sdmw: A Markov Chain Monte Carlo (MCMC) sampler for the panel spatial Durbin model (SDM) with unknown spatial weight matrix

Description

The sampler uses independent Normal-inverse-Gamma priors for the slope and variance parameters, as well as a four-parameter beta prior for the spatial autoregressive parameter $\rho$. It is a wrapper around W_sampler.

Usage

sdmw(
  Y,
  tt,
  X = matrix(0, nrow(Y), 0),
  Z = matrix(1, nrow(Y), 1),
  niter = 100,
  nretain = 50,
  W_prior = W_priors(n = nrow(Y)/tt),
  rho_prior = rho_priors(),
  beta_prior = beta_priors(k = ncol(X) * 2 + ncol(Z)),
  sigma_prior = sigma_priors()
)

Value

List with posterior samples for the slope parameters, $\rho$, $\sigma^2$, $W$, and average direct, indirect, and total effects.

Arguments

Y: numeric $N \times 1$ matrix containing the dependent variables, where $N = nT$ is the number of spatial ($n$) times the number of time observations ($T$, with tt=T). Note that the observations have organized such that $Y = [Y_1',...,Y_T']'$.
tt: single number greater or equal to 1. Denotes the number of time observations. $tt = T$.
X: numeric $N \times k_1$ design matrix of independent variables. These will be automatically spatially lagged. If no spatially lagged variable is included in the model a matrix with $N$ rows and zero columns should be supplied (the default value). Note: either $X$ or $Z$ has to be a matrix with at least one column.
Z: numeric $N \times k_3$ design matrix of independent variables which are not spatially lagged. The default value is a $N \times 1$ vector of ones (i.e. an intercept for the model). Note: either $X$ or $Z$ has to be a matrix with at least one column.
niter: single number greater or equal to 1, indicating the total number of draws. Will be automatically coerced to integer. The default value is 100.
nretain: single number greater or equal to 0, indicating the number of draws kept after the burn-in. Will be automatically coerced to integer. The default value is 50.
W_prior: list containing prior settings for estimating the spatial weight matrix $W$. Generated by the smart constructor W_priors.
rho_prior: list of prior settings for estimating $\rho$, generated by the smart constructor rho_priors
beta_prior: list containing priors for the slope coefficients $\beta$, generated by the smart constructor beta_priors. The ordering of the priors is: (1) priors of $X$, (2) priors of spatially lagged $X$, (3) priors of $Z$.
sigma_prior: list containing priors for the error variance $\sigma^2$, generated by the smart constructor sigma_priors

Details

The considered panel spatial Durbin model (SDM) with unknown ($n$ by $n$) spatial weight matrix $W$ takes the form:

$$ Y_t = \rho W Y_t + X_t \beta_1 + W X_t \beta_2 + Z \beta_3 + \varepsilon_t, $$

with $\varepsilon_t \sim N(0,I_n \sigma^2)$ and $W = f(\Omega)$. The $n$ by $n$ matrix $\Omega$ is an unknown binary adjacency matrix with zeros on the main diagonal and $f(\cdot)$ is the (optional) row-standardization function. $\rho$ is a scalar spatial autoregressive parameter.

$Y_t$ ($n \times 1$) collects the $n$ cross-sectional (spatial) observations for time $t=1,...,T$. $X_t$ ($n \times k_1$) and $Z_t$ ($n \times k_2$) are matrices of explanatory variables, where the former will also be spatially lagged. $\beta_1$ ($k_1 \times 1$), $\beta_2$ ($k_1 \times 1$) and $\beta_3$ ($k_2 \times 1$) are unknown slope parameter vectors.

After vertically staking the $T$ cross-sections $Y=[Y_1',...,Y_T']'$ ($N \times 1$), $X=[X_1',...,X_T']'$ ($N \times k_1$) and $Z=[Z_1', ..., Z_T']'$ ($N \times k_2$), with $N=nT$. The final model can be expressed as:

$$ Y = \rho \tilde{W}Y + X \beta_1 + \tilde{W} X \beta_2 + Z \beta_3 + \varepsilon, $$

where $\tilde{W}=I_T \otimes W$ and $\varepsilon \sim N(0,I_N \sigma^2)$. Note that the input data matrices have to be ordered first by the cross-sectional spatial units and then stacked by time.

Estimation usually even works well in cases of $n >> T$. However, note that for applications with $n > 200$ the estimation process becomes computationally demanding and slow. Consider in this case reducing niter and nretain and carefully check whether the posterior chains have converged.

Examples

Run this code

n = 20; tt = 10
dgp_dat = sim_dgp(n =n, tt = tt, rho = .75, beta1 = c(.5,1),beta2 = c(-1,.5),
            beta3 = c(1.5), sigma2 = .05,n_neighbor = 3,intercept = TRUE)
res = sdmw(Y = dgp_dat$Y,tt = tt,X = dgp_dat$X,Z = dgp_dat$Z,niter = 20,nretain = 10)

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