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Convert VAR model to VEC model

If any of the time series in a vector autoregression (VAR) model are cointegrated, then the VAR model is nonstationary. You can determine the error-correction coefficient by converting the VAR model to a vector error-correction (VEC) model. The error-correction coefficient matrix determines, on average, how the time series react to deviations from their long-run averages. The rank of the error-correction coefficient determines how many cointegrating relations there exist in the model.

Because `estimate`

is
suitable for estimating VAR models in reduced form, you can convert
an estimated VAR model to its VEC model equivalent using `var2vec`

.

```
[VEC,C]
= var2vec(VAR)
```

`[`

returns the coefficient
matrices (`VEC`

,`C`

]
= var2vec(`VAR`

)`VEC`

) and the error-correction coefficient
matrix (`C`

) of the vector error-correction model
equivalent to the vector autoregressive model with coefficient matrices
(`VAR`

). If the number of lags in the input vector
autoregressive model is *p*, then the number of lags
in the output vector error-correction model is *q* = *p* –
1.

To accommodate structural VAR models, specify the input argument

`VAR`

as a`LagOp`

lag operator polynomial.To access the cell vector of the lag operator polynomial coefficients of the output argument

`VEC`

, enter`toCellArray(VEC)`

.To convert the model coefficients of the output argument from lag operator notation to the model coefficients in difference-equation notation, enter

VECDEN = toCellArray(reflect(VEC));

`VECDEN`

is a cell vector containing*p*coefficients corresponding to the differenced response terms in`VEC.Lags`

in difference-equation notation. The first element is the coefficient of*Δy*, the second element is the coefficient of_{t}*Δy*_{t–1}, and so on.Consider converting a VAR(

*p*) model to a VEC(*q*) model. If the error-correction coefficient matrix (`C`

) has:Rank zero, then the converted VEC model is a stable VAR(

*p*– 1) model in terms of*Δy*._{t}Full rank, then the VAR(

*p*) model is stable (i.e., has no unit roots) [2].Rank

*r*, such that 0 <*r*<*n*, then the stable VEC model has*r*cointegrating relations.

The constant offset of the converted VEC model is the same as the constant offset of the VAR model.

`var2vec`

does not impose stability requirements on the coefficients. To check for stability, use`isStable`

.`isStable`

requires a`LagOp`

lag operator polynomial as an input argument. For example, to check whether`VAR`

, the cell array of-by-`n`

numeric matrices, composes a stable time series, enter`n`

varLagOp = LagOp([eye(

*n*) VAR]); isStable(varLagOp)A

`0`

indicates that the polynomial is not stable.

[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.

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