Likelihood ratio test for restrictions on alpha

This function estimates a restricted VAR, where the restrictions are base upon \(\bold{\alpha}\), i.e. the loading vectors. The test statistic is distributed as \(\chi^2\) with \(r(p-m)\) degrees of freedom, with \(m\) equal to the columns of the restricting matrix \(\bold{A}\).

Usage

alrtest(z, A, r)

Arguments

z: An object of class ca.jo.
A: The \((p \times m)\) matrix containing the restrictions on \(\bold{\alpha}\).
r: The count of cointegration relationships;
inferred from summary(ca.jo-object).

Details

The orthogonal matrix to \(\bold{A}\) can be accessed as object@B. The restricted \(\bold{\alpha}\) matrix is normalised with respect to the first variable.

Value

An object of class cajo.test.

References

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.

Author

Bernhard Pfaff

Examples

data(denmark)
sjd <- denmark[, c("LRM", "LRY", "IBO", "IDE")]
sjd.vecm <- ca.jo(sjd, ecdet = "const", type="eigen", K=2, spec="longrun",
season=4)
DA <- matrix(c(1,0,0,0), c(4,1))
summary(alrtest(sjd.vecm, A=DA, r=1))
#> 
#> ###################### 
#> # Johansen-Procedure # 
#> ###################### 
#> 
#> Estimation and testing under linear restrictions on beta 
#> 
#> The VECM has been estimated subject to: 
#> beta=H*phi and/or alpha=A*psi
#> 
#>      [,1]
#> [1,]    1
#> [2,]    0
#> [3,]    0
#> [4,]    0
#> 
#> Eigenvalues of restricted VAR (lambda):
#> [1] 0.3573 0.0000 0.0000 0.0000 0.0000
#> 
#> The value of the likelihood ratio test statistic:
#> 6.66 distributed as chi square with 3 df.
#> The p-value of the test statistic is: 0.08 
#> 
#> Eigenvectors, normalised to first column
#> of the restricted VAR:
#> 
#>                [,1]
#> RK.LRM.l2    1.0000
#> RK.LRY.l2   -0.9585
#> RK.IBO.l2    4.7641
#> RK.IDE.l2   -2.5708
#> RK.constant -6.5825
#> 
#> Weights W of the restricted VAR:
#> 
#>         [,1]
#> [1,] -0.2543
#> [2,]  0.0000
#> [3,]  0.0000
#> [4,]  0.0000
#>