# [R] calculating recursive sequences

From: Vaidotas Zemlys <mpiktas_at_gmail.com>
Date: Thu 29 Dec 2005 - 23:33:32 EST

Hi,

I was trying to repeat the estimation of threshold GARCH models from the book "Analysis of Financial Time Series" by Ruey S. Tsay, and I was succesfull, but I had to use "for" loop, which is quite slow. The loop is necessary, since you need to calculate recursive sequence. Is there a faster way to do this in R, without using loops?

The model is such:
r_t = \mu + \alpha_2 r_{t-2} + a_t
a_t = \sigma_t\varepsilon_t

\sigma_t^2 =
\beta_1a_{t-1}^2+\beta_2\sigma_{t-1}^2+
1_{\{a_{t-1}>0\}}(\gamma_0+
\gamma_1a_{t-1}^2+\gamma_2\sigma^2_{t-1})

It is asummed that \varepsilon_t are iid and normal with zero mean and variance one. The data given is r_t, and you have to estimate variables, \mu, \alpha, \beta and \gamma. Since

\varepsilon_t=\frac{a_t}{\sqrt{sigma_t}}

using the equations we calculate a_t and \sigma_t and estimate the variables using maximum likelihood method. a_t can be estimated directly using first equation and rt. \sigma_t^2 depends on sigma_{t-1}^2, so it should be calculated recursively.

The function calculating negative log-likelihood of this problem I wrote:

garchln <- function(p,rt) {

n <- length(rt)

at <- rt[4:n]-p-p*rt[4:n-2]
u <- as.numeric(at>0)
h <- rep(0,length(at))
# h is \sigma_t^2
for(i in 1:(length(h)-1)) {

h[i+1] <- p*at[i]^2+p*h[i]+u[i]*(p+p*at[i]^2+p*h[i])     }

#Maximum likelihood function
sum(log(h[-1])+(at[-1]^2)/h[-1])/2
#list(h=h[-1],at=at[-1])
}

For fitting I used optim, with methods "Nelder-Mead" and "BFGS",

Initial parameter values from the book are  0.03 -0.03 0.10 0.60 0.10 0.05 0.10 The fitted values from the book are
0.043 -0.022 0.098 0.954 0.060 -0.052 -0.069.

The link to the data used:
http://www.gsb.uchicago.edu/fac/ruey.tsay/teaching/fts/d-ibmln99.dat

For this problem recursive sequence is linear, so it is possible to calculate it as a linear equations solution, but it is easy to think of the case where the recursion is non-linear. Is the speed-up possible only by writing C or Fortran code with loops?

Vaidotas Zemlys

--
Doctorate student, http://www.mif.vu.lt/katedros/eka/katedra/zemlys.php
Vilnius University

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