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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_uni_garch_asym1_estim (g13fa)

## Purpose

nag_tsa_uni_garch_asym1_estim (g13fa) estimates the parameters of either a standard univariate regression GARCH process, or a univariate regression-type I $\text{AGARCH}\left(p,q\right)$ process (see Engle and Ng (1993)).

## Syntax

[theta, se, sc, covr, hp, et, ht, lgf, ifail] = g13fa(dist, yt, x, ip, iq, nreg, mn, isym, theta, hp, copts, maxit, tol, 'num', num, 'npar', npar)
[theta, se, sc, covr, hp, et, ht, lgf, ifail] = nag_tsa_uni_garch_asym1_estim(dist, yt, x, ip, iq, nreg, mn, isym, theta, hp, copts, maxit, tol, 'num', num, 'npar', npar)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 25: nreg was made optional

## Description

A univariate regression-type I $\text{AGARCH}\left(p,q\right)$ process, with $q$ coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$, $p$ coefficients ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$, and $k$ linear regression coefficients ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$, can be represented by:
 $yt = bo + xtT b + εt$ (1)
 $ht=α0+∑i=1qαi εt-i+γ 2+∑i=1pβiht-i, t=1,2,…,T$ (2)
where ${\epsilon }_{t}\mid {\psi }_{t-1}=N\left(0,{h}_{t}\right)$ or ${\epsilon }_{t}\mid {\psi }_{t-1}={S}_{t}\left(\mathit{df},{h}_{t}\right)$. Here ${S}_{t}$ is a standardized Student's $t$-distribution with $\mathit{df}$ degrees of freedom and variance ${h}_{t}$, $T$ is the number of terms in the sequence, ${y}_{t}$ denotes the endogenous variables, ${x}_{t}$ the exogenous variables, ${b}_{o}$ the regression mean, $b$ the regression coefficients, ${\epsilon }_{t}$ the residuals, ${h}_{t}$ the conditional variance, $\mathit{df}$ the number of degrees of freedom of the Student's $t$-distribution, and ${\psi }_{t}$ the set of all information up to time $t$.
nag_tsa_uni_garch_asym1_estim (g13fa) provides an estimate for $\stackrel{^}{\theta }$, the parameter vector $\theta =\left({b}_{o},{b}^{\mathrm{T}},{\omega }^{\mathrm{T}}\right)$ where ${b}^{\mathrm{T}}=\left({b}_{1},\dots ,{b}_{k}\right)$, ${\omega }^{\mathrm{T}}=\left({\alpha }_{0},{\alpha }_{1},\dots ,{\alpha }_{q},{\beta }_{1},\dots ,{\beta }_{p},\gamma \right)$ when ${\mathbf{dist}}=\text{'N'}$ and ${\omega }^{\mathrm{T}}=\left({\alpha }_{0},{\alpha }_{1},\dots ,{\alpha }_{q},{\beta }_{1},\dots ,{\beta }_{p},\gamma ,\mathit{df}\right)$ when ${\mathbf{dist}}=\text{'T'}$.
isym, mn and nreg can be used to simplify the $\text{GARCH}\left(p,q\right)$ expression in (1) as follows:
No Regression and No Mean
• ${y}_{t}={\epsilon }_{t}$,
• ${\mathbf{isym}}=0$,
• ${\mathbf{mn}}=0$,
• ${\mathbf{nreg}}=0$ and
• $\theta$ is a $\left(p+q+1\right)$ vector when ${\mathbf{dist}}=\text{'N'}$ and a $\left(p+q+2\right)$ vector when ${\mathbf{dist}}=\text{'T'}$.
No Regression
• ${y}_{t}={b}_{o}+{\epsilon }_{t}$,
• ${\mathbf{isym}}=0$,
• ${\mathbf{mn}}=1$,
• ${\mathbf{nreg}}=0$ and
• $\theta$ is a $\left(p+q+2\right)$ vector when ${\mathbf{dist}}=\text{'N'}$ and a $\left(p+q+3\right)$ vector when ${\mathbf{dist}}=\text{'T'}$.
Note:  if the ${y}_{t}=\mu +{\epsilon }_{t}$, where $\mu$ is known (not to be estimated by nag_tsa_uni_garch_asym1_estim (g13fa)) then (1) can be written as ${y}_{t}^{\mu }={\epsilon }_{t}$, where ${y}_{t}^{\mu }={y}_{t}-\mu$. This corresponds to the case No Regression and No Mean, with ${y}_{t}$ replaced by ${y}_{t}-\mu$.
No Mean
• ${y}_{t}={x}_{t}^{\mathrm{T}}b+{\epsilon }_{t}$,
• ${\mathbf{isym}}=0$,
• ${\mathbf{mn}}=0$,
• ${\mathbf{nreg}}=k$ and
• $\theta$ is a $\left(p+q+k+1\right)$ vector when ${\mathbf{dist}}=\text{'N'}$ and a $\left(p+q+k+2\right)$ vector when ${\mathbf{dist}}=\text{'T'}$.

