Appendix — Solutions — PS 5.1 (Sorting, Double-Sorting, and the Fama–MacBeth Horse Race in Fama & French 1992)

Problem set: book/weeks/week-05/ps5.1.md (PS 5.1, Week 5). Chapter: Ch 5.1 — Reader’s Guide: The Cross-Section of Expected Stock Returns (Fama & French 1992).

These are full worked solutions, not just answers. Notation follows CONVENTIONS.md and locks to Ch 5.1: portfolio sorts and return spreads; the Reader’s-Guide anatomy (Research question · Identification strategy · Data · Table-by-table reading order · What’s clever · What’s vulnerable · Three replication exercises); the Fama–MacBeth two-pass machine of Lab 2 with \hat\gamma_j=\frac1T\sum_t\hat\gamma_{j,t} and FM standard error \widehat{\operatorname{se}}(\hat\gamma_j)=\operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T; the descriptive/predictive (not causal) reading; the Week-1 multiple-testing bound P=1-(1-\alpha)^m; the Week-2/3 errors-in-variables → attenuation result; and look-ahead/survivorship bias. Every number here is illustrative and constructed — Fama and French’s actual table values are not reproduced. All return figures are percent per month unless annualized. Numerical results were confirmed in Python (spread arithmetic, the t-statistics, the multiple-testing probabilities, and the inverted FM-SE); verifying notes appear where useful. Cite the paper by name: Fama & French (1992).

Problem 1 — Why sort? Portfolios versus individual stocks (12 points)

(a) (4 pts) Write a single small stock’s monthly return as a true mean plus idiosyncratic noise, r_{it}=\mu_i+\varepsilon_{it}, where \varepsilon_{it} is the firm-specific shock (lawsuit, recall, earnings surprise) with some large variance \sigma^2. For one stock the signal \mu_i is buried under \sigma. Now average n such stocks into a portfolio. The portfolio return is \bar r_t=\frac1n\sum_{i=1}^n r_{it}, and if the idiosyncratic shocks are roughly independent across firms, \operatorname{Var}(\bar r_t)=\operatorname{Var}\!\Big(\tfrac1n\sum_i \varepsilon_{it}\Big)=\frac{1}{n^2}\sum_i \sigma^2=\frac{\sigma^2}{n}. Averaging n roughly-independent pieces shrinks the idiosyncratic variance by about 1/n, so the standard deviation of the portfolio’s monthly return falls by about 1/\sqrt n. With a few hundred stocks in a decile, the firm-specific churn that dominates any single name is averaged down by an order of magnitude, while the common part of the return — the part the sort is built to isolate — survives. That is exactly what lets a real premium under one percent per month poke above the noise: the signal stays put while the noise collapses as \sigma/\sqrt n. On a single stock that same sub-1% premium is invisible against monthly swings of tens of percent.

(b) (4 pts) The single pooled regression \mathbb{E}[r_{it}]=\gamma_0+\gamma_1 x_i assumes the relationship is linear — every one-unit change in x moves expected return by the same \gamma_1, everywhere along x. The decile sort assumes almost nothing about shape: it only assumes you can rank firms by x and that the bucket means are informative. It lets the ten decile means trace out whatever curve the data want — flat in the middle and steep at one end, monotone, hump-shaped, anything. Concrete reveal: suppose the size premium lived almost entirely in the smallest decile — D1 at 1.62 %/mo with D2 through D10 nearly flat around 1.0. A linear regression would average that one extreme bucket against nine flat ones and report a modest, possibly insignificant \hat\gamma_1, hiding the fact that the action is a single corner of the distribution. The sort shows it at a glance: nine flat dots and one that jumps. Sorts let the data show their shape before any model is forced on them.

(c) (4 pts) The section is §4, “Table-by-table reading order.” Its rule: read the sort tables before the regression tables, because the sorts “are the most honest thing in the paper — they assume almost nothing,” so you can see the result (beta flat within size; value and size live) before a single coefficient or functional-form choice has been imposed. For Sam: leading with the kitchen-sink regression means the first thing he sees is filtered through a linear specification and the choice of which regressors to include — if that regression were misspecified or a variable mismeasured, he would have no model-free benchmark to catch it. Reading the sorts first gives him the nonparametric truth to which the regression must then conform; a regression that disagreed with a clean monotone sort would be the regression he distrusts, not the sort.

Problem 2 — Read a univariate decile sort and lift the premium off it (16 points)

(a) (5 pts) Size premium. “Small” is the D1 end (smallest market equity), and small firms earn more in Fama & French (1992), so the premium is small minus big: \widehat{\text{size premium}} = \bar r_{\text{D1}} - \bar r_{\text{D10}} = 1.62 - 0.78 = \mathbf{0.84\ \%/\text{mo}} \approx 0.84\times 12 \approx \mathbf{10.1\ \%/\text{yr}}. The sign convention is small-minus-big because the live pattern is “smaller firms, higher returns,” so we subtract the lower-return big-firm decile to get a positive premium.

Value premium. “Value” is the high-BE/ME end, B10, and value beats growth, so the premium is value minus growth: \widehat{\text{value premium}} = \bar r_{\text{B10}} - \bar r_{\text{B1}} = 1.95 - 0.72 = \mathbf{1.23\ \%/\text{mo}} \approx 1.23\times 12 \approx \mathbf{14.8\ \%/\text{yr}}. These are §4’s two live patterns: average return rises as you move toward small firms (size), and average return rises across BE/ME from growth to value (book-to-market). (Annualizing by \times 12 is the crude simple-sum convention; it ignores compounding, but it is the right order-of-magnitude move for reading a sort.)

(b) (4 pts) Both columns are strictly monotone: the size column falls at every single step (1.62, 1.48, 1.39, …, 0.78 — each entry below the one before), and the BE/ME column rises at every single step (0.72, 0.86, …, 1.95). A referee trusts a smooth monotone gradient more than a sort where only the extreme bucket is unusual, because monotonicity is hard to fake by chance or by one outlier portfolio: it says the characteristic orders returns all along its range, not that one weird corner is dragging the spread. A single-bucket jump could be a handful of firms, a data error, or a small-sample fluke in one decile; ten dots marching in order is a pattern. The BE/ME sort shows the steeper and cleaner gradient: its endpoint spread (1.23) is larger than size’s (0.84), and it climbs without hesitation — consistent with §5’s remark that book-to-market “does enormous work, arguably more than size.”

Verify: size steps are all negative; BE/ME steps all positive — confirmed both monotone in Python.

