# irf

Impulse response function (IRF) of state-space model

## Syntax

## Description

`irf`

returns a numeric array representing the IRFs of the state and measurement variables in a state-space model. To plot the IRFs instead, use `irfplot`

. Other state-space model tools to characterize the dynamics of a specified system include:

The forecast error variance decomposition (FEVD), computed by

`fevd`

, provides information about the relative importance of each state disturbance in affecting the forecast error variance of all measurement variables in the system.Model-implied temporal correlations, computed by

`corr`

for a standard state-space model, measure the association between present and past state or measurement variables, as prescribed by the form of the model.

### Fully Specified State-Space Model

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

= irf(`Mdl`

,`Name,Value`

)`'NumPeriods',10,'Cumulative',true`

specifies a 10-period cumulative IRF starting at time 1, during which `irf`

applies the shock to a state-disturbance variable in the system, and ending at period 10.

### Partially Specified State-Space Model and Confidence Interval Estimation

`[`

also returns, for each period, the lower and upper 95% Monte Carlo confidence bounds of each measurement variable IRF ([`ResponseY`

,`ResponseX`

,`LowerY`

,`UpperY`

,`LowerX`

,`UpperX`

] = irf(___,`'Params'`

,estParams,`'EstParamCov'`

,EstParamCov)`LowerY`

,`UpperY`

]) and each state variable IRF ([`LowerX`

,`UpperX`

]). `EstParamCov`

specifies the estimated covariance matrix of the parameter estimates, as returned by the `estimate`

function, and is required for confidence interval estimation.

## Examples

### Compute Measurement Variable IRF

Explicitly create the state-space model

$$\begin{array}{l}{x}_{t}=0.5{x}_{t-1}+0.2{u}_{t}\\ {y}_{t}=2{x}_{t}+0.01{\epsilon}_{t}.\end{array}$$

A = 0.5; B = 0.2; C = 2; D = 0.01; Mdl = ssm(A,B,C,D)

Mdl = State-space model type: ssm State vector length: 1 Observation vector length: 1 State disturbance vector length: 1 Observation innovation vector length: 1 Sample size supported by model: Unlimited State variables: x1, x2,... State disturbances: u1, u2,... Observation series: y1, y2,... Observation innovations: e1, e2,... State equation: x1(t) = (0.50)x1(t-1) + (0.20)u1(t) Observation equation: y1(t) = (2)x1(t) + (0.01)e1(t) Initial state distribution: Initial state means x1 0 Initial state covariance matrix x1 x1 0.05 State types x1 Stationary

`Mdl`

is a `ssm`

model object. Because all parameters have known values, the object is fully specified.

Compute the IRF of the measurement variable.

responseY = irf(Mdl)

`responseY = `*20×1*
0.4000
0.2000
0.1000
0.0500
0.0250
0.0125
0.0063
0.0031
0.0016
0.0008
⋮

`responseY`

is a 20-by-1 vector representing the 20-period IRF of the measurement variable ${\mathit{y}}_{\mathit{t}}$. `responseY(5)`

is `0.0250`

, which means that the response of ${\mathit{y}}_{\mathit{t}}$ at period 5, to a unit shock to the state disturbance ${\mathit{u}}_{\mathit{t}}$ at period 1, is `0.0250`

.

### Specify Number of Periods

Explicitly create the multivariate diffuse state-space model

$$\begin{array}{l}{x}_{1,t}={x}_{1,t-1}+0.2{u}_{1,t}\\ {x}_{2,t}={x}_{1,t-1}+0.3{x}_{2,t-1}+{u}_{2,t}\\ {y}_{1,t}={x}_{1,t}+{\epsilon}_{1,t}\\ {y}_{2,t}={x}_{1,t}+{x}_{2,t}+{\epsilon}_{2,t}.\end{array}$$

```
A = [1 0; 1 0.3];
B = [0.2 0; 0 1];
C = [1 0; 1 1];
D = eye(2);
Mdl = dssm(A,B,C,D,'StateType',[2 2])
```

