---
title: "Most-used equations and models: finance, economics, AI"
description: "The most-mentioned equations and models in the tracked finance, accounting, economics, management, operations research and AI journals over the last 30 days were Scaled dot-product attention (Transformer) (203), Difference-in-differences (135), Logistic regression (logit) (123), Linear and integer programming (116) and Instrumental variables (two-stage least squares) (115)."
url: https://scholar4ai.com/rankings/equations
markdown_url: https://scholar4ai.com/rankings/equations.md
license: CC BY-NC 4.0 (https://creativecommons.org/licenses/by-nc/4.0/)
data_url: https://scholar4ai.com/data/rankings.json
updated: 2026-10-26T12:13:05+09:00
window: 2026-09-27 to 2026-10-26
source: OpenAlex (CC0)
---

# Most-used equations and models

A paper counts once for each named equation or model it mentions, in its title, abstract, keywords or full text; a paper that mentions several counts for each of them. Each row gives the formula and the name. Window: 2026-09-27 – 2026-10-26 (30 days).

## Key findings

- The most-mentioned equations and models in the tracked finance, accounting, economics, management, operations research and AI journals over the last 30 days were Scaled dot-product attention (Transformer) (203), Difference-in-differences (135), Logistic regression (logit) (123), Linear and integer programming (116) and Instrumental variables (two-stage least squares) (115).
- Most-mentioned in each journal category: Finance: Difference-in-differences (47); Accounting: Instrumental variables (two-stage least squares) (6); Economics: Difference-in-differences (64); Management: Structural equation modeling (SEM) (62); Operations Research: Linear and integer programming (100); AI: Scaled dot-product attention (Transformer) (175).
- 1,365 of the 3,332 papers (41%) mention at least one of the 112 watched equations and models.

