diff --git a/Main_Monte_Carlo.R b/Main_Monte_Carlo.R
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+#install.packages("telescope")
+#installed.packages("extraDistr")
+#
+rm(list = ls())
+#
+library("telescope")
+library("stats")
+library("extraDistr")       # Package for the "dbnbinom" probability mass function
+
+source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sampleUniNormMixture.R")
+source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/priorOnK_spec.R")
+source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/prior_alpha_e0.R")
+source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sample_e0_alpha.R")
+source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sampleK.R")
+source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/identifyMixture.R")
+
+############################
+## (1) Set the Parameters ##
+############################
+set.seed(123)
+TT <- 50             # Number of time series observations
+N <- 50            # Number of cross-section observations
+MC <- 50               # Number of Monte Carlo iterations
+#
+sigma_Z <- 1        # standard deviation of the covariates Z
+beta <- 0           # value of the common parameter
+## Parameters of the mixture:
+K <- 3              # Number of the mixture components
+w_true <- c(1/3,1/3,1/3) # Vector of weights of each component of the mixture
+alpha_true <- c(0,5,-5)  # True values of the incidental parameter for the K mixture components
+var_u <- c(1,1,1)          # True values of the variance of the model error for the K mixture components
+## Parameters of the MCMC sampler:
+Mmax <- 10000   # the maximum number of iterations
+thin <- 1       # the thinning imposed to reduce auto-correlation in the chain by only recording every `thined' observation
+burnin <- 100   # the number of burn-in iterations not recorded
+M <- Mmax/thin  # the number of recorded observations.
+
+Kmax <- 50      # the maximum number of components possible during sampling
+Kinit <- 10     # the initial number of filled components
+
+## Initial values:
+# we use initial values corresponding to equal component sizes and half of the value of `C0` for the variances.
+
+eta_0 <- rep(1/Kinit, Kinit)
+r <- 1    # dimension
+c0 <- 2 + (r-1)/2
+g0 <- 0.2 + (r-1) / 2
+
+# Initial value for beta
+beta_0 <- beta
+
+###############
+## (2) Prior ##
+###############
+# Static specification for the weights with a fixed prior on $e_0$ where the value is set to 1.
+G <- "MixStatic" 
+
+priorOn_n0 <- priorOnE0_spec("e0const", 1)        # Prior on the hyperparameter of the Dirichlet prior
+
+priorOn_K <- priorOnK_spec("BNB_143", 30)          # Prior on K. This gives the function to compute log(PMF)-log(.5).
+
+# Prior on the common coefficient beta: N(beta_prior,Omega_prior)
+beta_prior <- 0
+Omega_prior <- 1
+
+# Initialization of matrices to store the results
+theta_alpha <- array(,dim=c(M,Kmax,MC))
+theta_Sigma2 <- array(,dim=c(M,Kmax,MC))
+Beta <- matrix(,M,MC)
+Eta <- array(,dim=c(M,Kmax,MC))
+S_label <- array(,dim=c(M,N,MC))
+Nk <- array(,dim=c(M,Kmax,MC))
+K_MC <- matrix(,M,MC)
+#
+Kplus <- matrix(,M,MC) 
+Kplus_stat <- matrix(,MC,4)
+K_stat <- matrix(,MC,5)
+#
+# Estimators
+theta_alpha_hat <- matrix(,MC,Kmax)
+theta_Sigma2_hat <- matrix(,MC,Kmax)
+Eta_hat <- matrix(,MC,Kmax)
+Beta_hat <- matrix(,MC,3)
+
+#####################
+## (3) Monte Carlo ##
+#####################
+for (mc in 1:MC){
+  #print(mc)
+  cat("mc = ",mc)
+  ## (3.1) Simulate the Z and Y 
+  #
+  ZZ <- rnorm(N*TT, mean = 0, sd = sigma_Z)
+  Z <- matrix(ZZ,TT,N)
+  
+  rm(ZZ)
+  S_unif <- runif(N, min = 0, max = 1)
+  S <- matrix(,N,1)               # S is the latent allocation variable 
+  Y <- matrix(,TT,N)
+  
+  for (ii in 1:N){
+    # Fill the latent allocation variable S
+    if (S_unif[ii] >= 0 & S_unif[ii] < w_true[1]) {
+      S[ii,1] <- 1
+    } else if (S_unif[ii] >= w_true[1] & S_unif[ii] < (w_true[1] + w_true[2])) {
+      S[ii,1] <- 2
+    } else {
+      S[ii,1] <- 3
+    }
+    u <- rnorm(TT, mean = 0, sd = sqrt(var_u[S[ii]]))
+    Y[,ii] <- alpha_true[S[ii]] + beta*Z[,ii] + u
+  }
+  
+  rm(S,S_unif,u)
+  Y_mean <- colMeans(Y)
+  
+  ## (3.2) Initial values of the other parameters (not specified outside the loop):
+  # Component-specific priors on \alpha_k (i.e. N(b0,B0)) and \sigma_k^2 (i.e. IG(c_0,C_0)) following Richardson and Green (1997).
