OLS regression of VECM weighting matrix

This functions estimates the $\bold{\alpha}$ matrix of a VECM. The following OLS regression of the R-form of the VECM is hereby utilised: $$\bold{R}_{0t} = \bold{\alpha}\bold{\beta}\prime \bold{R}_{kt} + \bold{\varepsilon}_t \qquad t=1, \dots, T$$

Usage

alphaols(z, reg.number = NULL)

Arguments

z: An object of class ca.jo.
reg.number: The number of the equation in the R-form that should be estimated or if set to NULL (the default), all equations within the R-form are estimated.

Details

The cointegrating relations, i.e. $\bold{R}_{kt}\prime \bold{\beta}$ are calculated by using z@RK and z@V.

Value

Returns an object of class lm.

References

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.

Author

Bernhard Pfaff

Examples

data(denmark)
sjd <- denmark[, c("LRM", "LRY", "IBO", "IDE")]
sjd.vecm1 <- ca.jo(sjd, ecdet = "const", type="eigen", K=2, spec="longrun",
season=4)
summary(alphaols(sjd.vecm1))
#> Response R0.LRM.d :
#> 
#> Call:
#> lm(formula = R0.LRM.d ~ V.RK.LRM.l2 + V.RK.LRY.l2 + V.RK.IBO.l2 + 
#>     V.RK.IDE.l2 + V.RK.constant - 1, data = data.mat)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -0.039482 -0.014437 -0.005498  0.013169  0.051973 
#> 
#> Coefficients:
#>                 Estimate Std. Error t value Pr(>|t|)    
#> V.RK.LRM.l2   -2.130e-01  6.039e-02  -3.526 0.000938 ***
#> V.RK.LRY.l2   -4.815e-03  4.028e-02  -0.120 0.905341    
#> V.RK.IBO.l2    3.501e-02  2.184e-02   1.603 0.115453    
#> V.RK.IDE.l2    2.029e-03  2.815e-03   0.721 0.474633    
#> V.RK.constant -1.172e-12  3.123e-02   0.000 1.000000    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.02001 on 48 degrees of freedom
#> Multiple R-squared:  0.2446,	Adjusted R-squared:  0.1659 
#> F-statistic: 3.108 on 5 and 48 DF,  p-value: 0.01647
#> 
#> 
#> Response R0.LRY.d :
#> 
#> Call:
#> lm(formula = R0.LRY.d ~ V.RK.LRM.l2 + V.RK.LRY.l2 + V.RK.IBO.l2 + 
#>     V.RK.IDE.l2 + V.RK.constant - 1, data = data.mat)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -0.03448 -0.01507 -0.00100  0.01115  0.05666 
#> 
#> Coefficients:
#>                 Estimate Std. Error t value Pr(>|t|)  
#> V.RK.LRM.l2    1.150e-01  6.189e-02   1.859   0.0692 .
#> V.RK.LRY.l2    1.975e-02  4.128e-02   0.478   0.6345  
#> V.RK.IBO.l2    4.994e-02  2.238e-02   2.231   0.0304 *
#> V.RK.IDE.l2    1.109e-03  2.885e-03   0.384   0.7025  
#> V.RK.constant -3.486e-13  3.200e-02   0.000   1.0000  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.0205 on 48 degrees of freedom
#> Multiple R-squared:  0.1551,	Adjusted R-squared:  0.06707 
#> F-statistic: 1.762 on 5 and 48 DF,  p-value: 0.1387
#> 
#> 
#> Response R0.IBO.d :
#> 
#> Call:
#> lm(formula = R0.IBO.d ~ V.RK.LRM.l2 + V.RK.LRY.l2 + V.RK.IBO.l2 + 
#>     V.RK.IDE.l2 + V.RK.constant - 1, data = data.mat)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -0.0232722 -0.0044453 -0.0000303  0.0048527  0.0176373 
#> 
#> Coefficients:
#>                 Estimate Std. Error t value Pr(>|t|)
#> V.RK.LRM.l2    2.318e-02  2.403e-02   0.965    0.340
#> V.RK.LRY.l2   -1.060e-02  1.602e-02  -0.661    0.512
#> V.RK.IBO.l2    3.480e-03  8.688e-03   0.401    0.691
#> V.RK.IDE.l2   -1.574e-03  1.120e-03  -1.405    0.166
#> V.RK.constant  2.287e-14  1.242e-02   0.000    1.000
#> 
#> Residual standard error: 0.00796 on 48 degrees of freedom
#> Multiple R-squared:  0.068,	Adjusted R-squared:  -0.02908 
#> F-statistic: 0.7004 on 5 and 48 DF,  p-value: 0.6258
#> 
#> 
#> Response R0.IDE.d :
#> 
#> Call:
#> lm(formula = R0.IDE.d ~ V.RK.LRM.l2 + V.RK.LRY.l2 + V.RK.IBO.l2 + 
#>     V.RK.IDE.l2 + V.RK.constant - 1, data = data.mat)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -0.0091249 -0.0028761 -0.0000153  0.0024579  0.0148999 
#> 
#> Coefficients:
#>                 Estimate Std. Error t value Pr(>|t|)   
#> V.RK.LRM.l2    2.941e-02  1.524e-02   1.930  0.05949 . 
#> V.RK.LRY.l2   -3.023e-02  1.016e-02  -2.975  0.00458 **
#> V.RK.IBO.l2   -2.812e-03  5.510e-03  -0.510  0.61222   
#> V.RK.IDE.l2   -4.768e-05  7.104e-04  -0.067  0.94677   
#> V.RK.constant  1.248e-13  7.879e-03   0.000  1.00000   
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.005048 on 48 degrees of freedom
#> Multiple R-squared:  0.211,	Adjusted R-squared:  0.1288 
#> F-statistic: 2.568 on 5 and 48 DF,  p-value: 0.03881
#> 
#> 
summary(alphaols(sjd.vecm1, reg.number=1))
#> 
#> Call:
#> lm(formula = substitute(form1), data = data.mat)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -0.039482 -0.014437 -0.005498  0.013169  0.051973 
#> 
#> Coefficients:
#>                 Estimate Std. Error t value Pr(>|t|)    
#> V.RK.LRM.l2   -2.130e-01  6.039e-02  -3.526 0.000938 ***
#> V.RK.LRY.l2   -4.815e-03  4.028e-02  -0.120 0.905341    
#> V.RK.IBO.l2    3.501e-02  2.184e-02   1.603 0.115453    
#> V.RK.IDE.l2    2.029e-03  2.815e-03   0.721 0.474633    
#> V.RK.constant -1.172e-12  3.123e-02   0.000 1.000000    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.02001 on 48 degrees of freedom
#> Multiple R-squared:  0.2446,	Adjusted R-squared:  0.1659 
#> F-statistic: 3.108 on 5 and 48 DF,  p-value: 0.01647
#>