NAG Library Function Document

nag_rand_egarch (g05pgc)

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)

3 Description

An exponential

GARCH (p, q)

process is represented by:

l n (h_{t}) = α_{0} + \sum_{i = 1}^{q} α_{i} z_{t - i} + \sum_{i = 1}^{q} ϕ_{i} (|z_{t - i}| - E [|z_{t - i}|]) + \sum_{j = 1}^{p} β_{j} l n (h_{t - j}), t = 1, 2, \dots, T;

where

z_{t} = \frac{ε_{t}}{\sqrt{h_{t}}}

E [|z_{t - i}|]

denotes the expected value of

|z_{t - i}|

, and

ε_{t} ∣ ψ_{t - 1} = N (0, h_{t})

ε_{t} ∣ ψ_{t - 1} = S_{t} (df, h_{t})

. Here

S_{t}

is a standardized Student's

t

-distribution with

df

degrees of freedom and variance

h_{t}

T

is the number of observations in the sequence,

ε_{t}

is the observed value of the

GARCH (p, q)

process at time

t

h_{t}

is the conditional variance at time

t

, and

ψ_{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).

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

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

5 Arguments

1: dist – Nag_ErrorDistnInput

On entry: the type of distribution to use for

ε_{t}

$dist = Nag_NormalDistn$: A Normal distribution is used.
$dist = Nag_Tdistn$: A Student's $t$ -distribution is used.

Constraint:

dist = Nag_NormalDistn

Nag_Tdistn

2: num – IntegerInput

On entry:

T

, the number of terms in the sequence.

Constraint:

num \geq 0

3: ip – IntegerInput

On entry: the number of coefficients,

β_{i}

, for

i = 1, 2, \dots, p

Constraint:

ip \geq 0

4: iq – IntegerInput

On entry: the number of coefficients,

α_{i}

, for

i = 1, 2, \dots, q

Constraint:

iq \geq 1

5: theta[ $2 \times iq + ip + 1$ ] – const doubleInput

On entry: the initial parameter estimates for the vector

θ

. The first element must contain the coefficient

α_{o}

and the next iq elements must contain the autoregressive coefficients

α_{i}

, for

i = 1, 2, \dots, q

. The next iq elements must contain the coefficients

ϕ_{i}

, for

i = 1, 2, \dots, q

. The next ip elements must contain the moving average coefficients

β_{j}

, for

j = 1, 2, \dots, p

Constraints:

$\sum_{i = 1}^{p} β_{i} \neq 1.0$ ;
$\frac{α_{0}}{1 - \sum_{i = 1}^{p} β_{i}} \leq - \log (nag_real_safe_small_number)$ .

6: df – IntegerInput

On entry: the number of degrees of freedom for the Student's

t

-distribution.

dist = Nag_NormalDistn

, df is not referenced.

Constraint: if

dist = Nag_Tdistn

df > 2

7: ht[num] – doubleOutput

On exit: the conditional variances

h_{t}

, for

t = 1, 2, \dots, T

, for the

GARCH (p, q)

sequence.

8: et[num] – doubleOutput

On exit: the observations

ε_{t}

, for

t = 1, 2, \dots, T

, for the

GARCH (p, q)

sequence.

9: fcall – Nag_BooleanInput

On entry: if

fcall = Nag_TRUE

, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in r.

10: r[lr] – doubleInput/Output

On entry: the array contains information required to continue a sequence if

fcall = Nag_FALSE

On exit: contains information that can be used in a subsequent call of nag_rand_egarch (g05pgc), with

fcall = Nag_FALSE

11: lr – IntegerInput

On entry: the dimension of the array r.

Constraint:

lr \geq 2 \times (ip + 2 \times iq + 2)

12: state[ $\dim$ ] – 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.

13: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $df = 〈value〉$ .
Constraint: $df \geq 3$ .; On entry, $ip = 〈value〉$ .
Constraint: $ip \geq 0$ .; On entry, $iq = 〈value〉$ .
Constraint: $iq \geq 1$ .; On entry, lr is not large enough, $lr = 〈value〉$ : minimum length required $= 〈value〉$ .; On entry, $num = 〈value〉$ .
Constraint: $num \geq 0$ .
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 $ip = 〈value〉$ and $ip = 〈value〉$ .
Previous value of $iq = 〈value〉$ and $iq = 〈value〉$ .
NE_REAL_ARRAY: Invalid sequence generated, use different parameters.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

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