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NAG Toolbox: nag_rand_times_garch_asym2 (g05pe)

Purpose

nag_rand_times_garch_asym2 (g05pe) generates a given number of terms of a type II AGARCH(p,q)AGARCH(p,q) process (see Engle and Ng (1993)).

Syntax

[ht, et, r, state, ifail] = g05pe(dist, num, ip, iq, theta, gamma, df, fcall, r, state, 'lr', lr)
[ht, et, r, state, ifail] = nag_rand_times_garch_asym2(dist, num, ip, iq, theta, gamma, df, fcall, r, state, 'lr', lr)

Description

A type II AGARCH(p,q)AGARCH(p,q) process can be represented by:
q p
ht = α0 + αi (|εti| + γεti)2 + βi hti ,  t = 1,2,,T;
i = 1 i = 1
ht = α0 + i=1 q αi (|εt-i|+γεt-i) 2 + i=1 p βi ht-i ,   t=1,2,,T ;
where εt ψt1 = N (0,ht) εt ψ t-1 = N (0,ht)  or εt ψt1 = St (df,ht) εt ψt-1= St (df,ht) . Here StSt is a standardized Student's tt-distribution with dfdf degrees of freedom and variance htht, TT is the number of observations in the sequence, εtεt is the observed value of the GARCH(p,q)GARCH(p,q) process at time tt, htht is the conditional variance at time tt, and ψtψt the set of all information up to time tt. Symmetric GARCH sequences are generated when γγ is zero, otherwise asymmetric GARCH sequences are generated with γγ specifying the amount by which positive (or negative) shocks are to be enhanced.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_times_garch_asym2 (g05pe).

References

Bollerslev T (1986) Generalised autoregressive conditional heteroskedasticity Journal of Econometrics 31 307–327
Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation Econometrica 50 987–1008
Engle R and Ng V (1993) Measuring and testing the impact of news on volatility Journal of Finance 48 1749–1777
Hamilton J (1994) Time Series Analysis Princeton University Press

Parameters

Compulsory Input Parameters

1:     dist – string (length ≥ 1)
The type of distribution to use for εtεt.
dist = 'N'dist='N'
A Normal distribution is used.
dist = 'T'dist='T'
A Student's tt-distribution is used.
Constraint: dist = 'N'dist='N' or 'T''T'.
2:     num – int64int32nag_int scalar
TT, the number of terms in the sequence.
Constraint: num0num0.
3:     ip – int64int32nag_int scalar
The number of coefficients, βiβi, for i = 1,2,,pi=1,2,,p.
Constraint: ip0ip0.
4:     iq – int64int32nag_int scalar
The number of coefficients, αiαi, for i = 1,2,,qi=1,2,,q.
Constraint: iq1iq1.
5:     theta(iq + ip + 1iq+ip+1) – double array
The first element must contain the coefficient αoαo, the next iq elements must contain the coefficients αiαi, for i = 1,2,,qi=1,2,,q. The remaining ip elements must contain the coefficients βjβj, for j = 1,2,,pj=1,2,,p.
Constraints:
  • i = 2iq + ip + 1 theta(i) < 1.0 i=2 iq+ip+1 thetai<1.0;
  • theta(i)0.0thetai0.0, for i = 2,3,,ip + iq + 1i=2,3,,ip+iq+1.
6:     gamma – double scalar
The asymmetry parameter γγ for the GARCH(p,q)GARCH(p,q) sequence.
7:     df – int64int32nag_int scalar
The number of degrees of freedom for the Student's tt-distribution.
If dist = 'N'dist='N', df is not referenced.
Constraint: if dist = 'T'dist='T', df > 2df>2.
8:     fcall – logical scalar
If fcall = truefcall=true, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in r.
9:     r(lr) – double array
lr, the dimension of the array, must satisfy the constraint lr2 × (ip + iq + 2)lr2×(ip+iq+2).
The array contains information required to continue a sequence if fcall = falsefcall=false.
10:   state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

Optional Input Parameters

1:     lr – int64int32nag_int scalar
Default: The dimension of the array r.
The dimension of the array r as declared in the (sub)program from which nag_rand_times_garch_asym2 (g05pe) is called.
Constraint: lr2 × (ip + iq + 2)lr2×(ip+iq+2).

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     ht(num) – double array
The conditional variances htht, for t = 1,2,,Tt=1,2,,T, for the GARCH(p,q)GARCH(p,q) sequence.
2:     et(num) – double array
The observations εtεt, for t = 1,2,,Tt=1,2,,T, for the GARCH(p,q)GARCH(p,q) sequence.
3:     r(lr) – double array
Contains information that can be used in a subsequent call of nag_rand_times_garch_asym2 (g05pe), with fcall = falsefcall=false.
4:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
5:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,dist'N'dist'N' or 'T''T'.
  ifail = 2ifail=2
On entry,num < 0num<0.
  ifail = 3ifail=3
On entry,ip < 0ip<0.
  ifail = 4ifail=4
On entry,iq < 1iq<1.
  ifail = 7ifail=7
On entry,dist = 'T'dist='T' and df2df2.
  ifail = 11ifail=11
The value of ip or iq is not the same as when r was set up in a previous call.
  ifail = 12ifail=12
On entry,lr < 2 × (ip + iq + 2)lr<2×(ip+iq+2).
  ifail = 13ifail=13
On entry,state vector was not initialized or has been corrupted.

