This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

Plot the Impulse Response Function

Moving Average Model

This example shows how to calculate and plot the impulse response function for a moving average (MA) model. The MA(q) model is given by

yt=μ+θ(L)εt,

where θ(L) is a q-degree MA operator polynomial, (1+θ1L++θqLq).

The impulse response function for an MA model is the sequence of MA coefficients, 1,θ1,,θq.

Step 1. Specify the MA model.

Specify a zero-mean MA(3) model with coefficients θ1=0.8, θ2=0.5, and θ3=-0.1.

modelMA = arima('Constant',0,'MA',{0.8,0.5,-0.1});

Step 2. Plot the impulse response function.

impulse(modelMA)

For an MA model, the impulse response function cuts off after q periods. For this example, the last nonzero coefficient is at lag q = 3.

Autoregressive Model

This example shows how to compute and plot the impulse response function for an autoregressive (AR) model. The AR(p) model is given by

yt=μ+ϕ(L)-1εt,

where ϕ(L) is a p-degree AR operator polynomial, (1-ϕ1L--ϕpLp).

An AR process is stationary provided that the AR operator polynomial is stable, meaning all its roots lie outside the unit circle. In this case, the infinite-degree inverse polynomial, ψ(L)=ϕ(L)-1, has absolutely summable coefficients, and the impulse response function decays to zero.

Step 1. Specify the AR model.

Specify an AR(2) model with coefficients ϕ1=0.5 and ϕ2=-0.75.

modelAR = arima('AR',{0.5,-0.75});

Step 2. Plot the impulse response function.

Plot the impulse response function for 30 periods.

impulse(modelAR,30)

The impulse function decays in a sinusoidal pattern.

ARMA Model

This example shows how to plot the impulse response function for an autoregressive moving average (ARMA) model. The ARMA(p, q) model is given by

yt=μ+θ(L)ϕ(L)εt,

where θ(L) is a q-degree MA operator polynomial, (1+θ1L++θqLq), and ϕ(L) is a p-degree AR operator polynomial, (1-ϕ1L--ϕpLp).

An ARMA process is stationary provided that the AR operator polynomial is stable, meaning all its roots lie outside the unit circle. In this case, the infinite-degree inverse polynomial, ψ(L)=θ(L)/ϕ(L) , has absolutely summable coefficients, and the impulse response function decays to zero.

Step 1. Specify an ARMA model.

Specify an ARMA(2,1) model with coefficients ϕ1 = 0.6, ϕ2=-0.3, and θ1=0.4.

modelARMA = arima('AR',{0.6,-0.3},'MA',0.4);

Step 2. Plot the impulse response function.

Plot the impulse response function for 10 periods.

impulse(modelARMA,10)

See Also

| | | | |

Related Topics