## ----setup, include = FALSE---------------------------------------------------
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  echo = TRUE, include = TRUE, message = FALSE, warning = FALSE, eval = FALSE
)

## -----------------------------------------------------------------------------
# library(kimfilter)
# library(data.table)
# library(maxLik)
# library(ggplot2)
# library(gridExtra)
# data(sw_dcf)
# data = sw_dcf[, colnames(sw_dcf) != "dcoinc", with = F]
# vars = colnames(data)[colnames(data) != "date"]
# 
# #State space model for the Stock and Watson Markov Switching Dynamic Common Factor model
# msdcf_ssm = function(par, yt, n_states = NULL){
#   #Get the number of states
#   n_states = length(unique(unlist(lapply(strsplit(names(par)[grepl("p_", names(par))], "p_"), function(x){substr(x[2], 1, 1)}))))
# 
#   #Get the parameters
#   vars = dimnames(yt)[which(unlist(lapply(dimnames(yt), function(x){!is.null(x)})))][[1]]
#   phi = par[grepl("phi", names(par))]
#   names(phi) = gsub("phi", "", names(phi))
#   gamma = par[grepl("gamma_", names(par))]
#   names(gamma) = gsub("gamma_", "", names(gamma))
#   psi = par[grepl("psi_", names(par))]
#   names(psi) = gsub("psi_", "", names(psi))
#   sig = par[grepl("sigma_", names(par))]
#   names(sig) = gsub("sigma_", "", names(sig))
#   mu = par[grepl("mu", names(par))]
#   names(mu) = gsub("mu_", "", names(mu))
#   pr = par[grepl("p_", names(par))]
#   names(pr) = gsub("p_", "", names(pr))
#   states = sort(unique(substr(names(pr), 1, 1)))
# 
#   #Steady state probabilities
#   Pm = matrix(NA, nrow = n_states, ncol = n_states)
#   rownames(Pm) = colnames(Pm) = unique(unlist(lapply(names(pr), function(x){strsplit(x, "")[[1]][2]})))
#   for(j in names(pr)){
#     Pm[strsplit(j, "")[[1]][2], strsplit(j, "")[[1]][1]] = pr[j]
#   }
#   for(j in 1:ncol(Pm)){
#     Pm[which(is.na(Pm[, j])), j] = 1 - sum(Pm[, j], na.rm = TRUE)
#   }
# 
#   #Build the transition matrix
#   phi_dim = max(c(length(phi)), length(unique(sapply(strsplit(names(gamma), "\\."), function(x){x[2]}))))
#   psi_dim = sapply(unique(sapply(strsplit(names(psi), "\\."), function(x){x[1]})), function(x){
#     max(as.numeric(sapply(strsplit(names(psi)[grepl(paste0("^", x), names(psi))], "\\."), function(x){x[2]})))
#   })
#   Fm = matrix(0, nrow = phi_dim + length(psi), ncol = phi_dim + length(psi),
#               dimnames = list(
#                 c(paste0("ct.", 0:(phi_dim - 1)),
#                   unlist(lapply(names(psi_dim), function(x){paste0("e_", x, ".", 0:(psi_dim[x] - 1))}))),
#                 c(paste0("ct.", 1:phi_dim),
#                   unlist(lapply(names(psi_dim), function(x){paste0("e_", x, ".", 1:psi_dim[x])})))
#               ))
#   Fm["ct.0", paste0("ct.", names(phi))] = phi
#   for(i in 1:length(vars)){
#     Fm[paste0("e_", i, ".0"),
#        paste0("e_", names(psi)[grepl(paste0("^", i), names(psi))])] = psi[grepl(paste0("^", i), names(psi))]
#   }
#   diag(Fm[intersect(rownames(Fm), colnames(Fm)), intersect(rownames(Fm), colnames(Fm))]) = 1
#   Fm = array(Fm, dim = c(nrow(Fm), ncol(Fm), n_states), dimnames = list(rownames(Fm), colnames(Fm), states))
# 
#   #Build the observation matrix
#   Hm = matrix(0, nrow = nrow(yt), ncol = nrow(Fm), dimnames = list(rownames(yt), rownames(Fm)))
#   for(i in 1:length(vars)){
#     