NAG Library Routine Document

G05PGF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     DIST – CHARACTER(1)Input

On entry: 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:     NUM – INTEGERInput

On entry: $T$, the number of terms in the sequence.

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

3:     IP – INTEGERInput

On entry: the number of coefficients, ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$.

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

4:     IQ – INTEGERInput

On entry: the number of coefficients, ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

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

5:     THETA($2\times \mathbf{IQ}+\mathbf{IP}+1$) – REAL (KIND=nag_wp) arrayInput

On entry: 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:

• ${\displaystyle \sum _{\mathit{i}=1}^p}{\beta }_i\ne 1.0$;
• $\frac{{\alpha }_0}{1-{\displaystyle \sum _{\mathit{i}=1}^p}{\beta }_i}\le -\log \left(\mathbf{X02AMF}\right)$.

6:     DF – INTEGERInput

On entry: 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:     HT(NUM) – REAL (KIND=nag_wp) arrayOutput

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

8:     ET(NUM) – REAL (KIND=nag_wp) arrayOutput

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

9:     FCALL – LOGICALInput

On entry: if $\mathbf{FCALL}=\mathrm{.TRUE.}$, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in R.

10:   R(LR) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the array contains information required to continue a sequence if $\mathbf{FCALL}=\mathrm{.FALSE.}$.

On exit: contains information that can be used in a subsequent call of G05PGF, with $\mathbf{FCALL}=\mathrm{.FALSE.}$.

11:   LR – INTEGERInput

On entry: the dimension of the array R as declared in the (sub)program from which G05PGF is called.

Constraint: $\mathbf{LR}\ge 2\times \left(\mathbf{IP}+2\times \mathbf{IQ}+2\right)$.

12:   STATE($*$) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

13:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,

$\mathbf{DIST}\ne \text{'N'}$ or $\text{'T'}$.

$\mathbf{IFAIL}=2$

On entry,

$\mathbf{NUM}<0$.

$\mathbf{IFAIL}=3$

On entry,

$\mathbf{IP}<0$.

$\mathbf{IFAIL}=4$

On entry,

$\mathbf{IQ}<1$.

$\mathbf{IFAIL}=6$

On entry,

$\mathbf{DIST}=\text{'T'}$ and $\mathbf{DF}\le 2$.

$\mathbf{IFAIL}=10$

The value of IP or IQ is not the same as when R was set up in a previous call.

$\mathbf{IFAIL}=11$

On entry,

$\mathbf{LR}<2\times \left(\mathbf{IP}+\mathbf{IQ}+2\right)$.

$\mathbf{IFAIL}=12$

On entry,

STATE vector was not initialized or has been corrupted.

$\mathbf{IFAIL}=20$

Invalid sequence generated, use different parameters.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (g05pgfe.f90)

9.2  Program Data

Program Data (g05pgfe.d)

9.3  Program Results

Program Results (g05pgfe.r)