# filter

Filter disturbances through regression model with ARIMA errors

## Description

returns a 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 regression model with
ARIMA errors `Mdl`

. `Z`

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

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 that drives
`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,X=Pred,Z0=PSZ)`

specifies the
predictor data `Pred`

for the model regression component and the
observed errors in the presample period `PSZ`

to initialize the
model.

## Examples

### Filter Disturbance Vector to Compute Impulse Response Function

Compute the impulse response function (IRF) of an innovation shock to the regression model with ARMA(2,1) errors. Supply the innovation shock as a vector.

The IRF assesses the dynamic behavior of a system to a one-time shock. Typically, the magnitude of the shock is 1. Alternatively, it might be more meaningful to examine an IRF of an innovation shock with a magnitude of one standard deviation.

In regression models with ARIMA errors,

The IRF is invariant to the behavior of the predictors and the intercept.

The IRF of the model is defined as the impulse response of the unconditional disturbances as governed by the ARIMA error component.

Create the following regression model with ARMA(2,1) errors:

$$\begin{array}{c}{y}_{t}={u}_{t}\\ {u}_{t}=0.5{u}_{t-1}-0.8{u}_{t-2}+{\epsilon}_{t}-0.5{\epsilon}_{t-1},\end{array}$$

where $${\epsilon}_{t}$$ is Gaussian with variance 0.1.

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

When you construct an impulse response function for a regression model with ARIMA errors, you must set `Intercept`

to 0.

Simulate the first 30 responses of the impulse response function by generating a error series with a one-time impulse with magnitude equal to one standard deviation, and then filter it. Also, use `impulse`

to compute the IRF.

```
z = [sqrt(Mdl.Variance); zeros(29,1)]; % Shock of 1 std
yFltr = filter(Mdl,z);
yImpls = impulse(Mdl,30);
```

When you construct an IRF of a regression model with ARIMA errors containing a regression component, do not specify the predictor matrix, `X`

, in `filter`

.

Plot the IRFs.

figure tiledlayout(2,1) nexttile stem((0:numel(yFltr)-1)',yFltr,"filled") title("Impulse Response to Shock of One Standard Deviation") nexttile stem((0:numel(yImpls)-1)',yImpls,"filled") title("Impulse Response to Unit Shock")

The IRF given a shock of one standard deviation is a scaled version of the IRF returned by `impulse`

.

### Compute Step Response

Compute the step response function of a regression model with ARMA(2,1) errors.

The step response assesses the dynamic behavior of a system to a persistent shock. Typically, the magnitude of the shock is 1. Alternatively, it might be more meaningful to examine a step response of a persistent innovation shock with a magnitude of one standard deviation. This example plots the step response of a persistent innovations shock in a model without an intercept and predictor matrix for regression. However, note that `filter`

is flexible in that it accepts a persistent innovations or predictor shock that you construct using any magnitude, then filters it through the model.

Specify the following regression model with ARMA(2,1) errors:

$$\begin{array}{c}{y}_{t}={u}_{t}\\ {u}_{t}=0.5{u}_{t-1}-0.8{u}_{t-2}+{\epsilon}_{t}-0.5{\epsilon}_{t-1},\end{array}$$

where $${\epsilon}_{t}$$ is Gaussian with variance 0.1.

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

Compute the first 30 responses to a sequence of unit errors by generating an error series of one standard deviation, and then filtering it.

z = sqrt(Mdl.Variance)*ones(30,1); % Persistent shock of one std y = filter(Mdl,z); y = y/y(1); % Normalize relative to y(1)

Plot the step response function.

figure stem((0:numel(y)-1)',y,"filled") title("Step Response for Persistent Shock of One STD")

The step response settles around 0.4.

### Filter Timetable of Disturbances Through Estimated Model

Fit a regression model with ARMA(1,1) errors by regressing the US consumer price index (CPI) quarterly changes onto the US gross domestic product (GDP) growth rate. Supply a timetable of data and specify the series for the fit. Then, filter paths of disturbances in a timetable through the fitted model.

**Load and Transform Data**

Load the US macroeconomic data set. Compute the series of GDP quarterly growth rates and CPI quarterly changes.

load Data_USEconModel DTT = price2ret(DataTimeTable,DataVariables="GDP"); DTT.GDPRate = 100*DTT.GDP; DTT.CPIDel = diff(DataTimeTable.CPIAUCSL); T = height(DTT)

T = 248

figure tiledlayout(2,1) nexttile plot(DTT.Time,DTT.GDPRate) title("GDP Rate") ylabel("Percent Growth") nexttile plot(DTT.Time,DTT.CPIDel) title("Index")

The series appear stationary, albeit heteroscedastic.

