g13fg estimates the parameters of a univariate regression-exponential

GARCH (p, q)

process (see Engle and Ng (1993)).

Syntax

C#
public static void g13fg( string dist, double[] yt, double[,] x, int num, int ip, int iq, int nreg, int mn, int npar, double[] theta, double[] se, double[] sc, double[,] covr, ref double hp, double[] et, double[] ht, out double lgf, bool copts, int maxit, double tol, out int ifail )

public static void g13fg(
	string dist,
	double[] yt,
	double[,] x,
	int num,
	int ip,
	int iq,
	int nreg,
	int mn,
	int npar,
	double[] theta,
	double[] se,
	double[] sc,
	double[,] covr,
	ref double hp,
	double[] et,
	double[] ht,
	out double lgf,
	bool copts,
	int maxit,
	double tol,
	out int ifail
)

Visual Basic
Public Shared Sub g13fg ( _ dist As String, _ yt As Double(), _ x As Double(,), _ num As Integer, _ ip As Integer, _ iq As Integer, _ nreg As Integer, _ mn As Integer, _ npar As Integer, _ theta As Double(), _ se As Double(), _ sc As Double(), _ covr As Double(,), _ ByRef hp As Double, _ et As Double(), _ ht As Double(), _ <OutAttribute> ByRef lgf As Double, _ copts As Boolean, _ maxit As Integer, _ tol As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g13fg ( _
	dist As String, _
	yt As Double(), _
	x As Double(,), _
	num As Integer, _
	ip As Integer, _
	iq As Integer, _
	nreg As Integer, _
	mn As Integer, _
	npar As Integer, _
	theta As Double(), _
	se As Double(), _
	sc As Double(), _
	covr As Double(,), _
	ByRef hp As Double, _
	et As Double(), _
	ht As Double(), _
	<OutAttribute> ByRef lgf As Double, _
	copts As Boolean, _
	maxit As Integer, _
	tol As Double, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g13fg( String^ dist, array<double>^ yt, array<double,2>^ x, int num, int ip, int iq, int nreg, int mn, int npar, array<double>^ theta, array<double>^ se, array<double>^ sc, array<double,2>^ covr, double% hp, array<double>^ et, array<double>^ ht, [OutAttribute] double% lgf, bool copts, int maxit, double tol, [OutAttribute] int% ifail )

Visual C++

public:
static void g13fg(
	String^ dist, 
	array<double>^ yt, 
	array<double,2>^ x, 
	int num, 
	int ip, 
	int iq, 
	int nreg, 
	int mn, 
	int npar, 
	array<double>^ theta, 
	array<double>^ se, 
	array<double>^ sc, 
	array<double,2>^ covr, 
	double% hp, 
	array<double>^ et, 
	array<double>^ ht, 
	[OutAttribute] double% lgf, 
	bool copts, 
	int maxit, 
	double tol, 
	[OutAttribute] int% ifail
)

F#
static member g13fg : dist : string * yt : float[] * x : float[,] * num : int * ip : int * iq : int * nreg : int * mn : int * npar : int * theta : float[] * se : float[] * sc : float[] * covr : float[,] * hp : float byref * et : float[] * ht : float[] * lgf : float byref * copts : bool * maxit : int * tol : float * ifail : int byref -> unit

static member g13fg : 
        dist : string * 
        yt : float[] * 
        x : float[,] * 
        num : int * 
        ip : int * 
        iq : int * 
        nreg : int * 
        mn : int * 
        npar : int * 
        theta : float[] * 
        se : float[] * 
        sc : float[] * 
        covr : float[,] * 
        hp : float byref * 
        et : float[] * 
        ht : float[] * 
        lgf : float byref * 
        copts : bool * 
        maxit : int * 
        tol : float * 
        ifail : int byref -> unit

Parameters

dist

Type: System..::..String

On entry: the type of distribution to use for

e_{t}

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

Constraint:

dist = "N"

"T"

yt: Type: array<System..::..Double>[]()[][]
An array of size [num]
On entry: the sequence of observations, $y_{t}$ , for $t = 1, 2, \dots, T$ .

x: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $dim1 \geq num$
Note: the second dimension of the array x must be at least $nreg$ .
On entry: row $t$ of x must contain the time dependent exogenous vector $x_{t}$ , where $x_{t}^{T} = (x_{t}^{1}, \dots, x_{t}^{k})$ , for $t = 1, 2, \dots, T$ .

num

Type: System..::..Int32

On entry:

T

, the number of terms in the sequence.

