impulse

Class: regARIMA

Impulse response of regression model with ARIMA errors

Syntax

impulse(Mdl)
impulse(Mdl,numObs)
Y = impulse(___)


Description

impulse(Mdl) plots a discrete stem plot of the impulse response function for the regression model with ARIMA time series errors, Mdl, in the current figure window.

impulse(Mdl,numObs) plots the impulse response function for numObs periods.

Y = impulse(___) returns the impulse response in a column vector for any of the previous input arguments.

Input Arguments

Mdl	Regression model with ARIMA errors, as created by regARIMA or estimate.
numObs	Number of observations to include in the impulse response, specified as a positive integer. numObs is the number of periods for which impulse computes the impulse response. Default: impulse determines the number of observations using the polynomial division algorithm of the underlying lag operator polynomials, mldivide.

Output Arguments

Y	Impulse responses of the model Mdl, specified as a column vector. If you specify numObs, then Y is numObs-by-1. If you do not specify numObs, the underlying lag operator polynomial division algorithm returns an impulse response of generally unknown length.

Examples

Plot an Impulse Response Function

Specify the following regression model with ARMA(2,1) errors:

$$\begin{array}{l}\begin{array}{c}{y}_t=X_t\left[\begin{array}{c}0.1\\ -0.2\end{array}\right]+u_t\\ u_t=0.5u_{t-1}-0.8u_{t-2}+{\epsilon }_t-0.5{\epsilon }_{t-1},\end{array}\end{array}$$

where $${\epsilon }_t$$ is Gaussian with variance 0.1.

Mdl = regARIMA('Intercept',0, 'AR', {0.5 -0.8}, ...
    'MA',-0.5,'Beta',[0.1 -0.2], 'Variance',0.1);

Time the calculation of and plot the impulse response function without specifying the number of observations.

tic
impulse(Mdl)

toc

Elapsed time is 0.169437 seconds.

The model is stationary; the impulse response function decays in a sinusoidal pattern.

Time the calculation of and plot the impulse response function using 45 observations.

tic
impulse(Mdl,45)

toc

Elapsed time is 0.176164 seconds.

There are more observations represented in this plot than the one generated in the previous step. However, the impulse response function and the plot took less time to generate in this step than the previous. This is because the software did not calculate the impulse response function using an infinite-degree moving average as in the previous step.

Store an Impulse Response Function

Specify the following regression model with ARMA(2,1) errors:

$$\begin{array}{l}\begin{array}{c}{y}_t=X_t\left[\begin{array}{c}0.1\\ -0.2\end{array}\right]+u_t\\ u_t=0.5u_{t-1}-0.8u_{t-2}+{\epsilon }_t-0.5{\epsilon }_{t-1},\end{array}\end{array}$$

where $${\epsilon }_t$$ is Gaussian with variance 0.1.

Mdl = regARIMA('Intercept',0, 'AR', {0.5 -0.8}, ...
    'MA',-0.5,'Beta',[0.1 -0.2], 'Variance',0.1);

Store the impulse response function for 15 periods.

Y = impulse(Mdl,15)

Y = 15×1

    1.0000
         0
   -0.8000
   -0.4000
    0.4400
    0.5400
   -0.0820
   -0.4730
   -0.1709
    0.2930
      ⋮

The length of the output impulse response series is numObs.

More About

Impulse Response Function

The impulse response function for regression models with ARIMA errors is the dynamic response of the system to a single impulse, or innovation shock, of unit size. The specific impulse response calculated by impulse is the dynamic multiplier, defined as the partial derivative of the output response with respect to an innovation shock at time 0.

For a regression model with ARIMA errors, y_t, unconditional disturbances u_t, and innovation series ε_t, the impulse response at time j, Ψ_j, is given by

$${\psi }_j=\frac{\partial y_j}{\partial {\epsilon }_0}=\frac{\partial u_j}{\partial {\epsilon }_0}.$$

Expressed as a function of time, the sequence of dynamic multipliers, Ψ₁, Ψ₂,..., measures the sensitivity of the process to a purely transitory change in the innovation process. impulse computes the impulse response function by shocking the system with a unit impulse ε₀ = 1, with all past observations of y_t and all future shocks of ε_t set to 0. The impulse response function is the partial derivative of the ARIMA process with respect to an innovation shock at time 0. Because of this, the presence of an intercept or a regression component corresponding to predictors in the model has no effect on the output.

This impulse response is sometimes called the forecast error impulse response, because the innovations, ε_t, can be interpreted as the one-step-ahead forecast errors.

Tips

• To improve performance of the filtering algorithm, specify the number of observations, numObs, to include in the impulse response.

Algorithms

• If you specify the number of observations, numObs, impulse computes the impulse response by filtering a unit shock followed by an appropriate length vector of 0s. The filtering algorithm is very fast and results in an impulse response of known (numObs) length.

• If you do not specify numObs, then impulse converts the error model to a truncated, infinite-degree moving average using the relatively slow lag operator polynomial division algorithm. This produces an impulse response of generally unknown length.

References

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

[2] Enders, W. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, 1995.

[3] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[4] Lütkepohl, Helmut. New Introduction to Multiple Time Series Analysis. New York, NY: Springer-Verlag, 2007.