## 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
Hamilton J (1994) Time Series Analysis Princeton University Press

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{dist}$ – string (length ≥ 1)
The type of distribution to use for ${e}_{t}$.
${\mathbf{dist}}=\text{'N'}$
A Normal distribution is used.
${\mathbf{dist}}=\text{'T'}$
A Student's $t$-distribution is used.
Constraint: ${\mathbf{dist}}=\text{'N'}$ or $\text{'T'}$.
2:     $\mathrm{yt}\left({\mathbf{num}}\right)$ – double array
The sequence of observations, ${y}_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.
3:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x must be at least ${\mathbf{num}}$.
The second dimension of the array x must be at least ${\mathbf{nreg}}$.
Row $\mathit{t}$ of x must contain the time dependent exogenous vector ${x}_{\mathit{t}}$, 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:     $\mathrm{ip}$int64int32nag_int scalar
The number of coefficients, ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$.
Constraint: ${\mathbf{ip}}\ge 0$ (see also npar).
5:     $\mathrm{iq}$int64int32nag_int scalar
The number of coefficients, ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
Constraint: ${\mathbf{iq}}\ge 1$ (see also npar).
6:     $\mathrm{nreg}$int64int32nag_int scalar
$k$, the number of regression coefficients.
Constraint: ${\mathbf{nreg}}\ge 0$ (see also npar).
7:     $\mathrm{mn}$int64int32nag_int scalar
If ${\mathbf{mn}}=1$, the mean term ${b}_{0}$ will be included in the model.
Constraint: ${\mathbf{mn}}=0$ or $1$.
8:     $\mathrm{isym}$int64int32nag_int scalar
If ${\mathbf{isym}}=1$, the asymmetry term $\gamma$ will be included in the model.
Constraint: ${\mathbf{isym}}=0$ or $1$.
9:     $\mathrm{theta}\left({\mathbf{npar}}\right)$ – double array
The initial parameter estimates for the vector $\theta$.
The first element must contain the coefficient ${\alpha }_{o}$ and the next iq elements must contain the coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
The next ip elements must contain the coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
If ${\mathbf{isym}}=1$, the next element must contain the asymmetry parameter $\gamma$.
If ${\mathbf{dist}}=\text{'T'}$, the next element must contain $\mathit{df}$, the number of degrees of freedom of the Student's $t$-distribution.
If ${\mathbf{mn}}=1$, the next element must contain the mean term ${b}_{o}$.
If ${\mathbf{copts}}\left(2\right)=\mathit{false}$, the remaining nreg elements are taken as initial estimates of the linear regression coefficients ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.
10:   $\mathrm{hp}$ – double scalar
If ${\mathbf{copts}}\left(2\right)=\mathit{false}$, hp is the value to be used for the pre-observed conditional variance; otherwise hp is not referenced.
11:   $\mathrm{copts}\left(2\right)$ – logical array
The options to be used by nag_tsa_uni_garch_asym1_estim (g13fa).
${\mathbf{copts}}\left(1\right)=\mathit{true}$
Stationary conditions are enforced, otherwise they are not.
${\mathbf{copts}}\left(2\right)=\mathit{true}$
The function provides initial parameter estimates of the regression terms, otherwise these are to be provided by you.
12:   $\mathrm{maxit}$int64int32nag_int scalar
The maximum number of iterations to be used by the optimization function when estimating the $\text{GARCH}\left(p,q\right)$ parameters. If maxit is set to $0$, the standard errors, score vector and variance-covariance are calculated for the input value of $\theta$ in theta when ${\mathbf{dist}}=\text{'N'}$; however the value of $\theta$ is not updated.
Constraint: ${\mathbf{maxit}}\ge 0$.
13:   $\mathrm{tol}$ – double scalar
The tolerance to be used by the optimization function when estimating the $\text{GARCH}\left(p,q\right)$ parameters.