Constraints:

$num \geq \max (ip, iq)$ ;
$num \geq nreg + mn$ .

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

Constraint: $ip \geq 0$ (see also npar).

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

Constraint: $iq \geq 1$ (see also npar).

nreg: Type: System..::..Int32
On entry: $k$ , the number of regression coefficients.

Constraint: $nreg \geq 0$ (see also npar).

mn: Type: System..::..Int32
On entry: if $mn = 1$ , the mean term $b_{0}$ will be included in the model.

Constraint: $mn = 0$ or $1$ .

npar: Type: System..::..Int32
On entry: the number of parameters to be included in the model. $npar = 1 + 2 \times iq + ip + mn + nreg$ when $dist = "N"$ and $npar = 2 + 2 \times iq + ip + mn + nreg$ when $dist = "T"$ .

Constraint: $npar < 20$ .

theta: Type: array<System..::..Double>[]()[][]
An array of size [npar]
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 contain the coefficients $ϕ_{i}$ , for $i = 1, 2, \dots, q$ .

The next ip elements must contain the moving average coefficients $β_{i}$ , for $i = 1, 2, \dots, p$ .

If $dist = "T"$ , the next element must contain an estimate for $df$ , the number of degrees of freedom of the Student's $t$ -distribution.

If $mn = 1$ , the next element must contain the mean term $b_{o}$ .

If $copts = false$ , the remaining nreg elements are taken as initial estimates of the linear regression coefficients $b_{i}$ , for $i = 1, 2, \dots, k$ .

On exit: the estimated values $\hat{θ}$ for the vector $θ$ .
The first element contains the coefficient $α_{o}$ and the next iq elements contain the coefficients $α_{i}$ , for $i = 1, 2, \dots, q$ .

The next iq elements contain the coefficients $ϕ_{i}$ , for $i = 1, 2, \dots, q$ .

The next ip elements are the moving average coefficients $β_{i}$ , for $i = 1, 2, \dots, p$ .

If $dist = "T"$ , the next element contains an estimate for $df$ then the number of degrees of freedom of the Student's $t$ -distribution.

If $mn = 1$ , the next element contains an estimate for the mean term $b_{o}$ .

The final nreg elements are the estimated linear regression coefficients $b_{i}$ , for $i = 1, 2, \dots, k$ .

se: Type: array<System..::..Double>[]()[][]
An array of size [npar]
On exit: the standard errors for $\hat{θ}$ .
The first element contains the standard error for $α_{o}$ and the next iq elements contain the standard errors for $α_{i}$ , for $i = 1, 2, \dots, q$ . The next iq elements contain the standard errors for $ϕ_{i}$ , for $i = 1, 2, \dots, q$ . The next ip elements are the standard errors for $β_{j}$ , for $j = 1, 2, \dots, p$ .

If $dist = "T"$ , the next element contains the standard error for $df$ , the number of degrees of freedom of the Student's $t$ -distribution.

If $mn = 1$ , the next element contains the standard error for $b_{o}$ .

The final nreg elements are the standard errors for $b_{j}$ , for $j = 1, 2, \dots, k$ .

sc: Type: array<System..::..Double>[]()[][]
An array of size [npar]
On exit: the scores for $\hat{θ}$ .
The first element contains the scores for $α_{o}$ , the next iq elements contain the scores for $α_{i}$ , for $i = 1, 2, \dots, q$ , the next iq elements contain the scores for $ϕ_{i}$ , for $i = 1, 2, \dots, q$ , the next ip elements are the scores for $β_{j}$ , for $j = 1, 2, \dots, p$ .

If $dist = "T"$ , the next element contains the scores for $df$ , the number of degrees of freedom of the Student's $t$ -distribution.

If $mn = 1$ , the next element contains the score for $b_{o}$ .

The final nreg elements are the scores for $b_{j}$ , for $j = 1, 2, \dots, k$ .

covr: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, npar]
Note: dim1 must satisfy the constraint: $dim1 \geq npar$
On exit: the covariance matrix of the parameter estimates $\hat{θ}$ , that is the inverse of the Fisher Information Matrix.

hp: Type: System..::..Double%
On entry: if $copts = false$ then hp is the value to be used for the pre-observed conditional variance, otherwise hp is not referenced.
On exit: if $copts = true$ then hp is the estimated value of the pre-observed conditional variance.

