Problem Set 6.2 — 10-K Cosine Similarity & TNIC Mini-Build

Covers Chapter 6.2 (Reader’s Guide: Hoberg & Phillips (2016)). This set is a replication-flavored walk through the measurement machinery of Hoberg and Phillips (2016), “Text-Based Network Industries and Endogenous Product Differentiation.” Methods through Ch 6.2 only: turning a firm’s 10-K product description (Item 1) into a bag-of-words vector; comparing two firm vectors with cosine similarity — the same cosine you met in Chapter 1.2 when we said correlation is a cosine, now pointed at word counts instead of returns; the TF–IDF reweighting that down-weights ubiquitous words; and the construction of TNIC (Text-based Network Industry Classifications) from a similarity matrix and a threshold \tau, with its three signature properties — firm-specific, time-varying, intransitive — that a fixed SIC code cannot represent. Notation follows the Conventions: firm i’s description is a vector \mathbf{w}_i with one entry per dictionary word j; w_{ij} is the weight of word j in firm i; the cosine similarity between firms i and k is \text{sim}(i,k) \;=\; \cos\theta \;=\; \frac{\mathbf{w}_i \cdot \mathbf{w}_k}{\lVert\mathbf{w}_i\rVert\,\lVert\mathbf{w}_k\rVert} \;=\; \frac{\sum_j w_{ij}\,w_{kj}}{\sqrt{\sum_j w_{ij}^2}\;\sqrt{\sum_j w_{kj}^2}}.

Six problems, escalating, 100 points total. Every numerical input is supplied so you can work the arithmetic by hand — no computer needed for Problems 1–4. The companion notebook nb6.2 (notebooks/week-06/nb6.2-10k-text-similarity.ipynb) lets you scale the same operations to a 16-firm corpus with scikit-learn’s CountVectorizer, TfidfVectorizer, and cosine_similarity, and watch the industry blocks light up on a heatmap — but the understanding is built here, by hand, on three short descriptions.

The grading rule of this set: a number reported without saying what it means — a cosine without the words “shared vocabulary,” a TNIC neighbor list without naming the property it demonstrates (firm-specific, intransitive) — earns half credit at most. As everywhere in this book, this is a measurement paper, so the skill is reading the measure: where it comes from, what choice each number depends on, and where it is a noisy or biased signal of what the firm actually does. Whenever you name a vulnerability, name the construction choice that mitigates it.

A standing warning on numbers and text. Every word count, cosine, IDF weight, similarity matrix, and product description in this problem set is illustrative — invented to make the arithmetic clean and the reasoning visible. None of it is from Hoberg and Phillips’s tables or the TNIC data library. Do not quote any figure here as if it were theirs. The qualitative facts (two same-industry firms score high; cosine normalizes away length; common words must be down-weighted; TNIC is firm-specific, time-varying, and intransitive where SIC is not) are well established and safe to state; any precise number belongs in your own replication in nb6.2, not in memory. Citation by name: Hoberg & Phillips (2016).

Problems 1–3 share one small world, defined once here so you can carry the vectors between them.

The illustrative three-firm corpus (the working world). After a firm files its 10-K, we extract Item 1 (the product description), lowercase it, and drop stop-words (ultra-common function words like we, and, to, into, the that carry no product information). What survives is a short list of content words. Here are three deliberately tiny descriptions, already tokenized and stop-word-stripped, with each content word’s count:

Firm What it is Content words (with counts)

A an oil firm oil×2, crude×1, refine×1, fuel×1

B an oil firm oil×2, refine×1, fuel×1, barrels×1

C an airline flights×1, airline×1, tickets×1, travel×1

Firm	What it is	Content words (with counts)
A	an oil firm	`oil`×2, `crude`×1, `refine`×1, `fuel`×1
B	an oil firm	`oil`×2, `refine`×1, `fuel`×1, `barrels`×1
C	an airline	`flights`×1, `airline`×1, `tickets`×1, `travel`×1

The union dictionary across the three firms is the set of all content words that appear in any of them: {airline, barrels, crude, flights, fuel, oil, refine, tickets, travel} — nine words. Firm i’s bag-of-words vector \mathbf{w}_i has one entry per dictionary word, equal to that word’s count in firm i (zero if absent). You will build all three vectors in Problem 1.

Problem 1 — Vectorize three product descriptions and compute cosine similarity by hand (24 points)

This is the factory floor of the whole paper: prose in, geometry out. You will turn three product descriptions into vectors and measure who is near whom — without ever telling the arithmetic what an industry is.

