# Marginal effects for spatial probit and Tobit models (SAR probit, SAR Tobit) Estimate marginal effects (average direct, indirect and total impacts) for the SAR probit and SAR Tobit model. ```r ## S3 method for class 'sarprobit' marginal.effects(object, o = 100, ...) ## S3 method for class 'sartobit' marginal.effects(object, o = 100, ...) ## S3 method for class 'sarprobit' impacts(obj, file=NULL, digits = max(3, getOption("digits")-3), ...) ## S3 method for class 'sartobit' impacts(obj, file=NULL, digits = max(3, getOption("digits")-3), ...) ``` ## Arguments - `object`: Estimated model of class `sarprobit` or `sartobit` - `obj`: Estimated model of class `sarprobit` or `sartobit` - `o`: maximum value for the power $tr(W^i), i=1,...,o$ to be estimated - `digits`: number of digits for printing - `file`: Output to file or console - `...`: additional parameters ## Details `impacts()` will extract and print the marginal effects from a fitted model, while `marginal.effects(x)` will estimate the marginal effects anew for a fitted model. In spatial models, a change in some explanatory variable $x_{ir}$ for observation $i$ will not only affect the observations $y_i$ directly (direct impact), but also affect neighboring observations $y_j$ (indirect impact). These impacts potentially also include feedback loops from observation $i$ to observation $j$ and back to $i$. (see LeSage (2009), section 2.7 for interpreting parameter estimates in spatial models). For the $r$-th non-constant explanatory variable, let $S_r(W)$ be the $n x n$ matrix that captures the impacts from observation $i$ to $j$. The direct impact of a change in $x_{ir}$ on its own observation $y_i$ can be written as $$ \frac{\partial y_i}{\partial x_{ir}} = S_r(W)_{ii}d y_i / d x_{ir} = S_r(W)_{ii} $$ and the indirect impact from observation $j$ to observation $i$ as $$ \frac{\partial y_i}{\partial x_{jr}} = S_r(W)_{ij}d y_i / d x_{jr} = S_r(W)_{ij} $$ . LeSage(2009) proposed summary measures for direct, indirect and total effects, e.g. averaged direct impacts across all $n$ observations. See LeSage(2009), section 5.6.2., p.149/150 for marginal effects estimation in general spatial models and section 10.1.6, p.293 for marginal effects in SAR probit models. We implement these three summary measures: 1. average direct impacts: $$ M_r(D) = \bar{S_r(W)_{ii}} = n^{-1} tr(S_r(W))M_r(D) = average(S_r(W)_{ii}) = n^{-1} tr(S_r(W)) $$ 2. average total impacts: $$ M_r(T) = n^{-1} 1'_n S_r(W) 1_n $$ 3. average indirect impacts: $$ M_r(I) = M_r(T) - M_r(D) $$ The average direct impact is the average of the diagonal elements, the average total impacts is the mean of the row (column) sums. For the average direct impacts $M_r(D)$, there are efficient approaches available, see LeSage (2009), chapter 4, pp.114/115. The computation of the average total effects $M_r(T)$ and hence also the average indirect effects $M_r(I)$ are more subtle, as $S_r(W)$ is a dense $n x n$ matrix. In the LeSage Spatial Econometrics Toolbox for MATLAB (March 2010), the implementation in `sarp_g` computes the matrix inverse of $S= (I_n - \rho W)$ which all the negative consequences for large n. We implemented $n^{-1} 1'_n S_r(W) 1_n$ via a QR decomposition of $S = (I_n - \rho W)$ (already available from a previous step) and solving a linear equation, which is less costly and will work better for large $n$. SAR probit model Specifically, for the SAR probit model the $n