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LFS_model_with_GT_cycle.R
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LFS_model_with_GT_cycle.R
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# This R-script is party based on "An Introduction to State Space Models" by Marc Wildi.
# The function "KF_slopes_mixed_factor" 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 (9x1 vector).
# 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=167).
# opti: if TRUE, optimizes the function.
# k: Tx5 matrix of thestandard errors of the GREG estimates.
# delta: autocorrelation coefficient of the survey errors.
# outofsample: if TRUE, computes the loglikelihood based on the out-of-sample forecast errors.
# parP10: large number for the diffuse initialization.
# nstates: number of state variables in the model.
# lambda: nx1 vector of estimated, by PCA, factor's loading: ONLY 1 FACTOR IS ESTIMATED FOR THE GOOGLE TRENDS.
# 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.
# sARIMA: vector with the estimated coefficients of an ARIMA(3,1,1) model for the Google Trends' factor.
# The lines that are not commented here, are commented on /LFS_model.R and /LFS_model_with_GT.R.
# Packages required to run the scripts:
library(magic)
KF_slopes_mixed_factor <- function(par,y,opti,k,delta,outofsample,parP10,nstates,lambda,H,ns.id,sARIMA){
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)
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))
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), 0, exp(sigma_Rx), 0, exp(sigma_Rx), diag(sqrt(diag(H)[ns.id])))
gamma <- par[9]
R[32,2] <- tanh(gamma)
R[2,32] <- tanh(gamma)
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)
Tylambda <- diag(4)
TyE <- rbind(matrix(0,9,5), cbind(diag(4), c(0,0,0,0)))
delta <- delta
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 <- adiag(matrix(c(1,1,0,0,0,sARIMA[1],1,1,0,sARIMA[2],0,0,0,0,sARIMA[3]/sARIMA[2],0),4,4,byrow=T),diag(1,ncol=(length(ns.id)), nrow=(length(ns.id)))) # include the ARIMA estimated coefficients in the transition matrix for the Google Trends' factor.
Tmatrix <- adiag(Ty, Tx)
Rt <- adiag(diag(nstates-length(ns.id)-4),diag(c(0,1,0,sARIMA[4])),diag(length(ns.id))) # R matrix that premultiplies the states' innovations in the transition matrix.
# Initialization of loglikelihood:
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)))
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 component.
Zx <- diag(one.ns.id)
Zx <- Zx[,which(!apply(Zx,2,FUN = function(x){all(x == 0)}))]
Zx <- cbind(as.matrix(lambda,length(lambda),1)%*%matrix(c(1,1,0,0),1,4,byrow=T),Zx)
Z <- adiag(Zy,Zx)
epshatoutofsample <- 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) + adiag(diag(0,ncol(waves),ncol(waves)),diag(diag.H,nrow(y)-ncol(waves),nrow(y)-ncol(waves)))
if ((NaN %in% Fmatrix)==T){
logl<- -P10[1]
} else {
svdFmatrix <- svd(Fmatrix)
Kg <- Pttm1[[i]]%*%t(Z)%*%svdFmatrix$v%*%diag(1/svdFmatrix$d)%*%t(svdFmatrix$u)
if (is.na(epshatoutofsample[1,])){
Kg[,c(1:ncol(waves))] <- matrix(0,nstates,ncol(waves))
}
if (is.na(epshatoutofsample[ncol(waves)+1,])){
Kg[,c((ncol(waves)+1):ncol(Kg))] <- matrix(0,nstates,ncol(Kg)-(ncol(waves)))
}
epshatoutofsample <- ifelse(is.na(epshatoutofsample), 0, epshatoutofsample)
xtt[,i] <- xttm1[,i]+Kg%*%epshatoutofsample
epshatinsample <- y[,i]-Z%*%xtt[,i]
epshatinsample <- ifelse(is.na(epshatinsample), 0, epshatinsample)
Ptt[[i]] <- Pttm1[[i]]-Kg%*%Z%*%Pttm1[[i]]
Pttm1[[i+1]] <- Tmatrix%*%Ptt[[i]]%*%t(Tmatrix)+Rt%*%Q%*%t(Rt)
xttm1[,i+1] <- Tmatrix%*%xtt[,i]
# The optimization criterion:
if (outofsample) {
if (i <= (30-13) ){
logl <- logl - nrow(y)/2*log(2*pi)
} else if (i > (30-13) ){
logl <- logl - nrow(y)/2*log(2*pi) - 1/2*log(det(Fmatrix)) - 1/2*t(epshatoutofsample)%*%svdFmatrix$v%*%diag(1/svdFmatrix$d)%*%t(svdFmatrix$u)%*%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)%*%svdFmatrix$v%*%diag(1/svdFmatrix$d)%*%t(svdFmatrix$u)%*%epshatinsample
if ((NaN %in% logl)==T){
logl<- -P10[1]
}
}
}
}
}
if (opti) {
return(-logl)
}
else {
return(list(logl=-logl, xtt=xtt,xttm1=xttm1,Pttm1=Pttm1,Ptt=Ptt))
}
}
objopt <- optim(par=c(log(2000),log(0.02),log(900),log(1.07),log(0.99*(1-0.21^2)),
log(1.01*(1-0.21^2)),log(1.13*(1-0.21^2)),log(1.06*(1-0.21^2)),0),
KF_slopes_mixed_factor,y=y,opti=T,k=k,delta=0.21,outofsample=T,
parP10=1000000000000,nstates=34+length(rev(tis[,2])[j.star:s]),lambda=Lambda.hat.max[,1],H=H, ns.id=sort(rev(tis[,2])[j.star:s]), sARIMA=sARIMA, hessian=F, method="L-BFGS-B")
par <- objopt$par
obj <- KF_slopes_mixed_factor(par=objopt$par,y=y,opti=F,k=k,delta=0.21,outofsample=T,parP10=1000000000000,
nstates=34+length(rev(tis[,2])[j.star:s]),lambda=Lambda.hat.max[,1],H=H, ns.id=sort(rev(tis[,2])[j.star:s]), sARIMA=sARIMA)