(a) [6 pts] Write out the three bag-of-words vectors \mathbf{w}_A, \mathbf{w}_B, \mathbf{w}_C over the nine-word union dictionary, in the fixed alphabetical order [airline, barrels, crude, flights, fuel, oil, refine, tickets, travel]. (Each vector is nine numbers.)

(b) [8 pts] Compute the cosine similarity \text{sim}(A,B) between the two oil firms. Show the three pieces explicitly: the dot product \mathbf{w}_A\cdot\mathbf{w}_B = \sum_j w_{Aj}w_{Bj}, and the two norms \lVert\mathbf{w}_A\rVert = \sqrt{\sum_j w_{Aj}^2} and \lVert\mathbf{w}_B\rVert. Then assemble the cosine. Report it as an exact fraction and as a decimal to three places.

(c) [4 pts] Compute \text{sim}(A,C) and \text{sim}(B,C) (oil firm vs. airline). You should find both are exactly 0 — explain in one sentence why the cosine is exactly zero here, in terms of the words A and C share (or do not share), and what angle \theta a cosine of 0 corresponds to.

(d) [4 pts] You just produced a 3\times3 similarity matrix purely from word counts, and the two oil firms came out near-twins while each oil firm and the airline came out strangers — with no industry labels supplied anywhere. State in two sentences what this demonstrates about the central idea of Hoberg and Phillips (2016): that an industry can be discovered from text rather than assigned by a code.

(e) [2 pts] The Chapter 1.2 connection. In Chapter 1.2 you learned that the correlation between two mean-centered data vectors is the cosine of the angle between them. State in one sentence what is the same and what is different between that correlation-as-cosine and the similarity you just computed (Hint: same formula; what plays the role of the “data vector,” and is the word-count vector mean-centered?).

Problem 2 — TF–IDF: why down-weight common words (16 points)

The raw bag-of-words count treats every word as equally informative. It is not. This problem shows why the standard fix — TF–IDF, term frequency times inverse document frequency — is necessary, and makes the down-weighting concrete.

Recall the inverse-document-frequency weight for word j, with D the number of documents (firms) and d_j the number of documents containing word j: \text{idf}_j = \log\!\frac{D}{d_j}. The TF–IDF entry for word j in firm i is then w_{ij} = \text{tf}_{ij}\cdot\text{idf}_j, where \text{tf}_{ij} is the count of word j in firm i.

Consider a corpus of D = 4 firms with these document frequencies:

Word	d_j (docs containing it)
`company`	4
`products`	3
`oil`	2
`lithium`	1

(a) [6 pts] Compute \text{idf}_j (using natural log) for each of the four words. Report all four values. Which word is completely silenced (its IDF weight is exactly zero), and what is the one-sentence intuition for why that word deserves to be silenced in an industry-similarity measure?

(b) [6 pts] The spurious-similarity demonstration. Suppose you forgot to strip stop-words, so a boilerplate word — say company — survives in both an oil firm’s vector and an airline’s vector. Concretely, take the oil firm oil×2, crude×1, refine×1, fuel×1, company×1 and the airline flights×1, airline×1, tickets×1, travel×1, company×2. Their only shared word is company. Compute their raw count cosine similarity. (You should get a strictly positive number even though they share no product vocabulary.) Then explain in one or two sentences how giving company its IDF weight — which, if company appears in every document, is \log(D/D) = 0 — repairs this: what does the spurious similarity become?

(c) [4 pts] State the general principle in two sentences: TF–IDF up-weights words that are distinctive (rare across firms, like lithium) and down-weights words that are ubiquitous (common across firms, like company). Connect this to a vulnerability the Chapter 6.2 guide flags — boilerplate and legal language — and say in one clause why down-weighting common words is a partial, not a complete, defense against it.

Problem 3 — Build a TNIC from a similarity matrix (20 points)

Now scale up from a pair to a network. You are given the full pairwise cosine-similarity matrix for five firms (illustrative, symmetric, rounded to two decimals — built the same way as Problem 1, just at larger scale). The firms are two cloud-software firms (SoftA, SoftB), a retailer (RetA), a bank (Bank1), and a cross-over firm OmniMart that sells retail goods and runs a cloud platform (an Amazon-like straddler — the same role OmniMart plays in nb6.2).

