# irf

## Syntax

## Description

The `irf`

function returns the dynamic response, or the impulse response
function (IRF), to a one-standard-deviation shock to each variable in a VEC(*p* – 1)
model. A fully specified `vecm`

model object
characterizes the VEC model.

IRFs trace the effects of an innovation shock to one variable on the response of all
variables in the system. In contrast, the forecast error variance decomposition (FEVD)
provides information about the relative importance of each innovation in affecting all
variables in the system. To estimate the FEVD of a VEC model characterized by a
`vecm`

model object, see `fevd`

.

You can supply optional data, such as a presample, as a numeric array, table, or
timetable. However, all specified input data must be the same data type. When the input model
is estimated (returned by `estimate`

), supply
the same data type as the data used to estimate the model. The data type of the outputs
matches the data type of the specified input data.

returns a numeric array containing the orthogonalized IRF of the response variables that
compose the VEC(`Response`

= irf(`Mdl`

)*p* – 1) model `Mdl`

characterized by a
fully specified `vecm`

model object.
`irf`

shocks variables at time 0, and returns the IRF for
times 0 through 19.

If `Mdl`

is an estimated model (returned by `estimate`

) fit to
a numeric matrix of input response data, this syntax applies.

returns numeric arrays when all optional input data are numeric arrays. For example,
`Response`

= irf(`Mdl`

,`Name,Value`

)`irf(Mdl,NumObs=10,Method="generalized")`

specifies estimating a
generalized IRF for 10 time points starting at time 0, during which
`irf`

applies the shock.

If `Mdl`

is an estimated model (returned by `estimate`

) fit to
a numeric matrix of input response data, this syntax applies.

`[`

returns numeric arrays of lower `Response`

,`Lower`

,`Upper`

] = irf(___)`Lower`

and upper
`Upper`

95% confidence bounds for confidence intervals on the true
IRF, for each period and variable in the IRF, using any input argument combination in the
previous syntaxes. By default, `irf`

estimates confidence
bounds by conducting Monte Carlo simulation.

If `Mdl`

is an estimated model fit to a numeric matrix of input
response data, this syntax applies.

If `Mdl`

is a custom `vecm`

model object
(an object not returned by `estimate`

or
modified after estimation), `irf`

can require a sample size for
the simulation `SampleSize`

or presample responses
`Y0`

.

returns the table or
timetable `Tbl`

= irf(___)`Tbl`

containing the IRFs and, optionally, corresponding 95%
confidence bounds, of the response variables that compose the VEC(*p* –
1) model `Mdl`

. The IRF of the corresponding response is a variable in
`Tbl`

containing a matrix with columns corresponding to the variables
in the system shocked at time 0.* (since R2022b)*

If you set at least one name-value argument that controls the 95% confidence bounds on
the IRF, `Tbl`

also contains a variable for each of the lower and upper
bounds. For example, `Tbl`

contains confidence bounds when you set the
`NumPaths`

name-value argument.

If `Mdl`

is an estimated model fit to a table or timetable of input
response data, this syntax applies.

## Examples

### Specify Data in Numeric Matrix When Plotting IRF

Fit a 4-D VEC(2) model with two cointegrating relations to Danish money and income rate series data in a numeric matrix. Then, estimate the orthogonalized IRF from the estimated model.

Load the Danish money and income data set.

`load Data_JDanish`

The data set includes four time series in the table `DataTable`

. For more details on the data set, enter `Description`

at the command line.

Create a `vecm`

model object that represents a 4-D VEC(2) model with two cointegrating relations. Specify the variable names.

Mdl = vecm(4,2,2); Mdl.SeriesNames = DataTable.Properties.VariableNames;

`Mdl`

is a `vecm`

model object specifying the structure of a 4-D VEC(2) model; it is a template for estimation.

Fit the VEC(2) model to the numeric matrix of time series data `Data`

.

EstMdl = estimate(Mdl,Data);

`Mdl`

is a fully specified `vecm`

model object representing an estimated 4-D VEC(2) model.

Estimate the orthogonalized IRF from the estimated VEC(2) model.

Response = irf(EstMdl);

`Response`

is a 20-by-4-by-4 array representing the IRF of `Mdl`

. Rows correspond to consecutive time points from time 0 to 19, columns correspond to variables receiving a one-standard-deviation innovation shock at time 0, and pages correspond to responses of variables to the variable being shocked. `Mdl.SeriesNames`

specifies the variable order.

Display the IRF of the bond rate (variable 3, `IB`

) when the log of real income (variable 2, `Y`

) is shocked at time 0.

Response(:,2,3)

`ans = `*20×1*
0.0021
0.0057
0.0064
0.0067
0.0064
0.0061
0.0057
0.0056
0.0057
0.0058
⋮

The `armairf`

function plots the IRF of VAR models characterized by AR coefficient matrices. Plot the IRF of a VEC model by:

Expressing the VEC(2) model as a VAR(3) model by passing

`Mdl`

to`varm`

Passing the VAR model AR coefficients and innovations covariance matrix to

`armairf`

Plot the VEC(2) model IRF for 40 periods.

```
VARMdl = varm(EstMdl);
armairf(VARMdl.AR,[],InnovCov=VARMdl.Covariance, ...
NumObs=40);
```

Each plot shows the four IRFs of a variable when all other variables are shocked at time 0. `Mdl.SeriesNames`

specifies the variable order.