*Most-used equations and models · 2026-09-27 – 2026-10-26 (30 days)*

| # | Equation or model | Formula | Area | Papers | By journal category | Last 7 days |
|---:|---|---|---|---:|---|---:|
| 1 | [Scaled dot-product attention (Transformer)](https://scholar4ai.com/equations/scaled-dot-product-attention-transformer.md) | $\mathrm{Attention}(Q, K, V) = \mathrm{softmax}\Big(\frac{Q K^\top}{\sqrt{d_k}}\Big) V$ | Machine learning | 203 | AI 175 · Operations Research 11 · Management 9 · Finance 7 · Economics 1 | 16 |
| 2 | [Difference-in-differences](https://scholar4ai.com/equations/difference-in-differences.md) | $Y_{it} = \alpha_i + \lambda_t + \delta\,(\mathit{Treat}_i \times \mathit{Post}_t) + \varepsilon_{it}$ | Causal inference | 135 | Economics 64 · Finance 47 · Management 17 · Accounting 5 · Operations Research 1 · AI 1 | 8 |
| 3 | [Logistic regression (logit)](https://scholar4ai.com/equations/logistic-regression-logit.md) | $\Pr(y = 1 \mid x) = \frac{1}{1 + e^{-x^\top\beta}}$ | Regression models | 123 | Economics 36 · AI 29 · Management 21 · Finance 18 · Operations Research 17 · Accounting 2 | 16 |
| 4 | [Linear and integer programming](https://scholar4ai.com/equations/linear-and-integer-programming.md) | $\min_{x}\ c^\top x\ \ \text{s.t.}\ \ A x \le b,\ x \ge 0,\ x_j \in \mathbb{Z}\ (j \in I)$ | Efficiency and operations | 116 | Operations Research 100 · AI 11 · Economics 4 · Finance 1 | 7 |
| 5 | [Instrumental variables (two-stage least squares)](https://scholar4ai.com/equations/instrumental-variables-two-stage-least-squares.md) | $\hat\beta_{\mathrm{2SLS}} = (X^\top P_Z X)^{-1} X^\top P_Z\, y,\quad P_Z = Z (Z^\top Z)^{-1} Z^\top$ | Causal inference | 115 | Economics 50 · Finance 38 · Management 18 · Accounting 6 · Operations Research 2 · AI 1 | 12 |
| 6 | [Convolutional neural network (CNN)](https://scholar4ai.com/equations/convolutional-neural-network-cnn.md) | $(I * K)(i, j) = \sum_{m}\sum_{n} I(i + m,\, j + n)\, K(m, n)$ | Machine learning | 114 | AI 99 · Finance 5 · Management 4 · Economics 3 · Operations Research 3 | 11 |
| 7 | [Ordinary least squares (OLS)](https://scholar4ai.com/equations/ordinary-least-squares-ols.md) | $\hat\beta = (X^\top X)^{-1} X^\top y$ | Regression models | 113 | Finance 41 · Economics 41 · Management 24 · Accounting 4 · AI 2 · Operations Research 1 | 20 |
| 8 | [Graph neural network (graph convolution)](https://scholar4ai.com/equations/graph-neural-network-graph-convolution.md) | $H^{(l+1)} = \sigma\big(\tilde D^{-1/2} \tilde A\, \tilde D^{-1/2} H^{(l)} W^{(l)}\big)$ | Machine learning | 110 | AI 99 · Operations Research 5 · Management 4 · Finance 2 | 8 |
| 9 | [Structural equation modeling (SEM)](https://scholar4ai.com/equations/structural-equation-modeling-sem.md) | $\eta = B\eta + \Gamma\xi + \zeta,\quad y = \Lambda_y\,\eta + \varepsilon$ | Regression models | 81 | Management 62 · Finance 9 · Accounting 4 · AI 3 · Operations Research 2 · Economics 1 | 7 |
| 10 | [Contrastive loss (InfoNCE)](https://scholar4ai.com/equations/contrastive-loss-infonce.md) | $\mathcal{L} = -\log \frac{\exp(\mathrm{sim}(z_i, z_j)/\tau)}{\sum_{k \ne i} \exp(\mathrm{sim}(z_i, z_k)/\tau)}$ | Machine learning | 80 | AI 77 · Accounting 1 · Management 1 · Operations Research 1 | 3 |
| 11 | [Long short-term memory (LSTM)](https://scholar4ai.com/equations/long-short-term-memory-lstm.md) | $c_t = f_t \odot c_{t-1} + i_t \odot \tilde c_t,\quad h_t = o_t \odot \tanh(c_t)$ | Machine learning | 73 | AI 44 · Management 8 · Operations Research 8 · Economics 7 · Finance 6 | 9 |
| 12 | [Mediation model (indirect effect)](https://scholar4ai.com/equations/mediation-model-indirect-effect.md) | $M = \alpha_0 + a X + e_1,\quad Y = \beta_0 + c' X + b M + e_2,\quad \text{indirect effect} = a\,b$ | Causal inference | 72 | Management 32 · Finance 26 · Economics 11 · Accounting 2 · Operations Research 1 | 7 |
| 13 | [Gradient boosting (XGBoost)](https://scholar4ai.com/equations/gradient-boosting-xgboost.md) | $\mathcal{L} = \sum_i l(y_i, \hat y_i) + \sum_k \Omega(f_k),\quad \Omega(f) = \gamma T + \tfrac{1}{2}\lambda \lVert w \rVert^2$ | Machine learning | 68 | AI 28 · Finance 13 · Economics 11 · Management 10 · Operations Research 5 · Accounting 1 | 10 |
| 14 | [Random forest](https://scholar4ai.com/equations/random-forest.md) | $\hat f(x) = \frac{1}{B}\sum_{b=1}^{B} T_b(x)$ | Machine learning | 65 | AI 29 · Finance 11 · Management 11 · Economics 8 · Operations Research 6 | 9 |
| 15 | [Two-way fixed-effects panel regression](https://scholar4ai.com/equations/two-way-fixed-effects-panel-regression.md) | $y_{it} = \alpha_i + \lambda_t + x_{it}^\top\beta + \varepsilon_{it}$ | Regression models | 63 | Finance 25 · Economics 24 · Management 11 · Operations Research 2 · AI 1 | 7 |
