R/midasreg.R

In mpiktas/midasr: Mixed Data Sampling Regression

##' Estimate unrestricted MIDAS regression
##'
##' Estimate unrestricted MIDAS regression using OLS. This function is a wrapper for \code{lm}.
##'
##' @param formula MIDAS regression model formula
##' @param data  a named list containing data with mixed frequencies
##' @param ... further arguments, which could be passed to \code{\link{lm}} function.
##' @return \code{\link{lm}} object.
##' @author Virmantas Kvedaras, Vaidotas Zemlys
##' @references Kvedaras V., Zemlys, V. \emph{Testing the functional constraints on parameters in regressions with variables of different frequency} Economics Letters 116 (2012) 250-254
##' @examples
##' ##The parameter function
##' theta_h0 <- function(p, dk, ...) {
##'    i <- (1:dk-1)/100
##'    pol <- p[3]*i + p[4]*i^2
##'    (p[1] + p[2]*i)*exp(pol)
##' }
##'
##' ##Generate coefficients
##' theta0 <- theta_h0(c(-0.1,10,-10,-10),4*12)
##'
##' ##Plot the coefficients
##' ##Do not run
##' #plot(theta0)
##'
##' ##' ##Generate the predictor variable
##' xx <- ts(arima.sim(model = list(ar = 0.6), 600 * 12), frequency = 12)
##'
##' ##Simulate the response variable
##' y <- midas_sim(500, xx, theta0)
##'
##' x <- window(xx, start=start(y))
##'
##' ##Create low frequency data.frame
##' ldt <- data.frame(y=y,trend=1:length(y))
##'
##' ##Create high frequency data.frame
##'
##' hdt <- data.frame(x=window(x, start=start(y)))
##'
##' ##Fit unrestricted model
##' mu <- midas_u(y~fmls(x,2,12)-1, list(ldt, hdt))
##'
##' ##Include intercept and trend in regression
##'
##' mu_it <- midas_u(y~fmls(x,2,12)+trend, list(ldt, hdt))
##'
##' ##Pass data as partialy named list
##'
##' mu_it <- midas_u(y~fmls(x,2,12)+trend, list(ldt, x=hdt$x))
##'
##' @details MIDAS regression has the following form:
##'
##' \deqn{y_t = \sum_{j=1}^p\alpha_jy_{t-j} +\sum_{i=0}^{k}\sum_{j=0}^{l_i}\beta_{j}^{(i)}x_{tm_i-j}^{(i)} + u_t,}
##'
##' where \eqn{x_\tau^{(i)}}, \eqn{i=0,...k} are regressors of higher (or similar) frequency than \eqn{y_t}.
##' Given certain assumptions the coefficients can be estimated using usual OLS and they have the familiar properties associated with simple linear regression.
##'
##' @export
midas_u <- function(formula, data, ...) {
  Zenv <- new.env(parent = environment(formula))

  if (missing(data)) {
    ee <- NULL
  }
  else {
    ee <- data_to_env(data)
    parent.env(ee) <- parent.frame()
  }

  assign("ee", ee, Zenv)
  mf <- match.call(expand.dots = TRUE)
  mf <- mf[-4]
  mf[[1L]] <- as.name("lm")
  mf[[3L]] <- as.name("ee")

  if (is.null(ee)) {
    yy <- eval(formula[[2]], Zenv)
  } else {
    yy <- eval(formula[[2]], ee)
  }

  if (inherits(yy, "ts")) {
    y_index <- 1:length(yy)
    if (!is.null(attr(mf, "na.action"))) {
      y_index <- y_index[-attr(mf, "na.action")]
    }
    if (length(y_index) > 1) {
      if (sum(abs(diff(y_index) - 1)) > 0) warning("There are NAs in the middle of the time series")
    }
    ysave <- yy[y_index]
    class(ysave) <- class(yy)
    attr(ysave, "tsp") <- c(time(yy)[range(y_index)], frequency(yy))
  } else {
    ysave <- yy
  }

  out <- eval(mf, Zenv)
  out$Zenv <- Zenv
  out$midas_coefficients <- out$coefficients
  out$lhs <- ysave
  class(out) <- c("midas_u", class(out))

