Lexicon Sentiment and the Loughran–McDonald Dictionary

Overview & Motivation

Take the $1{,}500$ synthetic financial filings of the linear classifiers topic, labeled negative, neutral, or positive in tone, and score each one with a word list: count its positive words, subtract its negative words, divide by its length. With two cutpoints tuned on the training filings, this net tone gets $0.637$ of the test filings right. Logistic regression on all the word counts got $0.739$ . The dictionary uses no labels at all, apart from its two cutpoints.

Finance learned to distrust such lists the hard way. Loughran and McDonald (2011) found that in a large sample of 10-K filings, almost three-fourths of the words that the widely used Harvard-IV dictionary calls negative are not negative in a financial context: tax, cost, capital, liability, crude. Their finance word lists replaced Harvard-IV across the literature. This page treats a dictionary as what the previous topic would call it, a linear classifier with fixed weights, and asks what the fixing costs. In order:

The category simplex. A dictionary places a document at its rates of positive and negative words, and net tone keeps one direction of that plane.
Length. How many words a document needs before the sign of its score can be trusted.
Contamination. What counting tax as negative does, exactly.
What no word list reads. Directions, the valence trap, and negation.
The dictionary as a prior. Shrinking logistic regression toward the dictionary instead of toward zero.
Inducing a dictionary. Propagating polarity from a few seed words, and where that stops.

The corpus has a hand-written stand-in lexicon (positive, negative, uncertainty, and litigious lists of $15$ to $18$ words each). It is not derived from the Loughran–McDonald dictionary, whose license allows academic research but not redistribution here; every number on this page comes from the stand-in.

A dictionary reads a filing

Here are two sentences from one positive test filing, an MD&A section, with the stand-in’s words marked: “Operating margin in networkpoint was resilient. Liquidity in efficiency operations and multistores surpassed expectations.” Two positive hits, no negative ones. The same filing also says “Management pointed to loss in free cash flow,” one negative hit for a fact that is bad news, and “Estimates of fluctuations could change, and the claimbase outlook remains uncertain given variability in overpipe,” five uncertainty hits that say nothing about direction.

Definition 1 (Lexicon, counts, and net tone).

A lexicon is a family of disjoint word sets, here positive $P$ , negative $N$ , uncertainty, and litigious. For a document with count vector $x$ and length $L = \sum_j x_j$ , the category counts are $n_+ = \sum_{j \in P} x_j$ and $n_- = \sum_{j \in N} x_j$ , the rates are $\hat p_\pm = n_\pm / L$ , the weight vector is $w_L = \mathbf 1_P - \mathbf 1_N$ , the raw score is $s = w_L \cdot x = n_+ - n_-$ , and the net tone is $s/L = \hat p_+ - \hat p_-$ .

The raw score is a linear function of the counts with weights in $\{-1, 0, 1\}$ , so a lexicon is a point of the previous topic’s hypothesis class that was chosen without data. Net tone divides by the length, which is Loughran and McDonald’s proportional measure, and the cutpoints turn it into a three-way decision.

The category simplex

Discard everything about a document except its rates. Then the document is a point $(\hat p_+, \hat p_-)$ in a small triangle, the simplex whose third corner is “everything else”. Net tone measures that point along the diagonal direction $(1, -1)$ , so a net-tone rule cuts the triangle with lines parallel to $\hat p_+ = \hat p_-$ . Whether that is the right cut depends on how the tones place their documents.

Suppose each tone $t$ draws every token’s category independently with rates $q_t = (q_{t+}, q_{t-}, q_{t0})$ , where $q_{t0}$ covers every token that is in neither list, and draws the word within its category from a distribution that does not depend on the tone. Call this a category world. For two tones $t$ and $t'$ , write

$u_c = \log\frac{q_{tc}}{q_{t'c}} - \log\frac{q_{t0}}{q_{t'0}} \quad (c = +, -), \qquad v = \log\frac{q_{t0}}{q_{t'0}}.$

Proposition 1 (The Bayes rule of a category world).

Assume a category world with equal tone priors.

(a) The posterior depends on the document only through its category counts: $\log P(t \mid x) = n_+ \log q_{t+} + n_- \log q_{t-} + n_0 \log q_{t0} + c(x)$ with $n_0 = L - n_+ - n_-$ and $c(x)$ free of $t$ . Divided by $L$ , every tone’s score is affine in $(\hat p_+, \hat p_-)$ , so the Bayes regions are cut by lines.

