HMMpanelRE {MCMCpack}

Markov Chain Monte Carlo for the Hidden Markov Random-effects Model

Description

HMMpanelRE generates a sample from the posterior distribution of the hidden Markov random-effects model discussed in Park (2011). The code works for panel data with the same starting point. The sampling of panel parameters is based on Algorithm 2 of Chib and Carlin (1999). This model uses a multivariate Normal prior for the fixed effects parameters and varying individual effects, an Inverse-Wishart prior on the random-effects parameters, 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

HMMpanelRE(
  subject.id,
  time.id,
  y,
  X,
  W,
  m = 1,
  mcmc = 1000,
  burnin = 1000,
  thin = 1,
  verbose = 0,
  b0 = 0,
  B0 = 0.001,
  c0 = 0.001,
  d0 = 0.001,
  r0,
  R0,
  a = NULL,
  b = NULL,
  seed = NA,
  beta.start = NA,
  sigma2.start = NA,
  D.start = NA,
  P.start = NA,
  marginal.likelihood = c("none", "Chib95"),
  ...
)

Arguments

subject.id

A numeric vector indicating the group number. It should start from 1.

time.id

A numeric vector indicating the time unit. It should start from 1.

y

The dependent variable

X

The model matrix of the fixed-effects

W

The model matrix of the random-effects. W should be a subset of X.

m

The number of changepoints.

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.

r0

The shape parameter for the Inverse-Wishart prior on variance matrix for the random effects. Set r=q for an uninformative prior where q is the number of random effects

R0

The scale matrix for the Inverse-Wishart prior on variance matrix for the random effects. This must be a square q-dimension matrix. Use plausible variance regarding random effects for the diagonal of R.

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.

beta.start

The starting values for the beta vector. This can either be a scalar or a column vector with dimension equal to the number of betas. The default value of NA will use draws from the Uniform distribution with the same boundary with the data as the starting value. If this is a scalar, that value will serve as the starting value mean for all of the betas. When there is no covariate, the log value of means should be used.

sigma2.start

The starting values for \sigma^2. This can either be a scalar or a column vector with dimension equal to the number of states.

D.start

The starting values for the beta vector. This can either be a scalar or a column vector with dimension equal to the number of betas. The default value of NA will use draws from the Uniform distribution with the same boundary with the data as the starting value. If this is a scalar, that value will serve as the starting value mean for all of the betas. When there is no covariate, the log value of means should be used.

P.start

The starting values for the transition matrix. A user should provide a square matrix with dimension equal to the number of states. By default, draws from the Beta(0.9, 0.1) are used to construct a proper transition matrix for each raw except the last raw.

marginal.likelihood

How should the marginal likelihood be calculated? Options are: none in which case the marginal likelihood will not be calculated and Chib95 in which case the method of Chib (1995) is used.

...

further arguments to be passed

Details

HMMpanelRE simulates from the random-effect hidden Markov panel model introduced by Park (2011).

The model takes the following form:

y_i = X_i \beta_m + W_i b_i + \varepsilon_i\;\; m = 1, \ldots, M

Where each group i have k_i observations. Random-effects parameters are assumed to be time-varying at the system level:

b_i \sim \mathcal{N}_q(0, D_m)

\varepsilon_i \sim \mathcal{N}(0, \sigma^2_m I_{k_i})

And the errors: We assume standard, conjugate priors:

\beta \sim \mathcal{N}_p(b0, B0)

And:

\sigma^{2} \sim \mathcal{IG}amma(c0/2, d0/2)

And:

D \sim \mathcal{IW}ishart(r0, R0)

See Chib and Carlin (1999) for more details.

And:

p_{mm} \sim \mathcal{B}eta(a, b),\;\; m = 1, \ldots, M

Where M is the number of states.

NOTE: We do not provide default parameters for the priors on the precision matrix for the random effects. When fitting one of these models, it is of utmost importance to choose a prior that reflects your prior beliefs about the random effects. Using the dwish and rwish functions might be useful in choosing these values.

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 prob.state storage matrix that contains the probability of state_i for each period, and the log-marginal likelihood of the model (logmarglike).

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. doi:10.18637/jss.v042.i09.

Examples

## Not run: 
  ## data generating
  set.seed(1977)
  Q <- 3
  true.beta1   <-  c(1, 1, 1) ; true.beta2   <-  c(-1, -1, -1)
  true.sigma2 <-  c(2, 5); true.D1 <- diag(.5, Q); true.D2 <- diag(2.5, Q)
  N=30; T=100;
  NT <- N*T
  x1 <- runif(NT, 1, 2)
  x2 <- runif(NT, 1, 2)
  X <- cbind(1, x1, x2);   W <- X;   y <- rep(NA, NT)

  ## true break numbers are one and at the center
  break.point = rep(T/2, N); break.sigma=c(rep(1, N));
  break.list <- rep(1, N)
  id  <-  rep(1:N, each=NT/N)
  K <-  ncol(X);
  ruler <- c(1:T)

  ## compute the weight for the break
  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)

  ## data generating by weighting two means and variances
  j = 1
  for (i in 1:N){
    Xi <- X[j:(j+T-1), ]
    Wi <- W[j:(j+T-1), ]
    true.V1 <- true.sigma2[1]*diag(T) + Wi%*%true.D1%*%t(Wi)
    true.V2 <- true.sigma2[2]*diag(T) + Wi%*%true.D2%*%t(Wi)
    true.mean1 <- Xi%*%true.beta1
    true.mean2 <- Xi%*%true.beta2
    weight <- Weight[j:(j+T-1)]
    y[j:(j+T-1)] <- (1-weight)*true.mean1 + (1-weight)*chol(true.V1)%*%rnorm(T) +
      weight*true.mean2 + weight*chol(true.V2)%*%rnorm(T)
    j <- j + T
  }
  ## model fitting
  subject.id <- c(rep(1:N, each=T))
  time.id <- c(rep(1:T, N))

  ## model fitting
  G <- 100
  b0  <- rep(0, K) ; B0  <- solve(diag(100, K))
  c0  <- 2; d0  <- 2
  r0  <- 5; R0  <- diag(c(1, 0.1, 0.1))
  subject.id <- c(rep(1:N, each=T))
  time.id <- c(rep(1:T, N))
  out1 <- HMMpanelRE(subject.id, time.id, y, X, W, m=1,
                     mcmc=G, burnin=G, thin=1, verbose=G,
                     b0=b0, B0=B0, c0=c0, d0=d0, r0=r0, R0=R0)

  ## latent state changes
  plotState(out1)

  ## print mcmc output
  summary(out1)




## End(Not run)