  out
}

##' Restricted MIDAS regression
##'
##' Estimate restricted MIDAS regression using non-linear least squares.
##'
##' @param formula formula for restricted MIDAS regression or \code{midas_r} object. Formula must include \code{\link{fmls}} function
##' @param data a named list containing data with mixed frequencies
##' @param start the starting values for optimisation. Must be a list with named elements.
##' @param Ofunction the list with information which R function to use for optimisation. The list must have element named \code{Ofunction} which contains character string of chosen R function. Other elements of the list are the arguments passed to this function.  The default optimisation function is \code{\link{optim}} with argument \code{method="BFGS"}. Other supported functions are \code{\link{nls}}
##' @param weight_gradients a named list containing gradient functions of weights. The weight gradient function must return the matrix with dimensions
##' \eqn{d_k \times q}, where \eqn{d_k} and \eqn{q} are the number of coefficients in unrestricted and restricted regressions correspondingly.
##' The names of the list should coincide with the names of weights used in formula.
##' The default value is NULL, which means that the numeric approximation of weight function gradient is calculated. If the argument is not NULL, but the
##' name of the weight used in formula is not present, it is assumed that there exists an R function which has
##' the name of the weight function appended with \code{_gradient}.
##' @param ... additional arguments supplied to optimisation function
##' @return a \code{midas_r} object which is the list with the following elements:
##'
##' \item{coefficients}{the estimates of parameters of restrictions}
##' \item{midas_coefficients}{the estimates of MIDAS coefficients of MIDAS regression}
##' \item{model}{model data}
##' \item{unrestricted}{unrestricted regression estimated using \code{\link{midas_u}}}
##' \item{term_info}{the named list. Each element is a list with the information about the term, such as its frequency, function for weights, gradient function of weights, etc.}
##' \item{fn0}{optimisation function for non-linear least squares problem solved in restricted MIDAS regression}
##' \item{rhs}{the function which evaluates the right-hand side of the MIDAS regression}
##' \item{gen_midas_coef}{the function which generates the MIDAS coefficients of MIDAS regression}
##' \item{opt}{the output of optimisation procedure}
##' \item{argmap_opt}{the list containing the name of optimisation function together with arguments for optimisation function}
##' \item{start_opt}{the starting values used in optimisation}
##' \item{start_list}{the starting values as a list}
##' \item{call}{the call to the function}
##' \item{terms}{terms object}
##' \item{gradient}{gradient of NLS objective function}
##' \item{hessian}{hessian of NLS objective function}
##' \item{gradD}{gradient function of MIDAS weight functions}
##' \item{Zenv}{the environment in which data is placed}
##' \item{use_gradient}{TRUE if user supplied gradient is used, FALSE otherwise}
##' \item{nobs}{the number of effective observations}
##' \item{convergence}{the convergence message}
##' \item{fitted.values}{the fitted values of MIDAS regression}
##' \item{residuals}{the residuals of MIDAS regression}
##'
##' @author Virmantas Kvedaras, Vaidotas Zemlys
##' @references Clements, M. and Galvao, A., \emph{Macroeconomic Forecasting With Mixed-Frequency Data: Forecasting Output Growth in the United States}, Journal of Business and Economic Statistics, Vol.26 (No.4), (2008) 546-554
##' @rdname midas_r
##' @examples
##' ##The parameter function
##' theta_h0 <- function(p, dk, ...) {
##'    i <- (1:dk-1)/100
##'    pol <- p[3]*i + p[4]*i^2
##'    (p[1] + p[2]*i)*exp(pol)
##' }
##'
##' ##Generate coefficients
##' theta0 <- theta_h0(c(-0.1,10,-10,-10),4*12)
##'
##' ##Plot the coefficients
##' plot(theta0)
##'
##' ##Generate the predictor variable
##' xx <- ts(arima.sim(model = list(ar = 0.6), 600 * 12), frequency = 12)
##'
##' ##Simulate the response variable
##' y <- midas_sim(500, xx, theta0)
##'
##' x <- window(xx, start=start(y))
##'
##' ##Fit restricted model
##' mr <- midas_r(y~fmls(x,4*12-1,12,theta_h0)-1,
##'               list(y=y,x=x),
##'               start=list(x=c(-0.1,10,-10,-10)))
##'
##' ##Include intercept and trend in regression
##' mr_it <- midas_r(y~fmls(x,4*12-1,12,theta_h0)+trend,
##'                  list(data.frame(y=y,trend=1:500),x=x),
##'                  start=list(x=c(-0.1,10,-10,-10)))
##'
##' data("USrealgdp")
##' data("USunempr")
##'
##' y.ar <- diff(log(USrealgdp))
##' xx <- window(diff(USunempr), start = 1949)
##' trend <- 1:length(y.ar)
##'
##' ##Fit AR(1) model
##' mr_ar <- midas_r(y.ar ~ trend + mls(y.ar, 1, 1) +
##'                  fmls(xx, 11, 12, nealmon),
##'                  start = list(xx = rep(0, 3)))
##'
##' ##First order MIDAS-AR* restricted model
##' mr_arstar <-  midas_r(y.ar ~ trend + mls(y.ar, 1, 1, "*")
##'                      + fmls(xx, 11, 12, nealmon),
##'                      start = list(xx = rep(0, 3)))
##'
##' @details Given MIDAS regression:
##'
##' \deqn{y_t = \sum_{j=1}^p\alpha_jy_{t-j} +\sum_{i=0}^{k}\sum_{j=0}^{l_i}\beta_{j}^{(i)}x_{tm_i-j}^{(i)} + u_t,}
##'
##' estimate the parameters of the restriction
##'
##' \deqn{\beta_j^{(i)}=g^{(i)}(j,\lambda).}
##'
##' Such model is a generalisation of so called ADL-MIDAS regression. It is not required that all the coefficients should be restricted, i.e the function \eqn{g^{(i)}}
##' might be an identity function. Model with no restrictions is called U-MIDAS model. The regressors \eqn{x_\tau^{(i)}} must be of higher
##' (or of the same) frequency as the dependent variable \eqn{y_t}.
##'
##' MIDAS-AR* (a model with a common factor, see (Clements and Galvao, 2008)) can be estimated by specifying additional argument, see an example.
##'
##' The restriction function must return the restricted coefficients of
##' the MIDAS regression.
##'
##' @importFrom stats as.formula formula model.matrix model.response terms lsfit time
##' @importFrom zoo index index2char
##' @export
midas_r <- function(formula, data, start, Ofunction = "optim", weight_gradients = NULL, ...) {
  Zenv <- new.env(parent = environment(formula))

  if (missing(data)) {
    ee <- NULL
  }
  else {
    ee <- data_to_env(data)
    parent.env(ee) <- parent.frame()
  }

