# filter

Filter disturbances using univariate ARIMA or ARIMAX model

## Description

returns the numeric array of one or more response series `Y`

= filter(`Mdl`

,`Z`

)`Y`

resulting
from filtering the numeric array of one or more underlying disturbance series
`Z`

through the fully specified, univariate ARIMA model
`Mdl`

. `Z`

is associated with the model innovations
process that drives the specified ARIMA model.

returns the table or timetable `Tbl2`

= filter(`Mdl`

,`Tbl1`

)`Tbl2`

containing the results from
filtering the paths of disturbances in the input table or timetable
`Tbl1`

through `Mdl`

. The disturbance variable in
`Tbl1`

is associated with the model innovations process through
`Mdl`

. * (since R2023b)*

`filter`

selects the variable
`Mdl.SeriesName`

, or the sole variable in `Tbl1`

, as
the disturbance variable to filter through the model. To select a different variable in
`Tbl1`

to filter through the model, use the
`DisturbanceVariable`

name-value argument.

`[___] = filter(___,`

specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
`Name,Value`

)`filter`

returns the output argument combination for the
corresponding input arguments. For example, `filter(Mdl,Z,Z0=PS,X=Pred)`

filters the
numeric vector of disturbances `Z`

through the ARIMAX
`Mdl`

, and specifies the numeric vector of presample disturbance data
`PS`

to initialize the model and the exogenous predictor data
`X`

for the regression component.

## Examples

### Filter Vector of Disturbances Through Model

Compute the impulse response function (IRF) of an ARMA model by filtering a vector of zeros, representing disturbances, through the model.

Specify a mean zero ARMA(2,0,1) model.

```
Mdl = arima(Constant=0,AR={0.5 -0.8},MA=-0.5, ...
Variance=0.1);
```

Simulate the first 20 responses of the IRF. Generate a disturbance series with a one-time, unit impulse, and then filter it.

z = [1; zeros(19,1)]; y = filter(Mdl,z);

`y`

is a 20-by-1 response path resulting from filtering the disturbance path `z`

through the model. y represents the IRF. The `filter`

function requires one presample observation to initialize the model. By default, `filter`

uses the unconditional mean of the process, which is `0`

.

y = y/y(1);

Normalize the IRF such that the first element is 1.

Plot the impulse response function.

figure stem((0:numel(y)-1)',y,"filled"); title("Impulse Response")

The impulse response assesses the dynamic behavior of a system to a one-time, unit impulse.

Alternatively, you can use the `impulse`

function to plot the IRF for an ARIMA process.

### Simulate and Filter Multiple Paths

Filter a matrix of disturbance paths. Return the paths of responses and innovations, which drive the data-generating processes.

Create a mean zero ARIMA(2,0,1) model.

```
Mdl = arima(Constant=0,AR={0.5,-0.8},MA=-0.5, ...
Variance=0.1);
```

Generate 20 random, length 100 paths from the model.

rng(1,"twister"); % For reproducibility [ySim,eSim,vSim] = simulate(Mdl,100,NumPaths=20);

`ySim`

, `eSim`

, and `vSim`

are 100-by-20 matrices of 20 simulated response, innovation, and conditional variance paths of length 100, respectively. Because Mdl does not have a conditional variance model, `vSim`

is a matrix completely composed of the value of `Mdl.Variance`

.

Obtain disturbance paths by standardizing the simulated innovations.

zSim = eSim./sqrt(vSim);

Filter the disturbance paths through the model.

[yFil,eFil] = filter(Mdl,zSim);

`yFil`

and `eFil`

are 100-by-20 matrices. The columns are independent paths generated from filtering corresponding disturbance paths in `zSim`

through the model `Mdl`

.

Confirm that the outputs of `simulate`

and `filter`

are identical.

sameE = norm(eSim - eFil) < eps

`sameE = `*logical*
1

sameY = norm(ySim - yFil) < eps

`sameY = `*logical*
1

The logical values `1`

confirm the outputs are effectively identical.

### Filter Disturbance Path in Timetable

*Since R2023b*

Fit an ARIMA(1,1,1) model to the weekly average NYSE closing prices. Supply a timetable of data and specify the series for the fit. Then, filter randomly generated Gaussian noise paths through the estimated model to simulate responses and innovations.

**Load Data**

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. Set the response series name to `NYSE`

.

