NAG Library Routine Document

G13FHF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     NUM – INTEGERInput

On entry: the number of terms in the arrays HT and ET from the modelled sequence.

Constraint: $\max \left(\mathbf{IP},\mathbf{IQ}\right)\le \mathbf{NUM}$.

2:     NT – INTEGERInput

On entry: $\xi$, the forecast horizon.

Constraint: $\mathbf{NT}>0$.

3:     IP – INTEGERInput

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

Constraints:

• $\max \left(\mathbf{IP},\mathbf{IQ}\right)\le 20$;
• $\mathbf{IP}\ge 0$.

4:     IQ – INTEGERInput

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

Constraints:

• $\max \left(\mathbf{IP},\mathbf{IQ}\right)\le 20$;
• $\mathbf{IQ}\ge 1$.

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 $\mathit{j}=1,2,\dots ,p$.

6:     FHT(NT) – REAL (KIND=nag_wp) arrayOutput

On exit: the forecast values of the conditional variance, $h_t$, for $\mathit{t}=T+1,\dots ,T+\xi$.

7:     HT(NUM) – REAL (KIND=nag_wp) arrayInput

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

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

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

9:     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,	$\mathbf{NUM}<\max \left(\mathbf{IP},\mathbf{IQ}\right)$,
or	$\mathbf{IQ}<1$,
or	$\mathbf{IP}<0$,
or	$\max \left(\mathbf{IP},\mathbf{IQ}\right)>20$,
or	$\mathbf{NT}\le 0$.

7  Accuracy

8  Further Comments

9  Example