x n$ matrix of marginal effects is $$ S_r(W) = \frac{\partial E[y | x_r]}{\partial x_{r}'} = \phi\left((I_n - \rho W)^{-1} \bar{x}_r \beta_r \right) \odot (I_n - \rho W)^{-1} I_n \beta_rS_r(W) = d E[y | x_r] / d x_r' = phi((I_n - rho W)^{-1} I_n mean(x_r) beta_r) * (I_n - rho W)^{-1} I_n beta_r $$ SAR Tobit model Specifically, for the SAR Tobit model the $n x n$ matrix of marginal effects is $$ S_r(W) = \frac{\partial E[y | x_r]}{\partial x_{r}'} = \Phi\left((I_n - \rho W)^{-1} \bar{x}_r \beta_r / \sigma \right) \odot (I_n - \rho W)^{-1} I_n \beta_rS_r(W) = d E[y | x_r] / d x_r' = Phi((I_n - rho W)^{-1} I_n mean(x_r) beta_r / sigma) * (I_n - rho W)^{-1} I_n beta_r $$ ## Returns This function returns a list with 6 elements: 'direct' for direct effects, 'indirect' for indirect effects, 'total' for total effects, and 'summary_direct', 'summary_indirect', 'summary_total' for the summary of direct, indirect and total effects. ## Warning 1. Although the direct impacts can be efficiently estimated, the computation of the indirect effects require the inversion of a $n x n$ matrix and will break down for large $n$. 2. $tr(W^i)$ is determined with simulation, so different calls to this method will produce different estimates. ## References LeSage, J. and Pace, R. K. (2009), **Introduction to Spatial Econometrics**, CRC Press ## Author(s) Stefan Wilhelm ## See Also `marginal.effects.sartobit` ## Examples ```r require(spatialprobit) # number of observations n <- 100 # true parameters beta <- c(0, 1, -1) rho <- 0.75 # design matrix with two standard normal variates as "covariates" X <- cbind(intercept=1, x=rnorm(n), y=rnorm(n)) # sparse identity matrix I_n <- sparseMatrix(i=1:n, j=1:n, x=1) # number of nearest neighbors in spatial weight matrix W m <- 6 # spatial weight matrix with m=6 nearest neighbors # W must not have non-zeros in the main diagonal! W <- kNearestNeighbors(x = rnorm(n), y = rnorm(n), k = m) # innovations eps <- rnorm(n=n, mean=0, sd=1) # generate data from model S <- I_n - rho * W z <- solve(qr(S), X %*% beta + eps) y <- as.vector(z >= 0) # 0 or 1, FALSE or TRUE # estimate SAR probit model set.seed(12345) sarprobit.fit1 <- sar_probit_mcmc(y, X, W, ndraw=500, burn.in=100, thinning=1, prior=NULL, computeMarginalEffects=TRUE) summary(sarprobit.fit1) # print impacts impacts(sarprobit.fit1) ################################################################################ # # Example from LeSage/Pace (2009), section 10.3.1, p. 302-304 # ################################################################################ # Value of "a" is not stated in book! # Assuming a=-1 which gives approx. 50% censoring library(spatialprobit) a <- -1 # control degree of censored observation n <- 1000 rho <- 0.7 beta <- c(0, 2) sige <- 0.5 I_n <- sparseMatrix(i=1:n, j=1:n, x=1) x <- runif(n, a, 1) X <- cbind(1, x) eps <- rnorm(n, sd=sqrt(sige)) param <- c(beta, sige, rho) # random locational coordinates and 6 nearest neighbors lat <- rnorm(n) long <- rnorm(n) W <- kNearestNeighbors(lat, long, k=6) y <- as.double(solve(I_n - rho * W) %*% (X %*% beta + eps)) table(y > 0) # set negative values to zero to reflect sample truncation ind <- which(y <=0) y[ind] <- 0 # Fit SAR Tobit (with approx. 50% censored observations) fit_sartobit <- sartobit(y ~ x, W, ndraw=1000, burn.in=200, computeMarginalEffects=TRUE, showProgress=TRUE) # print impacts impacts(fit_sartobit) ```

marginal.effects.sarprobit function