NAG FL Interface
g13fgf (uni_​garch_​exp_​estim)

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1 Purpose

g13fgf estimates the parameters of a univariate regression-exponential GARCH(p,q) process (see Engle and Ng (1993)).

2 Specification

Fortran Interface
Subroutine g13fgf ( dist, yt, x, ldx, num, ip, iq, nreg, mn, npar, theta, se, sc, covr, ldcovr, pht, et, ht, lgf, copts, maxit, tol, work, lwork, ifail)
Integer, Intent (In) :: ldx, num, ip, iq, nreg, mn, npar, ldcovr, maxit, lwork
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: yt(num), x(ldx,*), tol
Real (Kind=nag_wp), Intent (Inout) :: theta(npar), covr(ldcovr,npar), pht
Real (Kind=nag_wp), Intent (Out) :: se(npar), sc(npar), et(num), ht(num), lgf, work(lwork)
Logical, Intent (In) :: copts
Character (1), Intent (In) :: dist
C Header Interface
#include <nag.h>
void  g13fgf_ (const char *dist, const double yt[], const double x[], const Integer *ldx, const Integer *num, const Integer *ip, const Integer *iq, const Integer *nreg, const Integer *mn, const Integer *npar, double theta[], double se[], double sc[], double covr[], const Integer *ldcovr, double *pht, double et[], double ht[], double *lgf, const logical *copts, const Integer *maxit, const double *tol, double work[], const Integer *lwork, Integer *ifail, const Charlen length_dist)
The routine may be called by the names g13fgf or nagf_tsa_uni_garch_exp_estim.

3 Description

A univariate regression-exponential GARCH(p,q) process, with q coefficients αi, for i=1,2,,q, q coefficients ϕi, for i=1,2,,q, p coefficients, βi, for i=1,2,,p, and k linear regression coefficients bi, for i=1,2,,k, can be represented by:
yt = bo + xtT b + εt ln(ht) = α0+ i=1 q αi zt-i + i=1 q ϕi (|zt-i|-E[|zt-i|]) + i=1 p βi ln (ht-i) ,   t=1,2,,T (1)
where zt= εt ht , E[|zt-i|] denotes the expected value of |zt-i| and εtψt-1=N(0,ht) or εtψt-1=St(df,ht). Here St is a standardized Student's t-distribution with df degrees of freedom and variance ht, T is the number of terms in the sequence, yt denotes the endogenous variables, xt the exogenous variables, bo the regression mean, b the regression coefficients, εt the residuals, ht the conditional variance, df the number of degrees of freedom of the Student's t-distribution, and ψt the set of all information up to time t.
g13fgf provides an estimate θ^, for the vector θ = (bo,bT,ωT) where bT = (b1,,bk) , ωT = (α0,α1,,αq,ϕ1,,ϕq,β1,,βp,γ) when dist='N', and ωT = (α0,α1,,αq,ϕ1,,ϕq,β1,,βp,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
No Regression
Note:  if the yt=μ+εt, where μ is known (not to be estimated by g13fgf) then (1) can be written as ytμ=εt, where ytμ=yt-μ. This corresponds to the case No Regression and No Mean, with yt replaced by yt-μ.
No Mean

