Markov Chain Monte Carlo for the Hidden Markov Fixed-effects Model
HMMpanelFE generates a sample from the posterior distribution of the fixed-effects model with varying individual effects model discussed in Park (2011). The code works for both balanced and unbalanced panel data as long as there is no missing data in the middle of each group. This model uses a multivariate Normal prior for the fixed effects parameters and varying individual effects, an Inverse-Gamma prior on the residual error variance, and Beta prior for transition probabilities. The user supplies data and priors, and a sample from the posterior distribution is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.
HMMpanelFE( subject.id, y, X, m, mcmc = 1000, burnin = 1000, thin = 1, verbose = 0, b0 = 0, B0 = 0.001, c0 = 0.001, d0 = 0.001, delta0 = 0, Delta0 = 0.001, a = NULL, b = NULL, seed = NA, ... )
subject.id |
A numeric vector indicating the group number. It should start from 1. |
y |
The response variable. |
X |
The model matrix excluding the constant. |
m |
A vector of break numbers for each subject in the panel. |
mcmc |
The number of MCMC iterations after burn-in. |
burnin |
The number of burn-in iterations for the sampler. |
thin |
The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value. |
verbose |
A switch which determines whether or not the
progress of the sampler is printed to the screen. If
|
b0 |
The prior mean of β. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the prior mean for all of the betas. |
B0 |
The prior precision of β. This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of beta. Default value of 0 is equivalent to an improper uniform prior for beta. |
c0 |
c_0/2 is the shape parameter for the inverse Gamma prior on σ^2 (the variance of the disturbances). The amount of information in the inverse Gamma prior is something like that from c_0 pseudo-observations. |
d0 |
d_0/2 is the scale parameter for the inverse Gamma prior on σ^2 (the variance of the disturbances). In constructing the inverse Gamma prior, d_0 acts like the sum of squared errors from the c_0 pseudo-observations. |
delta0 |
The prior mean of α. |
Delta0 |
The prior precision of α. |
a |
a is the shape1 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states. |
b |
b is the shape2 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states. |
seed |
The seed for the random number generator. If NA, current R system seed is used. |
... |
further arguments to be passed |
HMMpanelFE simulates from the fixed-effect hidden Markov
pbject level:
\varepsilon_{it} \sim \mathcal{N}(α_{im}, σ^2_{im})
We assume standard, semi-conjugate priors:
β \sim \mathcal{N}(b_0,B_0^{-1})
And:
σ^{-2} \sim \mathcal{G}amma(c_0/2, d_0/2)
And:
α \sim \mathcal{N}(delta_0,Delta_0^{-1})
β, α and σ^{-2} are assumed a priori independent.
And:
p_{mm} \sim \mathcal{B}eta(a, b),\;\; m = 1, …, M
Where M is the number of states.
OLS estimates are used for starting values.
An mcmc object that contains the posterior sample. This
object can be summarized by functions provided by the coda
package. The object contains an attribute sigma storage
matrix that contains time-varying residual variance, an attribute
state storage matrix that contains posterior samples of
hidden states, and an attribute delta storage matrix
containing time-varying intercepts.
Jong Hee Park, 2012. “Unified Method for Dynamic and Cross-Sectional Heterogeneity: Introducing Hidden Markov Panel Models.” American Journal of Political Science.56: 1040-1054. <doi: 10.1111/j.1540-5907.2012.00590.x>
Siddhartha Chib. 1998. “Estimation and comparison of multiple change-point models.” Journal of Econometrics. 86: 221-241. <doi: 10.1016/S0304-4076(97)00115-2>
## Not run:
## data generating
set.seed(1974)
N <- 30
T <- 80
NT <- N*T
## true parameter values
true.beta <- c(1, 1)
true.sigma <- 3
x1 <- rnorm(NT)
x2 <- runif(NT, 2, 4)
## group-specific breaks
break.point = rep(T/2, N); break.sigma=c(rep(1, N));
break.list <- rep(1, N)
X <- as.matrix(cbind(x1, x2), NT, );
y <- rep(NA, NT)
id <- rep(1:N, each=NT/N)
K <- ncol(X);
true.beta <- as.matrix(true.beta, K, 1)
## compute the break probability
ruler <- c(1:T)
W.mat <- matrix(NA, T, N)
for (i in 1:N){
W.mat[, i] <- pnorm((ruler-break.point[i])/break.sigma[i])
}
Weight <- as.vector(W.mat)
## draw time-varying individual effects and sample y
j = 1
true.sigma.alpha <- 30
true.alpha1 <- true.alpha2 <- rep(NA, N)
for (i in 1:N){
Xi <- X[j:(j+T-1), ]
true.mean <- Xi %*% true.beta
weight <- Weight[j:(j+T-1)]
true.alpha1[i] <- rnorm(1, 0, true.sigma.alpha)
true.alpha2[i] <- -1*true.alpha1[i]
y[j:(j+T-1)] <- ((1-weight)*true.mean + (1-weight)*rnorm(T, 0, true.sigma) +
(1-weight)*true.alpha1[i]) +
(weight*true.mean + weight*rnorm(T, 0, true.sigma) + weight*true.alpha2[i])
j <- j + T
}
## extract the standardized residuals from the OLS with fixed-effects
FEols <- lm(y ~ X + as.factor(id) -1 )
resid.all <- rstandard(FEols)
time.id <- rep(1:80, N)
## model fitting
G <- 100
BF <- testpanelSubjectBreak(subject.id=id, time.id=time.id,
resid= resid.all, max.break=3, minimum = 10,
mcmc=G, burnin = G, thin=1, verbose=G,
b0=0, B0=1/100, c0=2, d0=2, Time = time.id)
## get the estimated break numbers
estimated.breaks <- make.breaklist(BF, threshold=3)
## model fitting
out <- HMMpanelFE(subject.id = id, y, X=X, m = estimated.breaks,
mcmc=G, burnin=G, thin=1, verbose=G,
b0=0, B0=1/100, c0=2, d0=2, delta0=0, Delta0=1/100)
## print out the slope estimate
## true values are 1 and 1
summary(out)
## compare them with the result from the constant fixed-effects
summary(FEols)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.