heavyModel: HEAVY Model estimation

Description

This function calculatest the High frEquency bAsed VolatilitY (HEAVY) model proposed in Shephard and Sheppard (2010). This function is used as a predictive volatility model built to exploit highfrequency data.

Usage

heavyModel(
  data,
  p = matrix(c(0, 0, 1, 1), ncol = 2),
  q = matrix(c(1, 0, 0, 1), ncol = 2),
  startingvalues = NULL,
  LB = NULL,
  UB = NULL,
  backcast = NULL,
  compconst = FALSE
)

Arguments

data

a (T x K) matrix containing the data, with T the number of days. For the traditional HEAVY model: K = 2, the first column contains the squared daily demeaned returns, the second column contains the realized measures.

a (K x K) matrix containing the lag length for the model innovations. Position (i, j) in the matrix indicates the number of lags in equation i of the model for the innovations in data column j. For the traditional heavy model p is given by matrix(c(0,0,1,1), ncol = 2) (default).

a (K x K) matrix containing the lag length for the conditional variances. Position (i, j) in the matrix indicates the number of lags in equation i of the model for conditional variances corresponding to series j. For the traditional heavy model introduced above q is given by matrix( c(1,0,0,1),ncol=2 ) (default).

startingvalues

a vector containing the starting values to be used in the optimization to find the optimal parameters estimates.

a vector of length K indicating the lower bounds to be used in the estimation. If NULL it is set to a vector of zeros by default.

a vector of length K indicating the upper bounds to be used in the estimation. If NULL it is set to a vector of Inf by default.

backcast

a vector of length K used to initialize the estimation. If NULL the unconditional estimates are taken.

compconst

a boolean variable. In case TRUE, the omega values are estimated in the optimization. In case FALSE, volatility targeting is done and omega is just 1 minus the sum of all relevant alpha's and beta's multiplied by the unconditional variance.

Value

A list with the following values: (i) loglikelihood: the log likelihood evaluated at the parameter estimates. (ii) likelihoods: an xts object of length T containing the log likelihoods per day. (iii) condvar: a (T x K) xts object containing the conditional variances (iv) estparams: a vector with the parameter estimates. The order in which the parameters are reported is as follows: First the estimates for omega then the estimates for the non-zero alpha's with the most recent lags first in case max(p) > 1, then the estimates for the non-zero beta's with the most recent lag first in case max(q) > 1. (v) convergence: an integer code indicating the successfulness of the optimization. See optim for more information.

Details

Assume there are $T$ daily returns and realized measures in the period $t$ . Let $r_{i}$ and $R M_{i}$ be the $i^{t h}$ daily return and daily realized measure respectively (with $i = 1, \dots, T$ ).

The most basic heavy model is the one with lag matrices p of $(\begin{array}{ccc} 0 & 1 \\ 0 & 1 \end{array})$ and q of $(\begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array})$ . This can be reprensented by the following equations: $var (r_{t}) = h_{t} = w + α R M_{t - 1} + β h_{t - 1}; w, α \geq 0, β \in [0, 1]$ $E (R M_{t}) = μ_{t} = w_{R} + α_{R} R M_{t - 1} + β_{R} μ_{t - 1}; w_{R}, α_{R}, β_{R} \geq 0, α_{R} + β_{R} \in [0, 1]$

Equivalently, they can be presented in terms of matrix notation as below: $(\begin{array}{ccc} h_{t} \\ μ_{t} \end{array}) = (\begin{array}{ccc} w \\ w_{R} \end{array}) + (\begin{array}{ccc} 0 & α \\ 0 & α_{R} \end{array}) (\begin{array}{ccc} r_{t - 1}^{2} \\ R M_{t - 1} \end{array}) + (\begin{array}{ccc} β & 0 \\ 0 & β_{R} \end{array}) (\begin{array}{ccc} h_{t - 1} \\ μ_{t - 1} \end{array})$

In this version, the parameters vector to be estimated is $(w, w_{R}, α, α_{R}, β, β_{R})$ .

In terms of startingvalues, Shephard and Sheppard recommend for this version of the Heavy model to set $β$ be around 0.6 and sum of $α$ + $β$ to be close to but slightly less than one. In general, the lag length for the model innovation and the conditional covariance can be greater than 1. Consider, for example, matrix p is $(\begin{array}{ccc} 0 & 2 \\ 0 & 1 \end{array})$ and matrix q is the same as above. Matrix notation will be as below: $(\begin{array}{ccc} h_{t} \\ μ_{t} \end{array}) = (\begin{array}{ccc} w \\ w_{R} \end{array}) + (\begin{array}{ccc} 0 & α_{1} \\ 0 & α_{R} \end{array}) (\begin{array}{ccc} r_{t - 1}^{2} \\ R M_{t - 1} \end{array}) + (\begin{array}{ccc} 0 & α_{2} \\ 0 & 0 \end{array}) (\begin{array}{ccc} r_{t - 2}^{2} \\ R M_{t - 2} \end{array}) + (\begin{array}{ccc} β & 0 \\ 0 & β_{R} \end{array}) (\begin{array}{ccc} h_{t - 1} \\ μ_{t - 1} \end{array})$

In this version, the parameters vector to be estimated is $(w, w_{R}, α_{1}, α_{R}, α_{2}, β, β_{R})$ .

References

Shephard, N. and K. Sheppard (2010). Realising the future: forecasting with high frequency based volatility (heavy) models. Journal of Applied Econometrics 25, 197-231.

Examples

Run this code

# NOT RUN {
# Implementation of the heavy model on DJI:
returns <-  realized_library$open_to_close
bv      <-  realized_library$bv
returns <- returns[!is.na(bv)]
bv <- bv[!is.na(bv)] # Remove NA's
data <- cbind( returns^2, bv) # Make data matrix with returns and realized measures
backcast <- matrix(c(var(returns), mean(bv)), ncol = 1)

#For traditional (default) version:
startvalues <- c(0.004,0.02,0.44,0.41,0.74,0.56) # Initial values
output <- heavyModel(data = as.matrix(data,ncol=2), compconst=FALSE,
                     startingvalues = startvalues, backcast=backcast)
#For general version:
startvalues <- c(0.004, 0.02, 0.44, 0.4, 0.41, 0.74, 0.56) # Initial values;
p <- matrix(c(2, 0, 0, 1), ncol = 2)
q <- matrix(c(1, 0, 0, 1), ncol = 2)

heavy_model <- heavyModel(data = as.matrix(data, ncol = 2), p = p, q = q, compconst = FALSE,
                      startingvalues = startvalues, backcast = backcast)

# }

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