NAG Toolbox: nag_tsa_uni_garch_asym2_estim (g13fc)

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 coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

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

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 contains 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 $\mathit{i}=1,2,\dots ,k$.

9:     $\mathrm{hp}$ – 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.

10:   $\mathrm{copts}\left(2\right)$ – logical array

The options to be used by nag_tsa_uni_garch_asym2_estim (g13fc).

$\mathbf{copts}\left(1\right)=\mathit{true}$

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.

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. 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: $\mathbf{maxit}\ge 0$.

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. $\mathbf{npar}=2+\mathbf{iq}+\mathbf{ip}+\mathbf{mn}+\mathbf{nreg}$ when $\mathbf{dist}=\text{'N'}$ and $\mathbf{npar}=3+\mathbf{iq}+\mathbf{ip}+\mathbf{mn}+\mathbf{nreg}$ 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 ${\alpha }_o$, 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 $\mathit{j}=1,2,\dots ,p$.

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 $\hat{\theta }$.

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 ip 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 $\mathbf{dist}=\text{'T'}$, 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 score for ${\alpha }_o$ and 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 $\mathit{j}=1,2,\dots ,p$.

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 $\hat{\theta }$, 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 $\hat{\theta }$.

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	$\mathit{ldcovr}<\mathbf{npar}$,
or	$\mathit{ldx}<\mathbf{num}$,
or	$\mathbf{dist}\ne \text{'N'}$, and $\mathbf{dist}\ne \text{'T'}$,
or	$\mathbf{maxit}<0$,
or	$\mathbf{num}<\mathbf{nreg}+\mathbf{mn}$,
or	npar has an invalid value $\mathbf{num}<\max \left(\mathbf{ip},\mathbf{iq}\right)$.

   $\mathbf{ifail}=2$

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: g13fc_example

function g13fc_example


fprintf('g13fc example results\n\n');

mn  = int64(1);
nreg = int64(2);
yt = [ 8.87;  9.82;  9.02;  9.24;  9.46;
       8.93; 10.20;  9.19;  8.27;  9.08;
       9.11;  9.95;  8.11;  9.13;  9.49;
      10.08;  9.74; 10.72;  8.94; 10.10;
      10.19;  9.68;  9.09;  9.88;  9.55;
       9.52;  8.45;  9.14;  9.52;  9.27;
       9.50;  9.93;  9.86;  9.16;  9.00;
       9.28;  9.83;  9.86;  9.55; 10.12;
       8.47; 10.10;  8.70;  9.44;  9.10;
       7.54;  8.08;  9.47; 12.32; 10.75;
      11.66; 10.59; 10.93; 10.21;  9.39;
       9.74; 10.91;  9.46; 10.32; 11.00;
       9.47;  8.14;  9.88; 11.15; 11.21;
      10.06;  9.50;  9.56;  9.23; 10.88;
      10.93;  9.89;  9.89;  9.37; 10.44;
       9.52;  9.92;  7.44; 10.36;  7.73;
      10.53;  9.38; 11.14; 10.73; 10.02;
      10.36; 10.18;  9.52;  9.59; 12.73;
       9.38;  8.69;  9.78; 11.85;  9.23;
      10.13; 10.77;  8.68; 10.39;  9.74];
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];
dist = 't';
ip = int64(1);
iq = int64(1);
copts = [true; true];
maxit = int64(200);
tol = 0.00001;
hp = 0;

% Theta is [alpha_0; alpha_1; beta_1; gamma; df; b_0]
theta = [ 0.05; 0.05; 0.4; -0.2; 2.6; 1.5; 0; 0];
nt = int64(4);
% Fit the GARCH model
[theta, se, sc, covar, hp, et, ht, lgf, ifail] = ...
  g13fc( ...
         dist, yt, x, ip, iq, mn, theta, hp, copts, maxit, tol);

% Extract the estimate of the asymmetry parameter from theta
gamma = theta(4);

% Calculate the volatility forecast
[fht, ifail] = g13fd( ...
                      nt, ip, iq, theta, gamma, 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
fprintf('\n Gamma %16.2f%16.2f\n', theta(l), se(l));
l = l+1;

% 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 = %12.2f\n', fht(nt));

g13fc example results


               Parameter        Standard
               estimates         errors
Alpha0             6.82            1.68
Alpha1             0.00            1.00

 Beta1             0.00            3.17

 Gamma            -0.36            1.01

    DF             2.10            0.33

    B0           -25.14            4.80
    B1            -0.95            0.90
    B2            14.41            2.08

Volatility forecast =         6.82