simulate

Monte Carlo simulation of conditional variance models

Syntax

V = simulate(Mdl,numobs)

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

[V,Y] = simulate(___)

Tbl = simulate(Mdl,numobs,Presample=Presample,Name=Value)

Description

V = simulate(Mdl,numobs) returns the numeric array V containing a random numobs-period path of a conditional variance series from the fully specified conditional variance model Mdl. Mdl can be a garch, egarch, or gjr model.

V = simulate(Mdl,numobs,Name=Value) specifies options using one or more name-value arguments. For example, simulate(Mdl,100,NumPaths=1000,V0=v0) returns a numeric matrix of 1000, 100-period simulated conditional variance paths from Mdl and specifies the numeric vector of presample conditional variances v0 to initialize each returned path.

To produce a conditional simulation, specify response data in the simulation horizon by using the YF name-value argument.

[V,Y] = simulate(___) also returns simulated response series paths using any of the input arguments in the previous syntaxes.

Tbl = simulate(Mdl,numobs,Presample=Presample,Name=Value) returns the table or timetable Tbl containing the random conditional variance and response series, which results from simulating the model Mdl. simulate uses the table or timetable of innovations or conditional variance presample data Presample to initialize the model. If you specify Presample, you must specify the variable containing the presample innovation or conditional variance data by using the PresampleInnovationVariable or PresampleVarianceVariable name-value argument.

Examples

Return Matrices of Simulated GARCH Model Conditional Variances and Responses

Simulate conditional variance and response paths from a GARCH(1,1) model. Return results in numeric matrices.

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);

V and Y are 100-by-500 matrices of 500 simulated paths of conditional variances and responses, respectively.

figure
tiledlayout(2,1)
nexttile
plot(V)
title("Simulated Conditional Variances")
nexttile
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% Confidence 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
tiledlayout(2,1)
nexttile
plot(V)
title("Simulated Conditional Variances")
nexttile
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% Confidence 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
tiledlayout(2,1)
nexttile
plot(V)
title("Simulated Conditional Variances")
nexttile
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% Confidence 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

Since R2023a

Simulate conditional variances of the average weekly closing NASDAQ returns for 100 weeks. 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. Supply timetables of presample data.

Load the U.S. equity indices data Data_EquityIdx.mat.

load Data_EquityIdx

The timetable DataTimeTable contains the daily NASDAQ closing prices, among other indices.

Compute the weekly average closing prices of all timetable variables.

DTTW = convert2weekly(DataTimeTable,Aggregation="mean");

Compute the weekly returns.

DTTRet = price2ret(DTTW);
DTTRet.Interval = [];
T = height(DTTRet)

T = 626

When you plan to supply a timetable, you must ensure it has all the following characteristics:

• The selected response variable is numeric and does not contain any missing values.

• The timestamps in the Time variable are regular, and they are ascending or descending.

Remove all missing values from the timetable, relative to the NASDAQ returns series.

DTTRet = rmmissing(DTTRet,DataVariables="NASDAQ");
numobs = height(DTTRet)

numobs = 626

Because all sample times have observed NASDAQ returns, rmmissing does not remove any observations.

Determine whether the sampling timestamps have a regular frequency and are sorted.

areTimestampsRegular = isregular(DTTRet,"weeks")

areTimestampsRegular = logical
   1

areTimestampsSorted = issorted(DTTRet.Time)

areTimestampsSorted = logical
   1

areTimestampsRegular = 1 indicates that the timestamps of DTTRet represent a regular weekly sample. areTimestampsSorted = 1 indicates that the timestamps are sorted.

Fit GARCH(1,1), EGARCH(1,1), and GJR(1,1) models to the entire data set.

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,DTTRet,ResponseVariable="NASDAQ", ...
    Display="off"),Mdl,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.

Simulate 1000 samples paths with 100 observations each. Infer conditional variances and residuals to use as a presample for the forecast simulation.

T0 = 100;
DTTSim = cell(3,1); % Preallocation
PS = cell(3,1);

for j = 1:3
    rng("default") % For reproducibility
    PS{j} = infer(EstMdl{j},DTTRet,ResponseVariable="NASDAQ");
    DTTSim{j} = simulate(EstMdl{j},T0,NumPaths=1000, ...
        Presample=PS{j},PresampleInnovationVariable="Y_Residual", ...
        PresampleVarianceVariable="Y_Variance");
end

DTTSim is a 3-by-1 cell vector, and each cell contains a 100-by-2 timetable of 1000 simulated paths of conditional variances and responses 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.Y_Variance,2.5,2),DTTSim,UniformOutput=false);
upper = cellfun(@(x)prctile(x.Y_Variance,97.5,2),DTTSim,UniformOutput=false);
mn = cellfun(@(x)mean(x.Y_Variance,2),DTTSim,UniformOutput=false);
datesPlot = DTTRet.Time(end - 50:end);
datesFH = DTTRet.Time(end) + caldays(1:100)';

h = zeros(3,4);

figure
for j = 1:3
    col = zeros(1,3);
    col(j) = 1;    
    h(j,1) = plot(datesPlot,PS{j}.Y_Variance(end-50: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]),hGCA.YLim,"k--");
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. numobs is the number of random observations to generate per output path.

