impulse(Mdl)
impulse(Mdl,numObs)
Y = impulse(___)
impulse(
plots
a discrete stem plot of the impulse response function for
the regression model with ARIMA time series errors, Mdl
)Mdl
,
in the current figure window.
impulse(
plots
the impulse response function for Mdl
,numObs
)numObs
periods.
returns
the impulse response in a column vector for any of the previous input
arguments.Y
= impulse(___)
To improve performance of the filtering algorithm,
specify the number of observations, numObs
, to
include in the impulse response.

Regression model with ARIMA errors, as created by 

Number of observations to include in the impulse response, specified
as a positive integer. Default: 

Impulse responses of the model

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, 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 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 onestepahead forecast errors.
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, infinitedegree moving average using
the relatively slow lag operator polynomial division algorithm. This
produces an impulse response of generally unknown length.
[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.