### Optional Input Parameters

1:     $\mathrm{num}$int64int32nag_int scalar
Default: the dimension of the array yt and the first dimension of the array x. (An error is raised if these dimensions are not equal.)
$T$, the number of terms in the sequence.
Constraints:
• ${\mathbf{num}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)$;
• ${\mathbf{num}}\ge {\mathbf{nreg}}+{\mathbf{mn}}$.
2:     $\mathrm{npar}$int64int32nag_int scalar
Default: the dimension of the array theta.
The number of parameters to be included in the model. ${\mathbf{npar}}=1+{\mathbf{iq}}+{\mathbf{ip}}+{\mathbf{isym}}+{\mathbf{mn}}+{\mathbf{nreg}}$ when ${\mathbf{dist}}=\text{'N'}$, and ${\mathbf{npar}}=2+{\mathbf{iq}}+{\mathbf{ip}}+{\mathbf{isym}}+{\mathbf{mn}}+{\mathbf{nreg}}$ when ${\mathbf{dist}}=\text{'T'}$.
Constraint: ${\mathbf{npar}}<20$.

### Output Parameters

1:     $\mathrm{theta}\left({\mathbf{npar}}\right)$ – double array
The estimated values $\stackrel{^}{\theta }$ for the vector $\theta$.
The first element contains the coefficient ${\alpha }_{o}$, the next iq elements contain the coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
The next ip elements are the coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
If ${\mathbf{isym}}=1$, the next element contains the estimate for the asymmetry parameter $\gamma$.
If ${\mathbf{dist}}=\text{'T'}$, the next element contains an estimate for $\mathit{df}$, the number of degrees of freedom of the Student's $t$-distribution.
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 $\mathit{i}=1,2,\dots ,k$.
2:     $\mathrm{se}\left({\mathbf{npar}}\right)$ – double array
The standard errors for $\stackrel{^}{\theta }$.
The first element contains the standard error for ${\alpha }_{o}$. The next iq elements contain the standard errors for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$. The next ip elements are the standard errors for ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
If ${\mathbf{isym}}=1$, the next element contains the standard error for $\gamma$.
If ${\mathbf{dist}}=\text{'T'}$, the next element contains the standard error for $\mathit{df}$, the number of degrees of freedom of the Student's $t$-distribution.
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 $\mathit{j}=1,2,\dots ,k$.
3:     $\mathrm{sc}\left({\mathbf{npar}}\right)$ – double array
The scores for $\stackrel{^}{\theta }$.
The first element contains the score for ${\alpha }_{o}$.
The next iq elements contain the score for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.
The next ip elements are the scores for ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.
If ${\mathbf{isym}}=1$, the next element contains the score for $\gamma$.
If ${\mathbf{dist}}=\text{'T'}$, the next element contains the score for $\mathit{df}$, the number of degrees of freedom of the Student's $t$-distribution.
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 $\mathit{j}=1,2,\dots ,k$.
4:     $\mathrm{covr}\left(\mathit{ldcovr},{\mathbf{npar}}\right)$ – double array
The covariance matrix of the parameter estimates $\stackrel{^}{\theta }$, that is the inverse of the Fisher Information Matrix.
5:     $\mathrm{hp}$ – double scalar
If ${\mathbf{copts}}\left(2\right)=\mathit{true}$, hp is the estimated value of the pre-observed conditional variance.
6:     $\mathrm{et}\left({\mathbf{num}}\right)$ – double array
The estimated residuals, ${\epsilon }_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.
7:     $\mathrm{ht}\left({\mathbf{num}}\right)$ – double array
The estimated conditional variances, ${h}_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.
8:     $\mathrm{lgf}$ – double scalar
The value of the log-likelihood function at $\stackrel{^}{\theta }$.
9:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_tsa_uni_garch_asym1_estim (g13fa) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{nreg}}<0$, or ${\mathbf{mn}}>1$, or ${\mathbf{mn}}<0$, or ${\mathbf{isym}}>1$, or ${\mathbf{isym}}<0$, or ${\mathbf{iq}}<1$, or ${\mathbf{ip}}<0$, or ${\mathbf{npar}}\ge 20$, or npar has an invalid value, or $\mathit{ldcovr}<{\mathbf{npar}}$, or $\mathit{ldx}<{\mathbf{num}}$, or ${\mathbf{dist}}\ne \text{'N'}$, or ${\mathbf{dist}}\ne \text{'T'}$, or ${\mathbf{maxit}}<0$, or ${\mathbf{num}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)$, or ${\mathbf{num}}<{\mathbf{nreg}}+{\mathbf{mn}}$.
${\mathbf{ifail}}=2$
 On entry, $\mathit{lwork}<\left({\mathbf{nreg}}+3\right)×{\mathbf{num}}+{\mathbf{npar}}+403$.
${\mathbf{ifail}}=3$
The matrix $X$ is not full rank.
${\mathbf{ifail}}=4$
The information matrix is not positive definite.
${\mathbf{ifail}}=5$
The maximum number of iterations has been reached.
W  ${\mathbf{ifail}}=6$
The log-likelihood cannot be optimized any further.
${\mathbf{ifail}}=7$
No feasible model parameters could be found.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Not applicable.