| 16 | [Principal component analysis (PCA)](https://scholar4ai.com/equations/principal-component-analysis-pca.md) | $\max_{\lVert w \rVert = 1}\ w^\top \Sigma\, w$ | Machine learning | 58 | AI 14 · Finance 13 · Economics 13 · Management 8 · Operations Research 7 · Accounting 3 | 6 |
| 17 | [Total factor productivity (Solow residual)](https://scholar4ai.com/equations/total-factor-productivity-solow-residual.md) | $\ln A_t = \ln Y_t - \alpha \ln K_t - (1 - \alpha) \ln L_t$ | Growth, trade and environment | 57 | Economics 28 · Finance 19 · Management 8 · Operations Research 1 · AI 1 | 3 |
| 18 | [Support vector machine (SVM)](https://scholar4ai.com/equations/support-vector-machine-svm.md) | $\min_{w, b, \xi}\ \tfrac{1}{2}\lVert w \rVert^2 + C \sum_i \xi_i\ \ \text{s.t.}\ \ y_i (w^\top x_i + b) \ge 1 - \xi_i,\ \xi_i \ge 0$ | Machine learning | 56 | AI 31 · Finance 8 · Operations Research 7 · Economics 5 · Management 5 | 6 |
| 18 | [Vector autoregression (VAR)](https://scholar4ai.com/equations/vector-autoregression-var.md) | $y_t = c + \sum_{i=1}^{p} A_i\, y_{t-i} + \varepsilon_t$ | Time series | 56 | Finance 25 · Economics 22 · Management 5 · Operations Research 3 · AI 1 | 4 |
| 20 | [Propensity score matching](https://scholar4ai.com/equations/propensity-score-matching.md) | $e(x) = \Pr(D = 1 \mid X = x)$ | Causal inference | 55 | Finance 22 · Economics 18 · Management 11 · Accounting 2 · Operations Research 1 · AI 1 | 2 |
| 21 | [Shapley value (SHAP)](https://scholar4ai.com/equations/shapley-value-shap.md) | $\phi_i(v) = \sum_{S \subseteq N \setminus \{i\}} \frac{\lvert S \rvert!\,(n - \lvert S \rvert - 1)!}{n!}\big[v(S \cup \{i\}) - v(S)\big]$ | Game theory and decision analysis | 49 | AI 24 · Management 8 · Operations Research 8 · Economics 6 · Finance 3 | 3 |
| 22 | [Quantile regression](https://scholar4ai.com/equations/quantile-regression.md) | $\hat\beta_\tau = \operatorname*{arg\,min}_{\beta} \sum_{i} \rho_\tau(y_i - x_i^\top\beta),\quad \rho_\tau(u) = u\,(\tau - \mathbf{1}\{u < 0\})$ | Regression models | 48 | Economics 20 · Finance 16 · Management 5 · AI 3 · Accounting 2 · Operations Research 2 | 5 |
| 23 | [Generalized method of moments (GMM)](https://scholar4ai.com/equations/generalized-method-of-moments-gmm.md) | $\hat\theta = \operatorname*{arg\,min}_{\theta}\ \bar g(\theta)^\top W\, \bar g(\theta),\quad \bar g(\theta) = \frac{1}{n}\sum_{i=1}^{n} g(w_i, \theta)$ | Regression models | 44 | Finance 16 · Economics 16 · Management 10 · Accounting 1 · Operations Research 1 | 1 |
| 24 | [Cross-entropy loss](https://scholar4ai.com/equations/cross-entropy-loss.md) | $\mathcal{L} = -\sum_{c=1}^{C} y_c \log \hat y_c$ | Machine learning | 42 | AI 34 · Finance 4 · Management 2 · Economics 1 · Operations Research 1 | 3 |
| 24 | [Softmax function](https://scholar4ai.com/equations/softmax-function.md) | $\mathrm{softmax}(z)_i = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}$ | Machine learning | 42 | AI 37 · Finance 3 · Operations Research 2 | 5 |
| 26 | [Probit model](https://scholar4ai.com/equations/probit-model.md) | $\Pr(y = 1 \mid x) = \Phi(x^\top\beta)$ | Regression models | 40 | Economics 17 · Finance 12 · Management 7 · Accounting 2 · Operations Research 2 | 6 |
| 27 | [Adam optimizer](https://scholar4ai.com/equations/adam-optimizer.md) | $\theta_t = \theta_{t-1} - \alpha\,\frac{\hat m_t}{\sqrt{\hat v_t} + \epsilon}$ | Machine learning | 39 | AI 28 · Finance 4 · Operations Research 4 · Management 3 | 5 |
| 27 | [GARCH](https://scholar4ai.com/equations/garch.md) | $\sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2$ | Time series | 39 | Finance 23 · Economics 13 · Management 1 · Operations Research 1 · AI 1 | 5 |
| 29 | [Gradient descent](https://scholar4ai.com/equations/gradient-descent.md) | $\theta_{t+1} = \theta_t - \eta\, \nabla_\theta \mathcal{L}(\theta_t)$ | Machine learning | 38 | AI 21 · Operations Research 14 · Economics 2 · Finance 1 | 7 |
| 30 | [ReLU activation](https://scholar4ai.com/equations/relu-activation.md) | $\mathrm{ReLU}(x) = \max(0, x)$ | Machine learning | 36 | AI 29 · Operations Research 3 · Finance 2 · Economics 1 · Management 1 | 2 |
| 31 | [Mean-variance portfolio (Markowitz)](https://scholar4ai.com/equations/mean-variance-portfolio-markowitz.md) | $\min_{w}\ w^\top \Sigma\, w\ \ \text{s.t.}\ \ w^\top \mu = \mu_p,\ \ w^\top \mathbf{1} = 1$ | Asset pricing and risk | 31 | Finance 14 · Operations Research 6 · AI 5 · Economics 4 · Management 2 | 2 |
