semw: A Markov Chain Monte Carlo (MCMC) sampler for the panel spatial error model (SEM) 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 $\rho$. This is a wrapper function calling sdemw with no spatially lagged exogenous variables.

Usage

semw(
  Y,
  tt,
  Z,
  niter = 100,
  nretain = 50,
  W_prior = W_priors(n = nrow(Y)/tt),
  rho_prior = rho_priors(),
  beta_prior = beta_priors(k = 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. $tt = T$.
Z: numeric $N \times k_3$ design matrix of independent variables. The default value is a $N \times 1$ vector of ones (i.e. an intercept for the model).
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.
sigma_prior: list containing priors for the error variance $\sigma^2$, generated by the smart constructor sigma_priors

Details

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

$$ Y_t = Z \beta + \varepsilon_t, $$

with $\varepsilon_t \sim N(0,(I_n - \rho W) \sigma^2)$ 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 $t=1,...,T$. $Z_t$ ($n \times k_3$) is a matrix of explanatory variables. $\beta$ ($k_3 \times 1$) is an unknown slope parameter vector.

After vertically staking the $T$ cross-sections $Y=[Y_1',...,Y_T']'$ ($N \times 1$), and $Z=[Z_1', ..., Z_T']'$ ($N \times k_3$), with $N=nT$. The final model can be expressed as:

$$ Y = Z \beta + \varepsilon, $$

where $\tilde{W}=I_T \otimes W$ and $\varepsilon \sim N(0,\sigma^2 (I_n \otimes (I_n - \rho W) )$. 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 = .5, beta3 = c(.5,1),
            sigma2 = .05,n_neighbor = 3,intercept = TRUE,spatial_error = TRUE)
res = semw(Y = dgp_dat$Y,tt = tt,Z = dgp_dat$Z,niter = 20,nretain = 10)

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