**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.

DTT = rmmissing(DTT); T_DTT = height(DTT)

T_DTT = 248

Because each sample time has an observation for all variables, `rmmissing`

does not remove any observations.

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

`areTimestampsRegular = isregular(DTT,"quarters")`

`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. Macroeconomic series in this example are timestamped at the end of the month. This quality induces an irregularly measured series.

Remedy the time irregularity by shifting all dates to the first day of the quarter.

dt = DTT.Time; dt = dateshift(dt,"start","quarter"); DTT.Time = dt; areTimestampsRegular = isregular(DTT,"quarters")

`areTimestampsRegular = `*logical*
1

`DTT`

is regular.

**Create Model Template for Estimation**

Suppose that a regression model of CPI quarterly changes onto the GDP rate, with ARMA(1,1) errors, is appropriate.

Create a model template for a regression model with ARMA(1,1) errors template.

Mdl = regARIMA(1,0,1)

Mdl = regARIMA with properties: Description: "ARMA(1,1) Error Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" Intercept: NaN Beta: [1×0] P: 1 Q: 1 AR: {NaN} at lag [1] SAR: {} MA: {NaN} at lag [1] SMA: {} Variance: NaN

`Mdl`

is a partially specified `regARIMA`

object.

**Fit Model to Data**

Fit a regression model with ARMA(1,1) errors to the data. Specify the entire series GDP rate and CPI quarterly changes series, and specify the response and predictor variable names.

EstMdl = estimate(Mdl,DTT,ResponseVariable="GDPRate", ... PredictorVariables="CPIDel");

Regression with ARMA(1,1) Error Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ __________ Intercept 0.0162 0.0016077 10.077 6.9995e-24 AR{1} 0.60515 0.089912 6.7305 1.6906e-11 MA{1} -0.16221 0.11051 -1.4678 0.14216 Beta(1) 0.002221 0.00077691 2.8587 0.0042532 Variance 0.000113 7.2753e-06 15.533 2.0838e-54

`EstMdl`

is a fully specified, estimated `regARIMA`

object.

**Filter Random Gaussian Disturbance Paths**

Generate 2 random, independent series of length `T_DTT`

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

.

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

`DTT`

contains a new variable called `Z`

containing a `T_DTT`

-by-2 matrix of two disturbance paths.

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

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

Time Interval GDP GDPRate CPIDel Z Y_Response Y_ErrorInnovation Y_RegressionInnovation _____ ________ ___________ __________ ______ ______________________ ______________________ __________________________ __________________________ Q2-07 91 0.00018278 0.018278 1.675 -0.36436 -0.7055 0.016068 0.0071243 -0.0038733 -0.0074997 -0.0001316 -0.0090757 Q3-07 91 0.00016916 0.016916 1.359 -0.093312 -0.3311 0.015757 0.0084046 -0.00099194 -0.0035197 -0.00044331 -0.0077954 Q4-07 94 6.1286e-05 0.0061286 3.355 0.48981 -1.5208 0.021299 -0.0041131 0.0052068 -0.016167 0.0050995 -0.020313 Q1-08 91 9.3272e-05 0.0093272 1.93 1.4014 0.16528 0.033339 0.0082868 0.014898 0.001757 0.017139 -0.0079132 Q2-08 91 0.00011103 0.011103 3.367 -0.27422 -0.48787 0.02124 0.00594 -0.0029151 -0.0051862 0.0050402 -0.01026 Q3-08 92 8.9585e-05 0.0089585 1.641 0.67582 0.58697 0.026907 0.017072 0.0071842 0.0062397 0.010707 0.00087209 Q4-08 92 -0.00016145 -0.016145 -7.098 0.19058 -0.90337 0.02354 0.0061124 0.0020259 -0.0096032 0.00734 -0.010088 Q1-09 90 -8.6878e-05 -0.0086878 1.137 0.67036 0.37101 0.027439 0.015597 0.0071262 0.003944 0.011239 -0.00060284

size(Tbl2)

`ans = `*1×2*
248 8

`Tbl2`

is a 248-by-8 timetable containing all variables in DTT, and the two filtered response paths `Y_Response`

, error model innovation paths `Y_ErrorInnovation`

, and unconditional disturbance paths `Y_RegressionInnovation`

.

