Integer, Intent (In)	::	num, ip, iq, df, lr
Integer, Intent (Inout)	::	state(*), ifail
Real (Kind=nag_wp), Intent (In)	::	theta(2*iq+ip+1)
Real (Kind=nag_wp), Intent (Inout)	::	r(lr)
Real (Kind=nag_wp), Intent (Out)	::	ht(num), et(num)
Logical, Intent (In)	::	fcall
Character (1), Intent (In)	::	dist

C Header Interface

#include <nag.h>

void	g05pgf_ (const char dist, const Integer num, const Integer ip, const Integer iq, const double theta[], const Integer df, double ht[], double et[], const logical fcall, double r[], const Integer lr, Integer state[], Integer ifail, const Charlen length_dist)

The routine may be called by the names g05pgf or nagf_rand_times_garch_exp.

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 routines g05kff (for a repeatable sequence if computed sequentially) or g05kgf (for a non-repeatable sequence) must be called prior to the first call to g05pgf.

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: $dist$ – Character(1) Input

On entry: the type of distribution to use for

ε_{t}

$dist ='N'$: A Normal distribution is used.
$dist ='T'$: A Student's $t$ -distribution is used.

Constraint:

dist ='N'

'T'

2: $num$ – Integer Input

On entry:

T

, the number of terms in the sequence.

Constraint:

num \geq 0

3: $ip$ – Integer Input

On entry: the number of coefficients,

β_{i}

, for

i = 1, 2, \dots, p

Constraint:

ip \geq 0

4: $iq$ – Integer Input

On entry: the number of coefficients,

α_{i}

, for

i = 1, 2, \dots, q

Constraint:

iq \geq 1

5: $theta (2 \times iq + ip + 1)$ – Real (Kind=nag_wp) array Input

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:

$\sum_{i = 1}^{p} β_{i} \neq 1.0$ ;
$\frac{α_{0}}{1 - \sum_{i = 1}^{p} β_{i}} \leq - \log (x02amf)$ .

6: $df$ – Integer Input

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

t

-distribution.

dist ='N'

, df is not referenced.

Constraint: if

dist ='T'

df > 2

7: $ht (num)$ – Real (Kind=nag_wp) array Output

On exit: the conditional variances

h_{t}

, for

t = 1, 2, \dots, T

, for the

GARCH (p, q)

sequence.

8: $et (num)$ – Real (Kind=nag_wp) array Output

On exit: the observations

ε_{t}

, for

t = 1, 2, \dots, T

, for the

GARCH (p, q)

sequence.

9: $fcall$ – Logical Input

On entry: if

fcall = .TRUE.

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

10: $r (lr)$ – Real (Kind=nag_wp) array Communication Array

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

fcall = .FALSE.

On exit: contains information that can be used in a subsequent call of g05pgf, with

fcall = .FALSE.

11: $lr$ – Integer Input

On entry: the dimension of the array r as declared in the (sub)program from which g05pgf is called.

Constraint:

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

12: $state (*)$ – Integer array Communication Array

Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.

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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, dist is not valid: $dist = ⟨ value ⟩$ .

$ifail = 2$: On entry, $num = ⟨ value ⟩$ .
Constraint: $num \geq 0$ .

$ifail = 3$: On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 0$ .

$ifail = 4$: On entry, $iq = ⟨ value ⟩$ .
Constraint: $iq \geq 1$ .

$ifail = 6$: On entry, $df = ⟨ value ⟩$ .
Constraint: $df \geq 3$ .

$ifail = 10$: 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 $iq = ⟨ value ⟩$ .

$ifail = 11$: On entry, lr is not large enough, $lr = ⟨ value ⟩$ : minimum length required $= ⟨ value ⟩$ .

$ifail = 12$: On entry, state vector has been corrupted or not initialized.

$ifail = 20$: Invalid sequence generated, use different parameters.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g05pgf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example first calls g05kff to initialize a base generator then calls g05pgf to generate two realizations, each consisting of ten observations, from an exponential

GARCH (1, 1)

model.

g05pg: FL CL CPP AD

NAG FL Interfaceg05pgf (times_​garch_​exp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g05pgf (times_garch_exp)