\text{sim}(i,k) SoftA SoftB OmniMart RetA Bank1

SoftA 1.00 0.62 0.34 0.05 0.08

SoftB 0.62 1.00 0.22 0.04 0.07

OmniMart 0.34 0.22 1.00 0.41 0.06

RetA 0.05 0.04 0.41 1.00 0.05

Bank1 0.08 0.07 0.06 0.05 1.00

\text{sim}(i,k)	SoftA	SoftB	OmniMart	RetA	Bank1
SoftA	1.00	0.62	0.34	0.05	0.08
SoftB	0.62	1.00	0.22	0.04	0.07
OmniMart	0.34	0.22	1.00	0.41	0.06
RetA	0.05	0.04	0.41	1.00	0.05
Bank1	0.08	0.07	0.06	0.05	1.00

The TNIC rule (Ch 6.2 §2, step four): firm i’s text-based industry is the set of other firms whose similarity to i clears a chosen threshold \tau. Use \tau = 0.20 throughout this problem.

(a) [6 pts] For each of the five firms, list its TNIC industry — the set of other firms with \text{sim} \ge 0.20. (Read off each firm’s row; ignore the diagonal 1.00, a firm is not its own competitor.) Write out all five neighbor sets.

(b) [6 pts] Firm-specific, not a partition. Compare SoftA’s industry to OmniMart’s industry from part (a). Show they are different sets — and note the asymmetry of membership in a set vs. having the same set: SoftA appears in OmniMart’s industry, yet OmniMart’s industry is not identical to SoftA’s. Explain in two sentences why this is impossible under a SIC code (which assigns each firm one box and declares everyone in that box mutual competitors) and what HP mean by calling TNIC firm-specific: each firm centers its own neighborhood.

(c) [6 pts] Intransitivity. Find a triple of firms X, Y, Z in this matrix such that \text{sim}(X,Y) \ge \tau and \text{sim}(Y,Z) \ge \tau but \text{sim}(X,Z) < \tau — i.e., X and Z are both neighbors of Y but are not neighbors of each other. State the triple and the three similarity values. Then explain in two sentences why this intransitivity is exactly the Amazon-style fact SIC cannot represent (Hint: Y is the cross-over firm; tie it to “neighbor to both Microsoft on cloud and Walmart on groceries, though those two are not neighbors”).

(d) [2 pts] The threshold is a knob. Recompute OmniMart’s neighbor count (number of firms in its industry) at four thresholds: \tau = 0.10,\ 0.30,\ 0.40,\ 0.50. State in one sentence what happens to a firm’s competitor count as \tau rises, and why this makes \tau a lever, not a fact — a measurement choice that must be disclosed and defended (Ch 6.2 §3, “knobs are levers”).

Problem 4 — TNIC vs. SIC on a cross-over firm (12 points)

This problem makes the horse race personal: take the one firm that breaks the official taxonomy and show how each system handles it. Use the same five-firm matrix and \tau = 0.20 from Problem 3.

OmniMart sells retail goods and operates a cloud platform. A government classifier must assign it one four-digit SIC code; suppose — because retail is its larger revenue segment — it lands in a retail SIC box, the same box as RetA. (This mirrors nb6.2, where the SIC-like foil forces each cross-over firm into a single box and it “loses half its identity.”)

(a) [4 pts] What SIC sees. Under the retail SIC box, who are OmniMart’s competitors, and who is invisible to it? State which of {SoftA, SoftB, RetA, Bank1} the SIC box counts as OmniMart’s industry, and which real competitor(s) — visible in the similarity matrix — the box misses entirely.

(b) [5 pts] What TNIC sees. From OmniMart’s TNIC industry (Problem 3a), state how many distinct SIC-style businesses OmniMart neighbors — i.e., does its text-based industry span retail and software? Name the neighbors and the business each represents. Then state in one sentence the precise sense in which TNIC recovers the half of OmniMart’s identity that the single SIC box threw away.

(c) [3 pts] The horse-race framing. The Chapter 6.2 guide insists this is a measurement paper: its central claim is validated by a horse race, not a causal identifying assumption. State in two sentences what the horse race is — what outcome would TNIC and SIC compete to explain on the same firms in the same years — and why “TNIC neighbors OmniMart with both a software firm and a retailer” is suggestive but not yet the validation (Hint: what would still have to be shown about outcomes like co-movement or margins?).

Problem 5 — What’s vulnerable: boilerplate, self-description, bag-of-words (16 points)

A measurement paper has measurement vulnerabilities — different from the causal threats of Week 5. The right skeptical question is not “what confounds the treatment?” but “what makes the text a noisy or biased signal of what the firm actually does?” This problem is pure reasoning; it tests whether you can push on the measure the way a referee would.

(a) [5 pts] Boilerplate and legal language. A 10-K is written by lawyers as much as managers; large stretches are formulaic and shared across firms (risk-factor language, disclaimers, accounting boilerplate). Explain in two or three sentences how shared boilerplate could make two firms in different businesses score “similar,” and name the two construction choices from Problems 2 and the working-world setup that limit this. Then state in one clause why the problem is worse for the time-varying comparisons TNIC is built to make (Hint: disclosure norms — and therefore the boilerplate baseline — drift over time).

