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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

$${y_t} = \mu  + \theta (L){\varepsilon _t},$$

where $\theta(L)$ is a q-degree MA operator polynomial, $(1 + {\theta _1}L +  \ldots  + {\theta _q}{L^q}).$

The impulse response function for an MA model is the sequence of MA coefficients, $1,{\theta _1}, \ldots ,{\theta _q}.$

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.

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

$${y_t} = \mu  + \phi {(L)^{ - 1}}{\varepsilon _t},$$

where $\phi(L)$ is a $p$-degree AR operator polynomial, $(1 - {\phi _1}L -  \ldots  - {\phi _p}{L^p})$.

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, $\psi (L) = \phi {(L)^{ - 1}}$, 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 $\phi_1 = 0.5$ and $\phi_2 = -0.75$.

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

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

$${y_t} = \mu  + \frac{{\theta (L)}}{{\phi (L)}}{\varepsilon _t},$$

where $\theta(L)$ is a q-degree MA operator polynomial, $(1 + {\theta _1}L +  \ldots  + {\theta _q}{L^q})$, and $\phi(L)$ is a p-degree AR operator polynomial, $(1 - {\phi _1}L -  \ldots  - {\phi _p}{L^p})$.

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, $\psi (L) = {{\theta (L)} \mathord{\left/ {\vphantom {{\theta (L)} {\phi (L)}}} \right. \kern-\nulldelimiterspace} {\phi (L)}}$ , 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 $\phi_1$ = 0.6, $\phi_2 = -0.3$, 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)

See Also

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