<p>I'm struggling with understand the difference between (and why there is this difference) the normalized cointegration vectors provided by the <code>ca.jo</code> function and the ones provided by <code>cajorls</code> and <code>VECM</code> (<code>tsDyn</code> package) functions.</p>
<p>Those are the results from <code>ca.jo</code> function, with the cointegration vectors normalized for the first variable <code>LSELIC</code> (the test returned r=2, <code>LSELIC.l1</code> and <code>LIBCBR.l1</code> are the two vectors of coefficients from that cointegrations relationships):
<a href="https://i.sstatic.net/5n40r7HO.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5n40r7HO.jpg" alt="enter image description here" /></a></p>
<p>The <code>cajorls</code> function &quot;Returns a list object with elements of class lm for the restricted VECM and a matrix object with the normalized cointegrating vectors&quot;.</p>
<p><a href="https://i.sstatic.net/bmDe1V2U.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bmDe1V2U.jpg" alt="enter image description here" /></a></p>
<p>Ultimately, the <code>VECM</code> provides the same results of <code>cajorls</code> function:
<a href="https://i.sstatic.net/JpnuAhm2.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JpnuAhm2.jpg" alt="enter image description here" /></a></p>
<p>So, what is the &quot;correct&quot; normalization? Or, what I believe, what are the interpretations of these two normalized cointegration vectors for the same variable?</p>

CC BY-SA 4.0
2024-06-11T07:13:39.880Z
Is the difference not that one is restricted and the other is unrestricted? That would seem to be an obvious answer.
Richard Hardy
CC BY-SA 4.0
2024-06-12T03:24:32.160Z
Right. I dind't understand the meaning of "restricted" in cajorls and VECM functions, since ca.jo function already provided a normalized vector for the first variable (LSELIC). Could you explain it?
Daniel Xavier
CC BY-SA 4.0
2024-06-12T06:53:17.207Z
I do not remember what that means (if I ever knew). If you cannot find the answer in the help file of `cajorls`, consider contacting the author of the R package.
Richard Hardy
<macroeconomics><applied-econometrics><r><co-integration>
What are the true normalized cointegration vectors in R (ca.jo, cajorls and VECM functions) and how to interpret them?
Daniel Xavier
# What are the true normalized cointegration vectors in R (ca.jo, cajorls and VECM functions) and how to interpret them?

(Asked by: username_0 on 2024-06-10)

I'm struggling with understand the difference between (and why there is this difference) the normalized cointegration vectors provided by the `ca.jo` function and the ones provided by `cajorls` and `VECM` (`tsDyn` package) functions.

Those are the results from `ca.jo` function, with the cointegration vectors normalized for the first variable `LSELIC` (the test returned r=2, `LSELIC.l1` and `LIBCBR.l1` are the two vectors of coefficients from that cointegrations relationships):
[![enter image description here](https://i.sstatic.net/5n40r7HO.jpg)](https://i.sstatic.net/5n40r7HO.jpg)

The `cajorls` function "Returns a list object with elements of class lm for the restricted VECM and a matrix object with the normalized cointegrating vectors".

[![enter image description here](https://i.sstatic.net/bmDe1V2U.jpg)](https://i.sstatic.net/bmDe1V2U.jpg)

Ultimately, the `VECM` provides the same results of `cajorls` function:
[![enter image description here](https://i.sstatic.net/JpnuAhm2.jpg)](https://i.sstatic.net/JpnuAhm2.jpg)

So, what is the "correct" normalization? Or, what I believe, what are the interpretations of these two normalized cointegration vectors for the same variable?

**Tags:** macroeconomics, applied-econometrics, r, co-integration
> **Comments:**
> **username_1**: Is the difference not that one is restricted and the other is unrestricted? That would seem to be an obvious answer.  
> **username_0**: Right. I dind't understand the meaning of "restricted" in cajorls and VECM functions, since ca.jo function already provided a normalized vector for the first variable (LSELIC). Could you explain it?  
> **username_1**: I do not remember what that means (if I ever knew). If you cannot find the answer in the help file of `cajorls`, consider contacting the author of the R package.  

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## 0 answers