(b) [5 pts] Firms’ incentives in self-description. This is the sharpest critique, specific to self-reported text. Firms choose how to describe themselves — vaguely to avoid tipping off rivals, grandly to flatter investors (“we are an AI company now”), conservatively to manage legal exposure. Explain in two or three sentences the reflexivity problem this creates specifically because differentiation is itself one of the paper’s outcomes: if a firm that merely talks differently is coded as differentiated, what exactly is the measure measuring? Pose the referee’s one-sentence question in the form “are we measuring , or ?”

(c) [4 pts] Bag-of-words ignores meaning. The representation throws away word order, synonymy, and context. Give the two canonical failures: (i) a negation example — write two short sentences that mean opposite things about competing in a market but produce nearly identical bags-of-words; and (ii) a synonym example — two words a human reads as the same product that the bag treats as unrelated. Then state in one sentence which later tool (Ch 6.5) is built precisely to close this gap, and what kind of vector it uses.

(d) [2 pts] Threat → guard, one per vulnerability. For each of (a), (b), (c), name in a phrase the discipline or check a careful replicator uses to address it (one per vulnerability). This is spec discipline (Conventions §4) pointed at a text-based measure: name the threat, then name the guard.

Problem 6 — Replication design: from this problem set to nb6.2 (12 points)

You will run this for real in nb6.2. This problem makes you design the replication before you touch the keyboard — stating the corpus, the pipeline, and the checks you would commit to in advance.

(a) [4 pts] The pipeline, in order. Name the four steps that turn a folder of product descriptions into a similarity matrix, in the order nb6.2 runs them (Ch 6.2 §2): (1) get/clean the text, (2) build vectors, (3) measure similarity, (4) build the network. For step (2), name the two vectorizers nb6.2 compares (CountVectorizer for plain counts vs. TfidfVectorizer for TF–IDF) and state in one sentence what swapping one for the other lets you see about the role of IDF (Hint: Problem 2b).

(b) [4 pts] The headline check. nb6.2 hands you a 16-firm corpus across three latent industries (software/cloud, banking/lending, retail/apparel) plus two deliberate cross-over firms. State the one quantitative comparison you would compute to confirm the measure “works” before trusting any TNIC neighbor list (Hint: compare the average similarity of firms in the same latent industry to the average across different industries — what ordering must hold, and what is the name for that ratio?). Then name the one picture that makes the same point visually (Hint: what lights up on the diagonal of a cluster-ordered heatmap?).

(c) [4 pts] Two network properties to reproduce, and the real-data pointer. Name the two network properties — beyond same-industry blocks — that nb6.2 asks you to demonstrate on the cross-over firm and (if two years of filings are supplied) on a firm that changed its business; tie each to the matching property of TNIC (intransitivity and time-variation). Finally, state where a real replication would get the text and the ready-made industries instead of re-parsing tens of thousands of filings yourself: name the source of the raw text (the public SEC filing system) and the public data library Hoberg and Phillips maintain so researchers can download the computed TNIC industries directly. [The exact library URL is left as a verify-before-printing item, consistent with the Ch 6.2 guide.]

End of Problem Set 6.2. Solutions: Appendix E, E-w6-ps6.2-solutions.md. The companion notebook nb6.2 (notebooks/week-06/nb6.2-10k-text-similarity.ipynb) lets you scale every operation here to a 16-firm corpus — tokenize and TF–IDF-vectorize the product descriptions, compute the full pairwise cosine matrix, print it as a cluster-ordered heatmap and watch the industry blocks light up on the diagonal, then build a TNIC at a threshold \tau and verify its firm-specific and intransitive structure on a cross-over firm — checking your hand arithmetic from Problems 1–4 against scikit-learn, with a “Your Turn” extension that adds a firm, sweeps the threshold, or switches to bigrams. The single discipline to carry forward: this is a measurement paper, so read every cosine as a choice-dependent estimate* of product overlap — clean enough to redraw the map of American industry from text alone, but only as good as the words a firm chooses, the stop-words you strip, and the threshold you pick. Citation by name: Hoberg & Phillips (2016).*