Mdl = State-space model type: dssm State vector length: 2 Observation vector length: 2 State disturbance vector length: 2 Observation innovation vector length: 2 Sample size supported by model: Unlimited State variables: x1, x2,... State disturbances: u1, u2,... Observation series: y1, y2,... Observation innovations: e1, e2,... State equations: x1(t) = x1(t-1) + (0.20)u1(t) x2(t) = x1(t-1) + (0.30)x2(t-1) + u2(t) Observation equations: y1(t) = x1(t) + e1(t) y2(t) = x1(t) + x2(t) + e2(t) Initial state distribution: Initial state means x1 x2 0 0 Initial state covariance matrix x1 x2 x1 Inf 0 x2 0 Inf State types x1 x2 Diffuse Diffuse

`Mdl`

is a `dssm`

model object.

Compute the 10-period IRFs of the measurement variables.

`ResponseY = irf(Mdl,'NumPeriods',10);`

`ResponseY`

is a 10-by-2-by-2 array representing the 10-period IRFs of the measurement variables. For example, `ResponseY(:,1,2)`

is the IRF of ${\mathit{y}}_{2,\mathit{t}}$ as a result of a shock applied to ${\mathit{u}}_{1,\mathit{t}}$.

ResponseY(:,1,2)

`ans = `*10×1*
0.2000
0.4000
0.4600
0.4780
0.4834
0.4850
0.4855
0.4857
0.4857
0.4857

### State Variable Cumulative IRFs of Estimated Model

Simulate data from a known model, fit the data to a state-space model, and then estimate the cumulative IRFs of the state variables.

Assume that the data generating process (DGP) is the AR(1) model

$${x}_{t}=1+0.9{x}_{t-2}+{u}_{t},$$

where ${\mathit{u}}_{\mathit{t}}$ is a series of independent and identically distributed Gaussian variables with mean 0 and variance 1.

Simulate 500 observations from the model.

rng(1); % For reproducibility DGP = arima('Constant',1,'AR',{0 0.9},'Variance',1); y = simulate(DGP,500);

Explicitly create a state-space model template for estimation that represents the model

$$\begin{array}{l}{x}_{t}=c+\varphi {x}_{t-2}+\eta {u}_{t}\\ {y}_{t}={x}_{t}.\end{array}$$

```
A = [0 NaN NaN; 0 1 0; 1 0 0];
B = [NaN; 0; 0];
C = [1 0 0];
D = 0;
Mdl = ssm(A,B,C,D,'StateType',[0 1 0]);
```

Fit the model template to the data. Specify a set of positive, random standard Gaussian starting values for the three model parameters.

EstMdl = estimate(Mdl,y,abs(randn(3,1)));

Method: Maximum likelihood (fminunc) Sample size: 500 Logarithmic likelihood: -2085.74 Akaike info criterion: 4177.49 Bayesian info criterion: 4190.13 | Coeff Std Err t Stat Prob --------------------------------------------------- c(1) | 0.36553 0.07967 4.58829 0.00000 c(2) | 0.70179 0.00738 95.13852 0 c(3) | 1.16649 0.02236 52.16929 0 | | Final State Std Dev t Stat Prob x(1) | 10.72536 0 Inf 0 x(2) | 1 0 Inf 0 x(3) | 6.66084 0 Inf 0

`EstMdl`

is a fully specified `ssm`

model object.

Estimate the cumulative IRFs of the state and measurement variables.

`[ResponseY,ResponseX] = irf(EstMdl,'Cumulative',true);`

`ResponseY`

is a 20-by-1 vector representing the measurement variable IRF. `ResponseX`

is a 20-by-1-by-3 array representing the IRF of the state variables.

