Equations and models used in research

Which equations and models research papers use: the 112 named ones the site looks for, in 11 areas, each with its formula, what it is, how many papers of the tracked journals mentioned it in their title, abstract or keywords (the search of earlier months' full texts is still under way) from Oct 2023 to Sep 2026, the trend by month and the fields that use it most. The 30-day ranking is on /rankings/equations; each name links to the papers mentioning it now. Also as Markdown.

Areas: Machine learning · Causal inference · Regression models · Efficiency and operations · Growth, trade and environment · Game theory and decision analysis · Time series · Asset pricing and risk · Labour, inequality and competition · Macroeconomics · Accounting · By category

Key findings#

The site looks for 112 named equations and models in 11 areas; 112 of them were mentioned by papers of the tracked journals from Oct 2023 to Sep 2026.
The most mentioned was Scaled dot-product attention (Transformer) (4,331 papers), followed by Graph neural network (graph convolution) (3,646), Linear and integer programming (2,919), Convolutional neural network (CNN) (2,568) and Contrastive loss (InfoNCE) (2,339).
Mentions grew fastest for Low-rank adaptation (LoRA) (145 papers in the last 12 months, 6.6× the yearly rate before), Ordinary least squares (OLS) (184 papers in the last 12 months, 2.5× the yearly rate before) and Proximal policy optimization (PPO) (124 papers in the last 12 months, 2.4× the yearly rate before).
Most used per field: finance, Mediation model (indirect effect); accounting, Difference-in-differences; economics, Difference-in-differences; health economics and policy, Logistic regression (logit); management, Structural equation modeling (SEM); marketing, Structural equation modeling (SEM).