(b) Tones $t$ and $t'$ are decided by a threshold on net tone if and only if $u_+ = -u_-$ .

(c) In the symmetric-tilt world with $q_0 = (m, m, 1 - 2m)$ , $q_+ = (m e^\beta, m e^{-\beta}, 1 - 2m)/Z$ , and $q_-$ its mirror image, where $Z = 1 + 2m(\cosh\beta - 1)$ , the Bayes rule is: positive if $s/L \gt \tau$ , negative if $s/L \lt -\tau$ , neutral otherwise, with $\tau = \log Z/\beta \gt 0$ . The positive-against-negative log-odds is $2\beta s$ .

Proof.

(a) A word $j$ of category $c$ has probability $q_{tc}\, r_c(j)$ under tone $t$ , so $\log P(x \mid t)$ is the multinomial coefficient, plus $\sum_c n_c \log q_{tc}$ , plus $\sum_j x_j \log r_{c(j)}(j)$ . Only the middle term depends on $t$ , and with equal priors the posterior is the softmax of $n \cdot \log q_t$ . Since $n_0/L = 1 - \hat p_+ - \hat p_-$ , each score divided by $L$ is affine in the rates, and the boundary between two tones is the line $u_+ \hat p_+ + u_- \hat p_- + v = 0$ .

(b) The pair’s log-odds divided by $L$ is $u_+\hat p_+ + u_-\hat p_- + v$ . It is a function of $\hat p_+ - \hat p_-$ alone on the open simplex if and only if its gradient $(u_+, u_-)$ is parallel to $(1, -1)$ , that is, if and only if $u_+ = -u_-$ .

(c) Positive against neutral has $u_+ = \beta$ , $u_- = -\beta$ , and $v = -\log Z$ , so positive wins exactly when $\beta s/L \gt \log Z$ ; negative against neutral is the mirror image. Positive against negative has $u_+ = 2\beta$ , $u_- = -2\beta$ , and $v = 0$ , so its log-odds is $2\beta s$ . Since $\cosh\beta \gt 1$ , we have $Z \gt 1$ and $\tau \gt 0$ , and the three pairwise rules agree on the stated regions.

∎

So net tone is the Bayes rule of a particular world, one in which a positive document trades negative words for positive ones symmetrically. With unequal priors the prior adds $\log \pi_t$ to each score, which becomes a length-dependent offset after dividing by $L$ , so even there a fixed cutpoint on net tone is only right for documents of one length.

The finance corpus is not that world. Positive filings have $1.79$ positive hits per $100$ tokens against $0.69$ in neutral ones, but $0.82$ negative hits against $0.68$ : a positive filing still reports some bad news. Fitting the category world’s rates to the training filings gives $(u_+, u_-) = (0.967, 0.191)$ for positive against neutral, far from antisymmetric. Neutral filings are the ones with few polar words of either kind, and a band of diagonal lines cannot separate them from filings with many polar words in balance. The plug-in Bayes rule of the category world, which uses both counts, scores $0.724$ on the test filings, against net tone’s $0.637$ , close to logistic regression’s $0.739$ on every word. On this split the projection is most of the gap between the dictionary and the learned model; over the previous topic’s six random splits into training, development, and test filings (this page’s is the first), it is, on average, all of it. There the plug-in rule scores $0.734 \pm 0.012$ and logistic regression $0.733 \pm 0.012$ , a difference of $0.001 \pm 0.021$ in the plug-in rule’s favor, which is ahead on three splits of six. The two counts also leave little to find: on $60{,}000$ fresh filings, the best rule on $(n_+, n_-)$ at all, a table of the commonest tone for each pair, scores $0.736$ on $60{,}000$ more, the plug-in rule $0.737$ , and net tone with its best cutpoints $0.638$ . What separates the dictionary from the learned model is the projection, not the word list. The Simplex tab draws the filings and both rules.

Each filing is a point of the category simplex: net tone keeps one direction of it

negative filingneutral filingpositive filing

net tone, tuned cutpoints

0.637

category-world Bayes

0.724

unigram logistic regression

0.739

slopes (u₊, u₋), positive vs neutral

0.967, 0.191

Net tone p̂₊ − p̂₋ can only cut along the dashed diagonals, at the cutpoints -0.619 and 0.75 per 100 tokens. Neutral filings sit near the origin, with few polar words of either kind, and a diagonal band cannot separate them from balanced filings with many. The Bayes rule's boundaries (Proposition 1) are lines too, but they are not parallel to the diagonal. A few filings beyond 3.2 hits per 100 tokens are off the chart.