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 et.
dist='N'
A Normal distribution is used.
dist='T'
A Student's t-distribution is used.
Constraint: dist='N' or 'T'.
2: yt(num) Real (Kind=nag_wp) array Input
On entry: the sequence of observations, yt, for t=1,2,,T.
3: x(ldx,*) Real (Kind=nag_wp) array Input
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 xt , where xtT = (xt1,,xtk) , for t=1,2,,T.
4: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g13fgf is called.
Constraint: ldxnum.
5: num Integer Input
On entry: T, the number of terms in the sequence.
Constraints:
  • nummax(ip,iq);
  • numnreg+mn.
6: ip Integer Input
On entry: the number of coefficients, βi, for i=1,2,,p.
Constraint: ip0 (see also npar).
7: iq Integer Input
On entry: the number of coefficients, αi, for i=1,2,,q.
Constraint: iq1 (see also npar).
8: nreg Integer Input
On entry: k, the number of regression coefficients.
Constraint: nreg0 (see also npar).
9: mn Integer Input
On entry: if mn=1, the mean term b0 will be included in the model.
Constraint: mn=0 or 1.
10: npar Integer Input
On entry: the number of parameters to be included in the model. npar=1+2×iq+ip+mn+nreg when dist='N' and npar=2+2×iq+ip+mn+nreg when dist='T'.
Constraint: npar<20.
11: theta(npar) Real (Kind=nag_wp) array Input/Output
On entry: the initial parameter estimates for the vector θ.
The first element must contain the coefficient αo and the next iq elements must contain the autoregressive coefficients αi, for i=1,2,,q.
The next iq elements contain the coefficients ϕi, for i=1,2,,q.
The next ip elements must contain the moving average coefficients βi, for i=1,2,,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 bo.
If copts=.FALSE., the remaining nreg elements are taken as initial estimates of the linear regression coefficients bi, for i=1,2,,k.
On exit: the estimated values θ^ for the vector θ.
The first element contains the coefficient αo and the next iq elements contain the coefficients αi, for i=1,2,,q.
The next iq elements contain the coefficients ϕi, for i=1,2,,q.
The next ip elements are the moving average coefficients βi, for i=1,2,,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 bo.
The final nreg elements are the estimated linear regression coefficients bi, for i=1,2,,k.
12: se(npar) Real (Kind=nag_wp) array Output
On exit: the standard errors for θ^.
The first element contains the standard error for αo and the next iq elements contain the standard errors for αi, for i=1,2,,q. The next iq elements contain the standard errors for ϕi, for i=1,2,,q. The next ip elements are the standard errors for βj, for j=1,2,,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 bo.
The final nreg elements are the standard errors for bj, for j=1,2,,k.
13: sc(npar) Real (Kind=nag_wp) array Output
On exit: the scores for θ^.
The first element contains the scores for αo, the next iq elements contain the scores for αi, for i=1,2,,q, the next iq elements contain the scores for ϕi, for i=1,2,,q, the next ip elements are the scores for βj, for j=1,2,,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 bo.
The final nreg elements are the scores for bj, for j=1,2,,k.
14: covr(ldcovr,npar) Real (Kind=nag_wp) array Output
On exit: the covariance matrix of the parameter estimates θ^, that is the inverse of the Fisher Information Matrix.
15: ldcovr Integer Input
On entry: the first dimension of the array covr as declared in the (sub)program from which g13fgf is called.
Constraint: ldcovrnpar.
16: pht Real (Kind=nag_wp) Input/Output
On entry: if copts=.FALSE. then pht is the value to be used for the pre-observed conditional variance, otherwise pht is not referenced.
On exit: if copts=.TRUE. then pht is the estimated value of the pre-observed conditional variance.
17: et(num) Real (Kind=nag_wp) array Output
On exit: the estimated residuals, εt, for t=1,2,,T.
18: ht(num) Real (Kind=nag_wp) array Output
On exit: the estimated conditional variances, ht, for t=1,2,,T.
19: lgf Real (Kind=nag_wp) Output
On exit: the value of the log-likelihood function at θ^.
20: copts Logical Input
On entry: if copts=.TRUE., the routine provides initial parameter estimates of the regression terms, otherwise these are provided by you.
21: maxit Integer Input
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) array Workspace
24: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which g13fgf is called.
Constraint: lwork(nreg+3)×num+3.
25: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 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 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value -1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. 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 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g13fgf may return useful information.
ifail=1
On entry, dist=value.
Constraint: dist='N' or 'T'.
On entry, ip=value.
Constraint: ip0.
On entry, iq=value.
Constraint: iq1.
On entry, ldcovr=value.
Constraint: ldcovrnpar.
On entry, ldx=value.
Constraint: ldxnum.
On entry, maxit=value.
Constraint: maxit>0.
On entry, mn=value.
Constraint: mn=0 or 1.
On entry, npar=value.
Constraint: if dist='N' then npar=1+2×iq+ip+mn+nreg, else npar=2+2×iq+ip+mn+nreg.
On entry, npar=value.
Constraint: npar<20.
On entry, nreg=value.
Constraint: nreg0.
On entry, num=value.
Constraint: max(ip,iq)num.
On entry, num=value.
Constraint: numnreg+mn.
ifail=2
On entry, lwork=value.
Constraint: lwork(nreg+3)×num+3.
ifail=3
On entry, 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 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

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, θ^, are obtained using g13fgf, and a four step ahead volatility estimate is computed using g13fhf.
The data was simulated using g05pgf.

10.1 Program Text

Program Text (g13fgfe.f90)

10.2 Program Data

Program Data (g13fgfe.d)

10.3 Program Results

Program Results (g13fgfe.r)