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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_times_garch_exp (g05pg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_rand_times_garch_exp (g05pg) generates a given number of terms of an exponential GARCHp,q process (see Engle and Ng (1993)).

Syntax

[ht, et, r, state, ifail] = g05pg(dist, num, ip, iq, theta, df, fcall, r, state, 'lr', lr)
[ht, et, r, state, ifail] = nag_rand_times_garch_exp(dist, num, ip, iq, theta, df, fcall, r, state, 'lr', lr)

Description

An exponential GARCHp,q process is represented by:
lnht=α0+i=1qαizt-i+i=1qϕizt-i-Ezt-i+j=1pβjlnht-j,  t=1,2,,T;  
where zt= εtht , Ezt-i denotes the expected value of zt-i, and εtψt-1=N0,ht or εtψt-1=Stdf,ht. Here St is a standardized Student's t-distribution with df degrees of freedom and variance ht, T is the number of observations in the sequence, εt is the observed value of the GARCHp,q process at time t, ht is the conditional variance at time t, and ψt the set of all information up to time t.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_times_garch_exp (g05pg).

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

Parameters

Compulsory Input Parameters

1:     dist – string (length ≥ 1)
The type of distribution to use for εt.
dist='N'
A Normal distribution is used.
dist='T'
A Student's t-distribution is used.
Constraint: dist='N' or 'T'.
2:     num int64int32nag_int scalar
T, the number of terms in the sequence.
Constraint: num0.
3:     ip int64int32nag_int scalar
The number of coefficients, βi, for i=1,2,,p.
Constraint: ip0.
4:     iq int64int32nag_int scalar
The number of coefficients, αi, for i=1,2,,q.
Constraint: iq1.
5:     theta2×iq+ip+1 – double array
The initial parameter estimates for the vector θ. The first element must contain the coefficient αo and the next iq elements must contain the autoregressive coefficients αi, for i=1,2,,q. The next iq elements must contain the coefficients ϕi, for i=1,2,,q. The next ip elements must contain the moving average coefficients βj, for j=1,2,,p.
Constraints:
  • i=1 p βi1.0;
  • α0 1- i=1 p βi - logx02am .
6:     df int64int32nag_int scalar
The number of degrees of freedom for the Student's t-distribution.
If dist='N', df is not referenced.
Constraint: if dist='T', df>2.
7:     fcall – logical scalar
If fcall=true, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in r.
8:     rlr – double array
The array contains information required to continue a sequence if fcall=false.
9:     state: int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

Optional Input Parameters

1:     lr int64int32nag_int scalar
Default: the dimension of the array r.
The dimension of the array r.
Constraint: lr2×ip+2×iq+2.

Output Parameters

1:     htnum – double array
The conditional variances ht, for t=1,2,,T, for the GARCHp,q sequence.
2:     etnum – double array
The observations εt, for t=1,2,,T, for the GARCHp,q sequence.
3:     rlr – double array
Contains information that can be used in a subsequent call of nag_rand_times_garch_exp (g05pg), with fcall=false.
4:     state: int64int32nag_int array
Contains updated information on the state of the generator.
5:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry, dist is not valid.
   ifail=2
Constraint: num0.
   ifail=3
Constraint: ip0.
   ifail=4
Constraint: iq1.
   ifail=6
Constraint: df3.
   ifail=10
ip or iq is not the same as when r was set up in a previous call.
   ifail=11
On entry, lr is not large enough, lr=_: minimum length required .
   ifail=12
On entry, state vector has been corrupted or not initialized.
   ifail=20
Invalid sequence generated, use different parameters.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

None.

Example

This example first calls nag_rand_init_repeat (g05kf) to initialize a base generator then calls nag_rand_times_garch_exp (g05pg) to generate two realizations, each consisting of ten observations, from an exponential GARCH1,1 model.
function g05pg_example


fprintf('g05pg example results\n\n');

% Initialize the generator to a repeatable sequence
seed = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

% Input parameters
dist  = 'N';
num   = int64(10);
ip    = int64(1);
iq    = int64(1);
theta = [0.1; -0.3; 0.1; 0.9];
df    = int64(0);
fcall = true;
r     = zeros(2*(2*ip+iq+2),1);

% Generate the first realisation
[ht, et, r, state, ifail] = g05pg( ...
                                   dist, num, ip, iq, ...
                                   theta, df, fcall, r, state);
% Display the results
fprintf('\n Realisation Number 1\n');
fprintf('   i            ht(i)            et(i)\n');
fprintf('  --------------------------------------\n');
for i=1:num
  fprintf('  %2d  %16.4f %16.4f\n', i, ht(i), et(i));
end

% Generate a second realisation
fcall = false;
[ht, et, r, state, ifail] = g05pg( ...
                                   dist, num, ip, iq, ...
                                   theta, df, fcall, r, state);
% Display the results
fprintf('\n Realisation Number 2\n');
fprintf('   i            ht(i)            et(i)\n');
fprintf('  --------------------------------------\n');
for i=1:num
  fprintf('  %2d  %16.4f %16.4f\n', i, ht(i), et(i));
end


g05pg example results


 Realisation Number 1
   i            ht(i)            et(i)
  --------------------------------------
   1            2.5098           0.5526
   2            2.1785          -1.8383
   3            3.3844           1.2180
   4            2.6780           1.3672
   5            2.0953          -1.8178
   6            3.2813          -0.0343
   7            2.9958          -0.5094
   8            3.0815           1.3978
   9            2.3961          -0.0070
  10            2.2445           0.6661

 Realisation Number 2
   i            ht(i)            et(i)
  --------------------------------------
   1            1.9327          -2.2795
   2            3.5577          -1.2249
   3            4.1461           0.6424
   4            3.4455          -2.9920
   5            5.9199           0.5777
   6            4.8221          -1.2894
   7            5.3174          -1.6473
   8            6.1095           6.1689
   9            3.1579           2.2935
  10            2.2189           0.1141

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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