Display the IRF of ${\mathit{x}}_{\mathit{t}}$, which is the first state variable in the system ${\mathit{x}}_{1,\mathit{t}}$.

irfx = ResponseX(:,:,1)

`irfx = `*20×1*
1.1665
1.1665
1.9851
1.9851
2.5596
2.5596
2.9628
2.9628
3.2458
3.2458
⋮

Verify that, because ${\mathit{y}}_{\mathit{t}}={\mathit{x}}_{\mathit{t}}$, `ResponseY = ResponseX(:,:,1)`

.

ver1 = sum(abs(ResponseY - ResponseX(:,:,1)))

ver1 = 0

Verify that, because ${\mathit{x}}_{1,\mathit{t}-1}={\mathit{x}}_{3,\mathit{t}}$, `ResponseX(1:(end-2),1,1) = ResponseX(2:(end-1),:,3)`

.

ver2 = sum(abs(ResponseX(1:(end-2),:,1) - ResponseX(2:(end-1),:,3)))

ver2 = 0

### Time-Varying IRF

Simulate data from a time-varying state-space model, fit a model to the data, and then estimate the time-varying IRF.

Consider the DGP represented by the system

$$\begin{array}{l}{x}_{t}=\{\begin{array}{ll}0.75{x}_{t-1}+{u}_{t};& t<11\\ -0.1{x}_{t-1}+3{u}_{t};& t\ge 11\end{array}\\ {y}_{t}=1.5{x}_{t}+2{\epsilon}_{t}.\end{array}$$

The function `timeVariantAR1ParamMap.m`

, stored in `mlr/examples/econ/main`

, specifies the model structure. `mlr`

is the value of `matlabroot`

.

`type timeVariantAR1ParamMap.m`

% Copyright 2020 The MathWorks, Inc. function [A,B,C,D] = timeVariantAR1ParamMap(params) % Time-varying state-space model parameter mapping function example. This % function maps the vector params to the state-space matrices (A, B, C, and % D). From periods 1 through 10, the state model is an AR(1)model, and from % periods 11 through 20, the state model is possibly a different AR(1) % model. The measurement equation is the same throughout the time span. A1 = {params(1)}; A2 = {params(2)}; varu1 = exp(params(3)); % Positive variance constraints varu2 = exp(params(4)); B1 = {sqrt(varu1)}; B2 = {sqrt(varu2)}; C = params(5); vare1 = exp(params(6)); D = sqrt(vare1); A = [repmat(A1,10,1); repmat(A2,10,1)]; B = [repmat(B1,10,1); repmat(B2,10,1)]; end

Implicitly create a partially specified state-space model representing the DGP. For this example, fix the measurement-sensitivity coefficient $\mathit{C}$ to `1.5`

.

C = 1.5; fixCParamMap = @(x)timeVariantAR1ParamMap([x(1:4), C, x(5)]); DGP = ssm(fixCParamMap);

Simulate 20 observations from the DGP. Because `DGP`

is partially specified, pass the true parameter values to `simulate`

by using the `'Params'`

name-value pair argument.

rng(10) % For reproducibility A1 = 0.75; A2 = -0.1; B1 = 1; B2 = 3; D = 2; trueParams = [A1 A2 2*log(B1) 2*log(B2) 2*log(D)]; % Transform variances for parameter map y = simulate(DGP,20,'Params',trueParams);

`y`

is a 20-by-1 vector of simulated measurements ${\mathit{y}}_{\mathit{t}}$ from the DGP.

Because `DGP`

is a partially specified, implicit model object, its parameters are unknown. Therefore, it can serve as a model template for estimation.

Fit the model to the simulated data. Specify random standard Gaussian draws for the initial parameter values. Return the parameter estimates.

[~,estParams] = estimate(DGP,y,randn(1,5),'Display','off')

`estParams = `*1×5*
0.6164 -0.1665 0.0135 1.6803 -1.5855

`estParams`

is a 1-by-5 vector of parameter estimates. The output argument list of the parameter mapping function determines the order of the estimates: `A{1}`

, `A{2}`

, `B{1}`

, `B{2}`

, and `D`

.

