Nothing
R/sfacross-unormal.R
In sfaR: Stochastic Frontier Analysis using R
Defines functions
cmarguninorm_Vu
cmarguninorm_Eu
cuninormeff
uninormAlgOpt
chessuninormlike
cgraduninormlike
cstuninorm
cuninormlike
########################################################
# #
# uniform + normal distributions #
# #
# #
########################################################
# Log-likelihood ----------
cuninormlike <- function(parm, nXvar, nuZUvar, nvZVvar, uHvar,
vHvar, Yvar, Xvar, S) {
beta <- parm[1:(nXvar)]
delta <- parm[(nXvar + 1):(nXvar + nuZUvar)]
phi <- parm[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)]
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- Yvar - as.numeric(crossprod(matrix(beta), t(Xvar)))
ll <- -Wu + log(pnorm((exp(Wu) + S * epsilon)/exp(Wv/2)) -
pnorm(S * epsilon/exp(Wv/2)))
return(ll)
}
# starting value for the log-likelihood ----------
cstuninorm <- function(olsObj, epsiRes, S, nuZUvar, uHvar, nvZVvar,
vHvar) {
m2 <- moment(epsiRes, order = 2)
m4 <- moment(epsiRes, order = 4)
if ((m2^2 - m4) < 0) {
theta <- (abs(120 * (3 * m2^2 - m4)))^(1/4)
varu <- theta^2/12
} else {
theta <- (120 * (3 * m2^2 - m4))^(1/4)
varu <- theta^2/12
}
if ((m2 - varu) < 0) {
varv <- abs(m2 - varu)
} else {
varv <- m2 - varu
}
dep_u <- 1/2 * log(((epsiRes^2 - varv) * 12)^2) * 1/2
dep_v <- 1/2 * log((epsiRes^2 - varu)^2)
reg_hetu <- if (nuZUvar == 1) {
lm(log(varu) ~ 1)
} else {
lm(dep_u ~ ., data = as.data.frame(uHvar[, 2:nuZUvar]))
}
if (any(is.na(reg_hetu$coefficients)))
stop("At least one of the OLS coefficients of 'uhet' is NA: ",
paste(colnames(uHvar)[is.na(reg_hetu$coefficients)],
collapse = ", "), ". This may be due to a singular matrix due to potential perfect multicollinearity",
call. = FALSE)
reg_hetv <- if (nvZVvar == 1) {
lm(log(varv) ~ 1)
} else {
lm(dep_v ~ ., data = as.data.frame(vHvar[, 2:nvZVvar]))
}
if (any(is.na(reg_hetv$coefficients)))
stop("at least one of the OLS coefficients of 'vhet' is NA: ",
paste(colnames(vHvar)[is.na(reg_hetv$coefficients)],
collapse = ", "), ". This may be due to a singular matrix due to potential perfect multicollinearity",
call. = FALSE)
delta <- coefficients(reg_hetu)
names(delta) <- paste0("Zu_", colnames(uHvar))
phi <- coefficients(reg_hetv)
names(phi) <- paste0("Zv_", colnames(vHvar))
if (names(olsObj)[1] == "(Intercept)") {
beta <- c(olsObj[1] + S * theta/2, olsObj[-1])
} else {
beta <- olsObj
}
return(c(beta, delta, phi))
}
# Gradient of the likelihood function ----------
cgraduninormlike <- function(parm, nXvar, nuZUvar, nvZVvar, uHvar,
vHvar, Yvar, Xvar, S) {
beta <- parm[1:(nXvar)]
delta <- parm[(nXvar + 1):(nXvar + nuZUvar)]
phi <- parm[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)]
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- Yvar - as.numeric(crossprod(matrix(beta), t(Xvar)))
epsiv <- S * (epsilon)/exp(Wv/2)
epsiu <- exp(Wu) + S * (epsilon)
epsiuv <- (epsiu)/exp(Wv/2)
depsiv <- dnorm(epsiv)
depsiuv <- dnorm(epsiuv)
pepsiv <- pnorm(epsiv)
pepsiuv <- pnorm(epsiuv)
sigx1 <- 0.5 * (S * depsiv * (epsilon)) - 0.5 * (depsiuv *
(epsiu))
sigx2 <- depsiv - depsiuv
sigx3 <- pepsiuv - pepsiv
sigx4 <- exp(Wv/2) * (sigx3)
depsiuvx2 <- depsiuv * exp(Wu)
gradll <- (cbind(sweep(Xvar, MARGIN = 1, STATS = S * (sigx2)/(sigx4),
