ThetaEstimator.R

#' @title Estimator for the value of the asymptotic variance of the Mr statistic. 
#' This is equivalent to a weighted sum of the asymptotic variances for each of 
#' the autocorrelation co-efficients under the null hypothesis.
#' 
#' @details Given a log price process, X, and a sampling interval, q, this 
#' method is used to estimate the asymptoticvariance of the Mr statistic in the 
#' presence of stochastic volatility. In other words, it is a heteroskedasticity 
#' consistent estimator of the variance of the Mr statistic. This parameter is 
#' used to estimate the probability that the given log price process was 
#' generated by a Brownian Motion model with drift and stochastic volatility. 
#' 
#' @param X vector :: A log price process.
#' @param q int :: The sampling interval for the estimator.
#' 
calibrateAsymptoticVariance <- function(X, q) {
  avar <- 0.0
  for (j in 1:(q - 1)) {
    theta <- calibrateDelta(X, q, j)
    avar <- avar + (2 * (q - j) / q) ^ 2 * theta
  }
  return(avar)
}


#' @title Helper function for the calibrateAsymptoticVariance function.
#' 
calibrateDelta <- function(X, q, j) { 
  # Get the estimate value for the drift component.
  mu.est <- calibrateMu(X, FALSE)
  
  # Estimate the asymptotice variance given q and j.
  n <- floor(length(X)/q)
  numerator <- 0.0
  for (k in (j + 2):(n * q)) {
    t1 <- (X[k] - X[k - 1] - mu.est)^2
    t2 <- (X[k - j] - X[k - j - 1] - mu.est)^2
    numerator <- numerator + (t1 * t2)
  }
  denominator <- 0.0
  for (k in 2:(n * q))
    denominator <- denominator + (X[k] - X[k - 1] - mu.est)^2
  
  # Compute and return the statistic.
  thetaJ <- (n * q * numerator) / (denominator^2)
  return(thetaJ)
}