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LFS_model_with_GT.R
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LFS_model_with_GT.R
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# This R-script is party based on "An Introduction to State Space Models" by Marc Wildi.
# The function "LFS_GT" performs the estimation of the LFS model, with the multivariate auxiliary series of Google Trends. It requires the following arguments:
# par: initial values for the model's parameters ((9+r)x1 vector)g; r is the number of Google Trends' factors.
# y: (5+n)xT matrix of the unemployed labour force and the n Google trends (the first 5 series are the unemployed labour force) (T=184).
# opti: if TRUE, optimizes the function.
# k: Tx5 matrix of the standard errors of the GREG estimates.
# outofsample: if TRUE, computes the log-likelihood based on the out-of-sample forecast errors.
# parP10: large number for the diffuse initialization.
# nfactors: number of Google Trends' factors (r).
# nstates: number of state variables in the model.
# lambda: nxr vector of estimated, by PCA, factors' loadings.
# H: the nxn estimated covariance matrix, by PCA, of the idiosyncratic components of the Google Trends.
# ns.id: vector of length equal to the number of nonstationary idiosyncratic components. It should contain the ordered numbers corresponing to the nonstationary idiosyncratic components.
# The lines that are not commented here, are commented on /LFS_model.R.
# Packages required to run the scripts:
library(magic)
library(ucminf)
is.scalar <- function(x) is.atomic(x) && length(x) == 1L
LFS_GT <- function(par,y,opti,k,outofsample,parP10,nfactors,nstates,lambda,H,ns.id){
len <- length(y[1,])
sigma_Ry <- par[1]
sigma_omegay <- par[2]
sigma_lambda <- par[3]
sd_nu <- diag(exp(c(par[4], par[5], par[6], par[7], par[8])), 5,5)
sigma_Rx <- log(1) # variance of the Google Trends' factors' innovations (it is fixed).
x10 <- rep(0,nstates)
Pttm1 <- lapply(seq_len(len+1), function(X) matrix(0,nstates,nstates))
Ptt <- lapply(seq_len(len), function(X) matrix(0,nstates,nstates))
delta <- par[9]
P10 <- diag(c(rep(parP10[1],17),c(1,rep((1-delta^2),4),1,rep((1-delta^2),3),1,rep((1-delta^2),3)),rep(parP10[1],nstates-30)),nstates,nstates)
Pttm1[[1]] <- P10
xtt <- matrix(0,nstates,(len))
xttm1 <- matrix(0,nstates,(len+1))
xttm1[,1] <- x10
R <- diag(1,nstates,nstates)
D <- adiag(0, exp(sigma_Ry), exp(sigma_omegay)*diag(11), exp(sigma_lambda)*diag(4), sd_nu, diag(0,8,8), diag(rep(exp(sigma_Rx),nfactors)), diag(sqrt(diag(H)[ns.id])))
for (j in 1:nfactors){
R[(30+j),2] <- tanh(par[9+j]) # correlation between the LFS solpe's and the Google Trends' factors' innovations.
R[2,(30+j)] <- tanh(par[9+j])
}
Q <- D%*%R%*%D
# Build T (the transition matrix):
Tymu <- matrix(c(1,1,0,1),2,2, byrow=T)
C <- array(0,dim=c(2,2,5))
for (l in 1:5){
C[,,l] <- matrix(c(cos((pi*l)/6), sin((pi*l)/6), -sin((pi*l)/6), cos((pi*l)/6)),2,2,byrow=TRUE)
}
Tyomega <- adiag(C[,,1],C[,,2],C[,,3],C[,,4],C[,,5],-1)
ncol(Tyomega)
nrow(Tyomega)
Tylambda <- diag(4)
TyE <- rbind(matrix(0,9,5), cbind(diag(4), c(0,0,0,0)))
TyE <- cbind(TyE, rbind(c(0,0,0,0),diag(delta,nrow=4,ncol=4),matrix(0,8,4)))
TyE <- cbind(TyE, rbind(matrix(0,5,4),diag(4),matrix(0,4,4)))
Ty <- adiag(Tymu, Tyomega, Tylambda, TyE)
Tx <- diag(1,ncol=(length(ns.id)+nfactors), nrow=(length(ns.id)+nfactors)) # transition matrix of the Google Trends' factors and the nonstationary idiosyncratic components.
Tmatrix <- adiag(Ty, Tx)
# initialization of log-likelihood:
logl <- 0
# Start of KF recursions:
for (i in 1:len){
# Bulild Z:
Zy <- c(1,0)
Zy <- rep(Zy,6)
Zy <- c(Zy,1)
Zy <- rbind(Zy,Zy,Zy,Zy,Zy)
Zy <- cbind(Zy,rbind(c(0,0,0,0),diag(4)))
if (is.na(k[i,])) {
k[i,which(is.na(k[i,]))] <- 0
}
Zy <- cbind(Zy, diag(as.numeric(k[i,]), nrow=5, ncol=5), matrix(0, nrow=5, ncol=8))
one.ns.id <- rep(0, nrow(H))
one.ns.id[ns.id] <- 1 # one if element of vector corresponds to nonstationary idiosyncratic components.