# Problem Set 6.2 — 10-K Cosine Similarity & TNIC Mini-Build **Covers Chapter 6.2 (Reader's Guide: Hoberg & Phillips (2016)).** This set is a *replication-flavored* walk through the measurement machinery of Hoberg and Phillips (2016), "Text-Based Network Industries and Endogenous Product Differentiation." Methods through Ch 6.2 only: turning a firm's 10-K **product description** (Item 1) into a **bag-of-words vector**; comparing two firm vectors with **cosine similarity** — the *same* cosine you met in Chapter 1.2 when we said correlation *is* a cosine, now pointed at word counts instead of returns; the **TF–IDF** reweighting that down-weights ubiquitous words; and the construction of **TNIC** (Text-based Network Industry Classifications) from a similarity matrix and a threshold $\tau$, with its three signature properties — **firm-specific, time-varying, intransitive** — that a fixed SIC code cannot represent. Notation follows the Conventions: firm $i$'s description is a vector $\mathbf{w}_i$ with one entry per dictionary word $j$; $w_{ij}$ is the weight of word $j$ in firm $i$; the cosine similarity between firms $i$ and $k$ is $$ \text{sim}(i,k) \;=\; \cos\theta \;=\; \frac{\mathbf{w}_i \cdot \mathbf{w}_k}{\lVert\mathbf{w}_i\rVert\,\lVert\mathbf{w}_k\rVert} \;=\; \frac{\sum_j w_{ij}\,w_{kj}}{\sqrt{\sum_j w_{ij}^2}\;\sqrt{\sum_j w_{kj}^2}}. $$ Six problems, escalating, **100 points total**. Every numerical input is supplied so you can work the arithmetic by hand — no computer needed for Problems 1–4. The companion notebook **nb6.2** (`notebooks/week-06/nb6.2-10k-text-similarity.ipynb`) lets you scale the same operations to a 16-firm corpus with `scikit-learn`'s `CountVectorizer`, `TfidfVectorizer`, and `cosine_similarity`, and watch the industry blocks light up on a heatmap — but the *understanding* is built here, by hand, on three short descriptions. **The grading rule of this set:** a number reported *without saying what it means* — a cosine without the words "shared vocabulary," a TNIC neighbor list without naming the property it demonstrates (firm-specific, intransitive) — earns half credit at most. As everywhere in this book, this is a **measurement** paper, so the skill is *reading the measure*: where it comes from, what choice each number depends on, and where it is a noisy or biased signal of what the firm actually does. Whenever you name a vulnerability, name the construction choice that mitigates it. > **A standing warning on numbers and text.** Every word count, cosine, IDF weight, similarity matrix, and > product description in this problem set is **illustrative** — invented to make the arithmetic clean and > the reasoning visible. **None of it is from Hoberg and Phillips's tables or the TNIC data library.** Do > not quote any figure here as if it were theirs. The *qualitative* facts (two same-industry firms score > high; cosine normalizes away length; common words must be down-weighted; TNIC is firm-specific, > time-varying, and intransitive where SIC is not) are well established and safe to state; any precise > number belongs in your own replication in nb6.2, not in memory. Citation by name: Hoberg & Phillips > (2016). Problems 1–3 share one small world, defined once here so you can carry the vectors between them. > **The illustrative three-firm corpus (the working world).** After a firm files its 10-K, we extract > Item 1 (the product description), lowercase it, and **drop stop-words** (ultra-common function words like > *we, and, to, into, the* that carry no product information). What survives is a short list of **content > words**. Here are three deliberately tiny descriptions, already tokenized and stop-word-stripped, with > each content word's count: > | Firm | What it is | Content words (with counts) | > |:---|:---|:---| > | **A** | an oil firm | `oil`×2, `crude`×1, `refine`×1, `fuel`×1 | > | **B** | an oil firm | `oil`×2, `refine`×1, `fuel`×1, `barrels`×1 | > | **C** | an airline | `flights`×1, `airline`×1, `tickets`×1, `travel`×1 | > The **union dictionary** across the three firms is the set of all content words that appear in any of > them: `{airline, barrels, crude, flights, fuel, oil, refine, tickets, travel}` — nine words. Firm $i$'s > bag-of-words vector $\mathbf{w}_i$ has one entry per dictionary word, equal to that word's count in firm > $i$ (zero if absent). You will build all three vectors in Problem 1. --- ## Problem 1 — Vectorize three product descriptions and compute cosine similarity by hand (24 points) This is the factory floor of the whole paper: prose in, geometry out. You will turn three product descriptions into vectors and measure who is near whom — *without ever telling the arithmetic what an industry is.