Conducts the Johansen procedure on a given data set. The
`"trace"`

or `"eigen"`

statistics are reported and the
matrix of eigenvectors as well as the loading matrix.

`cointegration_johansen(x, ...)`

- x
Data matrix to be investigated for cointegration.

- ...
Additional arguments passed to

`urca::ca.jo()`

.

An object of class `ca.jo`

.

Given a general VAR of the form:

$$\bold{X}_t = \bold{\Pi}_1 \bold{X}_{t-1} + \dots + \bold{\Pi}_k \bold{X}_{t-k} + \bold{\mu} + \bold{\Phi D}_t + \bold{\varepsilon}_t , \quad (t = 1, \dots, T),$$

the following two specifications of a VECM exist:

$$\Delta \bold{X}_t = \bold{\Gamma}_1 \Delta \bold{X}_{t-1} + \dots + \bold{\Gamma}_{k-1} \Delta \bold{X}_{t-k+1} + \bold{\Pi X}_{t-k} + \bold{\mu} + \bold{\Phi D}_t + \bold{\varepsilon}_t$$

where

$$\bold{\Gamma}_i = - (\bold{I} - \bold{\Pi}_1 - \dots - \bold{\Pi}_i), \quad (i = 1, \dots , k-1),$$

and

$$\bold{\Pi} = -(\bold{I} - \bold{\Pi}_1 - \dots - \bold{\Pi}_k)$$

The \(\bold{\Gamma}_i\) matrices contain the cumulative long-run
impacts, hence if `spec="longrun"`

is choosen, the above VECM is
estimated.

The other VECM specification is of the form:

$$\Delta \bold{X}_t = \bold{\Gamma}_1 \Delta \bold{X}_{t-1} + \dots + \bold{\Gamma}_{k-1} \Delta \bold{X}_{t-k+1} + \bold{\Pi X}_{t-1} + \bold{\mu} + \bold{\Phi D}_t + \bold{\varepsilon}_t$$

where

$$\bold{\Gamma}_i = - (\bold{\Pi}_{i+1} + \dots + \bold{\Pi}_k), \quad(i = 1, \dots , k-1),$$

and

$$\bold{\Pi} = -(\bold{I} - \bold{\Pi}_1 - \dots - \bold{\Pi}_k).$$

The \(\bold{\Pi}\) matrix is the same as in the first specification.
However, the \(\bold{\Gamma}_i\) matrices now differ, in the sense
that they measure transitory effects, hence by setting
`spec="transitory"`

the second VECM form is estimated. Please note
that inferences drawn on \(\bold{\Pi}\) will be the same, regardless
which specification is choosen and that the explanatory power is the
same, too.

If `"season"`

is not NULL, centered seasonal dummy variables are
included.

If `"dumvar"`

is not NULL, a matrix of dummy variables is included
in the VECM. Please note, that the number of rows of the matrix
containing the dummy variables must be equal to the row number of
`x`

.

Critical values are only reported for systems with less than 11 variables and are taken from Osterwald-Lenum.

Johansen, S. (1988), Statistical Analysis of Cointegration Vectors,
*Journal of Economic Dynamics and Control*, **12**, 231–254.

Johansen, S. and Juselius, K. (1990), Maximum Likelihood Estimation and
Inference on Cointegration – with Applications to the Demand for
Money, *Oxford Bulletin of Economics and Statistics*, **52,
2**, 169–210.

Johansen, S. (1991), Estimation and Hypothesis Testing of
Cointegration Vectors in Gaussian Vector Autoregressive Models,
*Econometrica*, **Vol. 59, No. 6**, 1551–1580.

Osterwald-Lenum, M. (1992), A Note with Quantiles of the Asymptotic
Distribution of the Maximum Likelihood Cointegration Rank Test
Statistics, *Oxford Bulletin of Economics and Statistics*,
**55, 3**, 461–472.

```
cointegration_johansen(cbind(mdeaths, fdeaths))
#> $johansen_stat
#> R<=1 R=0
#> 13.57827 20.34401
#>
#> $johansen_pvalue
#> R<=1 R=0
#> 0.01 0.01
#>
```