How long a document must be

Even in a world where net tone is right, a short document does not contain enough hits to decide. Model the tokens of a positive document as independent, each a positive hit with probability $a$ , a negative hit with probability $b \lt a$ , and neither otherwise, and let $S_L$ be the raw score of $L$ tokens.

Theorem 1 (Length and the sign of the score).

Let $\rho = 1 - (\sqrt a - \sqrt b)^2$ . Then (a) $P(S_L \le 0) \le \rho^L$ for every $L$ , and (b) $\frac1L \log P(S_L \le 0) \to \log\rho$ as $L \to \infty$ .

Proof.

(a) For $t \ge 0$ , Markov’s inequality gives $P(S_L \le 0) = P(e^{-tS_L} \ge 1) \le E\,e^{-tS_L} = M(t)^L$ with $M(t) = a e^{-t} + b e^{t} + 1 - a - b$ . The function $M$ is strictly convex, with $M'(t) = 0$ where $e^{2t} = a/b$ , at $t^\ast = \tfrac12 \log(a/b) \gt 0$ , and there $M(t^\ast) = 2\sqrt{ab} + 1 - a - b = 1 - (\sqrt a - \sqrt b)^2$ .

(b) Cramér’s theorem for the mean $S_L/L$ (formalStatistics: large deviations, Theorem 6; Dembo and Zeitouni, Theorem 2.2.3) has rate function $I(y) = \sup_\lambda [\lambda y - \log M(-\lambda)]$ . Because $E X = a - b \gt 0$ , the function $I$ is convex, vanishes at $a - b$ , and is finite and continuous on $(-1, 1)$ , so its infimum over $(-\infty, 0)$ equals its infimum over $(-\infty, 0]$ , which is $I(0) = -\log\min_t M(t) = -\log\rho$ . The upper and lower bounds of Cramér’s theorem therefore meet at $-\log \rho$ .

∎

The factor $\rho = 2\sqrt{ab} + (1 - a - b)$ is the Bhattacharyya coefficient between the hit distributions of positive and negative documents, so $1 - \rho = (\sqrt a - \sqrt b)^2$ is their squared Hellinger distance, and the error is small once $L(\sqrt a - \sqrt b)^2$ is large. For the stand-in on positive filings, $a = 0.0179$ and $b = 0.0082$ , so $\rho = 0.998$ . Positive filings average $210.5$ tokens, about $5.5$ expected hits, and at that length the bound is $0.671$ . That is loose: the exact error of the independent model is $0.253$ . The corpus is not independent tokens, since its tone lives in five to eight aspect sentences, yet on this split the measured share of positive filings with a raw score of zero or below is $0.252$ , which the independent model matches. That match is particular to this split. Over the six splits the measured share is $0.217 \pm 0.020$ , and this split’s $0.252$ is the largest of the six, while the independent model, at each filing’s own length, predicts $0.252 \pm 0.010$ . On average it overpredicts the error, because the corpus’s scores are less spread out than independent tokens would make them: their variance is $0.76 \pm 0.06$ of the independent model’s. Almost all of that comes from the positive count, whose variance is $2.15$ against $3.91$ for independent tokens. The two counts are slightly negatively correlated (covariance $-0.19$ , against $+0.07$ for independent tokens), and since the score is their difference, that widens it rather than narrowing it. Still, one positive filing in four or five is misread before any subtlety of language enters. A lexicon ten times denser would bring the error at the corpus’s $210$ tokens down to $0.003$ , which is why finance dictionaries are long. For small rates, ten times the density and ten times the length change the exponent $L(\sqrt a - \sqrt b)^2$ identically, and the exact errors agree: $0.0028$ at ten times the rates, $0.0030$ at ten times the length. The Length tab plots the exact error and the bound for three presets.

When is a liability not a liability

§ Finance case study: a general-purpose list in a 10-K

A general-purpose negative list reads tax, cost, capital, liability, and crude as bad news. In a filing they are the vocabulary of accounting. The corpus plants these five words and their plurals, eight forms in all, in neutral sentences that report a tax expense, a lease liability, or a cost of revenue, and the general list adds them to the stand-in’s negative list.

Proposition 2 (Contamination by tone-independent words).

