1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
|
########################################################################
##
## Copyright (C) 1995-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{y} =} arch_rnd (@var{a}, @var{b}, @var{t})
## Simulate an ARCH sequence of length @var{t} with AR coefficients @var{b} and
## CH coefficients @var{a}.
##
## The result @math{y(t)} follows the model
## @c Set example in small font to prevent overfull line
##
## @smallexample
## y(t) = b(1) + b(2) * y(t-1) + @dots{} + b(lb) * y(t-lb+1) + e(t),
## @end smallexample
##
## @noindent
## where @math{e(t)}, given @var{y} up to time @math{t-1}, is
## @math{N(0, h(t))}, with
## @c Set example in small font to prevent overfull line
##
## @smallexample
## h(t) = a(1) + a(2) * e(t-1)^2 + @dots{} + a(la) * e(t-la+1)^2
## @end smallexample
## @end deftypefn
function y = arch_rnd (a, b, t)
if (nargin != 3)
print_usage ();
endif
if (! ((min (size (a)) == 1) && (min (size (b)) == 1)))
error ("arch_rnd: A and B must both be scalars or vectors");
endif
if (! (isscalar (t) && (t > 0) && (rem (t, 1) == 0)))
error ("arch_rnd: T must be a positive integer");
endif
if (! (a(1) > 0))
error ("arch_rnd: A(1) must be positive");
endif
## perhaps add a test for the roots of a(z) here ...
la = length (a);
a = reshape (a, 1, la);
if (la == 1)
a = [a, 0];
la += 1;
endif
lb = length (b);
b = reshape (b, 1, lb);
if (lb == 1)
b = [b, 0];
lb += 1;
endif
m = max ([la, lb]);
e = zeros (t, 1);
h = zeros (t, 1);
y = zeros (t, 1);
h(1) = a(1);
e(1) = sqrt (h(1)) * randn ();
y(1) = b(1) + e(1);
for t = 2:m
ta = min ([t, la]);
h(t) = a(1) + a(2:ta) * e(t-ta+1:t-1).^2;
e(t) = sqrt (h(t)) * randn ();
tb = min ([t, lb]);
y(t) = b(1) + b(2:tb) * y(t-tb+1:t-1) + e(t);
endfor
if (t > m)
for t = m+1:t
h(t) = a(1) + a(2:la) * e(t-la+1:t-1).^2;
e(t) = sqrt (h(t)) * randn ();
y(t) = b(1) + b(2:lb) * y(t-tb+1:t-1) + e(t);
endfor
endif
y = y(1:t);
endfunction
|
