I want to figure out how to read the cointegration relation between 5 cointegrated variables. I present the results down here with r=number of cointegrated equations is 3. Variables are normalized respect to A.
A B C D A 1 1 1 1 B 5 -3 -5 -16 C 4 20 8 -14 D -2 6 9 -15 Please can you help by confirming me that this is the right way to write the 3 cointegrated equations:
$A=-5B-4C+2D$; $B=A+3*B-20*C-6*D$; $C=A+5*B-8*C-9*D$ with lag length=1 But in this case we have $B$ in both sides and it doesn't help. Please me help me to find how to write the equations.
Best Answer
I will use the data(Canada) from vars package in R for illustration.
library(urca) library(vars) data(Canada) vecm<-ca.jo(Canada[,c("rw","prod","e","U")],type="trace",ecdet="trend",K=3,spec="transitory") vecm.r1<-cajorls(vecm, r = 3) > vecm.r1 $rlm Call: lm(formula = substitute(form1), data = data.mat) Coefficients: rw.d prod.d e.d U.d ect1 -5.994e-02 -1.020e-01 -6.503e-02 4.080e-02 ect2 -2.090e-01 -1.051e-01 9.748e-02 -4.554e-02 ect3 -1.388e-01 1.690e-01 1.715e-01 -1.393e-01 constant 2.445e+02 -7.725e+01 -1.761e+02 1.354e+02 rw.dl1 -7.313e-02 7.385e-02 -5.179e-03 -4.171e-02 prod.dl1 5.465e-02 2.048e-01 8.275e-02 -6.216e-02 e.dl1 -3.970e-01 -3.514e-01 5.886e-01 -4.852e-01 U.dl1 3.790e-01 -1.130e+00 -3.101e-01 2.840e-02 rw.dl2 -2.140e-01 -1.304e-01 -2.924e-02 2.745e-02 prod.dl2 -1.437e-01 2.308e-02 -2.634e-02 2.960e-02 e.dl2 2.877e-01 -4.913e-01 -6.056e-01 4.686e-02 U.dl2 -2.829e-02 -3.648e-01 -3.482e-01 -6.446e-02 $beta ect1 ect2 ect3 rw.l1 1.000000e+00 0.000000e+00 0.000000e+00 prod.l1 -8.586881e-17 1.000000e+00 5.757114e-17 e.l1 3.848918e-18 -1.298874e-16 1.000000e+00 U.l1 1.962269e+00 -3.513510e-01 3.405232e+00 trend.l1 -6.061442e-01 -1.935128e-01 -1.843491e-01 alpha<-coef(vecm.r1$rlm)[1,] > alpha rw.d prod.d e.d U.d -0.05993772 -0.10200406 -0.06502751 0.04080300 beta<-vecm.r1$beta beta ect1 ect2 ect3 rw.l1 1.000000e+00 0.000000e+00 0.000000e+00 prod.l1 -8.586881e-17 1.000000e+00 5.757114e-17 e.l1 3.848918e-18 -1.298874e-16 1.000000e+00 U.l1 1.962269e+00 -3.513510e-01 3.405232e+00 trend.l1 -6.061442e-01 -1.935128e-01 -1.843491e-01 The output under coefficients give you vecm; there are 4 variables and so 4 equations. Since you find three cointegrating equations, there are three one period lagged error correction terms indicated by ect1,ect2,ect3.
The long run equilibrium equation is given by output under beta. They are lagged here, but for interpretation as long run equation you have to forward those equations by one period. Cointegrating eqn 1 is obtained by normalising on rw and hence 1.000, equation 2 on prod, and eqn 3 on e. You need to go back and read some literature on your area to decide on which variables to normalise and whether there is long run relationship between these variables.
Please go through the text book of Walter Enders for more clarifications.
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