highfrequency (version 0.4)

heavyModelC: HEAVY Model estimation using C code

Description

This function is the same as heavyModel function, except for using C code to speed up the calculation process. 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

heavyModelC(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.

p

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

q

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 traditionalheavy model introduced

startingvalues

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

LB

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.

UB

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 $RM_i$ be the $i^{th}$ daily return and daily realized measure respectively (with $i=1, \ldots,T$). The most basic heavy model is the one with lag matrices p of $\left( \begin{array}{ccc} 0 & 1 \ 0 & 1 \end{array} \right)$ and q of $\left( \begin{array}{ccc} 1 & 0 \ 0 & 1 \end{array} \right)$. This can be reprensented by the following equations: $$\mbox{var}{\left(r_t \right)} = h_t = w + \alpha RM_{t-1} + \beta h_{t-1}; w,\alpha \geq 0, \beta \in [0,1]$$ $$\mbox{E}{\left(RM_t \right)} = \mu_t = w_R + \alpha_R RM_{t-1} + \beta_R \mu_{t-1}; w_R,\alpha_R, \beta_R \geq 0, \alpha_R+\beta_R \in [0,1]$$ Equivalently, they can be presented in terms of matrix notation as below: $$\left( \begin{array}{ccc} h_t \ \mu_t \end{array} \right) = \left( \begin{array}{ccc} w \ w_R \end{array} \right) + \left( \begin{array}{ccc} 0 & \alpha \ 0 & \alpha_R \end{array} \right) \left( \begin{array}{ccc} r^2_{t-1} \ RM_{t-1} \end{array} \right) + \left( \begin{array}{ccc} \beta & 0 \ 0 & \beta_R \end{array} \right) \left( \begin{array}{ccc} h_{t-1} \ \mu_{t-1} \end{array} \right)$$ In this version, the parameters vector to be estimated is $\left( w, w_R,\alpha, \alpha_R, \beta, \beta_R \right)$. In terms of startingvalues, Shephard and Sheppard recommend for this version of the Heavy model to set $\beta$ be around 0.6 and sum of $\alpha$+$\beta$ 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 $\left( \begin{array}{ccc} 0 & 2 \ 0 & 1 \end{array} \right)$ and matrix q is the same as above. Matrix notation will be as below: $$\left( \begin{array}{ccc} h_t \ \mu_t \end{array} \right) = \left( \begin{array}{ccc} w \ w_R \end{array} \right) + \left( \begin{array}{ccc} 0 & \alpha_1 \ 0 & \alpha_R \end{array} \right) \left( \begin{array}{ccc} r^2_{t-1} \ RM_{t-1} \end{array} \right) +\left( \begin{array}{ccc} 0 & \alpha_2 \ 0 & 0 \end{array} \right) \left( \begin{array}{ccc} r^2_{t-2} \ RM_{t-2} \end{array} \right) + \left( \begin{array}{ccc} \beta & 0 \ 0 & \beta_R \end{array} \right) \left( \begin{array}{ccc} h_{t-1} \ \mu_{t-1} \end{array} \right)$$ In this version, the parameters vector to be estimated is $\left( w, w_R,\alpha_1, \alpha_R, \alpha_2, \beta, \beta_R \right)$.

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

# Implementation of the heavy model on DJI:
 data("realized_library");
 returns =  realized_library$Dow.Jones.Industrials.Returns; 
 rk      =  realized_library$Dow.Jones.Industrials.Realized.Kernel; 
 returns = returns[!is.na(rk)];  rk = rk[!is.na(rk)]; # Remove NA's 
 data = cbind( returns^2, rk ); # Make data matrix with returns and realized measures
 backcast = matrix( c(var(returns),mean(rk)) ,ncol=1);
 
 #For traditional (default) version:
 startvalues = c(0.004,0.02,0.44,0.41,0.74,0.56); # Initial values;
 output = heavyModelC( 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);
 
 output = heavyModelC( data = as.matrix(data,ncol=2), p=p, q=q, compconst=FALSE, 
                    startingvalues = startvalues, backcast=backcast);