FUN = "*"), sweep(uHvar, MARGIN = 1, STATS = (depsiuvx2/(sigx4) -
1), FUN = "*"), sweep(vHvar, MARGIN = 1, STATS = (sigx1)/(sigx4),
FUN = "*")))
return(gradll)
}
# Hessian of the likelihood function ----------
chessuninormlike <- function(parm, nXvar, nuZUvar, nvZVvar, uHvar,
vHvar, Yvar, Xvar, S) {
beta <- parm[1:(nXvar)]
delta <- parm[(nXvar + 1):(nXvar + nuZUvar)]
phi <- parm[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)]
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- Yvar - as.numeric(crossprod(matrix(beta), t(Xvar)))
epsiv <- S * (epsilon)/exp(Wv/2)
epsiu <- exp(Wu) + S * (epsilon)
epsiuv <- (epsiu)/exp(Wv/2)
depsiv <- dnorm(epsiv)
depsiuv <- dnorm(epsiuv)
pepsiv <- pnorm(epsiv)
pepsiuv <- pnorm(epsiuv)
sigx1 <- 0.5 * (S * depsiv * (epsilon)) - 0.5 * (depsiuv *
(epsiu))
sigx2 <- depsiv - depsiuv
sigx3 <- pepsiuv - pepsiv
sigx4 <- exp(Wv/2) * (sigx3)
depsiuvx2 <- depsiuv * exp(Wu)
sigx5 <- exp(Wv/2)^3 * (sigx3)
hessll <- matrix(nrow = nXvar + nuZUvar + nvZVvar, ncol = nXvar +
nuZUvar + nvZVvar)
hessll[1:nXvar, 1:nXvar] <- crossprod(sweep(Xvar, MARGIN = 1,
STATS = S^2 * ((S * depsiv * (epsilon) - depsiuv * (epsiu))/(sigx5) -
(sigx2)^2/(sigx4)^2), FUN = "*"), Xvar)
hessll[1:nXvar, (nXvar + 1):(nXvar + nuZUvar)] <- crossprod(sweep(Xvar,
MARGIN = 1, STATS = S * ((epsiu)/(sigx5) - (sigx2)/(sigx4)^2) *
depsiuvx2, FUN = "*"), uHvar)
hessll[1:nXvar, (nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)] <- crossprod(sweep(Xvar,
MARGIN = 1, STATS = S * ((0.5 * (depsiv * (S^2 * (epsilon)^2/exp(Wv/2)^2 -
1)) - 0.5 * (((epsiu)^2/exp(Wv/2)^2 - 1) * depsiuv))/(sigx4) -
(sigx1) * (sigx2)/(sigx4)^2), FUN = "*"), vHvar)
hessll[(nXvar + 1):(nXvar + nuZUvar), (nXvar + 1):(nXvar +
nuZUvar)] <- crossprod(sweep(uHvar, MARGIN = 1, STATS = ((1 -
exp(Wu) * epsiuv/exp(Wv/2))/(sigx4) - depsiuvx2/(sigx4)^2) *
depsiuvx2, FUN = "*"), uHvar)
hessll[(nXvar + 1):(nXvar + nuZUvar), (nXvar + nuZUvar + 1):(nXvar +
nuZUvar + nvZVvar)] <- crossprod(sweep(uHvar, MARGIN = 1,
STATS = -(((sigx1)/(sigx4)^2 + 0.5 * ((1 - epsiuv^2)/(sigx4))) *
depsiuvx2), FUN = "*"), vHvar)
hessll[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar), (nXvar +
nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)] <- crossprod(sweep(vHvar,
MARGIN = 1, STATS = ((0.25 * (S^3 * depsiv * (epsilon)^3) -
0.25 * (depsiuv * (epsiu)^3))/(sigx5) - (0.5 * (sigx4) +
sigx1) * (sigx1)/(sigx4)^2), FUN = "*"), vHvar)
hessll[lower.tri(hessll)] <- t(hessll)[lower.tri(hessll)]
# hessll <- (hessll + (hessll))/2
return(hessll)
}
# Optimization using different algorithms ----------
uninormAlgOpt <- function(start, olsParam, dataTable, S, nXvar,
uHvar, nuZUvar, vHvar, nvZVvar, Yvar, Xvar, method, printInfo,
itermax, stepmax, tol, gradtol, hessianType, qac) {
startVal <- if (!is.null(start))
start else cstuninorm(olsObj = olsParam, epsiRes = dataTable[["olsResiduals"]],
S = S, uHvar = uHvar, nuZUvar = nuZUvar, vHvar = vHvar,
nvZVvar = nvZVvar)
startLoglik <- sum(cuninormlike(startVal, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar, vHvar = vHvar,
Yvar = Yvar, Xvar = Xvar, S = S))
if (method %in% c("bfgs", "bhhh", "nr", "nm")) {
maxRoutine <- switch(method, bfgs = function(...) maxBFGS(...),
bhhh = function(...) maxBHHH(...), nr = function(...) maxNR(...),
nm = function(...) maxNM(...))