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

`Mdl`

is a partially specified `arima`

model object.

**Fit Model to Data**

Fit an ARIMA(1,1,1) model to weekly average NYSE closing prices. Specify the entire series.

EstMdl = estimate(Mdl,DTTW);

ARIMA(1,1,1) Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ ___________ Constant 0.86386 0.46496 1.8579 0.06318 AR{1} -0.37582 0.22719 -1.6542 0.09809 MA{1} 0.47221 0.21741 2.172 0.029858 Variance 55.89 1.832 30.507 2.1199e-204

`EstMdl`

is a fully specified, estimated `arima`

model object. By default, `estimate`

backcasts for the required `Mdl.P = 2`

presample responses.

**Filter Random Gaussian Disturbance Paths**

Generate 2 random, independent series of length `T_DTTW`

from the standard Gaussian distribution. Store the matrix of series as one variable in `DTTW`

.

rng(1,"twister") % For reproducibility DTTW.Z = randn(T_DTTW,2);

DTTW contains a new variable called Z containing a `T_DTTW`

-by-2 matrix of two disturbance paths.

Filter the paths of disturbances through the estimated ARIMA model. Specify the table variable name containing the disturbance paths.

```
Tbl2 = filter(EstMdl,DTTW,DisturbanceVariable="Z");
tail(Tbl2)
```

Time NYSE NASDAQ Z NYSE_Response NYSE_Innovation NYSE_Variance ___________ ______ ______ _____________________ ________________ ___________________ ______________ 16-Nov-2001 577.11 1886.9 -1.8948 0.41292 358.78 433.57 -14.166 3.087 55.89 55.89 23-Nov-2001 583 1898.3 1.3583 0.27051 367.95 436.63 10.155 2.0223 55.89 55.89 30-Nov-2001 581.41 1925.8 -0.9118 1.1119 363.35 445.61 -6.8165 8.3125 55.89 55.89 07-Dec-2001 584.96 1998.1 -0.14964 -2.418 361.61 428.95 -1.1187 -18.077 55.89 55.89 14-Dec-2001 574.03 1981 -0.40114 0.98498 359.6 434.9 -2.9989 7.3636 55.89 55.89 21-Dec-2001 582.1 1967.9 -0.57758 0.0039243 355.48 437.04 -4.318 0.029338 55.89 55.89 28-Dec-2001 590.28 1967.2 2.0039 -0.92415 370.84 430.2 14.981 -6.9089 55.89 55.89 04-Jan-2002 589.8 1950.4 -0.50964 -0.43856 369.19 427.09 -3.8101 -3.2787 55.89 55.89

size(Tbl2)

`ans = `*1×2*
627 6

`Tbl2`

is a 627-by-6 timetable containing all variables in `DTTW`

, and the two filtered response paths `NYSE_Response`

, innovation paths `NYSE_Innovation`

, and constant variance paths `NYSE_Variance`

(`Mdl.Variance = 55.89`

).

### Supply Presample Responses

Assess the dynamic behavior of a system to a persistent change in a variable by plotting a step response. Supply presample responses to initialize the model.

Specify a mean zero ARIMA(2,0,1) process.

```
Mdl = arima(Constant=0,AR={0.5 -0.8},MA=-0.5, ...
Variance=0.1);
```

Simulate the first 20 responses to a sequence of unit disturbances. Generate a disturbance series of ones, and then filter it. Set all presample observations equal to zero.

Z = ones(20,1); Y = filter(Mdl,Z,Y0=zeros(Mdl.P,1)); Y = Y/Y(1);

The last step normalizes the step response function to ensure that the first element is 1.

Plot the step response function.

figure stem((0:numel(Y)-1)',Y,"filled"); title("Step Response")

### Simulate Responses from ARIMAX Model

Create models for the response and predictor series. Set an ARIMAX(2,1,3) model to the response `MdlY`

, and an AR(1) model to the `MdlX`

.

```
MdlY = arima(AR={0.1 0.2},D=1,MA={-0.1 0.1 0.05}, ...
Constant=1,Variance=0.5,Beta=2);
MdlX = arima(AR=0.5,Constant=0,Variance=0.1);
```

Simulate a length 100 predictor series `x`

and a series of iid normal disturbances `z`

having mean zero and variance 1.

```
rng(1,"twister")
z = randn(100,1);
x = simulate(MdlX,100);
```