Hm[i, paste0("ct.", sapply(strsplit(names(gamma)[grepl(paste0("^", i), names(gamma))], "\\."), function(x){x[2]}))] =
#       gamma[grepl(paste0("^", i), names(gamma))]
#   }
#   diag(Hm[, paste0("e_", 1:length(vars), ".0")]) = 1
#   Hm = array(Hm, dim = c(nrow(Hm), ncol(Hm), n_states), dimnames = list(rownames(Hm), colnames(Hm), states))
# 
#   #Build the state covariance matrix
#   #Set the dynamic common factor standard deviation to 1
#   Qm = matrix(0, ncol = ncol(Fm), nrow = nrow(Fm), dimnames = list(rownames(Fm), rownames(Fm)))
#   Qm["ct.0", "ct.0"] = 1
#   for(i in 1:length(vars)){
#     Qm[paste0("e_", i, ".0"), paste0("e_", i, ".0")] = sig[names(sig) == i]^2
#   }
#   Qm = array(Qm, dim = c(nrow(Qm), ncol(Qm), n_states), dimnames = list(rownames(Qm), colnames(Qm), states))
# 
#   #Build the observation equation covariance matrix
#   Rm = matrix(0, ncol = nrow(Hm), nrow = nrow(Hm), dimnames = list(vars, vars))
#   Rm = array(Rm, dim = c(nrow(Rm), ncol(Rm), n_states), dimnames = list(rownames(Rm), colnames(Rm), states))
# 
#   #State intercept matrix: the Markov switching mean matrix
#   Dm = matrix(0, nrow = nrow(Fm), ncol = 1, dimnames = list(rownames(Fm), NULL))
#   Dm = array(Dm, dim = c(nrow(Dm), 1, n_states), dimnames = list(rownames(Fm), NULL, states))
#   for(i in names(mu)){
#     Dm["ct.0", , i] = mu[i]
#   }
# 
#   #Observation equation intercept matrix
#   Am = matrix(0, nrow = nrow(Hm), ncol = 1)
#   Am = array(Am, dim = c(nrow(Am), ncol(Am), n_states), dimnames = list(vars, NULL, states))
# 
#   #Initialize the filter for each state
#   B0 = matrix(0, nrow(Fm), 1)
#   P0 = diag(nrow(Fm))
#   B0 = array(B0, dim = c(nrow(B0), ncol(B0), n_states), dimnames = list(rownames(Fm), NULL, states))
#   P0 = array(P0, dim = c(nrow(P0), ncol(P0), n_states), dimnames = list(rownames(B0), colnames(B0), states))
#   for(i in states){
#     B0[,,i] = solve(diag(ncol(Fm)) - Fm[,,i]) %*% Dm[,,i]
#     VecP_ll = solve(diag(prod(dim(Fm[,,i]))) - kronecker(Fm[,,i], Fm[,,i])) %*% matrix(as.vector(Qm[,,i]), ncol = 1)
#     P0[,,i] = matrix(VecP_ll[, 1], ncol = ncol(Fm))
#   }
# 
#   return(list(B0 = B0, P0 = P0, Am = Am, Dm = Dm, Hm = Hm, Fm = Fm, Qm = Qm, Rm = Rm, Pm = Pm))
# }
# 
# #Log the data
# data.log = copy(data)
# data.log[, c(vars) := lapply(.SD, log), .SDcols = c(vars)]
# 
# #Difference the data
# data.logd = copy(data.log)
# data.logd[, c(vars) := lapply(.SD, function(x){
#   x - shift(x, type = "lag", n = 1)
# }), .SDcols = c(vars)]
# 
# #Center the data
# data.logds = copy(data.logd)
# data.logds[, c(vars) := lapply(.SD, scale, scale = FALSE), .SDcols = c(vars)]
# 
# #Transpose the data
# yt = t(data.logds[, c(vars), with = F])
# 
# #Set the initial values
# init = c(phi1 = 0.8760, phi2 = -0.2171,
#          mu_u = 0.2802, mu_d = -1.5700,
#          p_dd = 0.8406, p_uu = 0.9696,
#          psi_1.1 = 0.0364, psi_1.2 = -0.0008,
#          psi_2.1 = -0.2965, psi_2.2 = -0.0657,
#          psi_3.1 = -0.3959, psi_3.2 = -0.1903,
#          psi_4.1 = -0.2436, psi_4.2 = 0.1281,
#          gamma_1.0 = 0.0058, gamma_1.1 = -0.0033,
#          gamma_2.0 = 0.0011,
#          gamma_3.0 = 0.0051, gamma_3.1 = -0.0033 ,
#          gamma_4.0 = 0.0012, gamma_4.1 = -0.0005, gamma_4.2 = 0.0001, gamma_4.3 = 0.0002,
#          sigma_1 = 0.0048, sigma_2 = 0.0057, sigma_3 = 0.0078, sigma_4 = 0.0013)
# 
# #Set the constraints