Add to the negative list a word set, disjoint from $P$ and $N$ , whose tokens occur at rate $c$ in every tone, taking that mass from the non-polar tokens. Per token, in positive documents (hit rates $a$ , $b$ ) and negative ones ( $b$ , $a$ ): (i) the gap between the two tones’ means stays $2(a - b)$ ; (ii) the variances rise by exactly $c(1 - c + 2(a - b))$ and $c(1 - c - 2(a - b))$ ; (iii) if $a \ne b$ and $c + 2\lvert a - b\rvert \lt 1$ , Fisher’s ratio $(\mu_+ - \mu_-)^2 / \tfrac12(\sigma_+^2 + \sigma_-^2)$ falls strictly. The same holds for documents, with means and variances multiplied by $L$ .

Proof.

A token in a positive document now scores $+1$ with probability $a$ and $-1$ with probability $b + c$ . Its mean is $a - b - c$ and its second moment is $a + b + c$ , so its variance is $a + b + c - (a - b - c)^2$ , and subtracting the clean variance $a + b - (a - b)^2$ leaves $c - c^2 + 2c(a - b)$ . In a negative document the mean is $b - a - c$ , and the same computation gives $c - c^2 - 2c(a - b)$ . Both means move by $-c$ , so the gap is unchanged. Under the stated condition both increments are positive, so the ratio’s denominator grows while its numerator stays fixed. Over $L$ independent tokens the means and variances scale by $L$ .

∎

Re-tuning the cutpoints moves them with the means and absorbs the shift. The added variance cannot be tuned away. On the corpus the polysemes add $1.815$ , $1.77$ , and $1.837$ negative hits per $100$ tokens to negative, neutral, and positive filings, nearly the same rate in every tone, as Proposition 2 assumes. Every mean slides left, by about that amount. Fisher’s ratio between the polar tones falls from $3.653$ to $2.457$ ; over the six splits the ratio of the general list’s to the stand-in’s is $0.619 \pm 0.029$ , near Proposition 2’s independent-token prediction of $0.593$ . The share of polar filings whose score has the right sign at a cut of zero falls from $0.759$ to $0.600$ , but that cut pays the mean shift, which re-tuning absorbs. With one cut re-tuned on the training filings, the share falls only from $0.824$ to $0.799$ ; over the six splits, from $0.834 \pm 0.012$ to $0.787 \pm 0.011$ , against $0.781$ to $0.591$ at zero. About a quarter of the drop at zero is the spread. Three-class accuracy with re-tuned cutpoints falls from $0.637$ to $0.572$ . In positive filings the general list’s negative rate ( $0.0265$ ) now exceeds the positive rate ( $0.0179$ ), so Theorem 1 gives no bound at all, and the raw score’s sign gets worse with length. The Contamination tab shows the three histograms slide and spread.

The real filings committed with the corpus show the same words. In three short MD&A excerpts from public 10-Ks, the finance stand-in finds $2$ negative words, while a general list would add $4$ more: cost, liabilities, and tax from ExxonMobil, taxes from Microsoft. The counts are small because the excerpts are, but the direction is Loughran and McDonald’s.

What no word list reads

The corpus’s gold annotation records each aspect a sentence evaluates and its polarity for the company, and its templates tell us how each sentence was written. Restrict to the $5{,}171$ test sentences whose aspects share one nonzero polarity, and ask whether the sign of the sentence’s raw score matches it.

construction	sentences	right
adjective (“margins were solid”)	$1{,}205$	$1.000$
direction with a comment (“rose 4 percent, a trend executives characterized as solid”)	$1{,}746$	$1.000$
bare direction (“costs rose 5 percent”)	$595$	$0$ (abstains)
negated adjective (“revenue was not weak”)	$302$	$0$
verb, up is good (“sales beat expectations”)	$446$	$1.000$
verb, cost-like (“costs exceeded expectations”)	$352$	$0$
noun phrase (“Management pointed to strength in demand”)	$525$	$1.000$

Overall the lexicon gets $0.758$ of these sentences right and abstains on $0.115$ . The failures are of two kinds. A bare direction and a cost-like “exceeded expectations” are the previous topic’s exclusive-or of aspect valence and direction: “costs rose” is bad news and “revenue rose” good news, written with the same word, and no fixed weight on single words can read both. Negation is different, because it can be patched with a scope rule.

Definition 2 (Negation scope).

With scope $k$ , a positive or negative hit at position $i$ counts for the other side if a negator (not, no, never) occurs among positions $i - k, \dots, i - 1$ .