* **(a)** [6 pts] Write out the three bag-of-words vectors $\mathbf{w}_A, \mathbf{w}_B, \mathbf{w}_C$ over the nine-word union dictionary, in the fixed alphabetical order `[airline, barrels, crude, flights, fuel, oil, refine, tickets, travel]`. (Each vector is nine numbers.) **(b)** [8 pts] Compute the cosine similarity $\text{sim}(A,B)$ between the two oil firms. Show the three pieces explicitly: the **dot product** $\mathbf{w}_A\cdot\mathbf{w}_B = \sum_j w_{Aj}w_{Bj}$, and the two **norms** $\lVert\mathbf{w}_A\rVert = \sqrt{\sum_j w_{Aj}^2}$ and $\lVert\mathbf{w}_B\rVert$. Then assemble the cosine. Report it as an exact fraction and as a decimal to three places. **(c)** [4 pts] Compute $\text{sim}(A,C)$ and $\text{sim}(B,C)$ (oil firm vs. airline). You should find both are exactly $0$ — explain in one sentence *why* the cosine is exactly zero here, in terms of the words A and C share (or do not share), and what angle $\theta$ a cosine of $0$ corresponds to. **(d)** [4 pts] You just produced a $3\times3$ similarity matrix purely from word counts, and the two oil firms came out near-twins while each oil firm and the airline came out strangers — *with no industry labels supplied anywhere.* State in two sentences what this demonstrates about the central idea of Hoberg and Phillips (2016): that an industry can be **discovered from text** rather than assigned by a code. **(e)** [2 pts] **The Chapter 1.2 connection.** In Chapter 1.2 you learned that the correlation between two mean-centered data vectors *is* the cosine of the angle between them. State in one sentence what is the *same* and what is *different* between that correlation-as-cosine and the similarity you just computed (Hint: same formula; what plays the role of the "data vector," and is the word-count vector mean-centered?). --- ## Problem 2 — TF–IDF: why down-weight common words (16 points) The raw bag-of-words count treats every word as equally informative. It is not. This problem shows *why* the standard fix — **TF–IDF**, term frequency times inverse document frequency — is necessary, and makes the down-weighting concrete. Recall the inverse-document-frequency weight for word $j$, with $D$ the number of documents (firms) and $d_j$ the number of documents containing word $j$: $$ \text{idf}_j = \log\!\frac{D}{d_j}. $$ The TF–IDF entry for word $j$ in firm $i$ is then $w_{ij} = \text{tf}_{ij}\cdot\text{idf}_j$, where $\text{tf}_{ij}$ is the count of word $j$ in firm $i$. Consider a corpus of $D = 4$ firms with these document frequencies: | Word | $d_j$ (docs containing it) | |:---|:---:| | `company` | 4 | | `products` | 3 | | `oil` | 2 | | `lithium` | 1 | **(a)** [6 pts] Compute $\text{idf}_j$ (using natural log) for each of the four words. Report all four values. Which word is **completely silenced** (its IDF weight is exactly zero), and what is the one-sentence intuition for *why* that word deserves to be silenced in an industry-similarity measure? **(b)** [6 pts] **The spurious-similarity demonstration.** Suppose you *forgot* to strip stop-words, so a boilerplate word — say `company` — survives in both an oil firm's vector and an airline's vector. Concretely, take the oil firm `oil`×2, `crude`×1, `refine`×1, `fuel`×1, `company`×1 and the airline `flights`×1, `airline`×1, `tickets`×1, `travel`×1, `company`×2. Their *only* shared word is `company`. Compute their raw **count** cosine similarity. (You should get a strictly positive number even though they share no product vocabulary.) Then explain in one or two sentences how giving `company` its IDF weight — which, if `company` appears in *every* document, is $\log(D/D) = 0$ — repairs this: what does the spurious similarity become? **(c)** [4 pts] State the general principle in two sentences: TF–IDF up-weights words that are *distinctive* (rare across firms, like `lithium`) and down-weights words that are *ubiquitous* (common across firms, like `company`). Connect this to a vulnerability the Chapter 6.2 guide flags — **boilerplate and legal language** — and say in one clause why down-weighting common words is a partial, not a complete, defense against it. --- ## Problem 3 — Build a TNIC from a similarity matrix (20 points) Now scale up from a pair to a network. You are given the full pairwise cosine-similarity matrix for **five** firms (illustrative, symmetric, rounded to two decimals — built the same way as Problem 1, just at larger scale). The firms are two cloud-software firms (**SoftA**, **SoftB**), a retailer (**RetA**), a bank (**Bank1**), and a cross-over firm **OmniMart** that sells retail goods *and* runs a cloud platform (an Amazon-like straddler — the same role OmniMart plays in nb6.2). > | $\text{sim}(i,k)$ | SoftA | SoftB | OmniMart | RetA | Bank1 | > |:---|:---:|:---:|:---:|:---:|:---:| > | **SoftA** | 1.00 | 0.62 | 0.34 | 0.05 | 0.08 | > | **SoftB** | 0.62 | 1.00 | 0.22 | 0.04 | 0.07 | > | **OmniMart** | 0.34 | 0.22 | 1.00 | 0.41 | 0.06 | > | **RetA** | 0.05 | 0.04 | 0.41 | 1.00 | 0.05 | > | **Bank1** | 0.08 | 0.07 | 0.06 | 0.05 | 1.00 | The TNIC rule (Ch 6.2 §2, step four): firm $i$'s **text-based industry** is the set of *other* firms whose similarity to $i$ clears a chosen threshold $\tau$. Use $\tau = 0.20$ throughout this problem. **(a)** [6 pts] For each of the five firms, list its TNIC industry — the set of other firms with $\text{sim} \ge 0.20$. (Read off each firm's row; ignore the diagonal $1.00$, a firm is not its own competitor.) Write out all five neighbor sets. **(b)** [6 pts] **Firm-specific, not a partition.