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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

$${y}_{t}=\mu +\theta (L){\epsilon}_{t},$$

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

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

**Step 1. Specify the MA model.**

Specify a zero-mean MA(3) model with coefficients $${\theta}_{1}=0.8$$, $${\theta}_{2}=0.5$$, and $${\theta}_{3}=-0.1.$$

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.

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

$${y}_{t}=\mu +\varphi (L{)}^{-1}{\epsilon}_{t},$$

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

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, $$\psi (L)=\varphi (L{)}^{-1}$$, 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 $${\varphi}_{1}=0.5$$ and $${\varphi}_{2}=-0.75$$.

`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.

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

$${y}_{t}=\mu +\frac{\theta (L)}{\varphi (L)}{\epsilon}_{t},$$

where $$\theta (L)$$ is a *q*-degree MA operator polynomial, $$(1+{\theta}_{1}L+\dots +{\theta}_{q}{L}^{q})$$, and $$\varphi (L)$$ is a *p*-degree AR operator polynomial, $$(1-{\varphi}_{1}L-\dots -{\varphi}_{p}{L}^{p})$$.

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, $$\psi (L)=\theta (L)/\varphi (L)$$ , 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 $${\varphi}_{1}$$ = 0.6, $${\varphi}_{2}=-0.3$$, and $${\theta}_{1}=0.4$$.

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)

`LagOp`

| `arima`

| `cell2mat`

| `impulse`

| `isStable`

| `toCellArray`