Filter the disturbances `z`

using `MdlY`

to produce the response series `y`

. Plot `y`

.

y = filter(MdlY,z,X=x); figure plot(y); xlabel("Time") ylabel("Response")

### Filter Disturbances Through Composite Conditional Mean and Variance Model

Create the composite AR(1)/GARCH(1,1) model

$$\begin{array}{l}{y}_{t}=1+0.5{y}_{t-1}+{\epsilon}_{t}\\ {\epsilon}_{t}={\sigma}_{t}{z}_{t}\\ {\sigma}_{t}^{2}=0.2+0.1{\sigma}_{t-1}^{2}+0.05{\epsilon}_{t-1}^{2}\\ {z}_{t}\sim N(0,1).\end{array}$$

Create the composite model.

CVMdl = garch(Constant=0.2,GARCH=0.1,ARCH=0.05); Mdl = arima(Constant=1,AR=0.5,Variance=CVMdl)

Mdl = arima with properties: Description: "ARIMA(1,0,0) Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 1 D: 0 Q: 0 Constant: 1 AR: {0.5} at lag [1] SAR: {} MA: {} SMA: {} Seasonality: 0 Beta: [1×0] Variance: [GARCH(1,1) Model]

`Mdl`

is an `arima`

object. The property `Mdl.Variance`

contains a `garch`

object that represents the conditional variance model.

Generate a random series of 100 standard Gaussian of disturbances.

rng(1,"twister") % For reproducibility z = randn(100,1);

Filter the disturbances through the model. Return and plot the simulated conditional variances.

[y,e,v] = filter(Mdl,z); plot(z)

## Input Arguments

`Z`

— Disturbance series paths *z*_{t}

numeric column vector | numeric matrix

_{t}

Underlying disturbance paths *z _{t}*, specified
as a

`numobs`

-by-1 numeric column vector or
`numobs`

-by-`numpaths`

numeric matrix.
`numObs`

is the length of the time series (sample size).
`numpaths`

is the number of separate, independent disturbance
paths.*z _{t}* drives the innovation process

*ε*. For a variance process

_{t}*σ*

_{t}^{2}, the innovation process is

$${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$

Each row corresponds to a sampling time. The last row contains the latest set of disturbances.

Each column corresponds to a separate, independent path of disturbances.
`filter`

assumes that disturbances across any row occur
simultaneously.

`Z`

is the continuation of the presample disturbances
`Z0`

.

**Data Types: **`double`

`Tbl1`

— Time series data

table | timetable

*Since R2023b*

Time series data containing the observed disturbance variable
*z _{t}*, associated with the model innovations
process

*ε*, and, optionally, predictor variables

_{t}*x*, specified as a table or timetable with

_{t}`numvars`

variables and `numobs`

rows.
You can optionally select the disturbance variable or `numpreds`

predictor variables by using the `DisturbanceVariable`

or
`PredictorVariables`

name-value arguments, respectively.For a variance process
*σ _{t}*

^{2}, the innovation process is

$${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$

Each row is an observation, and measurements in each row occur simultaneously. The
selected disturbance variable is a single path (`numobs`

-by-1 vector)
or multiple paths (`numobs`

-by-`numpaths`

matrix) of
`numobs`

observations of disturbance data.

Each path (column) of the selected disturbance variable is independent of the other
paths, but path

of all presample and
in-sample variables correspond, for `j`

=
1,…,`j`

`numpaths`

. Each selected predictor variable is a
`numobs`

-by-1 numeric vector representing one path. The
`filter`

function includes all predictor variables in the
model when it filters each disturbance path. Variables in `Tbl1`

represent the continuation of corresponding variables in
`Presample`

.

If `Tbl1`

is a timetable, it must represent a sample with a
regular datetime time step (see `isregular`

), and the datetime vector `Tbl1.Time`

must be
strictly ascending or descending.

If `Tbl1`

is a table, the last row contains the latest
observation.

### 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: **`filter(Mdl,Z,Z0=PS,X=Pred)`

specifies the numeric vector of
presample disturbance data `PS`

to initialize the model and the exogenous
predictor data `X`

for the regression component.