  if (missing(start)) {
    stop("Please supply starting values.")
  }

  assign("ee", ee, Zenv)
  formula <- as.formula(formula)
  cl <- match.call()
  mf <- match.call(expand.dots = FALSE)
  mf$formula <- formula
  m <- match(c("formula", "data"), names(mf), 0L)
  mf <- mf[c(1L, m)]
  mf[[1L]] <- as.name("model.frame")
  mf[[3L]] <- as.name("ee")
  mf[[4L]] <- as.name("na.omit")
  names(mf)[c(2, 3, 4)] <- c("formula", "data", "na.action")

  itr <- checkARstar(terms(eval(mf[[2]], Zenv)))
  if (!is.null(itr$lagsTable)) {
    mf[[2]] <- itr$x
  }

  mf <- eval(mf, Zenv)
  mt <- attr(mf, "terms")
  args <- list(...)
  y <- as.numeric(model.response(mf, "numeric"))
  X <- model.matrix(mt, mf)

  # Save ts/zoo information
  if (is.null(ee)) {
    yy <- eval(formula[[2]], Zenv)
  } else {
    yy <- eval(formula[[2]], ee)
  }

  y_index <- 1:length(yy)
  if (!is.null(attr(mf, "na.action"))) {
    y_index <- y_index[-attr(mf, "na.action")]
  }
  if (length(y_index) > 1) {
    if (sum(abs(diff(y_index) - 1)) > 0) warning("There are NAs in the middle of the time series")
  }

  ysave <- yy[y_index]

  if (inherits(yy, "ts")) {
    class(ysave) <- class(yy)
    attr(ysave, "tsp") <- c(time(yy)[range(y_index)], frequency(yy))
  }

  if (inherits(yy, c("zoo", "ts"))) {
    y_start <- index2char(index(ysave)[1], frequency(ysave))
    y_end <- index2char(index(ysave)[length(ysave)], frequency(ysave))
  } else {
    y_start <- y_index[1]
    y_end <- y_index[length(y_index)]
  }

  prepmd <- prepmidas_r(y, X, mt, Zenv, cl, args, start, Ofunction, weight_gradients, itr$lagsTable)

  prepmd <- c(prepmd, list(lhs = ysave, lhs_start = y_start, lhs_end = y_end))

  class(prepmd) <- "midas_r"

  midas_r.fit(prepmd)
}

##' @method update midas_r
##' @importFrom stats getCall update.formula setNames
##' @export
update.midas_r <- function(object, formula., ..., evaluate = TRUE) {
  if (is.null(call <- getCall(object))) {
    stop("need an object with call component")
  }
  extras <- match.call(expand.dots = FALSE)$...
  if (!missing(formula.)) {
    call$formula <- update.formula(formula(object), formula.)
  }

  if (length(extras)) {
    existing <- !is.na(match(names(extras), names(call)))
    for (a in names(extras)[existing]) call[[a]] <- extras[[a]]
    if (any(!existing)) {
      call <- c(as.list(call), extras[!existing])
      call <- as.call(call)
    }
  }

  ## 1. If no start is supplied update the start from the call
  ## 2. If start is supplied intersect it with already fitted values.

  cf <- coef(object)
  ustart <- lapply(object$term_info, function(x) cf[x[["coef_index"]]])

  redo <- FALSE
  if (!("start" %in% names(extras))) {
    if (!("start" %in% names(call) && is.null(call$start))) {
      call$start <- ustart
      object$start_opt <- cf
    }
  } else {
    if (is.null(extras$start)) {
      ## If start is null, we want to fit unrestricted midas model, this means that we need to call midas_r
      call["start"] <- list(NULL)
      redo <- TRUE
    } else {
      cstart <- eval(call$start, object$Zenv)
      ustart[names(cstart)] <- cstart
      call$start <- ustart
      object$start_opt <- unlist(ustart)
    }
  }
  if (evaluate) {
    if (!missing(formula.) || any(c("data", "weight_gradients", "start") %in% names(extras)) || redo) {
      eval(call, parent.frame())
    } else {
      ## If we got here, we assume that we do not need to reevaluate terms.
      if (!is.null(extras$Ofunction)) {
        Ofunction <- eval(extras$Ofunction)
        extras$Ofunction <- NULL
      } else {
        Ofunction <- object$argmap_opt$Ofunction
      }
      dotargnm <- names(extras)
      if (length(dotargnm) > 0) {
        offending <- dotargnm[!dotargnm %in% names(formals(Ofunction))]
        if (length(offending) > 0) {
          stop(paste("The function ", Ofunction, " does not have the following arguments: ",
            paste(offending, collapse = ", "),
            sep = ""
          ))
        }
      }
      else {
        extras <- NULL
      }
      if (Ofunction != object$argmap_opt$Ofunction) {
        argmap <- c(list(Ofunction = Ofunction), extras)
      }
      else {
        argmap <- object$argmap_opt
        argmap$Ofunction <- NULL
        argnm <- union(names(argmap), names(extras))
        marg <- vector("list", length(argnm))
        names(marg) <- argnm
        marg[names(extras)] <- extras
        oldarg <- setdiff(names(argmap), names(extras))
        marg[oldarg] <- argmap[oldarg]
        argmap <- c(list(Ofunction = Ofunction), marg)
      }
      object$call <- call
      object$argmap_opt <- argmap
      midas_r.fit(object)
    }
  }
  else {
    call
  }
}


