# simulate

Monte Carlo simulation of univariate ARIMA or ARIMAX models

## Syntax

``Y = simulate(Mdl,numobs)``
``Y = simulate(Mdl,numobs,Name=Value)``
``[Y,E,V] = simulate(___)``
``Tbl = simulate(Mdl,numobs,Presample=Presample,PresampleResponseVariable=PresampleResponseVariable)``
``Tbl = simulate(Mdl,numobs,InSample=InSample,PredictorVariables=PredictorVariables)``
``Tbl = simulate(Mdl,numobs,Presample=Presample,PresampleResponseVariable=PresampleResponseVariable,InSample=InSample,PredictorVariables=PredictorVariables)``
``Tbl = simulate(___,Name=Value)``

## Description

example

````Y = simulate(Mdl,numobs)` returns the numeric vector `Y` containing a random `numobs`-period response path from simulating the fully specified ARIMA model `Mdl`.```

example

````Y = simulate(Mdl,numobs,Name=Value)` uses additional options specified by one or more name-value arguments. `simulate` returns numeric arrays when all optional input data are numeric arrays. For example, `simulate(Mdl,100,NumPaths=1000,Y0=PS)` returns a numeric array of 1000, 100-period simulated response paths from `Mdl` and specifies the numeric array of presample response data `PS`.```

example

````[Y,E,V] = simulate(___)` uses any input-argument combination in the previous syntaxes to return numeric arrays of one or more independent series of model innovations `E` and, when `Mdl` represents a composite conditional mean and variance model, conditional variances `V`, resulting from simulating the ARIMA model.```

example

````Tbl = simulate(Mdl,numobs,Presample=Presample,PresampleResponseVariable=PresampleResponseVariable)` returns the table or timetable `Tbl` containing a variable for each of the random paths of response, innovation, and conditional variance series resulting from simulating the ARIMA model `Mdl`. `simulate` uses the response variable `PresampleResponseVariable` in the table or timetable of presample data `Presample` to initialize the response series. (since R2023b)To initialize the model using presample innovation or conditional variance data, replace the `PresampleResponseVariable` name-value argument with `PresampleInnovationVariable` or `PresampleVarianceVariable` name-value argument.```

example

````Tbl = simulate(Mdl,numobs,InSample=InSample,PredictorVariables=PredictorVariables)` specifies the variables `PredictorVariables` in the in-sample table or timetable of data `InSample` containing the predictor data for the exogenous regression component in the ARIMA model `Mdl`. (since R2023b)```

example

````Tbl = simulate(Mdl,numobs,Presample=Presample,PresampleResponseVariable=PresampleResponseVariable,InSample=InSample,PredictorVariables=PredictorVariables)` specifies presample response data to initialize the model and in-sample predictor data for the exogenous regression component. (since R2023b)```

example

````Tbl = simulate(___,Name=Value)` uses additional options specified by one or more name-value arguments, using any input argument combination in the previous three syntaxes. (since R2023b)For example, `simulate(Mdl,100,NumPaths=1000,Presample=PSTbl,PresampleResponseVariables="GDP")` returns a timetable containing a variable for each of the response, innovations, and conditional variance series. Each variable is a 100-by-1000 matrix representing 1000, 100-period paths simulated from the ARIMA model. `simulate` initializes the model by using the presample data in the `GDP` variable of the timetable `PSTbl`.```

## Examples

collapse all

Simulate a response path from an ARIMA model. Return the path in a vector.

Consider the ARIMA(4,1,1) model

`$\left(1-0.75{L}^{4}\right)\left(1-L\right){y}_{t}=2+\left(1+0.1L\right){\epsilon }_{t},$`

where ${\epsilon }_{\mathit{t}}$ is a Gaussian innovations series with a mean of 0 and a variance of 1.

Create the ARIMA(4,1,1) model.

`Mdl = arima(AR=-0.75,ARLags=4,MA=0.1,Constant=2,Variance=1)`
```Mdl = arima with properties: Description: "ARIMA(4,0,1) Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 4 D: 0 Q: 1 Constant: 2 AR: {-0.75} at lag  SAR: {} MA: {0.1} at lag  SMA: {} Seasonality: 0 Beta: [1×0] Variance: 1 ```

`Mdl` is a fully specified `arima` object representing the ARIMA(4,1,1) model.