Estimate the IRFs of the measurement and state variables by supplying `DGP`

(not the estimated model) and the estimated parameters using the `'Params'`

name-value pair argument.

```
[responseY,responseX] = irf(DGP,'Params',estParams);
table(responseY,responseX)
```

`ans=`*20×2 table*
responseY responseX
___________ ___________
1.5101 1.0068
0.93091 0.6206
0.57385 0.38257
0.35374 0.23583
0.21806 0.14537
0.13442 0.089615
0.082863 0.055242
0.05108 0.034054
0.031488 0.020992
0.019411 0.01294
-0.0032311 -0.0021541
0.00053785 0.00035857
-8.9531e-05 -5.9687e-05
1.4903e-05 9.9356e-06
-2.4808e-06 -1.6539e-06
4.1296e-07 2.7531e-07
⋮

`responseY`

and `responseX`

are time-varying IRFs. The first 10 periods correspond to the IRF of the first state equation. During period 11, the remainder of the shock transfers to the second state equation and filters through that system until it diminishes.

### Estimate IRF Confidence Bounds

Assume that the data generating process (DGP) is the AR(1) model

$${x}_{t}=1+0.9{x}_{t-2}+{u}_{t},$$

where ${\mathit{u}}_{\mathit{t}}$ is a series of independent and identically distributed Gaussian variables with mean 0 and variance 1.

Simulate 500 observations from the model.

rng(1); % For reproducibility DGP = arima('Constant',1,'AR',{0 0.9},'Variance',1); y = simulate(DGP,500);

Explicitly create a diffuse state-space model template for estimation that represents the model. Fit the model to the data, and return parameter estimates and their corresponding estimated covariance matrix.

```
A = [0 NaN NaN; 0 1 0; 1 0 0];
B = [NaN; 0; 0];
C = [1 0 0];
D = 0;
Mdl = dssm(A,B,C,D,'StateType',[0 1 0]);
[~,estParams,EstParamCov] = estimate(Mdl,y,abs(randn(3,1)));
```

Method: Maximum likelihood (fminunc) Effective Sample size: 500 Logarithmic likelihood: -2085.74 Akaike info criterion: 4177.49 Bayesian info criterion: 4190.13 | Coeff Std Err t Stat Prob --------------------------------------------------- c(1) | 0.36553 0.07967 4.58829 0.00000 c(2) | 0.70179 0.00738 95.13852 0 c(3) | 1.16649 0.02236 52.16929 0 | | Final State Std Dev t Stat Prob x(1) | 10.72536 0 Inf 0 x(2) | 1 0 Inf 0 x(3) | 6.66084 0 Inf 0

`Mdl`

is an `ssm`

model template for estimation. `estParams`

is a 3-by-1 vector of estimated coefficients. `EstParamCov`

is a 3-by-3 estimated covariance matrix of the coefficient estimates.

Estimate the IRFs of the state and measurement variables with 95% confidence intervals.

[ResponseY,ResponseX,LowerY,UpperY,LowerX,UpperX] = irf(Mdl,'Params',estParams,... 'EstParamCov',EstParamCov);

`ResponseY`

, `LowerY`

, and `UpperY`

are 20-by-1 vectors representing the measurement variable IRF and corresponding lower and upper confidence bounds. `ResponseX`

, `LowerX`

, and `UpperX`

are 20-by-1-by-3 arrays representing the IRF and corresponding lower and upper confidence bounds of the state variables.

Display a table containing the IRF and confidence bounds of the first state, which represents the AR(2) model.

table(LowerX(:,1,1),ResponseX(:,1,1),UpperX(:,1,1),... 'VariableNames',["LowerIRFx" "IRFX" "UpperIRFX"])

`ans=`*20×3 table*
LowerIRFx IRFX UpperIRFX
_________ ________ _________
1.1214 1.1665 1.209
0 0 0
0.78826 0.81864 0.84833
0 0 0
0.54845 0.57452 0.60214
0 0 0
0.37964 0.40319 0.42929
0 0 0
0.2609 0.28296 0.30597
0 0 0
0.17908 0.19858 0.21954
0 0 0
0.12339 0.13936 0.15655
0 0 0
0.084751 0.097803 0.11184
0 0 0
⋮

The model has only one lag term (lag 2). Therefore, as the shock filters through the system, it impacts the first state variable during odd periods only.

