NAG Toolbox: nag_tsa_uni_garch_exp_estim (g13fg)

Purpose

Syntax

Description

References

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:     $\mathrm{theta}\left(\mathbf{npar}\right)$ – 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 autoregressive coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next iq elements contain the coefficients ${\varphi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next ip elements must contain the moving average coefficients ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$.

If $\mathbf{dist}=\text{'T'}$, the next element must contain an estimate for $\mathit{df}$, 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 $\mathbf{copts}=\mathit{false}$, the remaining nreg elements are taken as initial estimates of the linear regression coefficients $b_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.

9:     $\mathrm{hp}$ – double scalar

If $\mathbf{copts}=\mathit{false}$ then hp is the value to be used for the pre-observed conditional variance, otherwise hp is not referenced.

10:   $\mathrm{copts}$ – logical scalar

If $\mathbf{copts}=\mathit{true}$, the function provides initial parameter estimates of the regression terms, otherwise these are provided by you.

11:   $\mathrm{maxit}$ – 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.

Constraint: $\mathbf{maxit}>0$.

12:   $\mathrm{tol}$ – 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:

• $\mathbf{num}\ge \max \left(\mathbf{ip},\mathbf{iq}\right)$;
• $\mathbf{num}\ge \mathbf{nreg}+\mathbf{mn}$.

2:     $\mathrm{npar}$ – int64int32nag_int scalar

Default: the dimension of the array theta.

The number of parameters to be included in the model. $\mathbf{npar}=1+2\times \mathbf{iq}+\mathbf{ip}+\mathbf{mn}+\mathbf{nreg}$ when $\mathbf{dist}=\text{'N'}$ and $\mathbf{npar}=2+2\times \mathbf{iq}+\mathbf{ip}+\mathbf{mn}+\mathbf{nreg}$ when $\mathbf{dist}=\text{'T'}$.

Constraint: $\mathbf{npar}<20$.

Output Parameters

1:     $\mathrm{theta}\left(\mathbf{npar}\right)$ – double array

The estimated values $\hat{\theta }$ for the vector $\theta$.

The first element contains the coefficient ${\alpha }_o$ and the next iq elements contain the coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next iq elements contain the coefficients ${\varphi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next ip elements are the moving average coefficients ${\beta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,p$.

If $\mathbf{dist}=\text{'T'}$, the next element contains an estimate for $\mathit{df}$ then 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 $\hat{\theta }$.

The first element contains the standard error for ${\alpha }_o$ and the next iq elements contain the standard errors for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$. The next iq elements contain the standard errors for ${\varphi }_{\mathit{i}}$, 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 $\mathbf{dist}=\text{'T'}$, 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 $\hat{\theta }$.

The first element contains the scores for ${\alpha }_o$, the next iq elements contain the scores for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$, the next iq elements contain the scores for ${\varphi }_{\mathit{i}}$, 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 $\mathbf{dist}=\text{'T'}$, the next element contains the scores 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 $\hat{\theta }$, that is the inverse of the Fisher Information Matrix.

5:     $\mathrm{hp}$ – double scalar

If $\mathbf{copts}=\mathit{true}$ then 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 $\hat{\theta }$.

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

   $\mathbf{ifail}=1$

On entry,	$\mathbf{nreg}<0$,
or	$\mathbf{mn}>1$,
or	$\mathbf{mn}<0$,
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	$\mathbf{maxit}\le 0$,
or	$\mathbf{num}<\max \left(\mathbf{ip},\mathbf{iq}\right)$,
or	$\mathbf{num}<\mathbf{nreg}+\mathbf{mn}$.

   $\mathbf{ifail}=2$

On entry,

$\mathit{lwork}<\left(\mathbf{nreg}+3\right)\times \mathbf{num}+3$.

   $\mathbf{ifail}=3$

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.

   $\mathbf{ifail}=6$

The log-likelihood cannot be optimized any further.

   $\mathbf{ifail}=7$

No feasible model parameters could be found.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g13fg_example

function g13fg_example


fprintf('g13fg example results\n\n');

num = 100;
mn  = int64(1);
nreg = int64(2);

% The series
yt = [7.53; 6.64; 7.39; 7.15; 6.42; 6.32; 6.98; 7.09; 6.63; 6.93;
      7.01; 5.30; 7.86; 6.73; 7.39; 5.61; 7.02; 6.04; 7.46; 4.33;
      6.02; 6.37; 3.93; 7.24; 8.58; 5.70; 9.13; 7.99; 7.79; 6.13;
      8.78; 6.52; 6.79; 7.77; 7.31; 7.58; 8.78; 7.39; 8.00; 7.07;
      7.65; 9.15; 8.32; 7.32; 7.58; 9.78; 8.17; 9.26; 7.79; 7.03;
      7.45; 7.09; 8.06; 7.06; 9.91; 7.01; 8.32; 6.41; 8.59; 8.55;
      7.77; 8.04; 9.54; 8.28; 7.97; 8.42; 8.30; 7.98; 7.60; 8.77;
      7.54; 7.40; 9.26; 7.30; 9.33; 9.54; 8.08; 6.93; 4.27; 2.65;
      5.03; 0.91;12.63;10.87; 9.26; 8.30; 6.85; 7.48; 9.67; 9.54;
      7.33; 8.84; 7.75; 8.12; 7.29; 8.58; 7.80; 3.07; 9.33;16.91];

% The exogenous variables
x = [ 2.40, 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; 2.41, 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; 2.42, 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; 2.43, 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; 2.44, 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; 2.45, 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; 2.46, 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; 2.47, 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; 2.48, 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; 2.49, 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; 2.50, 0.64];

% Details of the model to fit
dist = 't';
ip = int64(1);
iq = int64(2);

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

% Initial parameter estimates
theta = [0.05; -0.15; -0.05; 0.05; 0.15; 0.35; 3.25; 1.5; 0; 0];

% Forecast horizon
nt = int64(4);

% Fit the GARCH model
[theta, se, sc, covar, hp, et, ht, lgf, ifail] = ...
  g13fg( ...
         dist, yt, x, ip, iq, mn, theta, 0, ...
         copts, maxit, tol, 'ifail', int64(-1));

% Calculate the volatility forecast
[fht, ifail] = g13fh( ...
                      nt, ip, iq, theta, 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 psi_i
for i = l:l+iq-1
  fprintf('  Psi%d %16.2f%16.2f\n', i-l+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 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));

g13fg example results


               Parameter        Standard
               estimates         errors
Alpha0             0.17            0.19
Alpha1            -0.65            0.23
Alpha2            -0.44            0.24
  Psi1            -0.06            0.22
  Psi2             0.35            0.25

 Beta1             0.42            0.17

    DF             5.59            3.75

    B0           128.75           42.09
    B1           -51.74           17.78
    B2            13.01            3.40

Volatility forecast =       1.3404