Das and Chen (2007) mark negated words in this way, and VADER (Hutto and Gilbert 2014) looks three tokens back. On this corpus, whose only negation template is “not” followed by an adjective, scope $k = 1$ repairs all $302$ negated sentences, raising sentence accuracy to $0.817$ and three-class document accuracy from $0.637$ to $0.696$ . Wider scopes change nothing until $k = 8$ , where $36$ flips land on words in the next sentence and document accuracy slips to $0.694$ . Real text negates in more ways and at more distances, so the scope is a trade-off there, not a free fix. The Constructions tab shows the table with and without the scope.

Loughran and McDonald’s other lists measure something other than polarity. Uncertainty words occur at $1.618$ , $1.618$ , and $1.565$ per $100$ tokens in negative, neutral, and positive filings: the same rate in every tone. Litigious words occur only in the 8-K filings, at $1.330$ per $100$ tokens. The rule “any uncertainty word means neutral” scores $0.414$ , far below net tone, and a logistic model on all four category rates scores $0.725$ , the same accuracy it gets from the two polarity rates alone. The two models are not identical: their predictions differ on $3.8\%$ of the test filings, and those differences cancel exactly in accuracy. Uncertainty is not neutrality; it is a separate dimension of the text.

The dictionary as a prior

A dictionary is free and a labeled filing is not. The previous topic’s logistic regression shrinks its weights toward zero; we can shrink them toward the dictionary instead. Let $A$ be a $V \times r$ matrix of dictionary directions, either $A = w_L$ (net tone, $r = 1$ ) or $A = [\mathbf 1_P, \mathbf 1_N]$ (the two counts, $r = 2$ ), and give class $k$ the logit $x \cdot \Delta_k + (A^\top x) \cdot \beta_k + b_k$ , so that its weight vector is $W_k = \Delta_k + A\beta_k$ . Fit by minimizing the mean cross-entropy plus $\frac\lambda2 \sum_k \lVert\Delta_k\rVert^2$ , with no penalty on $\beta$ or $b$ .

Theorem 2 (Shrinking toward a dictionary).

Let $A$ have full column rank, let $P_\perp$ be the orthogonal projection onto the complement of its column space, and let $G = \frac1n \sum_i \lVert x_i \rVert$ .

(a) The fit is equivalent to minimizing the cross-entropy of $W$ plus $\frac\lambda2 \sum_k \lVert P_\perp W_k \rVert^2$ , ridge on the part of each class’s weights orthogonal to the dictionary’s directions, with $\Delta_k = P_\perp W_k$ . In particular $A^\top \Delta_k = 0$ .

(b) At the optimum, $\lVert \Delta_k \rVert \le G/\lambda$ for every class $k$ .

(c) If the limiting problem, softmax regression on the dictionary counts $A^\top x$ alone, has a nondegenerate minimizer (after fixing the common shift of the softmax), then the fitted probabilities converge to its probabilities as $\lambda \to \infty$ , uniformly on bounded sets, at rate $O(1/\lambda)$ .

Proof.

(a) The cross-entropy depends on $(\Delta, \beta)$ only through $W$ . For a fixed $W_k$ , minimizing $\lVert\Delta_k\rVert^2$ over the decompositions $W_k = \Delta_k + A\beta_k$ is least squares in $\beta_k$ , with minimum $\lVert P_\perp W_k \rVert^2$ attained at $\Delta_k = P_\perp W_k$ .

(b) Setting the gradient in $\Delta_k$ to zero gives $\nabla_{W_k}\mathrm{CE} + \lambda\Delta_k = 0$ . (Setting the gradient in $\beta_k$ to zero gives $A^\top \nabla_{W_k}\mathrm{CE} = 0$ , which is the orthogonality of (a) again.) Since $\nabla_{W_k}\mathrm{CE} = \frac1n \sum_i (p_{ik} - y_{ik})\, x_i$ and $\lvert p_{ik} - y_{ik} \rvert \le 1$ , we get $\lVert\Delta_k\rVert \le G/\lambda$ .

