<p>They are both OLS estimators but generally <span class="math-container">$\beta_1 \neq b_1$</span>.</p>
<p>In statistics broadly speaking estimator is just some rule, method, or criterion for arriving at an estimate of the value of a parameter. According to Stachurski: A Primer in Econometric Theory, pp 217:</p>
<blockquote>
<p>An <strong>estimator</strong> is a statistic used to infer some feature <span class="math-container">$\gamma(P)$</span> of an unknown distribution <span class="math-container">$P$</span>. Thus a statistic becomes an estimator when paired with and compared to feature of the distribution. Nothe that there is nothing in the definition of an estimator that implies it will be a sensible estimator of the target feature, let alone a good one.</p>
</blockquote>
<p>Consequently, by definition if you use the following formulas:</p>
<p><span class="math-container">$$b_1= (X_1’M_1X_1)^{-1} X_1’M_2y$$</span></p>
<p><span class="math-container">$$\beta_1= (X_1’M_2y)^{-1} (X_1’M_1X_1)$$</span></p>
<p>for an inference they become estimators. Moreover, both of these look like OLS estimator. So in that sense they are the same <em>type</em> of estimator.</p>
<p>However, running the same estimator will not give you the same result when you change position of dependent and independent variable (in fact OLS estimator will be biased if there is reverse causality so it does not make any sense to do it in the first place). Consequently generally speaking <span class="math-container">$\beta_1 \neq b_2$</span>.</p>
<p>In fact in your specific case:</p>
<p><span class="math-container">$$y= X_1b_1+X_2b_2 + u \implies X_1=y\frac{1}{b_1}-X_2\frac{b_2}{b_1}-\frac{u}{b_1}$$</span></p>
<p>Consequently, we have:</p>
<p><span class="math-container">$$\beta_1 = \frac{1}{b_1}; \beta_2 = -\frac{b_2}{b_1}; v = -\frac{u}{b_1}$$</span></p>
<p>Trivially since <span class="math-container">$\beta_1 = \frac{1}{b_1}$</span>,   <span class="math-container">$\beta_1 \neq b_1$</span>, save for some special cases such as when <span class="math-container">$b_1=1$</span> for example.</p>

CC BY-SA 4.0
2020-12-07T18:56:02.977Z
2020-12-07T20:46:02.047Z
2020-12-07T20:46:02.047Z
1muflon1
<p>I have a question so basically.</p>
<p>I have two model <span class="math-container">$$y= X_1b_1+X_2b_2+u$$</span>
And <span class="math-container">$$X_2=y\beta_1+X_2\beta_2+v$$</span></p>
<p>I want to show <span class="math-container">$b_1=\beta_1$</span></p>
<p>Are these estimators are equivalent?</p>
<p><span class="math-container">$$b_1= (X_1’M_1X_1)^{-1} X_1’M_2y$$</span></p>
<p>And</p>
<p><span class="math-container">$$\beta_1= (X_1’M_2y)^{-1} (X_1’M_1X_1)$$</span></p>

CC BY-SA 4.0
2020-12-07T16:59:44.743Z
Seems like $\beta b = I$
Dayne
CC BY-SA 4.0
2020-12-07T17:06:49.263Z
Usually the dimensionality of $X_1^\top M_2 y$ would be $K \times 1$ and $X_1^\top M_1 X_1$ would be $K \times K$ so the multiplication $(X_1^\top M_2 y)^{-1}(X_1^\top M_1 X_1)$ does not make sense also the inverse $(X_1^\top M_2 y)^{-1}$ does not make sense. So you should probably try to define the "stuff" you are writing. I suggest to close this question, as it stands it is non-sensical.
Jesper Hybel
CC BY-SA 4.0
2020-12-07T17:50:16.687Z
@JesperHybel I add more info please look at
studentp
CC BY-SA 4.0
2020-12-07T17:50:32.587Z
@Dayne I add more info please look at
studentp
CC BY-SA 4.0
2020-12-07T18:58:32.150Z
I assume $X_2$ and that it should be $X_1$ in the second equation is a typo otherwise you would be regressing variable on itself
1muflon1
CC BY-SA 4.0
2020-12-07T20:41:06.807Z
As an off chance, you might be thinking of: https://en.wikipedia.org/wiki/Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem
Walrasian Auctioneer
CC BY-SA 4.0
2020-12-08T19:22:40.143Z
Usually $M_1X_1=0$ and so does $X_1'M_1X_1=0$. I am also not convinced that your question makes sense.
Bertrand
<econometrics><self-study>
These two estimators are equal?
studentp
# These two estimators are equal?