Machine learning#

Machine learning: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Scaled dot-product attention (Transformer)	$\mathrm{Attention}(Q, K, V) = \mathrm{softmax}\Big(\frac{Q K^\top}{\sqrt{d_k}}\Big) V$	The core operation of Transformer models: each token's output is a weighted average of values, weighted by query-key similarity.	4,331	AI (4,041), operations research (148)
Graph neural network (graph convolution)	$H^{(l+1)} = \sigma\big(\tilde D^{-1/2} \tilde A\, \tilde D^{-1/2} H^{(l)} W^{(l)}\big)$	Updates each node's representation from its neighbours' representations.	3,646	AI (3,379), operations research (163)
Convolutional neural network (CNN)	$(I * K)(i, j) = \sum_{m}\sum_{n} I(i + m,\, j + n)\, K(m, n)$	Networks built from learned convolution filters, standard for images and other grid data.	2,568	AI (2,334), operations research (95)
Contrastive loss (InfoNCE)	$\mathcal{L} = -\log \frac{\exp(\mathrm{sim}(z_i, z_j)/\tau)}{\sum_{k \ne i} \exp(\mathrm{sim}(z_i, z_k)/\tau)}$	Pulls representations of matching pairs together and pushes others apart.	2,339	AI (2,282), operations research (27)
Long short-term memory (LSTM)	$c_t = f_t \odot c_{t-1} + i_t \odot \tilde c_t,\quad h_t = o_t \odot \tanh(c_t)$	Recurrent network cell whose gates decide what to keep, add and output at each step.	1,531	AI (1,059), operations research (157)
Gradient boosting (XGBoost)	$\mathcal{L} = \sum_i l(y_i, \hat y_i) + \sum_k \Omega(f_k),\quad \Omega(f) = \gamma T + \tfrac{1}{2}\lambda \lVert w \rVert^2$	Adds trees one at a time, each fitted to the errors of the ensemble so far, with a penalty on tree complexity.	1,012	AI (458), operations research (163)
Generative adversarial network (GAN)	$\min_G \max_D\ E_x\big[\log D(x)\big] + E_z\big[\log\big(1 - D(G(z))\big)\big]$	A generator and a discriminator trained against each other.	998	AI (911), operations research (44)
Random forest	$\hat f(x) = \frac{1}{B}\sum_{b=1}^{B} T_b(x)$	Averages many decision trees grown on bootstrap samples with random feature subsets.	989	AI (406), operations research (184)
Support vector machine (SVM)	$\min_{w, b, \xi}\ \tfrac{1}{2}\lVert w \rVert^2 + C \sum_i \xi_i\ \ \text{s.t.}\ \ y_i (w^\top x_i + b) \ge 1 - \xi_i,\ \xi_i \ge 0$	Finds the separating hyperplane with the widest margin; support vector regression adapts it to regression.	820	AI (514), operations research (123)
Principal component analysis (PCA)	$\max_{\lVert w \rVert = 1}\ w^\top \Sigma\, w$	Finds the orthogonal directions of largest variance in the data.	569	AI (233), operations research (76)
Q-learning	$Q(s, a) \leftarrow Q(s, a) + \alpha\big[r + \gamma \max_{a'} Q(s', a') - Q(s, a)\big]$	Reinforcement learning of action values from rewards; deep Q-networks approximate Q with a neural network.	549	AI (361), operations research (163)
k-means clustering	$\min_{C_1, \dots, C_k} \sum_{j=1}^{k} \sum_{x \in C_j} \lVert x - \mu_j \rVert^2$	Partitions observations into k clusters around their means.	486	AI (270), operations research (103)
Gated recurrent unit (GRU)	$h_t = (1 - z_t) \odot h_{t-1} + z_t \odot \tilde h_t$	A simpler gated recurrent cell with update and reset gates.	481	AI (366), operations research (35)
Variational autoencoder (ELBO)	$\log p_\theta(x) \ge E_{q_\phi(z \mid x)}\big[\log p_\theta(x \mid z)\big] - D_{\mathrm{KL}}\big(q_\phi(z \mid x) \parallel p(z)\big)$	Generative model trained by maximizing the evidence lower bound on the data likelihood.	447	AI (405), operations research (23)
Gradient descent	$\theta_{t+1} = \theta_t - \eta\, \nabla_\theta \mathcal{L}(\theta_t)$	Moves parameters a step against the gradient of the loss; stochastic gradient descent uses mini-batches.	395	AI (267), operations research (84)
Denoising diffusion model	$\mathcal{L} = E_{t, x_0, \epsilon}\Big[\left\lVert \epsilon - \epsilon_\theta\big(\sqrt{\bar\alpha_t}\, x_0 + \sqrt{1 - \bar\alpha_t}\, \epsilon,\ t\big) \right\rVert^2\Big]$	Generative model that learns to remove noise step by step; the basis of current image generators.	283	AI (274), operations research (4)
Kullback-Leibler divergence	$D_{\mathrm{KL}}(P \parallel Q) = \sum_x P(x) \log \frac{P(x)}{Q(x)}$	How much one probability distribution differs from another.	240	AI (159), statistics (40)
Proximal policy optimization (PPO)	$L(\theta) = E_t\big[\min\big(r_t(\theta)\hat A_t,\ \mathrm{clip}(r_t(\theta), 1 - \epsilon, 1 + \epsilon)\,\hat A_t\big)\big]$	Policy-gradient reinforcement learning with a clipped objective that keeps updates small.	226	AI (142), operations research (70)
Low-rank adaptation (LoRA)	$W = W_0 + B A,\quad B \in \mathbb{R}^{d \times r},\ A \in \mathbb{R}^{r \times k},\ r \ll \min(d, k)$	Fine-tunes a large model by learning a small low-rank update to its weights.	189	AI (181), operations research (4)
Shannon entropy	$H(X) = -\sum_x p(x) \log p(x)$	The average information content, or uncertainty, of a random variable.	185	AI (120), operations research (21)
Cross-entropy loss	$\mathcal{L} = -\sum_{c=1}^{C} y_c \log \hat y_c$	Standard loss for classification: the negative log-probability of the true class.	167	AI (139), operations research (12)
Softmax function	$\mathrm{softmax}(z)_i = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}$	Turns a vector of scores into probabilities.	115	AI (109), finance (2)
ReLU activation	$\mathrm{ReLU}(x) = \max(0, x)$	The most common activation function in deep networks.	97	AI (75), operations research (13)
Adam optimizer	$\theta_t = \theta_{t-1} - \alpha\,\frac{\hat m_t}{\sqrt{\hat v_t} + \epsilon}$	Gradient descent with running averages of the gradient (m) and its square (v).	58	AI (49), operations research (5)

