g13fh forecasts the conditional variances,

h_{t}, t = T + 1, \dots, T + ξ

from an exponential

GARCH (p, q)

sequence, where

ξ

is the forecast horizon and

T

is the current time (see Engle and Ng (1993)).

Syntax

C#
public static void g13fh( int num, int nt, int ip, int iq, double[] theta, double[] fht, double[] ht, double[] et, out int ifail )

Visual Basic
Public Shared Sub g13fh ( _ num As Integer, _ nt As Integer, _ ip As Integer, _ iq As Integer, _ theta As Double(), _ fht As Double(), _ ht As Double(), _ et As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g13fh ( _
	num As Integer, _
	nt As Integer, _
	ip As Integer, _
	iq As Integer, _
	theta As Double(), _
	fht As Double(), _
	ht As Double(), _
	et As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g13fh( int num, int nt, int ip, int iq, array<double>^ theta, array<double>^ fht, array<double>^ ht, array<double>^ et, [OutAttribute] int% ifail )

F#
static member g13fh : num : int * nt : int * ip : int * iq : int * theta : float[] * fht : float[] * ht : float[] * et : float[] * ifail : int byref -> unit

Parameters

num: Type: System..::..Int32
On entry: the number of terms in the arrays ht and et from the modelled sequence.

Constraint: $\max (ip, iq) \leq num$ .

nt: Type: System..::..Int32
On entry: $ξ$ , the forecast horizon.

Constraint: $nt > 0$ .

ip

Type: System..::..Int32

On entry: the number of coefficients,

β_{i}

, for

i = 1, 2, \dots, p

Constraints:

$\max (ip, iq) \leq 20$ ;
$ip \geq 0$ .

iq

Type: System..::..Int32

On entry: the number of coefficients,

α_{i}

, for

i = 1, 2, \dots, q

Constraints:

$\max (ip, iq) \leq 20$ ;
$iq \geq 1$ .

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 $j = 1, 2, \dots, p$ .

fht: Type: array<System..::..Double>[]()[][]
An array of size [nt]
On exit: the forecast values of the conditional variance, $h_{t}$ , for $t = T + 1, \dots, T + ξ$ .

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

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

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

Assume the

GARCH (p, q)

process represented by:

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

where

ε_{t} ∣ ψ_{t - 1} = N (0, h_{t})

ε_{t} ∣ ψ_{t - 1} = S_{t} (df, h_{t})

, and

z_{t} = \frac{ε_{t}}{\sqrt{h_{t}}}

E [|z_{t - i}|]

denotes the expected value of

|z_{t - i}|

, has been modelled by g13fg, and the estimated conditional variances and residuals are contained in the arrays ht and et respectively.

g13fh will then use the last

\max (p, q)

elements of the arrays ht and et to estimate the conditional variance forecasts,

h_{t} ∣ ψ_{T}

, where

t = T + 1, \dots, T + ξ

and

ξ

is the forecast horizon.

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,	$num < \max (ip, iq)$ ,
or	$iq < 1$ ,
or	$ip < 0$ ,
or	$\max (ip, iq) > 20$ ,
or	$nt \leq 0$ .

$ifail = -9000$: 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