| 32 | [Value at risk (VaR)](https://scholar4ai.com/equations/value-at-risk-var.md) | $\mathrm{VaR}_\alpha(L) = \inf\{\ell \in \mathbb{R} : \Pr(L > \ell) \le 1 - \alpha\}$ | Asset pricing and risk | 30 | Operations Research 8 · Finance 7 · Economics 6 · Management 6 · AI 3 | 5 |
| 33 | [Generative adversarial network (GAN)](https://scholar4ai.com/equations/generative-adversarial-network-gan.md) | $\min_G \max_D\ E_x\big[\log D(x)\big] + E_z\big[\log\big(1 - D(G(z))\big)\big]$ | Machine learning | 28 | AI 21 · Operations Research 4 · Finance 1 · Economics 1 · Management 1 | 0 |
| 33 | [Nash equilibrium](https://scholar4ai.com/equations/nash-equilibrium.md) | $u_i(s_i^{*}, s_{-i}^{*}) \ge u_i(s_i, s_{-i}^{*})\quad \forall i,\ \forall s_i$ | Game theory and decision analysis | 28 | Economics 13 · Operations Research 7 · AI 5 · Finance 2 · Management 1 | 2 |
| 35 | [Cointegration and error correction (VECM)](https://scholar4ai.com/equations/cointegration-and-error-correction-vecm.md) | $\Delta y_t = \Pi\, y_{t-1} + \sum_{i=1}^{p-1}\Gamma_i\,\Delta y_{t-i} + \varepsilon_t$ | Time series | 26 | Economics 14 · Finance 9 · Management 3 | 4 |
| 35 | [LASSO](https://scholar4ai.com/equations/lasso.md) | $\hat\beta = \operatorname*{arg\,min}_{\beta}\ \lVert y - X\beta \rVert_2^2 + \lambda \lVert \beta \rVert_1$ | Regression models | 26 | Economics 8 · Finance 7 · AI 6 · Operations Research 3 · Management 2 | 6 |
| 37 | [ARIMA](https://scholar4ai.com/equations/arima.md) | $\phi(L)\,(1 - L)^{d}\, y_t = \theta(L)\,\varepsilon_t$ | Time series | 25 | Economics 7 · AI 7 · Finance 5 · Operations Research 3 · Management 2 · Accounting 1 | 5 |
| 37 | [Q-learning](https://scholar4ai.com/equations/q-learning.md) | $Q(s, a) \leftarrow Q(s, a) + \alpha\big[r + \gamma \max_{a'} Q(s', a') - Q(s, a)\big]$ | Machine learning | 25 | AI 15 · Operations Research 8 · Economics 2 | 3 |
| 39 | [Expected shortfall (CVaR)](https://scholar4ai.com/equations/expected-shortfall-cvar.md) | $\mathrm{ES}_\alpha(L) = \frac{1}{1 - \alpha}\int_\alpha^{1} \mathrm{VaR}_u(L)\,du$ | Asset pricing and risk | 24 | Operations Research 8 · Finance 7 · Economics 5 · AI 4 | 1 |
| 39 | [Gravity model of trade](https://scholar4ai.com/equations/gravity-model-of-trade.md) | $X_{ij} = G\,\frac{Y_i^{\beta_1}\, Y_j^{\beta_2}}{D_{ij}^{\beta_3}}$ | Growth, trade and environment | 24 | AI 10 · Economics 7 · Finance 4 · Operations Research 2 · Management 1 | 2 |
| 41 | [Autoregressive distributed lag (ARDL)](https://scholar4ai.com/equations/autoregressive-distributed-lag-ardl.md) | $y_t = \alpha + \sum_{i=1}^{p}\phi_i\, y_{t-i} + \sum_{j=0}^{q}\beta_j^\top x_{t-j} + \varepsilon_t$ | Time series | 23 | Economics 11 · Finance 7 · Management 4 · Operations Research 1 | 3 |
| 41 | [k-means clustering](https://scholar4ai.com/equations/k-means-clustering.md) | $\min_{C_1, \dots, C_k} \sum_{j=1}^{k} \sum_{x \in C_j} \lVert x - \mu_j \rVert^2$ | Machine learning | 23 | AI 12 · Operations Research 6 · Economics 4 · Management 1 | 4 |
| 43 | [Data envelopment analysis (DEA)](https://scholar4ai.com/equations/data-envelopment-analysis-dea.md) | $\max_{u, v}\ \frac{\sum_r u_r\, y_{r0}}{\sum_i v_i\, x_{i0}}\ \ \text{s.t.}\ \ \frac{\sum_r u_r\, y_{rj}}{\sum_i v_i\, x_{ij}} \le 1\ \ \forall j$ | Efficiency and operations | 21 | Operations Research 10 · Economics 5 · Finance 4 · AI 2 | 0 |
| 43 | [Stackelberg game](https://scholar4ai.com/equations/stackelberg-game.md) | $\max_{x_L}\ \pi_L\big(x_L, x_F^{*}(x_L)\big),\quad x_F^{*}(x_L) = \operatorname*{arg\,max}_{x_F}\ \pi_F(x_L, x_F)$ | Game theory and decision analysis | 21 | Operations Research 14 · Economics 6 · Management 1 | 4 |
| 45 | [Kullback-Leibler divergence](https://scholar4ai.com/equations/kullback-leibler-divergence.md) | $D_{\mathrm{KL}}(P \parallel Q) = \sum_x P(x) \log \frac{P(x)}{Q(x)}$ | Machine learning | 19 | AI 14 · Operations Research 3 · Finance 1 · Economics 1 | 3 |
| 45 | [Sharpe ratio](https://scholar4ai.com/equations/sharpe-ratio.md) | $S_p = \frac{E[R_p] - R_f}{\sigma_p}$ | Asset pricing and risk | 19 | Finance 9 · Economics 5 · AI 3 · Management 1 · Operations Research 1 | 4 |
| 47 | [Denoising diffusion model](https://scholar4ai.com/equations/denoising-diffusion-model.md) | $\mathcal{L} = E_{t, x_0, \epsilon}\Big[\left\lVert \epsilon - \epsilon_\theta\big(\sqrt{\bar\alpha_t}\, x_0 + \sqrt{1 - \bar\alpha_t}\, \epsilon,\ t\big) \right\rVert^2\Big]$ | Machine learning | 18 | AI 16 · Economics 1 · Management 1 | 2 |
| 47 | [Gated recurrent unit (GRU)](https://scholar4ai.com/equations/gated-recurrent-unit-gru.md) | $h_t = (1 - z_t) \odot h_{t-1} + z_t \odot \tilde h_t$ | Machine learning | 18 | AI 13 · Finance 3 · Management 2 | 1 |