### Estimate Generalized IRF of VEC Model

Consider the 4-D VEC(2) model with two cointegrating relations in Specify Data in Numeric Matrix When Plotting IRF. Estimate the generalized IRF of the system for 50 periods.

Load the Danish money and income data set, then estimate the VEC(2) model.

```
load Data_JDanish
Mdl = vecm(4,2,2);
Mdl.SeriesNames = DataTable.Properties.VariableNames;
Mdl = estimate(Mdl,DataTable.Series);
```

Estimate the generalized IRF from the estimated VEC(2) model.

`Response = irf(Mdl,Method="generalized",NumObs=50);`

`Response`

is a 50-by-4-by-4 array representing the generalized IRF of `Mdl`

.

Plot the generalized IRF of the bond rate when real income is shocked at time 0.

figure; plot(0:49,Response(:,2,3)) title("IRF of IB When Y Is Shocked") xlabel("Observation Time") ylabel("Response") grid on

When real income is shocked, the bond rate reacts and then settles at approximately 0.0032 after 15 periods.

### Specify Data in Timetables When Computing IRF and Confidence Intervals

*Since R2022b*

Fit a 4-D VEC(2) model with two cointegrating relations to Danish money and income rate series data in a numeric matrix. Then, estimate and plot the orthogonalized IRF and corresponding confidence intervals from the estimated model.

Load the Danish money and income data set.

`load Data_JDanish`

Create a `vecm`

model object that represents a 4-D VEC(2) model with two cointegrating relations. Specify the variable names.

Mdl = vecm(4,2,2); Mdl.SeriesNames = DataTable.Properties.VariableNames;

`Mdl`

is a `vecm`

model object specifying the structure of a 4-D VEC(2) model; it is a template for estimation.

Fit the VEC(2) model to the data set.

EstMdl = estimate(Mdl,DataTimeTable);

`EstMdl`

is a fully specified `vecm`

model object representing an estimated 4-D VEC(2) model.

Estimate the orthogonalized IRF and corresponding 95% confidence intervals from the estimated VEC(2) model. To return confidence intervals, you must set a name-value argument that controls confidence intervals, for example, `Confidence`

. Set `Confidence`

to `0.95`

.

```
rng(1); % For reproducibility
Tbl = irf(EstMdl,Confidence=0.95);
Tbl.Time(1)
```

`ans = `*datetime*
01-Oct-1974

size(Tbl)

`ans = `*1×2*
20 12

`Tbl`

is a timetable with 20 rows, representing the periods in the IRF, and 12 variables. Each variable is a 20-by-4 matrix of the IRF or confidence bound associated with a variable in the model `EstMdl`

. For example, `Tbl.M2_IRF(:,2)`

is the IRF of `M2`

resulting from a 1-standard-deviation shock on `01-Jul-1974`

(period 0) to `Mdl.SeriesNames(2)`

, which is the variable `Y`

. [`Tbl.M2_IRF_LowerBound(:,2)`

,`Tbl.M2_IRF_UpperBound(:,2)`

] are the corresponding 95% confidence intervals. By default, `irf`

uses the H1 Johansen form, which is the same default form that `estimate`

uses.

Plot the IRF of M2 and its 95% confidence interval resulting from a 1-standard-deviation shock on `01-Jul-1974`

(period 0) to `Mdl.SeriesNames(2)`

, which is the variable `Y`

.

idxM2 = startsWith(Tbl.Properties.VariableNames,"M2"); M2IRF = Tbl(:,idxM2); shockIdx = 2; figure hold on plot(M2IRF.Time,M2IRF.M2_IRF(:,shockIdx),"-o") plot(M2IRF.Time,[M2IRF.M2_IRF_LowerBound(:,shockIdx) ... M2IRF.M2_IRF_UpperBound(:,shockIdx)],"-o",Color="r") legend("IRF","95% confidence interval") title('M2 IRF, Shock to Y') hold off

### Monte Carlo Confidence Intervals on True IRF

Consider the 4-D VEC(2) model with two cointegrating relations in Specify Data in Numeric Matrix When Plotting IRF. Estimate and plot its orthogonalized IRF and 95% Monte Carlo confidence intervals on the true IRF.

Load the Danish money and income data set, then estimate the VEC(2) model.

```
load Data_JDanish
Mdl = vecm(4,2,2);
Mdl.SeriesNames = DataTable.Properties.VariableNames;
Mdl = estimate(Mdl,DataTable.Series);
```

Estimate the IRF and corresponding 95% Monte Carlo confidence intervals from the estimated VEC(2) model.

```
rng(1); % For reproducibility
[Response,Lower,Upper] = irf(Mdl);
```

`Response`

, `Lower`

, and `Upper`

are 20-by-4-by-4 arrays representing the orthogonalized IRF of `Mdl`

and corresponding lower and upper bounds of the confidence intervals. For all arrays, rows correspond to consecutive time points from time 0 to 19, columns correspond to variables receiving a one-standard-deviation innovation shock at time 0, and pages correspond to responses of variables to the variable being shocked. `Mdl.SeriesNames`

specifies the variable order.