### Filter Error Paths Through Regression Model with SARIMA Errors

Simulate 100 independent paths of responses by filtering 100 independent paths of errors ${\mathit{z}}_{\mathit{t}}$, where innovations ${\epsilon}_{\mathit{t}}=\sigma \text{\hspace{0.17em}}{\mathit{z}}_{\mathit{t}}$, through the following regression model with SARIMA${\left(2,1,1\right)}_{12}$ errors.

$$\begin{array}{c}{y}_{t}=X\left[\begin{array}{c}1.5\\ -2\end{array}\right]+{u}_{t}\\ \left(1-0.2L-0.1{L}^{2}\right)\left(1-L\right)\left(1-0.01{L}^{12}\right)\left(1-{L}^{12}\right){u}_{t}=\left(1+0.5L\right)\left(1+0.02{L}^{12}\right){\epsilon}_{t},\end{array}$$

where $${\epsilon}_{t}$$ follows a $\mathit{t}$-distribution with 15 degrees of freedom.

Distribution = struct("Name","t","DoF",15); Mdl = regARIMA(AR={0.2 0.1},SAR=0.01,SARLags=12, ... MA=0.5,SMA=0.02,SMALags=12,D=1,Seasonality=12, ... Beta=[1.5; -2],Intercept=0,Variance=0.1, ... Distribution=Distribution)

Mdl = regARIMA with properties: Description: "Regression with ARIMA(2,1,1) Error Model Seasonally Integrated with Seasonal AR(12) and MA(12) (t Distribution)" SeriesName: "Y" Distribution: Name = "t", DoF = 15 Intercept: 0 Beta: [1.5 -2] P: 27 D: 1 Q: 13 AR: {0.2 0.1} at lags [1 2] SAR: {0.01} at lag [12] MA: {0.5} at lag [1] SMA: {0.02} at lag [12] Seasonality: 12 Variance: 0.1

Simulate a length 25 path of data from the standard bivariate normal distribution for the predictor variables in the regression component.

rng(1,"twister") % For reproducibility numObs = 25; Pred = randn(numObs,2);

Simulate 100 independent paths of errors of length 25 from the standard normal distribution.

numPaths = 100; Z = randn(numObs,numPaths);

Simulate 100 independent response paths from model by filtering the paths of errors through the model. Supply the predictor data for the regression component.

```
Y = filter(Mdl,Z,X=Pred);
figure
plot(Y)
title("Simulated Response Paths")
```

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(1:25,lower,"r:",1:25,middle,"k", ... 1:25,upper,"r:") title("Monte Carlo Summary of Responses") legend("95% Interval","Median",Location="best")

### Compare Responses from `filter`

and `simulate`

Simulate responses using `filter`

and `simulate`

. Then compare the simulated responses.

Both `filter`

and `simulate`

filter a series of errors to produce output responses `y`

, innovations `e`

, and unconditional disturbances `u`

. The difference is that `simulate`

generates errors from `Mdl.Distribution`

, whereas `filter`

accepts a random array of errors that you generate from any distribution.

Specify the following regression model with ARMA(2,1) errors:

$$\begin{array}{l}\begin{array}{c}{y}_{t}={X}_{t}\left[\begin{array}{c}0.1\\ -0.2\end{array}\right]+{u}_{t}\\ {u}_{t}=0.5{u}_{t-1}-0.8{u}_{t-2}+{\epsilon}_{t}-0.5{\epsilon}_{t-1},\end{array}\end{array}$$

where $${\epsilon}_{t}$$ is Gaussian with variance 0.1.

```
Mdl = regARIMA(Intercept=0,AR={0.5 -0.8},MA=-0.5, ...
Beta=[0.1 -0.2],Variance=0.1);
```

`Mdl`

is a fully specified `regARIMA`

object.

Simulate a one path of bivariate standard normal data for the predictor variables. Then, simulate a path of responses and innovations from the regression model with ARMA(2,1) errors. Supply the simulated predictor data to `simulate`

for the regression component.

rng(1,"twister") % For reproducibility Pred = randn(100,2); % Simulate predictor data [ySim,eSim] = simulate(Mdl,100,X=Pred);

`ySim`

and `eSIM`

are 100-by-1 vectors of simulated responses and innovations, respectively, from the model `Mdl`

.

Produce model errors by standardizing the simulated innovations. Filter the simulated errors through the model. Supply the predictor data to `filter`

.

z1 = eSim./sqrt(Mdl.Variance); yFlt1 = filter(Mdl,z1,X=Pred);

`yFlt1`

is a 100-by-1 vector of responses resulting from filtering the simulated errors `z1`

through the model `Mdl`

.