** Compare SoftA's industry to OmniMart's industry from part (a). Show they are **different sets** — and note the asymmetry of *membership in a set* vs. *having the same set*: SoftA appears in OmniMart's industry, yet OmniMart's industry is not identical to SoftA's. Explain in two sentences why this is impossible under a SIC code (which assigns each firm one box and declares everyone in that box mutual competitors) and what HP mean by calling TNIC **firm-specific**: each firm centers its *own* neighborhood. **(c)** [6 pts] **Intransitivity.** Find a triple of firms $X, Y, Z$ in this matrix such that $\text{sim}(X,Y) \ge \tau$ and $\text{sim}(Y,Z) \ge \tau$ but $\text{sim}(X,Z) < \tau$ — i.e., $X$ and $Z$ are *both* neighbors of $Y$ but are *not* neighbors of each other. State the triple and the three similarity values. Then explain in two sentences why this **intransitivity** is exactly the Amazon-style fact SIC cannot represent (Hint: $Y$ is the cross-over firm; tie it to "neighbor to both Microsoft on cloud and Walmart on groceries, though those two are not neighbors"). **(d)** [2 pts] **The threshold is a knob.** Recompute OmniMart's neighbor *count* (number of firms in its industry) at four thresholds: $\tau = 0.10,\ 0.30,\ 0.40,\ 0.50$. State in one sentence what happens to a firm's competitor count as $\tau$ rises, and why this makes $\tau$ a **lever**, not a fact — a measurement choice that must be disclosed and defended (Ch 6.2 §3, "knobs are levers"). --- ## Problem 4 — TNIC vs. SIC on a cross-over firm (12 points) This problem makes the horse race personal: take the one firm that breaks the official taxonomy and show how each system handles it. Use the same five-firm matrix and $\tau = 0.20$ from Problem 3. OmniMart sells retail goods *and* operates a cloud platform. A government classifier must assign it **one** four-digit SIC code; suppose — because retail is its larger revenue segment — it lands in a **retail** SIC box, the same box as RetA. (This mirrors nb6.2, where the SIC-like foil forces each cross-over firm into a single box and it "loses half its identity.") **(a)** [4 pts] **What SIC sees.** Under the retail SIC box, who are OmniMart's competitors, and who is *invisible* to it? State which of {SoftA, SoftB, RetA, Bank1} the SIC box counts as OmniMart's industry, and which real competitor(s) — visible in the similarity matrix — the box *misses entirely*. **(b)** [5 pts] **What TNIC sees.** From OmniMart's TNIC industry (Problem 3a), state how many *distinct* SIC-style businesses OmniMart neighbors — i.e., does its text-based industry span retail *and* software? Name the neighbors and the business each represents. Then state in one sentence the precise sense in which TNIC *recovers the half of OmniMart's identity* that the single SIC box threw away. **(c)** [3 pts] **The horse-race framing.** The Chapter 6.2 guide insists this is a **measurement** paper: its central claim is validated by a *horse race*, not a causal identifying assumption. State in two sentences what the horse race is — what outcome would TNIC and SIC compete to explain on the *same* firms in the *same* years — and why "TNIC neighbors OmniMart with both a software firm and a retailer" is suggestive but **not yet** the validation (Hint: what would still have to be shown about *outcomes* like co-movement or margins?). --- ## Problem 5 — What's vulnerable: boilerplate, self-description, bag-of-words (16 points) A measurement paper has measurement vulnerabilities — different from the causal threats of Week 5. The right skeptical question is not "what confounds the treatment?" but **"what makes the text a noisy or biased signal of what the firm actually does?"** This problem is pure reasoning; it tests whether you can push on the measure the way a referee would. **(a)** [5 pts] **Boilerplate and legal language.** A 10-K is written by lawyers as much as managers; large stretches are formulaic and shared across firms (risk-factor language, disclaimers, accounting boilerplate). Explain in two or three sentences how shared boilerplate could make two firms in *different* businesses score "similar," and name the *two* construction choices from Problems 2 and the working-world setup that limit this. Then state in one clause why the problem is *worse* for the time-varying comparisons TNIC is built to make (Hint: disclosure norms — and therefore the boilerplate baseline — drift over time). **(b)** [5 pts] **Firms' incentives in self-description.