Plot the orthogonalized IRF with its confidence bounds of the bond rate when real income is shocked at time 0.

irfshock2resp3 = Response(:,2,3); IRFCIShock2Resp3 = [Lower(:,2,3) Upper(:,2,3)]; figure; h1 = plot(0:19,irfshock2resp3); hold on h2 = plot(0:19,IRFCIShock2Resp3,"r--"); legend([h1 h2(1)],["IRF" "95% Confidence Interval"]) xlabel("Time Index"); ylabel("Response"); title("IRF of IB When Y Is Shocked"); grid on hold off

When real income is shocked, the bond rate reacts and then settles between –0.002 and 0.0095 with 95% confidence.

### Bootstrap Confidence Intervals on True IRF

Consider the 4-D VEC(2) model with two cointegrating relations in Specify Data in Numeric Matrix When Plotting IRF. Estimate and plot its orthogonalized IRF and 90% bootstrap confidence intervals on the true IRF.

Load the Danish money and income data set, then estimate the VEC(2) model. Return the residuals from model estimation.

load Data_JDanish Mdl = vecm(4,2,2); Mdl.SeriesNames = DataTable.Properties.VariableNames; [Mdl,~,~,Res] = estimate(Mdl,DataTable.Series); T = size(DataTable,1) % Total sample size

T = 55

`n = size(Res,1) % Effective sample size`

n = 52

`Res`

is a 52-by-4 array of residuals. Columns correspond to the variables in `Mdl.SeriesNames`

. The `estimate`

function requires `Mdl.P`

= 3 observations to initialize a VEC(2) model for estimation. Because presample data (`Y0`

) is unspecified, `estimate`

takes the first three observations in the specified response data to initialize the model. Therefore, the resulting effective sample size is `T`

– `Mdl.P`

= 52, and rows of `Res`

correspond to the observation indices 4 through `T`

.

Estimate the orthogonalized IRF and corresponding 90% bootstrap confidence intervals from the estimated VEC(2) model. Draw 500 paths of length `n`

from the series of residuals.

rng(1) % For reproducibility [Response,Lower,Upper] = irf(Mdl,E=Res,NumPaths=500, ... Confidence=0.9);

Plot the orthogonalized IRF with its confidence bounds of the bond rate when real income is shocked at time 0.

irfshock2resp3 = Response(:,2,3); IRFCIShock2Resp3 = [Lower(:,2,3) Upper(:,2,3)]; figure; h1 = plot(0:19,irfshock2resp3); hold on h2 = plot(0:19,IRFCIShock2Resp3,"r--"); legend([h1 h2(1)],["IRF" "90% Confidence Interval"]) xlabel("Time Index"); ylabel("Response"); title("IRF of IB When Y Is Shocked"); grid on hold off

When real income is shocked, the bond rate reacts and then settles between approximately 0 and 0.010 with 90% confidence.

## Input Arguments

### 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: **`'NumObs',10,'Method',"generalized"`

specifies estimating a
generalized IRF for 10 time points starting at time 0, during which
`irf`

applies the shock, and ending at period 9.

**Options for All IRFs**

`NumObs`

— Number of periods

`20`

(default) | positive integer

Number of periods for which `irf`

computes
the IRF, specified as a positive integer.
`NumObs`

specifies the number of
observations to include in the IRF (the number of rows in
`Response`

or
`Tbl`

).

**Example: **`NumObs=10`

specifies the inclusion of 10
time points in the IRF starting at time 0, during which
`irf`

applies the shock,
and ending at period 9.

**Data Types: **`double`

`Method`

— IRF computation method

`"orthogonalized"`

(default) | `"generalized"`

| character vector

IRF computation method, specified as a value in this table.

Value | Description |
---|---|

`"orthogonalized"` | Compute impulse responses using orthogonalized,
one-standard-deviation innovation shocks.
`irf` uses the Cholesky factorization of
`Mdl.Covariance` for orthogonalization. |

`"generalized"` | Compute impulse responses using one-standard-deviation innovation shocks. |

**Example: **`Method="generalized"`

**Data Types: **`string`

| `char`

`Model`

— Johansen form of VEC(*p* – 1) model deterministic terms

`"H1"`

(default) | `"H2"`

| `"H1*"`

| `"H*"`

| `"H"`

| character vector

Johansen form of the VEC(*p* – 1) model deterministic terms [2], specified as a value
in this table (for variable definitions, see Vector Error-Correction Model).

