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