None.

## Example

This example fits a $\text{GARCH}\left(1,1\right)$ model with Student's $t$-distributed residuals to some simulated data.
The process parameter estimates, $\stackrel{^}{\theta }$, are obtained using nag_tsa_uni_garch_asym1_estim (g13fa), and a four step ahead volatility estimate is computed using nag_tsa_uni_garch_asym1_forecast (g13fb).
The data was simulated using nag_rand_times_garch_asym1 (g05pd).
```function g13fa_example

fprintf('g13fa example results\n\n');

num  = 100;

% Series
yt = [ 9.04;  9.49;  9.12;  9.23;  9.35;
9.09;  9.75;  9.23;  8.76;  9.17;
9.20;  9.64;  8.74;  9.23;  9.42;
9.70;  9.55; 10.00;  9.18;  9.77;
9.80;  9.56;  9.28;  9.68;  9.51;
9.51;  8.97;  9.30;  9.52;  9.41;
9.53;  9.75;  9.72;  9.38;  9.28;
9.42;  9.74;  9.75;  9.60;  9.90;
9.06;  9.92;  9.21;  9.57;  9.42;
8.65;  8.85;  9.61; 10.77; 10.19;
10.47; 10.10; 10.21;  9.96;  9.66;
9.79; 10.30;  9.68; 10.08; 10.38;
9.69;  9.02;  9.89; 10.46; 10.47;
9.99;  9.76;  9.78;  9.62; 10.43;
10.42;  9.95;  9.95;  9.70; 10.24;
9.78;  9.98;  8.73; 10.23;  9.10;
10.27;  9.85; 10.44; 10.30; 10.08;
10.20; 10.14;  9.89;  9.90; 11.33;
9.71;  9.40;  9.97; 10.92;  9.76;
10.16; 10.43;  9.60; 10.29; 10.03];

% Exogenous variables
x = [0.12, 2.40; 0.12, 2.40; 0.13, 2.40; 0.14, 2.40; 0.14, 2.40;
0.15, 2.40; 0.16, 2.40; 0.16, 2.40; 0.17, 2.40; 0.18, 2.41;
0.19, 2.41; 0.19, 2.41; 0.20, 2.41; 0.21, 2.41; 0.21, 2.41;
0.22, 2.41; 0.23, 2.41; 0.23, 2.41; 0.24, 2.41; 0.25, 2.42;
0.25, 2.42; 0.26, 2.42; 0.26, 2.42; 0.27, 2.42; 0.28, 2.42;
0.28, 2.42; 0.29, 2.42; 0.30, 2.42; 0.30, 2.42; 0.31, 2.43;
0.32, 2.43; 0.32, 2.43; 0.33, 2.43; 0.33, 2.43; 0.34, 2.43;
0.35, 2.43; 0.35, 2.43; 0.36, 2.43; 0.37, 2.43; 0.37, 2.44;
0.38, 2.44; 0.38, 2.44; 0.39, 2.44; 0.39, 2.44; 0.40, 2.44;
0.41, 2.44; 0.41, 2.44; 0.42, 2.44; 0.42, 2.44; 0.43, 2.45;
0.43, 2.45; 0.44, 2.45; 0.45, 2.45; 0.45, 2.45; 0.46, 2.45;
0.46, 2.45; 0.47, 2.45; 0.47, 2.45; 0.48, 2.45; 0.48, 2.46;
0.49, 2.46; 0.49, 2.46; 0.50, 2.46; 0.50, 2.46; 0.51, 2.46;
0.51, 2.46; 0.52, 2.46; 0.52, 2.46; 0.53, 2.46; 0.53, 2.47;
0.54, 2.47; 0.54, 2.47; 0.54, 2.47; 0.55, 2.47; 0.55, 2.47;
0.56, 2.47; 0.56, 2.47; 0.57, 2.47; 0.57, 2.47; 0.57, 2.48;
0.58, 2.48; 0.58, 2.48; 0.59, 2.48; 0.59, 2.48; 0.59, 2.48;