Accuracy

Not applicable.

Further Comments

None.

Example

function nag_rand_times_garch_asym2_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);

dist = 'N';
num = 10;
ip = 1;
iq = 1;
theta = [0.08; 0.2; 0.7];
gamma = -0.4;
df = int64(0);
fcall = true;
r = zeros(2*(ip+iq+2),1);
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);

% Generate the first realisation
[ht, et, r, state, ifail] = ...
    nag_rand_times_garch_asym2(dist, int64(num), int64(ip), int64(iq), theta, gamma, df, fcall, r, state);
% Display the results
if ifail == 0
  fprintf('\n Realisation Number 1\n');
  fprintf('   I            HT(I)            ET(I)\n');
  fprintf('  --------------------------------------\n');
  for i=1:num
    fprintf('  %2d  %16.4f %16.4f\n', i, ht(i), et(i));
  end
else
  fprintf('\n nag_rand_times_garch_asym2 exited with ifail = %d \n', ifail);
end

% Generate a second realisation
fcall = false;
[ht, et, r, state, ifail] = ...
    nag_rand_times_garch_asym2(dist, int64(num), int64(ip), int64(iq), theta, gamma, df, fcall, r, state);
% Display the results
if ifail == 0
  fprintf('\n Realisation Number 2\n');
  fprintf('   I            HT(I)            ET(I)\n');
  fprintf('  --------------------------------------\n');
  for i=1:num
    fprintf('  %2d  %16.4f %16.4f\n', i, ht(i), et(i));
  end
else
  fprintf('\n nag_rand_times_garch_asym2 exited with ifail = %d \n', ifail);
end
 

 Realisation Number 1
   I            HT(I)            ET(I)
  --------------------------------------
   1            0.6400           0.2790
   2            0.5336          -0.9098
   3            0.7780           0.5840
   4            0.6491           0.6731
   5            0.5670          -0.9456
   6            0.8275          -0.0172
   7            0.6593          -0.2390
   8            0.5639           0.5980
   9            0.5005          -0.0032
  10            0.4303           0.2917

 Realisation Number 2
   I            HT(I)            ET(I)
  --------------------------------------
   1            0.3874          -1.0205
   2            0.7594          -0.5659
   3            0.7371           0.2709
   4            0.6013          -1.2499
   5            1.1133           0.2505
   6            0.8638          -0.5457
   7            0.8014          -0.6395
   8            0.8013           2.2341
   9            1.0003           1.2908
  10            0.9002           0.0727

function g05pe_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);

dist = 'N';
num = 10;
ip = 1;
iq = 1;
theta = [0.08; 0.2; 0.7];
gamma = -0.4;
df = int64(0);
fcall = true;
r = zeros(2*(ip+iq+2),1);
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);

% Generate the first realisation
[ht, et, r, state, ifail] = ...
     g05pe(dist, int64(num), int64(ip), int64(iq), theta, gamma, df, fcall, r, state);
% Display the results
if ifail == 0
  fprintf('\n Realisation Number 1\n');
  fprintf('   I            HT(I)            ET(I)\n');
  fprintf('  --------------------------------------\n');
  for i=1:num
    fprintf('  %2d  %16.4f %16.4f\n', i, ht(i), et(i));
  end
else
  fprintf('\n g05pe exited with ifail = %d \n', ifail);
end

% Generate a second realisation
fcall = false;
[ht, et, r, state, ifail] = ...
     g05pe(dist, int64(num), int64(ip), int64(iq), theta, gamma, df, fcall, r, state);
% Display the results
if ifail == 0
  fprintf('\n Realisation Number 2\n');
  fprintf('   I            HT(I)            ET(I)\n');
  fprintf('  --------------------------------------\n');
  for i=1:num
    fprintf('  %2d  %16.4f %16.4f\n', i, ht(i), et(i));
  end
else
  fprintf('\n g05pe exited with ifail = %d \n', ifail);
end
 

 Realisation Number 1
   I            HT(I)            ET(I)
  --------------------------------------
   1            0.6400           0.2790
   2            0.5336          -0.9098
   3            0.7780           0.5840
   4            0.6491           0.6731
   5            0.5670          -0.9456
   6            0.8275          -0.0172
   7            0.6593          -0.2390
   8            0.5639           0.5980
   9            0.5005          -0.0032
  10            0.4303           0.2917

 Realisation Number 2
   I            HT(I)            ET(I)
  --------------------------------------
   1            0.3874          -1.0205
   2            0.7594          -0.5659
   3            0.7371           0.2709
   4            0.6013          -1.2499
   5            1.1133           0.2505
   6            0.8638          -0.5457
   7            0.8014          -0.6395
   8            0.8013           2.2341
   9            1.0003           1.2908
  10            0.9002           0.0727


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