| 47 | [Granger causality](https://scholar4ai.com/equations/granger-causality.md) | $y_t = \sum_{i=1}^{p} a_i\, y_{t-i} + \sum_{j=1}^{p} b_j\, x_{t-j} + \varepsilon_t,\quad H_0: b_1 = \dots = b_p = 0$ | Time series | 18 | Finance 8 · Economics 5 · Management 3 · AI 2 | 2 |
| 47 | [Variational autoencoder (ELBO)](https://scholar4ai.com/equations/variational-autoencoder-elbo.md) | $\log p_\theta(x) \ge E_{q_\phi(z \mid x)}\big[\log p_\theta(x \mid z)\big] - D_{\mathrm{KL}}\big(q_\phi(z \mid x) \parallel p(z)\big)$ | Machine learning | 18 | AI 13 · Operations Research 4 · Finance 1 | 2 |
| 51 | [Gini coefficient](https://scholar4ai.com/equations/gini-coefficient.md) | $G = \frac{\sum_{i=1}^{n}\sum_{j=1}^{n} \lvert x_i - x_j \rvert}{2 n^2 \bar x}$ | Labour, inequality and competition | 17 | Economics 7 · Operations Research 4 · Finance 3 · Management 2 · AI 1 | 1 |
| 51 | [Proximal policy optimization (PPO)](https://scholar4ai.com/equations/proximal-policy-optimization-ppo.md) | $L(\theta) = E_t\big[\min\big(r_t(\theta)\hat A_t,\ \mathrm{clip}(r_t(\theta), 1 - \epsilon, 1 + \epsilon)\,\hat A_t\big)\big]$ | Machine learning | 17 | AI 9 · Operations Research 7 · Economics 1 | 1 |
| 51 | [Regression discontinuity design](https://scholar4ai.com/equations/regression-discontinuity-design.md) | $\tau = \lim_{x \downarrow c} E[Y \mid X = x] - \lim_{x \uparrow c} E[Y \mid X = x]$ | Causal inference | 17 | Economics 15 · Accounting 1 · AI 1 | 1 |
| 51 | [Spatial autoregressive models (spatial Durbin)](https://scholar4ai.com/equations/spatial-autoregressive-models-spatial-durbin.md) | $y = \rho\, W y + X\beta + W X\theta + \varepsilon$ | Regression models | 17 | Economics 10 · Finance 5 · Management 2 | 1 |
| 55 | [Fama-French five-factor model](https://scholar4ai.com/equations/fama-french-five-factor-model.md) | $R_{it} - R_{ft} = \alpha_i + b_i\,\mathit{MKT}_t + s_i\,\mathit{SMB}_t + h_i\,\mathit{HML}_t + r_i\,\mathit{RMW}_t + c_i\,\mathit{CMA}_t + \varepsilon_{it}$ | Asset pricing and risk | 14 | Finance 11 · Management 2 · Economics 1 | 1 |
| 55 | [Low-rank adaptation (LoRA)](https://scholar4ai.com/equations/low-rank-adaptation-lora.md) | $W = W_0 + B A,\quad B \in \mathbb{R}^{d \times r},\ A \in \mathbb{R}^{r \times k},\ r \ll \min(d, k)$ | Machine learning | 14 | AI 13 · Management 1 | 0 |
| 57 | [Capital asset pricing model (CAPM)](https://scholar4ai.com/equations/capital-asset-pricing-model-capm.md) | $E[R_i] = R_f + \beta_i\,(E[R_m] - R_f)$ | Asset pricing and risk | 13 | Finance 7 · Economics 2 · Management 2 · Accounting 1 · Operations Research 1 | 1 |
| 57 | [Cobb-Douglas production function](https://scholar4ai.com/equations/cobb-douglas-production-function.md) | $Y = A\, K^{\alpha} L^{1 - \alpha}$ | Growth, trade and environment | 13 | Economics 8 · Finance 5 | 0 |
| 57 | [Markov-switching model](https://scholar4ai.com/equations/markov-switching-model.md) | $y_t = \mu_{s_t} + \varepsilon_t,\quad \Pr(s_t = j \mid s_{t-1} = i) = p_{ij}$ | Time series | 13 | Finance 6 · Economics 4 · Management 1 · Operations Research 1 · AI 1 | 1 |
| 57 | [Shannon entropy](https://scholar4ai.com/equations/shannon-entropy.md) | $H(X) = -\sum_x p(x) \log p(x)$ | Machine learning | 13 | AI 12 · Finance 1 | 0 |
| 57 | [Stochastic frontier analysis](https://scholar4ai.com/equations/stochastic-frontier-analysis.md) | $\ln y_i = f(x_i; \beta) + v_i - u_i,\quad u_i \ge 0$ | Efficiency and operations | 13 | Economics 6 · Finance 3 · Operations Research 2 · Accounting 1 · Management 1 | 0 |
| 62 | [Herfindahl-Hirschman index](https://scholar4ai.com/equations/herfindahl-hirschman-index.md) | $\mathit{HHI} = \sum_{i=1}^{N} s_i^2$ | Labour, inequality and competition | 12 | Finance 8 · Management 3 · Operations Research 1 | 0 |
| 62 | [Kalman filter](https://scholar4ai.com/equations/kalman-filter.md) | $\hat x_{t \mid t} = \hat x_{t \mid t-1} + K_t\,(y_t - H\,\hat x_{t \mid t-1})$ | Time series | 12 | AI 6 · Finance 3 · Operations Research 2 · Economics 1 | 2 |
| 64 | [Event study (abnormal returns)](https://scholar4ai.com/equations/event-study-abnormal-returns.md) | $AR_{it} = R_{it} - (\hat\alpha_i + \hat\beta_i R_{mt}),\quad CAR_i = \sum_{t=t_1}^{t_2} AR_{it}$ | Causal inference | 11 | Finance 8 · Accounting 1 · Management 1 · Operations Research 1 | 1 |
| 64 | [Tobin's q](https://scholar4ai.com/equations/tobin-s-q.md) | $q = \frac{\text{market value of assets}}{\text{replacement cost of assets}}$ | Asset pricing and risk | 11 | Finance 7 · Management 3 · Economics 1 | 0 |
