g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_agarchI (g05pdc)

## 1  Purpose

nag_rand_agarchI (g05pdc) generates a given number of terms of a type I $\text{AGARCH}\left(p,q\right)$ process (see Engle and Ng (1993)).

## 2  Specification

 #include #include
 void nag_rand_agarchI (Nag_ErrorDistn dist, Integer num, Integer ip, Integer iq, const double theta[], double gamma, Integer df, double ht[], double et[], Nag_Boolean fcall, double r[], Integer lr, Integer state[], NagError *fail)

## 3  Description

A type I $\text{AGARCH}\left(p,q\right)$ process can be represented by:
 $ht = α0 + ∑i=1q αi εt-i + γ 2 + ∑i=1p βi ht-i , t=1,2,…,T ;$
where ${\epsilon }_{t}\mid {\psi }_{t-1}=N\left(0,{h}_{t}\right)$ or ${\epsilon }_{t}\mid {\psi }_{t-1}={S}_{t}\left(\mathit{df},{h}_{t}\right)$. Here ${S}_{t}$ is a standardized Student's $t$-distribution with $\mathit{df}$ degrees of freedom and variance ${h}_{t}$, $T$ is the number of observations in the sequence, ${\epsilon }_{t}$ is the observed value of the $\text{GARCH}\left(p,q\right)$ process at time $t$, ${h}_{t}$ is the conditional variance at time $t$, and ${\psi }_{t}$ the set of all information up to time $t$. Symmetric GARCH sequences are generated when $\gamma$ is zero, otherwise asymmetric GARCH sequences are generated with $\gamma$ specifying the amount by which positive (or negative) shocks are to be enhanced.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_agarchI (g05pdc).

## 4  References

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
Hamilton J (1994) Time Series Analysis Princeton University Press

## 5  Arguments

1:     distNag_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:     numIntegerInput
On entry: $T$, the number of terms in the sequence.
Constraint: ${\mathbf{num}}\ge 0$.
3:     ipIntegerInput
On entry: the number of coefficients, ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$.
Constraint: ${\mathbf{ip}}\ge 0$.
4:     iqIntegerInput
On entry: the number of coefficients, ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
Constraint: ${\mathbf{iq}}\ge 1$.
5:     theta[${\mathbf{iq}}+{\mathbf{ip}}+1$]const doubleInput
On entry: the first element must contain the coefficient ${\alpha }_{o}$, 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 $\mathit{j}=1,2,\dots ,p$.
Constraints:
• $\sum _{\mathit{i}=2}^{{\mathbf{iq}}+{\mathbf{ip}}+1}{\mathbf{theta}}\left[\mathit{i}-1\right]<1.0$;
• ${\mathbf{theta}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=2,3,\dots ,{\mathbf{ip}}+{\mathbf{iq}}+1$.
On entry: the asymmetry parameter $\gamma$ for the $\text{GARCH}\left(p,q\right)$ sequence.
7:     dfIntegerInput
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 ${\mathbf{dist}}=\mathrm{Nag_Tdistn}$, ${\mathbf{df}}>2$.
8:     ht[num]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.
9:     et[num]doubleOutput
On exit: the observations ${\epsilon }_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$, for the $\text{GARCH}\left(p,q\right)$ sequence.
10:   fcallNag_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.
11:   r[lr]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_agarchI (g05pdc), with ${\mathbf{fcall}}=\mathrm{Nag_FALSE}$.
12:   lrIntegerInput
On entry: the dimension of the array r.
Constraint: ${\mathbf{lr}}\ge 2×\left({\mathbf{ip}}+{\mathbf{iq}}+2\right)$.
13:   state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions 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.
14:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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 $\text{}=〈\mathit{\text{value}}〉$.
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.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
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
On entry, sum of ${\mathbf{theta}}\left[\mathit{i}-1\right]$, for $\mathit{i}=2,3,\dots ,{\mathbf{ip}}+{\mathbf{iq}}+1$ is $\text{}\ge 1.0$: $\mathrm{sum}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{theta}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{theta}}\left[i-1\right]\ge 0.0$.

Not applicable.

None.

## 9  Example

This example first calls nag_rand_init_repeatable (g05kfc) to initialize a base generator then calls nag_rand_agarchI (g05pdc) to generate two realizations, each consisting of ten observations, from a symmetric $\mathrm{GARCH}\left(1,1\right)$ model.

### 9.1  Program Text

Program Text (g05pdce.c)

None.

### 9.3  Program Results

Program Results (g05pdce.r)