When you have models with parameters (known or estimated), you can examine the predictions of the models. For information on creating VAR models, see Multivariate Time Series Model Creation. For information on estimating models, see VAR Model Estimation.
This list describes the main forecasting methods.
Generate minimum mean square error forecasts and corresponding mean square error matrices. For an example, see Forecast VAR Model.
Generate conditional forecasts and corresponding mean square error matrices given some future response values in the forecast horizon. For an example, see Forecast VAR Model Conditional Responses.
Generate many random conditional future response paths given some future response values in the forecast horizon. For an example, see Simulate VAR Model Conditional Responses.
These functions base their forecasts on a fully specified model object and initial data. The functions differ in their innovations processes:
forecast assumes zero-valued
forecast yields a deterministic
forecast, conditional or otherwise.
simulate assumes the multivariate
innovations are jointly Gaussian distributed with covariance matrix Σ.
pseudorandom, Monte Carlo sample paths.
filter requires innovations process
filter yields a sample path that is deterministically
based on the specified innovations process paths.
forecast is faster and requires less memory
than generating many sample paths using
forecast is not as flexible as
For example, suppose you transform some time series before making
a model, and want to undo the transformation when examining forecasts.
The error bounds given by transforms of
bounds are not valid bounds. In contrast, the error bounds given by
the statistics of transformed simulations are valid.
For unconditional forecasting,
A deterministic forecast time series based on 0 innovations
Time series of forecast mean square error matrices based on the Σ, the innovations covariance matrix.
For conditional forecasting:
forecast requires an array of future response data that contains a mix of
NaN) and known values.
forecast generates forecasts for the missing
values conditional on the known values.
The forecasts generated by
forecast are also deterministic, but the mean
square error matrices are based on Σ and the
known response values in the forecast horizon.
forecast uses the Kalman filter to generate forecasts. Specifically:
forecast represents the VAR model
as a state-space model (
object) without observation error.
forecast filters the forecast data
through the state-space model. That is, at period
t in the forecast horizon, any
unknown response is
s < t, is the
filtered estimate of y from period
s in the forecast horizon.
forecast uses presample values
for periods before the forecast horizon.
For either type of forecast, To initialize the VAR(p)
model in the forecast horizon,
forecast requires p presample
observations. You can optionally specify more than one path of presample
data. If you do specify multiple paths,
a three-dimensional array of forecasted responses, with each page
corresponding to a path of presample values.
For unconditional simulation,
Generates random time series based on the model using random paths of multivariate Gaussian innovations distributed with a mean of zero and a covariance of Σ
Filters the random paths of innovations through the model
For conditional simulation:
requires an array of future response data that contains a mix of missing
and known values, and generates values for the missing responses.
simulate performs conditional
simulation using this process. At each time
the forecast horizon:
simulate infers (or, inverse
filters) the innovations (
from the known future responses.
For missing future innovations,
Z1, which is the random,
standard Gaussian distribution disturbances conditional on the known
Z1 by the lower triangular
Cholesky factor of the conditional covariance matrix. That is,
the covariance of the conditional Gaussian distribution.
Z2 in place of the corresponding
missing values in
For the missing values in the future response data,
the corresponding random innovations through the VAR model
For either type of simulation:
simulate does not require presample observations. For details on the
default values of the presample data, see
To carry out inference, generate 1000s of response
paths, and then estimate sample statistics from the generated paths
at each time in the forecast horizon. For example, suppose
a three-dimensional array of forecasted paths. Monte Carlo point and
interval estimates of the forecast at time
the forecast horizon is
MCPointEst = mean(Y(t,:,:),3); MCPointInterval = quantile(Y(t,:,:),[0.025 0.975],3);
That is, the Monte Carlo point estimate is the mean across pages and the Monte Carlo interval estimate is composed of the 2.5th and the 97.5th percentiles computed across paths. Observe that Monte Carlo estimates are subject to Monte Carlo error, and so estimates differ each time you run the analysis under the same conditions, but using a different random number seed.
If you scaled any time series before fitting a model, you can unscale the resulting time series to understand its predictions more easily.
If you scaled a series with
log, transform predictions of the corresponding
If you scaled a series with
diff(log) or, equivalently,
predictions of the corresponding model with
cumsum is the inverse of
diff; it calculates
cumulative sums. As in integration, you must choose an appropriate additive
constant for the cumulative sum. For example, take the log of the final
entry in the corresponding data series, and use it as the first term in the
series before applying
You can examine the effect of impulse responses to
armairf. An impulse
response is the deterministic response of a time series model to an
innovations process that has the value of one standard deviation in
one component at the initial time, and zeros in all other components
and times. The main component of the impulse response function are
the dynamic multipliers, that is, the coefficients
of the VMA representation of the VAR model. For more details, see Impulse Response Function.
Given a fully specified
varm model, you must
supply the autoregression coefficients to
armairf sends a unit shock through
the system, which results in the forecast error impulse
response. You can optionally supply the innovations covariance
matrix and choose whether to generate generalized or orthogonalized impulse
responses. Generalized impulse responses amount to filtering a shock
of one standard error of each innovation though the VAR model. Orthogonalized
impulse responses scale the dynamic multipliers by the lower triangular
Cholesky factor of the innovations covariance. For more details, see .
For an example, see Generate VAR Model Impulse Responses.
 Lütkepohl, H. New Introduction to Multiple Time Series Analysis. Berlin: Springer, 2005.
 Pesaran, H. H. and Y. Shin. “Generalized Impulse Response Analysis in Linear Multivariate Models.” Economic Letters. Vol. 58, 1998, 17–29.