Simulate a 100-period random response path from the ARIMA(4,1,1) model.

```rng(1,"twister") % For reproducibility y = simulate(Mdl,100);```

`y` is a 100-by-1 vector containing the random response path.

Plot the simulated path.

```plot(y) ylabel("y") xlabel("Time")``` Simulate three predictor series and a response series.

Specify and simulate a path of length 20 for each of the three predictor series modeled by

`$\left(1-0.2L\right){x}_{it}=2+\left(1+0.5L-0.3{L}^{2}\right){\eta }_{it},$`

where ${\eta }_{it}$ follows a Gaussian distribution with mean 0 and variance 0.01, and $i$ = {1,2,3}.

```[MdlX1,MdlX2,MdlX3] = deal(arima(AR=0.2,MA={0.5 -0.3}, ... Constant=2,Variance=0.01)); rng(4,"twister"); % For reproducibility simX1 = simulate(MdlX1,20); simX2 = simulate(MdlX2,20); simX3 = simulate(MdlX3,20); SimX = [simX1 simX2 simX3];```

Specify and simulate a path of length 20 for the response series modeled by

`$\left(1-0.05L+0.02{L}^{2}-0.01{L}^{3}\right)\left(1-L{\right)}^{1}{y}_{t}=0.05+{x}_{t}^{\prime }\left[\begin{array}{c}0.5\\ -0.03\\ -0.7\end{array}\right]+\left(1+0.04L+0.01{L}^{2}\right){\epsilon }_{t},$`

where ${\epsilon }_{t}$ follows a Gaussian distribution with mean 0 and variance 1.

```MdlY = arima(AR={0.05 -0.02 0.01},MA={0.04 0.01}, ... D=1,Constant=0.5,Variance=1,Beta=[0.5 -0.03 -0.7]); simY = simulate(MdlY,20,X=SimX);```

Plot the series together.

```figure plot([SimX simY]) title("Simulated Series") legend("x_1","x_2","x_3","y")``` Forecast the daily NASDAQ Composite Index using Monte Carlo simulations. Supply presample observations to initialize the model.

Load the NASDAQ data included with the toolbox. Extract the first 1500 observations for fitting.

```load Data_EquityIdx nasdaq = DataTable.NASDAQ(1:1500); T = length(nasdaq);```

Fit an ARIMA(1,1,1) model.

```NasdaqModel = arima(1,1,1); NasdaqFit = estimate(NasdaqModel,nasdaq);```
``` ARIMA(1,1,1) Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0.43031 0.18555 2.3191 0.020392 AR{1} -0.074391 0.081985 -0.90737 0.36421 MA{1} 0.31126 0.077266 4.0284 5.6158e-05 Variance 27.826 0.63625 43.735 0 ```

Simulate 1000 paths with 500 observations each. Use the observed data as presample data.

```rng(1,"twister"); Y = simulate(NasdaqFit,500,NumPaths=1000,Y0=nasdaq);```

Plot the simulation mean forecast and approximate 95% forecast intervals.

```lower = prctile(Y,2.5,2); upper = prctile(Y,97.5,2); mn = mean(Y,2); x = T + (1:500); figure plot(nasdaq,Color=[.7,.7,.7]) hold on h1 = plot(x,lower,"r:",LineWidth=2); plot(x,upper,"r:",LineWidth=2) h2 = plot(x,mn,"k",LineWidth=2); legend([h1 h2],"95% confidence interval","Simulation mean", ... Location="NorthWest") title("NASDAQ Composite Index Forecast") hold off``` Simulate response and innovation paths from a multiplicative seasonal model.

Specify the model

`$\left(1-L\right)\left(1-{L}^{12}\right){y}_{t}=\left(1-0.5L\right)\left(1+0.3{L}^{12}\right){\epsilon }_{t},$`

where ${\epsilon }_{t}$ follows a Gaussian distribution with mean 0 and variance 0.1.

```Mdl = arima(MA=-0.5,SMA=0.3,SMALags=12,D=1, ... Seasonality=12,Variance=0.1,Constant=0);```

Simulate 500 paths with 100 observations each.