(c) (4 pts) The bias is look-ahead bias, one of §6’s four vulnerabilities. Market equity is price \times shares, and a firm’s contemporaneous (this-month) market equity is built from this-month’s price — but this-month’s price is part of what produces this-month’s return. A stock whose price just dropped is mechanically sorted into a smaller-size bucket and mechanically has a low return that month; a stock whose price just jumped is sorted larger and has a high return. Sorting on the within-month price thus aligns the bucket assignment with the very return being measured, manufacturing a “small firms earn more” spread that is pure mechanical contamination, not a premium any investor could have traded on (you cannot sort on a price you only learn at month’s end). The §3 July-to-June timing gap prevents exactly this: it uses lagged size (formed before the return window) so the sorting variable is genuinely knowable before the returns it is asked to explain — the explanatory variable cannot peek at the outcome. That is why the honest lagged sort gives a modest 0.84 %/mo while the contaminated contemporaneous sort balloons.

(d) (3 pts) A spread between two portfolio means is a single realized difference; without a measure of how much it would bounce around in another sample, you cannot tell a real 0.84 %/mo premium from a draw that happened to land at 0.84 by luck. You need a standard error to form a t-statistic and ask whether the spread is distinguishable from zero. The object from Lab 2 / §2 that supplies the honest one, when the premium is re-estimated as a regression slope, is the Fama–MacBeth standard error, \operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T — built from the month-to-month scatter of the cross-sectional slopes, which correctly accounts for the within-month correlation a naive pooled SE would ignore. (Problem 4 does the computing.)

Problem 3 — The double sort: was beta’s premium just size in disguise? (20 points)

(a) (5 pts) Naive High-minus-Low beta spread: 1.33 - 1.01 = \mathbf{0.32\ \%/\text{mo}}. Taken at face value this looks like support for Sam’s “riskier pays more” CAPM prior — higher beta, higher average return, a positive premium of about 0.32 %/mo (~3.9 %/yr). But look at the size-mix columns. As you move from the Low-beta bucket to the High-beta bucket, the bucket goes from 60% big / 10% small to 10% big / 60% small. So beta is not the only thing changing across the columns — firm size is changing in lockstep: the high-beta bucket is dominated by small firms, the low-beta bucket by big firms. The “beta premium” might be nothing but the size premium wearing a beta costume, because small firms (which earn more, per Problem 2) have piled into the high-beta column.

(b) (7 pts)

Within-size High-minus-Low beta spreads, row by row: \text{Small: } 1.49-1.50 = -0.01,\qquad \text{Mid: } 1.17-1.18 = -0.01,\qquad \text{Big: } 0.86-0.84 = +0.02. Holding size roughly fixed, sorting on beta buys essentially no spread — the three numbers are within \pm 0.02 %/mo of zero, economically nothing and one of them the wrong sign. This is §4’s phrase made literal: you “watch beta die.” The flat within-size beta row is the headline of the entire paper, found in a table with no regression coefficient in it.
Within-beta Small-minus-Big size spreads, column by column: \text{Low beta: } 1.50-0.84 = +0.66,\qquad \text{Mid beta: } 1.53-0.81 = +0.72,\qquad \text{High beta: } 1.49-0.86 = +0.63. Holding beta roughly fixed, the size premium is large and positive in every beta column — about 0.6–0.7 %/mo of extra return for small over big, whichever beta group you look within. Size is a live pattern; beta is a dead one.
Reconciliation: the one-way beta sort showed +0.32 %/mo only because the high-beta bucket is 60% small firms while the low-beta bucket is 60% big firms (the size-mix columns). When you average each beta bucket’s return, the high-beta average is pulled up by the small firms crowding it and the low-beta average is pulled down by the big firms crowding it — so the “beta spread” is really the size spread leaking in through the correlated size composition. The double sort breaks the tangle by comparing High vs. Low beta within a fixed size row, where the size mix is held constant, and there the spread vanishes. Beta’s apparent premium was size in disguise.

Verify (Python): count-weighted marginal beta returns reproduce 1.01 / 1.19 / 1.33 (spread +0.32); within-size beta spreads -0.01, -0.01, +0.02; within-beta size spreads +0.66, +0.72, +0.63 — confirmed.

(c) (5 pts) The mapping to “controlling for a confounder”: the outcome is average return; the treatment-like variable whose effect we want is beta; the confounder is size, which is correlated with beta (small firms are high-beta) and independently moves returns (small firms earn more). Forming size–beta cells achieves with buckets what a control achieves with a slope, because within a single size row every firm has roughly the same size, so size can no longer drive the within-row return differences — any remaining spread across the beta columns must be beta’s own doing, exactly as adding size to a regression strips size’s variation out before reading beta’s coefficient. The one thing the bucket version buys that the linear control does not is that it is nonparametric (§2’s word): it does not assume return is linear in beta or in size, nor that size’s effect is the same at every beta level. The buckets let the data show their shape — including, here, the fact that beta’s row is flat everywhere, not just on average — without imposing a functional form that could hide a nonlinearity.

(d) (3 pts) The legitimate kernel: coarse bins average over heterogeneity inside each bin, so a relationship that is flat across three broad groups could in principle be non-flat across, say, ten finer beta groups if the action lived between the bins. That is a real limit of any coarse sort. But the §5 reason it does not rescue beta here is the flatness across the whole row, not just the endpoints: every within-size beta spread is within \pm 0.02 %/mo of zero, and one is the wrong sign. For finer bins to manufacture a beta premium, the relationship would have to be flat at the coarse level yet strongly sloped at the fine level — a contrived, non-monotone pattern with no economic story. When the coarse spread is essentially zero with a near-zero gradient, “finer bins would rescue beta” is a strained hope, whereas if the coarse spread were merely attenuated (small but clearly positive) the objection would carry more force.

Problem 4 — Read the Fama–MacBeth horse race (22 points)

(a) (5 pts) t=\hat\gamma_j/\widehat{\operatorname{se}}(\hat\gamma_j):

Spec	Regressor	\hat\gamma_j	FM SE	t
A	\beta	+0.18	0.14	+0.18/0.14 = \mathbf{+1.3}
B	\log\text{size}	-0.15	0.035	-0.15/0.035 = \mathbf{-4.3}
	\log\text{BE/ME}	+0.42	0.085	+0.42/0.085 = \mathbf{+4.9}
C	\beta	-0.02	0.13	-0.02/0.13 = \mathbf{-0.2}
	\log\text{size}	-0.14	0.036	-0.14/0.036 = \mathbf{-3.9}
	\log\text{BE/ME}	+0.40	0.090	+0.40/0.090 = \mathbf{+4.4}

Against a conventional |t|>2 bar: in spec A beta does not clear it (|t|=1.3). In spec B both \log\text{size} (|t|=4.3) and \log\text{BE/ME} (|t|=4.9) clear it strongly. In spec C beta again fails (|t|=0.2), while \log\text{size} (|t|=3.9) and \log\text{BE/ME} (|t|=4.4) both keep clearing it easily.

