# NAG Library Routine Document

## 1Purpose

g13fff forecasts the conditional variances, ${h}_{t}$, for $\mathit{t}=T+1,\dots ,T+\xi$ from a GJR $\text{GARCH}\left(p,q\right)$ sequence, where $\xi$ is the forecast horizon and $T$ is the current time (see Glosten et al. (1993)).

## 2Specification

Fortran Interface
 Subroutine g13fff ( num, nt, ip, iq, fht, ht, et,
 Integer, Intent (In) :: num, nt, ip, iq Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: theta(iq+ip+1), gamma, ht(num), et(num) Real (Kind=nag_wp), Intent (Out) :: fht(nt)
#include nagmk26.h
 void g13fff_ (const Integer *num, const Integer *nt, const Integer *ip, const Integer *iq, const double theta[], const double *gamma, double fht[], const double ht[], const double et[], Integer *ifail)

## 3Description

Assume the $\text{GARCH}\left(p,q\right)$ process can be represented by:
 $ht = α0 + ∑ i=1 q αi + γ It-i ε t-i 2 + ∑ i=1 p βi ht-i , t=1,2,…,T .$
where ${\epsilon }_{t}\mid {\psi }_{t-1}=N\left(0,{h}_{t}\right)$ or ${\epsilon }_{t}\mid {\psi }_{t-1}={S}_{t}\left(\mathit{df},{h}_{t}\right)$, and ${I}_{t}=1$, if ${\epsilon }_{t}<0$, or ${I}_{t}=0$, if ${\epsilon }_{t}\ge 0$, has been modelled by g13fef, and the estimated conditional variances and residuals are contained in the arrays ht and et respectively.
g13fff will then use the last $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(p,q\right)$ elements of the arrays ht and et to estimate the conditional variance forecasts, ${h}_{t}\mid {\psi }_{T}$, where $t=T+1,\dots ,T+\xi$ and $\xi$ is the forecast horizon.

## 4References

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
Glosten L, Jagannathan R and Runkle D (1993) Relationship between the expected value and the volatility of nominal excess return on stocks Journal of Finance 48 1779–1801
Hamilton J (1994) Time Series Analysis Princeton University Press

## 5Arguments

1:     $\mathbf{num}$ – IntegerInput
On entry: the number of terms in the arrays ht and et from the modelled sequence.
Constraint: $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le {\mathbf{num}}$.
2:     $\mathbf{nt}$ – IntegerInput
On entry: $\xi$, the forecast horizon.
Constraint: ${\mathbf{nt}}>0$.
3:     $\mathbf{ip}$ – IntegerInput
On entry: the number of coefficients, ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$.
Constraints:
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le 20$;
• ${\mathbf{ip}}\ge 0$.
4:     $\mathbf{iq}$ – IntegerInput
On entry: the number of coefficients, ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
Constraints:
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le 20$;
• ${\mathbf{iq}}\ge 1$.
5:     $\mathbf{theta}\left({\mathbf{iq}}+{\mathbf{ip}}+1\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the first element must contain the coefficient ${\alpha }_{o}$ and the next iq elements must contain the coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$. The remaining ip elements must contain the coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
6:     $\mathbf{gamma}$ – Real (Kind=nag_wp)Input
On entry: the asymmetry parameter $\gamma$ for the $\text{GARCH}\left(p,q\right)$ sequence.
7:     $\mathbf{fht}\left({\mathbf{nt}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the forecast values of the conditional variance, ${h}_{t}$, for $\mathit{t}=T+1,\dots ,T+\xi$.
8:     $\mathbf{ht}\left({\mathbf{num}}\right)$ – 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 $\mathit{t}=1,2,\dots ,T$.
9:     $\mathbf{et}\left({\mathbf{num}}\right)$ – 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 $\mathit{t}=1,2,\dots ,T$.
10:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{num}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)$, or ${\mathbf{iq}}<1$, or ${\mathbf{ip}}<0$, or $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)>20$, or ${\mathbf{nt}}\le 0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable

## 8Parallelism and Performance

g13fff is not threaded in any implementation.