OLS regression of VECM
cajorls.RdThis function returns the OLS regressions of a restricted VECM,
i.e. it returns a list object with elements of class `lm'
containing the restricted VECM and a matrix object with the normalised
cointegrating relationships. The user can provide a certain number of
which equation in the VECM should be estimated and reported, or if
"reg.number = NULL" each equation in the VECM will be estimated
and its results are reported. Furthermore, the cointegratioon rank has
to be supplied too.
Details
The cointegration space is normalised as \(\bold{\beta}_c = \bold{\beta}(S'\bold{\beta})^{-1}\), with \(S' = (I_r, 0)\).
Value
Returns a list object with elements of class lm for the
restricted VECM and a matrix object with the normalised cointegrating
vectors.
References
Johansen, S. (1995), Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press, Oxford.
Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.
See also
ca.jo, cajools, lm,
ca.jo-class and urca-class.
Examples
data(finland)
sjf <- finland
sjf.vecm <- ca.jo(sjf, ecdet = "none", type = "eigen", K = 2,
spec = "longrun", season = 4)
sjf.vecm.rls <- cajorls(sjf.vecm, r = 2)
summary(sjf.vecm.rls$rlm)
#> Response lrm1.d :
#>
#> Call:
#> lm(formula = lrm1.d ~ ect1 + ect2 + constant + sd1 + sd2 + sd3 +
#> lrm1.dl1 + lny.dl1 + lnmr.dl1 + difp.dl1 - 1, data = data.mat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.167337 -0.031168 -0.000444 0.032038 0.134665
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> ect1 0.013062 0.027131 0.481 0.6313
#> ect2 -0.005718 0.033439 -0.171 0.8646
#> constant 0.034544 0.063420 0.545 0.5873
#> sd1 0.039661 0.021929 1.809 0.0737 .
#> sd2 0.037178 0.015135 2.456 0.0159 *
#> sd3 0.100957 0.016637 6.068 2.7e-08 ***
#> lrm1.dl1 -0.144815 0.110924 -1.306 0.1949
#> lny.dl1 -0.282268 0.141776 -1.991 0.0494 *
#> lnmr.dl1 -0.092989 0.136421 -0.682 0.4971
#> difp.dl1 -0.175051 0.455241 -0.385 0.7015
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.05088 on 94 degrees of freedom
#> Multiple R-squared: 0.5241, Adjusted R-squared: 0.4735
#> F-statistic: 10.35 on 10 and 94 DF, p-value: 1.415e-11
#>
#>
#> Response lny.d :
#>
#> Call:
#> lm(formula = lny.d ~ ect1 + ect2 + constant + sd1 + sd2 + sd3 +
#> lrm1.dl1 + lny.dl1 + lnmr.dl1 + difp.dl1 - 1, data = data.mat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.063049 -0.022580 0.001169 0.020766 0.060946
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> ect1 0.016827 0.016423 1.025 0.30816
#> ect2 -0.014446 0.020241 -0.714 0.47719
#> constant 0.050219 0.038389 1.308 0.19401
#> sd1 0.043686 0.013274 3.291 0.00141 **
#> sd2 0.082752 0.009161 9.033 2.09e-14 ***
#> sd3 0.095593 0.010070 9.492 2.21e-15 ***
#> lrm1.dl1 0.036994 0.067144 0.551 0.58297
#> lny.dl1 -0.669425 0.085819 -7.800 8.35e-12 ***
#> lnmr.dl1 -0.067311 0.082577 -0.815 0.41706
#> difp.dl1 -0.194686 0.275563 -0.707 0.48162
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.0308 on 94 degrees of freedom
#> Multiple R-squared: 0.7709, Adjusted R-squared: 0.7465
#> F-statistic: 31.63 on 10 and 94 DF, p-value: < 2.2e-16
#>
#>
#> Response lnmr.d :
#>
#> Call:
#> lm(formula = lnmr.d ~ ect1 + ect2 + constant + sd1 + sd2 + sd3 +
#> lrm1.dl1 + lny.dl1 + lnmr.dl1 + difp.dl1 - 1, data = data.mat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.13378 -0.02073 -0.00116 0.01910 0.09730
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> ect1 0.100381 0.019373 5.182 1.25e-06 ***
#> ect2 -0.114265 0.023877 -4.786 6.32e-06 ***
#> constant 0.227298 0.045285 5.019 2.45e-06 ***
#> sd1 0.008791 0.015659 0.561 0.5758
#> sd2 0.012457 0.010807 1.153 0.2520
#> sd3 0.020114 0.011879 1.693 0.0937 .
#> lrm1.dl1 -0.156875 0.079205 -1.981 0.0506 .
#> lny.dl1 -0.010677 0.101234 -0.105 0.9162
#> lnmr.dl1 -0.231381 0.097411 -2.375 0.0196 *
#> difp.dl1 0.424736 0.325063 1.307 0.1945
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.03633 on 94 degrees of freedom
#> Multiple R-squared: 0.3987, Adjusted R-squared: 0.3347
#> F-statistic: 6.232 on 10 and 94 DF, p-value: 2.984e-07
#>
#>
#> Response difp.d :
#>
#> Call:
#> lm(formula = difp.d ~ ect1 + ect2 + constant + sd1 + sd2 + sd3 +
#> lrm1.dl1 + lny.dl1 + lnmr.dl1 + difp.dl1 - 1, data = data.mat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.022935 -0.007270 -0.001109 0.006187 0.044326
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> ect1 -0.011799 0.006208 -1.901 0.0604 .
#> ect2 0.017539 0.007651 2.292 0.0241 *
#> constant -0.030559 0.014511 -2.106 0.0379 *
#> sd1 0.001724 0.005018 0.344 0.7319
#> sd2 -0.007526 0.003463 -2.173 0.0323 *
#> sd3 -0.008354 0.003807 -2.195 0.0307 *
#> lrm1.dl1 0.001332 0.025381 0.052 0.9583
#> lny.dl1 0.028501 0.032440 0.879 0.3819
#> lnmr.dl1 0.023617 0.031215 0.757 0.4512
#> difp.dl1 -0.793653 0.104164 -7.619 1.99e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.01164 on 94 degrees of freedom
#> Multiple R-squared: 0.5132, Adjusted R-squared: 0.4615
#> F-statistic: 9.911 on 10 and 94 DF, p-value: 3.783e-11
#>
#>
sjf.vecm.rls$beta
#> ect1 ect2
#> lrm1.l2 1.000000e+00 0.00000
#> lny.l2 1.002732e-16 1.00000
#> lnmr.l2 -2.137582e+01 -14.63113
#> difp.l2 -9.078461e+01 -85.79672