method <- "maxLikAlgo"
}
mleObj <- switch(method, ucminf = ucminf(par = startVal,
fn = function(parm) -sum(cuninormlike(parm, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar,
vHvar = vHvar, Yvar = Yvar, Xvar = Xvar, S = S)),
gr = function(parm) -colSums(cgraduninormlike(parm, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar,
vHvar = vHvar, Yvar = Yvar, Xvar = Xvar, S = S)),
hessian = 0, control = list(trace = if (printInfo) 1 else 0, maxeval = itermax,
stepmax = stepmax, xtol = tol, grtol = gradtol)),
maxLikAlgo = maxRoutine(fn = cuninormlike, grad = cgraduninormlike,
hess = chessuninormlike, start = startVal, finalHessian = if (hessianType ==
2) "bhhh" else TRUE, control = list(printLevel = if (printInfo) 2 else 0,
iterlim = itermax, reltol = tol, tol = tol, qac = qac),
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), sr1 = trust.optim(x = startVal, fn = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gr = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), method = "SR1", control = list(maxit = itermax,
cgtol = gradtol, stop.trust.radius = tol, prec = tol,
report.level = if (printInfo) 2 else 0, report.precision = 1L)),
sparse = trust.optim(x = startVal, fn = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gr = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), hs = function(parm) as(-chessuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), "dgCMatrix"), method = "Sparse", control = list(maxit = itermax,
cgtol = gradtol, stop.trust.radius = tol, prec = tol,
report.level = if (printInfo) 2 else 0, report.precision = 1L,
preconditioner = 1L)), mla = mla(b = startVal, fn = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gr = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), hess = function(parm) -chessuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), print.info = printInfo, maxiter = itermax,
epsa = gradtol, epsb = gradtol), nlminb = nlminb(start = startVal,
objective = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gradient = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), hessian = function(parm) -chessuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), control = list(iter.max = itermax, trace = if (printInfo) 1 else 0,
eval.max = itermax, rel.tol = tol, x.tol = tol)))
if (method %in% c("ucminf", "nlminb")) {
mleObj$gradient <- colSums(cgraduninormlike(mleObj$par,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S))
}
mlParam <- if (method %in% c("ucminf", "nlminb")) {
mleObj$par
} else {
if (method == "maxLikAlgo") {
mleObj$estimate
} else {
if (method %in% c("sr1", "sparse")) {
names(mleObj$solution) <- names(startVal)
mleObj$solution
} else {
if (method == "mla") {
mleObj$b
}
}
}
}
if (hessianType != 2) {
if (method %in% c("ucminf", "nlminb"))
mleObj$hessian <- chessuninormlike(parm = mleObj$par,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)
if (method == "sr1")
mleObj$hessian <- chessuninormlike(parm = mleObj$solution,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)
}
mleObj$logL_OBS <- cuninormlike(parm = mlParam, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar, vHvar = vHvar,
Yvar = Yvar, Xvar = Xvar, S = S)
mleObj$gradL_OBS <- cgraduninormlike(parm = mlParam, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar, vHvar = vHvar,
