# simulate

Monte Carlo simulation of conditional variance models

## Syntax

## Description

specifies options using one or more name-value arguments. For example,
`V`

= simulate(`Mdl`

,`numobs`

,`Name=Value`

)`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.

returns the table or timetable `Tbl`

= simulate(`Mdl`

,`numobs`

,Presample=`Presample`

,`Name=Value`

)`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 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. Supply timetables of presample data.

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

```
load Data_EquityIdx
DTTRet = price2ret(DataTimeTable);
DTTRet.Interval = [];
T = height(DTTRet);
```

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 500 observations each. Infer conditional variances and residuals to use as a presample for the forecast simulation.

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},500,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 500-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 - 250:end); datesFH = DTTRet.Time(end) + caldays(1:500)'; 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-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]),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

`Presample`

— Presample data

table | timetable

*Since R2023a*

Presample data for innovations
*ε _{t}* or conditional variances

*σ*

_{t}^{2}to initialize the model, specified as a table or timetable, the same type as

`Tbl1`

, with `numprevars`

variables
and `numpreobs`

rows. Use `Presample`

only when you supply a table or timetable of data
`Tbl1`

.`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 `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.

`E0`

— Presample innovation paths *ε*_{t}

numeric column vector | numeric matrix

_{t}

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*_{t}^{2}

numeric column vector | numeric matrix

_{t}

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

_{t}

*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(`

selects variable) = true`j`

from`j`

`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*_{t}^{2}

string scalar | character vector | integer | logical vector

_{t}

*Since R2023a*

Variable of `Presample`

containing data for the presample conditional variances *σ _{t}*

^{2}, 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`PresampleVarianceVariable(`

selects variable) = true`j`

from`j`

`Presample.Properties.VariableNames`

, and`sum(PresampleVarianceVariable)`

is`1`

The selected variable must be a numeric vector and cannot contain missing values (`NaN`

).

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*_{t}^{2}

numeric column vector | numeric matrix

_{t}

Simulated conditional variance paths
*σ _{t}*

^{2}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

_{t}

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*_{t}^{2} and
response *y*_{t} paths

table | timetable

_{t}

_{t}

Simulated conditional variance
*σ _{t}*

^{2}and response

*y*paths, returned as a table or timetable, the same data type as

_{t}`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`

, where_Variance`responseName`

is`responseName`

`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`

, where_Response`responseName`

is`responseName`

`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`

.

## See Also

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