Value | Error-Correction Term | Description |
---|---|---|

`"H2"` |
| No intercepts or trends are present in the cointegrating relations, and no deterministic trends are present in the levels of the data. Specify this model only when all response series have a mean of zero. |

`"H1*"` |
| Intercepts are present in the cointegrating relations, and no deterministic trends are present in the levels of the data. |

`"H1"` |
| Intercepts are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data. |

`"H*"` | A(B´y_{t−1}+c_{0}+d_{0}t)+c_{1} | Intercepts and linear trends are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data. |

`"H"` | A(B´y_{t−1}+c_{0}+d_{0}t)+c_{1}+d_{1}t | Intercepts and linear trends are present in the cointegrating relations, and deterministic quadratic trends are present in the levels of the data. If quadratic trends are not present in the data, this model can produce good in-sample fits but poor out-of-sample forecasts. |

For more details on the Johansen forms, see `estimate`

.

**Tip**

A best practice is to maintain model consistency during the simulation that
estimates confidence bounds. Therefore, if `Mdl`

is an estimated
`vecm`

model
object (an object returned by `estimate`

and unmodified thereafter), incorporate any constraints imposed during estimation by
deferring to the default value of `Model`

.

**Example: **`Model="H1*"`

**Data Types: **`string`

| `char`

**Options for Confidence Bound Estimation**

`NumPaths`

— Number of sample paths

`100`

(default) | positive integer

Number of sample paths (trials) to generate, specified as a positive integer.

**Example: **`NumPaths=1000`

generates `1000`

sample
paths from which the software derives the confidence bounds.

**Data Types: **`double`

`SampleSize`

— Number of observations for Monte Carlo simulation or bootstrap per sample path

positive integer

Number of observations for the Monte Carlo simulation or bootstrap per sample path, specified as a positive integer.

**Example: **If you specify `SampleSize=100`

and do not specify the
`E`

name-value argument, the software estimates confidence bounds
from `NumPaths`

random paths of length `100`

from
`Mdl`

.

**Example: **If you specify `SampleSize=100,E=Res`

, the software
resamples, with replacement, `100`

observations (rows) from
`Res`

to form a sample path of innovations to filter through
`Mdl`

. The software forms `NumPaths`

random
sample paths from which it derives confidence bounds.

**Data Types: **`double`

`Y0`

— Presample response data

numeric matrix

Presample response data that provides initial values for model estimation during the
simulation, specified as a `numpreobs`

-by-`numseries`

numeric matrix. Use `Y0`

only in the following situations:

`numpreobs`

is the number of presample observations.
`numseries`

is `Mdl.NumSeries`

, the dimensionality
of the input model.

Each row is a presample observation, and measurements in each row occur
simultaneously. The last row contains the latest presample observation.
`numpreobs`

is the number of specified presample responses and it
must be at least `Mdl.P`

. If you supply more rows than necessary,
`irf`

uses the latest `Mdl.P`

observations
only.

`numseries`

is the dimensionality of the input VEC model
`Mdl.NumSeries`

. Columns must correspond to the response variables
in `Mdl.SeriesNames`

.

The following situations determine the default or whether presample response data is required.

**Data Types: **`double`

`Presample`

— Presample data

table | timetable

*Since R2022b*

Presample data that provide initial values for the model `Mdl`

,
specified as a table or timetable with `numprevars`

variables and
`numpreobs`

rows. Use `Presample`

only in the
following situations:

Each row is a presample observation, and measurements in each row occur
simultaneously. The last row contains the latest presample observation.
`numpreobs`

is the number of specified presample responses and it
must be at least `Mdl.P`

. If you supply more rows than necessary,
`irf`

uses the latest `Mdl.P`

observations
only.

Each variable is a `numpreobs`

numeric vector representing one path.
To control presample variable selection, see the optional
`PresampleResponseVariables`

name-value argument.

If `Presample`

is a timetable, all the following conditions must be true:

`Presample`

must represent a sample with a regular datetime time step (see`isregular`

).The datetime vector of sample timestamps

`Presample.Time`

must be ascending or descending.

If `Presample`

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

The following situations determine the default or whether presample response data is required.

If

`Mdl`

is an unmodified estimated model,`irf`

sets`Presample`

to the presample response data used for estimation by default (see the`Presample`

name-value argument of`estimate`

).If

`Mdl`

is a custom model (for example, you modify a model after estimation by using dot notation) and you return confidence bounds in the table or timetable`Tbl`

, you must specify`Presample`

.

`PresampleResponseVariables`

— Variables to select from `Presample`

to use for presample response data

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

*Since R2022b*

Variables to select from `Presample`

to use for presample data, specified
as one of the following data types:

String vector or cell vector of character vectors containing

`numseries`

variable names in`Presample.Properties.VariableNames`

A length

`numseries`

vector of unique indices (integers) of variables to select from`Presample.Properties.VariableNames`

A length

`numprevars`

logical vector, where`PresampleResponseVariables(`

selects variable) = true`j`

from`j`

`Presample.Properties.VariableNames`

, and`sum(PresampleResponseVariables)`

is`numseries`

`PresampleResponseVariables`

applies only when you specify
`Presample`

.

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

).

`PresampleResponseNames`

does not need to contain the same names as in
`Mdl.SeriesNames`

; `irf`

uses the data in
selected variable `PresampleResponseVariables(`

as a presample for * j*)

`Mdl.SeriesNames(``j`

)

.If the number of variables in `Presample`

matches
`Mdl.NumSeries`

, the default specifies all variables in
`Presample`

. If the number of variables in `Presample`

exceeds `Mdl.NumSeries`

, the default matches variables in
`Presample`

to names in `Mdl.SeriesNames`

.