Yvar = Yvar, Xvar = Xvar, S = S)
return(list(startVal = startVal, startLoglik = startLoglik,
mleObj = mleObj, mlParam = mlParam))
}
# Conditional efficiencies estimation ----------
cuninormeff <- function(object, level) {
beta <- object$mlParam[1:(object$nXvar)]
delta <- object$mlParam[(object$nXvar + 1):(object$nXvar +
object$nuZUvar)]
phi <- object$mlParam[(object$nXvar + object$nuZUvar + 1):(object$nXvar +
object$nuZUvar + object$nvZVvar)]
Xvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 1)
uHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 2)
vHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 3)
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- model.response(model.frame(object$formula, data = object$dataTable)) -
as.numeric(crossprod(matrix(beta), t(Xvar)))
theta <- exp(Wu)
u1 <- -exp(Wv/2) * ((dnorm((theta + object$S * epsilon)/exp(Wv/2)) -
dnorm(object$S * epsilon/exp(Wv/2)))/(pnorm((theta +
object$S * epsilon)/exp(Wv/2)) - pnorm(object$S * epsilon/exp(Wv/2)))) -
object$S * epsilon
u2 <- exp(Wv/2) * (dnorm(object$S * epsilon/exp(Wv/2))/(1 -
pnorm(object$S * epsilon/exp(Wv/2))) - object$S * epsilon/exp(Wv/2)) # when theta/sigmav ---> Infty
uLB <- exp(Wv/2) * qnorm((1 - level)/2 * pnorm((theta + object$S *
epsilon)/exp(Wv/2)) + (1 - (1 - level)/2) * pnorm(object$S *
epsilon/exp(Wv/2))) - object$S * epsilon
uUB <- exp(Wv/2) * qnorm((1 - (1 - level)/2) * pnorm((theta +
object$S * epsilon)/exp(Wv/2)) + (1 - level)/2 * pnorm(object$S *
epsilon/exp(Wv/2))) - object$S * epsilon
m <- ifelse(-theta < object$S * epsilon & object$S * epsilon <
0, -object$S * epsilon, ifelse(object$S * epsilon >=
0, 0, theta))
if (object$logDepVar == TRUE) {
teJLMS1 <- exp(-u1)
teJLMS2 <- exp(-u2)
teMO <- exp(-m)
teBC1 <- exp(object$S * epsilon + exp(Wv)/2) * (pnorm((object$S *
epsilon + theta)/exp(Wv/2) + exp(Wv/2)) - pnorm(object$S *
epsilon/exp(Wv/2) + exp(Wv/2)))/(pnorm((theta + object$S *
epsilon)/exp(Wv/2)) - pnorm(object$S * epsilon/exp(Wv/2)))
teBC2 <- exp(object$S * epsilon + exp(Wv)/2) * (1 - pnorm(object$S *
epsilon/exp(Wv/2) + exp(Wv/2)))/(1 - pnorm(object$S *
epsilon/exp(Wv/2)))
teBCLB <- exp(-uUB)
teBCUB <- exp(-uLB)
res <- bind_cols(u1 = u1, u2 = u2, uLB = uLB, uUB = uUB,
teJLMS1 = teJLMS1, teJLMS2 = teJLMS2, m = m, teMO = teMO,
teBC1 = teBC1, teBC2 = teBC2, teBCLB = teBCLB, teBCUB = teBCUB)
} else {
res <- bind_cols(u1 = u1, u2 = u2, uLB = uLB, uUB = uUB,
m = m)
}
return(res)
}
# Marginal effects on inefficiencies ----------
cmarguninorm_Eu <- function(object) {
delta <- object$mlParam[(object$nXvar + 1):(object$nXvar +
object$nuZUvar)]
uHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 2)
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
margEff <- kronecker(matrix(delta[2:object$nuZUvar], nrow = 1),
matrix(sqrt(3)/2 * exp(Wu/2), ncol = 1))
colnames(margEff) <- paste0("Eu_", colnames(uHvar)[-1])
return(margEff)
}
cmarguninorm_Vu <- function(object) {
delta <- object$mlParam[(object$nXvar + 1):(object$nXvar +
object$nuZUvar)]
uHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 2)
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
margEff <- kronecker(matrix(delta[2:object$nuZUvar], nrow = 1),
matrix(exp(Wu), ncol = 1))
colnames(margEff) <- paste0("Vu_", colnames(uHvar)[-1])
