g13 Chapter Contents
g13 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_estimate_garchGJR (g13fec)

## 1  Purpose

nag_estimate_garchGJR (g13fec) estimates the arguments of a univariate regression-GJR GARCH$\left(p,q\right)$ process (see Glosten et al. (1993)).

## 2  Specification

 #include #include
 void 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$\left(p,q\right)$ process, with $p$ coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$, $q$ coefficients, ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$, mean ${b}_{o}$, and $k$ linear regression coefficients ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$, can be represented by:
 $y t = b o + xtT b + ε t$ (1)
 $ε t ∣ ψ t-1 ∼ N 0, h t$
 $h t = α 0 + ∑ i=1 q α i + γ S t-i ε t-i 2 + ∑ i=1 p β i h t-i , t = 1 , … , T .$
where ${S}_{t}=1$, if ${\epsilon }_{t}<0$, and ${S}_{t}=0$, if ${\epsilon }_{t}\ge 0$. 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, ${\epsilon }_{t}$ the residuals, $\gamma$ is the asymmetry parameter, ${h}_{t}$ is the conditional variance, and ${\psi }_{t}$ the information set of all information up to time $t$.
nag_estimate_garchGJR (g13fec) provides an estimate for $\stackrel{^}{\theta }$, the $\left(p+q+k+3\right)×1$ parameter vector $\theta =\left({b}_{o},{b}^{\mathrm{T}},{\omega }^{\mathrm{T}}\right)$ where ${\omega }^{\mathrm{T}}=\left({\alpha }_{0},{\alpha }_{1},\dots ,{\alpha }_{q},{\beta }_{1},\dots ,{\beta }_{p},\gamma \right)$ and ${b}^{\mathrm{T}}=\left({b}_{1},\dots ,{b}_{k}\right)$.
mn, nreg can be used to simplify the GARCH$\left(p,q\right)$ expression in equation (1) as follows:
No Regression or Mean
• ${y}_{t}={\epsilon }_{t}$,
• ${\mathbf{mn}}=0$,
• ${\mathbf{nreg}}=0$, and
• $\theta$ is a $\left(p+q+2\right)×1$ vector.
No Regression
• ${y}_{t}={b}_{o}+{\epsilon }_{t}$,
• ${\mathbf{mn}}=1$,
• ${\mathbf{nreg}}=0$, and
• $\theta$ is a $\left(p+q+3\right)×1$ vector.
Note: if the ${y}_{t}=\mu +{\epsilon }_{t}$, where $\mu$ is known (not to be estimated by nag_estimate_garchGJR (g13fec)) then equation (1) can be written as ${y}_{t}^{\mu }={\epsilon }_{t}$, where ${y}_{t}^{\mu }={y}_{t}-\mu$. This corresponds to the case No Regression or Mean, with ${y}_{t}$ replaced by ${y}_{t}-\mu$.
No Mean
• ${y}_{t}={x}_{t}^{\mathrm{T}}b+{\epsilon }_{t}$,
• ${\mathbf{mn}}=0$,
• ${\mathbf{nreg}}=k$ and
• $\theta$ is a $\left(p+q+k+2\right)×1$ 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 $2+{\mathbf{q}}+{\mathbf{p}}+{\mathbf{mn}}+{\mathbf{nreg}}$ representing the number of model parameters.
2:     yt[num]const doubleInput
On entry: the sequence of observations, ${y}_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.
3:     x[${\mathbf{num}}×{\mathbf{tdx}}$]const doubleInput
Note: the $i$th element of the $j$th vector $X$ is stored in ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{tdx}}+j-1\right]$.
On entry: row $t$ of x must contain the time dependent exogenous vector ${x}_{t}$, where ${x}_{\mathit{t}}^{\mathrm{T}}=\left({{x}_{\mathit{t}}}^{1},\dots ,{{x}_{\mathit{t}}}^{k}\right)$, for $\mathit{t}=1,2,\dots ,T$.
4:     tdxIntegerInput
On entry: the stride separating matrix column elements in the array x.
Constraint: ${\mathbf{tdx}}\ge {\mathbf{nreg}}$.
5:     numIntegerInput
On entry: the number of terms in the sequence, $T$.
Constraint: ${\mathbf{num}}\ge \mathit{npar}$.
6:     pIntegerInput
On entry: the GARCH$\left(p,q\right)$ parameter $p$.
Constraint: ${\mathbf{p}}\ge 0$.
7:     qIntegerInput
On entry: the GARCH$\left(p,q\right)$ parameter $q$.
Constraint: ${\mathbf{q}}\ge 1$.
8:     nregIntegerInput
On entry: $k$, the number of regression coefficients.
Constraint: ${\mathbf{nreg}}\ge 0$.
9:     mnIntegerInput
On entry: if ${\mathbf{mn}}=1$, the mean term ${b}_{0}$ will be included in the model.
Constraint: ${\mathbf{mn}}=0$ or $1$.
10:   theta[$\mathit{npar}$]doubleInput/Output
On entry: the initial parameter estimates for the vector $\theta$.
The first element contains the coefficient ${\alpha }_{o}$, the next q elements contain the autoregressive coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
The next p elements are the moving average coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
The next element contains the asymmetry parameter $\gamma$.
If ${\mathbf{est_opt}}=\mathrm{Nag_Garch_Est_Initial_False}$, (when ${\mathbf{mn}}=1$) 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 ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.
On exit: the estimated values $\stackrel{^}{\theta }$ for the vector $\theta$.
The first element contains the coefficient ${\alpha }_{o}$, the next q elements contain the coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
The next p elements are the coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
The next element contains the estimate for the asymmetry parameter $\gamma$.
If ${\mathbf{mn}}=1$, the next element contains an estimate for the mean term ${b}_{o}$.
The final nreg elements are the estimated linear regression coefficients ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.
11:   se[$\mathit{npar}$]doubleOutput
On exit: the standard errors for $\stackrel{^}{\theta }$.
