Generates \(u_t = \Delta^{-d}\epsilon_t\), with
\(\epsilon_t\) i.i.d. \((0, \sigma^2)\), through a
truncated \(MA(\infty)\) expansion of the fractional-differencing
operator, for use as sim_psy1(..., e = sim_fi(...)).
Arguments
- n
Number of innovations to generate.
- d
Long-memory (fractional differencing) parameter, in (0, 0.5) so that \(u_t\) itself is stationary.
- sigma
A positive scalar indicating the standard deviation of the innovations.
- seed
An object specifying if and how the random number generator (rng) should be initialized. It is either NULL or an integer, which is passed to
set.seedbefore the simulation. If you set it, the value is saved as the "seed" attribute of the returned value. The default, NULL, leaves the state of the rng unchanged and returns .Random.seed as the "seed" attribute. Results are reproducible across the parallel and the non-parallel option when you use the same seed.
Details
$$\Delta^{-d} = \sum_{j=0}^\infty \psi_j L^j,\quad \psi_0 = 1,\quad \psi_j = \psi_{j-1}\frac{j-1+d}{j}$$
The expansion is truncated at max(200, n) lags with a matching burn-in,
which is dropped before the function returns, to limit the truncation bias in the
early observations.
References
Lui, Y.L., Phillips, P.C.B. & Yu, J. (2024). "Robust testing for explosive behavior with strongly dependent errors."