return(margEff)
}
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Defines functions cmarguninorm_Vu cmarguninorm_Eu cuninormeff uninormAlgOpt chessuninormlike cgraduninormlike cstuninorm cuninormlike
########################################################
# #
# uniform + normal distributions #
# #
# #
########################################################
# Log-likelihood ----------
cuninormlike <- function(parm, nXvar, nuZUvar, nvZVvar, uHvar,
vHvar, Yvar, Xvar, S) {
beta <- parm[1:(nXvar)]
delta <- parm[(nXvar + 1):(nXvar + nuZUvar)]
phi <- parm[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)]
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- Yvar - as.numeric(crossprod(matrix(beta), t(Xvar)))
ll <- -Wu + log(pnorm((exp(Wu) + S * epsilon)/exp(Wv/2)) -
pnorm(S * epsilon/exp(Wv/2)))
return(ll)
}
# starting value for the log-likelihood ----------
cstuninorm <- function(olsObj, epsiRes, S, nuZUvar, uHvar, nvZVvar,
vHvar) {
m2 <- moment(epsiRes, order = 2)
m4 <- moment(epsiRes, order = 4)
if ((m2^2 - m4) < 0) {
theta <- (abs(120 * (3 * m2^2 - m4)))^(1/4)
varu <- theta^2/12
} else {
theta <- (120 * (3 * m2^2 - m4))^(1/4)
varu <- theta^2/12
}
if ((m2 - varu) < 0) {
varv <- abs(m2 - varu)
} else {
varv <- m2 - varu
}
dep_u <- 1/2 * log(((epsiRes^2 - varv) * 12)^2) * 1/2
dep_v <- 1/2 * log((epsiRes^2 - varu)^2)
reg_hetu <- if (nuZUvar == 1) {
lm(log(varu) ~ 1)
} else {
lm(dep_u ~ ., data = as.data.frame(uHvar[, 2:nuZUvar]))
}
if (any(is.na(reg_hetu$coefficients)))
stop("At least one of the OLS coefficients of 'uhet' is NA: ",
paste(colnames(uHvar)[is.na(reg_hetu$coefficients)],
collapse = ", "), ". This may be due to a singular matrix due to potential perfect multicollinearity",
call. = FALSE)
reg_hetv <- if (nvZVvar == 1) {
lm(log(varv) ~ 1)
} else {
lm(dep_v ~ ., data = as.data.frame(vHvar[, 2:nvZVvar]))
}
if (any(is.na(reg_hetv$coefficients)))
stop("at least one of the OLS coefficients of 'vhet' is NA: ",
paste(colnames(vHvar)[is.na(reg_hetv$coefficients)],
collapse = ", "), ". This may be due to a singular matrix due to potential perfect multicollinearity",
call. = FALSE)
delta <- coefficients(reg_hetu)
names(delta) <- paste0("Zu_", colnames(uHvar))
phi <- coefficients(reg_hetv)
names(phi) <- paste0("Zv_", colnames(vHvar))
if (names(olsObj)[1] == "(Intercept)") {
beta <- c(olsObj[1] + S * theta/2, olsObj[-1])
} else {
beta <- olsObj
}
return(c(beta, delta, phi))
}
# Gradient of the likelihood function ----------
cgraduninormlike <- function(parm, nXvar, nuZUvar, nvZVvar, uHvar,
vHvar, Yvar, Xvar, S) {
beta <- parm[1:(nXvar)]
delta <- parm[(nXvar + 1):(nXvar + nuZUvar)]
phi <- parm[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)]
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- Yvar - as.numeric(crossprod(matrix(beta), t(Xvar)))
epsiv <- S * (epsilon)/exp(Wv/2)
epsiu <- exp(Wu) + S * (epsilon)
epsiuv <- (epsiu)/exp(Wv/2)
depsiv <- dnorm(epsiv)
depsiuv <- dnorm(epsiuv)
pepsiv <- pnorm(epsiv)
pepsiuv <- pnorm(epsiuv)
sigx1 <- 0.5 * (S * depsiv * (epsilon)) - 0.5 * (depsiuv *
(epsiu))
sigx2 <- depsiv - depsiuv
sigx3 <- pepsiuv - pepsiv
sigx4 <- exp(Wv/2) * (sigx3)
depsiuvx2 <- depsiuv * exp(Wu)
gradll <- (cbind(sweep(Xvar, MARGIN = 1, STATS = S * (sigx2)/(sigx4),
