Simulate univariate GARCH(P,Q) process with Gaussian innovations
[U, H] = ugarchsim(Kappa, Alpha, Beta, NumSamples)
 Scalar constant term [[KAPPA]] of the GARCH process. 




 Positive, scalar integer indicating the number of samples
of the innovations 
[U, H] = ugarchsim(Kappa, Alpha, Beta, NumSamples)
simulates
a univariate GARCH(P,Q) process with Gaussian innovations.
U
is a number of samples (NUMSAMPLES
)by1
vector
of innovations (ɛ_{t}),
representing a meanzero, discretetime stochastic process. The innovations
time series U
is designed to follow the GARCH(P,Q)
process specified by the inputs Kappa
, Alpha
,
and Beta
.
H
is a NUMSAMPLES
by1
vector of the conditional variances (
) corresponding to the innovations
vector U
. Note that U
and H
are
the same length, and form a "matching" pair of vectors.
As shown in the following equation, $${\sigma}_{t}^{2}$$ (that
is, H(t)
) represents the time series inferred from
the innovations time series {ɛ_{t}}
(that is, U
).
The timeconditional variance, $${\sigma}_{t}^{2}$$, of a GARCH(P,Q) process is modeled as
$${\sigma}_{t}^{2}=K+{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}{\sigma}_{ti}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}{\epsilon}_{tj}^{2}},$$
where α represents the argument Alpha
,
β represents Beta
, and the GARCH(P,Q) coefficients
{Κ, α, β}
are subject to the following constraints.
$$\begin{array}{l}{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}}<1\\ K>0\\ \begin{array}{cc}{\alpha}_{i}\ge 0& i=1,2,\dots ,P\\ {\beta}_{j}\ge 0& j=1,2,\dots ,Q.\end{array}\end{array}$$
Note that U
is a vector of residuals or innovations
(ɛ_{t})
of an econometric model, representing a meanzero, discretetime stochastic
process.
Although $${\sigma}_{t}^{2}$$ is generated using the equation above, ɛ_{t} and $${\sigma}_{t}^{2}$$ are related as
$${\epsilon}_{t}={\sigma}_{t}{\upsilon}_{t},$$
where $$\left\{{\upsilon}_{t}\right\}$$ is an independent, identically distributed (iid) sequence ~ N(0,1).
The output vectors U
and H
are
designed to be steadystate sequences in which transients have arbitrarily
small effect. The (arbitrary) metric used by ugarchsim
strips
the first N
samples of U
and H
such
that the sum of the GARCH coefficients, excluding Kappa
,
raised to the N
th power, does not exceed 0.01.
0.01 = (sum(Alpha) + sum(Beta))^N
Thus
N = log(0.01)/log((sum(Alpha) + sum(Beta)))
Note
The Econometrics Toolbox™ software provides a comprehensive
and integrated computing environment for the analysis of volatility
in time series. For information see the Econometrics Toolbox documentation
or the financial products Web page at 
This example simulates a GARCH(P,Q) process with P
= 2
and Q = 1
.
% Set the random number generator seed for reproducability. rng('default') % Set the simulation parameters of GARCH(P,Q) = GARCH(2,1) process. Kappa = 0.25; %a positive scalar. Alpha = [0.2 0.1]'; %a column vector of nonnegative numbers (P = 2). Beta = 0.4; % Q = 1. NumSamples = 500; % number of samples to simulate. % Now simulate the process. [U , H] = ugarchsim(Kappa, Alpha, Beta, NumSamples); % Estimate the process parameters. P = 2; % Model order P (P = length of Alpha). Q = 1; % Model order Q (Q = length of Beta). [k, a, b] = ugarch(U , P , Q); disp(' ') disp(' Estimated Coefficients:') disp(' ') disp([k; a; b]) disp(' ') % Forecast the conditional variance using the estimated % coefficients. NumPeriods = 10; % Forecast out to 10 periods. [VarianceForecast, H1] = ugarchpred(U, k, a, b, NumPeriods); disp(' Variance Forecasts:') disp(' ') disp(VarianceForecast) disp(' ')
When the above code is executed, the screen output looks like the display shown.
____________________________________________________________ Diagnostic Information Number of variables: 4 Functions Objective: ugarchllf Gradient: finitedifferencing Hessian: finitedifferencing (or QuasiNewton) Constraints Nonlinear constraints: do not exist Number of linear inequality constraints: 1 Number of linear equality constraints: 0 Number of lower bound constraints: 4 Number of upper bound constraints: 0 Algorithm selected mediumscale: SQP, QuasiNewton, linesearch ____________________________________________________________ End diagnostic information Max Line search Directional Firstorder Iter Fcount f(x) constraint steplength derivative optimality Procedure 0 5 625.735 0.05 1 12 611.683 0.05614 0.25 146 411 2 18 609.396 0.0375 0.5 98.1 94.8 3 24 600.17 0.07969 0.5 61.9 56.1 4 31 598.818 0.05977 0.25 24 27.9 5 38 598.308 0.04482 0.25 12.4 42.9 6 43 598.247 0 1 15.7 23.9 7 48 598.02 0 1 22.3 3.59 8 57 598.01 0.02759 0.0625 2.03 3.89 9 62 597.999 0.01749 1 1.48 0.249 10 67 597.999 0.01638 1 0.15 0.0365 11 72 597.999 0.0164 1 0.0269 0.00774 Hessian modified Local minimum possible. Constraints satisfied. fmincon stopped because the predicted change in the objective function is less than the default value of the function tolerance and constraints are satisfied to within the default value of the constraint tolerance. No active inequalities. Estimated Coefficients:  0.2868 0.1984 0.0164 0.4076 Variance Forecasts:  0.4130 0.5453 0.6240 0.6740 0.7055 0.7254 0.7380 0.7460 0.7510 0.7542
James D. Hamilton, Time Series Analysis, Princeton University Press, 1994