# filter

Filter disturbances using ARIMA or ARIMAX model

## Description

uses additional
options specified by one or more name-value arguments. For example, `Y`

= filter(`Mdl`

,`Z`

,`Name,Value`

)`filter(Mdl,Z,'X',X,'Z0',Z0)`

specifies
exogenous predictor data `X`

and presample disturbances
`Z0`

to initialize the model.

## Examples

### Compute Impulse Response Function

Specify a mean zero ARIMA(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 impulse response function (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 Step Response

Assess the dynamic behavior of a system to a persistent change in a variable by plotting a step response.

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); z = randn(100,1); x = simulate(MdlX,100);

Filter the disturbances `z`

using `MdlY`

to produce the response series `y`

, and plot `y`

.

y = filter(MdlY,z,'X',x); figure; plot(y); title 'Filter to simulate ARIMA(2,1,3)'; xlabel 'Time'; ylabel 'Response';

### Simulate and Filter Multiple Paths

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 paths from the model.

rng(1); % 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 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)" 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) % 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`

— Underlying disturbance series

numeric column vector | numeric matrix

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

`numObs`

numeric column vector or a
`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 period. 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`

### 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,'X',X,'Z0',Z0)`

specifies exogenous predictor
data `X`

and presample disturbances `Z0`

to initialize
the model.

`Y0`

— Presample response data

numeric column vector | numeric matrix

Presample response data providing initial values for the model, specified as a numeric column vector or a numeric matrix.

Each row of `Y0`

corresponds to a period in the presample. The
following conditions apply:

The last row contains the latest presample responses.

To initialize the AR components,

`Y0`

must have at least`Mdl.P`

rows.If

`Y0`

has more rows than is required to initialize the model,`filter`

uses only the latest required rows.

Each column of `Y0`

corresponds to a separate, independent
presample path. The following conditions apply:

If

`Y0`

is a column vector,`filter`

applies it to each path.If

`Y0`

is a matrix,`filter`

applies`Y0(:,`

to initialize path)`j`

.`j`

`Y0`

must have at least`numPaths`

columns;`filter`

uses only the first`numPaths`

columns of`Y0`

.

By default, `filter`

sets any necessary presample
observations by using one of the following methods:

For a model with a stationary AR process and without a regression component,

`filter`

sets all presample responses to the unconditional mean of the model.For a model that represents a nonstationary process or that contains a regression component,

`filter`

sets all necessary presample responses to zero.

**Data Types: **`double`

`Z0`

— Presample disturbance data

numeric column vector | numeric matrix

Presample disturbances providing initial values for the input disturbance series
`Z`

, specified as a numeric column vector or a numeric
matrix.

Each row of `Z0`

corresponds to a period in the presample. The
following conditions apply:

The last row contains the latest presample disturbances.

To initialize the MA components,

`Z0`

must have at least`Mdl.Q`

rows.If

`Mdl.Variance`

is a conditional variance model (for example, a`garch`

model object),`Z0`

can require more rows than`Mdl.Q`

to initialize the model.If

`Z0`

has more rows than is required to initialize the model,`filter`

uses only the latest required rows.

Each column of `Z0`

corresponds to a separate, independent
presample path. The following conditions apply:

If

`Z0`

is a column vector,`filter`

applies it to each path.If

`Z0`

is a matrix,`filter`

applies`Z0(:,`

to initialize path)`j`

.`j`

`Z0`

must have at least`numPaths`

columns;`filter`

uses only the first`numPaths`

columns of`Y0`

.

By default, `filter`

sets the necessary presample
disturbances to 0.

**Data Types: **`double`

`V0`

— Presample conditional variance data

positive numeric column vector | positive numeric matrix

Presample conditional variance data used to initialize the conditional variance
model, specified as a positive numeric column vector or a positive numeric matrix. If
the conditional variance `Mdl.Variance`

is constant,
`filter`

ignores `V0`

.

Each row of `V0`

corresponds to a period in the presample. The
following conditions apply:

The last row contains the latest presample conditional variances.

To initialize the conditional variance model,

`V0`

must have at least`Mdl.Q`

rows. For details, see the`filter`

function of conditional variance models.If

`V0`

has more rows than is required to initialize the conditional variance model,`filter`

uses only the latest required rows.

Each column of `V0`

corresponds to a separate, independent
presample path. The following conditions apply:

If

`V0`

is a column vector,`filter`

applies it to each simulated path.If

`V0`

is a matrix,`filter`

applies`V0(:,`

to initialize simulating path)`j`

.`j`

`V0`

must have at least`numPaths`

columns;`filter`

uses only the first`numPaths`

columns of`V0`

.

By default, `filter`

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

**Data Types: **`double`

`X`

— Exogenous predictor data

numeric matrix

Exogenous predictor data for the regression component in the model, specified as a
`numObs`

-by-`numPreds`

numeric matrix.

`numPreds`

is the number of predictor variables
(`numel(Mdl.Beta)`

).

Each row of `X`

corresponds to the corresponding period in
`Z`

(period for which `filter`

filters
disturbances; the period after the presample). The following conditions apply:

The last row contains the latest predictor data.

If the specified predictor data has more then

`numObs`

rows,`filter`

uses only the latest`numObs`

rows.`filter`

does not use the regression component in the presample period.

Each column of `X`

corresponds to a separate predictor
variable.

`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`

**Note**

`NaN`

s in input data indicate missing values.`filter`

uses*listwise deletion*to delete all sampled times (rows) in the input data containing at least one missing value. Specifically,`filter`

performs these steps:Synchronize, or merge, the presample data sets

`Y0`

,`Z0`

, and`V0`

to create the set`Presample`

, in other words,`PreSample`

=`[Y0 Z0 V0]`

.Synchronize the filter data

`Z`

and`X`

to create the set`Data`

, in other words,`Data`

=`[Z X]`

.Remove all rows from

`Presample`

and`Data`

containing at least one`NaN`

.

Listwise deletion applied to the filter data can reduce the sample size and create irregular time series.

`filter`

merges data sets that have a different number of rows according to the latest observation, and then pads the shorter series with enough default values to form proper matrices.`filter`

assumes that you synchronize the data sets so that the latest observations occur simultaneously. The software also assumes that you synchronize the presample series similarly.

## 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.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}*
with conditional variances $${\sigma}_{t}^{2}$$ (

`V`

), returned as a length
`numObs`

column vector or a
`numObs`

-by-`numPaths`

numeric matrix. 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

numeric column vector | numeric matrix

Conditional variance paths $${\sigma}_{t}^{2}$$, returned as a length `numObs`

column vector or
`numObs`

-by-`numPaths`

numeric matrix. 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`

.

## 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**

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