FUN = "*"), sweep(uHvar, MARGIN = 1, STATS = (depsiuvx2/(sigx4) -
1), FUN = "*"), sweep(vHvar, MARGIN = 1, STATS = (sigx1)/(sigx4),
FUN = "*")))
return(gradll)
}
# Hessian of the likelihood function ----------
chessuninormlike <- function(parm, nXvar, nuZUvar, nvZVvar, uHvar,
vHvar, Yvar, Xvar, S) {
beta <- parm[1:(nXvar)]
delta <- parm[(nXvar + 1):(nXvar + nuZUvar)]
phi <- parm[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)]
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- Yvar - as.numeric(crossprod(matrix(beta), t(Xvar)))
epsiv <- S * (epsilon)/exp(Wv/2)
epsiu <- exp(Wu) + S * (epsilon)
epsiuv <- (epsiu)/exp(Wv/2)
depsiv <- dnorm(epsiv)
depsiuv <- dnorm(epsiuv)
pepsiv <- pnorm(epsiv)
pepsiuv <- pnorm(epsiuv)
sigx1 <- 0.5 * (S * depsiv * (epsilon)) - 0.5 * (depsiuv *
(epsiu))
sigx2 <- depsiv - depsiuv
sigx3 <- pepsiuv - pepsiv
sigx4 <- exp(Wv/2) * (sigx3)
depsiuvx2 <- depsiuv * exp(Wu)
sigx5 <- exp(Wv/2)^3 * (sigx3)
hessll <- matrix(nrow = nXvar + nuZUvar + nvZVvar, ncol = nXvar +
nuZUvar + nvZVvar)
hessll[1:nXvar, 1:nXvar] <- crossprod(sweep(Xvar, MARGIN = 1,
STATS = S^2 * ((S * depsiv * (epsilon) - depsiuv * (epsiu))/(sigx5) -
(sigx2)^2/(sigx4)^2), FUN = "*"), Xvar)
hessll[1:nXvar, (nXvar + 1):(nXvar + nuZUvar)] <- crossprod(sweep(Xvar,
MARGIN = 1, STATS = S * ((epsiu)/(sigx5) - (sigx2)/(sigx4)^2) *
depsiuvx2, FUN = "*"), uHvar)
hessll[1:nXvar, (nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)] <- crossprod(sweep(Xvar,
MARGIN = 1, STATS = S * ((0.5 * (depsiv * (S^2 * (epsilon)^2/exp(Wv/2)^2 -
1)) - 0.5 * (((epsiu)^2/exp(Wv/2)^2 - 1) * depsiuv))/(sigx4) -
(sigx1) * (sigx2)/(sigx4)^2), FUN = "*"), vHvar)
hessll[(nXvar + 1):(nXvar + nuZUvar), (nXvar + 1):(nXvar +
nuZUvar)] <- crossprod(sweep(uHvar, MARGIN = 1, STATS = ((1 -
exp(Wu) * epsiuv/exp(Wv/2))/(sigx4) - depsiuvx2/(sigx4)^2) *
depsiuvx2, FUN = "*"), uHvar)
hessll[(nXvar + 1):(nXvar + nuZUvar), (nXvar + nuZUvar + 1):(nXvar +
nuZUvar + nvZVvar)] <- crossprod(sweep(uHvar, MARGIN = 1,
STATS = -(((sigx1)/(sigx4)^2 + 0.5 * ((1 - epsiuv^2)/(sigx4))) *
depsiuvx2), FUN = "*"), vHvar)
hessll[(nXvar + nuZUvar + 1):(nXvar + nuZUvar + nvZVvar), (nXvar +
nuZUvar + 1):(nXvar + nuZUvar + nvZVvar)] <- crossprod(sweep(vHvar,
MARGIN = 1, STATS = ((0.25 * (S^3 * depsiv * (epsilon)^3) -
0.25 * (depsiuv * (epsiu)^3))/(sigx5) - (0.5 * (sigx4) +
sigx1) * (sigx1)/(sigx4)^2), FUN = "*"), vHvar)
hessll[lower.tri(hessll)] <- t(hessll)[lower.tri(hessll)]
# hessll <- (hessll + (hessll))/2
return(hessll)
}
# Optimization using different algorithms ----------
uninormAlgOpt <- function(start, olsParam, dataTable, S, nXvar,
uHvar, nuZUvar, vHvar, nvZVvar, Yvar, Xvar, method, printInfo,
itermax, stepmax, tol, gradtol, hessianType, qac) {
startVal <- if (!is.null(start))
start else cstuninorm(olsObj = olsParam, epsiRes = dataTable[["olsResiduals"]],
S = S, uHvar = uHvar, nuZUvar = nuZUvar, vHvar = vHvar,
nvZVvar = nvZVvar)
startLoglik <- sum(cuninormlike(startVal, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar, vHvar = vHvar,
Yvar = Yvar, Xvar = Xvar, S = S))
if (method %in% c("bfgs", "bhhh", "nr", "nm")) {
maxRoutine <- switch(method, bfgs = function(...) maxBFGS(...),
bhhh = function(...) maxBHHH(...), nr = function(...) maxNR(...),
nm = function(...) maxNM(...))
