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A common objective of time series modeling is generating forecasts
for a process over a future time horizon. That is, given an observed
series *y*_{1}, *y*_{2},...,*y _{N}* and
a forecast horizon

Let $${\widehat{y}}_{t+1}$$ denote a forecast for the process
at time *t* + 1, conditional on the history of the
process up to time *t*, *H _{t}*,
and the exogenous covariate series up to time

$$E{\left({y}_{t+1}-{\widehat{y}}_{t+1}|{H}_{t},{X}_{t+1}\right)}^{2}.$$

Minimizing this loss function yields the MMSE forecast,

$${\widehat{y}}_{t+1}=E({y}_{t+1}|{H}_{t},{X}_{t+1}).$$

The `forecast`

method generates MMSE forecasts
recursively. When you call `forecast`

, you can specify
presample observations (`Y0`

), innovations (`E0`

),
conditional variances (`V0`

), and exogenous covariate
data (`X0`

) using name-value arguments. If you include
presample exogenous covariate data, then you must also specify exogenous
covariate forecasts (`XF`

).

To begin forecasting from the end of an observed series, say `Y`

,
use the last few observations of `Y`

as presample
responses `Y0`

to initialize the forecast. There
are several points to keep in mind when you specify presample data:

The minimum number of responses needed to initialize forecasting is stored in the property

`P`

of an`arima`

model. If you provide too few presample observations,`forecast`

returns an error.If you do not provide any presample responses, then

`forecast`

assigns default values:For models that are stationary

*and*do not contain a regression component, all presample observations are set to the unconditional mean of the process.For nonstationary models

*or*models with a regression component, all presample observations are set to zero.

If you forecast a model with an MA component, then

`forecast`

requires presample innovations. The number of innovations needed is stored in the property`Q`

of an`arima`

model. If you also have a conditional variance model, you must additionally account for any presample innovations it requires. If you specify presample innovations, but not enough,`forecast`

returns an error.If you forecast a model with a regression component, then

`forecast`

requires presample exogenous covariate data. The number of presample exogenous covariate data needed is at least the number of presample responses minus`P`

. If you provide presample exogenous covariate data, but not enough, then`forecast`

returns an error.If you do not specify any presample innovations, but specify sufficient presample

*responses*(at least`P`

+`Q`

) and exogenous covariate data (at least the number of presample responses minus`P`

), then`forecast`

automatically infers presample innovations. In general, the longer the presample response series you provide, the better the inferred presample innovations will be. If you provide presample responses and exogenous covariate data, but not enough,`forecast`

sets presample innovations equal to zero.

Consider generating forecasts for an AR(2) process,

$${y}_{t}=c+{\varphi}_{1}{y}_{t-1}+{\varphi}_{2}{y}_{t-2}+{\epsilon}_{t}.$$

Given presample observations $${y}_{N-1}$$ and $${y}_{N},$$ forecasts are recursively generated as follows:

$${\widehat{y}}_{N+1}=c+{\varphi}_{1}{y}_{N}+{\varphi}_{2}{y}_{N-1}$$

$${\widehat{y}}_{N+2}=c+{\varphi}_{1}{\widehat{y}}_{N+1}+{\varphi}_{2}{y}_{N}$$

$${\widehat{y}}_{N+3}=c+{\varphi}_{1}{\widehat{y}}_{N+2}+{\varphi}_{2}{\widehat{y}}_{N+1}$$

$$\vdots $$

For a stationary AR process, this recursion converges to the unconditional mean of the process,

$$\mu =\frac{c}{\left(1-{\varphi}_{1}-{\varphi}_{2}\right)}.$$

For an MA(12) process, e.g.,

$${y}_{t}=\mu +{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+\mathrm{...}+{\theta}_{12}{\epsilon}_{t-12},$$

you need 12 presample
innovations to initialize the forecasts. All innovations from time *N* +
1 and greater are set to their expectation, zero. Thus, for an MA(12)
process, the forecast for any time more than 12 steps in the future
is the unconditional mean, *μ*.

The forecast mean square error for an *s*-step
ahead forecast is given by

$$MSE=E{\left({y}_{t+s}-{\widehat{y}}_{t+s}|{H}_{t+s-1},{X}_{t+s}\right)}^{2}.$$

Consider a conditional mean model given by

$${y}_{t}=\mu +{x}_{t}^{\prime}\beta +\psi (L){\epsilon}_{t},$$

where $$\psi (L)=1+{\psi}_{1}L+{\psi}_{2}{L}^{2}+\dots $$.
Sum the variances of the lagged innovations to get the *s*-step *MSE*,

$$(1+{\psi}_{1}^{2}+{\psi}_{2}^{2}+\dots +{\psi}_{s-1}^{2}){\sigma}_{\epsilon}^{2},$$

where $${\sigma}_{\epsilon}^{2}$$ denotes the innovation variance.

For stationary processes, the coefficients of the infinite lag
operator polynomial are absolutely summable, and the *MSE* converges
to the unconditional variance of the process.

For nonstationary processes, the series does not converge, and the forecast error grows over time.

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