simulate

Monte Carlo simulation of conditional variance models

Syntax

V = simulate(Mdl,numObs)

V = simulate(Mdl,numObs,Name,Value)

[V,Y] = simulate(___)

Description

V = simulate(Mdl,numObs) simulates a numObs-period conditional variance path from the fully specified conditional variance model Mdl. Mdl can be a garch, egarch, or gjr model.

V = simulate(Mdl,numObs,Name,Value) simulates conditional variance paths with additional options specified by one or more Name,Value pair arguments. For example, you can generate multiple sample paths or specify presample innovation paths.

[V,Y] = simulate(___) additionally simulates response paths using any of the input arguments in the previous syntaxes.

Examples

Simulate GARCH Model Conditional Variances and Responses

Simulate conditional variance and response paths from a GARCH(1,1) model.

Specify a GARCH(1,1) model with known parameters.

Mdl = garch('Constant',0.01,'GARCH',0.7,'ARCH',0.2);

Simulate 500 sample paths, each with 100 observations.

rng default; % For reproducibility
[V,Y] = simulate(Mdl,100,'NumPaths',500);

figure
subplot(2,1,1)
plot(V)
title('Simulated Conditional Variances')

subplot(2,1,2)
plot(Y)
title('Simulated Responses')

The simulated responses look like draws from a stationary stochastic process.

Plot the 2.5th, 50th (median), and 97.5th percentiles of the simulated conditional variances.

lower = prctile(V,2.5,2);
middle = median(V,2);
upper = prctile(V,97.5,2);

figure
plot(1:100,lower,'r:',1:100,middle,'k',...
		 1:100,upper,'r:','LineWidth',2)
legend('95% Interval','Median')
title('Approximate 95% Intervals')

The intervals are asymmetric due to positivity constraints on the conditional variance.

Simulate EGARCH Model Conditional Variances and Responses

Simulate conditional variance and response paths from an EGARCH(1,1) model.

Specify an EGARCH(1,1) model with known parameters.

Mdl = egarch('Constant',0.001,'GARCH',0.7,'ARCH',0.2,...
            'Leverage',-0.3);

Simulate 500 sample paths, each with 100 observations.

rng default; % For reproducibility
[V,Y] = simulate(Mdl,100,'NumPaths',500);

figure
subplot(2,1,1)
plot(V)
title('Simulated Conditional Variances')

subplot(2,1,2)
plot(Y)
title('Simulated Responses (Innovations)')

The simulated responses look like draws from a stationary stochastic process.

Plot the 2.5th, 50th (median), and 97.5th percentiles of the simulated conditional variances.

lower = prctile(V,2.5,2);
middle = median(V,2);
upper = prctile(V,97.5,2);

figure
plot(1:100,lower,'r:',1:100,middle,'k',...
     1:100, upper,'r:','LineWidth',2)
legend('95% Interval','Median')
title('Approximate 95% Intervals')

The intervals are asymmetric due to positivity constraints on the conditional variance.

Simulate GJR Model Conditional Variances and Responses

Simulate conditional variance and response paths from a GJR(1,1) model.

Specify a GJR(1,1) model with known parameters.

Mdl = gjr('Constant',0.001,'GARCH',0.7,'ARCH',0.2,...
    'Leverage',0.1);

Simulate 500 sample paths, each with 100 observations.

rng default; % For reproducibility
[V,Y] = simulate(Mdl,100,'NumPaths',500);

figure
subplot(2,1,1)
plot(V)
title('Simulated Conditional Variances')

subplot(2,1,2)
plot(Y)
title('Simulated Responses (Innovations)')

The simulated responses look like draws from a stationary stochastic process.

Plot the 2.5th, 50th (median), and 97.5th percentiles of the simulated conditional variances.

lower = prctile(V,2.5,2);
middle = median(V,2);
upper = prctile(V,97.5,2);

figure
plot(1:100,lower,'r:',1:100,middle,'k',...
     1:100, upper,'r:','LineWidth',2)
legend('95% Interval','Median')
title('Approximate 95% Intervals')

The intervals are asymmetric due to positivity constraints on the conditional variance.

