<p>I have seen two different versions of the limit distribution of trace test. </p>

<p>If <span class="math-container">$B_u$</span> is a standard Brownian motion, then I have seen</p>

<p><span class="math-container">$tr\left\{\int_0^1(dB)B'\left[\int_0^1BB'du\right]\int_0^1B(dB')\right\} $</span> from 'Likelihood-Based Inference in Cointegrated Vector Autoregressive Models' by Søren Johansen </p>

<p>and <span class="math-container">$tr\left\{\left(\int_0^1B'(dB)\right)'\left[\int_0^1BB'du\right]\int_0^1B(dB')\right\} $</span> from 'New Introduction to Multiple Time Series Analysis' by Helmut Lütkepohl.</p>

<p>I don't understand why they are equal. I guess that <span class="math-container">$\int_0^1(dB)B'=\left(\int_0^1B'(dB)\right)'$</span>, but I don't see why. </p>

<p>Any help is really appreciated. </p>

CC BY-SA 4.0
2020-06-14T17:32:30.097Z
I’m voting to close this question because this belong to the mathematics SE as the question is about mathematical operations not about economics or econometrics.
1muflon1
CC BY-SA 4.0
2020-06-15T04:55:08.823Z
It's just notation, used in econometrics (not mathematics). The expression $∫_0^1(dB)B′$ has no mathematical meaning. $∫_0^1 B′dB$ is well-defined. It's a matrix whose ij-th entry is the Ito integral $∫_0^1 B_i' dB_j$, where $B_i$ is the i-th component of $B$. You should take the LHS as simply notation for the RHS.
Michael
<econometrics>
Two different notations of the limit distribution of the trace test
Chrisoe
# Two different notations of the limit distribution of the trace test

(Asked by: username_0 on 2020-06-14)

I have seen two different versions of the limit distribution of trace test.

If $B\_u$ is a standard Brownian motion, then I have seen

$tr\left\{\int\_0^1(dB)B'\left[\int\_0^1BB'du\right]\int\_0^1B(dB')\right\} $ from 'Likelihood-Based Inference in Cointegrated Vector Autoregressive Models' by Søren Johansen

and $tr\left\{\left(\int\_0^1B'(dB)\right)'\left[\int\_0^1BB'du\right]\int\_0^1B(dB')\right\} $ from 'New Introduction to Multiple Time Series Analysis' by Helmut Lütkepohl.

I don't understand why they are equal. I guess that $\int\_0^1(dB)B'=\left(\int\_0^1B'(dB)\right)'$, but I don't see why.

Any help is really appreciated.

**Tags:** econometrics

> **Comments:**
> **username_1**: I’m voting to close this question because this belong to the mathematics SE as the question is about mathematical operations not about economics or econometrics.  
> **username_2**: It's just notation, used in econometrics (not mathematics). The expression $∫_0^1(dB)B′$ has no mathematical meaning. $∫_0^1 B′dB$ is well-defined. It's a matrix whose ij-th entry is the Ito integral $∫_0^1 B_i' dB_j$, where $B_i$ is the i-th component of $B$. You should take the LHS as simply notation for the RHS.  

---

## 0 answers

