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Simulate a univariate GARCH(P,Q) process with Gaussian innovations
Syntax
Arguments
Description
[U, H] = ugarchsim(Kappa, Alpha, Beta, NumSamples)
simulates a univariate GARCH(P,Q) process with Gaussian innovations.
U
is a number of samples (NUMSAMPLES
)-by-1 vector of innovations (t), representing a mean-zero, discrete-time 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
-by-1 vector of the conditional variances (t2) 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, t2 (i.e.,
H(t)
) represents the time series inferred from the innovations time series {t} (i.e.,
U)
.
The time-conditional variance, t2, of a GARCH(P,Q) process is modeled as
where represents the argument
Alpha
, represents
Beta
, and the GARCH(P,Q) coefficients {,
,
} are subject to the following constraints.
Note that U
is a vector of residuals or innovations (t) of an econometric model, representing a mean-zero, discrete-time stochastic process.
Although t2 is generated using the equation above,
t and
t2 are related as
where {vt} is an independent, identically distributed (i.i.d.) sequence ~ N(0,1).
The output vectors U
and H
are designed to be steady-state 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.
Note
ugarchsim corresponds generally to the GARCH Toolbox function garchsim . The GARCH Toolbox provides a comprehensive and integrated computing environment for the analysis of volatility in time series. For information see the GARCH Toolbox User's Guide or the financial products Web page at http://www.mathworks.com/products/finprod/.
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Examples
This example simulates a GARCH(P,Q) process with P = 2
and Q = 1
.
% Set the random number generator seed for reproducability. randn('seed', 10) % 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: finite-differencing Hessian: finite-differencing (or Quasi-Newton) 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 medium-scale%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
End diagnostic information max Directional Iter F-count f(x) constraint Step-size derivative Procedure 1 5 699.185 -0.125 1 -2.97e+006 2 22 658.224 -0.1249 0.000488 -64.6 3 28 610.181 0 1 -49.4 4 35 590.888 0 0.5 -38.9 5 42 583.961 -0.03317 0.5 -29.8 6 49 583.224 -0.02756 0.5 -31.8 7 57 582.947 -0.02067 0.25 -7.28 8 63 578.182 0 1 -2.43 9 71 578.138 -0.09145 0.25 -0.55 10 77 577.898 -0.04452 1 -0.148 11 84 577.882 -0.06128 0.5 -0.0488 12 90 577.859 -0.07117 1 -0.000758 13 96 577.858 -0.07033 1 -0.000305 Hessian modified 14 102 577.858 -0.07042 1 -3.32e-005 Hessian modified 15 108 577.858 -0.0707 1 -1.29e-006 Hessian modified 16 114 577.858 -0.07077 1 -1.29e-007 Hessian modified 17 120 577.858 -0.07081 1 -1.97e-007 Hessian modified Optimization Converged Successfully Magnitude of directional derivative in search direction less than 2*options.TolFun and maximum constraint violation is less than options.TolCon No Active Constraints Estimated Coefficients: ---------------------- 0.2520 0.0708 0.1623 0.4000 Variance Forecasts: ------------------ 1.3243 0.9594 0.9186 0.8402 0.7966 0.7634 0.7407 0.7246 0.7133 0.7054
See Also
ugarch
, ugarchpred
, and the GARCH Toolbox function garchsim
References
James D. Hamilton, Time Series Analysis, Princeton University Press, 1994
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