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

$y_{t} = μ + θ (L) ε_{t},$

where $θ (L)$ is a q-degree MA operator polynomial, $(1 + θ_{1} L + \dots + θ_{q} L^{q}) .$

The impulse response function for an MA model is the sequence of MA coefficients, $1, θ_{1}, \dots, θ_{q} .$

Create MA Model

Create a zero-mean MA(3) model with coefficients $θ_{1} = 0.8$ , $θ_{2} = 0.5$ , and $θ_{3} = - 0.1 .$

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

Step 2. Plot the impulse response function.

impulse(Mdl)

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

Open Live Script

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

$y_{t} = μ + ϕ (L)^{- 1} ε_{t},$

where $ϕ (L)$ is a $p$ -degree AR operator polynomial, $(1 - ϕ_{1} L - \dots - ϕ_{p} L^{p})$ .

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 $ϕ_{1} = 0.5$ and $ϕ_{2} = - 0.75$ .

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

Open Live Script

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

$y_{t} = μ + \frac{θ (L)}{ϕ (L)} ε_{t},$

where $θ (L)$ is a q-degree MA operator polynomial, $(1 + θ_{1} L + \dots + θ_{q} L^{q})$ , and $ϕ (L)$ is a p-degree AR operator polynomial, $(1 - ϕ_{1} L - \dots - ϕ_{p} L^{p})$ .

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.

Specify ARMA Model

Specify an ARMA(2,1) model with coefficients $ϕ_{1}$ = 0.6, $ϕ_{2} = - 0.3$ , and $θ_{1} = 0.4$ .

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

Plot Impulse Response Function

Plot the impulse response function for 10 periods.

impulse(Mdl,10)

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