g05pg generates a given number of terms of an exponential

GARCH (p, q)

process (see Engle and Ng (1993)).

Syntax

C#
public static void g05pg( string dist, int num, int ip, int iq, double[] theta, int df, double[] ht, double[] et, bool fcall, double[] r, G05..::..G05State g05state, out int ifail )

Visual Basic
Public Shared Sub g05pg ( _ dist As String, _ num As Integer, _ ip As Integer, _ iq As Integer, _ theta As Double(), _ df As Integer, _ ht As Double(), _ et As Double(), _ fcall As Boolean, _ r As Double(), _ g05state As G05..::..G05State, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g05pg ( _
	dist As String, _
	num As Integer, _
	ip As Integer, _
	iq As Integer, _
	theta As Double(), _
	df As Integer, _
	ht As Double(), _
	et As Double(), _
	fcall As Boolean, _
	r As Double(), _
	g05state As G05..::..G05State, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g05pg( String^ dist, int num, int ip, int iq, array<double>^ theta, int df, array<double>^ ht, array<double>^ et, bool fcall, array<double>^ r, G05..::..G05State^ g05state, [OutAttribute] int% ifail )

Visual C++

public:
static void g05pg(
	String^ dist, 
	int num, 
	int ip, 
	int iq, 
	array<double>^ theta, 
	int df, 
	array<double>^ ht, 
	array<double>^ et, 
	bool fcall, 
	array<double>^ r, 
	G05..::..G05State^ g05state, 
	[OutAttribute] int% ifail
)

F#
static member g05pg : dist : string * num : int * ip : int * iq : int * theta : float[] * df : int * ht : float[] * et : float[] * fcall : bool * r : float[] * g05state : G05..::..G05State * ifail : int byref -> unit

static member g05pg : 
        dist : string * 
        num : int * 
        ip : int * 
        iq : int * 
        theta : float[] * 
        df : int * 
        ht : float[] * 
        et : float[] * 
        fcall : bool * 
        r : float[] * 
        g05state : G05..::..G05State * 
        ifail : int byref -> unit

Parameters

dist

Type: System..::..String

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"

num: Type: System..::..Int32
On entry: $T$ , the number of terms in the sequence.

Constraint: $num \geq 0$ .

ip: Type: System..::..Int32
On entry: the number of coefficients, $β_{i}$ , for $i = 1, 2, \dots, p$ .

Constraint: $ip \geq 0$ .

iq: Type: System..::..Int32
On entry: the number of coefficients, $α_{i}$ , for $i = 1, 2, \dots, q$ .

Constraint: $iq \geq 1$ .

theta

Type: array<System..::..Double>[]()[][]

An array of size [

2 \times iq + ip + 1

]

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:

$\sum_{i = 1}^{p} β_{i} \neq 1.0$ ;
$\frac{α_{0}}{1 - \sum_{i = 1}^{p} β_{i}} \leq - \log (x02am)$ .

df: Type: System..::..Int32
On entry: the number of degrees of freedom for the Student's $t$ -distribution.
If $dist = "N"$ , df is not referenced.

Constraint: if $dist = "T"$ , $df > 2$ .

ht: Type: array<System..::..Double>[]()[][]
An array of size [num]
On exit: the conditional variances $h_{t}$ , for $t = 1, 2, \dots, T$ , for the $GARCH (p, q)$ sequence.

et: Type: array<System..::..Double>[]()[][]
An array of size [num]
On exit: the observations $ε_{t}$ , for $t = 1, 2, \dots, T$ , for the $GARCH (p, q)$ sequence.

fcall: Type: System..::..Boolean
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.

r: Type: array<System..::..Double>[]()[][]
An array of size [lr]
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 g05pg, with $fcall = false$ .

g05state: Type: NagLibrary..::..G05..::..G05State
An Object of type G05.G05State.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

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 methods (G05KFF not in this release) (for a repeatable sequence if computed sequentially) or (G05KGF not in this release) (for a non-repeatable sequence) must be called prior to the first call to g05pg.

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,

dist \neq "N"

"T"

$ifail = 2$

On entry,

num < 0

$ifail = 3$

On entry,

ip < 0

$ifail = 4$

On entry,

iq < 1

$ifail = 6$

On entry,

dist = "T"

and

df \leq 2

$ifail = 10$: The value of ip or iq is not the same as when r was set up in a previous call.

$ifail = 11$

On entry,

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

$ifail = 12$

On entry,

state vector was not initialized or has been corrupted.

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

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

Not applicable.

Parallelism and Performance

None.

Further Comments

None.

Example

Example program (C#): g05pge.cs

Example program data: g05pge.d

Example program results: g05pge.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also