NAG Library Function Document

nag_rand_egarch (g05pgc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{dist}$ – Nag_ErrorDistnInput

On entry: the type of distribution to use for ${\epsilon }_t$.

$\mathbf{dist}=\mathrm{Nag\_NormalDistn}$

A Normal distribution is used.

$\mathbf{dist}=\mathrm{Nag\_Tdistn}$

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

Constraint: $\mathbf{dist}=\mathrm{Nag\_NormalDistn}$ or $\mathrm{Nag\_Tdistn}$.

2:    $\mathbf{num}$ – IntegerInput

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

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

3:    $\mathbf{ip}$ – IntegerInput

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

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

4:    $\mathbf{iq}$ – IntegerInput

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

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

5:    $\mathbf{theta}\left[2\times \mathbf{iq}+\mathbf{ip}+1\right]$ – const doubleInput

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:

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

6:    $\mathbf{df}$ – IntegerInput

On entry: the number of degrees of freedom for the Student's $t$-distribution.

If $\mathbf{dist}=\mathrm{Nag\_NormalDistn}$, df is not referenced.

Constraint: if $\mathbf{dist}=\mathrm{Nag\_Tdistn}$, $\mathbf{df}>2$.

7:    $\mathbf{ht}\left[\mathbf{num}\right]$ – doubleOutput

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:    $\mathbf{et}\left[\mathbf{num}\right]$ – doubleOutput

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

9:    $\mathbf{fcall}$ – Nag_BooleanInput

On entry: if $\mathbf{fcall}=\mathrm{Nag\_TRUE}$, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in r.

10:  $\mathbf{r}\left[\mathbf{lr}\right]$ – doubleInput/Output

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

On exit: contains information that can be used in a subsequent call of nag_rand_egarch (g05pgc), with $\mathbf{fcall}=\mathrm{Nag\_FALSE}$.

11:  $\mathbf{lr}$ – IntegerInput

On entry: the dimension of the array r.

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

12:  $\mathbf{state}\left[\mathit{dim}\right]$ – IntegerCommunication Array

Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

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:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

On entry, $\mathbf{df}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{df}\ge 3$.

On entry, $\mathbf{ip}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ip}\ge 0$.

On entry, $\mathbf{iq}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{iq}\ge 1$.

On entry, lr is not large enough, $\mathbf{lr}=〈\mathit{\text{value}}〉$: minimum length required $=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{num}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{num}\ge 0$.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_INVALID_STATE

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

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

NE_PREV_CALL

ip or iq is not the same as when r was set up in a previous call.
Previous value of $\mathbf{ip}=〈\mathit{\text{value}}〉$ and $\mathbf{ip}=〈\mathit{\text{value}}〉$.
Previous value of $\mathbf{iq}=〈\mathit{\text{value}}〉$ and $\mathbf{iq}=〈\mathit{\text{value}}〉$.

NE_REAL_ARRAY

Invalid sequence generated, use different parameters.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g05pgce.c)

10.2

Program Data

10.3

Program Results

Program Results (g05pgce.r)