The first element contains the standard error for ${\alpha }_{o}$.
The next q elements contain the standard errors for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
The next p elements are the standard errors for ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
The next element contains the standard error for $\gamma$.
If ${\mathbf{mn}}=1$, the next element contains the standard error for ${b}_{o}$.
The final nreg elements are the standard errors for ${b}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,k$.
12:   sc[$\mathit{npar}$]doubleOutput
On exit: the scores for $\stackrel{^}{\theta }$.
The first element contains the score for ${\alpha }_{o}$, the next q elements contain the score for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
The next p elements are the scores for ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
The next element contains the score for $\gamma$.
If ${\mathbf{mn}}=1$, the next element contains the score for ${b}_{o}$.
The final nreg elements are the scores for ${b}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,k$.
13:   covar[$\mathit{npar}×{\mathbf{tdc}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{covar}}\left[\left(i-1\right)×{\mathbf{tdc}}+j-1\right]$.
On exit: the covariance matrix of the parameter estimates $\stackrel{^}{\theta }$, that is the inverse of the Fisher Information Matrix.
14:   tdcIntegerInput
On entry: the stride separating matrix column elements in the array covar.
Constraint: ${\mathbf{tdc}}\ge \mathit{npar}$.
15:   hpdouble *Input/Output
On entry: if ${\mathbf{est_opt}}=\mathrm{Nag_Garch_Est_Initial_False}$, hp is the value to be used for the pre-observed conditional variance.
If ${\mathbf{est_opt}}=\mathrm{Nag_Garch_Est_Initial_True}$, hp is not referenced.
On exit: if ${\mathbf{est_opt}}=\mathrm{Nag_Garch_Est_Initial_True}$, hp is the estimated value of the pre-observed of the conditional variance.
16:   et[num]doubleOutput
On exit: the estimated residuals, ${\epsilon }_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.
17:   ht[num]doubleOutput
On exit: the estimated conditional variances, ${h}_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.
18:   lgfdouble *Output
On exit: the value of the log likelihood function at $\stackrel{^}{\theta }$.
19:   stat_optNag_Garch_Stationary_TypeInput
On entry: if ${\mathbf{stat_opt}}=\mathrm{Nag_Garch_Stationary_True}$, Stationary conditions are enforced.
If ${\mathbf{stat_opt}}=\mathrm{Nag_Garch_Stationary_False}$, Stationary conditions are not enforced.
Constraint: ${\mathbf{stat_opt}}=\mathrm{Nag_Garch_Stationary_True}$ or $\mathrm{Nag_Garch_Stationary_False}$.
20:   est_optNag_Garch_Est_Initial_TypeInput
On entry: if ${\mathbf{est_opt}}=\mathrm{Nag_Garch_Est_Initial_True}$, the function provides initial parameter estimates of the regression terms $\left({b}_{0},{b}^{\mathrm{T}}\right)$.
If ${\mathbf{est_opt}}=\mathrm{Nag_Garch_Est_Initial_False}$, you must supply the initial estimations of the regression parameters $\left({b}_{0},{b}^{\mathrm{T}}\right)$.
Constraint: ${\mathbf{est_opt}}=\mathrm{Nag_Garch_Est_Initial_True}$ or $\mathrm{Nag_Garch_Est_Initial_False}$.
21:   max_iterIntegerInput
On entry: the maximum number of iterations to be used by the optimization function when estimating the GARCH$\left(p,q\right)$ parameters. If max_iter is set to 0, the standard errors, score vector and variance-covariance are calculated for the input value of $\theta$ in theta; however the value of $\theta$ is not updated.
Constraint: ${\mathbf{max_iter}}\ge 0$.
22:   toldoubleInput
On entry: the tolerance to be used by the optimization function when estimating the GARCH$\left(p,q\right)$ parameters.
23:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, ${\mathbf{num}}=〈\mathit{\text{value}}〉$ while $2+{\mathbf{q}}+{\mathbf{p}}+{\mathbf{mn}}+{\mathbf{nreg}}=〈\mathit{\text{value}}〉$. These parameters must satisfy ${\mathbf{num}}\ge 2+{\mathbf{q}}+{\mathbf{p}}+{\mathbf{mn}}+{\mathbf{nreg}}$.
On entry, ${\mathbf{tdc}}=〈\mathit{\text{value}}〉$ while $2+{\mathbf{q}}+{\mathbf{p}}+{\mathbf{mn}}+{\mathbf{nreg}}=〈\mathit{\text{value}}〉$. These parameters must satisfy ${\mathbf{tdc}}\ge 2+{\mathbf{q}}+{\mathbf{p}}+{\mathbf{mn}}+{\mathbf{nreg}}$.
On entry, ${\mathbf{tdx}}=〈\mathit{\text{value}}〉$ while ${\mathbf{nreg}}=〈\mathit{\text{value}}〉$. These parameters must satisfy ${\mathbf{tdx}}\ge {\mathbf{nreg}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
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: ${\mathbf{max_iter}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{nreg}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nreg}}\ge 0$.
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 0$.
On entry, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{q}}\ge 1$.
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 $〈\mathit{\text{value}}〉$ 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.

Not applicable.

None.

## 9  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).

### 9.1  Program Text

Program Text (g13fece.c)

None.

### 9.3  Program Results

Program Results (g13fece.r)