| 66 | [Bellman equation](https://scholar4ai.com/equations/bellman-equation.md) | $V(s) = \max_{a}\left\{ r(s, a) + \beta \sum_{s'} P(s' \mid s, a)\, V(s') \right\}$ | Game theory and decision analysis | 10 | Economics 5 · Finance 2 · Operations Research 2 · AI 1 | 0 |
| 66 | [Heckman selection model](https://scholar4ai.com/equations/heckman-selection-model.md) | $E[y_i \mid x_i, s_i = 1] = x_i^\top\beta + \rho\,\sigma_\varepsilon\,\lambda(z_i^\top\gamma),\quad \lambda(\cdot) = \frac{\phi(\cdot)}{\Phi(\cdot)}$ | Causal inference | 10 | Management 4 · Finance 3 · Economics 3 | 2 |
| 68 | [Environmental Kuznets curve](https://scholar4ai.com/equations/environmental-kuznets-curve.md) | $\ln E_{it} = \beta_0 + \beta_1 \ln Y_{it} + \beta_2 (\ln Y_{it})^2 + \varepsilon_{it}$ | Growth, trade and environment | 9 | Economics 5 · Finance 2 · Management 2 | 1 |
| 68 | [Synthetic control method](https://scholar4ai.com/equations/synthetic-control-method.md) | $\hat Y^{N}_{1t} = \sum_{j=2}^{J+1} w_j^{*}\, Y_{jt},\quad w_j^{*} \ge 0,\ \sum_j w_j^{*} = 1$ | Causal inference | 9 | Economics 5 · Finance 2 · Accounting 1 · Management 1 | 0 |
| 70 | [Bayes' theorem](https://scholar4ai.com/equations/bayes-theorem.md) | $P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}$ | Game theory and decision analysis | 8 | Economics 3 · Finance 2 · Management 1 · Operations Research 1 · AI 1 | 0 |
| 70 | [Carhart four-factor model](https://scholar4ai.com/equations/carhart-four-factor-model.md) | $R_{it} - R_{ft} = \alpha_i + b_i\,\mathit{MKT}_t + s_i\,\mathit{SMB}_t + h_i\,\mathit{HML}_t + m_i\,\mathit{UMD}_t + \varepsilon_{it}$ | Asset pricing and risk | 8 | Finance 5 · Management 2 · Economics 1 | 2 |
| 70 | [New Keynesian Phillips curve](https://scholar4ai.com/equations/new-keynesian-phillips-curve.md) | $\pi_t = \beta\,E_t[\pi_{t+1}] + \kappa\, x_t$ | Macroeconomics | 8 | Economics 4 · Finance 2 · Management 2 | 1 |
| 70 | [Solow growth model](https://scholar4ai.com/equations/solow-growth-model.md) | $\dot k = s\, f(k) - (n + \delta)\, k$ | Growth, trade and environment | 8 | Economics 6 · Finance 2 | 1 |
| 70 | [TOPSIS](https://scholar4ai.com/equations/topsis.md) | $C_i = \frac{d_i^{-}}{d_i^{+} + d_i^{-}}$ | Game theory and decision analysis | 8 | AI 5 · Finance 1 · Economics 1 · Operations Research 1 | 3 |
| 75 | [Analytic hierarchy process (AHP)](https://scholar4ai.com/equations/analytic-hierarchy-process-ahp.md) | $A\, w = \lambda_{\max}\, w,\quad CR = \frac{(\lambda_{\max} - n)/(n - 1)}{RI}$ | Game theory and decision analysis | 7 | AI 4 · Finance 2 · Operations Research 1 | 0 |
| 75 | [Entropy weight method](https://scholar4ai.com/equations/entropy-weight-method.md) | $e_j = -\frac{1}{\ln n}\sum_{i=1}^{n} p_{ij}\ln p_{ij},\quad w_j = \frac{1 - e_j}{\sum_k (1 - e_k)}$ | Game theory and decision analysis | 7 | Economics 3 · Finance 2 · Operations Research 1 · AI 1 | 2 |
| 77 | [Nash bargaining solution](https://scholar4ai.com/equations/nash-bargaining-solution.md) | $\max_{x}\ \big(u_1(x) - d_1\big)\big(u_2(x) - d_2\big)$ | Game theory and decision analysis | 6 | Economics 4 · Operations Research 2 | 2 |
| 78 | [Consumption Euler equation](https://scholar4ai.com/equations/consumption-euler-equation.md) | $u'(c_t) = \beta\,E_t\big[(1 + r_{t+1})\,u'(c_{t+1})\big]$ | Macroeconomics | 5 | Economics 3 · Finance 2 | 0 |
| 78 | [Fama-French three-factor model](https://scholar4ai.com/equations/fama-french-three-factor-model.md) | $R_{it} - R_{ft} = \alpha_i + b_i\,\mathit{MKT}_t + s_i\,\mathit{SMB}_t + h_i\,\mathit{HML}_t + \varepsilon_{it}$ | Asset pricing and risk | 5 | Finance 3 · Management 2 | 1 |
| 78 | [Modified Jones model (discretionary accruals)](https://scholar4ai.com/equations/modified-jones-model-discretionary-accruals.md) | $\frac{TA_{it}}{A_{i,t-1}} = \alpha_1\frac{1}{A_{i,t-1}} + \alpha_2\frac{\Delta REV_{it} - \Delta REC_{it}}{A_{i,t-1}} + \alpha_3\frac{PPE_{it}}{A_{i,t-1}} + \varepsilon_{it}$ | Accounting | 5 | Management 3 · Finance 2 | 1 |
| 78 | [Taylor rule](https://scholar4ai.com/equations/taylor-rule.md) | $i_t = r^{*} + \pi_t + 0.5\,(\pi_t - \pi^{*}) + 0.5\,(y_t - \bar y_t)$ | Macroeconomics | 5 | Economics 3 · Finance 2 | 0 |
| 82 | [Amihud illiquidity ratio](https://scholar4ai.com/equations/amihud-illiquidity-ratio.md) | $\mathit{ILLIQ}_i = \frac{1}{D_i}\sum_{d=1}^{D_i}\frac{\lvert R_{id} \rvert}{\mathit{DVOL}_{id}}$ | Asset pricing and risk | 4 | Finance 4 | 0 |
| 82 | [Best-worst method](https://scholar4ai.com/equations/best-worst-method.md) | $\min\ \xi\ \ \text{s.t.}\ \ \lvert w_B - a_{Bj}\, w_j \rvert \le \xi,\ \ \lvert w_j - a_{jW}\, w_W \rvert \le \xi,\ \ \sum_j w_j = 1$ | Game theory and decision analysis | 4 | AI 2 · Finance 1 · Operations Research 1 | 0 |
