VECM interpretation in Matlab?

Question

After testing for cointegration with Johansen test using the following code

[h,pValue,stat,cValue,mles] = jcitest(ret,'model','H1*','lags',4,'display','params')

I received this output:

Data: prices
Effective sample size: 56
Model: H1*
Lags: 4
Statistic: trace
Significance level: 0.05


r  h  stat      cValue   pValue   eigVal   
----------------------------------------
0  1  40.5214   35.1929  0.0161   0.3350  
1  0  14.6781   20.2619  0.1453   0.1667  
2  0  5.4670    9.1644   0.1580   0.1091 

So it rejects no cointegration. So I am running the VECM with rank 1 restriction:

B = mles.r1.paramVals.B % Cointegrating relations with rank = 1 restriction

I get the following output:

r = 1
------
A =
-0.6259
-0.2261
-0.0222

B =
0.7081
1.6282
-2.4581

B1 =
0.0579 1.0824 -0.8718
0.1182 0.4993 -0.5415
0.1050 0.1667 -0.1600

B2 =
-0.5462 2.2436 -1.7723
-0.1518 0.6605 -0.6169
-0.1610 0.5966 -0.5366

c0 =
2.2351

c1 =-0.0366
0.0872
0.1444

How would I interpret this output and how do I move forward after estimating the parameters? (I have only displayed the r=1, and these are not the actual results, by I couldn't find any explanation behind them)

Answer 1

Based on the official documentation, you have found that there is one cointegration relationship with 95% probability. Given your selection of Model: H1* and Lags: 4, the model you are estimating is:

$$ \Delta y_t = A(B´y_{t−1}+c_0) + B_1\Delta y_{t-1} + B_2\Delta y_{t-2} + B_3\Delta y_{t-3} + B_4\Delta y_{t-4} + \epsilon_t $$

The matrix B is of dimension $n \times r$, and it contains as columns the cointegration vector(s). In your case, $r=1$, and the cointegration vector is

B =
0.7081
1.6282
-2.4581 

The Matrix A represent the "error-correction" matrix. For a system to converge towards the equilibrium cointegration relation (i.e. to be stable), you want these numbers to be in between -2 and 0 (as in your case).

c0 is just a constant of the cointegration relationship (like in a standard function $y=c0+bx$).

The matrices Bi show how previous period shocks translate through the system over time.