Supported Models¶
Cross Sectional Models¶
OLS¶
\[y = X\beta + \epsilon\]
SLX¶
\[y = X\beta + WX\theta + \epsilon\]
SAR¶
\[y = \rho Wy + X\beta + \epsilon\]
SEM¶
\[y = X\beta + u, \quad u = \lambda Wu + \epsilon\]
SDM¶
\[y = \rho Wy + X\beta + WX\theta + \epsilon\]
SDEM¶
\[y = X\beta + WX\theta + u, \quad u = \lambda Wu + \epsilon\]
Panel Models¶
OLS panel¶
\[y_{it} = x_{it}' \beta + a_i + \tau_t + \epsilon_{it}\]
SAR panel¶
\[y_{it} = \rho Wy_{it} + x_{it}' \beta + a_i + \tau_t + \epsilon_{it}\]
SEM panel¶
\[y_{it} = x_{it}' \beta + a_i + \tau_t + u_{it}, \quad u_{it} = \lambda Wu_{it} + \epsilon_{it}\]
SDM panel¶
\[y_{it} = \rho Wy_{it} + x_{it}' \beta + Wx_{it}' \theta + a_i + \tau_t + \epsilon_{it}\]
SDEM panel¶
\[y_{it} = x_{it}' \beta + Wx_{it}' \theta + a_i + \tau_t + u_{it}, \quad u_{it} = \lambda Wu_{it} + \epsilon_{it}\]
SLX panel¶
\[y_{it} = x_{it}' \beta + Wx_{it}' \theta + a_i + \tau_t + \epsilon_{it}\]
OLS panel (Random Effects)¶
\[y_{it} = x_{it}' \beta + \alpha_i + \tau_t + \epsilon_{it}, \quad \alpha_i \sim N(0, \sigma_\alpha^2)\]
SAR panel (Random Effects)¶
\[y_{it} = \rho W y_{it} + x_{it}' \beta + \alpha_i + \tau_t + \epsilon_{it}, \quad \alpha_i \sim N(0, \sigma_\alpha^2)\]
SEM panel (Random Effects)¶
\[y_{it} = x_{it}' \beta + \alpha_i + \tau_t + u_{it}, \quad u_{it} = \lambda W u_{it} + \epsilon_{it}, \quad \alpha_i \sim N(0, \sigma_\alpha^2)\]
SDEM panel (Random Effects)¶
\[y_{it} = x_{it}' \beta + W x_{it}' \theta + \alpha_i + u_{it}, \quad u_{it} = \lambda W u_{it} + \epsilon_{it}, \quad \alpha_i \sim N(0, \sigma_\alpha^2)\]
Dynamic Panel Models¶
OLSPanelDynamic (Dynamic Linear Model)¶
\[y_{it} = \phi y_{i,t-1} + x_{it}' \beta + a_i + \tau_t + \epsilon_{it}\]
SDMRPanelDynamic (Dynamic Restricted Spatial Durbin)¶
\[y_{it} = \phi y_{i,t-1} + \rho W y_{it} - \rho \phi W y_{i,t-1} + x_{it}' \beta + W x_{it}' \theta + a_i + \tau_t + \epsilon_{it}\]
SDMUPanelDynamic (Dynamic Unrestricted Spatial Durbin)¶
\[y_{it} = \phi y_{i,t-1} + \rho W y_{it} + \theta W y_{i,t-1} + x_{it}' \beta + W x_{it}' \theta + a_i + \tau_t + \epsilon_{it}\]
SARPanelDynamic (Dynamic SAR)¶
\[y_{it} = \phi y_{i,t-1} + \rho W y_{it} + x_{it}' \beta + a_i + \tau_t + \epsilon_{it}\]
SEMPanelDynamic (Dynamic SEM)¶
\[y_{it} = \phi y_{i,t-1} + x_{it}' \beta + a_i + \tau_t + u_{it}, \quad u_{it} = \lambda W u_{it} + \epsilon_{it}\]
SDEMPanelDynamic (Dynamic SDEM)¶
\[y_{it} = \phi y_{i,t-1} + x_{it}' \beta + W x_{it}' \theta + a_i + \tau_t + u_{it}, \quad u_{it} = \lambda W u_{it} + \epsilon_{it}\]
SLXPanelDynamic (Dynamic SLX)¶
\[y_{it} = \phi y_{i,t-1} + x_{it}' \beta + W x_{it}' \theta + a_i + \tau_t + \epsilon_{it}\]
Non-Linear Models¶
Spatial Probit¶
\[y^* = \rho W y^* + X\beta + a + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, I), \quad y_i = \mathbf{1}[y_i^* > 0]\]
Tobit (SAR Tobit)¶
\[y_i = \max(c, y_i^*), \quad y^* = \rho W y^* + X\beta + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I)\]
Tobit (SEM Tobit)¶
\[y_i = \max(c, y_i^*), \quad y^* = X\beta + u, \quad u = \lambda Wu + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I)\]
Tobit (SDM Tobit)¶
\[y_i = \max(c, y_i^*), \quad y^* = \rho W y^* + X\beta + WX\theta + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I)\]
Panel Tobit (SAR)¶
\[y_{it} = \max(c, y_{it}^*), \quad y_t^* = \rho W y_t^* + X_t\beta + \varepsilon_t\]
Panel Tobit (SEM)¶
\[y_{it} = \max(c, y_{it}^*), \quad y_t^* = X_t\beta + u_t, \quad u_t = \lambda W u_t + \varepsilon_t\]
Flow Models¶
Vectorize the origin-destination flow matrix to \(y \in \mathbb{R}^{N}\) with \(N = n^2\), and define destination, origin, and network weight matrices as \(W_d\), \(W_o\), and \(W_w\).