##' Fit restricted MIDAS regression
##'
##' Workhorse function for fitting restricted MIDAS regression
##'
##' @param x \code{midas_r} object
##' @return \code{\link{midas_r}} object
##' @author Vaidotas Zemlys
midas_r.fit <- function(x) {
  args <- x$argmap_opt
  function.opt <- args$Ofunction
  args$Ofunction <- NULL

  if (!(function.opt %in% c("optim", "spg", "optimx", "lm", "nls", "dry_run"))) {
    stop("The optimisation function is not in the supported functions list. Please see the midasr:::midas_r.fit code for the supported function list")
  }
  if (function.opt == "optim" | function.opt == "spg") {
    args$par <- x$start_opt
    args$fn <- x$fn0
    if (x$use_gradient) {
      args$gr <- x$gradient
    }
    opt <- try(do.call(function.opt, args), silent = TRUE)
    if (inherits(opt, "try-error")) {
      stop("The optimisation algorithm of MIDAS regression failed with the following message:\n", opt, "\nPlease try other starting values or a different optimisation function")
    }
    par <- opt$par
    names(par) <- names(coef(x))
    x$convergence <- opt$convergence
  }
  if (function.opt == "optimx") {
    args$par <- x$start_opt
    args$fn <- x$fn0
    if (x$use_gradient) {
      args$gr <- x$gradient
    }
    opt <- try(do.call(function.opt, args), silent = TRUE)
    if (inherits(opt, "try-error")) {
      stop("The optimisation algorithm of MIDAS regression failed with the following message:\n", opt, "\nPlease try other starting values or a different optimisation function")
    }
    bmet <- which.min(opt$value)
    par <- as.numeric(opt[bmet, 1:length(args$par)])
    names(par) <- names(coef(x))
    x$convergence <- opt$convcode[bmet]
  }
  if (function.opt == "lm") {
    if (is.null(x$unrestricted)) stop("Not possible to estimate MIDAS model, more parameters than observations")
    par <- coef(x$unrestricted)
    names(par) <- names(coef(x))
    opt <- NULL
    x$convergence <- 0
  }
  if (function.opt == "nls") {
    rhs <- x$rhs
    if (x$use_gradient) {
      orhs <- rhs
      rhs <- function(p) {
        res <- orhs(p)
        attr(res, "gradient") <- x$model[, -1] %*% x$gradD(p)
        res
      }
    }
    y <- x$model[, 1]
    args$formula <- formula(y ~ rhs(p))
    args$start <- list(p = x$start_opt)
    opt <- try(do.call("nls", args), silent = TRUE)
    if (inherits(opt, "try-error")) {
      stop("The optimisation algorithm of MIDAS regression failed with the following message:\n", opt, "\nPlease try other starting values or a different optimisation function")
    }
    par <- coef(opt)
    names(par) <- names(coef(x))
    x$convergence <- opt$convInfo$stopCode
  }
  if (function.opt == "dry_run") {
    opt <- NULL
    par <- x$start_opt
  }

  x$opt <- opt
  x$coefficients <- par
  names(par) <- NULL
  x$midas_coefficients <- x$gen_midas_coef(par)
  x$fitted.values <- as.vector(x$model[, -1] %*% x$midas_coefficients)
  x$residuals <- as.vector(x$model[, 1] - x$fitted.values)
  x
}

## Prepare necessary objects for fitting of the MIDAS regression
##
## y the response
## X the model matrix
## mt the terms of the formula
## Zenv the environment to evaluate the formula
## cl call of the function
## args additional argument
## start starting values
## Ofunction the optimisation function
## weight_gradients a list of gradient functions for weights
## lagsTable the lagstable from checkARstar
## unrestricted the unrestricted model
## guess_start if TRUE, get the initial values for non-MIDAS terms via OLS, if FALSE, initialize them with zero.
## Vaidotas Zemlys
prepmidas_r <- function(y, X, mt, Zenv, cl, args, start, Ofunction, weight_gradients, lagsTable, unrestricted = NULL, guess_start = TRUE, tau = NULL) {
  start <- start[!sapply(start, is.null)]
  if (is.null(weight_gradients)) {
    use_gradient <- FALSE
  } else {
    use_gradient <- TRUE
  }

  if (!is.null(args$guess_start)) {
    guess_start <- args$guess_start
    args$guess_start <- NULL
  }
  terms.lhs <- as.list(attr(mt, "variables"))[-2:-1]

  dterm <- function(fr, ltb = NULL) {
    term_name <- as.character(fr)[1]
    weight_name <- ""
    rf <- function(p) p
    grf <- function(p) diag(1)
    start <- 0
    freq <- 1
    lagstruct <- 0
    if (term_name %in% c("mls", "fmls", "dmls", "mlsd")) {
      type <- term_name
      term_name <- as.character(fr[[2]])

      wpos <- 5
      if (type == "mlsd") {
        freq <- NA
      } else {
        freq <- eval(fr[[4]], Zenv)
      }


      lags <- eval(fr[[3]], Zenv)
      nol <- switch(type,
        fmls = lags + 1,
        dmls = lags + 1,
        mls = length(lags),
        mlsd = length(lags)
      )
      lagstruct <- switch(type,
        fmls = 0:lags,
        dmls = 0:lags,
        mls = lags,
        mlsd = lags
      )
      start <- rep(0, nol)
      grf <- function(p) diag(nol)
      if (length(fr) > wpos - 1 && fr[[wpos]] != "*") {
        mf <- fr[-wpos]
        mf[[1]] <- fr[[wpos]]
        weight_name <- as.character(fr[[wpos]])

        ## Since we allow stars and other stuff in mls, maybe allow to
        ## specify the multiplicative property in a call to mls?