```rng(1,"twister") % For reproducibility [Y,E] = simulate(Mdl,100,NumPaths=500); figure tiledlayout(2,1) nexttile plot(Y) title("Simulated Response") nexttile plot(E) title("Simulated Innovations")``` Plot the 2.5th, 50th (median), and 97.5th percentiles of the simulated response paths.

```lower = prctile(Y,2.5,2); middle = median(Y,2); upper = prctile(Y,97.5,2); figure plot(1:100,lower,"r:",1:100,middle,"k", ... 1:100,upper,"r:") legend("95% confidence interval","Median")``` Compute statistics across the second dimension (across paths) to summarize the sample paths.

Plot a histogram of the simulated paths at time 100.

```figure histogram(Y(100,:),10) title("Response Distribution at Time 100")``` Fit an ARIMA(1,1,1) model to the weekly average NYSE closing prices. Supply a timetable of presample responses to initialize the model and return a timetable of simulated values from the model.

Load the US equity index data set `Data_EquityIdx`.

```load Data_EquityIdx T = height(DataTimeTable)```
```T = 3028 ```

The timetable `DataTimeTable` includes the time series variable `NYSE`, which contains daily NYSE composite closing prices from January 1990 through December 2001.

Plot the daily NYSE price series.

```figure plot(DataTimeTable.Time,DataTimeTable.NYSE) title("NYSE Daily Closing Prices: 1990 - 2001")``` Prepare Timetable for Estimation

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 NYSE price series.

```DTT = rmmissing(DataTimeTable,DataVariables="NYSE"); T_DTT = height(DTT)```
```T_DTT = 3028 ```

Because all sample times have observed NYSE prices, `rmmissing` does not remove any observations.

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

`areTimestampsRegular = isregular(DTT,"days")`
```areTimestampsRegular = logical 0 ```
`areTimestampsSorted = issorted(DTT.Time)`
```areTimestampsSorted = logical 1 ```

`areTimestampsRegular = 0` indicates that the timestamps of `DTT` are irregular. `areTimestampsSorted = 1` indicates that the timestamps are sorted. Business day rules make daily macroeconomic measurements irregular.

Remedy the time irregularity by computing the weekly average closing price series of all timetable variables.

```DTTW = convert2weekly(DTT,Aggregation="mean"); areTimestampsRegular = isregular(DTTW,"weeks")```
```areTimestampsRegular = logical 1 ```
`T_DTTW = height(DTTW)`
```T_DTTW = 627 ```

`DTTW` is regular.

```figure plot(DTTW.Time,DTTW.NYSE) title("NYSE Daily Closing Prices: 1990 - 2001")``` Create Model Template for Estimation

Suppose that an ARIMA(1,1,1) model is appropriate to model NYSE composite series during the sample period.

Create an ARIMA(1,1,1) model template for estimation.

`Mdl = arima(1,1,1);`

`Mdl` is a partially specified `arima` model object.

Fit Model to Data

`infer` requires `Mdl.P` presample observations to initialize the model. `infer` backcasts for necessary presample responses, but you can provide a presample.

Partition the data into presample and in-sample, or estimation sample, observations.

```T0 = Mdl.P; DTTW0 = DTTW(1:T0,:); DTTW1 = DTTW((T0+1):end,:);```

Fit an ARIMA(1,1,1) model to the in-sample weekly average NYSE closing prices. Specify the response variable name, presample timetable, and the presample response variable name.

```EstMdl = estimate(Mdl,DTTW1,ResponseVariable="NYSE", ... Presample=DTTW0,PresampleResponseVariable="NYSE");```
``` ARIMA(1,1,1) Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ ___________ Constant 0.83623 0.453 1.846 0.064892 AR{1} -0.32862 0.23526 -1.3969 0.16246 MA{1} 0.42703 0.22612 1.8885 0.058962 Variance 56.065 1.8433 30.416 3.3814e-203 ```

`EstMdl` is a fully specified, estimated `arima` model object.

