g13fec (const double yt[], const double x[], Integer tdx, Integer num, Integer p, Integer q, Integer nreg, Integer mn, double theta[], double se[], double sc[], double covar[], Integer tdc, double *hp, double et[], double ht[], double *lgf, Nag_Garch_Stationary_Type stat_opt, Nag_Garch_Est_Initial_Type est_opt, Integer max_iter, double tol, NagError *fail)

The function may be called by the names: g13fec, nag_tsa_uni_garch_gjr_estim or nag_estimate_garchgjr.

3 Description

A univariate regression-GJR GARCH

(p, q)

process, with

p

coefficients

α_{i}

, for

i = 1, 2, \dots, p

q

coefficients,

β_{i}

, for

i = 1, 2, \dots, q

, mean

b_{o}

, and

k

linear regression coefficients

b_{i}

, for

i = 1, 2, \dots, k

, can be represented by:

y_{t} = b_{o} + x_{t}^{T} b + ε_{t}

(1)

ε_{t} ∣ ψ_{t - 1} \sim N (0, h_{t})

h_{t} = α_{0} + \sum_{i = 1}^{q} (α_{i} + γ S_{t - i}) ε_{t - i}^{2} + \sum_{i = 1}^{p} β_{i} h_{t - i}, t = 1, \dots, T .

where

S_{t} = 1

, if

ε_{t} < 0

, and

S_{t} = 0

, if

ε_{t} \geq 0

. Here

T

is the number of terms in the sequence,

y_{t}

denotes the endogenous variables,

x_{t}

the exogenous variables,

b_{o}

the mean,

b

the regression coefficients,

ε_{t}

the residuals,

γ

is the asymmetry parameter,

h_{t}

is the conditional variance, and

ψ_{t}

the information set of all information up to time

t

g13fec provides an estimate for

\hat{θ}

, the

(p + q + k + 3) \times 1

parameter vector

θ = (b_{o}, b^{T}, ω^{T})

where

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

and

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

mn, nreg can be used to simplify the GARCH

(p, q)

expression in equation (1) as follows:

No Regression or Mean

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

No Regression

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

Note: if the

y_{t} = μ + ε_{t}

, where

μ

is known (not to be estimated by g13fec) then equation (1) can be written as

y_{t}^{μ} = ε_{t}

, where

y_{t}^{μ} = y_{t} - μ

. This corresponds to the case No Regression or 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 $(p + q + k + 2) \times 1$ vector.

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

Note: for convenience npar will be used here to denote the expression

2 + q + p + mn + nreg

representing the number of model parameters.

1: $yt [num]$ – const double Input: On entry: the sequence of observations, $y_{t}$ , for $t = 1, 2, \dots, T$ .
2: $x [num \times tdx]$ – const double Input: Note: $i$ th element of the $j$ th vector $X$ is stored in $x [(i - 1) \times tdx + j - 1]$ .

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 $t = 1, 2, \dots, T$ .
3: $tdx$ – Integer Input: On entry: the stride separating matrix column elements in the array x.

Constraint: $tdx \geq nreg$ .
4: $num$ – Integer Input: On entry: the number of terms in the sequence, $T$ .

Constraint: $num \geq npar$ .
5: $p$ – Integer Input: On entry: the GARCH $(p, q)$ parameter $p$ .

Constraint: $p \geq 0$ .
6: $q$ – Integer Input: On entry: the GARCH $(p, q)$ parameter $q$ .

Constraint: $q \geq 1$ .
7: $nreg$ – Integer Input: On entry: $k$ , the number of regression coefficients.

Constraint: $nreg \geq 0$ .
8: $mn$ – Integer Input: On entry: if $mn = 1$ , the mean term $b_{0}$ will be included in the model.

Constraint: $mn = 0$ or $1$ .
9: $theta [npar]$ – double Input/Output: On entry: the initial parameter estimates for the vector $θ$ .
The first element contains the coefficient $α_{o}$ , the next q elements contain the autoregressive coefficients $α_{i}$ , for $i = 1, 2, \dots, q$ .

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

The next element contains the asymmetry parameter $γ$ .

If $est_opt = Nag_Garch_Est_Initial_False$ , (when $mn = 1$ ) the next term contains an initial estimate of the mean term $b_{o}$ and 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 $α_{o}$ , the next q elements contain the coefficients $α_{i}$ , for $i = 1, 2, \dots, q$ .

The next p elements are the coefficients $β_{j}$ , for $j = 1, 2, \dots, p$ .

The next element contains the estimate for the asymmetry parameter $γ$ .

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 $i = 1, 2, \dots, k$ .
10: $se [npar]$ – double Output: On exit: the standard errors for $\hat{θ}$ .
The first element contains the standard error for $α_{o}$ .