Confirm that the simulated responses from `simulate`

and `filter`

are identical by plotting the two series.

figure h1 = plot(ySim); hold on h2 = plot(yFlt1,"."); title("Filtered and Simulated Responses") legend([h1 h2],["Simulate" "Filter"],Location="best") hold off

Alternatively, simulate responses by randomly generating your own errors and passing them into `filter`

.

rng(1,"twister") Pred = randn(100,2); z2 = randn(100,1); yFlt2 = filter(Mdl,z2,X=Pred); figure h1 = plot(ySim); hold on h2 = plot(yFlt2,"."); title("Filtered and Simulated Responses") legend([h1 h2],["Simulate" "Filter"],Location="best") hold off

This plot is the same as the previous plot, confirming that both simulation methods are equivalent.

`filter`

multiplies the error, `Z`

, by `sqrt(Mdl.Variance)`

before filtering `Z`

through the model. Therefore, if you want to specify a different distribution, set `Mdl.Variance`

to 1, and then generate your own errors using, for example, `random("unif",a,b)`

for the Uniform(*a*, *b*) distribution.

## Input Arguments

`Z`

— Error model disturbance series *z*_{t} that drives the innovations process *ε*_{t}

numeric column vector | numeric matrix

_{t}

_{t}

Error model disturbance series *z _{t}* that
drives the error model innovations process

*ε*, specified as a

_{t}`numobs`

-by-1 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.
The innovations process *ε*=

_{t}*σz*, where

_{t}*σ*=

`sqrt(Mdl.Variance)`

, the standard deviation of the
innovations.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 error model 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 error model disturbance series
*z _{t}* that drives the error 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. The
innovations process *ε*=

_{t}*σz*, where

_{t}*σ*=

`sqrt(Mdl.Variance)`

, the standard deviation of the
innovations.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,X=Pred,Z0=PSZ)`

specifies the predictor data
`Pred`

for the model regression component and the observed errors in
the presample period `PSZ`

to initialize the model.

`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`

`X`

— Predictor data

numeric matrix

Predictor data for the model regression component, specified as a
`numobs`

-by-`numpreds`

numeric matrix.
`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`

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

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

_{t}

Predictor variables *x _{t}* to select from

`Tbl1`

containing the predictor data for the model 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`

`Z0`

— Presample disturbance data *z*_{t}

numeric column vector | numeric matrix

_{t}

Presample disturbance data *z _{t}* to
initialize the error model, 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 error model moving average (MA) component. 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`

`U0`

— Presample regression innovation data (unconditional disturbances) *u*_{t}

numeric column vector | numeric matrix

_{t}

Presample regression innovation data (unconditional disturbances)
*u _{t}* to initialize the error model,
specified as a

`numpreobs`

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

-by-`numprepaths`

numeric matrix. Use
`U0`

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.P`

to initialize
the error model autoregressive (AR) component. If `numpreobs`

is
larger than required, `filter`

uses the latest required
observations only.

Columns of `U0`

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

If

`U0`

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

applies it to each path.If

`U0`

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

applies`U0(:,`

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
unconditional disturbances to 0.

**Data Types: **`double`

`Presample`

— Presample data

table | timetable

*Since R2023b*

Presample data containing paths of disturbance
*z _{t}* or regression innovation (unconditional
disturbance)

*u*series to initialize the model, specified as a table or timetable, the same type as

_{t}`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
error model disturbance or regression innovation series for
`DisturbanceVariable`

, the selected error model 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 regression innovations to initialize the error model AR componentAt least

`Mdl.Q`

when`Presample`

provides only presample error model disturbances to initialize the error model MA componentAt least

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

otherwise

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 necessary presample error model
disturbances and regression innovations to zero.

If you specify the `Presample`

, you must specify the presample
error model disturbance or regression innovation variable name by using the
`PresampleDisturbanceVariable`

or
`PresampleRegressionDisturbanceVariable`

name-value
argument.

`PresampleDisturbanceVariable`

— Error model disturbance variable *z*_{t} to select from `Presample`

string scalar | character vector | integer | logical vector

_{t}

*Since R2023b*

Error model disturbance variable *z _{t}* to
select from

`Presample`

containing the presample error model
disturbance data, specified as one of the following data types:String scalar or character vector containing the variable name to select from

`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 vector and cannot contain missing values
(`NaN`

s).

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

name-value argument, you must specify
`PresampleDisturbanceVariable`

.

