Interpreting Johansen Cointegration Eigenvectors Versus OLS Coefficients

# Understanding output from Johansen Cointegration test


# Understanding output from Johansen Cointegration test












I have a VECM model that Im using to determine the revenues for a firm, based on factors like Interest rates, S&P 500 and company specific variables, as follows:

Stage 1: $$z_t= a+ bX_t+e_t$$

Stage 2: $$ \Delta z_t = \alpha_1 (z_{t-1}-a-bX_{t-1}) + \gamma_{11} \Delta z_{t-1} + \gamma_{12} \Delta X_{t-1} +\varepsilon_{t} $$

where t is in months and $z_t$ are Firm revenues

After running a Johansen Cointegration Test in R I get the following result.

```
          test 10pct  5pct  1pct
r <= 3 |  1.26  7.52  9.24 12.97
r <= 2 | 9.15 11.41 15.53 23.21
r <= 1 | 20.07 19.37 21.10 23.71
r = 0  | 47.23 22.32 25.23 36.33
```

Based on this there is only one linear combination that results in a stationary series.

The eigenvector associated with the highest eigenvalue, based on the output in R, "Eigenvectors, normalised to first column: (These are the cointegration relations)" is:

```
1.0000
-0.732
0.6174
-5.123
6.1542
```

The value in the first row is just the value associated with Z.

Based on this answer, which says that the linear combination of the eigenvector associated with the highest eigenvalue will result in a stationary series, I was assuming this vector would be the same as the coefficients I estimated via OLS. However, this doesn't seem to be the case and I was just wondering why this might be.

Coefficients from OLS:

```
0.729905
−0.048841
4.224540
−6.281800
```

Comparing the 2nd to 5th rows of the eigenvector from the Johansen test to the coefficients from the OLS it can be seen that the magnitudes are fairly close but the 2nd row is quite far off (the signs would be the same by taking into account the fact that the eigenvector would be on the other side of the equation for the eigenvector combination). Is there a reason for this difference?

Thanks