**Example: **`PresampleResponseVariables=["GDP" "CPI"]`

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

or
`PresampleResponseVariable=[1 3]`

selects the first and third table
variables for presample data.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`X`

— Predictor data

numeric matrix

Predictor data *x _{t}* for
estimating the model regression component during the simulation,
specified as a numeric matrix containing

`numpreds`

columns. Use
`X`

only in the following situations:

`numpreds`

is the number of predictor variables
(`size(Mdl.Beta,2)`

).

Each row corresponds to an observation, and measurements in each row
occur simultaneously. The last row contains the latest
observation. `X`

must have at least
`SampleSize`

rows. If you supply
more rows than necessary, `irf`

uses
only the latest observations. `irf`

does not use the regression component in the presample
period.

Columns correspond to individual predictor variables. All predictor variables are present in the regression component of each response equation.

To maintain model consistency when `irf`

estimates the confidence bounds, a good practice is to specify predictor data when
`Mdl`

has a regression component. If `Mdl`

is an
estimated model, specify the predictor data used during model estimation (see the `X`

name-value argument of `estimate`

).

By default, `irf`

excludes the regression
component from confidence bound estimation, regardless of its
presence in `Mdl`

.

**Data Types: **`double`

`E`

— Series of residuals from which to draw bootstrap samples

numeric matrix

Series of residuals from which to draw bootstrap samples, specified as a
`numperiods`

-by-`numseries`

numeric matrix.
`irf`

assumes that `E`

is free of serial
correlation. Use `E`

only in the following situations:

Each column is the residual series corresponding to the response series names in
`Mdl.SeriesNames`

.

Each row corresponds to a period in the FEVD and the corresponding confidence bounds.

If `Mdl`

is an estimated `vecm`

model
object (an object returned by `estimate`

),
you can specify `E`

as the inferred residuals from estimation (see the
`E`

output argument of `estimate`

or `infer`

).

By default, `irf`

derives confidence bounds by conducting a
Monte Carlo simulation.

**Data Types: **`double`

`InSample`

— Time series data

table | timetable

*Since R2022b*

Time series data containing `numvars`

variables,
including `numseries`

variables of residuals
*e _{t}* to
bootstrap or

`numpreds`

predictor variables
*x*for the model regression component, specified as a table or timetable. Use

_{t}`InSample`

only in the
following situations:

Each variable is a single path of observations, which
`irf`

applies to all
`NumPaths`

sample paths. If you
specify `Presample`

you must specify which
variables are residuals and predictors, see the
`ResidualVariables`

and
`PredictorVariables`

name-value
arguments.

Each row is an observation, and measurements in each row occur
simultaneously. `InSample`

must have at least
`SampleSize`

rows. If you supply
more rows than necessary, `irf`

uses
only the latest observations.

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 strictly ascending or descending.`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, `irf`

derives confidence bounds
by conducting a Monte Carlo simulation and does not use model
the regression component, regardless of its presence in
`Mdl`

.

`ResidualVariables`

— Variables to select from `InSample`

to treat as residuals *e*_{t} for bootstrapping

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

_{t}

*Since R2022b*

Variables to select from `InSample`

to treat as residuals for
bootstrapping, specified as one of the following data types:

String vector or cell vector of character vectors containing

`numseries`

variable names in`InSample.Properties.VariableNames`

A length

`numseries`

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

A length

`numvars`

logical vector, where`ResidualVariables(`

selects variable) = true`j`

from`j`

`InSample.Properties.VariableNames`

, and`sum(ResidualVariables)`

is`numseries`

Regardless, selected residual variable

is the residual series for
`j`

`Mdl.SeriesNames(`

.* j*)

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

).

By default, `irf`

derives confidence bounds by conducting a
Monte Carlo simulation.