method <- "maxLikAlgo"
}
mleObj <- switch(method, ucminf = ucminf(par = startVal,
fn = function(parm) -sum(cuninormlike(parm, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar,
vHvar = vHvar, Yvar = Yvar, Xvar = Xvar, S = S)),
gr = function(parm) -colSums(cgraduninormlike(parm, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar,
vHvar = vHvar, Yvar = Yvar, Xvar = Xvar, S = S)),
hessian = 0, control = list(trace = if (printInfo) 1 else 0, maxeval = itermax,
stepmax = stepmax, xtol = tol, grtol = gradtol)),
maxLikAlgo = maxRoutine(fn = cuninormlike, grad = cgraduninormlike,
hess = chessuninormlike, start = startVal, finalHessian = if (hessianType ==
2) "bhhh" else TRUE, control = list(printLevel = if (printInfo) 2 else 0,
iterlim = itermax, reltol = tol, tol = tol, qac = qac),
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), sr1 = trust.optim(x = startVal, fn = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gr = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), method = "SR1", control = list(maxit = itermax,
cgtol = gradtol, stop.trust.radius = tol, prec = tol,
report.level = if (printInfo) 2 else 0, report.precision = 1L)),
sparse = trust.optim(x = startVal, fn = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gr = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), hs = function(parm) as(-chessuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), "dgCMatrix"), method = "Sparse", control = list(maxit = itermax,
cgtol = gradtol, stop.trust.radius = tol, prec = tol,
report.level = if (printInfo) 2 else 0, report.precision = 1L,
preconditioner = 1L)), mla = mla(b = startVal, fn = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gr = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), hess = function(parm) -chessuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), print.info = printInfo, maxiter = itermax,
epsa = gradtol, epsb = gradtol), nlminb = nlminb(start = startVal,
objective = function(parm) -sum(cuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), gradient = function(parm) -colSums(cgraduninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)), hessian = function(parm) -chessuninormlike(parm,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S), control = list(iter.max = itermax, trace = if (printInfo) 1 else 0,
eval.max = itermax, rel.tol = tol, x.tol = tol)))
if (method %in% c("ucminf", "nlminb")) {
mleObj$gradient <- colSums(cgraduninormlike(mleObj$par,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S))
}
mlParam <- if (method %in% c("ucminf", "nlminb")) {
mleObj$par
} else {
if (method == "maxLikAlgo") {
mleObj$estimate
} else {
if (method %in% c("sr1", "sparse")) {
names(mleObj$solution) <- names(startVal)
mleObj$solution
} else {
if (method == "mla") {
mleObj$b
}
}
}
}
if (hessianType != 2) {
if (method %in% c("ucminf", "nlminb"))
mleObj$hessian <- chessuninormlike(parm = mleObj$par,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)
if (method == "sr1")
mleObj$hessian <- chessuninormlike(parm = mleObj$solution,
nXvar = nXvar, nuZUvar = nuZUvar, nvZVvar = nvZVvar,
uHvar = uHvar, vHvar = vHvar, Yvar = Yvar, Xvar = Xvar,
S = S)
}
mleObj$logL_OBS <- cuninormlike(parm = mlParam, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar, vHvar = vHvar,
Yvar = Yvar, Xvar = Xvar, S = S)
mleObj$gradL_OBS <- cgraduninormlike(parm = mlParam, nXvar = nXvar,
nuZUvar = nuZUvar, nvZVvar = nvZVvar, uHvar = uHvar, vHvar = vHvar,