(c) For fixed $\Delta$ , the optimal $(\beta, b)$ minimizes the convex function $\mathrm{CE}(\Delta, \cdot, \cdot)$ , and at $\Delta = 0$ its stationarity equations are those of the limiting problem. Nondegeneracy and the implicit function theorem give a smooth branch $(\beta(\Delta), b(\Delta))$ of minimizers near $\Delta = 0$ , and by convexity the optimum at $\Delta_\lambda$ lies on it, so $(\beta_\lambda, b_\lambda)$ is within $O(\lVert \Delta_\lambda \rVert)$ of the limit. Every logit then moves by $O(\lVert x\rVert\,\lVert\Delta_\lambda\rVert)$ , and the softmax is Lipschitz, so by (b) the probabilities move by $O(1/\lambda)$ .

∎

On the full training set, $G = 26.364$ , while for the prior toward net tone $\lambda \max_k \lVert\Delta_k\rVert$ is at most $0.725$ , so the constant in (b) is loose by a factor of $36$ at $\lambda = 1{,}000$ , its tightest, and about $97$ at $\lambda = 1$ ; for the two counts the factor is $95$ to $147$ . At $\lambda = 1{,}000$ the slack has two sources: cancellation of the terms $(p_{ik} - y_{ik})\,x_i$ across documents, a factor of about $14$ , and residuals $\lvert p_{ik} - y_{ik}\rvert$ well below $1$ , about $2.5$ . The largest gap between the fitted and limiting test probabilities falls from $0.4674$ at $\lambda = 1$ to $0.1591$ , $0.0215$ , and $0.0022$ at $\lambda = 10$ , $100$ , and $1{,}000$ : one decade of accuracy per decade of $\lambda$ .

The learning curves set the price of the dictionary in labeled filings. Each column is a number of labeled training filings $n$ ; small $n$ average ten random samples, and every penalty is chosen on $600$ development filings.

method	$10$	$100$	$300$	$1{,}000$	$1{,}500$
dictionary, net tone	$0.547$	$0.620$	$0.630$	$0.639$	$0.639$
dictionary, two counts	$0.623$	$0.712$	$0.719$	$0.717$	$0.717$
logistic regression on the $35$ lexicon words	$0.471$	$0.639$	$0.706$	$0.729$	$0.739$
logistic regression on every word	$0.399$	$0.532$	$0.614$	$0.719$	$0.739$
prior toward net tone	$0.549$	$0.701$	$0.723$	$0.751$	$0.766$
prior toward two counts	$0.616$	$0.712$	$0.725$	$0.748$	$0.757$

The dictionary rows are flat, because they fit only a scale and two cutpoints. On this grid, plain logistic regression first matches the dictionary at $n = 1{,}000$ , for both forms of it. But the grid jumps from $300$ to $1{,}000$ , and its $1{,}000$ and $1{,}500$ columns are one sample each. A finer grid, with ten random samples at each $n$ , counts the samples on which logistic regression matches or beats each form of the dictionary:

labeled filings $n$	$300$	$400$	$500$	$600$	$700$	$800$	$1{,}000$	$1{,}200$
matches or beats net tone	$5$	$9$	$10$	$10$	$10$	$10$	$10$	$10$
matches or beats the two counts	$0$	$0$	$0$	$0$	$0$	$0$	$3$	$9$

So net tone is worth $300$ to $400$ labeled filings. The two counts are worth $1{,}000$ to $1{,}200$ on this split and about $1{,}500$ on average: over the six splits, at all $1{,}500$ filings, logistic regression only ties them, ahead by $0.0006 \pm 0.021$ . Restricting the regression to the dictionary’s words, as Jegadeesh and Wu (2013) do with market reactions, climbs faster and ends at the same $0.739$ . The priors are never more than $0.01$ below the better of their two ends, and at $1{,}500$ filings the prior toward net tone beats plain logistic regression, $0.766$ to $0.739$ . That lead is not an artifact of the grid. Plain logistic regression chose $\lambda = 0.01$ , the top of its grid, on all six splits, but on a wider grid, from $1$ to $0.001$ , it scores $0.734 \pm 0.009$ , unchanged, and the prior toward net tone still leads it by $0.031 \pm 0.014$ , on all six. As a prior the dictionary is worth about a thousand labeled filings: with $300$ of them, the prior toward net tone scores $0.731$ on the same ten samples, which plain logistic regression reaches between $1{,}200$ and $1{,}500$ . Along the penalty path at $n = 1{,}500$ , the prior’s accuracy rises from the dictionary’s $0.639$ at $\lambda = 10^4$ to $0.766$ at $\lambda = 0.1$ , then falls to $0.731$ at $\lambda = 10^{-3}$ . Even there it is not ordinary ridge logistic regression, which scores $0.721$ at that $\lambda$ : the prior never penalizes its dictionary directions. At equal $\lambda$ the prior leads at each of $0.1$ ( $0.766$ against $0.730$ ), $0.01$ ( $0.753$ against $0.739$ ), and $0.001$ ( $0.731$ against $0.721$ ). All six methods remain well below the corpus’s full-text Bayes accuracy of $0.890$ , which needs the aspect structure that no bag of words sees. The Prior tab draws the curves and the path.

