This functions computes the quantity f(δ) defined in (24) of
Kley et al. (2019) when the underlying process follows an tvAR(p) process.
Recall that, to apply Theorem 3.1 in Kley et al. (2019), the function
f(δ) is required to be positive, which can be verified with the
numbers returned from this function.
The function returns a vector with elements f(δ)
for each δ in which.deltas, with f(δ)
defined as
The local autocovariances γk(u) are defined as the lag-k
autocovariances of an AR(p) process which has coefficients
a1(u),…,ap(u) and innovations with variance σ(u)2,
because the underlying model is assumed to be tvAR(p)
Yt,T=∑j=1paj(t/T)Yt−j,T+σ(t/T)εt,
where a1,…,ap are real valued functions (defined on [0,1]) and σ is a
positive function (defined on [0,1]).
Value
Returns a vector with the values f(δ), as defined in
(24) of Kley et al. (2019), where it is now denoted by q(δ), for each δ in
which.delta.
Examples
## Not run:
## because computation is quite time-consuming.
n <- 100
a <- list( function(u) {return(0.8+0.19*sin(4*pi*u))} )
sigma <- function (u) {return(1)}
Ns <- seq( floor((n/2)^(4/5)), floor(n^(4/5)),
ceiling((floor(n^(4/5)) - floor((n/2)^(4/5)))/25) )
which.deltas <- c(0, 0.01, 0.05, 0.1, 0.15, 0.2, 0.4, 0.6)
P_max <- 7
H <- 1
m <- floor(n^(.85)/4)
# now replicate some results from Table 4 in Kley et al. (2019)
f( which.deltas, P_max, h = 1, n - m, Ns, m, a, sigma )
f( which.deltas, P_max, h = 5, n - m, Ns, m, a, sigma )
## End(Not run)