The function may be called by the names: g13fac, nag_tsa_uni_garch_asym1_estim or nag_estimate_agarchi.
3Description
When , g13fac models a standard univariate regression-GARCH process, with coefficients , for , coefficients, , for , mean , and linear regression coefficients , for , which can be represented by:
(1)
When , g13fac models an asymmetric GARCH process where the conditional variance is given by:
Here the number of terms in the sequence, denotes the endogenous variables, the exogenous variables, the mean, the regression coefficients, the residuals, the asymmetry parameter, the conditional variance, and the information set of all information up to time .
When , g13fac provides an estimate for , the parameter vector where and .
isym, mn, nreg can be used to simplify the GARCH expression in (1) as follows:
No Regression or Mean
,
,
,
, and
is a vector.
No Regression
,
,
,
, and
is a vector.
Note: if the , where is known (not to be estimated by g13fac) then (1) can be written as , where . This corresponds to the case No Regression or Mean, with replaced by .
No Mean
,
,
,
and
is a vector.
4References
Bollerslev T (1986) Generalised autoregressive conditional heteroskedasticity Journal of Econometrics31 307–327
Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation Econometrica50 987–1008
Engle R and Ng V (1993) Measuring and testing the impact of news on volatility Journal of Finance48 1749–1777
Hamilton J (1994) Time Series Analysis Princeton University Press
5Arguments
Note: for convenience npar will be used here to denote the expression
representing the number of model parameters.
1: – const doubleInput
On entry: the sequence of observations, , for .
2: – const doubleInput
Note: th element of the th vector is stored in .
On entry: row of x must contain the time dependent exogenous vector , where , for .
3: – IntegerInput
On entry: the stride separating matrix column elements in the array x.
Constraint:
.
4: – IntegerInput
On entry: the number of terms in the sequence, .
Constraint:
.
5: – IntegerInput
On entry: the GARCH parameter .
Constraint:
.
6: – IntegerInput
On entry: the GARCH parameter .
Constraint:
.
7: – IntegerInput
On entry: , the number of refression coefficients.
Constraint:
.
8: – IntegerInput
On entry: if , the mean term will be included in the model.
Constraint:
or .
9: – IntegerInput
On entry: if , the asymmetry term will be included in the model.
Constraint:
or .
10: – doubleInput/Output
On entry: the initial parameter estimates for the vector .
The first element contains the coefficient , the next q elements contain the autoregressive coefficients , for .
The next p elements are the moving average coefficients , for .
The next element contains the asymmetry parameter .
If , (when ) the next term contains an initial estimate of the mean term and the remaining nreg elements are taken as initial estimates of the linear regression coefficients , for .
On exit: the estimated values for the vector .
The first element contains the coefficient , the next q elements contain the coefficients , for .
On exit: if , hp is the estimated value of the pre-observed of the conditional variance.
16: – doubleOutput
On exit: the estimated residuals, , for .
17: – doubleOutput
On exit: the estimated conditional variances, , for .
18: – double *Output
On exit: the value of the log-likelihood function at .
19: – Nag_Garch_Stationary_TypeInput
On entry: if , Stationary conditions are enforced.
If , Stationary conditions are not enforced.
Constraint:
or .
20: – Nag_Garch_Est_Initial_TypeInput
On entry: if , the function provides initial parameter estimates of the regression terms .
If , you must supply the initial estimations of the regression parameters .
Constraint:
or .
21: – IntegerInput
On entry: the maximum number of iterations to be used by the optimization function when estimating the GARCH parameters. If max_iter is set to , the standard errors, score vector and variance-covariance are calculated for the input value of in theta; however the value of is not updated.
Constraint:
.
22: – doubleInput
On entry: the tolerance to be used by the optimization function when estimating the GARCH parameters.
23: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_INT_RANGE_2
Value given to isym is not valid. Correct range is 0 to 1.
Value given to mn is not valid. Correct range is 0 to 1.
NE_MAT_NOT_FULL_RANK
Matrix does not give a model of full rank.
NE_MAT_NOT_POS_DEF
Attempt to invert the second derivative matrix needed in the calculation of the covariance matrix of the parameter estimates has failed. The matrix is not positive definite, possibly due to rounding errors.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g13fac is not threaded in any implementation.
9Further Comments
None.
10Example
This example program illustrates the use of g13fac to model a GARCH(1,1) sequence generated by g05pdc, a three step forecast is then calculated using g13fbc.