**Example: **```
ResidualVariables=["GDP_Residuals"
"CPI_Residuals"]
```

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

or
`ResidualVariable=[1 3]`

selects the first and third table variables as the
disturbance variables.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`PredictorVariables`

— Variables to select from `InSample`

to treat as exogenous predictor variables *x*_{t}

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

_{t}

*Since R2022b*

Variables to select from `InSample`

to treat as exogenous predictor
variables *x _{t}*, 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 length

`numpreds`

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

A length

`numvars`

logical vector, where`PredictorVariables(`

selects variable) = true`j`

from`j`

`InSample.Properties.VariableNames`

, and`sum(PredictorVariables)`

is`numpreds`

Regardless, selected predictor variable

corresponds to the coefficients
`j`

`Mdl.Beta(:,`

.* j*)

`PredictorVariables`

applies only when you specify
`InSample`

.

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

).

By default, `irf`

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 as
the response variables.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`Confidence`

— Confidence level

`0.95`

(default) | numeric scalar in [0,1]

Confidence level for the confidence bounds, specified as a numeric scalar in the interval [0,1].

For each period, randomly drawn confidence intervals cover the true response `100*Confidence`

% of the time.

The default value is `0.95`

, which implies that the confidence bounds represent 95% confidence intervals.

**Example: **`Confidence=0.9`

specifies 90% confidence
intervals.

**Data Types: **`double`

**Note**

`NaN`

values in`Y0`

,`X`

, and`E`

indicate missing data.`irf`

removes missing data from these arguments by list-wise deletion. For each argument, if a row contains at least one`NaN`

,`irf`

removes the entire row.List-wise deletion reduces the sample size, can create irregular time series, and can cause

`E`

and`X`

to be unsynchronized.`irf`

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

## Output Arguments

`Response`

— IRF

numeric array

IRF of each variable, returned as a
`numobs`

-by-`numseries`

-by-`numseries`

numeric array. `numobs`

is the value of `NumObs`

.
Columns and pages correspond to the response variables in
`Mdl.SeriesNames`

.

`irf`

returns `Response`

only in the
following situations:

You supply optional data inputs as numeric matrices.

`Mdl`

is an estimated model fit to a numeric matrix of response data.

`Response(`

is the
impulse response of variable * t* +
1,

*,*

`j`

*)*

`k`

`k`

to a
one-standard-deviation innovation shock to variable
`j`

at time 0, for
`t`

= 0, 1, ...,
`numObs`

– 1, `j`

=
1,2,...,`numseries`

, and
`k`

=
1,2,...,`numseries`

.`Lower`

— Lower confidence bounds

numeric array

Lower confidence bounds, returned as a
`numobs`

-by-`numseries`

-by-`numseries`

numeric array. Elements of `Lower`

correspond to elements of
`Response`

.

`irf`

returns `Lower`

only in the
following situations:

You supply optional data inputs as numeric matrices.

`Mdl`

is an estimated model fit to a numeric matrix of response data.

`Lower(`

is the
lower bound of the * t* +
1,

*,*

`j`

*)*

`k`

`100*Confidence`

-th percentile interval on the
true impulse response of variable `k`

to a
one-standard-deviation innovation shock to variable
`j`

at time 0.`Upper`

— Upper confidence bounds

numeric array

Upper confidence bounds, returned as a
`numobs`

-by-`numseries`

-by-`numseries`

numeric array. Elements of `Upper`

correspond to elements of
`Response`

.

`irf`

returns `Upper`

only in the
following situations:

You supply optional data inputs as numeric matrices.

`Mdl`

is an estimated model fit to a numeric matrix of response data.

`Upper(`

is the
upper bound of the * t* +
1,

*,*

`j`

*)*

`k`

`100*Confidence`

-th percentile interval on the
true impulse response of variable `k`

to a
one-standard-deviation innovation shock to variable
`j`

at time 0.`Tbl`

— IRF and confidence bounds

table | timetable

*Since R2022b*

IRF and confidence bounds, returned as a table or timetable with
`numobs`

rows. `irf`

returns
`Tbl`

only in the following situations:

You supply optional data inputs as tables or timetables.

`Mdl`

is an estimated model fit to response data in a table or timetable.

Regardless, the data type of `Tbl`

is the same as the
data type of specified data.

`Tbl`

contains the following variables:

The IRF of each series in

*y*. Each IRF variable in_{t}`Tbl`

is a`numobs`

-by-`numseries`

numeric matrix, where`numobs`

is the value of`NumObs`

and`numseries`

is the value of`Mdl.NumSeries`

.`irf`

names the IRF of response variable

in`ResponseJ`

`Mdl.SeriesNames`

. For example, if_IRF`ResponseJ`

`Mdl.Series(`

is)`j`

`GDP`

,`Tbl`

contains a variable for the corresponding IRF with the name`GDP_IRF`

._IRF`ResponseJ`

`(`

is the impulse response of variable+ 1,`t`

)`k`

to a one-standard-deviation innovation shock to variable`ResponseJ`

at time 0, for`k`

= 0,1, ...,`t`

`numObs`

– 1,

= 1,2,...,`J`

`numseries`

, and

= 1,2,...,`k`

`numseries`