# ineqA = matrix(0, nrow = 20, ncol = length(init), dimnames = list(NULL, names(init)))
# #Stationarity constraints
# ineqA[c(1, 2), c("phi1", "phi2")] = rbind(c(1, 1), c(-1, -1))
# ineqA[c(3, 4), grepl("psi_1", colnames(ineqA))] = rbind(c(1, 1), c(-1, -1))
# ineqA[c(5, 6), grepl("psi_2", colnames(ineqA))] = rbind(c(1, 1), c(-1, -1))
# ineqA[c(7, 8), grepl("psi_3", colnames(ineqA))] = rbind(c(1, 1), c(-1, -1))
# ineqA[c(9, 10), grepl("psi_4", colnames(ineqA))] = rbind(c(1, 1), c(-1, -1))
# #Non-negativity constraints
# diag(ineqA[c(11, 12, 13, 14), grepl("sigma_", colnames(ineqA))]) = 1
# ineqA[c(15, 16), "p_dd"] = c(1, -1)
# ineqA[c(17, 18), "p_uu"] = c(1, -1)
# #Up/down states must be positive/negative
# ineqA[19, "mu_u"] = 1
# ineqA[20, "mu_d"] = -1
# ineqB = matrix(c(rep(1, 10),
#                  rep(0, 4),
#                  c(0, 1),
#                  c(0, 1),
#                  rep(0, 2)), nrow = nrow(ineqA), ncol = 1)
# 
# #Define the objective function
# objective = function(par, yt){
#   ssm = msdcf_ssm(par, yt)
#   return(kim_filter(ssm, yt, smooth = FALSE)$lnl)
# }
# 
# #Solve the model
# solve = maxLik(logLik = objective, start = init, method = "BFGS",
#                finalHessian = FALSE, hess = NULL,
#                control = list(printLevel = 2, iterlim = 10000),
#                constraints = list(ineqA = ineqA, ineqB = ineqB),
#                yt = yt)
# 
# #Get the estimated state space model
# ssm = msdcf_ssm(solve$estimate, yt)
# 
# #Get the column means and standard deviations
# M = matrix(unlist(data.logd[, lapply(.SD, mean, na.rm = TRUE), .SDcols = c(vars)]),
#            ncol = 1, dimnames = list(vars, NULL))
# 
# #Get the steady state coefficient matrices
# Pm = matrix(ss_prob(ssm[["Pm"]]), ncol = 1, dimnames = list(rownames(ssm[["Pm"]]), NULL))
# Hm = Reduce("+", lapply(dimnames(ssm[["Hm"]])[[3]], function(x){
#   Pm[x, ]*ssm[["Hm"]][,, x]
# }))
# Fm = Reduce("+", lapply(dimnames(ssm[["Fm"]])[[3]], function(x){
#   Pm[x, ]*ssm[["Fm"]][,, x]
# }))
# 
# #Final K_t is approximation to steady state K
# filter = kim_filter(ssm, yt, smooth = TRUE)
# K = filter$K_t[,, dim(filter$K_t)[3]]
# W = solve(diag(nrow(K)) - (diag(nrow(K)) - K %*% Hm) %*% Fm) %*% K
# d = (W %*% M)[1, 1]
# 
# #Get the intercept terms
# gamma = Hm[, grepl("ct", colnames(Hm))]
# D = M - gamma %*% matrix(rep(d, ncol(gamma)))
# 
# #Initialize first element of the dynamic common factor
# Y1 = t(data.log[, c(vars), with = F][1, ])
# initC = function(par){
#   return(sum((Y1 - D - gamma %*% par)^2))
# }
# C10 = optim(par = Y1, fn = initC, method = "BFGS", control = list(trace = FALSE))$par[1]
# Ctt = rep(C10, ncol(yt))
# 
# #Build the rest of the level of the dynamic common factor
# ctt = filter$B_tt[which(rownames(Fm) == "ct.0"), ]
# for(j in 2:length(Ctt)){
#   Ctt[j] = ctt[j] + Ctt[j - 1] + c(d)
# }
# Ctt = data.table(date = data$date, dcf = Ctt, d.dcf = ctt)
# prob = data.table(date = data$date, data.table(filter$Pr_tt))
# colnames(prob) = c("date", paste0("pr_", dimnames(ssm$Dm)[[3]]))
# uc = merge(Ctt, prob, by = "date", all = TRUE)
# 
# #Plot the outputs
# g1 = ggplot(melt(data.log, id.vars = "date")[, "value" := scale(value), by = "variable"]) +
#   ggtitle("Data", subtitle = "Log Levels (Rescaled)") +
#   scale_y_continuous(name = "Value") +
#   scale_x_date(name = "") +
#   