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

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

lgf: Type: System..::..Double%
On exit: the value of the log-likelihood function at $\hat{θ}$ .

copts: Type: System..::..Boolean
On entry: if $copts = true$ , the method provides initial parameter estimates of the regression terms, otherwise these are provided by you.

maxit: Type: System..::..Int32
On entry: the maximum number of iterations to be used by the optimization method when estimating the $GARCH (p, q)$ parameters.

Constraint: $maxit > 0$ .

tol: Type: System..::..Double
On entry: the tolerance to be used by the optimization method when estimating the $GARCH (p, q)$ parameters.

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

A univariate regression-exponential

GARCH (p, q)

process, with

q

coefficients

α_{i}

, for

i = 1, 2, \dots, q

q

coefficients

ϕ_{i}

, for

i = 1, 2, \dots, q

p

coefficients,

β_{i}

, for

i = 1, 2, \dots, p

, and

k

linear regression coefficients

b_{i}

, for

i = 1, 2, \dots, k

, can be represented by:

\begin{matrix} y_{t} = b_{o} + x_{t}^{T} b + ε_{t} \\ 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_{i = 1}^{p} β_{i} l n (h_{t - i}), t = 1, 2, \dots, T \end{matrix}

(1)

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 terms in the sequence,

y_{t}

denotes the endogenous variables,

x_{t}

the exogenous variables,

b_{o}

the regression mean,

b

the regression coefficients,

ε_{t}

the residuals,

h_{t}

the conditional variance,

df

the number of degrees of freedom of the Student's

t

-distribution, and

ψ_{t}

the set of all information up to time

t

g13fg provides an estimate

\hat{θ}

, for the vector

θ = (b_{o}, b^{T}, ω^{T})

where

b^{T} = (b_{1}, \dots, b_{k})

ω^{T} = (α_{0}, α_{1}, \dots, α_{q}, ϕ_{1}, \dots, ϕ_{q}, β_{1}, \dots, β_{p}, γ)

when

dist = "N"

, and

ω^{T} = (α_{0}, α_{1}, \dots, α_{q}, ϕ_{1}, \dots, ϕ_{q}, β_{1}, \dots, β_{p}, γ, df)

when

dist = "T"

mn, nreg can be used to simplify the

GARCH (p, q)

expression in (1) as follows:

No Regression and No Mean

$y_{t} = ε_{t}$ ,
$mn = 0$ ,
$nreg = 0$ and
$θ$ is a $(2 \times q + p + 1)$ vector when $dist = "N"$ , and a $(2 \times q + p + 2)$ vector, when $dist = "T"$ .

No Regression

$y_{t} = b_{o} + ε_{t}$ ,
$mn = 1$ ,
$nreg = 0$ and
$θ$ is a $(2 \times q + p + 2)$ vector when $dist = "N"$ and a $(2 \times q + p + 3)$ vector, when $dist = "T"$ .

Note: if the

y_{t} = μ + ε_{t}

, where

μ

is known (not to be estimated by g13fg) then (1) can be written as

y_{t}^{μ} = ε_{t}

, where

y_{t}^{μ} = y_{t} - μ

. This corresponds to the case No Regression and No Mean, with

y_{t}

replaced by

y_{t} - μ

No Mean

$y_{t} = x_{t}^{T} b + ε_{t}$ ,
$mn = 0$ ,
$nreg = k$ and
$θ$ is a $(2 \times q + p + 1 + k)$ vector when $dist = "N"$ and a $(2 \times q + p + 2 + k)$ vector, when $dist = "T"$ .

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

Note: g13fg may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDX, LDCOVR) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$

On entry,	$nreg < 0$ ,
or	$mn > 1$ ,
or	$mn < 0$ ,
or	$iq < 1$ ,
or	$ip < 0$ ,
or	$npar \geq 20$ ,
or	npar has an invalid value,
or	$dist \neq "N"$ ,
or	$dist \neq "T"$ ,
or	$maxit \leq 0$ ,
or	$num < \max (ip, iq)$ ,
or	$num < nreg + mn$ .

$ifail = 2$

On entry,

lwork < (nreg + 3) \times num + 3

$ifail = 3$: The matrix $X$ is not full rank.

$ifail = 4$: The information matrix is not positive definite.

$ifail = 5$: The maximum number of iterations has been reached.

$ifail = 6$: The log-likelihood cannot be optimized any further.

$ifail = 7$: No feasible model parameters could be found.

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

Not applicable.

Parallelism and Performance

None.

Further Comments

None.

Example

This example fits a

GARCH (1, 2)

model with Student's

t

-distributed residuals to some simulated data.

Example program (C#): g13fge.cs

Example program data: g13fge.d

Example program results: g13fge.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also