# Simulating Count Data Models with Social Interactions Under Rational Expectations `simcdnet` simulates the count data model with social interactions under rational expectations developed by Houndetoungan (2024). ```r simcdnet( formula, group, Glist, parms, lambda, Gamma, delta, Rmax, Rbar, cont.var, bin.var, tol = 1e-10, maxit = 500, data ) ``` ## Arguments - `formula`: A class object of class formula : a symbolic description of the model. `formula` should be specified, for example, as `y ~ x1 + x2 + gx1 + gx2`, where `y` is the endogenous vector and `x1`, `x2`, `gx1`, and `gx2` are control variables. These control variables can include contextual variables, such as averages among the peers. Peer averages can be computed using the function `peer.avg`. - `group`: A vector indicating the individual groups. By default, this assumes a common group. If there are 2 groups (i.e., `length(unique(group)) = 2`, such as `A` and `B`), four types of peer effects are defined: peer effects of `A` on `A`, `A` on `B`, `B` on `A`, and `B` on `B`. - `Glist`: An adjacency matrix or list of adjacency matrices. For networks consisting of multiple subnets (e.g., schools), `Glist` can be a list of subnet matrices, where the $m$-th element is an $n_m \times n_m$ adjacency matrix, with $n_m$ representing the number of nodes in the $m$-th subnet. For heterogeneous peer effects (`length(unique(group)) = h > 1`), the $m$-th element should be a list of $h^2$ $n_m \times n_m$ adjacency matrices corresponding to different network specifications (see Houndetoungan, 2024). For heterogeneous peer effects in a single large network, `Glist` should be a one-item list, where the item is a list of $h^2$ network specifications. The order of these networks is important and must match `sort(unique(group))` (see examples). - `parms`: A vector defining the true values of $\theta = (\lambda', \Gamma', \delta')'$ (see model specification in the details section). Each parameter $\lambda$, $\Gamma$, or $\delta$ can also be provided separately to the arguments `lambda`, `Gamma`, or `delta`. - `lambda`: The true value of the vector $\lambda$. - `Gamma`: The true value of the vector $\Gamma$. - `delta`: The true value of the vector $\delta$. - `Rmax`: An integer indicating the theoretical upper bound of `y` (see model specification in detail). - `Rbar`: An $L$-vector, where $L$ is the number of groups. For large `Rmax`, the cost function is assumed to be semi-parametric (i.e., nonparametric from 0 to $\bar{R}$ and quadratic beyond $\bar{R}$). The $l$-th element of `Rbar` indicates $\bar{R}$ for the $l$-th value of `sort(unique(group))` (see model specification in detail). - `cont.var`: A character vector of continuous variable names for which the marginal effects should be computed. - `bin.var`: A character vector of binary variable names for which the marginal effects should be computed. - `tol`: The tolerance value used in the Fixed Point Iteration Method to compute the expectancy of `y`. The process stops if the $\ell_1$-distance between two consecutive $E(y)$ is less than `tol`. - `maxit`: The maximum number of iterations in the Fixed Point Iteration Method. - `data`: An optional data frame, list, or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in `data`, the variables are taken from `environment(formula)`, typically the environment from which `simcdnet` is called. ## Returns A list consisting of: - **yst**: $y^{\ast}$, the latent variable. - **y**: the observed count variable. - **Ey**: $E(y)$, the expectation of y. - **GEy**: the average of $E(y)$ among peers. - **meff**: a list including average and individual marginal effects. - **Rmax**: infinite sums in the marginal effects are approximated by sums up to Rmax. - **iteration**: number of iterations performed by sub-network in the Fixed Point Iteration Method. ## Details The count variable $y_i$ takes the value $r$ with probability. $$ P_{ir} = F(\sum_{s = 1}^S \lambda_s \bar{y}_i^{e,s} + \mathbf{z}_i'\Gamma - a_{h(i),r}) - F(\sum_{s = 1}^S \lambda_s \bar{y}_i^{e,s} + \mathbf{z}_i'\Gamma - a_{h(i),r + 1}). $$ In this equation, $\mathbf{z}_i$ is a vector of control variables; $F$ is the distribution function of the standard normal distribution; $\bar{y}_i^{e,s}$ is the average of $E(y)$ among peers using the `s`-th network definition; $a_{h(i),r}$ is the `r`-th cut-point in the cost group $h(i)$. The following identification conditions have been introduced: $\sum_{s = 1}^S \lambda_s > 0$, $a_{h(i),0} = -\infty$, $a_{h(i),1} = 0$, and $a_{h(i),r} = \infty$ for any $r \geq R_{\text{max}} + 1$. The last condition implies that $P_{ir} = 0$ for any $r \geq R_{\text{max}} + 1$. For any $r \geq 1$, the distance between two cut-points is $a_{h(i),r+1} - a_{h(i),r} = \delta_{h(i),r} + \sum_{s = 1}^S \lambda_s$. As the number of cut-points can be large, a quadratic cost function is considered for $r \geq \bar{R}_{h(i)}$, where $\bar{R} = (\bar{R}_{1}, ..., \bar{R}_{L})$. With the semi-parametric cost function, $a_{h(i),r + 1} - a_{h(i),r} = \bar{\delta}_{h(i)} + \sum_{s = 1}^S \lambda_s$. The model parameters are: $\lambda = (\lambda_1, ..., \lambda_S)'$, $\Gamma$, and $\delta = (\delta_1', ..., \delta_L')'$, where $\delta_l = (\delta_{l,2}, ..., \delta_{l,\bar{R}_l}, \bar{\delta}_l)'$ for $l = 1, ..., L$. The number of single parameters in $\delta_l$ depends on $R_{\text{max}}$ and $\bar{R}_l$. The components $\delta_{l,2}, ..., \delta_{l,\bar{R}_l}$ or/and $\bar{\delta}_l$ must be removed in certain cases. If $R_{\text{max}} = \bar{R}_l \geq 2$, then $\delta_l = (\delta_{l,2}, ..., \delta_{l,\bar{R}_l})'$. If $R_{\text{max}} = \bar{R}_l = 1$ (binary models), then $\delta_l$ must be empty. If $R_{\text{max}} > \bar{R}_l = 1$, then $\delta_l = \bar{\delta}_l$. ## Examples ```r set.seed(123) M <- 5 # Number of sub-groups nvec <- round(runif(M, 100, 200)) # Random group sizes n <- sum(nvec) # Total number of individuals # Adjacency matrix for each group A <- list() for (m in 1:M) { nm <- nvec[m] # Size of group m Am <- matrix(0, nm, nm) # Empty adjacency matrix max_d <- 30 # Maximum number of friends for (i in 1:nm) { tmp <- sample((1:nm)[-i], sample(0:max_d, 1)) # Sample friends Am[i, tmp] <- 1 # Set friendship links } A[[m]] <- Am # Add to the list } Anorm <- norm.network(A) # Row-normalization of the adjacency matrices # Covariates (X) X <- cbind(rnorm(n, 1, 3), rexp(n, 0.4)) # Random covariates # Two groups based on first covariate group <- 1 * (X[,1] > 0.95) # Assign to groups based on x1 # Networks: Define peer effects based on group membership # The networks should capture: # - Peer effects of `0` on `0` # - Peer effects of `1` on `0` # - Peer effects of `0` on `1` # - Peer effects of `1` on `1` G <- list() cums <- c(0, cumsum(nvec)) # Cumulative indices for groups for (m in 1:M) { tp <- group[(cums[m] + 1):(cums[m + 1])] # Group membership for group m Am <- A[[m]] # Adjacency matrix for group m # Define networks based on peer effects G[[m]] <- norm.network(list(Am * ((1 - tp) %*% t(1 - tp)), Am * ((1 - tp) %*% t(tp)), Am * (tp %*% t(1 - tp)), Am * (tp %*% t(tp)))) } # Parameters for the model lambda <- c(0.2, 0.3, -0.15, 0.25) Gamma <- c(4.5, 2.2, -0.9, 1.5, -1.2) delta <- rep(c(2.6, 1.47, 0.85, 0.7, 0.5), 2) # Repeated values for delta # Prepare data for the model data <- data.frame(X, peer.avg(Anorm, cbind(x1 = X[,1], x2 = X[,2]))) colnames(data) = c("x1", "x2", "gx1", "gx2") # Set column names # Simulate outcomes using the `simcdnet` function ytmp <- simcdnet(formula = ~ x1 + x2 + gx1 + gx2, Glist = G, Rbar = rep(5, 2), lambda = lambda, Gamma = Gamma, delta = delta, group = group, data = data) y <- ytmp$y # Plot histogram of the simulated outcomes hist(y, breaks = max(y) + 1) # Display frequency table of the simulated outcomes table(y) ``` ## References Houndetoungan, A. (2024). Count Data Models with Heterogeneous Peer Effects. Available at SSRN 3721250, tools:::Rd_expr_doi("10.2139/ssrn.3721250") . ## See Also `cdnet`, `simsart`, `simsar`.

simcdnet function