HMMpanelFE: Markov Chain Monte Carlo for the Hidden Markov Fixed-effects Model

Description

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.

Usage

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,
  ...
)

Value

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.

Arguments

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 verbose is greater than 0, the iteration number and the posterior density samples are printed to the screen every verboseth iteration.
b0: The prior mean of $\beta$. 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 $\beta$. 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 $\sigma^2$ (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 $\sigma^2$ (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 $\alpha$.
Delta0: The prior precision of $\alpha$.
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

Details

HMMpanelFE simulates from the fixed-effect hidden Markov pbject level: $$\varepsilon_{it} \sim \mathcal{N}(\alpha_{im}, \sigma^2_{im})$$

We assume standard, semi-conjugate priors: $$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$ And: $$\sigma^{-2} \sim \mathcal{G}amma(c_0/2, d_0/2)$$ And: $$\alpha \sim \mathcal{N}(delta_0,Delta_0^{-1})$$ $\beta$, $\alpha$ and $\sigma^{-2}$ are assumed a priori independent.

And: $$p_{mm} \sim \mathcal{B}eta(a, b),\;\; m = 1, \ldots, M$$ Where $M$ is the number of states.

OLS estimates are used for starting values.

References

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>

Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. ``MCMCpack: Markov Chain Monte Carlo in R.'', Journal of Statistical Software. 42(9): 1-21. tools:::Rd_expr_doi("10.18637/jss.v042.i09").

Examples

Run this code


if (FALSE) {
  ## 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)
}

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