normalgamma {estimateW}

A Markov Chain Monte Carlo (MCMC) sampler for a linear panel model

Description

The sampler uses independent Normal-inverse-Gamma priors to estimate a linear panel data model. The function is used for an illustration on using the beta_sampler and sigma_sampler classes.

Usage

normalgamma(
  Y,
  tt,
  X = matrix(1, nrow(Y), 1),
  niter = 200,
  nretain = 100,
  beta_prior = beta_priors(k = ncol(X)),
  sigma_prior = sigma_priors()
)

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.

niter

single number greater or equal to 1, indicating the total number of draws. Will be automatically coerced to integer. The default value is 200.

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 100.

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 model takes the form:

Y_t = X_t \beta + \varepsilon_t,

with \varepsilon_t \sim N(0,I_n \sigma^2).

Y_t (n \times 1) collects the n cross-sectional observations for time t=1,...,T. X_t (n \times k_1) is a matrix of explanatory variables. \beta (k_1 \times 1) is an unknown slope parameter matrix.

After vertically staking the T cross-sections Y=[Y_1',...,Y_T']' (N \times 1), X=[X_1',...,X_T']' (N \times k), with N=nT, the final model can be expressed as:

Y = X \beta + \varepsilon,

where \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.

Examples

n = 20; tt = 10; k = 3
X = matrix(stats::rnorm(n*tt*k),n*tt,k)
Y = X %*% c(1,0,-1) + stats::rnorm(n*tt,0,.5)
res = normalgamma(Y,tt,X)