(b) (5 pts) Scanning the beta row across A → C the way §4 instructs: beta’s slope is a modest, already-insignificant +0.18 (t=+1.3) when run alone in spec A, and the moment \log\text{size} and \log\text{BE/ME} enter alongside it in spec C, the slope collapses to -0.02 (t=-0.2) — near zero and even the wrong sign for CAPM. Headline (one sentence): size and book-to-market capture the cross-sectional variation in average returns, and market beta adds essentially nothing once they are controlled for — beta, the field’s crown jewel, comes out flat. This is the finding precisely because it is an absence: under CAPM, beta’s slope \gamma_\beta is the number that must be large and positive (it is the whole theory), so when the number that should be big is the number that isn’t, the missing premium is the result. The paper did not just add a variable; it executed the reigning one with the field’s own preferred weapon.

(c) (4 pts) Yes — size and BE/ME keep their large, significant slopes from spec B to spec C: \log\text{size} moves only from -0.15 to -0.14 (|t| from 4.3 to 3.9) and \log\text{BE/ME} only from +0.42 to +0.40 (|t| from 4.9 to 4.4) when beta joins them. They survive together; beta does not. Signs in plain English: \hat\gamma_{\log\text{size}}<0 means larger firms earn lower average returns (equivalently, smaller firms earn more) — the size premium. \hat\gamma_{\log\text{BE/ME}}>0 means higher book-to-market (value) firms earn higher average returns than low-BE/ME (growth) firms — the value premium. These are the same two live patterns from the Problem 2 sorts: the negative size slope is the regression form of the downward-sloping size decile column (small > big), and the positive BE/ME slope is the regression form of the upward-sloping BE/ME decile column (value > growth). The regression quantifies, in one specification with the characteristics fighting each other, what the sorts showed nonparametrically.

(d) (5 pts)

The FM standard error is honest because of the within-month cross-sectional correlation of Ch 2.4 — the “time effect.” In any single month, every stock is shoved by the same market-wide shock, so the regression residuals within a month are violently correlated, not independent. A single pooled OLS over all \simstock-months would treat each stock-month as an independent observation and so vastly overcount the effective sample size, reporting a standard error far too small (a falsely huge t). Fama–MacBeth never compares across months inside a regression: it collapses each month to one slope \hat\gamma_{j,t}, then treats the T slopes as a small time series and takes their scatter, \operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T. The month-to-month bounce in the slopes already embeds the common shocks, so the SE built from that bounce is the honest one.
Invert the FM-SE formula. With \widehat{\operatorname{se}}=\operatorname{sd}_t/\sqrt T, \operatorname{sd}_t(\hat\gamma_{\log\text{BE/ME},t}) = \widehat{\operatorname{se}}\times\sqrt T = 0.090\times\sqrt{330} = 0.090\times 18.166 \approx \mathbf{1.63}. So the monthly BE/ME slope swings with a standard deviation of about 1.63 (in the same %/mo-per-log-unit units) month to month, while its average is a much tighter 0.40 — the \sqrt{330} in the denominator is what turns a noisy monthly slope into a precisely-estimated average. Verify: 0.090\times18.166=1.635, and a simulated slope series with sd 1.63 over T=330 reproduces FM SE \approx 0.090 — confirmed.

(e) (3 pts) The section is §4, “Table-by-table reading order,” which says to read the regression columns “as a sequence of horse races, not as one table.” Staging beta-alone → size-and-BE/ME-alone → all-three-together is more persuasive than reporting only the kitchen sink because it shows beta is weak even when given its best shot alone (spec A), shows size and BE/ME are strong on their own (spec B), and only then lets all three compete (spec C) — so the reader watches beta fail to survive the contest rather than having to trust a single multivariate number. The sequence rules out the rejoinder “beta only looks bad because you buried it among collinear regressors”: it was already insignificant by itself.

Problem 5 — Read like a referee: three named vulnerabilities (20 points)

(a) (7 pts) Data-snooping / multiple testing (Week 1).

With m=20 independent worthless predictors each tested at \alpha=0.05: P(\text{at least one ``significant''}) = 1-(1-0.05)^{20} = 1-(0.95)^{20} = 1-0.3585 \approx \mathbf{0.64}, and the expected number of false discoveries is \alpha m = 0.05\times 20 = \mathbf{1.0}. So even with nothing real, you have about a 64% chance of finding at least one “significant” predictor and expect one to clear the bar by pure chance. Verify: 1-0.95^{20}=0.6415 — confirmed.
A single reported Fama–MacBeth t>2 on BE/ME does not rule out that BE/ME is one such lucky winner because that t-statistic is computed as if BE/ME were the only hypothesis ever tested — it makes no adjustment for the hundreds of characteristics the field tried before landing on the survivors. A t that would be impressive in isolation is unremarkable as the maximum over many silent tests. The §6 skeptical question that actually settles it: would size and BE/ME survive in data the authors had never seen? — because true effects replicate out-of-sample while chance winners of a single-sample lottery do not.
The out-of-sample evidence §6 names as kinder to BE/ME than to most factors is international and post-publication / later-sample evidence (the value premium showing up in other countries’ markets and in data drawn after 1992). Surviving there is the right falsification test because that data was not part of the search that produced the original finding — a snooped chance correlation has no reason to reappear in a fresh sample, so re-appearance is genuine corroboration rather than the same coincidence re-counted.

(b) (7 pts) Errors-in-variables in beta (Week 2 / Week 3).

The Week-2 result: a regressor measured with random (classical) measurement error suffers attenuation bias — its estimated slope is pulled toward zero. Beta is not observed; it is the Pass-1 estimate \hat\beta_i = \beta_i + \text{noise}, and individual-stock betas are noisy. The skeptic’s argument: the noise in \hat\beta_i biases the cross-sectional slope on beta toward zero, so beta’s flat estimated slope might be a measurement artifact — beta could truly matter, with attenuation merely masking a real positive premium. The bias points toward zero because random noise in the regressor inflates its variance without adding any true covariance with returns, diluting the signal-to-noise ratio that the slope estimates (the classic \hat\gamma \to \gamma\cdot\frac{\sigma^2_\beta}{\sigma^2_\beta+\sigma^2_{\text{noise}}} shrinkage).
The fix: assign betas via portfolios, not individual stocks — a stock inherits the beta of the size–beta portfolio it belongs to, estimated on the portfolio’s longer, less noisy return series. A portfolio beta is less noisy because it is an average of its members’ betas, and the independent estimation errors in the individual betas largely cancel when averaged (the same 1/n variance reduction as Problem 1a, applied to estimation error rather than return noise). So the portfolio beta has a much smaller noise variance \sigma^2_{\text{noise}}, which pushes the attenuation factor back toward 1.
The §6 honest reading: attenuation is a strained rescue because the estimated beta slope is so close to zero (and sometimes the wrong sign) that, after the portfolio-grouping fix has already shrunk the noise a great deal, there is essentially no positive signal left for attenuation to be hiding — to “uncover” a real beta premium you would need an implausibly large residual measurement error to scale a near-zero estimate up to a meaningful number. The objection is the right one to raise (it is precisely why the authors went to portfolios), but the combination of the portfolio fix and the near-zero estimate means the data do not let attenuation carry the day.