## Input Arguments

`Mdl`

— State-space model

`ssm`

model object | `dssm`

model object

State-space model, specified as an `ssm`

model object returned by `ssm`

or its `estimate`

function, or a `dssm`

model object returned by `dssm`

or its `estimate`

function.

If `Mdl`

is partially specified (that is, it contains unknown parameters), specify estimates of the unknown parameters by using the `'Params'`

name-value argument. Otherwise, `irf`

issues an error.

`irf`

issues an error when `Mdl`

is a *dimension-varying model*, which is a time-varying model containing at least one variable that changes dimension during the sampling period (for example, a state variable drops out of the model).

**Tip**

If `Mdl`

is fully specified, you cannot estimate confidence bounds. To estimate confidence bounds:

Create a partially specified state-space model template for estimation

`Mdl`

.Estimate the model by using the

`estimate`

function and data. Return the estimated parameters`estParams`

and estimated parameter covariance matrix`EstParamCov`

.Pass the model template for estimation

`Mdl`

to`irf`

, and specify the parameter estimates and covariance matrix by using the`'Params'`

and`'EstParamCov'`

name-value arguments.For the

`irf`

function, return the appropriate output arguments for lower and upper confidence bounds.

### 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: **`'NumPeriods',10,'Cumulative',true`

specifies a 10-period cumulative IRF starting at time 1, during which `irf`

applies the shock to a state-disturbance variable in the system, and ending at period 10.

**IRF Options**

`NumPeriods`

— Number of periods

`20`

(default) | positive integer

Number of periods for which `irf`

computes the IRF, specified as a positive integer. Periods in the IRF start at time 1 and end at time `NumPeriods`

.

**Example: **
`'NumPeriods',10`

specifies the inclusion of 10 consecutive time points in the IRF starting at time 1, during which `irf`

applies the shock, and ending at time 10.

**Data Types: **`double`

`Params`

— Estimates of unknown parameters

numeric vector

Estimates of the unknown parameters in the partially specified state-space model `Mdl`

, specified as a numeric vector.

If `Mdl`

is partially specified (contains unknown parameters specified by `NaN`

s), you must specify `Params`

. The `estimate`

function returns parameter estimates of `Mdl`

in the appropriate form. However, you can supply custom estimates by arranging the elements of `Params`

as follows:

If

`Mdl`

is an explicitly created model (`Mdl.ParamMap`

is empty`[]`

), arrange the elements of`Params`

to correspond to hits of a column-wise search of`NaN`

s in the state-space model coefficient matrices, initial state mean vector, and covariance matrix.If

`Mdl`

is time invariant, the order is`A`

,`B`

,`C`

,`D`

,`Mean0`

, and`Cov0`

.If

`Mdl`

is time varying, the order is`A{1}`

through`A{end}`

,`B{1}`

through`B{end}`

,`C{1}`

through`C{end}`

,`D{1}`

through`D{end}`

,`Mean0`

, and`Cov0`

.

If

`Mdl`

is an implicitly created model (`Mdl.ParamMap`

is a function handle), the first input argument of the parameter-to-matrix mapping function determines the order of the elements of`Params`

.

If `Mdl`

is fully specified, `irf`

ignores `Params`

.

**Example: **Consider the state-space model `Mdl`

with `A = B = [NaN 0; 0 NaN]`

, `C = [1; 1]`

, `D = 0`

, and initial state means of 0 with covariance `eye(2)`

. `Mdl`

is partially specified and explicitly created. Because the model parameters contain a total of four `NaN`

s, `Params`

must be a 4-by-1 vector, where `Params(1)`

is the estimate of `A(1,1)`

, `Params(2)`

is the estimate of `A(2,2)`

, `Params(3)`

is the estimate of `B(1,1)`

, and `Params(4)`

is the estimate of `B(2,2)`

.

**Data Types: **`double`

`Cumulative`

— Flag for computing cumulative IRF

`false`

(default) | `true`

Flag for computing the cumulative IRF, specified as a value in this table.