Causal inference#

Causal inference: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Mediation model (indirect effect)	$M = \alpha_0 + a X + e_1,\quad Y = \beta_0 + c' X + b M + e_2,\quad \text{indirect effect} = a\,b$	Splits the effect of X on Y into a direct part and a part that runs through a mediator M.	1,667	finance (565), management (517)
Difference-in-differences	$Y_{it} = \alpha_i + \lambda_t + \delta\,(\mathit{Treat}_i \times \mathit{Post}_t) + \varepsilon_{it}$	Compares the change in outcomes of a group exposed to a policy or event with the change in an unexposed group; delta is the estimated effect.	1,289	economics (571), finance (399)
Instrumental variables (two-stage least squares)	$\hat\beta_{\mathrm{2SLS}} = (X^\top P_Z X)^{-1} X^\top P_Z\, y,\quad P_Z = Z (Z^\top Z)^{-1} Z^\top$	Uses instruments Z that move the regressors but not the error term to estimate causal effects.	826	economics (387), finance (256)
Event study (abnormal returns)	$AR_{it} = R_{it} - (\hat\alpha_i + \hat\beta_i R_{mt}),\quad CAR_i = \sum_{t=t_1}^{t_2} AR_{it}$	Measures returns beyond what a market model predicts around an event such as an announcement.	494	finance (353), economics (66)
Propensity score matching	$e(x) = \Pr(D = 1 \mid X = x)$	Matches treated and untreated units with similar probabilities of treatment given their characteristics.	461	finance (157), economics (107)
Regression discontinuity design	$\tau = \lim_{x \downarrow c} E[Y \mid X = x] - \lim_{x \uparrow c} E[Y \mid X = x]$	Estimates an effect from the jump in outcomes at a cutoff c of a running variable X that decides treatment.	223	economics (161), health economics and policy (21)
Synthetic control method	$\hat Y^{N}_{1t} = \sum_{j=2}^{J+1} w_j^{}\, Y_{jt},\quad w_j^{} \ge 0,\ \sum_j w_j^{*} = 1$	Builds the counterfactual for a treated unit as a weighted average of untreated units.	146	economics (107), finance (15)
Heckman selection model	$E[y_i \mid x_i, s_i = 1] = x_i^\top\beta + \rho\,\sigma_\varepsilon\,\lambda(z_i^\top\gamma),\quad \lambda(\cdot) = \frac{\phi(\cdot)}{\Phi(\cdot)}$	Corrects for non-random selection of the observed sample with the inverse Mills ratio from a first-stage probit.	74	finance (32), economics (18)

Regression models#

Regression models: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Structural equation modeling (SEM)	$\eta = B\eta + \Gamma\xi + \zeta,\quad y = \Lambda_y\,\eta + \varepsilon$	Estimates relations among latent constructs measured by several observed indicators; PLS-SEM is a common variant.	1,589	management (833), information systems (190)
Logistic regression (logit)	$\Pr(y = 1 \mid x) = \frac{1}{1 + e^{-x^\top\beta}}$	Models the probability of a binary outcome with the logistic function.	1,364	operations research (324), economics (254)
Two-way fixed-effects panel regression	$y_{it} = \alpha_i + \lambda_t + x_{it}^\top\beta + \varepsilon_{it}$	Panel regression with unit and time fixed effects that absorb time-invariant differences and common shocks.	964	economics (379), finance (293)
Quantile regression	$\hat\beta_\tau = \operatorname*{arg\,min}_{\beta} \sum_{i} \rho_\tau(y_i - x_i^\top\beta),\quad \rho_\tau(u) = u\,(\tau - \mathbf{1}\{u < 0\})$	Estimates how regressors shift a chosen quantile of the outcome rather than its mean.	754	economics (235), finance (231)
LASSO	$\hat\beta = \operatorname*{arg\,min}_{\beta}\ \lVert y - X\beta \rVert_2^2 + \lambda \lVert \beta \rVert_1$	Regression with an L1 penalty that shrinks coefficients and sets many of them to exactly zero.	377	statistics (132), AI (70)
Generalized method of moments (GMM)	$\hat\theta = \operatorname*{arg\,min}_{\theta}\ \bar g(\theta)^\top W\, \bar g(\theta),\quad \bar g(\theta) = \frac{1}{n}\sum_{i=1}^{n} g(w_i, \theta)$	Estimates parameters by making sample moment conditions as close to zero as possible; system and difference GMM are common in dynamic panels.	368	finance (134), economics (112)
Ordinary least squares (OLS)	$\hat\beta = (X^\top X)^{-1} X^\top y$	Linear regression coefficients that minimize the sum of squared residuals.	330	economics (106), finance (93)
Spatial autoregressive models (spatial Durbin)	$y = \rho\, W y + X\beta + W X\theta + \varepsilon$	Lets outcomes and regressors of neighbouring units, weighted by a spatial matrix W, affect each unit.	320	economics (123), finance (68)
Probit model	$\Pr(y = 1 \mid x) = \Phi(x^\top\beta)$	Models the probability of a binary outcome with the standard normal distribution function.	230	economics (86), finance (45)
Cox proportional hazards model	$h(t \mid x) = h_0(t)\,\exp(x^\top\beta)$	Models the hazard of an event over time as a baseline hazard scaled by the regressors.	118	statistics (69), AI (12)

