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Infer vector autoregression model (VAR) innovations

`E = infer(Mdl,Y)`

`E = infer(Mdl,Y,Name,Value)`

```
[E,logL]
= infer(___)
```

uses additional
options specified by one or more `E`

= infer(`Mdl`

,`Y`

,`Name,Value`

)`Name,Value`

pair arguments. For example, you can specify presample responses or exogenous
predictor data. uses additional
options specified by one or more `Name,Value`

pair arguments.

`infer`

infers innovations by evaluating the VAR model`Mdl`

with respect to the innovations using the supplied data`Y`

,`Y0`

, and`X`

. The inferred innovations are$${\widehat{\epsilon}}_{t}=\widehat{\Phi}(L){y}_{t}-\widehat{c}-\widehat{\beta}{x}_{t}-\widehat{\delta}t.$$

`infer`

uses this process to determine the time origin*t*_{0}of models that include linear time trends.If you do not specify

`Y0`

, then*t*_{0}= 0.Otherwise,

`infer`

sets*t*_{0}to`size(Y0,1)`

–`Mdl.P`

. Therefore, the times in the trend component are*t*=*t*_{0}+ 1,*t*_{0}+ 2,...,*t*_{0}+`numobs`

, where`numobs`

is the effective sample size (`size(Y,1)`

after`infer`

removes missing values). This convention is consistent with the default behavior of model estimation in which`estimate`

removes the first`Mdl.P`

responses, reducing the effective sample size. Although`infer`

explicitly uses the first`Mdl.P`

presample responses in`Y0`

to initialize the model, the total number of observations in`Y0`

and`Y`

(excluding missing values) determines*t*_{0}.

[1]
Hamilton, J. D. *Time Series Analysis*. Princeton, NJ: Princeton University Press, 1994.

[2]
Johansen, S. *Likelihood-Based Inference in Cointegrated Vector Autoregressive Models*. Oxford: Oxford University Press, 1995.

[3]
Juselius, K. *The Cointegrated VAR Model*. Oxford: Oxford University Press, 2006.

[4]
Lütkepohl, H. *New Introduction to Multiple Time Series Analysis*. Berlin: Springer, 2005.

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