`"trace"`

or `"eigen"`

statistics are reported and the
matrix of eigenvectors as well as the loading matrix.```
ca.jo(x, type = c("eigen", "trace"), ecdet = c("none", "const", "trend"), K = 2,
spec=c("longrun", "transitory"), season = NULL, dumvar = NULL)
```

x

Data matrix to be investigated for cointegration.

type

The test to be conducted, either or
.

`eigen`

`trace`

ecdet

Character, for no intercept in cointegration,
for constant term in cointegration and
for trend variable in cointegration.

`none`

`const`

`trend`

K

The lag order of the series (levels) in the VAR.

spec

Determines the specification of the VECM, see details below.

season

If seasonal dummies should be included, the data
frequency must be set accordingly, *i.e* 4 for quarterly
data.

dumvar

If dummy variables should be included, a matrix with
row dimension equal to

`x`

can be provided.- An object of class
`ca.jo`

.

`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.`plotres`

, `alrtest`

, `ablrtest`

,
`blrtest`

, `cajolst`

, `cajools`

,
`lttest`

, `ca.jo-class`

and `urca-class`

.data(denmark) sjd <- denmark[, c("LRM", "LRY", "IBO", "IDE")] sjd.vecm <- ca.jo(sjd, ecdet = "const", type="eigen", K=2, spec="longrun", season=4) summary(sjd.vecm) # data(finland) sjf <- finland sjf.vecm <- ca.jo(sjf, ecdet = "none", type="eigen", K=2, spec="longrun", season=4) summary(sjf.vecm)

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