Efficiency and operations#

Efficiency and operations: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Linear and integer programming	$\min_{x}\ c^\top x\ \ \text{s.t.}\ \ A x \le b,\ x \ge 0,\ x_j \in \mathbb{Z}\ (j \in I)$	Optimizes a linear objective under linear constraints, with some variables restricted to integers.	2,919	operations research (2,525), AI (310)
Data envelopment analysis (DEA)	$\max_{u, v}\ \frac{\sum_r u_r\, y_{r0}}{\sum_i v_i\, x_{i0}}\ \ \text{s.t.}\ \ \frac{\sum_r u_r\, y_{rj}}{\sum_i v_i\, x_{ij}} \le 1\ \ \forall j$	Scores the efficiency of each unit as its best ratio of weighted outputs to weighted inputs relative to all units.	673	operations research (383), economics (103)
Stochastic frontier analysis	$\ln y_i = f(x_i; \beta) + v_i - u_i,\quad u_i \ge 0$	Splits deviations from a production or cost frontier into noise v and inefficiency u.	162	economics (101), operations research (24)
Newsvendor model	$Q^{*} = F^{-1}\Big(\frac{c_u}{c_u + c_o}\Big)$	Optimal stock for one selling period under uncertain demand, set by the critical ratio of underage and overage costs.	88	operations research (76), AI (4)
Economic order quantity (EOQ)	$Q^{*} = \sqrt{\frac{2 D S}{H}}$	Order size that minimizes ordering plus holding costs.	31	operations research (29), management (1)
Bass diffusion model	$\frac{f(t)}{1 - F(t)} = p + q\, F(t)$	Adoption of a new product driven by innovation (p) and imitation (q).	20	operations research (7), management (7)
Little's law	$L = \lambda W$	Average number in a queueing system equals the arrival rate times the average time spent in it.	6	operations research (5), management (1)

Growth, trade and environment#

Growth, trade and environment: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Total factor productivity (Solow residual)	$\ln A_t = \ln Y_t - \alpha \ln K_t - (1 - \alpha) \ln L_t$	The part of output growth not explained by measured inputs.	839	economics (381), finance (339)
Gravity model of trade	$X_{ij} = G\,\frac{Y_i^{\beta_1}\, Y_j^{\beta_2}}{D_{ij}^{\beta_3}}$	Trade between two countries rises with their economic size and falls with the distance between them.	235	economics (158), operations research (31)
Environmental Kuznets curve	$\ln E_{it} = \beta_0 + \beta_1 \ln Y_{it} + \beta_2 (\ln Y_{it})^2 + \varepsilon_{it}$	Tests whether pollution first rises and then falls as income grows (beta_2 < 0).	134	economics (92), finance (20)
CES production function	$Y = A\big[\alpha K^{\rho} + (1 - \alpha) L^{\rho}\big]^{1/\rho}$	Production (or utility) with a constant elasticity of substitution 1/(1 - rho) between inputs.	43	economics (29), finance (7)
Cobb-Douglas production function	$Y = A\, K^{\alpha} L^{1 - \alpha}$	Output from capital and labour with constant output shares.	43	economics (18), operations research (12)
Solow growth model	$\dot k = s\, f(k) - (n + \delta)\, k$	Capital per worker grows with saving and shrinks with population growth and depreciation.	43	economics (33), finance (8)
STIRPAT model	$I_i = a\, P_i^{b} A_i^{c} T_i^{d} e_i$	Environmental impact as a function of population, affluence and technology, estimated in logs.	9	economics (5), AI (1)
Kaya identity	$\mathrm{CO_2} = P \times \frac{\mathit{GDP}}{P} \times \frac{E}{\mathit{GDP}} \times \frac{\mathrm{CO_2}}{E}$	Decomposes carbon emissions into population, income per person, energy intensity and carbon intensity.	4	economics (4)