.The lower and upper confidence bounds on the true IRF of the response series, when you set at least one name-value argument that controls the confidence bounds. Each confidence bound variable in

`Tbl`

is a`numobs`

-by-`numseries`

numeric matrix.

and_IRF_LowerBound`ResponseJ`

are the names of the lower and upper bound variables, respectively, of the confidence interval on the IRF of response variable_IRF_UpperBound`ResponseJ`

`Mdl.SeriesNames(`

=)`J`

. For example, if`ResponseJ`

`Mdl.SeriesNames(`

is)`j`

`GDP`

,`Tbl`

contains variables for the corresponding lower and upper bounds of the confidence interval with the name`GDP_IRF_LowerBound`

and`GDP_IRF_UpperBound`

.(

,_IRF_LowerBound(`ResponseJ`

,`t`

)`k`

) is the 95% confidence interval on the IRF of response variable_IRF_UpperBound(`ResponseJ`

,`t`

)`k`

attributable to a one-standard-deviation innovation shock to variable`ResponseJ`

at time`k`

, for`t`

= 0,1,…,`t`

`numobs`

– 1,

= 1,2,...,`J`

`numseries`

, and

= 1,2,...,`k`

`numseries`

.

If `Tbl`

is a timetable, the row order of `Tbl`

,
either ascending or descending, matches the row order of `InSample`

,
when you specify it. If you do not specify `InSample`

and you specify
`Presample`

, the row order of `Tbl`

is the
same as the row order of `Presample`

.

## More About

### Impulse Response Function

An *impulse response function* (IRF) of a time
series model (or *dynamic response of the system*) measures the changes
in the future responses of all variables in the system when a variable is shocked by an
impulse. In other words, the IRF at time *t* is the derivative of the
responses at time *t* with respect to an innovation at time
*t*_{0} (the time that innovation was shocked),
*t* ≥ *t*_{0}.

Consider a `numseries`

-D VEC(*p* – 1)
model for the multivariate response variable
*y _{t}*. In lag operator notation, the equivalent
VAR(

*p*) representation of a VEC(

*p*– 1) model is:

$$\Gamma (L){y}_{t}=c+dt+\beta {x}_{t}+{\epsilon}_{t},$$

where $$\Gamma (L)=I-{\Gamma}_{1}L-{\Gamma}_{2}{L}^{2}-\mathrm{...}-{\Gamma}_{p}{L}^{p}$$ and *I* is the
`numseries`

-by-`numseries`

identity matrix.

In lag operator notation, the infinite lag MA representation of
*y _{t}* is:

$$\begin{array}{c}{y}_{t}={\Gamma}^{-1}(L)\left(c+\beta {x}_{t}+dt\right)+{\Gamma}^{-1}(L){\epsilon}_{t}\\ =\Omega (L)\left(c+\beta {x}_{t}+dt\right)+\Omega (L){\epsilon}_{t}.\end{array}$$

The general form of the IRF of *y _{t}* shocked by
an impulse to variable

*j*by one standard deviation of its innovation

*m*periods into the future is:

$${\psi}_{j}(m)={C}_{m}{e}_{j}.$$

*e*is a selection vector of length_{j}`numseries`

containing a 1 in element*j*and zeros elsewhere.For the orthogonalized IRF, $${C}_{m}={\Omega}_{m}P,$$ where

*P*is the lower triangular factor in the Cholesky factorization of Σ, and Ω_{m}is the lag*m*coefficient of Ω(*L*).For the generalized IRF, $${C}_{m}={\sigma}_{j}^{-1}{\Omega}_{m}\Sigma ,$$ where σ

_{j}is the standard deviation of innovation*j*.The IRF is free of the model constant, regression component, and time trend.

### Vector Error-Correction Model

A *vector error-correction (VEC) model* is a
multivariate, stochastic time series model consisting of a system of *m* =
`numseries`

equations of *m* distinct, differenced
response variables. Equations in the system can include an *error-correction
term*, which is a linear function of the responses in levels used to
stabilize the system. The *cointegrating rank*
*r* is the number of *cointegrating relations* that
exist in the system.

Each response equation can include an autoregressive polynomial composed of first
differences of the response series (*short-run polynomial* of degree
*p* – 1), a constant, a time trend, exogenous predictor variables, and
a constant and time trend in the error-correction term.

A VEC(*p* – 1) model in *difference-equation
notation* and in *reduced form* can be expressed in two ways:

This equation is the

*component form*of a VEC model, where the cointegration adjustment speeds and cointegration matrix are explicit, whereas the impact matrix is implied.$$\begin{array}{c}\Delta {y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}\\ =c+dt+AB\prime {y}_{t-1}+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}.\end{array}$$

The cointegrating relations are

*B'**y*_{t – 1}+*c*_{0}+*d*_{0}*t*and the error-correction term is*A*(*B'**y*_{t – 1}+*c*_{0}+*d*_{0}*t*).This equation is the

*impact form*of a VEC model, where the impact matrix is explicit, whereas the cointegration adjustment speeds and cointegration matrix are implied.$$\begin{array}{c}\Delta {y}_{t}=\Pi {y}_{t-1}+A\left({c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}\\ =c+dt+\Pi {y}_{t-1}+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}.\end{array}$$

In the equations:

*y*is an_{t}*m*-by-1 vector of values corresponding to*m*response variables at time*t*, where*t*= 1,...,*T*.Δ

*y*=_{t}*y*–_{t}*y*_{t – 1}. The structural coefficient is the identity matrix.*r*is the number of cointegrating relations and, in general, 0 <*r*<*m*.*A*is an*m*-by-*r*matrix of adjustment speeds.*B*is an*m*-by-*r*cointegration matrix.Π is an

