impulse(Mdl)
impulse(Mdl,numObs)
Y = impulse(___)
impulse(
plots
a discrete stem plot of the impulse response function for
the univariate ARIMA model, Mdl
)Mdl
, in the current
figure window.
impulse(
plots
the impulse response function for Mdl
,numObs
)numObs
periods.
Y = impulse(___)
returns the
impulse response in a column vector for any of the previous input
arguments.
To improve performance of the filtering algorithm,
specify the number of observations to include in the impulse response, numObs
.
When you do not specify numObs
, impulse
computes
the impulse response by converting the input model to a truncated,
infinitedegree, moving average representation using the relatively
slow lag operator polynomial division algorithm. This results in an
impulse response of generally unknown length.
 

Positive integer indicating the number of observations to include
in the impulse response (the number of periods for which When you specify If you do not specify 

Column vector of impulse responses. If you specify 
The impulse response function for a univariate
ARIMA process 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 zero.
For a univariate ARIMA process, y_{t}, and innovation series ε_{t}, the impulse response at time j, Ψ_{j}, is given by
$${\psi}_{j}=\frac{\partial {y}_{j}}{\partial {\epsilon}_{0}}.$$
Expressed as a function of time, the sequence of
dynamic multipliers, Ψ_{1}, Ψ_{2},...,
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 ε_{0} =
1, with all past observations of y_{t} and
all future shocks of ε_{t} set
to zero. Because the impulse response function is the partial derivative
of the ARIMA process with respect to an innovation shock at time 0,
the presence of a constant 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 onestepahead forecast errors.
[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, H. New Introduction to Multiple Time Series Analayis. New York, NY: SpringerVerlag, 2007.