0.60, 2.48; 0.60, 2.48; 0.61, 2.48; 0.61, 2.48; 0.61, 2.49;
0.62, 2.49; 0.62, 2.49; 0.62, 2.49; 0.63, 2.49; 0.63, 2.49;
0.63, 2.49; 0.64, 2.49; 0.64, 2.49; 0.64, 2.49; 0.64, 2.50];

% Details of model to fit
dist = 't';
n1 = int64(1);
ip = n1;
iq = n1;
isym = n1;
mn   = n1;
nreg = 2*n1;

% Control parameters
copts = [true; true];
maxit = int64(200);
tol = 0.00001;

% Initial values
gammaval = -0.1;
theta = [0.05; 0.1; 0.15; gammaval; 2.6; 1.5; 0; 0];

% Forecast horizon
nt = 4*n1;

% Fit the GARCH model
[theta, se, sc, covar, hp, et, ht, lgf, ifail] = ...
g13fa( ...
dist, yt, x, ip, iq, mn, isym, theta, 0, copts, maxit, tol);

% Calculate the volatility forecast
[fht, ifail] = g13fb( ...
nt, ip, iq, theta, gammaval, ht, et);

% Output the results
fprintf('\n               Parameter        Standard\n');
fprintf('               estimates         errors\n');

% Output the coefficient alpha_0
fprintf('Alpha0 %16.2f%16.2f\n', theta(1), se(1));
l = 2;

% Output the coefficients alpha_i
for i = l:l+iq-1
fprintf('Alpha%d %16.2f%16.2f\n', i-1, theta(i), se(i));
end
l = l+iq;

% Output the coefficients beta_j
fprintf('\n');
for i = l:l+ip-1
fprintf(' Beta%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end
l = l+ip;

% Output the estimated asymmetry parameter, gamma
if (isym == 1)
fprintf('\n Gamma %16.2f%16.2f\n', theta(l), se(l));
l = l+1;
end
% Output the estimated degrees of freedom, df
if (dist == 't')
fprintf('\n    DF %16.2f%16.2f\n', theta(l), se(l));
l = l + 1;
end

% Output the estimated mean term, b_0
if (mn == 1)
fprintf('\n    B0 %16.2f%16.2f\n', theta(l), se(l));
l = l + 1;
end

% Output the estimated linear regression coefficients, b_i
for i = l:l+nreg-1
fprintf('    B%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end

% Display the volatility forecast
fprintf('\nVolatility forecast = %12.4f\n', fht(nt));

```
```g13fa example results

Parameter        Standard
estimates         errors
Alpha0             0.00            0.06
Alpha1             0.11            0.13

Beta1             0.66            0.23

Gamma            -0.62            0.62

DF             6.25            4.70

B0             3.85           24.11
B1             1.48            1.82
B2             2.15           10.16

Volatility forecast =       0.0626
```