Simulate Model

Simulate the fitted model 20 weeks beyond the final period. Specify the entire in-sample data as a presample and the presample response variable name in the in-sample timetable.

```rng(1,"twister") % For reproducibility numobs = 20; Tbl = simulate(EstMdl,numobs,Presample=DTTW1, ... PresampleResponseVariable="NYSE"); tail(Tbl)```
``` Time Y_Response Y_Innovation Y_Variance ___________ __________ ____________ __________ 05-Apr-2002 564.68 -11.302 56.065 12-Apr-2002 570.47 6.5582 56.065 19-Apr-2002 570.39 -1.8179 56.065 26-Apr-2002 571.72 1.249 56.065 03-May-2002 557.94 -14.716 56.065 10-May-2002 547.51 -9.5098 56.065 17-May-2002 556.51 8.7992 56.065 24-May-2002 573.34 15.194 56.065 ```
`size(Tbl)`
```ans = 1×2 20 3 ```

`Tbl` is a 20-by-3 timetable containing the simulated response path `NYSE_Response`, the corresponding simulated innovation path `NYSE_Innovation`, and the constant variance path `NYSE_Variance` (`Mdl.Variance = 56.065`).

Plot the simulated responses.

```figure plot(DTTW1.Time((end-50):end),DTTW1.NYSE((end-50):end), ... Color=[0.7 0.7 0.7],LineWidth=2); hold on plot(Tbl.Time,Tbl.Y_Response,LineWidth=2); legend("Observed","Simulated responses") title("Weekly Average NYSE CLosing Prices") hold off``` Fit an ARIMAX(1,1,1) model to the weekly average NYSE closing prices. Include an exogenous regression term identifying whether a measurement observation occurs during a recession. Supply timetables of presample and in-sample exogenous data. Simulate the weekly average closing prices over a 10-week horizon, and compare the forecasts to held out data.

Load the US equity index data set `Data_EquityIdx`.

```load Data_EquityIdx T = height(DataTimeTable)```
```T = 3028 ```

Remedy the time irregularity by computing the weekly average closing price series of all timetable variables.

```DTTW = convert2weekly(DataTimeTable,Aggregation="mean"); T_DTTW = height(DTTW)```
```T_DTTW = 627 ```

Load the US recessions data set.

```load Data_Recessions RDT = datetime(Recessions,ConvertFrom="datenum", ... Format="yyyy-MM-dd");```

Determine whether each sampling time in the data occurs during a US recession.

```isrecession = @(x)any(isbetween(x,RDT(:,1),RDT(:,2))); DTTW.IsRecession = arrayfun(isrecession,DTTW.Time)*1;```

`DTTW` contains a variable `IsRecession`, which represents the exogenous variable in the ARIMAX model.

Create an ARIMA(1,1,1) model template for estimation. Set the response series name to `NYSE`.

```Mdl = arima(1,1,1); Mdl.SeriesName = "NYSE";```

Partition the data into required presample, in-sample (estimation sample), and 10 holdout sample observations.

```T0 = Mdl.P; T2 = 10; DTTW0 = DTTW(1:T0,:); DTTW1 = DTTW((T0+1):(end-T2),:); DTTW2 = DTTW((end-T2+1):end,:);```

Fit an ARIMA(1,1,1) model to the in-sample weekly average NYSE closing prices. Specify the presample timetable, the presample response variable name, and the in-sample predictor variable name.

```EstMdl = estimate(Mdl,DTTW1,Presample=DTTW0,PresampleResponseVariable="NYSE", ... PredictorVariables="IsRecession");```
``` ARIMAX(1,1,1) Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ ___________ Constant 1.0635 0.50013 2.1264 0.03347 AR{1} -0.33827 0.2088 -1.6201 0.10521 MA{1} 0.43879 0.20045 2.189 0.028595 Beta(1) -2.425 1.0861 -2.2327 0.025569 Variance 55.527 1.9255 28.838 7.0991e-183 ```

