NAG Toolbox: nag_rand_times_garch_exp (g05pg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{dist}$ – string (length ≥ 1)

The type of distribution to use for ${\epsilon }_t$.

$\mathbf{dist}=\text{'N'}$

A Normal distribution is used.

$\mathbf{dist}=\text{'T'}$

A Student's $t$-distribution is used.

Constraint: $\mathbf{dist}=\text{'N'}$ or $\text{'T'}$.

2:     $\mathrm{num}$ – int64int32nag_int scalar

$T$, the number of terms in the sequence.

Constraint: $\mathbf{num}\ge 0$.

3:     $\mathrm{ip}$ – int64int32nag_int scalar

The number of coefficients, ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$.

Constraint: $\mathbf{ip}\ge 0$.

4:     $\mathrm{iq}$ – int64int32nag_int scalar

The number of coefficients, ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

Constraint: $\mathbf{iq}\ge 1$.

5:     $\mathrm{theta}\left(2\times \mathbf{iq}+\mathbf{ip}+1\right)$ – double array

The initial parameter estimates for the vector $\theta$. The first element must contain the coefficient ${\alpha }_o$ and the next iq elements must contain the autoregressive coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$. The next iq elements must contain the coefficients ${\varphi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$. The next ip elements must contain the moving average coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.

Constraints:

• ${\displaystyle \sum _{\mathit{i}=1}^p}{\beta }_i\ne 1.0$;
• $\frac{{\alpha }_0}{1-{\displaystyle \sum _{\mathit{i}=1}^p}{\beta }_i}\le -\log \left(\mathbf{x02am}\right)$.

6:     $\mathrm{df}$ – int64int32nag_int scalar

The number of degrees of freedom for the Student's $t$-distribution.

If $\mathbf{dist}=\text{'N'}$, df is not referenced.

Constraint: if $\mathbf{dist}=\text{'T'}$, $\mathbf{df}>2$.

7:     $\mathrm{fcall}$ – logical scalar

If $\mathbf{fcall}=\mathit{true}$, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in r.

8:     $\mathrm{r}\left(\mathbf{lr}\right)$ – double array

The array contains information required to continue a sequence if $\mathbf{fcall}=\mathit{false}$.

9:     $\mathrm{state}\left(:\right)$ – 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:     $\mathrm{lr}$ – int64int32nag_int scalar

Default: the dimension of the array r.

The dimension of the array r.

Constraint: $\mathbf{lr}\ge 2\times \left(\mathbf{ip}+2\times \mathbf{iq}+2\right)$.

Output Parameters

1:     $\mathrm{ht}\left(\mathbf{num}\right)$ – double array

The conditional variances $h_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$, for the $\text{GARCH}\left(p,q\right)$ sequence.

2:     $\mathrm{et}\left(\mathbf{num}\right)$ – double array

The observations ${\epsilon }_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$, for the $\text{GARCH}\left(p,q\right)$ sequence.

3:     $\mathrm{r}\left(\mathbf{lr}\right)$ – double array

Contains information that can be used in a subsequent call of nag_rand_times_garch_exp (g05pg), with $\mathbf{fcall}=\mathit{false}$.

4:     $\mathrm{state}\left(:\right)$ – int64int32nag_int array

Contains updated information on the state of the generator.

5:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry, dist is not valid.

   $\mathbf{ifail}=2$

Constraint: $\mathbf{num}\ge 0$.

   $\mathbf{ifail}=3$

Constraint: $\mathbf{ip}\ge 0$.

   $\mathbf{ifail}=4$

Constraint: $\mathbf{iq}\ge 1$.

   $\mathbf{ifail}=6$

Constraint: $\mathbf{df}\ge 3$.

   $\mathbf{ifail}=10$

ip or iq is not the same as when r was set up in a previous call.

   $\mathbf{ifail}=11$

On entry, lr is not large enough, $\mathbf{lr}=\_$: minimum length required .

   $\mathbf{ifail}=12$

On entry, state vector has been corrupted or not initialized.

   $\mathbf{ifail}=20$

Invalid sequence generated, use different parameters.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g05pg_example

function g05pg_example


fprintf('g05pg example results\n\n');

% Initialize the generator to a repeatable sequence
seed = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

% Input parameters
dist  = 'N';
num   = int64(10);
ip    = int64(1);
iq    = int64(1);
theta = [0.1; -0.3; 0.1; 0.9];
df    = int64(0);
fcall = true;
r     = zeros(2*(2*ip+iq+2),1);

% Generate the first realisation
[ht, et, r, state, ifail] = g05pg( ...
                                   dist, num, ip, iq, ...
                                   theta, df, fcall, r, state);
% Display the results
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

% Generate a second realisation
fcall = false;
[ht, et, r, state, ifail] = g05pg( ...
                                   dist, num, ip, iq, ...
                                   theta, df, fcall, r, state);
% Display the results
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

g05pg example results


 Realisation Number 1
   i            ht(i)            et(i)
  --------------------------------------
   1            2.5098           0.5526
   2            2.1785          -1.8383
   3            3.3844           1.2180
   4            2.6780           1.3672
   5            2.0953          -1.8178
   6            3.2813          -0.0343
   7            2.9958          -0.5094
   8            3.0815           1.3978
   9            2.3961          -0.0070
  10            2.2445           0.6661

 Realisation Number 2
   i            ht(i)            et(i)
  --------------------------------------
   1            1.9327          -2.2795
   2            3.5577          -1.2249
   3            4.1461           0.6424
   4            3.4455          -2.9920
   5            5.9199           0.5777
   6            4.8221          -1.2894
   7            5.3174          -1.6473
   8            6.1095           6.1689
   9            3.1579           2.2935
  10            2.2189           0.1141