| 82 | [CES production function](https://scholar4ai.com/equations/ces-production-function.md) | $Y = A\big[\alpha K^{\rho} + (1 - \alpha) L^{\rho}\big]^{1/\rho}$ | Growth, trade and environment | 4 | Finance 2 · Economics 2 | 0 |
| 82 | [CoVaR (systemic risk)](https://scholar4ai.com/equations/covar-systemic-risk.md) | $\Pr\big(X^{\mathrm{sys}} \le \mathrm{CoVaR}_q^{\mathrm{sys} \mid i} \mid X^{i} = \mathrm{VaR}_q^{i}\big) = q$ | Asset pricing and risk | 4 | Finance 2 · Economics 1 · AI 1 | 0 |
| 82 | [DEMATEL](https://scholar4ai.com/equations/dematel.md) | $T = X\,(I - X)^{-1}$ | Game theory and decision analysis | 4 | AI 2 · Finance 1 · Economics 1 | 1 |
| 82 | [Merton model (distance to default)](https://scholar4ai.com/equations/merton-model-distance-to-default.md) | $DD = \frac{\ln(V/D) + (\mu - \sigma_V^2/2)\,T}{\sigma_V\sqrt{T}}$ | Asset pricing and risk | 4 | Finance 2 · Economics 2 | 0 |
| 82 | [Newsvendor model](https://scholar4ai.com/equations/newsvendor-model.md) | $Q^{*} = F^{-1}\Big(\frac{c_u}{c_u + c_o}\Big)$ | Efficiency and operations | 4 | Economics 2 · Operations Research 2 | 0 |
| 89 | [Black-Scholes formula](https://scholar4ai.com/equations/black-scholes-formula.md) | $C = S_0\,N(d_1) - K e^{-rT} N(d_2),\quad d_{1,2} = \frac{\ln(S_0/K) + (r \pm \sigma^2/2)\,T}{\sigma\sqrt{T}}$ | Asset pricing and risk | 3 | Finance 3 | 0 |
| 89 | [Cox proportional hazards model](https://scholar4ai.com/equations/cox-proportional-hazards-model.md) | $h(t \mid x) = h_0(t)\,\exp(x^\top\beta)$ | Regression models | 3 | Finance 1 · Accounting 1 · Management 1 | 0 |
| 89 | [Dechow-Dichev accruals quality](https://scholar4ai.com/equations/dechow-dichev-accruals-quality.md) | $\Delta WC_{it} = \beta_0 + \beta_1\, CFO_{i,t-1} + \beta_2\, CFO_{it} + \beta_3\, CFO_{i,t+1} + \varepsilon_{it}$ | Accounting | 3 | Finance 2 · Accounting 1 | 1 |
| 92 | [Altman Z-score](https://scholar4ai.com/equations/altman-z-score.md) | $Z = 1.2X_1 + 1.4X_2 + 3.3X_3 + 0.6X_4 + 1.0X_5$ | Asset pricing and risk | 2 | Finance 2 | 0 |
| 92 | [Bass diffusion model](https://scholar4ai.com/equations/bass-diffusion-model.md) | $\frac{f(t)}{1 - F(t)} = p + q\, F(t)$ | Efficiency and operations | 2 | Economics 1 · Operations Research 1 | 0 |
| 92 | [Coupling coordination degree](https://scholar4ai.com/equations/coupling-coordination-degree.md) | $C = \frac{2\sqrt{U_1 U_2}}{U_1 + U_2},\quad T = \alpha U_1 + \beta U_2,\quad D = \sqrt{C\, T}$ | Game theory and decision analysis | 2 | Finance 2 | 0 |
| 92 | [Covered interest parity](https://scholar4ai.com/equations/covered-interest-parity.md) | $\frac{F_t}{S_t} = \frac{1 + i_t}{1 + i_t^{*}}$ | Macroeconomics | 2 | Finance 1 · Economics 1 | 0 |
| 92 | [Economic order quantity (EOQ)](https://scholar4ai.com/equations/economic-order-quantity-eoq.md) | $Q^{*} = \sqrt{\frac{2 D S}{H}}$ | Efficiency and operations | 2 | Operations Research 2 | 0 |
| 92 | [Hodrick-Prescott filter](https://scholar4ai.com/equations/hodrick-prescott-filter.md) | $\min_{\{\tau_t\}} \sum_{t=1}^{T}(y_t - \tau_t)^2 + \lambda \sum_{t=2}^{T-1}\big[(\tau_{t+1} - \tau_t) - (\tau_t - \tau_{t-1})\big]^2$ | Time series | 2 | Finance 1 · Economics 1 | 0 |
| 92 | [Uncovered interest parity](https://scholar4ai.com/equations/uncovered-interest-parity.md) | $E_t[s_{t+1}] - s_t = i_t - i_t^{*}$ | Macroeconomics | 2 | Finance 1 · Economics 1 | 0 |
| 99 | [Heston stochastic volatility model](https://scholar4ai.com/equations/heston-stochastic-volatility-model.md) | $dv_t = \kappa\,(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^{v}$ | Asset pricing and risk | 1 | Finance 1 | 0 |
| 99 | [Kaya identity](https://scholar4ai.com/equations/kaya-identity.md) | $\mathrm{CO_2} = P \times \frac{\mathit{GDP}}{P} \times \frac{E}{\mathit{GDP}} \times \frac{\mathrm{CO_2}}{E}$ | Growth, trade and environment | 1 | Economics 1 | 0 |
| 99 | [Lerner index](https://scholar4ai.com/equations/lerner-index.md) | $L = \frac{P - MC}{P}$ | Labour, inequality and competition | 1 | Finance 1 | 0 |
| 99 | [Little's law](https://scholar4ai.com/equations/little-s-law.md) | $L = \lambda W$ | Efficiency and operations | 1 | Operations Research 1 | 0 |
| 99 | [Residual income valuation (Ohlson model)](https://scholar4ai.com/equations/residual-income-valuation-ohlson-model.md) | $P_t = B_t + \sum_{\tau=1}^{\infty}\frac{E_t[NI_{t+\tau} - r\,B_{t+\tau-1}]}{(1 + r)^{\tau}}$ | Accounting | 1 | Operations Research 1 | 0 |
| 99 | [STIRPAT model](https://scholar4ai.com/equations/stirpat-model.md) | $I_i = a\, P_i^{b} A_i^{c} T_i^{d} e_i$ | Growth, trade and environment | 1 | Finance 1 | 0 |