Simulate 1000 paths of responses from the fitted model into a 10-week horizon. Specify the in-sample data as a presample, the presample response variable name, the predictor data in the forecast horizon, and the predictor variable name.

```rng(1,"twister") Tbl = simulate(EstMdl,T2,Presample=DTTW1,PresampleResponseVariable="NYSE", ... InSample=DTTW2,PredictorVariables="IsRecession",NumPaths=1000);```

`Tbl` is a 10-by-6 timetable containing all variables in `InSample`, and the following variables:

• `NYSE_Response`: a 10-by-1000 matrix of 1000 random response paths simulated from the model.

• `NYSE_Innovation`: a 10-by-1000 matrix of 1000 random innovation paths filtered through the model to product the response paths

• `NYSE_Variance`: a 10-by-1000 matrix containing only the constant `Mdl.Variance = 55.527`.

Plot the observed responses, median of the simulated responses, a 95% percentile intervals of the simulated values.

```SimStats = quantile(Tbl.NYSE_Response,[0.025 0.5 0.975],2); figure h1 = plot(DTTW.Time((end-50):end),DTTW.NYSE((end-50):end),"k",LineWidth=2); hold on plot(Tbl.Time,Tbl.NYSE_Response,Color=[0.8 0.8 0.8]) h2 = plot(Tbl.Time,SimStats(:,2),'r--'); h3 = plot(Tbl.Time,SimStats(:,[1 3]),'b--'); legend([h1 h2 h3(1)],["Observations" "Sim. medians" "95% percentile intervals"]) title("NYSE Weekly Average Price Series and Monte Carlo Forecasts")``` ## Input Arguments

collapse all

Fully specified ARIMA model, specified as an `arima` model object created by `arima` or `estimate`.

The properties of `Mdl` cannot contain `NaN` values.

Sample path length, specified as a positive integer. `numobs` is the number of random observations to generate per output path.

Data Types: `double`

Since R2023b

Presample data containing paths of responses yt, innovations εt, or conditional variances σt2 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 the same type as `Presample`. If `Presample` is a timetable, `Tbl` is a timetable that immediately follows `Presample` in time with respect to the sampling frequency.

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 responses, 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:

• At least `Mdl.P` when `Presample` provides only presample responses

• At least `Mdl.Q` when `Presample` provides only presample disturbances or conditional variances

• At least `max([Mdl.P Mdl.Q])` otherwise

When `Mdl.Variance` is a conditional variance model, `simulate` can require more than the minimum required number of presample values.

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 you specify `InSample`, `Presample` must immediately precede `InSample`, with respect to the sampling frequency.

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

By default, `simulate` sets the following values:

• For necessary presample responses:

• The unconditional mean of the model when `Mdl` represents a stationary AR process without a regression component

• Zero when `Mdl` represents a nonstationary process or when it contains a regression component.

• For necessary presample disturbances, zero.

• For necessary presample conditional variances, the unconditional variance of the conditional variance model n `Mdl.Variance`.

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

Since R2023b

Response variable yt to select from `Presample` containing presample response data, 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 ```PresampleResponseVariable(j) = true``` selects variable `j` from `Presample.Properties.VariableNames`

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

If you specify presample response data by using the `Presample` name-value argument, you must specify `PresampleResponseVariable`.

Example: `PresampleResponseVariable="Stock0"`

Example: `PresampleResponseVariable=[false false true false]` or `PresampleResponseVariable=3` selects the third table variable as the presample response variable.

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

Since R2023b

In-sample predictor data for the exogenous regression component of the model, specified as a table or timetable. `InSample` contains `numvars` variables, including `numpreds` predictor variables xt.

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

Each row corresponds to an observation in the simulation horizon, the first row is the earliest observation, and measurements in each row, among all paths, occur simultaneously. `InSample` must have at least `numobs` rows to cover the simulation horizon. If you supply more rows than necessary, `simulate` uses only the first `numobs` rows.

Each selected predictor variable is a numeric vector without missing values (`NaN`s). All predictor variables are present in the regression component of each response equation and apply to all response paths.

If `InSample` is a timetable, the following conditions apply:

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

• The datetime vector `InSample.Time` must be ascending or descending.

• If you specify `Presample`, `Presample` must immediately precede `InSample`, with respect to the sampling frequency.

If `InSample` is a table, the last row contains the latest observation.

By default, `simulate` does not include the regression component in the model, regardless of the value of `Mdl.Beta`.