`DisturbanceVariable`

— Disturbance variable *z*_{t} to select from `Tbl1`

string scalar | character vector | integer | logical vector

_{t}

*Since R2023b*

Disturbance variable *z _{t}* to select from

`Tbl1`

containing the disturbance data to filter through
`Mdl`

, specified as one of the following data types:String scalar or character vector containing a variable name in

`Tbl1.Properties.VariableNames`

Variable index (positive integer) to select from

`Tbl1.Properties.VariableNames`

A logical vector, where

`DisturbanceVariable(`

selects variable) = true`j`

from`j`

`Tbl1.Properties.VariableNames`

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

s).

If `Tbl1`

has one variable, the default specifies that variable.
Otherwise, the default matches the variable to names in
`Mdl.SeriesName`

.

**Example: **`DisturbanceVariable="StockRateDist"`

**Example: **`DisturbanceVariable=[false false true false]`

or
`DisturbanceVariable=3`

selects the third table variable as the
disturbance variable.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`Y0`

— Presample response data *y*_{t}

numeric column vector | numeric matrix

_{t}

Presample response data *y _{t}* to initialize
the model, specified as a

`numpreobs`

-by-1 numeric column vector or a
`numpreobs`

-by-`numprepaths`

numeric matrix. Use
`Y0`

only when you supply the numeric array of disturbance data
`Z`

.`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`

,
`filter`

uses the latest required 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.`filter`

applies it to each output path.If

`Y0`

is a matrix, each column represents a presample response path.`filter`

applies`Y0(:,`

to initialize path)`j`

.`j`

`numprepaths`

must be at least`numpaths`

. If`numprepaths`

>`numpaths`

,`filter`

uses the first`size(Z,2)`

columns only.

By default, `filter`

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 componentZero when

`Mdl`

represents a nonstationary process or when it contains a regression component

**Data Types: **`double`

`Z0`

— Presample disturbance data *z*_{t}

numeric column vector | numeric matrix

_{t}

Presample disturbance data *z _{t}* providing
initial values for the input disturbance series

`Z`

, specified as a
`numpreobs`

-by-1 numeric column vector or a
`numpreobs`

-by-`numprepaths`

numeric matrix. Use
`Z0`

only when you supply the numeric array of disturbance data
`Z`

.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),
`filter`

can require more rows than
`Mdl.Q`

. If `numpreobs`

is larger than required,
`filter`

uses the latest required observations only.

Columns of `Z0`

are separate, independent presample paths. The
following conditions apply:

If

`Z0`

is a column vector, it represents a single disturbance path.`filter`

applies it to each output path.If

`Z0`

is a matrix, each column represents a presample disturbance path.`filter`

applies`Z0(:,`

to initialize path)`j`

.`j`

`numprepaths`

must be at least`numpaths`

. If`numprepaths`

>`numpaths`

,`filter`

uses the first`size(Z,2)`

columns only.

By default, `filter`

sets the necessary presample
disturbances to zero.

**Data Types: **`double`

`V0`

— Presample conditional variance data *σ*_{t}^{2}

positive numeric column vector | positive numeric matrix

_{t}

Presample conditional variance data
*σ _{t}*

^{2}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,
`filter`

ignores `V0`

. Use
`V0`

only when you supply the numeric array of disturbance data
`Z`

.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 `filter`

function of conditional variance
models. If `numpreobs`

is larger than required,
`filter`

uses the latest required 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.`filter`

applies it to each output path.If

`V0`

is a matrix, each column represents a presample path of conditional variances.`filter`

applies`V0(:,`

to initialize path)`j`

.`j`

`numprepaths`

must be at least`numpaths`

. If`numprepaths`

>`numpaths`

,`filter`

uses the first`size(Z,2)`

columns only.

By default, `filter`

sets all necessary presample
conditional variances to the unconditional variance of the conditional variance
process.

**Data Types: **`double`

`Presample`

— Presample data

table | timetable

*Since R2023b*

Presample data containing paths of response
*y _{t}*, disturbance

*z*, or conditional variance

_{t}*σ*

_{t}^{2}series 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`

.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 the
response, disturbance, or conditional variance series for
`DisturbanceVariable`

, the selected disturbance variable in
`Tbl1`

.

Each row is a presample observation, and measurements in each row occur
simultaneously. `numpreobs`

must be one of the following values:

At least

`Mdl.P`

when`Presample`

provides only presample responsesAt least

`Mdl.Q`

when`Presample`

provides only presample disturbances or conditional variancesAt least

`max([Mdl.P Mdl.Q])`

otherwise

When `Mdl.Variance`

is a conditional variance model,
`filter`

can require more than the minimum required number
of presample values.