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

`true` | `irf` computes the cumulative IRF of all variables over the specified time range. |

`false` | `irf` computes the standard, period-by-period IRF of all variables over the specified time range. |

**Example: **`'Cumulative',true`

**Data Types: **`logical`

`Method`

— IRF estimation algorithm

`'repeated-multiplication'`

(default) | `'eigendecomposition'`

IRF estimation algorithm, specified as `'repeated-multiplication'`

or `'eigendecomposition'`

.

The IRF estimator of time *m* contains the factor *A*^{m}. This table describes the supported algorithms to compute the matrix power.

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

`'repeated-multiplication'` | `irf` uses recursive multiplication. |

`'eigendecomposition'` | `irf` attempts to use the spectral decomposition of A to compute the matrix power. Specify this value only when you suspect that the recursive multiplication algorithm might experience numerical issues. For more details, see Algorithms. |

**Data Types: **`string`

| `char`

**Confidence Bound Estimation Options**

`EstParamCov`

— Estimated covariance matrix of unknown parameters

positive semidefinite numeric matrix

Estimated covariance matrix of unknown parameters in the partially specified state-space model `Mdl`

, specified as a positive semidefinite numeric matrix.

`estimate`

returns the estimated parameter covariance matrix of `Mdl`

in the appropriate form. However, you can supply custom estimates by setting `EstParamCov(`

to the estimated covariance of the estimated parameters * i*,

*)*

`j`

`Params(``i`

)

and `Params(``j`

)

, regardless of whether `Mdl`

is time invariant or time varying.If `Mdl`

is fully specified, `irf`

ignores `EstParamCov`

.

By default, `irf`

does not estimate confidence bounds.

**Data Types: **`double`

`NumPaths`

— Number of Monte Carlo sample paths

`1000`

(default) | positive integer

Number of Monte Carlo sample paths (trials) to generate to estimate confidence bounds, specified as a positive integer.

**Example: **`'NumPaths',5000`

**Data Types: **`double`

`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.

**Data Types: **`double`

## Output Arguments

`ResponseY`

— IRF of measurement variables

numeric array

IRFs of the measurement variables *y*_{t}, returned as a `NumPeriods`

-by-*k*-by-*n* numeric array.

`ResponseY(`

is the dynamic response of measurement variable * t*,

*,*

`i`

*)*

`j`

`j`

at period `t`

, when a unit shock is applied to state-disturbance variable `i`

during period 1, for `t`

= 1,2,...,`NumPeriods`

, `i`

= 1,2,...,*k*, and

`j`

= 1,2,...,*n*.

`ResponseX`

— IRF of state variables

numeric array

IRFs of the state variables *x*_{t}, returned as a `NumPeriods`

-by-*k*-by-*m* numeric array.

`ResponseX(`

is the dynamic response of state variable * t*,

*,*

`i`

*)*

`j`

`j`

at period `t`

, when a unit shock is applied to state-disturbance variable `i`

during period 1, for `t`

= 1,2,...,`NumPeriods`

, `i`

= 1,2,...,*k*, and

`j`

= 1,2,...,*m*.

`LowerY`

— Pointwise lower confidence bounds of measurement variable IRF

numeric array

Pointwise lower confidence bounds of the measurement variable IRF, returned as a `NumPeriods`

-by-*k*-by-*n* numeric array.

`LowerY(`

is the lower bound of the * t*,

*,*

`i`

*)*

`j`

`100*Confidence`

% percentile interval on the true dynamic response of measurement variable `j`

at period `t`

, when a unit shock is applied to state-disturbance variable `i`

during period 1.`UpperY`

— Pointwise upper confidence bounds of measurement variable IRF

numeric array

Pointwise upper confidence bounds of the measurement variable IRF, returned as a `NumPeriods`

-by-*k*-by-*n* numeric array.

`UpperY(`

is the upper confidence bound corresponding to the lower confidence bound * t*,

*,*

`i`

*)*

`j`

`LowerY`

`(``t`

,`i`

,`j`

)

.`LowerX`

— Pointwise lower confidence bounds of state variable IRF

numeric array

Pointwise lower confidence bounds of the state variable IRF, returned as a `NumPeriods`

-by-*k*-by-*m* numeric array.