(c) (6 pts) Look-ahead and survivorship in the book-equity data.

Survivorship bias is the distortion that arises when your sample includes only (or disproportionately) the firms that survived long enough to be recorded, dropping those that died. Compustat’s historical coverage was backfilled, and backfilling tends to add firms that lasted long enough to be worth adding — so early-sample book equity over-represents survivors, which can flatter the value premium if, for example, the distressed high-BE/ME firms that didn’t survive are missing from the early data, making value’s average return look safer and higher than it really was in real time.
Look-ahead bias is using information in the sort or regression that an investor could not actually have known at the time the position was formed. Matching December book equity to that same December’s return would credit an investor with an accounting number — the firm’s annual report — that is not actually public until months later; you would manufacture “predictability” that no real-time investor could have traded on, because the predictor was unknowable when the trade had to be placed.
The concrete check in nb5.1: the “break it on purpose” demonstration — re-running the sort with contemporaneous market equity (and, by the same logic, mismatched accounting timing) instead of the properly lagged values and showing that the spread balloons. By exhibiting how much the look-ahead-contaminated sort inflates the premium relative to the honest lagged sort, the notebook demonstrates that the timing lag is doing real protective work rather than merely asserting the construction was careful.

Problem 6 — Design the replication pipeline for `nb5.1` (10 points)

(a) (4 pts) The sort pipeline. Ordered steps from a firm-month panel to the sorts:

Assemble characteristics with the right timing. For each firm, take market equity (price \times shares from CRSP) measured at the portfolio-formation date and book equity (from Compustat) from the fiscal year ending in calendar year t-1. Form BE/ME = (lagged book equity) / (lagged market equity).
Apply the July-to-June timing lag (the load-bearing step). Match returns from July of year t through June of year t+1 to the accounting data from the fiscal year ending in t-1, and to size measured at the end of June t. Why: the lag forces every sorting variable to be genuinely knowable before the return window it explains, so the explanatory variable cannot peek at the outcome. This defends against look-ahead bias (§6) — the same reason December book equity must never be matched to December returns.
Univariate decile sorts (Problem 2). Each year, rank firms into ten size deciles by the lagged size; separately rank into ten BE/ME deciles. Form the ten portfolios (e.g., equal-weight the firms in each), compute each portfolio’s monthly return over the July–June window, and average over the sample. Read the spreads D1-D10 and B10-B1.
Size×BE/ME double sort (§7 Ex. 2). Each year, independently rank firms into size buckets and BE/ME buckets, then intersect them into the 5\times5 (or 3\times3) grid; form each cell’s portfolio, compute and average its monthly return; print the grid. The timing lag of step 2 applies identically — both sort keys are the lagged values.

(b) (4 pts) The Fama–MacBeth pipeline.

Pass 1 (betas). For each firm (or, better, each size–beta portfolio), run a time-series regression of returns on the market return over a window ending before the return month being explained, to get a pre-formation beta. Assign each stock the beta of the portfolio it belongs to — portfolio-level (tying to Problem 5b) because individual-stock betas are too noisy and would attenuate beta’s slope toward zero; averaging within a portfolio cancels the independent estimation errors and gives a much less noisy beta.
Pass 2 (monthly cross-sections). For each month t, run one cross-sectional regression of that month’s returns on the characteristics (\beta, \log\text{size}, \log\text{BE/ME}), and collect that month’s slope vector \hat\gamma_{j,t}. After looping over all T months, form the point estimate \hat\gamma_j=\frac1T\sum_t\hat\gamma_{j,t} and the FM standard error \widehat{\operatorname{se}}(\hat\gamma_j)=\operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T from the scatter of the collected slopes; the t-statistic is \hat\gamma_j/\widehat{\operatorname{se}}(\hat\gamma_j).
Three specifications to stage the §4 horse race: (A) beta alone, (B) \log\text{size} and \log\text{BE/ME} alone, (C) all three together (the kitchen sink). Watch beta’s t collapse from A to C while size and BE/ME survive.

(c) (2 pts) Reproducibility and data discipline. (i) Ship a seeded synthetic fallback — a panel with the same schema as CRSP/Compustat (firm-month returns, market equity, book equity), generated with a fixed random seed and with the size and value premia planted — so the entire notebook runs end-to-end on a student laptop with no licensed data and identical results every run. (ii) Gate the real-data path behind a flag (e.g., RUN_PATH_A=False), and on that path pin the CRSP/Compustat snapshot date and note that licensed data stays read-only on GMU infrastructure (Conventions §5) — and do not print or hard-code any real CRSP/Compustat table values, so no fabricated numbers can leak into the prose. The deliverables (sorts, double sort, FM table) are produced from the synthetic panel, clearly labeled as illustrative.

End of solutions for PS 5.1. The arithmetic checks — size spread 0.84 %/mo, value spread 1.23 %/mo, naive beta spread +0.32 dissolved to \pm0.02 within size, the six FM t-statistics (1.3 / 4.3 / 4.9 / 0.2 / 3.9 / 4.4), the inverted slope sd \approx 1.63, and the multiple-testing P\approx 0.64 — were all confirmed in Python and reproduce in nb5.1 (notebooks/week-05/nb5.1-ff92-portfolio-sorts.ipynb). If your numbers match and you can state, in your own words, why beta dies in the double sort and the kitchen sink while size and BE/ME survive, you have read Fama & French (1992) the way a referee reads it.