        noarg <- length(formals(eval(fr[[wpos]], Zenv)))
        if (noarg < 2) stop("The weight function must have at least two arguments")
        mf <- mf[1:min(length(mf), noarg + 1)]
        if (length(mf) > 3) {
          ## If we are in mlsd we just need to ignore the third parameter,
          ## it cannot be passed to weight function
          start_eval <- 4
          if (type == "mlsd") start_eval <- 5
          if (length(mf) >= start_eval) {
            for (j in start_eval:length(mf)) {
              mf[[j]] <- eval(mf[[j]], Zenv)
            }
          }
        }
        mf[[3]] <- ifelse(is.null(ltb), nol, sum(ltb[, 1]))
        rf <- function(p) {
          mf[[2]] <- p
          eval(mf, Zenv)
        }
        if (use_gradient) {
          gmf <- mf
          if (weight_name %in% names(weight_gradients)) {
            weight_gradient_name <- paste0(as.character(fr[[2]]), "_tmp_gradient_fun")
            gmf[[1]] <- as.name(weight_gradient_name)
            assign(weight_gradient_name, weight_gradients[[weight_name]], Zenv)
          } else {
            gmf[[1]] <- as.name(paste0(weight_name, "_gradient"))
          }
          grf <- function(p) {
            gmf[[2]] <- p
            eval(gmf, Zenv)
          }
        } else {
          grf <- NULL
        }
      }
    }
    list(
      weight = rf,
      term_name = term_name,
      gradient = grf,
      start = start,
      weight_name = weight_name,
      frequency = freq,
      lag_structure = lagstruct
    )
  }
  if (is.null(lagsTable)) {
    ltb <- rep(list(NULL), length(terms.lhs))
  } else {
    ltb <- lagsTable
    if (attr(mt, "intercept") == 1) {
      ltb <- ltb[-1]
    }
  }
  rfd <- mapply(dterm, terms.lhs, ltb, SIMPLIFY = FALSE)

  if (attr(mt, "intercept") == 1) {
    intc <- dterm(expression(1))
    intc$term_name <- "(Intercept)"
    rfd <- c(list(intc), rfd)
  }

  rf <- lapply(rfd, "[[", "weight")
  names(rf) <- sapply(rfd, "[[", "term_name")

  ## Note this is a bit of misnomer. Variable weight_names is actualy a vector of term names which have MIDAS weights.
  ## It *is not* the same as actual name of weight function. This is a left-over from the old code.
  weight_names <- sapply(rfd, "[[", "weight_name")
  weight_inds <- which(weight_names != "")
  weight_names <- names(rf)[weight_names != ""]


  start_default <- lapply(rfd, "[[", "start")
  names(start_default) <- names(rf)

  ## If there are no weight functions, we have unrestricted MIDAS model.
  if (length(weight_names) == 0) {
    Ofunction <- "lm"
  } else {
    if (is.null(start)) {
      cl$formula <- update_weights(cl$formula, setNames(lapply(1:length(weight_names), function(x) ""), weight_names))
      warning("Since the start = NULL, it is assumed that U-MIDAS model is fitted")
      return(eval(cl, Zenv))
    } else {
      if (any(!weight_names %in% names(start))) stop("Starting values for weight parameters must be supplied")
    }
  }

  start_default[names(start)] <- start

  np <- cumsum(sapply(start_default, length))

  pinds <- build_indices(np, names(start_default))

  for (i in 1:length(start_default)) names(start_default[[i]]) <- NULL


  if (!is.null(lagsTable)) {
    inones <- function(ones, intro) {
      ones[ones == 1] <- intro
      ones
    }
    yname <- all.vars(mt[[2]])
    nms <- names(pinds)
    all_coef2 <- function(p) {
      pp <- lapply(pinds, function(x) p[x])
      cr <- c(1, -p[pinds[[yname]]])
      res <- mapply(function(fun, cf, tb) {
        restr <- fun(cf)
        if (is.null(tb)) {
          restr
        } else {
          mltp <- rowSums(apply(tb, 2, inones, restr) %*% diag(cr))
          mltp[rowSums(tb) != 0]
        }
      }, rf, pp, lagsTable, SIMPLIFY = FALSE)
      return(res)
    }
  } else {
    all_coef2 <- function(p) {
      pp <- lapply(pinds, function(x) p[x])
      res <- mapply(function(fun, param) fun(param), rf, pp, SIMPLIFY = FALSE)
      return(res)
    }
  }


  initial_midas_coef <- all_coef2(unlist(start_default))

  if (sum(is.na(unlist(initial_midas_coef))) > 0) stop("Check your starting values, NA in midas coefficients")

  npx <- cumsum(sapply(initial_midas_coef, length))
  xinds <- build_indices(npx, names(start_default))

  if (length(weight_names) > 0 && guess_start) {
    wi <- rep(FALSE, length(rf))
    wi[weight_inds] <- TRUE
    Xstart <- mapply(function(cf, inds, iswhgt) {
      if (iswhgt) {
        X[, inds, drop = FALSE] %*% cf
      }
      else {
        X[, inds, drop = FALSE]
      }
    }, initial_midas_coef, xinds, wi, SIMPLIFY = FALSE)

    npxx <- cumsum(sapply(Xstart, function(x) {
      ifelse(is.null(dim(x)), 1, ncol(x))
    }))
    xxinds <- build_indices(npxx, names(start_default))
    XX <- do.call("cbind", Xstart)
    ### If the starting values for the weight restriction are all zeros, then the weighted explanatory variable is zero.
    ### In this case lsfit gives a warning about colinear matrix, which we can ignore.
    prec <- suppressWarnings(lsfit(XX, y, intercept = FALSE))
    lmstart <- lapply(xxinds, function(x) coef(prec)[x])
    names(lmstart) <- names(xxinds)
    for (i in 1:length(lmstart)) names(lmstart[[i]]) <- NULL