Exogenous predictor variables xt to select from `InSample` containing predictor data for the regression component, specified as one of the following data types:

• String vector or cell vector of character vectors containing `numpreds` variable names in `InSample.Properties.VariableNames`

• A vector of unique indices (positive integers) of variables to select from `InSample.Properties.VariableNames`

• A logical vector, where `PredictorVariables(j) = true ` selects variable `j` from `InSample.Properties.VariableNames`

The selected variables must be numeric vectors and cannot contain missing values (`NaN`s).

By default, `simulate` excludes the regression component, regardless of its presence in `Mdl`.

Example: ```PredictorVariables=["M1SL" "TB3MS" "UNRATE"]```

Example: `PredictorVariables=[true false true false]` or `PredictorVariable=[1 3]` selects the first and third table variables to supply the predictor data.

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

### 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,10,NumPaths=1000,Y0=y0)` simulates `1000` sample paths of length `10` from the ARIMA model `Mdl`, and uses the observations in `y0` as a presample to initialize each generated path.

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

Example: `NumPaths=1000`

Data Types: `double`

Presample response data yt used as initial values for the model, specified as a `numpreobs`-by-1 numeric column vector or a `numpreobs`-by-`numprepaths` numeric matrix. Use `Y0` only when you supply optional data inputs as numeric arrays.

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

Each row is a presample observation (sampling time), and measurements in each row occur simultaneously. The last row contains the latest presample observation. `numpreobs` must be at least `Mdl.P` to initialize the AR model component. If `numpreobs` > `Mdl.P`, `simulate` uses the latest required number of observations only.

Columns of `Y0` are separate, independent presample paths. The following conditions apply:

• If `Y0` is a column vector, it represents a single response path. `simulate` applies it to each output path.

• If `Y0` is a matrix, `simulate` applies `Y0(:,j)` to initialize path `j`. `Y0` must have at least `NumPaths` columns; `simulate` uses only the first `NumPaths` columns of `Y0`.

By default, `simulate` sets any necessary presample responses to one of the following values:

• The unconditional mean of the model when `Mdl` represents a stationary AR process without a regression component

• Zero when `Mdl` represents a nonstationary process or when it contains a regression component

Data Types: `double`

Presample innovation data εt used to initialize either the moving average (MA) component of the ARIMA model or the conditional variance model, 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.

Each row is a presample observation (sampling time), and measurements in each row occur simultaneously. The last row contains the latest presample observation. `numpreobs` must be at least `Mdl.Q` to initialize the MA model component. If `Mdl.Variance` is a conditional variance model (for example, a `garch` model object), `simulate` can require more rows than `Mdl.Q`. If `numpreobs` is larger than required, `simulate` uses the latest required number of observations only.

Columns of `E0` are separate, independent presample paths. The following conditions apply:

• If `E0` is a column vector, it represents a single residual path. `simulate` applies it to each output path.

• If `E0` is a matrix, `simulate` applies `E0(:,j)` to initialize simulating path `j`. `E0` must have at least `NumPaths` columns; `simulate` uses only the first `NumPaths` columns of `E0`.

By default, `simulate` sets the necessary presample disturbances to zero.

Data Types: `double`

Presample conditional variance data σt2 used to initialize the conditional variance model, specified as a `numpreobs`-by-1 positive numeric column vector or a `numpreobs`-by-`numprepaths` positive numeric matrix. If the conditional variance `Mdl.Variance` is constant, `simulate` ignores `V0`. Use `V0` only when you supply optional data inputs as numeric arrays.

Each row is a presample observation (sampling time), and measurements in each row occur simultaneously. The last row contains the latest presample observation. `numpreobs` must be at least `Mdl.Q` to initialize the conditional variance model in `Mdl.Variance`. For details, see the `simulate` function of conditional variance models. If `numpreobs` is larger than required, `simulate` uses the latest required number of observations only.

Columns of `V0` are separate, independent presample paths. The following conditions apply:

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

• If `V0` is a matrix, `simulate` applies `V0(:,j)` to initialize simulating path `j`. `V0` must have at least `NumPaths` columns; `simulate` uses only the first `NumPaths` columns of `V0`.