Inducing a dictionary

Loughran and McDonald built their lists by reading words in context. The alternative is to induce a list from unlabeled text: start from a few seed words of known polarity and spread polarity to words that keep their company (Hatzivassiloglou and McKeown 1997; Turney 2002; Hamilton and colleagues 2016). One way to spread it is to solve a Dirichlet problem on a word graph.

Let $W$ be a symmetric nonnegative matrix of edge weights on a vertex set $S \cup U$ , with seed values $f_S \in \{-1, +1\}^S$ , degrees $d_v = \sum_j W_{vj}$ , and $D = \operatorname{diag}(d)$ .

Theorem 3 (Harmonic polarity).

Suppose every vertex of $U$ is joined to some seed by a path of positive weight. Then (a) $(D - W)_{UU}$ is positive definite, so the system $(D - W)_{UU} f_U = W_{US} f_S$ has exactly one solution; (b) that solution is harmonic, $f_v = \sum_j (W_{vj}/d_v) f_j$ for $v \in U$ , and satisfies $\min f_S \le f \le \max f_S$ ; (c) $f_v = 2h_v - 1$ , where $h_v$ is the probability that the random walk with steps $W_{vj}/d_v$ , started at $v$ , reaches a positive seed before a negative one; and (d) clamped label propagation, which repeats $f \leftarrow D^{-1}Wf$ and resets the seeds, converges to $f$ .

Proof.

(a) For $g$ on $U$ , $g^\top (D - W)_{UU}\, g = \tfrac12 \sum_{i, j \in U} W_{ij}(g_i - g_j)^2 + \sum_{i \in U} g_i^2 \sum_{s \in S} W_{is} \ge 0$ . If it is $0$ , then $g$ is constant on each connected component of $U$ and vanishes at every vertex adjacent to a seed; every component contains such a vertex, so $g = 0$ .

(b) The averaging identity is row $v$ of the system. Let $\mu = \max_U f$ and suppose $\mu \gt \max f_S$ . At a maximizer, $f_v$ is an average of neighbors’ values that are all at most $\mu$ , so every neighbor also equals $\mu$ ; following a path from $v$ to a seed reaches a seed of value $\mu \gt \max f_S$ , which is impossible. The minimum is symmetric.

(c) From every vertex the seeds are reachable, so the walk is absorbed almost surely. Conditioning on its first step gives $h_v = \sum_j (W_{vj}/d_v) h_j$ on $U$ , with $h = 1$ on positive seeds and $h = 0$ on negative ones. Then $2h - 1$ is harmonic on $U$ with the boundary values $\pm 1$ , and by (a) it equals $f$ .

(d) The iteration is the Jacobi method for the system of (a). Its matrix $D_{UU}^{-1} W_{UU}$ is substochastic, and mass leaks to the seeds from every component, so its spectral radius is below $1$ .

∎

Zhu, Ghahramani, and Lafferty (2003) introduced this solution for semi-supervised learning, and Doyle and Snell (1984) give the classical theory. On the corpus we build the graph from the training filings: one vertex per word type in at least $30$ filings ( $699$ types), edge weights equal to the positive part of the words’ pointwise mutual information across documents, and seeds strong, robust against weak, adverse. We report each word’s harmonic value centered at the median and divided by the standard deviation, a score $z$ .

The signs come out mostly right: $25$ of the $28$ other stand-in polar words in the graph land on their own side ( $0.893$ ), and over $200$ random choices of two positive and two negative seeds the median is $0.821$ . Positive filings contain more positive words, so positive words share documents with each other, and propagation finds that. The magnitudes are another matter. The polar words have mean $\lvert z \rvert$ of $1.085$ and $1.055$ , but the polysemes have $0.907$ : liability ( $-1.462$ ), taxes ( $-1.267$ ), crude ( $-1.143$ ), and tax ( $-0.911$ ) land on the negative side, which reproduces part of the general list’s mistake. And the ten most positive words in the whole graph include just one lexicon word, strength; the rest are mostly sector words such as therapeutics and biosciences, along with on and headquartered, that happen to share documents with the seeds. Propagation is good at scoring a candidate list. It cannot choose the list, and choosing it is the curation Loughran and McDonald did by hand. The Induction tab plots every stand-in word’s score.