    nms <- !(names(start_default) %in% names(start))
    start_default[nms] <- lmstart[nms]

    for (ww in which(wi)) {
      normalized <- FALSE
      if (rfd[[ww]]$weight_name %in% c("nealmon", "nbeta", "nbetaMT", "gompertzp", "nakagamip", "lcauchyp")) {
        normalized <- TRUE
      } else {
        normalized <- is_weight_normalized(rf[[ww]], start_default[[ww]])
      }
      if (normalized) {
        start_default[[ww]][1] <- lmstart[[ww]]
      }
    }
  }

  starto <- unlist(start_default)
  ## This is workaround for AR* model
  all_coef <- function(p) unlist(all_coef2(p))
  mdsrhs <- function(p) {
    coefs <- all_coef(p)
    X %*% coefs
  }

  # aa <- try(mdsrhs(starto))

  fn0 <- function(p, ...) {
    r <- y - mdsrhs(p)
    sum(r^2)
  }
  if (!is.null(tau)) {
    fn0 <- function(p, ...) {
      r <- y - mdsrhs(p)
      sum(tau * pmax(r, 0) + (tau - 1) * pmin(r, 0))
    }
  }

  if (!use_gradient) {
    gradD <- function(p) jacobian(all_coef, p)
    gr <- function(p) grad(fn0, p)
  }
  else {
    grf <- sapply(rfd, "[[", "gradient")
    ## Calculate the initial value to get the idea about the dimensions
    pp0 <- lapply(pinds, function(xx) starto[xx])
    grmat0 <- mapply(function(fun, param) fun(param), grf, pp0, SIMPLIFY = FALSE)
    colnos <- sapply(grmat0, ncol)
    rownos <- sapply(grmat0, nrow)
    np <- length(colnos)
    ccol <- cumsum(colnos)
    rrow <- cumsum(rownos)
    pindm <- cbind(
      c(1, rrow[-np] + 1), rrow,
      c(1, ccol[-np] + 1), ccol
    )
    pindm <- apply(pindm, 1, function(x) list(row = x[1]:x[2], col = x[3]:x[4]))
    if (is.null(lagsTable)) {
      gradD <- function(p) {
        pp <- lapply(pinds, function(x) p[x])
        grmat <- mapply(function(fun, param) fun(param), grf, pp, SIMPLIFY = FALSE)
        if (length(grmat) == 1) {
          res <- grmat[[1]]
        }
        else {
          res <- matrix(0, nrow = sum(rownos), ncol = sum(colnos))
          for (j in 1:length(grmat)) {
            ind <- pindm[[j]]
            res[ind$row, ind$col] <- grmat[[j]]
          }
        }
        res
      }
    } else {
      expandD <- function(grm, ltb, cr) {
        if (is.null(ltb)) {
          return(grm)
        } else {
          el <- lapply(data.frame(ltb), inones2, grm)
          mltp <- Reduce("+", mapply(`*`, el, cr, SIMPLIFY = FALSE))
          return(mltp[rowSums(ltb) != 0, ])
        }
      }

      inones2 <- function(ones, intro) {
        m <- matrix(0, nrow = length(ones), ncol = ncol(intro))
        if (sum(ones) != nrow(intro)) stop("Wrong gradient for AR* term")
        m[ones == 1, ] <- intro
        m
      }

      expandD2 <- function(fun, param, ltb, nparam = 1) {
        cf <- fun(param)
        if (is.null(ltb)) {
          return(matrix(0, nrow = length(cf), ncol = nparam))
        } else {
          mltp <- -apply(ltb, 2, inones, cf)
          return(mltp[rowSums(ltb) != 0, -1, drop = FALSE])
        }
      }
      dind <- which(names(pinds) == yname)
      cr <- c(1, -starto[pinds[[dind]]])
      pp <- lapply(pinds, function(x) starto[x])
      grmat1 <- mapply(function(fun, param) fun(param), grf, pp, SIMPLIFY = FALSE)
      egrmat1 <- mapply(expandD, grmat1, lagsTable, SIMPLIFY = FALSE, MoreArgs = list(cr))
      colnos <- sapply(egrmat1, ncol)
      rownos <- sapply(egrmat1, nrow)
      np <- length(colnos)
      ccol <- cumsum(colnos)
      rrow <- cumsum(rownos)
      pindm <- cbind(
        c(1, rrow[-np] + 1), rrow,
        c(1, ccol[-np] + 1), ccol
      )
      pindm <- apply(pindm, 1, function(x) list(row = x[1]:x[2], col = x[3]:x[4]))

      gradD <- function(p) {
        cr <- c(1, -p[pinds[[dind]]])
        pp <- lapply(pinds, function(x) p[x])
        grmat <- mapply(function(fun, param) fun(param), grf, pp, SIMPLIFY = FALSE)
        egrmat <- mapply(expandD, grmat, lagsTable, SIMPLIFY = FALSE, MoreArgs = list(cr))
        res <- matrix(0, nrow = sum(rownos), ncol = sum(colnos))
        gr_star <- do.call("rbind", mapply(expandD2, rf, pp, lagsTable, SIMPLIFY = FALSE, MoreArgs = list(length(pinds[[dind]]))))
        res[, pinds[[dind]]] <- gr_star
        for (j in 1:length(egrmat)) {
          ind <- pindm[[j]]
          res[ind$row, ind$col] <- egrmat[[j]]
        }
        res
      }
    }
    gr <- function(p) {
      XD <- X %*% gradD(p)
      resid <- y - X %*% all_coef(p)
      as.vector(-2 * apply(as.vector(resid) * XD, 2, sum))
    }
    ## Seems to work
  }
  hess <- function(x) numDeriv::hessian(fn0, x)