OLSFlow¶
\[y = X\beta + \varepsilon\]
PoissonFlow¶
\[y_{ij} \sim \operatorname{Poisson}(\lambda_{ij}), \quad \log \boldsymbol{\lambda} = X\beta\]
SARFlow¶
\[y = \rho_d W_d y + \rho_o W_o y + \rho_w W_w y + X\beta + \varepsilon\]
SARFlowSeparable¶
\[y = \rho_d W_d y + \rho_o W_o y - \rho_d \rho_o W_w y + X\beta + \varepsilon\]
PoissonSARFlow¶
\[y_{ij} \sim \operatorname{Poisson}(\lambda_{ij}), \quad \log \boldsymbol{\lambda} = A(\boldsymbol{\rho})^{-1} X\beta\]
PoissonSARFlowSeparable¶
\[y_{ij} \sim \operatorname{Poisson}(\lambda_{ij}), \quad \log \boldsymbol{\lambda} = A(\boldsymbol{\rho})^{-1} X\beta, \quad \rho_w = -\rho_d \rho_o\]
SEMFlow¶
\[y = X\beta + u, \quad u = \lambda_d W_d u + \lambda_o W_o u + \lambda_w W_w u + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I)\]
SEMFlowSeparable¶
\[y = X\beta + u, \quad u = \lambda_d W_d u + \lambda_o W_o u - \lambda_d \lambda_o W_w u + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I)\]
Panel Flow Models¶
Stack the flow models above across \(T\) periods in time-first order. The Poisson panel variants currently operate in pooled mode.
OLSFlowPanel¶
\[y_t = X_t\beta + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0, \sigma^2 I_N)\]
PoissonFlowPanel¶
\[y_{ij,t} \sim \operatorname{Poisson}(\lambda_{ij,t}), \quad \log \boldsymbol{\lambda}_t = X_t\beta\]
SARFlowPanel¶
\[y_t = \rho_d W_d y_t + \rho_o W_o y_t + \rho_w W_w y_t + X_t\beta + \varepsilon_t\]
SARFlowSeparablePanel¶
\[y_t = \rho_d W_d y_t + \rho_o W_o y_t - \rho_d \rho_o W_w y_t + X_t\beta + \varepsilon_t\]
PoissonSARFlowPanel¶
\[y_{ij,t} \sim \operatorname{Poisson}(\lambda_{ij,t}), \quad \log \boldsymbol{\lambda}_t = A(\boldsymbol{\rho})^{-1} X_t\beta\]
PoissonSARFlowSeparablePanel¶
\[y_{ij,t} \sim \operatorname{Poisson}(\lambda_{ij,t}), \quad \log \boldsymbol{\lambda}_t = A(\boldsymbol{\rho})^{-1} X_t\beta, \quad \rho_w = -\rho_d \rho_o\]
SEMFlowPanel¶
\[y_t = X_t\beta + u_t, \quad u_t = \lambda_d W_d u_t + \lambda_o W_o u_t + \lambda_w W_w u_t + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0, \sigma^2 I_N)\]
SEMFlowSeparablePanel¶
\[y_t = X_t\beta + u_t, \quad u_t = \lambda_d W_d u_t + \lambda_o W_o u_t - \lambda_d \lambda_o W_w u_t + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0, \sigma^2 I_N)\]