By default, `simulate` sets all necessary presample observations to the unconditional variance of the conditional variance process.

Data Types: `double`

Since R2023b

Residual variable et to select from `Presample` containing the presample residual data, 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 ```PresampleInnovationVariable(j) = true``` selects variable `j` from `Presample.Properties.VariableNames`

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

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

Example: `PresampleInnovationVariable="StockRateDist0"`

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`

Since R2023b

Conditional variance variable σt2 to select from `Presample` containing presample conditional variance data, 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 (`NaN`s).

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`

Exogenous predictor data for the regression component in the model, specified as a numeric matrix with `numpreds` columns. `numpreds` is the number of predictor variables (`numel(Mdl.Beta)`). Use `X` only when you supply optional data inputs as numeric arrays.

Each row of `X` corresponds to a period in the length `numobs` simulation sample (period for which `simulate` simulates observations; the period after the presample). `X` must have at least `numobs` rows. The last row contains the latest predictor data. If `X` has more than `numobs` rows, `simulate` uses only the latest `numobs` rows.

`simulate` does not use the regression component in the presample period.

Columns of `X` are separate predictor variables.

`simulate` applies `X` to each simulated path; that is, `X` represents one path of observed predictors.

By default, `simulate` excludes the regression component, regardless of its presence in `Mdl`.

Data Types: `double`

Note

• `NaN` values in `X`, `Y0`, `E0`, and `V0` indicate missing values. `simulate` removes missing values from specified data by list-wise deletion.

• For the presample, `simulate` horizontally concatenates the possibly jagged arrays `Y0`, `E0`, and `V0` with respect to the last rows, and then it removes any row of the concatenated matrix containing at least one `NaN`.

• For in-sample data, `simulate` removes any row of `X` 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.

## Output Arguments

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Simulated response paths yt, returned as a `numobs`-by-1 numeric column vector or a `numobs`-by-`NumPaths` numeric matrix. `simulate` returns `Y` by default and when you supply optional data in numeric arrays.

`Y` represents the continuation of the presample responses in `Y0`.

Each row corresponds to a period in the simulated series; the simulated series has the periodicity of `Mdl`. Each column is a separate simulated path.

Simulated model innovations paths εt, returned as a `numobs`-by-1 numeric column vector or a `numobs`-by-`NumPaths` numeric matrix. `simulate` returns `E` by default and when you supply optional data in numeric arrays

The dimensions of `E` correspond to the dimensions of `Y`.

Simulated conditional variance paths σt2 of the mean-zero innovations associated with `Y`, returned as a `numobs`-by-1 numeric column vector or a `numobs`-by-`NumPaths` numeric matrix. `simulate` returns `V` by default and when you supply optional data in numeric arrays

The dimensions of `V` correspond to the dimensions of `Y`.

Since R2023b

Simulated response yt, innovation εt, and conditional variance σt2 paths, returned as a table or timetable, the same data type as `Presample` or `InSample`. `simulate` returns `Tbl` only when you supply at least one of the inputs `Presample` and `InSample`.

`Tbl` contains the following variables:

• 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 path in `Presample`, or each path corresponds, in time, with the rows of `InSample`. `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`.

• The simulated innovation 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 path in `Presample`, or each path corresponds, in time, with the rows of `InSample`. `simulate` names the simulated innovation variable in `Tbl` `responseName_Innovation`, where `responseName` is `Mdl.SeriesName`. For example, if `Mdl.SeriesName` is `StockReturns`, `Tbl` contains a variable for the corresponding simulated innovation paths with the name `StockReturns_Innovation`.

• 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 presample path in `Presample`, or each path corresponds, in time, with the rows of `InSample`. `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`.

• When you supply `InSample`, `Tbl` contains all variables in `InSample`.

If `Tbl` is a timetable, the following conditions hold:

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

• If you specify `InSample`, row times `Tbl.Time` are `InSample.Time(1:numobs)`. Otherwise, `Tbl.Time(1)` is the next time after `Presample(end)` relative to the sampling frequency, and `Tbl.Time(2:numobs)` are the following times relative to the sampling frequency.

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

 Enders, Walter. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, Inc., 1995.

 Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.