nag_estimate_garchGJR (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)

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

nag_estimate_garchGJR (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 nag_estimate_garchGJR (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 doubleInput: On entry: the sequence of observations, $y_{t}$ , for $t = 1, 2, \dots, T$ .
2: $x [num \times tdx]$ – const doubleInput: Note: the $i$ 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: $tdx$ – IntegerInput: On entry: the stride separating matrix column elements in the array x.

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

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

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

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

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

Constraint: $mn = 0$ or $1$ .
9: $theta [npar]$ – doubleInput/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: $se [npar]$ – doubleOutput: 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: $sc [npar]$ – doubleOutput: 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: $covar [npar \times tdc]$ – doubleOutput: 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: $tdc$ – IntegerInput: 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: $et [num]$ – doubleOutput: On exit: the estimated residuals, $ε_{t}$ , for $t = 1, 2, \dots, T$ .
16: $ht [num]$ – doubleOutput: 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_TypeInput: 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_TypeInput: 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: $max_iter$ – IntegerInput: 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: $tol$ – doubleInput: 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 3.7 in How to Use the NAG Library and its Documentation).

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

nag_estimate_garchGJR (g13fec) is not threaded in any implementation.

9

Further Comments

None.

10

Example

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