  if (is.null(unrestricted)) {
    if (ncol(X) < nrow(X)) {
      if (attr(mt, "intercept") == 1) {
        unrestricted <- lm(y ~ ., data = data.frame(cbind(y, X[, -1]), check.names = FALSE))
      } else {
        unrestricted <- lm(y ~ . - 1, data = data.frame(cbind(y, X), check.names = FALSE))
      }
    }
  }

  control <- c(list(Ofunction = Ofunction), args)
  ## Override default method of optim. Use BFGS instead of Nelder-Mead
  if (!("method" %in% names(control)) & Ofunction == "optim") {
    control$method <- "BFGS"
  }
  term_info <- rfd
  names(term_info) <- sapply(term_info, "[[", "term_name")
  term_info <- mapply(function(term, pind, xind) {
    term$start <- NULL
    term$coef_index <- pind
    term$midas_coef_index <- xind
    term
  }, term_info, pinds[names(term_info)], xinds[names(term_info)], SIMPLIFY = FALSE)

  if (!is.null(tau)) {
    ## At the moment do not calculate the gradient and hessian for
    ## quantile regression, as it does not make sense
    gr <- NULL
    hess <- NULL
  }
  list(
    coefficients = starto,
    midas_coefficients = all_coef(starto),
    model = cbind(y, X),
    unrestricted = unrestricted,
    term_info = term_info,
    fn0 = fn0,
    rhs = mdsrhs,
    gen_midas_coef = all_coef,
    opt = NULL,
    argmap_opt = control,
    start_opt = starto,
    start_list = start,
    call = cl,
    terms = mt,
    gradient = gr,
    hessian = hess,
    gradD = gradD,
    Zenv = Zenv,
    use_gradient = use_gradient,
    nobs = nrow(X),
    tau = tau
  )
}

##' Restricted MIDAS regression
##'
##' Function for fitting MIDAS regression without the formula interface
##' @param y model response
##' @param X prepared matrix of high frequency variable lags
##' @param z additional low frequency variables
##' @param weight the weight function
##' @param grw the gradient of weight function
##' @param startx the starting values for weight function
##' @param startz the starting values for additional low frequency variables
##' @param method a method passed to \link{optimx}
##' @param ... additional parameters to \link{optimx}
##' @return an object similar to \code{midas_r} object
##' @author Virmantas Kvedaras, Vaidotas Zemlys
##' @import numDeriv
##' @import optimx
##' @importFrom stats na.omit
##' @examples
##'
##' data("USunempr")
##' data("USrealgdp")
##' y <- diff(log(USrealgdp))
##' x <- window(diff(USunempr),start=1949)
##' trend <- 1:length(y)
##'
##' X<-fmls(x,11,12)
##'
##' midas_r_plain(y,X,trend,weight=nealmon,startx=c(0,0,0))
##' @export
##'
midas_r_plain <- function(y, X, z = NULL, weight, grw = NULL, startx, startz = NULL, method = c("Nelder-Mead", "BFGS"), ...) {
  d <- ncol(X)
  nw <- length(startx)

  if (!is.null(z) && !is.matrix(z)) z <- matrix(z, ncol = 1)
  model <- na.omit(cbind(y, X, z))
  y <- model[, 1]
  XX <- model[, -1]

  if (is.null(z)) {
    all_coef <- function(p) {
      weight(p, d)
    }
    gradD <- function(p) grw(p, d)
    start <- startx
  }
  else {
    all_coef <- function(p) {
      c(weight(p[1:nw], d), p[-nw:-1])
    }
    nz <- ncol(z)
    if (is.null(startz)) {
      ZZ <- model[, 1 + 1:d] %*% weight(startx, d)
      Z <- model[, (d + 2):ncol(model)]
      prec <- suppressWarnings(lsfit(cbind(Z, ZZ), y, intercept = FALSE))
      startz <- coef(prec)[1:nz]
    }
    if (!is.null(grw)) {
      gradD <- function(p) {
        ww <- grw(p[1:nw], d)
        zr <- matrix(0, nrow = d, ncol = nz)
        zb <- matrix(0, nrow = nz, ncol = nw)
        rbind(cbind(ww, zr), cbind(zb, diag(nz)))
      }
    }
    else {
      gradD <- NULL
    }
    start <- c(startx, startz)
  }