Forecast Conditional Variances by Monte-Carlo Simulation

Simulate conditional variances of the daily NASDAQ Composite Index returns for 500 days. Use the simulations to make forecasts and approximate 95% forecast intervals. Compare the forecasts among GARCH(1,1), EGARCH(1,1), and GJR(1,1) fits.

Load the NASDAQ data included with the toolbox. Convert the index to returns.

load Data_EquityIdx
nasdaq = DataTable.NASDAQ;
r = price2ret(nasdaq);
T = length(r);

Fit GARCH(1,1), EGARCH(1,1), and GJR(1,1) models to the entire data set. Infer conditional variances to use as presample conditional variances for the forecast simulation.

Mdl = cell(3,1); % Preallocation
Mdl{1} = garch(1,1);
Mdl{2} = egarch(1,1);
Mdl{3} = gjr(1,1);

EstMdl = cellfun(@(x)estimate(x,r,'Display','off'),Mdl,...
    'UniformOutput',false);
v0 = cellfun(@(x)infer(x,r),EstMdl,'UniformOutput',false);

EstMdl is 3-by-1 cell vector. Each cell is a different type of estimated conditional variance model, e.g., EstMdl{1} is an estimated GARCH(1,1) model. V0 is a 3-by-1 cell vector, and each cell contains the inferred conditional variances from the corresponding, estimated model.

Simulate 1000 samples paths with 500 observations each. Use the observed returns and inferred conditional variances as presample data.

vSim = cell(3,1); % Preallocation

for j = 1:3
    rng default; % For reproducibility
    vSim{j} = simulate(EstMdl{j},500,'NumPaths',1000,'E0',r,'V0',v0{j});  
end

vSim is a 3-by-1 cell vector, and each cell contains a 500-by-1000 matrix of simulated conditional variances generated from the corresponding, estimated model.

Plot the simulation mean forecasts and approximate 95% forecast intervals, along with the conditional variances inferred from the data.

lower = cellfun(@(x)prctile(x,2.5,2),vSim,'UniformOutput',false);
upper = cellfun(@(x)prctile(x,97.5,2),vSim,'UniformOutput',false);
mn = cellfun(@(x)mean(x,2),vSim,'UniformOutput',false);
datesPlot = dates(end - 250:end);
datesFH = dates(end) + (1:500)';

h = zeros(3,4);

figure
for j = 1:3
    col = zeros(1,3);
    col(j) = 1;    
    h(j,1) = plot(datesPlot,v0{j}(end-250:end),'Color',col);
    hold on
    h(j,2) = plot(datesFH,mn{j},'Color',col,'LineWidth',3);
    h(j,3:4) = plot([datesFH datesFH],[lower{j} upper{j}],':',...
        'Color',col,'LineWidth',2);
end
hGCA = gca;
plot(datesFH(1)*[1 1],hGCA.YLim,'k--');
datetick;
axis tight;
h = h(:,1:3);
legend(h(:),'GARCH - Inferred','EGARCH - Inferred','GJR - Inferred',...
      'GARCH - Sim. Mean','EGARCH - Sim. Mean','GJR - Sim. Mean',...
      'GARCH - 95% Fore. Int.','EGARCH - 95% Fore. Int.',...
      'GJR - 95% Fore. Int.','Location','NorthEast')
title('Simulated Conditional Variance Forecasts')
hold off

Input Arguments

Mdl — Conditional variance model

garch model object | egarch model object | gjr model object

Conditional variance model without any unknown parameters, specified as a garch, egarch, or gjr model object.

Mdl cannot contain any properties that have NaN value.

numObs — Sample path length

positive integer

Sample path length, specified as a positive integer. That is, the number of random observations to generate per output path. V and Y have numObs rows.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'numPaths',1000,'E0',[0.5; 0.5] specifies to generate 1000 sample paths and to use [0.5; 0.5] as presample innovations per path.

'NumPaths' — Number of sample paths to generate

1 (default) | positive integer

Number of sample paths to generate, specified as the comma-separated pair consisting of 'NumPaths' and a positive integer. V and Y have NumPaths columns.

Example: 'NumPaths',1000

Data Types: double

'E0' — Presample innovations

numeric column vector | numeric matrix

Presample innovations, specified as the comma-separated pair consisting of 'E0' and a numeric column vector or matrix. The presample innovations provide initial values for the innovations process of the conditional variance model Mdl. The presample innovations derive from a distribution with mean 0.

