EstimCarmaHawkes function

Estimation Methods for a CARMA(p,q)-Hawkes Counting Process

The function provides two estimation procedures: Maximum Likelihood Estimation and Matching Empirical Correlation

EstimCarmaHawkes(yuima, start, est.method = "qmle", method = "BFGS", 
lower = NULL, upper = NULL, lags = NULL, display = FALSE)

Arguments

• yuima: a yuima object.
• start: initial values to be passed to the optimizer.
• est.method: The method used to estimate the parameters. The default est.method = "qmle" indicates the MLE while the alternative approach is based on the minimization of the empirical and theoretical autocorrelation.
• method: The optimization method to be used. See optim.
• lower: Lower Bounds.
• upper: Upper Bounds.
• lags: Number of lags used in the autocorrelation.
• display: you can see a progress of the estimation when display=TRUE.

Returns

The output contains the estimated parameters.

References

Mercuri, L., Perchiazzo, A., & Rroji, E. (2022). A Hawkes model with CARMA (p, q) intensity. tools:::Rd_expr_doi("10.48550/arXiv.2208.02659") .

Author(s)

The YUIMA Project Team

Contacts: Lorenzo Mercuri lorenzo.mercuri@unimi.it

Examples

## Not run:

## MLE For A CARMA(2,1)-Hawkes ##

# Inputs:
a <- c(3,2)
b <- c(1,0.3)
mu<-0.30

true.par<-c(mu,a,b)

# step 1) Model Definition => Constructor 'setCarmaHawkes'
p <- 2
q <- 1
mod1 <- setCarmaHawkes(p = p,q = q)

# step 2) Grid Construction => Constructor 'setSampling'
FinalTime <- 5000
t0 <- 0
samp <- setSampling(t0, FinalTime, n = FinalTime)

# step 3) Simulation => method 'simulate'
# We use method 'simulate' to generate our dataset. 
# For the estimation from real data, 
# we use the constructors 'setData' and 
#'setYuima' (input 'model' is an object of 
#           'yuima.CarmaHawkes-class'). 

names(true.par) <- c(mod1@info@base.Int, mod1@info@ar.par, mod1@info@ma.par) 

set.seed(1)
system.time(
sim1 <- simulate(object = mod1, true.parameter = true.par, 
    sampling = samp)
)

plot(sim1)

# step 4) Estimation using the likelihood function.
system.time(
  res <- EstimCarmaHawkes(yuima = sim1, 
    start = true.par)
)
## End(Not run)