If you supply more rows than necessary, `filter`

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 inputs

`Tbl1`

and`Presample`

must be consistent in time such that`Presample`

immediately precedes`Tbl1`

with respect to the sampling frequency and order.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.

By default, `filter`

sets the following values:

For necessary presample responses:

The unconditional mean of the model when

`Mdl`

represents a stationary AR process without a regression componentZero 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, disturbance, or conditional variance name by using the
`PresampleResponseVariable`

,
`PresampleDisturbanceVariable`

, or
`PresampleVarianceVariable`

name-value argument.

`PresampleResponseVariable`

— Response variable *y*_{t} to select from `Presample`

string scalar | character vector | integer | logical vector

_{t}

*Since R2023b*

Response variable *y _{t}* 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(`

selects variable) = true`j`

from`j`

`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`

`PresampleDisturbanceVariable`

— Disturbance variable *z*_{t} to select from `Presample`

string scalar | character vector | integer | logical vector

_{t}

*Since R2023b*

Disturbance variable *z _{t}* to select from

`Presample`

containing presample disturbance 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

`PresampleDisturbanceVariable(`

selects variable) = true`j`

from`j`

`Presample.Properties.VariableNames`

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

s).

If you specify presample disturbance data by using the
`Presample`

name-value argument, you must specify
`PresampleDisturbanceVariable`

.

**Example: **`PresampleDisturbanceVariable="StockRateDist0"`

**Example: **`PresampleDisturbanceVariable=[false false true false]`

or `PresampleDisturbanceVariable=3`

selects the third table variable
as the presample disturbance variable.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`PresampleVarianceVariable`

— Conditional variance variable *σ*_{t}^{2} to select from
`Presample`

string scalar | character vector | integer | logical vector

_{t}

*Since R2023b*

Conditional variance variable
*σ _{t}*

^{2}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(`

selects variable) = true`j`

from`j`

`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`

`X`

— Exogenous predictor data

numeric matrix

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 the numeric array of disturbance data
`Z`

.

`X`

must have at least `numobs`

rows. The last
row contains the latest predictor data. If `X`

has more than
`numobs`

rows, `filter`

uses only the
latest `numobs`

rows. Each row of `X`

corresponds to
each period in `Z`

(period for which
`filter`

filters errors; the period after the
presample).

`filter`

does not use the regression component in the
presample period.

Columns of `X`

are separate predictor variables.

`filter`

applies `X`

to each filtered
path; that is, `X`

represents one path of observed predictors.

By default, `filter`

excludes the regression component,
regardless of its presence in `Mdl`

.

**Data Types: **`double`

`PredictorVariables`

— Exogenous predictor variables *x*_{t} to select from `Tbl1`

string vector | cell vector of character vectors | vector of integers | logical vector

_{t}

*Since R2023b*

Exogenous predictor variables *x _{t}* to select
from

`Tbl1`

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`Tbl1.Properties.VariableNames`

A vector of unique indices (positive integers) of variables to select from

`Tbl1.Properties.VariableNames`

A logical vector, where

`PredictorVariables(`

selects variable) = true`j`

from`j`

`Tbl1.Properties.VariableNames`

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

s).

By default, `filter`

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`

**Note**

`NaN`

values in`Z`

,`X`

,`Y0`

,`Z0`

, and`V0`

indicate missing values.`filter`

removes missing values from specified data by list-wise deletion.For the presample,

`filter`

horizontally concatenates the possibly jagged arrays`Y0`

,`Z0`

, 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,

`filter`

horizontally concatenates the possibly jagged arrays`Z`

and`X`

, 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,

`filter`

assumes that you synchronize the presample data such that the latest observations occur simultaneously.`filter`

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

## Output Arguments

`Y`

— Simulated response paths *y*_{t}

numeric column vector | numeric matrix

_{t}

Simulated response paths *y _{t}*, returned as a
length

`numobs`

column vector or a
`numobs`

-by-`numpaths`

numeric matrix.
`filter`

returns `Y`

only when you supply
the input `Z`

.For each

= 1, …,
`t`

`numobs`

, the simulated response at time
`t`

`Y(`

corresponds to the filtered
disturbance at time * t*,:)

`t`

`Z(``t`

,:)

and response path
`j`

`Y(:,``j`

)

corresponds to the filtered
disturbance path `j`

`Z(:,``j`

)

.`Y`

represents the continuation of the presample response paths
in `Y0`

.