E0 must contain at least Mdl.Q elements or rows. If E0 contains extra rows, simulate uses the latest Mdl.Q only.

The last element or row contains the latest presample innovation.

• If E0 is a column vector, it represents a single path of the underlying innovation series. simulate applies E0 to each simulated path.

• If E0 is a matrix, then each column represents a presample path of the underlying innovation series. E0 must have at least NumPaths columns. If E0 has more columns than necessary, simulate uses the first NumPaths columns only.

The defaults are:

• For GARCH(P,Q) and GJR(P,Q) models, simulate sets any necessary presample innovations to an independent sequence of disturbances with mean zero and standard deviation equal to the unconditional standard deviation of the conditional variance process.

• For EGARCH(P,Q) models, simulate sets any necessary presample innovations to an independent sequence of disturbances with mean zero and variance equal to the exponentiated unconditional mean of the logarithm of the EGARCH variance process.

Example: 'E0',[0.5; 0.5]

'V0' — Positive presample conditional variance paths

numeric column vector | numeric matrix

Positive presample conditional variance paths, specified as a numeric vector or matrix. V0 provides initial values for the conditional variances in the model.

• If V0 is a column vector, then simulate applies it to each output path.

• If V0 is a matrix, then it must have at least NumPaths columns. If V0 has more columns than necessary, simulate uses the first NumPaths columns only.

• For GARCH(P,Q) and GJR(P,Q) models:

  • V0 must have at least Mdl.P rows to initialize the variance equation.

  • By default, simulate sets any necessary presample variances to the unconditional variance of the conditional variance process.

• For EGARCH(P,Q) models, simulate:

  • V0 must have at least max(Mdl.P,Mdl.Q) rows to initialize the variance equation.

  • By default, simulate sets any necessary presample variances to the exponentiated unconditional mean of the logarithm of the EGARCH variance process.

If the number of rows in V0 exceeds the number necessary, then simulate uses the latest, required number of observations only. The last element or row contains the latest observation.

Example: 'V0',[1; 0.5]

Data Types: double

Notes

• If E0 and V0 are column vectors, simulate applies them to every column of the outputs V and Y. This application allows simulated paths to share a common starting point for Monte Carlo simulation of forecasts and forecast error distributions.

• NaNs indicate missing values. simulate removes missing values. The software merges the presample data (E0 and V0), and then uses list-wise deletion to remove any rows containing at least one NaN. Removing NaNs in the data reduces the sample size. Removing NaNs can also create irregular time series.

• simulate assumes that you synchronize presample data such that the latest observation of each presample series occurs simultaneously.

Output Arguments

V — Simulated conditional variance paths

numeric column vector | numeric matrix

Simulated conditional variance paths of the mean-zero innovations associated with Y, returned as a numeric column vector or matrix.

V is a numObs-by-NumPaths matrix, in which each column corresponds to a simulated conditional variance path. Rows of V are periods corresponding to the periodicity of Mdl.

Y — Simulated response paths

numeric column vector | numeric matrix

Simulated response paths, returned as a numeric column vector or matrix. Y usually represents a mean-zero, heteroscedastic time series of innovations with conditional variances given in V (a continuation of the presample innovation series E0).

Y can also represent a time series of mean-zero, heteroscedastic innovations plus an offset. If Mdl includes an offset, then simulate adds the offset to the underlying mean-zero, heteroscedastic innovations so that Y represents a time series of offset-adjusted innovations.

Y is a numObs-by-NumPaths matrix, in which each column corresponds to a simulated response path. Rows of Y are periods corresponding to the periodicity of Mdl.

References

[1] Bollerslev, T. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics. Vol. 31, 1986, pp. 307–327.

[2] Bollerslev, T. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics. Vol. 69, 1987, pp. 542–547.

[3] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

[4] Enders, W. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, 1995.

[5] Engle, R. F. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica. Vol. 50, 1982, pp. 987–1007.

[6] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance. Vol. 48, No. 5, 1993, pp. 1779–1801.

[7] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[8] Nelson, D. B. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica. Vol. 59, 1991, pp. 347–370.