+  R <- diff(range(Y))
+  C0 <- diag(c(0.02*(R^2)), nrow = r)
+  sigma2_0 <- array(0, dim = c(1, 1, Kinit))
+  sigma2_0[1, 1, ] <- 0.5 * C0
+  G0 <- diag(10/(R^2), nrow = r)
+  B0 <- diag((R^2), nrow = r)                       # prior variance of alpha_k
+  b0 <- as.matrix((max(Y) + min(Y))/2, ncol = 1)    # prior mean of alpha_k 
+
+  ## (3.3) Initial partition of the data and initial parameter values to start the MCMC. 
+  # We use `kmeans()` to determine the initial partition $S_0$ and the initial component-specific means \mu_0.
+  
+  cl_y <- kmeans(Y_mean, centers = Kinit, nstart = 30)
+  S_0 <- cl_y$cluster
+  alpha_0 <- t(cl_y$centers)
+  
+  ## (3.3) MCMC sampling 
+  # We draw samples from the posterior using the telescoping sampler of Fruhwirth-Schnatter. 
+  estGibbs <- sampleUniNormMixture(
+    Y, Z, S_0, alpha_0, sigma2_0, eta_0, beta_0,
+    c0, g0, G0, C0, b0, B0, beta_prior, Omega_prior,
+    M, burnin, thin, Kmax,
+    G, priorOn_K, priorOn_n0)
+  
+  #The sampling function returns a named list where the sampled parameters and latent variables are contained. 
+  theta_alpha[,,mc] <- estGibbs$Mu
+  theta_Sigma2[,,mc] <- estGibbs$Sigma2
+  Beta[,mc] <- estGibbs$Beta
+  Eta[,,mc] <- estGibbs$Eta
+  S_label[,,mc] <- estGibbs$S
+  Nk[,,mc] <- estGibbs$Nk
+  K_MC[,mc] <- estGibbs$K
+  Kplus[,mc] <- estGibbs$Kp
+  nonnormpost_mode_list <- estGibbs$nonnormpost_mode_list
+  acc <- estGibbs$acc
+  e0 <- estGibbs$e0
+  alpha_Dir <- estGibbs$alpha  
+  
+  #########################################
+  ## Identification of the mixture model ##
+  #########################################
+  ## Step 1: Estimating $K_+$ and $K$
+  ## K_+ ##
+  Nk_mc <- Nk[,,mc]
+  Kplus_mc <- rowSums(Nk_mc != 0, na.rm = TRUE)  
+  p_Kplus <- tabulate(Kplus_mc, nbins = max(Kplus_mc))/M  
+  # The distribution of $K_+$ is characterized using the 1st and 3rd quartile as well as the median.
+  Kplus_stat[mc,1:3] <- quantile(Kplus_mc, probs = c(0.25, 0.5, 0.75))
+  
+  # Point estimate for K_+ by taking the mode and determine the number of MCMC draws where exactly K_+ 
+  # components were filled.
+  Kplus_hat <- which.max(p_Kplus)
+  Kplus_stat[mc,4] <- Kplus_hat
+  M0 <- sum(Kplus_mc == Kplus_hat)
+  
+  ## K ##
+  # We also determine the posterior distribution of the number of components K directly drawn using the 
+  # telescoping sampler.
+  K_mc <- K_MC[,mc]
+  p_K <- tabulate(K_mc, nbins = max(K_mc, na.rm = TRUE))/M
+  round(p_K[1:20], digits = 2)
+  
+  # The posterior mode can be determined as well as the posterior mean and quantiles of the posterior.
+  K_hat <- which.max(tabulate(K_mc, nbins = max(K_mc)))
+  K_stat[mc,4] <- K_hat
+  K_stat[mc,5] <- mean(K_mc)
+  K_stat[mc,1:3] <- quantile(K_mc, probs = c(0.25, 0.5, 0.75))
+  
+  ## Step 2: Extracting the draws with exactly \hat{K}_+ non-empty components
+  # Select those draws where the number of filled components was exactly \hat{K}_+:
+  index <- Kplus_mc == Kplus_hat
+  Nk_mc[is.na(Nk_mc)] <- 0
+  Nk_Kplus <- (Nk_mc[index, ] > 0)
+  
+  # We extract the cluster means, data cluster sizes and cluster assignments for the draws where exactly 
+  # \hat{K}_+ components were filled.
+  Mu_inter <- estGibbs$Mu[index, , , drop = FALSE]
+  Mu_Kplus <- array(0, dim = c(M0, 1, Kplus_hat)) 
+  Mu_Kplus[, 1, ] <- Mu_inter[, 1, ][Nk_Kplus]          # Here, 'Nk_Kplus' is used to subset the extracted 
+  # elements from Mu_inter. It selects specific rows  
+  # from the first column of Mu_inter based on the 
+  # indices in Nk_Kplus.