# Solutions — PS 5.1 (Sorting, Double-Sorting, and the Fama–MacBeth Horse Race in Fama & French 1992) **Problem set:** `book/weeks/week-05/ps5.1.md` (PS 5.1, Week 5). **Chapter:** Ch 5.1 — Reader's Guide: *The Cross-Section of Expected Stock Returns* (Fama & French 1992). These are full worked solutions, not just answers. Notation follows `CONVENTIONS.md` and locks to Ch 5.1: portfolio sorts and return *spreads*; the **Reader's-Guide anatomy** (Research question · Identification strategy · Data · Table-by-table reading order · What's clever · What's vulnerable · Three replication exercises); the Fama–MacBeth two-pass machine of Lab 2 with $\hat\gamma_j=\frac1T\sum_t\hat\gamma_{j,t}$ and FM standard error $\widehat{\operatorname{se}}(\hat\gamma_j)=\operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T$; the descriptive/predictive (not causal) reading; the Week-1 multiple-testing bound $P=1-(1-\alpha)^m$; the Week-2/3 errors-in-variables → attenuation result; and look-ahead/survivorship bias. **Every number here is illustrative and constructed** — Fama and French's actual table values are not reproduced. All return figures are percent per month unless annualized. Numerical results were confirmed in Python (spread arithmetic, the $t$-statistics, the multiple-testing probabilities, and the inverted FM-SE); verifying notes appear where useful. Cite the paper by name: **Fama & French (1992)**. --- ## Problem 1 — Why sort? Portfolios versus individual stocks (12 points) **(a) (4 pts)** Write a single small stock's monthly return as a true mean plus idiosyncratic noise, $r_{it}=\mu_i+\varepsilon_{it}$, where $\varepsilon_{it}$ is the firm-specific shock (lawsuit, recall, earnings surprise) with some large variance $\sigma^2$. For one stock the signal $\mu_i$ is buried under $\sigma$. Now average $n$ such stocks into a portfolio. The portfolio return is $\bar r_t=\frac1n\sum_{i=1}^n r_{it}$, and if the idiosyncratic shocks are roughly independent across firms, $$ \operatorname{Var}(\bar r_t)=\operatorname{Var}\!\Big(\tfrac1n\sum_i \varepsilon_{it}\Big)=\frac{1}{n^2}\sum_i \sigma^2=\frac{\sigma^2}{n}. $$ Averaging $n$ roughly-independent pieces shrinks the idiosyncratic variance by about $1/n$, so the *standard deviation* of the portfolio's monthly return falls by about $1/\sqrt n$. With a few hundred stocks in a decile, the firm-specific churn that dominates any single name is averaged down by an order of magnitude, while the *common* part of the return — the part the sort is built to isolate — survives. That is exactly what lets a real premium **under one percent per month** poke above the noise: the signal stays put while the noise collapses as $\sigma/\sqrt n$. On a single stock that same sub-1% premium is invisible against monthly swings of tens of percent. **(b) (4 pts)** The single pooled regression $\mathbb{E}[r_{it}]=\gamma_0+\gamma_1 x_i$ assumes the relationship is **linear** — every one-unit change in $x$ moves expected return by the same $\gamma_1$, everywhere along $x$. The decile sort assumes *almost nothing* about shape: it only assumes you can **rank** firms by $x$ and that the bucket means are informative. It lets the ten decile means trace out whatever curve the data want — flat in the middle and steep at one end, monotone, hump-shaped, anything. Concrete reveal: suppose the size premium lived almost entirely in the smallest decile — D1 at 1.62 %/mo with D2 through D10 nearly flat around 1.0. A linear regression would average that one extreme bucket against nine flat ones and report a modest, possibly insignificant $\hat\gamma_1$, *hiding* the fact that the action is a single corner of the distribution. The sort shows it at a glance: nine flat dots and one that jumps. Sorts let the data show their shape before any model is forced on them. **(c) (4 pts)** The section is **§4, "Table-by-table reading order."** Its rule: read the **sort tables before the regression tables**, because the sorts "are the most honest thing in the paper — they assume almost nothing," so you can see the result (beta flat within size; value and size live) before a single coefficient or functional-form choice has been imposed. For Sam: leading with the kitchen-sink regression means the *first* thing he sees is filtered through a linear specification and the choice of which regressors to include — if that regression were misspecified or a variable mismeasured, he would have no model-free benchmark to catch it. Reading the sorts first gives him the nonparametric truth to which the regression must then conform; a regression that *disagreed* with a clean monotone sort would be the regression he distrusts, not the sort. --- ## Problem 2 — Read a univariate decile sort and lift the premium off it (16 points) **(a) (5 pts)** **Size premium.** "Small" is the D1 end (smallest market equity), and small firms earn *more* in Fama & French (1992), so the premium is small minus big: $$ \widehat{\text{size premium}} = \bar r_{\text{D1}} - \bar r_{\text{D10}} = 1.62 - 0.78 = \mathbf{0.84\ \%/\text{mo}} \approx 0.84\times 12 \approx \mathbf{10.1\ \%/\text{yr}}. $$ The sign convention is small-minus-big because the live pattern is "smaller firms, higher returns," so we subtract the lower-return big-firm decile to get a positive premium. **Value premium.** "Value" is the high-BE/ME end, B10, and value beats growth, so the premium is value minus growth: $$ \widehat{\text{value premium}} = \bar r_{\text{B10}} - \bar r_{\text{B1}} = 1.95 - 0.72 = \mathbf{1.23\ \%/\text{mo}} \approx 1.23\times 12 \approx \mathbf{14.8\ \%/\text{yr}}. $$ These are §4's **two live patterns**: average return *rises as you move toward small firms* (size), and average return *rises across BE/ME from growth to value* (book-to-market). (Annualizing by $\times 12$ is the crude simple-sum convention; it ignores compounding, but it is the right order-of-magnitude move for reading a sort.) **(b) (4 pts)** Both columns are **strictly monotone**: the size column falls at every single step (1.62, 1.48, 1.39, …, 0.78 — each entry below the one before), and the BE/ME column rises at every single step (0.72, 0.86, …, 1.95). A referee trusts a smooth monotone gradient more than a sort where only the extreme bucket is unusual, because monotonicity is hard to fake by chance or by one outlier portfolio: it says the characteristic orders returns *all along its range*, not that one weird corner is dragging the spread. A single-bucket jump could be a handful of firms, a data error, or a small-sample fluke in one decile; ten dots marching in order is a pattern. The **BE/ME** sort shows the steeper *and* cleaner gradient: its endpoint spread (1.23) is larger than size's (0.84), and it climbs without hesitation — consistent with §5's remark that book-to-market "does enormous work, arguably more than size." *Verify:* size steps are all negative; BE/ME steps all positive — confirmed both monotone in Python. **(c) (4 pts)** The bias is **look-ahead bias**, one of §6's four vulnerabilities. Market equity is price $\times$ shares, and a firm's *contemporaneous* (this-month) market equity is built from this-month's price — but this-month's price is part of what produces this-month's *return*. A stock whose price just dropped is mechanically sorted into a *smaller*-size bucket *and* mechanically has a low return