geom_line(aes(x = date, y = value, group = variable, color = variable)) +
#   theme_minimal() + theme(legend.position = "bottom") + guides(color = guide_legend(title = NULL))
# 
# g2 = ggplot( melt(data.logds, id.vars = "date")) +
#   ggtitle("Data", subtitle = "Log Differenced & Standardized") +
#   scale_y_continuous(name = "Value") +
#   scale_x_date(name = "") +
#   geom_hline(yintercept = 0, color = "black") +
#   geom_line(aes(x = date, y = value, group = variable, color = variable)) +
#   theme_minimal() + theme(legend.position = "bottom") + guides(color = guide_legend(title = NULL))
# 
# toplot3 = melt(uc, id.vars = "date")
# d_range1 = range(toplot3[variable == "dcf", ]$value, na.rm = TRUE)
# p_range1 = range(toplot3[variable %in% colnames(uc)[grepl("pr_", colnames(uc))], ]$value, na.rm = TRUE)
# toplot3[variable %in% colnames(uc)[grepl("pr_", colnames(uc))], "value" := (value - p_range1[1])/diff(p_range1) * diff(d_range1) + d_range1[1], by = "variable"]
# g3 = ggplot() +
#   ggtitle("Dynamic Common Factor", subtitle = "Levels") +
#   scale_x_date(name = "") +
#   geom_hline(yintercept = 0, color = "grey") +
#   geom_line(data = toplot3[variable == "dcf", ],
#             aes(x = date, y = value, group = variable, color = variable)) +
#   theme_minimal() + theme(legend.position = "bottom") +
#   guides(color = guide_legend(title = NULL), fill = guide_legend(title = NULL)) +
#   scale_color_manual(values = "black") +
#   scale_y_continuous(name = "Value", limits = range(toplot3[variable == "dcf", ]$value, na.rm = TRUE),
#                      sec.axis = sec_axis(name = "Probability", ~((. - d_range1[1])/diff(d_range1) * diff(p_range1) + p_range1[1]) * 100)) +
#   geom_ribbon(data = toplot3[variable %in% "pr_d", ],
#               aes(x = date, ymin = d_range1[1], ymax = value, group = variable, fill = variable), alpha = 0.5) +
#   scale_fill_manual(values = c("red", "green"))
# 
# toplot4 = melt(uc, id.vars = "date")
# d_range2 = range(toplot4[variable %in% c("d.dcf"), ]$value, na.rm = TRUE)
# p_range2 = range(toplot4[variable %in% colnames(uc)[grepl("pr_", colnames(uc))], ]$value, na.rm = TRUE)
# toplot4[variable %in% colnames(uc)[grepl("pr_", colnames(uc))], "value" := (value - p_range2[1])/diff(p_range2) * diff(d_range2) + d_range2[1], by = "variable"]
# g4 = ggplot() +
#   ggtitle("Dynamic Common Factor", subtitle = "Differenced") +
#   scale_x_date(name = "") +
#   geom_hline(yintercept = 0, color = "grey") +
#   geom_line(data = toplot4[variable %in% c("d.dcf"), ],
#             aes(x = date, y = value, group = variable, color = variable)) +
#   theme_minimal() + theme(legend.position = "bottom") +
#   guides(color = guide_legend(title = NULL), fill = guide_legend(title = NULL)) +
#   scale_color_manual(values = "black") +
#   scale_y_continuous(name = "Value", limits = range(toplot4[variable %in% c("d.dcf"), ]$value, na.rm = TRUE),
#                      sec.axis = sec_axis(name = "Probability", ~((. - d_range2[1])/diff(d_range2) * diff(p_range2) + p_range2[1]) * 100)) +
#   geom_ribbon(data = toplot4[variable %in% "pr_d", ],
#               aes(x = date, ymin = d_range2[1], ymax = value, group = variable, fill = variable), alpha = 0.5) +
#   scale_fill_manual(values = c("red", "green"))
# 
# grid.arrange(g1, g2, g3, g4, layout_matrix = matrix(c(1, 3, 2, 4), nrow = 2))

