NAG Library Routine Document

G13FGF

INTEGER	LDX, NUM, IP, IQ, NREG, MN, NPAR, LDCOVR, MAXIT, LWORK, IFAIL
REAL (KIND=nag_wp)	YT(NUM), X(LDX,*), THETA(NPAR), SE(NPAR), SC(NPAR), COVR(LDCOVR,NPAR), HP, ET(NUM), HT(NUM), LGF, TOL, WORK(LWORK)
LOGICAL	COPTS
CHARACTER(1)	DIST

3 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

G13FGF 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 G13FGF) 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 $DIST ='T'$ .

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 Parameters

1: $DIST$ – CHARACTER(1)Input

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'

2: $YT (NUM)$ – REAL (KIND=nag_wp) arrayInput

On entry: the sequence of observations,

y_{t}

, for

t = 1, 2, \dots, T

3: $X (LDX *)$ – REAL (KIND=nag_wp) arrayInput

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

4: $LDX$ – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G13FGF is called.

Constraint:

LDX \geq NUM

5: $NUM$ – INTEGERInput

On entry:

T

, the number of terms in the sequence.

Constraints:

$NUM \geq \max (IP, IQ)$ ;
$NUM \geq NREG + MN$ .

6: $IP$ – INTEGERInput

On entry: the number of coefficients,

β_{i}

, for

i = 1, 2, \dots, p

Constraint:

IP \geq 0

(see also NPAR).

7: $IQ$ – INTEGERInput

On entry: the number of coefficients,

α_{i}

, for

i = 1, 2, \dots, q

Constraint:

IQ \geq 1

(see also NPAR).

8: $NREG$ – INTEGERInput

On entry:

k

, the number of regression coefficients.

Constraint:

NREG \geq 0

(see also NPAR).

9: $MN$ – INTEGERInput

On entry: if

MN = 1

, the mean term

b_{0}

will be included in the model.

Constraint:

MN = 0

1

10: $NPAR$ – INTEGERInput

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

11: $THETA (NPAR)$ – REAL (KIND=nag_wp) arrayInput/Output

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

DIST ='T'

, the next element must contain an estimate for

df

, the number of degrees of freedom of the Student's

t

-distribution.

MN = 1

, the next element must contain the mean term

b_{o}

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

DIST ='T'

, the next element contains an estimate for

df

then the number of degrees of freedom of the Student's

t

-distribution.

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

12: $SE (NPAR)$ – REAL (KIND=nag_wp) arrayOutput

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

DIST ='T'

, the next element contains the standard error for

df

, the number of degrees of freedom of the Student's

t

-distribution.

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

13: $SC (NPAR)$ – REAL (KIND=nag_wp) arrayOutput

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

DIST ='T'

, the next element contains the scores for

df

, the number of degrees of freedom of the Student's

t

-distribution.

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

14: $COVR (LDCOVR, NPAR)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the covariance matrix of the parameter estimates

\hat{θ}

, that is the inverse of the Fisher Information Matrix.

15: $LDCOVR$ – INTEGERInput

On entry: the first dimension of the array COVR as declared in the (sub)program from which G13FGF is called.

Constraint:

LDCOVR \geq NPAR

16: $HP$ – REAL (KIND=nag_wp)Input/Output

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.

17: $ET (NUM)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the estimated residuals,

ε_{t}

, for

t = 1, 2, \dots, T

18: $HT (NUM)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the estimated conditional variances,

h_{t}

, for

t = 1, 2, \dots, T

19: $LGF$ – REAL (KIND=nag_wp)Output

On exit: the value of the log-likelihood function at

\hat{θ}

20: $COPTS$ – LOGICALInput

On entry: if

COPTS = .TRUE.

, the routine provides initial parameter estimates of the regression terms, otherwise these are provided by you.

21: $MAXIT$ – INTEGERInput

On entry: the maximum number of iterations to be used by the optimization routine when estimating the

GARCH (p, q)

parameters.

Constraint:

MAXIT > 0

22: $TOL$ – REAL (KIND=nag_wp)Input

On entry: the tolerance to be used by the optimization routine when estimating the

GARCH (p, q)

parameters.

23: $WORK (LWORK)$ – REAL (KIND=nag_wp) arrayWorkspace

24: $LWORK$ – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which G13FGF is called.

Constraint:

LWORK \geq (NREG + 3) \times NUM + 3

25: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. 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).

Note: G13FGF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$

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	$LDCOVR < NPAR$ ,
or	$LDX < NUM$ ,
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.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

G13FGF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

G13FGF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

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 fits a

GARCH (1, 2)

model with Student's

t

-distributed residuals to some simulated data.

The process parameter estimates,

\hat{θ}

, are obtained using G13FGF, and a four step ahead volatility estimate is computed using G13FHF.

The data was simulated using G05PGF.

NAG Library Routine DocumentG13FGF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

G13FGF