Where this sits

Dictionary tone entered finance with Tetlock (2007), who found that the Harvard-IV pessimism of a Wall Street Journal column predicts market returns. Loughran and McDonald (2011) showed that the same dictionary misreads 10-Ks and built the finance lists that replaced it, and their survey (2016) argues that a transparent list is worth some accuracy in research that others must replicate. Their lists also come with a tf–idf weighting, which formalRAG’s vector-space model topic derives; outside finance, Taboada and colleagues (2011) build lexicons with intensifiers and graded negation.

The next topic, aspect-based sentiment, attaches polarity to the aspect a sentence evaluates, which is what this page’s bare directions and cost-like verbs need. The capstone scores earnings calls with a dictionary and a learned model side by side.

Honest caveats

⚠ Rigor flag

Proved and cited. Propositions 1–2 and Theorems 1–3 are proved here, except the lower bound in Theorem 1(b), which is Cramér’s theorem.
Independent tokens. Proposition 1 and Theorem 1 assume each token’s category is drawn independently given the tone. The corpus breaks this too. The independent model matches the measured sign error on this page’s split, but over six splits it overpredicts it, $0.252$ against $0.217$ , because the corpus’s scores are under-dispersed.
A planted general list. Proposition 2’s general-purpose list is the stand-in plus eight planted finance polysemes, not Harvard-IV. Loughran and McDonald’s “almost three-fourths” is quoted from their paper, and no number here comes from their dictionary.
A friendly negation template. A one-token scope is exact here only because the corpus negates in one way.
Loose constants, tuned penalties. Theorem 2(c) assumes a nondegenerate limit, and the constant $G$ of Theorem 2(b) is loose by a factor of $36$ at its tightest and about $97$ at $\lambda = 1$ . The learning curves choose every penalty on $600$ development filings, which is a luxury when labels are the scarce resource.
Induction on a synthetic corpus. Theorem 3’s graph comes from templates that decide which words share documents. The tf–idf weighting is cited, not tested.

Implementation

The companion notebook (notebookPath) imports lexicon_sentiment_loughran_mcdonald.py, which owns every number on this page. That module imports the linear classifiers topic’s module for its train, development, and test split, its features, and its trust-region Newton fits, and the shared finance corpus through it. A harness of $29$ tests runs in about $45$ seconds on a laptop CPU, and --slow adds four checks that refit on all six of the linear classifiers topic’s random splits, on the finer learning-curve grid, and on $60{,}000$ fresh filings, about five minutes more. It checks:

Proposition 1 against the literal multinomial likelihood in random category worlds and tilt worlds, and on the corpus; with --slow, the plug-in rule against logistic regression on six splits and against the best count reader on fresh filings.
Theorem 1’s bound against the exact law of the score on $250$ instances, its exponent’s convergence, its prediction on the corpus and on all six splits, and the dense preset at the corpus’s length.
Proposition 2’s exact moments on $500$ random instances, and the general list’s effect on the corpus, on all six splits with the cut at zero and re-tuned, and on the EDGAR excerpts.
The example filing’s quotations, the construction table, the negation sweep, and the uncertainty and litigious profiles.
Theorem 2’s orthogonality, bound, slack, and $1/\lambda$ convergence for both dictionary subspaces, the learning curves, and plain logistic regression at the prior’s $\lambda$ ; with --slow, the finer curves and the wide grids.
Theorem 3’s uniqueness, maximum principle, label propagation, and a Monte Carlo random walk, and the induction on the corpus.
That no file in the repository looks like the Loughran–McDonald dictionary.

The module also includes load_lm_dictionary(), which loads the real dictionary into the stand-in’s format if you supply a copy (set LM_MASTER_DICTIONARY, or opt in to a download into ~/.cache/formalnlp/lm/). Nothing on this page or in the harness depends on it. The laboratory reads its data from a JSON file that one of the harness’s tests requires to equal the module’s viz_payload() exactly, so the figure cannot drift from the code.

Overview & Motivation

A dictionary reads a filing

The category simplex

How long a document must be

When is a liability not a liability

What no word list reads

The dictionary as a prior

Inducing a dictionary

Where this sits

Honest caveats

Implementation

References & Further Reading