`E`

— Simulated paths of model innovations *ε*_{t}

numeric column vector | numeric matrix

_{t}

Simulated paths of model innovations *ε _{t}*,
returned as a length

`numobs`

column vector or a
`numobs`

-by-`numpaths`

numeric matrix.
`filter`

returns `E`

only when you supply
the input `Z`

. The dimensions of `Y`

and
`E`

correspond.Columns of `E`

are scaled disturbance paths (innovations) such
that, for a particular path

$${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$

`V`

— Conditional variance paths *σ*_{t}^{2}

numeric column vector | numeric matrix

_{t}

Conditional variance paths
*σ _{t}*

^{2}, returned as a length

`numobs`

column vector or
`numobs`

-by-`numpaths`

numeric matrix.
`filter`

returns `V`

only when you supply
the input `Z`

. The dimensions of `Y`

and
`V`

correspond.If `Z`

is a matrix, then the columns of `V`

are
the filtered conditional variance paths corresponding to the columns of
`Z`

.

Columns of `V`

are conditional variance paths of corresponding
paths of innovations *ε _{t}*
(

`E`

) such that, for a particular path$${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$

`V`

represents the continuation of the presample conditional
variance paths in `V0`

.

`Tbl2`

— Simulated response *y*_{t}, innovation *ε*_{t}, and conditional variance
*σ*_{t}^{2}
paths

table | timetable

_{t}

_{t}

_{t}

*Since R2023b*

Simulated response *y _{t}*, innovation

*ε*, and conditional variance

_{t}*σ*

_{t}^{2}paths, returned as a table or timetable, the same data type as

`Tbl1`

.
`filter`

returns `Tbl2`

only when you
supply the input `Tbl1`

.`Tbl2`

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 corresponding to the input observations and paths of the disturbance variable in`Tbl1`

.`filter`

names the simulated response variable in`Tbl2`

, where_Response`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

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 corresponding to the input observations and paths of the disturbance variable in`Tbl1`

.`filter`

names the simulated innovation variable in`Tbl2`

, where_Innovation`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

contains a variable for the corresponding simulated innovation paths with the name`StockReturns_Innovation`

.The simulated conditional variances paths, which are in a

`numobs`

-by-`numpaths`

numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths of the disturbance variable in`Tbl1`

.`filter`

names the simulated conditional variance variable in`Tbl2`

, where_Variance`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

contains a variable for the corresponding simulated conditional variance paths with the name`StockReturns_Variance`

.All variables

`Tbl1`

.

If `Tbl1`

is a timetable, row times of `Tbl1`

and `Tbl2`

are equal.

## Alternative Functionality

`filter`

generalizes `simulate`

; both functions filter a series of disturbances to produce output
responses, innovations, and conditional variances. However, `simulate`

autogenerates a series of mean zero, unit variance, independent and identically distributed
(iid) disturbances according to the distribution in `Mdl`

. In contrast,
`filter`

enables you to directly specify custom disturbances.

## References

[1] 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.

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

[3] Hamilton, James D. *Time Series Analysis*. Princeton, NJ: Princeton University Press, 1994.

## Version History

**Introduced in R2012b**

### R2023b: `filter`

accepts input data in tables and timetables, and returns results in tables and timetables

In addition to accepting input data (in-sample and presample) in numeric arrays,
`filter`

accepts input data in tables or regular timetables. When
you supply data in a table or timetable, the following conditions apply:

`filter`

chooses the default in-sample disturbance series and predictor data on which to operate, but you can use the specified optional name-value argument to select a different series.If you specify optional presample data to initialize the model, you must also specify the presample response, disturbance, or conditional variance series name.

`filter`

returns results in a table or timetable.

Name-value arguments to support tabular workflows include:

`DisturbanceVariable`

specifies the name of the disturbance series to select from the input data to filter through the model.`Presample`

specifies the input table or timetable of presample response, disturbance, and conditional variance data.`PresampleResponseVariable`

specifies the name of the response series to select from`Presample`

.`PresampleDisturbanceVariable`

specifies the name of the disturbance series to select from`Presample`

.`PresampleVarianceVariable`

specifies the name of the conditional variance series to select from`Presample`

.`PredictorVariables`

specifies the names of the predictor series to select from the input data for a model regression component.

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