Yvar = Yvar, Xvar = Xvar, S = S)
return(list(startVal = startVal, startLoglik = startLoglik,
mleObj = mleObj, mlParam = mlParam))
}
# Conditional efficiencies estimation ----------
cuninormeff <- function(object, level) {
beta <- object$mlParam[1:(object$nXvar)]
delta <- object$mlParam[(object$nXvar + 1):(object$nXvar +
object$nuZUvar)]
phi <- object$mlParam[(object$nXvar + object$nuZUvar + 1):(object$nXvar +
object$nuZUvar + object$nvZVvar)]
Xvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 1)
uHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 2)
vHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 3)
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
Wv <- as.numeric(crossprod(matrix(phi), t(vHvar)))
epsilon <- model.response(model.frame(object$formula, data = object$dataTable)) -
as.numeric(crossprod(matrix(beta), t(Xvar)))
theta <- exp(Wu)
u1 <- -exp(Wv/2) * ((dnorm((theta + object$S * epsilon)/exp(Wv/2)) -
dnorm(object$S * epsilon/exp(Wv/2)))/(pnorm((theta +
object$S * epsilon)/exp(Wv/2)) - pnorm(object$S * epsilon/exp(Wv/2)))) -
object$S * epsilon
u2 <- exp(Wv/2) * (dnorm(object$S * epsilon/exp(Wv/2))/(1 -
pnorm(object$S * epsilon/exp(Wv/2))) - object$S * epsilon/exp(Wv/2)) # when theta/sigmav ---> Infty
uLB <- exp(Wv/2) * qnorm((1 - level)/2 * pnorm((theta + object$S *
epsilon)/exp(Wv/2)) + (1 - (1 - level)/2) * pnorm(object$S *
epsilon/exp(Wv/2))) - object$S * epsilon
uUB <- exp(Wv/2) * qnorm((1 - (1 - level)/2) * pnorm((theta +
object$S * epsilon)/exp(Wv/2)) + (1 - level)/2 * pnorm(object$S *
epsilon/exp(Wv/2))) - object$S * epsilon
m <- ifelse(-theta < object$S * epsilon & object$S * epsilon <
0, -object$S * epsilon, ifelse(object$S * epsilon >=
0, 0, theta))
if (object$logDepVar == TRUE) {
teJLMS1 <- exp(-u1)
teJLMS2 <- exp(-u2)
teMO <- exp(-m)
teBC1 <- exp(object$S * epsilon + exp(Wv)/2) * (pnorm((object$S *
epsilon + theta)/exp(Wv/2) + exp(Wv/2)) - pnorm(object$S *
epsilon/exp(Wv/2) + exp(Wv/2)))/(pnorm((theta + object$S *
epsilon)/exp(Wv/2)) - pnorm(object$S * epsilon/exp(Wv/2)))
teBC2 <- exp(object$S * epsilon + exp(Wv)/2) * (1 - pnorm(object$S *
epsilon/exp(Wv/2) + exp(Wv/2)))/(1 - pnorm(object$S *
epsilon/exp(Wv/2)))
teBCLB <- exp(-uUB)
teBCUB <- exp(-uLB)
res <- bind_cols(u1 = u1, u2 = u2, uLB = uLB, uUB = uUB,
teJLMS1 = teJLMS1, teJLMS2 = teJLMS2, m = m, teMO = teMO,
teBC1 = teBC1, teBC2 = teBC2, teBCLB = teBCLB, teBCUB = teBCUB)
} else {
res <- bind_cols(u1 = u1, u2 = u2, uLB = uLB, uUB = uUB,
m = m)
}
return(res)
}
# Marginal effects on inefficiencies ----------
cmarguninorm_Eu <- function(object) {
delta <- object$mlParam[(object$nXvar + 1):(object$nXvar +
object$nuZUvar)]
uHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 2)
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
margEff <- kronecker(matrix(delta[2:object$nuZUvar], nrow = 1),
matrix(sqrt(3)/2 * exp(Wu/2), ncol = 1))
colnames(margEff) <- paste0("Eu_", colnames(uHvar)[-1])
return(margEff)
}
cmarguninorm_Vu <- function(object) {
delta <- object$mlParam[(object$nXvar + 1):(object$nXvar +
object$nuZUvar)]
uHvar <- model.matrix(object$formula, data = object$dataTable,
rhs = 2)
Wu <- as.numeric(crossprod(matrix(delta), t(uHvar)))
margEff <- kronecker(matrix(delta[2:object$nuZUvar], nrow = 1),
matrix(exp(Wu), ncol = 1))
colnames(margEff) <- paste0("Vu_", colnames(uHvar)[-1])
return(margEff)
}
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