Game theory and decision analysis#

Game theory and decision analysis: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Shapley value (SHAP)	$\phi_i(v) = \sum_{S \subseteq N \setminus \{i\}} \frac{\lvert S \rvert!\,(n - \lvert S \rvert - 1)!}{n!}\big[v(S \cup \{i\}) - v(S)\big]$	A player's average marginal contribution over all coalitions; SHAP uses it to explain machine-learning predictions.	943	AI (427), operations research (188)
Nash equilibrium	$u_i(s_i^{}, s_{-i}^{}) \ge u_i(s_i, s_{-i}^{*})\quad \forall i,\ \forall s_i$	No player can gain by changing strategy alone.	603	economics (239), operations research (209)
Stackelberg game	$\max_{x_L}\ \pi_L\big(x_L, x_F^{}(x_L)\big),\quad x_F^{}(x_L) = \operatorname*{arg\,max}_{x_F}\ \pi_F(x_L, x_F)$	A leader moves first anticipating the follower's best response; common in supply-chain models.	415	operations research (250), economics (58)
TOPSIS	$C_i = \frac{d_i^{-}}{d_i^{+} + d_i^{-}}$	Ranks alternatives by closeness to the ideal solution and distance from the worst one.	294	operations research (114), AI (107)
Analytic hierarchy process (AHP)	$A\, w = \lambda_{\max}\, w,\quad CR = \frac{(\lambda_{\max} - n)/(n - 1)}{RI}$	Derives criteria weights from pairwise comparisons; CR checks their consistency.	269	operations research (107), AI (72)
Bellman equation	$V(s) = \max_{a}\left\{ r(s, a) + \beta \sum_{s'} P(s' \mid s, a)\, V(s') \right\}$	Recursive condition for the value of a dynamic optimization or Markov decision problem.	168	AI (64), operations research (49)
DEMATEL	$T = X\,(I - X)^{-1}$	Maps cause-and-effect relations among factors from a normalized direct-influence matrix X.	159	operations research (83), management (25)
Entropy weight method	$e_j = -\frac{1}{\ln n}\sum_{i=1}^{n} p_{ij}\ln p_{ij},\quad w_j = \frac{1 - e_j}{\sum_k (1 - e_k)}$	Weights indicators by how much their values vary across units, often to build composite indices.	150	AI (49), operations research (31)
Best-worst method	$\min\ \xi\ \ \text{s.t.}\ \ \lvert w_B - a_{Bj}\, w_j \rvert \le \xi,\ \ \lvert w_j - a_{jW}\, w_W \rvert \le \xi,\ \ \sum_j w_j = 1$	Derives criteria weights from comparisons with the best and the worst criterion.	149	operations research (92), AI (33)
Nash bargaining solution	$\max_{x}\ \big(u_1(x) - d_1\big)\big(u_2(x) - d_2\big)$	Splits a surplus by maximizing the product of the parties' gains over their disagreement points.	81	operations research (42), economics (27)
Bayes' theorem	$P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}$	Updates the probability of a hypothesis A after observing evidence B.	75	economics (37), AI (14)
Coupling coordination degree	$C = \frac{2\sqrt{U_1 U_2}}{U_1 + U_2},\quad T = \alpha U_1 + \beta U_2,\quad D = \sqrt{C\, T}$	Measures how well two systems (for example the economy and the environment) develop together.	58	economics (20), finance (17)

Time series#

Time series: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Vector autoregression (VAR)	$y_t = c + \sum_{i=1}^{p} A_i\, y_{t-i} + \varepsilon_t$	Models several time series jointly, each as a function of the lags of all of them.	867	economics (405), finance (362)
GARCH	$\sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2$	Models time-varying volatility: today's variance depends on yesterday's shock and variance (GARCH(1,1)).	497	finance (283), economics (117)
Cointegration and error correction (VECM)	$\Delta y_t = \Pi\, y_{t-1} + \sum_{i=1}^{p-1}\Gamma_i\,\Delta y_{t-i} + \varepsilon_t$	Models series that share a long-run equilibrium; Pi holds the cointegrating relations and adjustment speeds.	299	economics (128), finance (107)
Granger causality	$y_t = \sum_{i=1}^{p} a_i\, y_{t-i} + \sum_{j=1}^{p} b_j\, x_{t-j} + \varepsilon_t,\quad H_0: b_1 = \dots = b_p = 0$	Tests whether past values of x help predict y beyond y's own past.	298	finance (118), economics (97)
Autoregressive distributed lag (ARDL)	$y_t = \alpha + \sum_{i=1}^{p}\phi_i\, y_{t-i} + \sum_{j=0}^{q}\beta_j^\top x_{t-j} + \varepsilon_t$	Regresses a series on its own lags and current and lagged regressors; the bounds test checks for a long-run relation.	278	economics (117), finance (83)
ARIMA	$\phi(L)\,(1 - L)^{d}\, y_t = \theta(L)\,\varepsilon_t$	Autoregressive integrated moving-average model of a time series, with lag operator L and differencing order d.	260	AI (71), economics (60)
Kalman filter	$\hat x_{t \mid t} = \hat x_{t \mid t-1} + K_t\,(y_t - H\,\hat x_{t \mid t-1})$	Recursively updates estimates of an unobserved state from noisy observations.	172	AI (101), operations research (20)
Markov-switching model	$y_t = \mu_{s_t} + \varepsilon_t,\quad \Pr(s_t = j \mid s_{t-1} = i) = p_{ij}$	Lets a model's parameters change with an unobserved regime that follows a Markov chain.	157	finance (68), economics (57)
Hodrick-Prescott filter	$\min_{\{\tau_t\}} \sum_{t=1}^{T}(y_t - \tau_t)^2 + \lambda \sum_{t=2}^{T-1}\big[(\tau_{t+1} - \tau_t) - (\tau_t - \tau_{t-1})\big]^2$	Separates a series into a smooth trend and a cyclical component.	24	economics (17), finance (4)

