# vartovec

(Not recommended) Vector autoregression (VAR) to vector error-correction model (VEC)

## Syntax

```[VEC,C] = vartovec(VAR)```

## Description

 Note:   `vartovec` will be removed in a future release. Use `var2vec` instead.

Given a vector autoregression (VAR) model, ```[VEC,C] = vartovec(VAR)``` converts VAR to an equivalent vector error-correction (VEC) model. A VAR(p) model of a time series y(t) has the form:

`${A}_{0}y\left(t\right)={A}_{1}y\left(t-1\right)+...+{A}_{p}y\left(t-p\right)+\epsilon \left(t\right)$`

The equivalent VEC(q) model, with q = p − 1, has the form:

`${B}_{0}z\left(t\right)={B}_{1}z\left(t-1\right)+...+{B}_{q}z\left(t-q\right)+Cy\left(t-1\right)+\epsilon \left(t\right)$`

where z(t) = y(t) − y(t − 1) and C is the error-correction coefficient.

## Input Arguments

 `VAR` The `VAR`(p) model to be converted to an equivalent `VEC`(q) model, with q = p − 1. VAR is specified by a (p + 1)-element cell vector of square matrices {`A0` `A1` ... `Ap`} associated with coefficients at lags 0, 1, ..., p. To represent a univariate model, `VAR` may be specified as a double-precision vector. Alternatively, `VAR` may be specified as a `LagOp` object or a `vgxset` object.

## Output Arguments

 `VEC` The `VEC` representation of the input `VAR` model. The data type and orientation of `VEC` is consistent with that of `VAR` `C` The error-correction coefficient. `C` is a square matrix the same size as the coefficients of the associated `VEC`.

collapse all

### Algorithms

• Written as a polynomial in the lag operator Ly(t) = y(t − 1), a VAR(p) model has the form:

`$\left({A}_{0}-{A}_{1}L-...-{A}_{p}{L}^{p}\right)y\left(t\right)=A\left(L\right)y\left(t\right)=\epsilon \left(t\right)$`

The equivalent VEC(q) model has the form:

`$\left({B}_{0}-{B}_{1}L-...-{B}_{q}{L}^{q}\right)z\left(t\right)=B\left(L\right)z\left(t\right)=Cy\left(t-1\right)+\epsilon \left(t\right)$`

Thus, if `VAR` is specified as a `LagOp` object `A`, coefficients of lagged values of y(t) must be represented by the opposite of their values in standard difference-equation form, and the output `VEC` will follow a similar sign convention

• If `VAR` is specified as a `vgxset` object, the conversion involves only the `AR0`, `AR`, and `nAR` components of the model. Other model components are unaffected.

## References

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

[2] Lutkepohl, H. "New Introduction to Multiple Time Series Analysis." Springer-Verlag, 2007.