`LowerX(`

is the lower bound of the * t*,

*,*

`i`

*)*

`j`

`100*Confidence`

% percentile interval on the true dynamic response of state variable `j`

at period `t`

, when a unit shock is applied to state-disturbance variable `i`

during period 1.`UpperX`

— Pointwise upper confidence bounds of state variable IRF

numeric array

Pointwise upper confidence bounds of the state variable IRF, returned as a `NumPeriods`

-by-*k*-by-*m* numeric array.

`UpperX(`

is the upper confidence bound corresponding to the lower bound * t*,

*,*

`i`

*)*

`j`

`LowerX`

`(``t`

,`i`

,`j`

)

.## More About

### Impulse Response Function

An *impulse response function* (IRF) of a state-space model (or *dynamic response of the system*) measures contemporaneous and future changes in the state and measurement variables when each state-disturbance variable is shocked by a unit impulse at period 1. In other words, the IRF at time *t* is the derivative of each state and measurement variable at time *t* with respect to a state-disturbance variable at time 1, for each *t* ≥ 1.

Consider the time-invariant state-space model

$$\begin{array}{l}{x}_{t}=A{x}_{t-1}+B{u}_{t}\\ {y}_{t}=C{x}_{t}+D{\epsilon}_{t},\end{array}$$

and consider an unanticipated unit shock at period 1, applied to state-disturbance variable *j*
*u _{j,t}*.

The *r*-step-ahead response of the state variables *x _{t}* to the shock is

$${\psi}_{xj}(r)={A}^{r}{b}_{j},$$

where *r*
> 0 and *b _{j}* is column

*j*of the state-disturbance-loading matrix

*B*.

The *r*-step-ahead response of the measurement variables *y _{t}* to the shock is

$${\psi}_{yj}(r)=C{A}^{r}{b}_{j}.$$

IRFs depend on the time interval over which they are computed. However, the IRF of a time-invariant state-space model is *time homogeneous*, which means that the IRF does not depend on the time at which the shock is applied. *Time-varying IRFs*, which are the IRFs of a time-varying but dimension-invariant system, have the form

$$\begin{array}{l}{\psi}_{xj}(r)={A}_{r}\cdots {A}_{2}{A}_{1}{b}_{1,j}\\ {\psi}_{yj}(r)={C}_{r}{A}_{r}\cdots {A}_{2}{A}_{1}{b}_{1,j},\end{array}$$

where *b*_{1,j} is column *j* of *B*_{1}, the period 1 state-disturbance-loading matrix. Time-varying IRFs depend on the time at which the shock is applied. `irf`

always applies the shock at period 1.

IRFs are independent of the initial state distribution.

## Algorithms

If you specify

`'eigendecomposition'`

for the`'Method'`

name-value pair argument,`irf`

attempts to diagonalize the state-transition matrix*A*by using the spectral decomposition.`irf`

resorts to recursive multiplication instead under at least one of these circumstances:An eigenvalue is complex.

The rank of the matrix of eigenvectors is less than the number of states

`Mdl`

is time varying.

If you do not supply

`'EstParamCov'`

, confidence bounds of each period overlap.`irf`

uses Monte Carlo simulation to compute confidence intervals.`irf`

randomly draws`NumPaths`

variates from the asymptotic sampling distribution of the unknown parameters in`Mdl`

, which is N_{p}(`Params`

,`EstParamCov`

), where*p*is the number of unknown parameters.For each randomly drawn parameter set

*j*,`irf`

:Creates a state-space model that is equal to

`Mdl`

, but substitutes in parameter set*j*Computes the random IRF of the resulting model

*ψ*_{j}(*t*), where*t*= 1 through`NumPaths`

For each time

*t*, the lower bound of the confidence interval is the`(1 –`

quantile of the simulated IRF at period)/2`c`

*t**ψ*(*t*), where

=`c`

`Confidence`

. Similarly, the upper bound of the confidence interval at time*t*is the`(1 –`

upper quantile of)/2`c`

*ψ*(*t*).

## Version History

**Introduced in R2020b**

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