*m*-by-*m*impact matrix with a rank of*r*.*c*_{0}is an*r*-by-1 vector of constants (intercepts) in the cointegrating relations.*d*_{0}is an*r*-by-1 vector of linear time trends in the cointegrating relations.*c*_{1}is an*m*-by-1 vector of constants (*deterministic linear trends*in*y*)._{t}*d*_{1}is an*m*-by-1 vector of linear time-trend values (*deterministic quadratic trends*in*y*)._{t}*c*=*A**c*_{0}+*c*_{1}and is the overall constant.*d*=*A**d*_{0}+*d*_{1}and is the overall time-trend coefficient.Φ

_{j}is an*m*-by-*m*matrix of short-run coefficients, where*j*= 1,...,*p*– 1 and Φ_{p – 1}is not a matrix containing only zeros.*x*is a_{t}*k*-by-1 vector of values corresponding to*k*exogenous predictor variables.*β*is an*m*-by-*k*matrix of regression coefficients.*ε*is an_{t}*m*-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively an*m*-by-*m*covariance matrix Σ. For*t*≠*s*,*ε*and_{t}*ε*are independent._{s}

Condensed and in lag operator notation, the system is

$$\begin{array}{c}\Phi (L)(1-L){y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+\beta {x}_{t}+{\epsilon}_{t}\\ =c+dt+AB\prime {y}_{t-1}+\beta {x}_{t}+{\epsilon}_{t}\end{array}$$

where $$\Phi (L)=I-{\Phi}_{1}-{\Phi}_{2}-\mathrm{...}-{\Phi}_{p-1}$$, *I* is the *m*-by-*m*
identity matrix, and *L**y _{t}* =

*y*

_{t – 1}.

If *m* = *r*, then the VEC model is a stable
VAR(*p*) model in the levels of the responses. If *r*
= 0, then the error-correction term is a matrix of zeros, and the VEC(*p* –
1) model is a stable VAR(*p* – 1) model in the first differences of the
responses.

## Algorithms

If

`Method`

is`"orthogonalized"`

, then the resulting IRF depends on the order of the variables in the time series model. If`Method`

is`"generalized"`

, then the resulting IRF is invariant to the order of the variables. Therefore, the two methods generally produce different results.If

`Mdl.Covariance`

is a diagonal matrix, then the resulting generalized and orthogonalized IRFs are identical. Otherwise, the resulting generalized and orthogonalized IRFs are identical only when the first variable shocks all variables (for example, all else being the same, both methods yield the same value of`Response(:,1,:)`

).The predictor data in

`X`

or`InSample`

represents a single path of exogenous multivariate time series. If you specify`X`

or`InSample`

and the model`Mdl`

has a regression component (`Mdl.Beta`

is not an empty array),`irf`

applies the same exogenous data to all paths used for confidence interval estimation.`irf`

conducts a simulation to estimate the confidence bounds`Lower`

and`Upper`

or associated variables in`Tbl`

.If you do not specify residuals by supplying

`E`

or using`InSample`

,`irf`

conducts a Monte Carlo simulation by following this procedure:Simulate

`NumPaths`

response paths of length`SampleSize`

from`Mdl`

.Fit

`NumPaths`

models that have the same structure as`Mdl`

to the simulated response paths. If`Mdl`

contains a regression component and you specify predictor data by supplying`X`

or using`InSample`

, then`irf`

fits the`NumPaths`

models to the simulated response paths and the same predictor data (the same predictor data applies to all paths).Estimate

`NumPaths`

IRFs from the`NumPaths`

estimated models.For each time point

*t*= 0,…,`NumObs`

, estimate the confidence intervals by computing 1 –`Confidence`

and`Confidence`

quantiles (upper and lower bounds, respectively).

Otherwise,

`irf`

conducts a nonparametric bootstrap by following this procedure:Resample, with replacement,

`SampleSize`

residuals from`E`

or`InSample`

. Perform this step`NumPaths`

times to obtain`NumPaths`

paths.Center each path of bootstrapped residuals.

Filter each path of centered, bootstrapped residuals through

`Mdl`

to obtain`NumPaths`

bootstrapped response paths of length`SampleSize`

.Complete steps 2 through 4 of the Monte Carlo simulation, but replace the simulated response paths with the bootstrapped response paths.

## References

[1]
Hamilton, James D. *Time Series Analysis*. Princeton, NJ: Princeton University Press, 1994.

[2]
Johansen, S. *Likelihood-Based Inference in Cointegrated Vector Autoregressive Models*. Oxford: Oxford University Press, 1995.

[3]
Juselius, K. *The Cointegrated VAR Model*. Oxford: Oxford University Press, 2006.

[4]
Pesaran, H. H., and Y. Shin. "Generalized Impulse Response
Analysis in Linear Multivariate Models." *Economic Letters.* Vol. 58, 1998,
pp. 17–29.

## Version History

**Introduced in R2019a**

### R2022b: `irf`

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

In addition to accepting input data in numeric arrays,
`irf`

accepts input data in tables and timetables. `irf`

chooses default series on which to operate, but you can use the following name-value arguments to select variables.

`Presample`

specifies the input table or regular timetable of presample response data.`PresampleResponseVariables`

specifies the response series names from`Presample`

.`Insample`

specifies the table or regular timetable of residual and predictor data to compute bootstrap estimates.`ResidualVariables`

specifies the residual series names in`InSample`

.`PredictorVariables`

specifies the predictor series in`InSample`

for a model regression component.

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