Asset pricing and risk#

Asset pricing and risk: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Value at risk (VaR)	$\mathrm{VaR}_\alpha(L) = \inf\{\ell \in \mathbb{R} : \Pr(L > \ell) \le 1 - \alpha\}$	The loss that is exceeded with probability at most 1 - alpha over a given horizon.	515	finance (209), operations research (151)
Expected shortfall (CVaR)	$\mathrm{ES}_\alpha(L) = \frac{1}{1 - \alpha}\int_\alpha^{1} \mathrm{VaR}_u(L)\,du$	Average loss in the worst 1 - alpha share of outcomes; also called conditional value at risk.	396	finance (147), operations research (143)
Mean-variance portfolio (Markowitz)	$\min_{w}\ w^\top \Sigma\, w\ \ \text{s.t.}\ \ w^\top \mu = \mu_p,\ \ w^\top \mathbf{1} = 1$	Chooses portfolio weights with the lowest variance for a target expected return.	317	finance (128), operations research (106)
Sharpe ratio	$S_p = \frac{E[R_p] - R_f}{\sigma_p}$	Excess return of a portfolio per unit of total risk.	257	finance (137), economics (55)
Tobin's q	$q = \frac{\text{market value of assets}}{\text{replacement cost of assets}}$	Market value of a firm relative to the replacement cost of its assets; a common proxy for firm value.	121	finance (65), management (18)
Capital asset pricing model (CAPM)	$E[R_i] = R_f + \beta_i\,(E[R_m] - R_f)$	Expected return of an asset as the risk-free rate plus its market beta times the market risk premium.	98	finance (69), economics (14)
Black-Scholes formula	$C = S_0\,N(d_1) - K e^{-rT} N(d_2),\quad d_{1,2} = \frac{\ln(S_0/K) + (r \pm \sigma^2/2)\,T}{\sigma\sqrt{T}}$	Price of a European call option on a stock with constant volatility and interest rate.	84	finance (41), economics (30)
CoVaR (systemic risk)	$\Pr\big(X^{\mathrm{sys}} \le \mathrm{CoVaR}_q^{\mathrm{sys} \mid i} \mid X^{i} = \mathrm{VaR}_q^{i}\big) = q$	Value at risk of the financial system conditional on an institution being in distress; Delta CoVaR measures its contribution.	53	finance (34), economics (12)
Fama-French five-factor model	$R_{it} - R_{ft} = \alpha_i + b_i\,\mathit{MKT}_t + s_i\,\mathit{SMB}_t + h_i\,\mathit{HML}_t + r_i\,\mathit{RMW}_t + c_i\,\mathit{CMA}_t + \varepsilon_{it}$	Adds profitability (RMW) and investment (CMA) factors to the three-factor model.	52	finance (37), economics (7)
Merton model (distance to default)	$DD = \frac{\ln(V/D) + (\mu - \sigma_V^2/2)\,T}{\sigma_V\sqrt{T}}$	Treats equity as a call option on firm assets V; distance to default measures how far assets are from the default point D.	50	finance (31), economics (14)
Heston stochastic volatility model	$dv_t = \kappa\,(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^{v}$	Option pricing model in which the variance follows a mean-reverting square-root process.	45	finance (30), economics (7)
Altman Z-score	$Z = 1.2X_1 + 1.4X_2 + 3.3X_3 + 0.6X_4 + 1.0X_5$	Bankruptcy predictor that weights five accounting ratios (working capital, retained earnings, EBIT, market equity, sales).	36	finance (17), management (7)
Fama-French three-factor model	$R_{it} - R_{ft} = \alpha_i + b_i\,\mathit{MKT}_t + s_i\,\mathit{SMB}_t + h_i\,\mathit{HML}_t + \varepsilon_{it}$	Explains excess returns with the market, size (SMB) and value (HML) factors.	32	finance (25), economics (4)
Carhart four-factor model	$R_{it} - R_{ft} = \alpha_i + b_i\,\mathit{MKT}_t + s_i\,\mathit{SMB}_t + h_i\,\mathit{HML}_t + m_i\,\mathit{UMD}_t + \varepsilon_{it}$	Adds a momentum factor (UMD) to the Fama-French three factors; widely used for fund performance.	26	finance (17), economics (4)
Nelson-Siegel yield curve	$y(\tau) = \beta_0 + \beta_1\frac{1 - e^{-\tau/\lambda}}{\tau/\lambda} + \beta_2\Big(\frac{1 - e^{-\tau/\lambda}}{\tau/\lambda} - e^{-\tau/\lambda}\Big)$	Fits the term structure of interest rates with level, slope and curvature factors.	21	finance (13), economics (5)
Amihud illiquidity ratio	$\mathit{ILLIQ}_i = \frac{1}{D_i}\sum_{d=1}^{D_i}\frac{\lvert R_{id} \rvert}{\mathit{DVOL}_{id}}$	Average absolute daily return per dollar of trading volume: the price impact of trading.	14	finance (13), accounting (1)
Jensen's alpha	$\alpha_p = R_p - \big[R_f + \beta_p\,(R_m - R_f)\big]$	A portfolio's return beyond what the CAPM predicts for its beta.	8	finance (7), management (1)
Dividend discount (Gordon growth) model	$P_0 = \frac{D_1}{r - g}$	Values a stock as next year's dividend divided by the discount rate minus the dividend growth rate.	6	finance (6)