that month; a stock whose price just jumped is sorted larger and has a high return. Sorting on the within-month price thus aligns the bucket assignment with the very return being measured, manufacturing a "small firms earn more" spread that is pure mechanical contamination, not a premium any investor could have traded on (you cannot sort on a price you only learn at month's end). The §3 **July-to-June timing gap** prevents exactly this: it uses *lagged* size (formed before the return window) so the sorting variable is genuinely *knowable* before the returns it is asked to explain — the explanatory variable cannot peek at the outcome. That is why the honest lagged sort gives a modest 0.84 %/mo while the contaminated contemporaneous sort balloons. **(d) (3 pts)** A spread between two portfolio means is a single realized difference; without a measure of how much it would bounce around in another sample, you cannot tell a real 0.84 %/mo premium from a draw that happened to land at 0.84 by luck. You need a **standard error** to form a $t$-statistic and ask whether the spread is distinguishable from zero. The object from Lab 2 / §2 that supplies the honest one, when the premium is re-estimated as a regression slope, is the **Fama–MacBeth standard error**, $\operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T$ — built from the month-to-month scatter of the cross-sectional slopes, which correctly accounts for the within-month correlation a naive pooled SE would ignore. (Problem 4 does the computing.) --- ## Problem 3 — The double sort: was beta's premium just size in disguise? (20 points) **(a) (5 pts)** Naive **High-minus-Low beta spread**: $$ 1.33 - 1.01 = \mathbf{0.32\ \%/\text{mo}}. $$ Taken at face value this *looks* like support for Sam's "riskier pays more" CAPM prior — higher beta, higher average return, a positive premium of about 0.32 %/mo (~3.9 %/yr). But look at the size-mix columns. As you move from the Low-beta bucket to the High-beta bucket, the bucket goes from **60% big / 10% small** to **10% big / 60% small**. So beta is not the only thing changing across the columns — **firm size is changing in lockstep**: the high-beta bucket is dominated by small firms, the low-beta bucket by big firms. The "beta premium" might be nothing but the size premium wearing a beta costume, because small firms (which earn more, per Problem 2) have piled into the high-beta column. **(b) (7 pts)** (i) Within-size **High-minus-Low beta spreads**, row by row: $$ \text{Small: } 1.49-1.50 = -0.01,\qquad \text{Mid: } 1.17-1.18 = -0.01,\qquad \text{Big: } 0.86-0.84 = +0.02. $$ Holding size roughly fixed, sorting on beta buys essentially **no spread** — the three numbers are within $\pm 0.02$ %/mo of zero, economically nothing and one of them the wrong sign. This is §4's phrase made literal: you **"watch beta die."** The flat within-size beta row is the headline of the entire paper, found in a table with no regression coefficient in it. (ii) Within-beta **Small-minus-Big size spreads**, column by column: $$ \text{Low beta: } 1.50-0.84 = +0.66,\qquad \text{Mid beta: } 1.53-0.81 = +0.72,\qquad \text{High beta: } 1.49-0.86 = +0.63. $$ Holding beta roughly fixed, the size premium is **large and positive in every beta column** — about 0.6–0.7 %/mo of extra return for small over big, whichever beta group you look within. Size is a live pattern; beta is a dead one. (iii) Reconciliation: the one-way beta sort showed $+0.32$ %/mo only because the high-beta bucket is **60% small firms** while the low-beta bucket is **60% big firms** (the size-mix columns). When you average each beta bucket's return, the high-beta average is pulled *up* by the small firms crowding it and the low-beta average is pulled *down* by the big firms crowding it — so the "beta spread" is really the *size* spread leaking in through the correlated size composition. The double sort breaks the tangle by comparing High vs. Low beta *within* a fixed size row, where the size mix is held constant, and there the spread vanishes. Beta's apparent premium was size in disguise. *Verify (Python):* count-weighted marginal beta returns reproduce 1.01 / 1.19 / 1.33 (spread $+0.32$); within-size beta spreads $-0.01, -0.01, +0.02$; within-beta size spreads $+0.66, +0.72, +0.63$ — confirmed. **(c) (5 pts)** The mapping to "controlling for a confounder": the **outcome** is average return; the **treatment-like variable** whose effect we want is **beta**; the **confounder** is **size**, which is correlated with beta (small firms are high-beta) and independently moves returns (small firms earn more). Forming size–beta cells achieves with buckets what a control achieves with a slope, because within a single size row every firm has roughly the same size, so size can no longer drive the within-row return differences — any remaining spread across the beta columns must be beta's own doing, exactly as adding `size` to a regression strips size's variation out before reading beta's coefficient. The one thing the bucket version buys that the linear control does not is that it is **nonparametric** (§2's word): it does not assume return is *linear* in beta or in size, nor that size's effect is the same at every beta level. The buckets let the data show their shape — including, here, the fact that beta's row is flat *everywhere*, not just on average — without imposing a functional form that could hide a nonlinearity. **(d) (3 pts)** The legitimate kernel: coarse bins **average over heterogeneity inside each bin**, so a relationship that is flat across three broad groups could in principle be non-flat across, say, ten finer beta groups if the action lived between the bins. That is a real limit of any coarse sort. But the §5 reason it does not rescue beta here is the **flatness across the *whole* row**, not just the endpoints: every within-size beta spread is within $\pm 0.02$ %/mo of zero, and one is the wrong sign. For finer bins to manufacture a beta premium, the relationship would have to be flat at the coarse level yet strongly sloped at the fine level — a contrived, non-monotone pattern with no economic story. When the coarse spread is essentially *zero with a near-zero gradient*, "finer bins would rescue beta" is a strained hope, whereas if the coarse spread were merely *attenuated* (small but clearly positive) the objection would carry more force. --- ## Problem 4 — Read the Fama–MacBeth horse race (22 points) **(a) (5 pts)** $t=\hat\gamma_j/\widehat{\operatorname{se}}(\hat\gamma_j)$: | Spec | Regressor | $\hat\gamma_j$ | FM SE | $t$ | |:----:|:---------:|:---:|:---:|:---:| | **A** | $\beta$ | $+0.18$ | $0.14$ | $+0.18/0.14 = \mathbf{+1.3}$ | | **B** | $\log\text{size}$ | $-0.15$ | $0.035$ | $-0.15/0.035 = \mathbf{-4.3}$ | | | $\log\text{BE/ME}$ | $+0.42$ | $0.085$ | $+0.42/0.085 = \mathbf{+4.9}$ | | **C** | $\beta$ | $-0.02$ | $0.13$ | $-0.02/0.13 = \mathbf{-0.2}$ | | | $\log\text{size}$ | $-0.14$ | $0.036$ | $-0.14/0.036 = \mathbf{-3.9}$ | | | $\log\text{BE/ME}$ | $+0.40$ | $0.090$ | $+0.40/0.090 = \mathbf{+4.4}$ | Against a conventional $|t|>2$ bar: in **spec A** beta does **not** clear it ($|t|=1.3$). In **spec B** both $\log\text{size}$ ($|t|=4.3$) and $\log\text{BE/ME}$ ($|t|=4.9$) clear it strongly. In **spec C** beta again **fails** ($|t|=0.2$), while $\log\text{size}$ ($|t|=3.9$) and $\log\text{BE/ME}$ ($|t|=4.4$) both keep clearing it easily. **(b) (5 pts)** Scanning the **beta row** across A → C the way §4 instructs: beta's slope is a modest, already-insignificant $+0.18$ ($t=+1.3$) when run *alone* in spec A, and the moment $\log\text{size}$ and $\log\text{BE/ME}$ enter alongside it in spec C, the slope **collapses to $-0.02$** ($t=-0.2$) — near zero and even the wrong sign for CAPM. **Headline (one sentence):** size and book-to-market capture the cross-sectional variation in average returns, and market beta adds essentially nothing once they are controlled for — beta, the field's crown jewel, comes out flat. This is the finding precisely because it is an *absence*: under CAPM, beta's slope $\gamma_\beta$ is the number that *must* be large and positive (it is the whole theory), so when the number that should be big is the number that isn't, the missing premium *is* the result. The paper did not just add a variable; it executed the reigning one with the field's own preferred weapon. **(c) (4 pts)** Yes — size and BE/ME **keep** their large, significant slopes from spec B to spec C: $\log\text{size}$ moves only from $-0.15$ to $-0.14$ ($|t|$ from 4.3 to 3.9) and $\log\text{BE/ME}$ only from $+0.42$ to $+0.40$ ($|t|$ from 4.9 to 4.4) when beta joins them. They survive together; beta does not. Signs in plain English: $\hat\gamma_{\log\text{size}}<0$ means **larger firms earn lower average returns** (equivalently, smaller firms earn more) — the size premium. $\hat\gamma_{\log\text{BE/ME}}>0$ means **higher book-to-market (value) firms earn higher average returns** than low-BE/ME (growth) firms — the value premium. These are the *same* two live patterns from the Problem 2 sorts: the negative size slope is the regression form of the downward-sloping size decile column (small > big), and the positive BE/ME slope is the regression form of the upward-sloping BE/ME decile column (value > growth). The regression quantifies, in one specification with the characteristics fighting each other, what the sorts showed nonparametrically. **(d) (5 pts)** (i) The FM standard error is honest because of the **within-month cross-sectional correlation** of Ch 2.4 — the "time effect." In any single month, every stock is shoved by the *same* market-wide shock, so the regression residuals within a month are violently correlated, not independent. A single pooled OLS over all $\sim$stock-months would treat each stock-month as an independent observation and so vastly overcount the effective sample size, reporting a standard error far too small (a falsely huge $t$). Fama–MacBeth never compares *across* months inside a regression: it collapses each month to *one* slope $\hat\gamma_{j,t}$, then treats the $T$ slopes as a small time series and takes their scatter, $\operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T$. The month-to-month bounce in the slopes *already embeds* the common shocks, so the SE built from that bounce is the honest one. (ii) Invert the FM-SE formula. With $\widehat{\operatorname{se}}=\operatorname{sd}_t/\sqrt T$, $$ \operatorname{sd}_t(\hat\gamma_{\log\text{BE/ME},t}) = \widehat{\operatorname{se}}\times\sqrt T = 0.090\times\sqrt{330} = 0.090\times 18.166 \approx \mathbf{1.63}. $$ So the monthly BE/ME slope swings with a standard deviation of about 1.63 (in the same %/mo-per-log-unit units) month to month, while its *average* is a much tighter 0.40 — the $\sqrt{330}$ in the denominator is what turns a noisy monthly slope into a precisely-estimated average. *Verify:* $0.090\times18.166=1.635$, and a simulated slope series with sd $1.63$ over $T=330$ reproduces FM SE $\approx 0.090$ — confirmed. **(e) (3 pts)** The section is **§4, "Table-by-table reading order,"** which says to read the regression columns "as a *sequence of horse races*, not as one table." Staging beta-alone → size-and-BE/ME-alone → all-three-together is more persuasive than reporting only the kitchen sink because it shows beta is weak *even when given its best shot alone* (spec A), shows size and BE/ME are strong *on their own* (spec B), and only then lets all three compete (spec C) — so the reader watches beta fail to survive the contest rather than having to trust a single multivariate number. The sequence rules out the rejoinder "beta only looks bad because you buried it among collinear regressors": it was already insignificant by itself. --- ## Problem 5 — Read like a referee: three named vulnerabilities (20 points) **(a) (7 pts) Data-snooping / multiple testing (Week 1).** (i) With $m=20$ independent worthless predictors each tested at $\alpha=0.05$: $$ P(\text{at least one ``significant''}) = 1-(1-0.05)^{20} = 1-(0.95)^{20} = 1-0.3585 \approx \mathbf{0.64}, $$ and the expected number of false discoveries is $\alpha m = 0.05\times 20 = \mathbf{1.0}$. So even with *nothing real*, you have about a **64% chance** of finding at least one "significant" predictor and *expect one* to clear the bar by pure chance. *Verify:* $1-0.95^{20}=0.6415$ — confirmed. (ii) A single reported Fama–MacBeth $t>2$ on BE/ME does not rule out that BE/ME is one such lucky winner because that $t$-statistic is computed *as if BE/ME were the only hypothesis ever tested* — it makes no adjustment for the hundreds of characteristics the *field* tried before landing on the survivors. A $t$ that would be impressive in isolation is unremarkable as the maximum over many silent tests. The §6 skeptical question that actually settles it: *would size and BE/ME survive in data the authors had never seen?* — because true effects replicate out-of-sample while chance winners of a single-sample lottery do not. (iii) The out-of-sample evidence §6 names as kinder to BE/ME than to most factors is **international and post-publication / later-sample evidence** (the value premium showing up in other countries' markets and in data drawn after 1992). Surviving there is the right falsification test because that data was *not part of the search* that produced the original finding — a snooped chance correlation has no reason to reappear in a fresh sample, so re-appearance is genuine corroboration rather than the same coincidence re-counted. **(b) (7 pts) Errors-in-variables in beta (Week 2 / Week 3).** (i) The Week-2 result: a regressor measured with **random (classical) measurement error** suffers **attenuation bias** — its estimated slope is pulled *toward zero*. Beta is not observed; it is the Pass-1 *estimate* $\hat\beta_i = \beta_i + \text{noise}$, and individual-stock betas are noisy. The skeptic's argument: the noise in $\hat\beta_i$ biases the cross-sectional slope on beta toward zero, so beta's *flat* estimated slope might be a **measurement artifact** — beta could truly matter, with attenuation merely masking a real positive premium. The bias points toward zero because random noise in the regressor inflates its variance without adding any true covariance with returns, diluting the signal-to-noise ratio that the slope estimates (the classic $\hat\gamma \to \gamma\cdot\frac{\sigma^2_\beta}{\sigma^2_\beta+\sigma^2_{\text{noise}}}$ shrinkage). (ii) The fix: assign betas via **portfolios**, not individual stocks — a stock inherits the beta of the size–beta portfolio it belongs to, estimated on the portfolio's longer, less noisy return series. A portfolio beta is less noisy because it is an *average* of its members' betas, and the independent estimation errors in the individual betas largely **cancel** when averaged (the same $1/n$ variance reduction as Problem 1a, applied to estimation error rather than return noise). So the portfolio beta has a much smaller noise variance $\sigma^2_{\text{noise}}$, which pushes the attenuation factor back toward 1. (iii) The §6 honest reading: attenuation is a *strained* rescue because the estimated beta slope is so close to zero (and sometimes the wrong sign) that, after the portfolio-grouping fix has already shrunk the noise a great deal, there is essentially no positive signal left for attenuation to be hiding — to "uncover" a real beta premium you would need an implausibly large residual measurement error to scale a near-zero estimate up to a meaningful number. The objection is the *right* one to raise (it is precisely why the authors went to portfolios), but the combination of the portfolio fix and the near-zero estimate means the data do not let attenuation carry the day. **(c) (6 pts) Look-ahead and survivorship in the book-equity data.** (i) **Survivorship bias** is the distortion that arises when your sample includes only (or disproportionately) the firms that *survived* long enough to be recorded, dropping those that died. Compustat's historical coverage was **backfilled**, and backfilling tends to add firms that lasted long enough to be worth adding — so early-sample book equity over-represents survivors, which can flatter the value premium if, for example, the distressed high-BE/ME firms that *didn't* survive are missing from the early data, making value's average return look safer and higher than it really was in real time. (ii) **Look-ahead bias** is using information in the sort or regression that an investor could not actually have known *at the time* the position was formed. Matching December book equity to that same December's return would credit an investor with an accounting number — the firm's annual report — that is not actually public until months later; you would manufacture "predictability" that no real-time investor could have traded on, because the predictor was unknowable when the trade had to be placed. (iii) The concrete check in `nb5.1`: the **"break it on purpose"** demonstration — re-running the sort with *contemporaneous* market equity (and, by the same logic, mismatched accounting timing) instead of the properly lagged values and showing that the spread *balloons*. By exhibiting how much the look-ahead-contaminated sort inflates the premium relative to the honest lagged sort, the notebook *demonstrates* that the timing lag is doing real protective work rather than merely asserting the construction was careful. --- ## Problem 6 — Design the replication pipeline for `nb5.1` (10 points) **(a) (4 pts) The sort pipeline.** Ordered steps from a firm-month panel to the sorts: 1. **Assemble characteristics with the right timing.** For each firm, take **market equity** (price $\times$ shares from CRSP) measured at the *portfolio-formation* date and **book equity** (from Compustat) from the **fiscal year ending in calendar year $t-1$**. Form **BE/ME** = (lagged book equity) / (lagged market equity). 2. **Apply the July-to-June timing lag (the load-bearing step).** Match returns from **July of year $t$ through June of year $t+1$** to the accounting data from the fiscal year ending in $t-1$, and to size measured at the end of June $t$. *Why:* the lag forces every sorting variable to be genuinely *knowable before* the return window it explains, so the explanatory variable cannot peek at the outcome. This defends against **look-ahead bias** (§6) — the same reason December book equity must never be matched to December returns. 3. **Univariate decile sorts (Problem 2).** Each year, rank firms into ten **size** deciles by the lagged size; separately rank into ten **BE/ME** deciles. Form the ten portfolios (e.g., equal-weight the firms in each), compute each portfolio's monthly return over the July–June window, and average over the sample. Read the spreads D1$-$D10 and B10$-$B1. 4. **Size×BE/ME double sort (§7 Ex. 2).** Each year, independently rank firms into size buckets and BE/ME buckets, then intersect them into the $5\times5$ (or $3\times3$) grid; form each cell's portfolio, compute and average its monthly return; print the grid. The timing lag of step 2 applies identically — both sort keys are the lagged values. **(b) (4 pts) The Fama–MacBeth pipeline.** - **Pass 1 (betas).** For each firm (or, better, each size–beta portfolio), run a **time-series regression of returns on the market return** over a window *ending before* the return month being explained, to get a **pre-formation beta**. Assign each stock the beta of the **portfolio** it belongs to — *portfolio-level* (tying to Problem 5b) because individual-stock betas are too noisy and would attenuate beta's slope toward zero; averaging within a portfolio cancels the independent estimation errors and gives a much less noisy beta. - **Pass 2 (monthly cross-sections).** For **each month $t$**, run one cross-sectional regression of that month's returns on the characteristics ($\beta$, $\log\text{size}$, $\log\text{BE/ME}$), and **collect that month's slope vector** $\hat\gamma_{j,t}$. After looping over all $T$ months, form the point estimate $\hat\gamma_j=\frac1T\sum_t\hat\gamma_{j,t}$ and the FM standard error $\widehat{\operatorname{se}}(\hat\gamma_j)=\operatorname{sd}_t(\hat\gamma_{j,t})/\sqrt T$ from the scatter of the collected slopes; the $t$-statistic is $\hat\gamma_j/\widehat{\operatorname{se}}(\hat\gamma_j)$. - **Three specifications** to stage the §4 horse race: **(A) beta alone**, **(B) $\log\text{size}$ and $\log\text{BE/ME}$ alone**, **(C) all three together** (the kitchen sink). Watch beta's $t$ collapse from A to C while size and BE/ME survive. **(c) (2 pts) Reproducibility and data discipline.** (i) Ship a **seeded synthetic fallback** — a panel with the *same schema* as CRSP/Compustat (firm-month returns, market equity, book equity), generated with a fixed random seed and with the size and value premia *planted* — so the entire notebook runs end-to-end on a student laptop with no licensed data and identical results every run. (ii) Gate the real-data path behind a flag (e.g., `RUN_PATH_A=False`), and on that path **pin the CRSP/Compustat snapshot date** and note that licensed data stays read-only on GMU infrastructure (Conventions §5) — and *do not print or hard-code any real CRSP/Compustat table values*, so no fabricated numbers can leak into the prose. The deliverables (sorts, double sort, FM table) are produced from the synthetic panel, clearly labeled as illustrative. --- *End of solutions for PS 5.1. The arithmetic checks — size spread 0.84 %/mo, value spread 1.23 %/mo, naive beta spread +0.32 dissolved to $\pm0.02$ within size, the six FM $t$-statistics (1.3 / 4.3 / 4.9 / 0.2 / 3.9 / 4.4), the inverted slope sd $\approx 1.63$, and the multiple-testing $P\approx 0.64$ — were all confirmed in Python and reproduce in `nb5.1` (`notebooks/week-05/nb5.1-ff92-portfolio-sorts.ipynb`). If your numbers match and you can state, in your own words, why beta dies in the double sort and the kitchen sink while size and BE/ME survive, you have read Fama & French (1992) the way a referee reads it.*