  n <- nrow(model)
  fn0 <- function(p) {
    sum((y - XX %*% all_coef(p))^2)
  }

  if (is.null(grw)) {
    gradD <- function(p) jacobian(all_coef, p)
    gr <- function(p) grad(fn0, p)
    gr0 <- NULL
  }
  else {
    gr <- function(p) {
      XD <- XX %*% gradD(p)
      resid <- y - XX %*% all_coef(p)
      as.vector(-2 * apply(as.vector(resid) * XD, 2, sum))
    }
    gr0 <- gr
  }
  opt <- optimx(start, fn0, gr0, method = method, ...)
  bmet <- which.min(opt$value)
  par <- as.numeric(opt[bmet, 1:length(start)])
  call <- match.call()
  fitted.values <- as.vector(XX %*% all_coef(par))
  list(
    coefficients = par,
    midas_coefficients = all_coef(par),
    model = model,
    weights = weight,
    fn0 = fn0,
    opt = opt,
    call = call,
    gradient = gr,
    hessian = function(x) numDeriv::hessian(fn0, x),
    gradD = gradD,
    fitted.values = fitted.values,
    residuals = as.vector(y - fitted.values)
  )
}
##' Updates weights in a expression with MIDAS term
##'
##' For a MIDAS term \code{fmls(x, 6, 1, nealmon)} change weight \code{nealmon} to another weight.
##' @title Updates weights in MIDAS regression formula
##' @param expr expression with MIDAS term
##' @param tb a named list with redefined weights
##' @return an expression with changed weights
##' @author Vaidotas Zemlys
##' @export
##' @examples
##'
##' update_weights(y~trend+mls(x,0:7,4,nealmon)+mls(z,0:16,12,nealmon),list(x = "nbeta", z = ""))
##'
update_weights <- function(expr, tb) {
  if (length(expr) == 3) {
    expr[[2]] <- update_weights(expr[[2]], tb)
    expr[[3]] <- update_weights(expr[[3]], tb)
  }
  if (length(expr) == 5) {
    fun <- as.character(expr[[1]])
    if (fun[[1]] %in% c("fmls", "mls", "dmls", "mlsd")) {
      end <- 4
      if (fun[1] == "mlsd") end <- 5
      term_name <- as.character(expr[[2]])
      if (term_name %in% names(tb)) {
        if (is.null(tb[[term_name]]) || tb[[term_name]] == "") {
          expr <- expr[1:end]
        } else {
          expr[[end + 1]] <- as.name(tb[[term_name]])
        }
      }
    }
    else {
      return(expr)
    }
  }
  return(expr)
}

## Check whether the MIDAS model is MIDAS-AR* model
##
## authored by Julius Vainora
checkARstar <- function(trms) {
  vars <- as.list(attr(trms, "variables"))[-2:-1]
  env <- environment(trms)
  idx <- which(sapply(vars, function(y) if (length(y) >= 2) y[[2]]) == trms[[2]])

  lagsTable <- NULL
  if (length(idx) > 0 && length(vars[[idx]]) >= 5 && vars[[idx]][[5]] == "*") {
    fs <- lapply(sapply(vars, function(y) if (length(y) >= 4) y[[4]]), eval, env)
    if (length(unique(unlist(fs))) > 1) {
      ## mls for y is assumed
      lags <- eval(vars[[idx]][[3]], env)
      push <- lapply(fs, "*", lags)

      lagsTable <- lapply(1:length(vars), function(w) {
        z <- vars[[w]]
        if (length(z) >= 4 && eval(z[[4]], env) != 1) {
          l <- eval(z[[3]], env)
          if (length(l) == 1 & as.character(z[1]) %in% c("fmls", "dmls")) {
            l <- 0:l
          }
          tp <- matrix(0, ncol = length(lags) + 1, nrow = max(l) + max(push[[w]]) + 1)
          tp[l + 1, 1] <- 1
          for (r in 2:ncol(tp)) {
            tp[l + push[[w]][r - 1] + 1, r] <- 1
          }
          tp
        }
      })

      shortSeq <- function(s) {
        wt <- which(!diff(s) == 1)
        idx <- c(1, 1 + c(wt, wt - 1), length(s))
        ams <- s[intersect(1:length(s), idx)]
        fc <- cumsum(c(TRUE, !round(diff(ams) / 2 + head(ams, -1)) %in% s))
        out <- lapply(split(ams, fc), function(x) {
          if (length(x) == 2) {
            do.call("call", c(":", as.list(x)))
          } else {
            x
          }
        })
        names(out) <- NULL
        out
      }

      vars <- lapply(1:length(vars), function(w) {
        z <- vars[[w]]

        if (length(z) >= 4 && eval(z[[4]], env) != 1) {
          fun <- as.character(z[1])
          l <- eval(z[[3]], env)
          if (fun %in% c("fmls", "dmls")) {
            if (length(l) == 1) {
              l <- 0:l
            } else {
              stop("fmls and dmls are not used with a vector of lag orders")
            }
          }
          nl <- sort(unique(l + rep(c(0, push[[w]]), each = length(l))))
          if (fun == "mls") {
            z[[3]] <- do.call("call", c("c", shortSeq(nl)))
          } else if (all(diff(nl) == 1)) {
            z[3] <- max(nl)
          } else if (fun == "fmls") {
            z[1] <- call("mls")
            z[[3]] <- do.call("call", c("c", shortSeq(nl)))
          } else {
            # Problem in case of dmls and not full lag vector
            stop("Use fmls or mls instead of dmls")
          }
        }
        z
      })
      icp <- attr(trms, "intercept") == 1
      trms <- formula(paste(trms[[2]], "~", paste(vars, collapse = " + ")), env)
      if (!icp) {
        trms <- update.formula(trms, . ~ . - 1)
      } else {
        lagsTable <- c(list(NULL), lagsTable)
      }
    }
  }
  list(x = trms, lagsTable = lagsTable)
}

build_indices_list <- function(l) {
  build_indices(cumsum(sapply(l, length)), names(l))
}

build_indices <- function(ci, nm) {
  inds <- cbind(c(1, ci[-length(ci)] + 1), ci)
  inds <- apply(inds, 1, function(x) list(x[1]:x[2]))
  inds <- lapply(inds, function(x) x[[1]])
  names(inds) <- nm
  inds
}