Zx <- cbind(as.matrix(lambda),diag(one.ns.id))
Zx <- Zx[,which(!apply(Zx,2,FUN = function(x){all(x == 0)}))]
Z <- cbind(adiag(Zy,as.matrix(Zx)))
ncol(Z)
nrow(Z)
W <- diag(1,length(y[,i]))
if (length(which(is.na(y[,i]))) > 0 && length(which(is.na(y[,i]))) < length(y[,i])){
W <- matrix(W[-which(is.na(y[,i])),], nrow=(length(y[,i])-length(which(is.na(y[,i])))), ncol=ncol(W))
Z <- W%*%Z
y[which(is.na(y[,i])),i] <- 0
}
if (length(which(is.na(y[,i]))) > 0 && length(which(is.na(y[,i]))) == length(y[,i])){
xtt[,i] <- xttm1[,i]
Ptt[[i]] <- Pttm1[[i]]
Pttm1[[i+1]] <- Tmatrix%*%Pttm1[[i]]%*%t(Tmatrix) + Q
xttm1[,i+1] <- Tmatrix%*%xttm1[,i]
} else {
epshatoutofsample <- W%*%y[,i] - Z%*%xttm1[,i]
diag.H <- diag(H)
diag.H[ns.id] <- 0 # only variances of stationary idiosyncratic components.
Fmatrix <- Z%*%Pttm1[[i]]%*%t(Z) + W%*%adiag(diag(0,ncol(waves),ncol(waves)),diag(diag.H,nrow(y)-ncol(waves),nrow(y)-ncol(waves)))%*%t(W)
if (is.scalar(Fmatrix) == TRUE){
Fmatrix.inv = 1/Fmatrix
} else {
svdFmatrix <- svd(Fmatrix)
Fmatrix.inv <- svdFmatrix$v%*%diag(1/svdFmatrix$d)%*%t(svdFmatrix$u)
}
Kg <- Tmatrix%*%Pttm1[[i]]%*%t(Z)%*%Fmatrix.inv
xtt[,i] <- xttm1[,i]+Pttm1[[i]]%*%t(Z)%*%Fmatrix.inv%*%epshatoutofsample
epshatinsample <- W%*%y[,i]-Z%*%xtt[,i]
Ptt[[i]] <- Pttm1[[i]]-Pttm1[[i]]%*%t(Z)%*%Fmatrix.inv%*%Z%*%Pttm1[[i]]
Pttm1[[i+1]] <- Tmatrix%*%Pttm1[[i]]%*%t(Tmatrix-Kg%*%Z)+Q
xttm1[,i+1] <- Tmatrix%*%xttm1[,i] + Kg%*%epshatoutofsample
}
# The optimization criterion:
if (outofsample) {
if (i <= (30-13) ){
logl <- logl - nrow(y)/2*log(2*pi)
} else if (i > (30-13) ){
logl <- logl - nrow(y)/2*log(2*pi) - 1/2*log(det(Fmatrix)) - 1/2*t(epshatoutofsample)%*%Fmatrix.inv%*%epshatoutofsample
if ((NaN %in% logl)==T){
logl<- -P10[1]
}
}
} else {
if (i <= (30-13) ){
logl <- logl - nrow(y)/2*log(2*pi)
} else if (i > (30-13) ){
logl <- logl - nrow(y)/2*log(2*pi) - 1/2*log(det(Fmatrix)) - 1/2*t(epshatinsample)%*%Fmatrix.inv%*%epshatinsample
if ((NaN %in% logl)==T){
logl<- -P10[1]
}
}
}
}
if (opti) {
return(-logl)
}
else {
return(list(logl=-logl, xtt=xtt,xttm1=xttm1,Pttm1=Pttm1,Ptt=Ptt))
}
}
init.val.GT <- c(log(2000),log(0.02),log(900),log(1.07),log(0.99*(1-0.21^2)),
log(1.01*(1-0.21^2)),log(1.13*(1-0.21^2)),log(1.06*(1-0.21^2)), 0.21, rep(0,nfactors))
objopt.GT <- ucminf(par=init.val.GT, LFS_GT,y,opti=T,k,outofsample=T,parP10=1000000000000,nfactors,nstates,lambda,H,
ns.id,hessian=2,control=list(grad="central", gradstep = c(1e-2, 1e-3), trace=T))
par.GT <- objopt.GT$par
obj <- LFS_GT(par=objopt.GT$par,y,opti=F,k,outofsample=T,parP10=1000000000000,nfactors,nstates,lambda,H,ns.id)