NAG Library Function Document

1Purpose

nag_rand_egarch (g05pgc) generates a given number of terms of an exponential $\text{GARCH}\left(p,q\right)$ process (see Engle and Ng (1993)).

2Specification

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

3Description

An exponential $\text{GARCH}\left(p,q\right)$ process is represented by:
 $lnht=α0+∑i=1qαizt-i+∑i=1qϕizt-i-Ezt-i+∑j=1pβjlnht-j, t=1,2,…,T;$
where ${z}_{t}=\frac{{\epsilon }_{t}}{\sqrt{{h}_{t}}}$, $E\left[\left|{z}_{t-i}\right|\right]$ denotes the expected value of $\left|{z}_{t-i}\right|$, and ${\epsilon }_{t}\mid {\psi }_{t-1}=N\left(0,{h}_{t}\right)$ 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 $t$.
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_egarch (g05pgc).

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:    $\mathbf{dist}$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:    $\mathbf{theta}\left[2×{\mathbf{iq}}+{\mathbf{ip}}+1\right]$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:
• $\sum _{\mathit{i}=1}^{p}{\beta }_{i}\ne 1.0$;
• $\frac{{\alpha }_{0}}{1-\sum _{\mathit{i}=1}^{p}{\beta }_{i}}\le -\mathrm{log}\left({\mathbf{nag_real_safe_small_number}}\right)$.
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: ${\mathbf{lr}}\ge 2×\left({\mathbf{ip}}+2×{\mathbf{iq}}+2\right)$.
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).

6Error 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.
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.
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.

Not applicable.

8Parallelism and Performance

nag_rand_egarch (g05pgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

10Example

This example first calls nag_rand_init_repeatable (g05kfc) to initialize a base generator then calls nag_rand_egarch (g05pgc) to generate two realizations, each consisting of ten observations, from an exponential $\mathrm{GARCH}\left(1,1\right)$ model.

10.1Program Text

Program Text (g05pgce.c)

None.

10.3Program Results

Program Results (g05pgce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017