Also watched, with no paper in this window: Basu asymmetric timeliness, Beneish M-score, Dividend discount (Gordon growth) model, Fisher equation, Jensen's alpha, Mincer earnings equation, Nelson-Siegel yield curve, Okun's law.

## How the equations are found

The site watches a fixed list of 112 named equations and models in 11 areas, each with the names papers use for it (full names, abbreviations and common spellings). A paper counts when one of them appears in its title, abstract or OpenAlex keywords, or, for papers whose full text OpenAlex has indexed, anywhere in the full text (found with OpenAlex full-text search).

In this window OpenAlex has the abstract of 863 of the 3,332 papers (26%) and searchable full text for 619 (19%); titles and keywords are always there.

The full-text lookups go a group of names at a time, the stalest first, as far as the day's OpenAlex request budget allows: each of the 69 groups was last looked up on 2026-10-26.

The counts are therefore lower bounds that lean toward papers with open full text, and a mention is not proof that a paper uses the equation or dataset. Abstracts are only read to spot the names; they are neither stored nor republished.

## Questions this page answers

### Which equations and models are used most in finance, accounting, economics, management, operations research and AI research?

Counting papers that mention them by name, the most frequent in the tracked finance, accounting, economics, management, operations research and AI journals over the last 30 days were Scaled dot-product attention (Transformer) (203), Difference-in-differences (135), Logistic regression (logit) (123), Linear and integer programming (116) and Instrumental variables (two-stage least squares) (115).

### Which equations and models are used most in finance research?

Counting papers that mention them by name, the most frequent in the tracked finance journals over the last 30 days were Difference-in-differences (47), Ordinary least squares (OLS) (41), Instrumental variables (two-stage least squares) (38), Mediation model (indirect effect) (26) and Two-way fixed-effects panel regression (25).

### Which equations and models are used most in accounting research?

Counting papers that mention them by name, the most frequent in the tracked accounting journals over the last 30 days were Instrumental variables (two-stage least squares) (6), Difference-in-differences (5), Ordinary least squares (OLS) (4), Structural equation modeling (SEM) (4) and Principal component analysis (PCA) (3).

### Which equations and models are used most in economics research?

Counting papers that mention them by name, the most frequent in the tracked economics journals over the last 30 days were Difference-in-differences (64), Instrumental variables (two-stage least squares) (50), Ordinary least squares (OLS) (41), Logistic regression (logit) (36) and Total factor productivity (Solow residual) (28).

### Which equations and models are used most in management research?

Counting papers that mention them by name, the most frequent in the tracked management journals over the last 30 days were Structural equation modeling (SEM) (62), Mediation model (indirect effect) (32), Ordinary least squares (OLS) (24), Logistic regression (logit) (21) and Instrumental variables (two-stage least squares) (18).

### Which equations and models are used most in operations research?

Counting papers that mention them by name, the most frequent in the tracked operations research journals over the last 30 days were Linear and integer programming (100), Logistic regression (logit) (17), Gradient descent (14), Stackelberg game (14) and Scaled dot-product attention (Transformer) (11).

### Which equations and models are used most in AI research?

Counting papers that mention them by name, the most frequent in the tracked AI journals over the last 30 days were Scaled dot-product attention (Transformer) (175), Convolutional neural network (CNN) (99), Graph neural network (graph convolution) (99), Contrastive loss (InfoNCE) (77) and Long short-term memory (LSTM) (44).