** This is the sharpest critique, specific to *self-reported* text. Firms *choose* how to describe themselves — vaguely to avoid tipping off rivals, grandly to flatter investors ("we are an AI company now"), conservatively to manage legal exposure. Explain in two or three sentences the **reflexivity** problem this creates *specifically because differentiation is itself one of the paper's outcomes*: if a firm that merely *talks* differently is coded as differentiated, what exactly is the measure measuring? Pose the referee's one-sentence question in the form "are we measuring ___, or ___?" **(c)** [4 pts] **Bag-of-words ignores meaning.** The representation throws away word order, synonymy, and context. Give the two canonical failures: (i) a negation example — write two short sentences that mean *opposite* things about competing in a market but produce nearly *identical* bags-of-words; and (ii) a synonym example — two words a human reads as the same product that the bag treats as unrelated. Then state in one sentence which later tool (Ch 6.5) is built precisely to close this gap, and what kind of vector it uses. **(d)** [2 pts] **Threat → guard, one per vulnerability.** For each of (a), (b), (c), name in a phrase the discipline or check a careful replicator uses to address it (one per vulnerability). This is spec discipline (Conventions §4) pointed at a text-based measure: name the threat, then name the guard. --- ## Problem 6 — Replication design: from this problem set to nb6.2 (12 points) You will run this for real in nb6.2. This problem makes you design the replication *before* you touch the keyboard — stating the corpus, the pipeline, and the checks you would commit to in advance. **(a)** [4 pts] **The pipeline, in order.** Name the four steps that turn a folder of product descriptions into a similarity matrix, in the order nb6.2 runs them (Ch 6.2 §2): (1) get/clean the text, (2) build vectors, (3) measure similarity, (4) build the network. For step (2), name the *two* vectorizers nb6.2 compares (`CountVectorizer` for plain counts vs. `TfidfVectorizer` for TF–IDF) and state in one sentence what swapping one for the other lets you *see* about the role of IDF (Hint: Problem 2b). **(b)** [4 pts] **The headline check.** nb6.2 hands you a 16-firm corpus across three latent industries (software/cloud, banking/lending, retail/apparel) plus two deliberate cross-over firms. State the **one quantitative comparison** you would compute to confirm the measure "works" *before* trusting any TNIC neighbor list (Hint: compare the *average* similarity of firms in the *same* latent industry to the average across *different* industries — what ordering must hold, and what is the name for that ratio?). Then name the one *picture* that makes the same point visually (Hint: what lights up on the diagonal of a cluster-ordered heatmap?). **(c)** [4 pts] **Two network properties to reproduce, and the real-data pointer.** Name the two network properties — beyond same-industry blocks — that nb6.2 asks you to demonstrate on the cross-over firm and (if two years of filings are supplied) on a firm that changed its business; tie each to the matching property of TNIC (**intransitivity** and **time-variation**). Finally, state where a real replication would get the text and the ready-made industries instead of re-parsing tens of thousands of filings yourself: name the **source of the raw text** (the public SEC filing system) and the **public data library** Hoberg and Phillips maintain so researchers can download the computed TNIC industries directly. [The exact library URL is left as a verify-before-printing item, consistent with the Ch 6.2 guide.] --- *End of Problem Set 6.2. Solutions: Appendix E, `E-w6-ps6.2-solutions.md`. The companion notebook **nb6.2** (`notebooks/week-06/nb6.2-10k-text-similarity.ipynb`) lets you scale every operation here to a 16-firm corpus — tokenize and TF–IDF-vectorize the product descriptions, compute the full pairwise cosine matrix, print it as a cluster-ordered heatmap and watch the industry blocks light up on the diagonal, then build a TNIC at a threshold $\tau$ and verify its firm-specific and intransitive structure on a cross-over firm — checking your hand arithmetic from Problems 1–4 against `scikit-learn`, with a "Your Turn" extension that adds a firm, sweeps the threshold, or switches to bigrams. The single discipline to carry forward: this is a **measurement** paper, so read every cosine as a *choice-dependent estimate* of product overlap — clean enough to redraw the map of American industry from text alone, but only as good as the words a firm chooses, the stop-words you strip, and the threshold you pick. Citation by name: Hoberg & Phillips (2016).*