Data Types: double

Presample — Presample data

table | timetable

Since R2023a

Presample data for innovations ε_t or conditional variances σ_t² to initialize the model, specified as a table or timetable with numprevars variables and numpreobs rows.

simulate returns the simulated variables in the output table or timetable Tbl, which is commensurate with Presample.

Each selected variable is a single path (numpreobs-by-1 vector) or multiple paths (numpreobs-by-numprepaths matrix) of numpreobs observations representing the presample of numpreobs observations of innovations or conditional variances.

Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation. numpreobs must be one of the following values:

• Mdl.Q when Presample provides only presample innovations.

• For GARCH(P,Q) and GJR(P,Q) models:

  • Mdl.P when Presample provides only presample conditional variances.

  • max([Mdl.P Mdl.Q]) when Presample provides both presample innovations and conditional variances

• For EGARCH(P,Q) models, max([Mdl.P Mdl.Q]) when Presample provides presample conditional variances

If numpreobs exceeds the minimum number, simulate uses the latest required number of observations only.

If numprepaths > NumPaths, simulate uses only the first NumPaths columns.

If Presample is a timetable, all the following conditions must be true:

• Presample must represent a sample with a regular datetime time step (see isregular).

• The datetime vector of sample timestamps Presample.Time must be ascending or descending.

If Presample is a table, the last row contains the latest presample observation.

The defaults are:

• For GARCH(P,Q) and GJR(P,Q) models, simulate sets any necessary presample innovations to the square root of the average squared value of the offset-adjusted response series Y.

• For EGARCH(P,Q) models, simulate sets any necessary presample innovations to zero.

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

If you specify the Presample, you must specify the presample innovation or conditional variance variable names by using the PresampleInnovationVariable or PresampleVarianceVariable name-value argument.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: simulate(Mdl,100,NumPaths=1000,E0=[0.5; 0.5]) specifies generating 1000 sample paths of length 100 from the model Mdl, and using [0.5; 0.5] as the presample of innovations per path.

NumPaths — Number of sample paths to generate

1 (default) | positive integer

Number of sample paths to generate, specified as a positive integer.

Example: NumPaths=1000

Data Types: double

E0 — Presample innovation paths ε_t

numeric column vector | numeric matrix

Presample innovation paths ε_t, specified as a numpreobs-by-1 numeric column vector or a numpreobs-by-numprepaths matrix. Use E0 only when you supply optional data inputs as numeric arrays.

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.

numpreobs is the number of presample observations. numprepaths is the number of presample paths.

Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation. numpreobs must be at least Mdl.Q. If numpreobs > Mdl.Q, simulate uses the latest required number of observations only. The last element or row contains the latest observation.

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

• If E0 is a matrix, each column represents a presample path of the underlying innovation series. numprepaths must be at least NumPaths. If numprepaths > NumPaths, 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 σ_t²

numeric column vector | numeric matrix

Positive presample conditional variance paths, specified as a numpreobs-by-1 positive column vector or numpreobs-by-numprepaths positive matrix.. V0 provides initial values for the conditional variances in the model. Use V0 only when you supply optional data inputs as numeric arrays.

Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation.

• For GARCH(P,Q) and GJR(P,Q) models, numpreobs must be at least Mdl.P.

• For EGARCH(P,Q) models,numpreobs must be at least max([Mdl.P Mdl.Q]).

If numpreobs exceeds the minimum number, simulate uses only the latest observations. The last element or row contains the latest observation.

• If V0 is a column vector, it represents a single path of the conditional variance series. simulate applies it to each output path.

• If V0 is a matrix, each column represents a presample path of the conditional variance series. numprepaths must be at least NumPaths. If numprepaths > NumPaths, simulate uses the first NumPaths columns only.

The defaults are:

• For GARCH(P,Q) and GJR(P,Q) models, simulate sets any necessary presample variances to the unconditional variance of the conditional variance process.

• For EGARCH(P,Q) models, simulate sets any necessary presample variances to the exponentiated unconditional mean of the logarithm of the EGARCH variance process.

Example: V0=[1; 0.5]

Data Types: double

PresampleInnovationVariable — Variable of Presample containing presample innovation paths ε_t

string scalar | character vector | integer | logical vector

Since R2023a

Variable of Presample containing presample innovation paths ε_t, specified as one of the following data types:

• String scalar or character vector containing a variable name in Presample.Properties.VariableNames

• Variable index (integer) to select from Presample.Properties.VariableNames

• A length numprevars logical vector, where PresampleInnovationVariable(j) = true selects variable j from Presample.Properties.VariableNames, and sum(PresampleInnovationVariable) is 1

The selected variable must be a numeric matrix and cannot contain missing values (NaN).