The next q elements contain the standard errors for $α_{i}$ , for $i = 1, 2, \dots, q$ .

The next p elements are the standard errors for $β_{j}$ , for $j = 1, 2, \dots, p$ .

The next element contains the standard error for $γ$ .

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 $j = 1, 2, \dots, k$ .
11: $sc [npar]$ – double Output: On exit: the scores for $\hat{θ}$ .
The first element contains the score for $α_{o}$ , the next q elements contain the score for $α_{i}$ , for $i = 1, 2, \dots, q$ .

The next p elements are the scores for $β_{j}$ , for $j = 1, 2, \dots, p$ .

The next element contains the score for $γ$ .

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

The final nreg elements are the scores for $b_{j}$ , for $j = 1, 2, \dots, k$ .
12: $covar [npar \times tdc]$ – double Output: Note: the $(i, j)$ th element of the matrix is stored in $covar [(i - 1) \times tdc + j - 1]$ .

On exit: the covariance matrix of the parameter estimates $\hat{θ}$ , that is the inverse of the Fisher Information Matrix.
13: $tdc$ – Integer Input: On entry: the stride separating matrix column elements in the array covar.

Constraint: $tdc \geq npar$ .
14: $hp$ – double * Input/Output: On entry: if $est_opt = Nag_Garch_Est_Initial_False$ , hp is the value to be used for the pre-observed conditional variance.
If $est_opt = Nag_Garch_Est_Initial_True$ , hp is not referenced.

On exit: if $est_opt = Nag_Garch_Est_Initial_True$ , hp is the estimated value of the pre-observed of the conditional variance.
15: $et [num]$ – double Output: On exit: the estimated residuals, $ε_{t}$ , for $t = 1, 2, \dots, T$ .
16: $ht [num]$ – double Output: On exit: the estimated conditional variances, $h_{t}$ , for $t = 1, 2, \dots, T$ .
17: $lgf$ – double * Output: On exit: the value of the log-likelihood function at $\hat{θ}$ .
18: $stat_opt$ – Nag_Garch_Stationary_Type Input: On entry: if $stat_opt = Nag_Garch_Stationary_True$ , Stationary conditions are enforced.
If $stat_opt = Nag_Garch_Stationary_False$ , Stationary conditions are not enforced.

Constraint: $stat_opt = Nag_Garch_Stationary_True$ or $Nag_Garch_Stationary_False$ .
19: $est_opt$ – Nag_Garch_Est_Initial_Type Input: On entry: if $est_opt = Nag_Garch_Est_Initial_True$ , the function provides initial parameter estimates of the regression terms $(b_{0}, b^{T})$ .
If $est_opt = Nag_Garch_Est_Initial_False$ , you must supply the initial estimations of the regression parameters $(b_{0}, b^{T})$ .

Constraint: $est_opt = Nag_Garch_Est_Initial_True$ or $Nag_Garch_Est_Initial_False$ .
20: $max_iter$ – Integer Input: On entry: the maximum number of iterations to be used by the optimization function when estimating the GARCH $(p, q)$ parameters. If max_iter is set to $0$ , the standard errors, score vector and variance-covariance are calculated for the input value of $θ$ in theta; however the value of $θ$ is not updated.

Constraint: $max_iter \geq 0$ .
21: $tol$ – double Input: On entry: the tolerance to be used by the optimization function when estimating the GARCH $(p, q)$ parameters.
22: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $num = 〈value〉$ while $2 + q + p + mn + nreg = 〈value〉$ . These parameters must satisfy $num \geq 2 + q + p + mn + nreg$ .

On entry, $tdc = 〈value〉$ while $2 + q + p + mn + nreg = 〈value〉$ . These parameters must satisfy $tdc \geq 2 + q + p + mn + nreg$ .

On entry, $tdx = 〈value〉$ while $nreg = 〈value〉$ . These parameters must satisfy $tdx \geq nreg$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, parameter est_opt had an illegal value.

On entry, parameter stat_opt had an illegal value.
NE_INT_ARG_LT: On entry, max_iter must not be less than 0: $max_iter = 〈value〉$ .

On entry, $nreg = 〈value〉$ .
Constraint: $nreg \geq 0$ .

On entry, $p = 〈value〉$ .
Constraint: $p \geq 0$ .

On entry, $q = 〈value〉$ .
Constraint: $q \geq 1$ .
NE_INTERNAL_ERROR: 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 $〈value〉$ given to mn is not valid. Correct range is 0 to 1.
NE_MAT_NOT_FULL_RANK: Matrix $X$ 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.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g13fec is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example program illustrates the use of g13fec to model a GARCH(1,1) sequence generated by g05pfc, a six step forecast is then calculated using g13ffc.