**Example: **`PresampleDisturbanceVariable="GDP_Z"`

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

or `PresampleDisturbanceVariable=3`

selects the third table variable
for presample error model disturbance data.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`PresampleRegressionDisturbanceVariable`

— Regression model innovation variable to select from `Presample`

string scalar | character vector | integer | logical vector

*Since R2023b*

Regression model innovation variable, associated with unconditional disturbances
*u _{t}*, to select from

`Presample`

containing data for the presample regression model
innovations, 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

`PresampleRegressionDisturbanceVariable(`

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 regression model innovation data by using the
`Presample`

name-value argument, you must specify
`PresampleRegressionDisturbanceVariable`

.

**Example: **`PresampleRegressionDisturbanceVariable="StockRateU"`

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

or `PresampleRegressionDisturbanceVariable=3`

selects the third table variable as the presample regression model innovation
data.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

**Note**

`NaN`

values in`Z`

,`X`

,`Z0`

and`U0`

indicate missing values.`filter`

removes missing values from specified data by listwise deletion.For the presample,

`filter`

horizontally concatenates the possibly jagged arrays`Z0`

and`U0`

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.All predictor variables (columns) in

`X`

are associated with each input error series to produce`numpaths`

output series.

## Output Arguments

`Y`

— Simulated response paths *y*_{t}

numeric column vector | numeric matrix

_{t}

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

`numobs`

-by-1 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 responses at time
`t`

`Y(`

correspond to the filtered errors
at time * t*,:)

`t`

`Z(``t`

,:)

and response path
`j`

`Y(:,``j`

)

corresponds to the filtered
disturbance path `j`

`Z(:,``j`

)

when `Z`

is a
matrix.`Y`

represents the continuation of presample inputs.

`E`

— Simulated, mean-zero innovations paths *ε*_{t}

numeric column vector | numeric matrix

_{t}

Simulated, mean-zero innovations paths *ε _{t}*
of the error model, returned as a

`numobs`

-by-1 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, *ε _{t}* =

*σ*

*z*.

_{t}`U`

— Simulated unconditional disturbance paths *u*_{t}

numeric column vector | numeric matrix

_{t}

`Tbl2`

— Simulated response *y*_{t}, error model innovation *ε*_{t}, and unconditional disturbance
*u*_{t} paths

table | timetable

_{t}

_{t}

_{t}

*Since R2023b*

Simulated response *y _{t}*, error model
innovation

*ε*, and unconditional disturbance

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

_{t}`Tbl1`

.
`filter`

returns `Tbl2`

only when you
supply the input `Tbl1`

.`Tbl2`

contains the following variables:

The filtered 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 error model 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 error model 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 error model disturbance variable in`Tbl1`

.`filter`

names the simulated error model innovation variable in`Tbl2`

, where_ErrorInnovation`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

contains a variable for the corresponding simulated error model innovation paths with the name`StockReturns_ErrorInnovation`

.The simulated unconditional disturbance 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 error model disturbance variable in`Tbl1`

.`filter`

names the simulated unconditional disturbance variable in`Tbl2`

, where_RegressionInnovation`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

contains a variable for the corresponding simulated unconditional disturbance paths with the name`StockReturns_RegressionInnovation`

.All variables

`Tbl1`

.

If `Tbl1`

is a timetable, row times of `Tbl1`

and `Tbl2`

are equal.

## Alternative Functionality

`filter`

generalizes `simulate`

. Both filter a series of errors to produce responses
`Y`

, innovations `E`

, and unconditional disturbances
`U`

. However, `simulate`

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

. In contrast,
`filter`

requires that you specify your own errors, which can come
from any distribution.

## References

[1] 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.

[2] Davidson, R., and J. G. MacKinnon. *Econometric Theory and Methods*. Oxford, UK: Oxford University Press, 2004.

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

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

[5] Pankratz, A. *Forecasting with Dynamic Regression Models.* John Wiley & Sons, Inc., 1991.

[6] Tsay, R. S. *Analysis of Financial Time Series*. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.

## Version History

**Introduced in R2013b**

### R2023b: `filter`

accepts input data in tables and timetables

In addition to accepting input data (in-sample and presample data) 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 error model disturbance series on which to operate, but you can use the specified optional name-value argument to select a different series.If you specify optional presample error model disturbance or regression model innovation data to initialize the model, you must also specify the appropriate presample variable names.

`filter`

returns results in a table or timetable.

Name-value arguments to support tabular workflows include:

`DisturbanceVariable`

specifies the name of the disturbance series in the input data to filter through the model.`PredictorVariables`

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

specifies the input table or timetable of presample error model disturbance or regression innovation data.`PresampleDisturbanceVariable`

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

.`PresampleRegressionDisturbanceVariable`

specifies the name of the regression model innovation series to select from`Presample`

.

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