(Asked by: username_0 on 2020-12-07)

I have a question so basically.

I have two model $$y= X\_1b\_1+X\_2b\_2+u$$
And $$X\_2=y\beta\_1+X\_2\beta\_2+v$$

I want to show $b\_1=\beta\_1$

Are these estimators are equivalent?

$$b\_1= (X\_1’M\_1X\_1)^{-1} X\_1’M\_2y$$

And

$$\beta\_1= (X\_1’M\_2y)^{-1} (X\_1’M\_1X\_1)$$

**Tags:** econometrics, self-study
> **Comments:**
> **username_1**: Seems like $\beta b = I$  
> **username_2**: Usually the dimensionality of $X_1^\top M_2 y$ would be $K \times 1$ and $X_1^\top M_1 X_1$ would be $K \times K$ so the multiplication $(X_1^\top M_2 y)^{-1}(X_1^\top M_1 X_1)$ does not make sense also the inverse $(X_1^\top M_2 y)^{-1}$ does not make sense. So you should probably try to define the "stuff" you are writing. I suggest to close this question, as it stands it is non-sensical.  
> **username_0**: @JesperHybel I add more info please look at  
> **username_0**: @Dayne I add more info please look at  
> **username_3**: I assume $X_2$ and that it should be $X_1$ in the second equation is a typo otherwise you would be regressing variable on itself  
> **username_4**: As an off chance, you might be thinking of: https://en.wikipedia.org/wiki/Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem  
> **username_5**: Usually $M_1X_1=0$ and so does $X_1'M_1X_1=0$. I am also not convinced that your question makes sense.  

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## 1 answers

### Answer by username_3 on 2020-12-07. Score: 1

They are both OLS estimators but generally $\beta\_1 \neq b\_1$.

In statistics broadly speaking estimator is just some rule, method, or criterion for arriving at an estimate of the value of a parameter. According to Stachurski: A Primer in Econometric Theory, pp 217:

> An **estimator** is a statistic used to infer some feature $\gamma(P)$ of an unknown distribution $P$. Thus a statistic becomes an estimator when paired with and compared to feature of the distribution. Nothe that there is nothing in the definition of an estimator that implies it will be a sensible estimator of the target feature, let alone a good one.

Consequently, by definition if you use the following formulas:

$$b\_1= (X\_1’M\_1X\_1)^{-1} X\_1’M\_2y$$

$$\beta\_1= (X\_1’M\_2y)^{-1} (X\_1’M\_1X\_1)$$

for an inference they become estimators. Moreover, both of these look like OLS estimator. So in that sense they are the same *type* of estimator.

However, running the same estimator will not give you the same result when you change position of dependent and independent variable (in fact OLS estimator will be biased if there is reverse causality so it does not make any sense to do it in the first place). Consequently generally speaking $\beta\_1 \neq b\_2$.

In fact in your specific case:

$$y= X\_1b\_1+X\_2b\_2 + u \implies X\_1=y\frac{1}{b\_1}-X\_2\frac{b\_2}{b\_1}-\frac{u}{b\_1}$$

Consequently, we have:

$$\beta\_1 = \frac{1}{b\_1}; \beta\_2 = -\frac{b\_2}{b\_1}; v = -\frac{u}{b\_1}$$

Trivially since $\beta\_1 = \frac{1}{b\_1}$, $\beta\_1 \neq b\_1$, save for some special cases such as when $b\_1=1$ for example.


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