+  
+  Sigma2_inter <- estGibbs$Sigma2[index, , , drop = FALSE]
+  Sigma2_Kplus <- array(0, dim = c(M0, 1, Kplus_hat)) 
+  Sigma2_Kplus[, 1, ] <- Sigma2_inter[, 1, ][Nk_Kplus]
+  
+  Eta_inter <- estGibbs$Eta[index, ]
+  Eta_Kplus <- matrix(Eta_inter[Nk_Kplus], ncol = Kplus_hat)
+  Eta_Kplus <- sweep(Eta_Kplus, 1, rowSums(Eta_Kplus), "/")     # it normalizes the weights eta
+  
+  w <- which(index)
+  S_Kplus <- matrix(0, M0, ncol(Y))
+  for (i in seq_along(w)) {
+    m <- w[i]
+    perm_S <- rep(0, Kmax)
+    perm_S[Nk_mc[m, ] != 0] <- 1:Kplus_hat         # It assigns the sequence '1:Kplus_hat' to the positions in 
+    # perm_S where Nk[m, ] is not zero.
+    S_Kplus[i, ] <- perm_S[S_label[m, ,mc]]              # 'perm_S[S[m, ]]' indexes the vector 'perm_S' using the 
+    # values in S[m, ]. This permutes or reassigns the values in 
+    # 'S[m, ]' based on the mapping in 'perm_S'.
+  }
+  
+  ## Step 3: Clustering and relabeling of the MCMC draws in the point process representation
+  # For model identification, we cluster the draws of the means where exactly \hat{K}_+ components were 
+  # filled in the point process representation using k-means clustering. 
+  Func_init <- nonnormpost_mode_list[[Kplus_hat]]$mu  
+  identified_Kplus <- identifyMixture(
+    Mu_Kplus, Mu_Kplus, Eta_Kplus, S_Kplus, Sigma2_Kplus, Func_init)
+  
+  #A named list is returned which contains the proportion of draws where the clustering did not result in a 
+  # permutation and hence no relabeling could be performed and the draws had to be omitted.
+  identified_Kplus$non_perm_rate
+  
+  
+  ## Step 4: Estimating data cluster specific parameters and determining the final partition 
+  # The relabeled draws are also returned which can be used to determine posterior mean values for data 
+  # cluster specific parameters.
+  Mu_mean <- colMeans(identified_Kplus$Mu)
+  theta_alpha_hat[mc,1:length(Mu_mean)] <- colMeans(identified_Kplus$Mu)
+  theta_Sigma2_hat[mc,1:length(Mu_mean)] <- colMeans(identified_Kplus$Sigma2)
+  Eta_hat[mc,1:length(colMeans(identified_Kplus$Eta))] <- colMeans(identified_Kplus$Eta)
+  Beta_hat[mc,1] <- mean(Beta[,mc])
+  Beta_hat[mc,2:3] <- quantile(Beta[,mc], probs = c(0.025,0.975), na.rm =TRUE)
+  
+  rm(R, C0, sigma2_0, G0, B0, b0, cl_y, S_0, alpha_0, estGibbs, nonnormpost_mode_list, acc, e0, alpha_Dir, 
+     Kplus_mc, Nk_mc, p_Kplus, Kplus_hat, K_mc, index, Mu_inter, Mu_Kplus, Nk_Kplus, Sigma2_inter, 
+     Sigma2_Kplus, Eta_inter, Eta_Kplus, w, S_Kplus, perm_S, Func_init, identified_Kplus, Mu_mean)
+}
+
+theta_alpha_hat2<-t(apply(theta_alpha_hat[,1:3],1,sort))
+theta_alpha_hat_means <- colMeans(theta_alpha_hat2)
+Beta_hat_mean <- colMeans(Beta_hat)
+MSE_beta = mean((Beta_hat[,1] - beta)^2)
+MSE_alpha = colMeans((theta_alpha_hat2[,1:3] - matrix(rep(sort(alpha_true),MC), ncol=3, byrow=TRUE))^2)
+RMSE_alpha <- sqrt(MSE_alpha)
+
+Eta_hat2 <- t(matrix(Eta_hat[order(row(theta_alpha_hat[,1:3]), theta_alpha_hat[,1:3])], byrow = FALSE, ncol = ncol(Eta_hat)))
+Eta_hat_means <- colMeans(Eta_hat2)
+MSE_eta = colMeans((Eta_hat2[,1:3] - matrix(rep(w_true,MC), ncol=3, byrow=TRUE))^2)
+
+theta_Sigma2_hat2 <- t(matrix(theta_Sigma2_hat[order(row(theta_alpha_hat[,1:3]), theta_alpha_hat[,1:3])], byrow = FALSE, ncol = ncol(theta_Sigma2_hat)))
+theta_Sigma2_hat_means <- colMeans(theta_Sigma2_hat2)
+Sigma2_sorted <- var_u[order(sort(alpha_true))]
+MSE_Sigma2 = colMeans((theta_Sigma2_hat2[,1:3] - matrix(rep(Sigma2_sorted,MC), ncol=3, byrow=TRUE))^2)
+
+#MSE_beta = mean(Beta_hat[,1] - beta)^2
+#MSE_alpha = colMeans((theta_alpha_hat[,1:3] - matrix(rep(alpha_true,MC), ncol=3, byrow=TRUE))^2)
+#MSE_eta = colMeans((Eta_hat[,1:3] - matrix(rep(w_true,MC), ncol=3, byrow=TRUE))^2)
+
+