If you specify presample innovation data by using the Presample name-value argument, you must specify PresampleInnovationVariable.

Example: PresampleInnovationVariable="StockRateInnov0"

Example: PresampleInnovationVariable=[false false true false] or PresampleInnovationVariable=3 selects the third table variable as the presample innovation variable.

Data Types: double | logical | char | cell | string

PresampleVarianceVariable — Variable of Presample containing data for the presample conditional variances σ_t²

string scalar | character vector | integer | logical vector

Since R2023a

Variable of Presample containing data for the presample conditional variances σ_t², specified as one of the following data types:

• String scalar or character vector containing a variable name in Presample.Properties.VariableNames

• Variable index (positive integer) to select from Presample.Properties.VariableNames

• A logical vector, where PresampleVarianceVariable(j) = true selects variable j from Presample.Properties.VariableNames

The selected variable must be a numeric vector and cannot contain missing values (NaNs).

If you specify presample conditional variance data by using the Presample name-value argument, you must specify PresampleVarianceVariable.

Example: PresampleVarianceVariable="StockRateVar0"

Example: PresampleVarianceVariable=[false false true false] or PresampleVarianceVariable=3 selects the third table variable as the presample conditional variance variable.

Data Types: double | logical | char | cell | string

Notes

• NaN values in E0, and V0 indicate missing values. simulate removes missing values from specified data by list-wise deletion. simulate horizontally concatenates E0 and V0, and then it removes any row of the concatenated matrix containing at least one NaN. This type of data reduction reduces the effective sample size and can create an irregular time series.

• For numeric data inputs, simulate assumes that you synchronize the presample data such that the latest observations occur simultaneously.

• simulate issues an error when any table or timetable input contains missing values.

• 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.

Output Arguments

V — Simulated conditional variance paths σ_t²

numeric column vector | numeric matrix

Simulated conditional variance paths σ_t² of the mean-zero innovations associated with Y, returned as a numobs-by-1 numeric column vector or numobs-by-NumPaths matrix. simulate returns V when you do not specify the input table or timetable Presample.

Each column of V corresponds to a simulated conditional variance path. Rows of V are periods corresponding to the periodicity of Mdl.

Y — Simulated response paths y_t

numeric column vector | numeric matrix

Simulated response paths y_t, returned as a numobs-by-1 numeric column vector or numobs-by-NumPaths matrix. simulate returns Y when you do not specify the input table or timetable Presample.

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.

Each column of Y corresponds to a simulated response path. Rows of Y are periods corresponding to the periodicity of Mdl.

Tbl — Simulated conditional variance σ_t² and response y_t paths

table | timetable

Since R2023a

Simulated conditional variance σ_t² and response y_t paths, returned as a table or timetable, the same data type as Presample. simulate returns Tbl only when you supply the input Presample.

Tbl contains the following variables:

• The simulated conditional variance paths, which are in a numobs-by-NumPaths numeric matrix, with rows representing observations and columns representing independent paths. Each path represents the continuation of the corresponding path of presample conditional variances in Presample. simulate names the simulated conditional variance variable in Tbl responseName_Variance, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is StockReturns, Tbl contains a variable for the corresponding simulated conditional variance paths with the name StockReturns_Variance.

• The simulated response paths, which are in a numobs-by-NumPaths numeric matrix, with rows representing observations and columns representing independent paths. Each path represents the continuation of the corresponding presample innovations path in Presample. simulate names the simulated response variable in Tbl responseName_Response, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is StockReturns, Tbl contains a variable for the corresponding simulated response paths with the name StockReturns_Response.

If Tbl is a timetable, the following conditions hold:

• The row order of Tbl, either ascending or descending, matches the row order of Preample.

• Tbl.Time(1) is the next time after Presample(end) relative the sampling frequency, and Tbl.Time(2:numobs) are the following times relative to the sampling frequency.

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.

Version History

Introduced in R2012a

R2023a: simulate accepts input data in tables and timetables, and returns results in tables and timetables

In addition to accepting presample data in numeric arrays, simulate accepts presample data in tables or regular timetables. When you supply data in a table or timetable, the following conditions apply:

• If you specify optional presample innovation or conditional variance data to initialize the model, you must also specify the presample innovation or conditional variance series name.

• simulate returns results in a table or timetable.

Name-value arguments to support tabular workflows include:

• Presample specifies the input table or regular timetable of presample innovations and conditional variance data.

• PresampleInnovationVariable specifies the variable name of the innovation paths to select from Presample.

• PresampleVarianceVariable specifies the variable name of the conditional variance paths to select from Presample.