R/AJ_09_variation.R

In YalDan/hf.econometrics: High Frequency Econometrics

#' Ait-Sahalia & Jacod Variation (2009)
#'
#' Calculate sigma_hat according to Ait-Sahalia & Jacod (2009) "Estimating the degree of activity of jumps in high frequency data"
#'
#' @import data.table
#'
#' @param DATA A data.table with structure as provided in the example.
#' @param k The degree of subsampling the frequency in the multipower variation. Defaults to 3.
#' @param p The power for computing the multipower variation. Defaults to 4.
#' @param gamma Value >1 for amplifying the truncation. Defaults to 2.
#' @param a The level of truncation expressed in terms of numbers of stdev of the continuous part.  Defaults to 10.
#'
#' @return Returns the variation estimation

#' @export
AJ_09_variation <- function(DATA, k = 3, p = 4, gamma = 2, a = 10){

  DT_ts_p <- DATA

  # initialize dX
  dX <- DT_ts_p[!is.na(log_ret), log_ret] # both X and dX have length n
  x0 <- log(DT_ts_p[!is.na(log_ret), p][1]) # initial value


  # specify parameter
  delta <- 1
  PERIOD <- length(unique(hour(DT_ts_p$t))) / 24
  T_AJ <- 1/(365.25*PERIOD) # 1/365.25 = one calendar day, 1/252 = one exchange trading day, 21/252 = one month, 1/4 = one quarter, 1 = one year
  nblagj <- delta
  deltaj <- delta/(24*60*60*365.25) # measured in years, this is the value of delta corresponding to that jindex

  # initialize X
  X <- x0 + cumsum(dX) # do this instead of X=log(price) to avoid including the large overnight returns
  n <- length(X)
  dXobsj <- sort(abs(X[seq(from = (nblagj+1), to = n, by = nblagj)] - X[seq(from = 1, to = (n-nblagj), by = nblagj)])) # length(dXobsj) is equal to nj-1

  # calculate sigma_hat (truncated realized bipower variation)
  sigma_hat_as <- sqrt( (1/T_AJ) * sum( ((dXobsj)^2) * ( (dXobsj) <= 3 * 0.60 * deltaj^(1/2) )) )
  sigmahat <- sigma_hat_as
  ## ##

  nblagjk <- nblagj * k
  dXobsjk <- sort(abs(X[seq(from = (nblagjk+1), to = n, by = nblagjk)] - X[seq(from = 1, to = (n-nblagjk), by = nblagjk)]))

  sigma_hat <- sum( ((dXobsjk)^p) * ( (dXobsjk) < gamma * a * sigmahat * deltaj^(1/2) ))
  ## ##

  return(sigma_hat)
}