Labour, inequality and competition#

Labour, inequality and competition: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Gini coefficient	$G = \frac{\sum_{i=1}^{n}\sum_{j=1}^{n} \lvert x_i - x_j \rvert}{2 n^2 \bar x}$	Inequality measure from 0 (equal) to 1 (one unit has everything).	226	economics (128), operations research (34)
Herfindahl-Hirschman index	$\mathit{HHI} = \sum_{i=1}^{N} s_i^2$	Market concentration as the sum of squared market shares.	65	economics (32), finance (16)
Lerner index	$L = \frac{P - MC}{P}$	Market power as the markup of price over marginal cost.	27	economics (13), finance (12)
Mincer earnings equation	$\ln w_i = \beta_0 + \beta_1 S_i + \beta_2 X_i + \beta_3 X_i^2 + \varepsilon_i$	Log wages as a function of years of schooling S and experience X.	7	economics (6), finance (1)

Macroeconomics#

Macroeconomics: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
New Keynesian Phillips curve	$\pi_t = \beta\,E_t[\pi_{t+1}] + \kappa\, x_t$	Links inflation to expected future inflation and the output gap (or real marginal cost) x.	94	economics (77), finance (15)
Taylor rule	$i_t = r^{} + \pi_t + 0.5\,(\pi_t - \pi^{}) + 0.5\,(y_t - \bar y_t)$	Sets the policy interest rate from inflation's gap to target and the output gap.	56	economics (42), finance (14)
Uncovered interest parity	$E_t[s_{t+1}] - s_t = i_t - i_t^{*}$	The expected depreciation of a currency equals the interest rate differential.	29	economics (15), finance (11)
Okun's law	$\Delta u_t = -\beta\,(g_t - \bar g)$	Links changes in unemployment to output growth relative to its trend.	18	economics (17), finance (1)
Covered interest parity	$\frac{F_t}{S_t} = \frac{1 + i_t}{1 + i_t^{*}}$	The forward premium of a currency equals the interest rate differential; deviations signal funding frictions.	16	economics (9), finance (7)
Consumption Euler equation	$u'(c_t) = \beta\,E_t\big[(1 + r_{t+1})\,u'(c_{t+1})\big]$	Optimal saving condition: the marginal utility of consuming today equals the discounted expected return of saving it.	13	economics (9), finance (2)
Fisher equation	$i_t \approx r_t + E_t[\pi_{t+1}]$	The nominal interest rate equals the real rate plus expected inflation.	11	economics (5), operations research (2)

