# A Test for the Subject-level Break using a Unitivariate Linear Regression Model with Breaks testpanelSubjectBreak fits a unitivariate linear regression model with parametric breaks using panel residuals to test the existence of subject-level breaks in panel residuals. The details are discussed in Park (2011). ```r testpanelSubjectBreak( subject.id, time.id, resid, max.break = 2, minimum = 10, mcmc = 1000, burnin = 1000, thin = 1, verbose = 0, b0, B0, c0, d0, a = NULL, b = NULL, seed = NA, Time = NULL, ps.out = FALSE, ... ) ``` ## 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. - `resid`: A vector of panel residuals. - `max.break`: An upper bound of break numbers for the test. - `minimum`: A minimum length of time series for the test. The test will skip a subject with a time series shorter than this. - `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 `verbose`th iteration. - `b0`: The prior mean of the residual mean. - `B0`: The prior precision of the residual variance - `c0`: $c_0/2$ is the shape parameter for the inverse Gamma prior on $\sigma^2$. 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 $\sigma^2$. - `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. - `Time`: Times of the observations. This will be used to find the time of the first observations in panel residuals. - `ps.out`: If ps.out == TRUE, state probabilities are exported. If the number of panel subjects is huge, users can turn it off to save memory. - `...`: further arguments to be passed ## Returns The returned object is a matrix containing log marginal likelihoods for all HMMs. The dimension of the returned object is the number of panel subjects by max.break + 1. If psout == TRUE, the returned object has an array attribute `psout` containing state probabilities for all HMMs. ## Details `testpanelSubjectBreak` fits a univariate linear regression model for subject-level residuals from a panel model. The details are discussed in Park (2011). The model takes the following form: $$ e_{it} = \alpha_{im} + \varepsilon_{it}\;\; m = 1, \ldots, M $$ The errors are assumed to be time-varying at the subject level: $$ \varepsilon_{it} \sim \mathcal{N}(0, \sigma^2_{im}) $$ We assume standard, semi-conjugate priors: $$ \beta \sim \mathcal{N}(b_0,B_0^{-1}) $$ And: $$ \sigma^{-2} \sim \mathcal{G}amma(c_0/2, d_0/2) $$ Where $\beta$ 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. ## Examples ```r ## Not run: 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 <- 1000 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) ## estimated break numbers ## thresho estimated.breaks <- make.breaklist(BF, threshold=3) ## print all posterior model probabilities print(attr(BF, "model.prob")) ## End(Not run) ``` ## 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. Siddhartha Chib. 1998. ``Estimation and comparison of multiple change-point models.'' **Journal of Econometrics**. 86: 221-241.

testpanelSubjectBreak function