller shocks. Please check this against the paper. If my recollection is right, the sentence probably wants to say that they found little evidence of a decline at all.

3. The discount-factor verdict at learning:1143

The lecture tests Sargent (1999)'s two δ findings at δ ∈ {0.90, 0.95, 0.98} and concludes that "the second finding holds, but the first does not". A reviewer extended the grid, with 8 paths per value and the lecture's escape criterion:

On these numbers the book's first finding, that recurrent stabilizations need δ near one, does hold once δ is well below 0.9. The lecture's explanation of why δ hardly changes the escape timing near one still stands. Recommendation: extend the grid to include 0.5 and 0.7, or limit the verdict to the range tested, for example "over δ ∈ [0.90, 0.98] the timing of escapes hardly depends on δ". I have not re-run the reviewer's numbers.

4. The Sims comparison in phillips_priors

At priors:701 the lecture compares mean inflation after each path's first escape with "the 0.61 that Sargent and Williams (2005) report". A reviewer read SW's 0.61 as a whole-sample mean, which includes the pre-escape periods, so the two numbers measure different things. Separately, priors:643 says the 10,000-period run follows "Sims and Sargent–Williams"; it is worth confirming that both used that length. Recommendation: compare like with like, for example by printing each Sims path's whole-sample mean next to its post-escape mean, or say that SW's figure is a whole-sample mean.

5. Theorem conditions for the escape-time limits

escaping_nash:376 and learning:372 state the CWS Theorem 5.3 limits unconditionally. A reviewer notes that CWS prove the limits ε log P → −S̄ and ε log E τ → S̄ for bounded shocks. For IID unbounded shocks, such as the Gaussian shocks the lectures simulate, they give only the bound lim sup ε log P ≤ −S̄. Numerically it is harmless, since CWS Table 1 has S̄ = 4.987e-4 for normal shocks against 4.985e-4 for binomial. Recommendation: add a clause stating the condition.

6. The SW comparison in the en_ex4 solution (escaping_nash:1150)

The solution says the simulated mean exit periods "agree closely with SW's" at ε = 0.01, 0.005 and 0.001 (164, 426 and 4629 against 165.7, 409.6 and 4486.7). It does not mention ε = 0.05, where the simulation gives 28 against SW's 20.6, about 35 percent higher. SW's ε = 0.002 is not on the grid, which uses 0.0025. Recommendation: add a sentence on the ε = 0.05 gap (for example, a different starting point or escape criterion), or add 0.002 to the grid.

7. Prose that will go stale as the data update

These lectures read FRED at build time, so some sentences can become false without any edit:

• two_stories:650: "as of this writing it has not been back to 2 percent since early 2021".
• two_stories:722: "inflation has not yet returned to its pre-2021 level".
• lost_conquest:747: "stays below half a percent through 2021". The printed 2021Q4 real-time value is 0.47, close enough to the threshold that a later data update could cross it.

The Baxter–King comparison in phillips_two_stories had the same problem and has been pinned to 2008 to 2019 on the branch. For these three the choice is about wording, so it is left to you. Options are to date-stamp the claim ("as of late 2026"), or to phrase it so the printed output carries the fact.

8. Chain length in phillips_drifts_volatilities_2

Part 2 runs 10,000 sweeps and discards the first 1,000 (drifts_volatilities_2:535). The autocorrelation time of tr(Q) is about 1,000 to 1,100 sweeps, so the burn-in is about one autocorrelation time and the effective sample size of tr(Q) is about 8 or 9. The lecture prints both numbers. The text already says that magnitudes depending on Q carry large Monte Carlo error, and it quotes only features that are stable across seeds. I checked each of those statements against a second run, and all of them held.

The cost is that the printed tables change from build to build. The interest-rate splice fix on the branch changes the post-2000 data by at most 2.5 basis points a quarter. It moved:

• the posterior mean of tr(Q) from 0.433 to 0.318;
• the 1970s peak of core inflation from 16.6 to 23.6 percent;
• the last year before 2001 with g(0) > 1 from 1980 to 1983.

Each new quarter of FRED data will move them in the same way. Recommendation: either accept this, perhaps with a sentence telling readers that the printed values vary across builds, or lengthen the chain. Part 2 already takes 4 to 5 minutes locally, so a chain long enough to make a difference would probably need its draws cached or computed offline.

9. "Inverse relation" in phillips_drifts_volatilities

drifts_volatilities:1851 and :2413 describe the learning models' prediction as "an inverse relation between the level and the persistence of inflation", in the words of Cogley and Sargent's footnote 28. A reviewer computed the correlation between level and persistence in the model as about 0.03: persistence is hump-shaped, peaking during the disinflation, rather than inversely related to the level. The preceding sentence ("persistence should peak during a disinflation") already describes this. Recommendation: keep the quotation but present it as Cogley and Sargent's label rather than as a description of the correlation. I have not re-checked the 0.03.

10. Smaller notation and wording points

These come from the reviewers and have not been re-checked:

• phillips_adaptive: β is both the LQ discount factor (175, 213) and the vector of Keynesian coefficients (411 onward).
• phillips_self_confirming: C is both the "classical" subscript in $X_{C,t}$ (84) and the public's gain (516 onward).
• phillips_misspecified: $g(0) = 1$ at 273 evaluates the filter at L = 0, while the nearby $G(0)$ at 257 is the spectral density at ω = 0, that is, at z = 1. A reader could take the two zeros to be the same point.
• self_confirming:780: "only moderately serially correlated" (wording kept from main). True inflation has autocorrelations of about 0.03 at lags 1 to 10, so "weakly" would be more accurate.

Context: #1105, with the review summary in the PR conversation.

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