Accounting#

Accounting: papers mentioning each, Oct 2023 to Sep 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)
Equation or model	Formula	What it is	Papers	Used most in
Modified Jones model (discretionary accruals)	$\frac{TA_{it}}{A_{i,t-1}} = \alpha_1\frac{1}{A_{i,t-1}} + \alpha_2\frac{\Delta REV_{it} - \Delta REC_{it}}{A_{i,t-1}} + \alpha_3\frac{PPE_{it}}{A_{i,t-1}} + \varepsilon_{it}$	Separates total accruals into normal and discretionary parts; the residual is used as a measure of earnings management.	61	finance (28), accounting (22)
Dechow-Dichev accruals quality	$\Delta WC_{it} = \beta_0 + \beta_1\, CFO_{i,t-1} + \beta_2\, CFO_{it} + \beta_3\, CFO_{i,t+1} + \varepsilon_{it}$	Measures how well working-capital accruals map into past, current and future cash flows.	9	accounting (6), finance (3)
Beneish M-score	$M = -4.84 + 0.92\,\mathit{DSRI} + 0.528\,\mathit{GMI} + 0.404\,\mathit{AQI} + 0.892\,\mathit{SGI} + 0.115\,\mathit{DEPI} - 0.172\,\mathit{SGAI} + 4.679\,\mathit{TATA} - 0.327\,\mathit{LVGI}$	Flags likely earnings manipulation from eight financial-statement ratios.	7	health economics and policy (2), finance (2)
Residual income valuation (Ohlson model)	$P_t = B_t + \sum_{\tau=1}^{\infty}\frac{E_t[NI_{t+\tau} - r\,B_{t+\tau-1}]}{(1 + r)^{\tau}}$	Values equity as book value plus the discounted earnings in excess of a normal return on book value.	3	finance (1), management (1)
Basu asymmetric timeliness	$\frac{EPS_{it}}{P_{i,t-1}} = \beta_0 + \beta_1 D_{it} + \beta_2 R_{it} + \beta_3 D_{it} R_{it} + \varepsilon_{it}$	Measures accounting conservatism as how much faster earnings reflect bad news (negative returns) than good news.	2	finance (1), management (1)

By category#

The five equations and models most mentioned in each category's journals, Oct 2023 to Sep 2026
Category	Most-mentioned equations and models
Finance	Mediation model (indirect effect) (565); Difference-in-differences (399); Vector autoregression (VAR) (362); Event study (abnormal returns) (353); Total factor productivity (Solow residual) (339)
Accounting	Difference-in-differences (45); Mediation model (indirect effect) (28); Event study (abnormal returns) (25); Modified Jones model (discretionary accruals) (22); Instrumental variables (two-stage least squares) (16)
Economics	Difference-in-differences (571); Vector autoregression (VAR) (405); Instrumental variables (two-stage least squares) (387); Total factor productivity (Solow residual) (381); Two-way fixed-effects panel regression (379)
Health Economics and Policy	Logistic regression (logit) (172); Difference-in-differences (81); Two-way fixed-effects panel regression (64); Propensity score matching (47); Instrumental variables (two-stage least squares) (46)
Management	Structural equation modeling (SEM) (833); Mediation model (indirect effect) (517); Two-way fixed-effects panel regression (121); Logistic regression (logit) (89); Difference-in-differences (86)
Marketing	Structural equation modeling (SEM) (117); Mediation model (indirect effect) (98); Logistic regression (logit) (23); Difference-in-differences (12); Gradient boosting (XGBoost) (12)
Information Systems	Structural equation modeling (SEM) (190); Mediation model (indirect effect) (69); Random forest (35); Scaled dot-product attention (Transformer) (32); Shapley value (SHAP) (32)
Tourism and Hospitality	Mediation model (indirect effect) (91); Structural equation modeling (SEM) (64); Logistic regression (logit) (13); Shapley value (SHAP) (11); Long short-term memory (LSTM) (10)
Higher Education and Research	Structural equation modeling (SEM) (88); Logistic regression (logit) (29); Mediation model (indirect effect) (22); Shapley value (SHAP) (21); Difference-in-differences (17)
Operations Research	Linear and integer programming (2,525); Data envelopment analysis (DEA) (383); Logistic regression (logit) (324); Stackelberg game (250); Nash equilibrium (209)
Statistics	Quantile regression (133); LASSO (132); Logistic regression (logit) (74); Cox proportional hazards model (69); Principal component analysis (PCA) (63)
AI	Scaled dot-product attention (Transformer) (4,041); Graph neural network (graph convolution) (3,379); Convolutional neural network (CNN) (2,334); Contrastive loss (InfoNCE) (2,282); Long short-term memory (LSTM) (1,059)

Cite this page

Scholar4AI. (2026). Equations and models used in research. Journal Research Rankings. Retrieved October 31, 2026, from https://scholar4ai.com/equations

@misc{scholar4ai-equations,
  author = {{Scholar4AI}},
  title = {Equations and models used in research},
  year = {2026},
  howpublished = {\url{https://scholar4ai.com/equations}},
  urldate = {2026-10-31},
  note = {Journal Research Rankings; data from OpenAlex (CC0); licensed CC BY-NC 4.0}
}

Free to reuse for non-commercial purposes with attribution (CC BY-NC 4.0).