<p>So basically the question is: <br>
If I know the average ($\hat{\mu}$) of the daily temperatures ($y_i$) of last year, does that tell me anything about how many people were born ($x_i$) each day?</p>

<p>Unsurprisingly the answer is no.</p>

<p>The most you can get is the average of the $x_i$ series if you have the parameters of an unbiased regression between $x_i$ and $y_i$.</p>

CC BY-SA 3.0
2017-02-11T21:06:18.247Z
I have expressed my question in an improper way. What I really want to ask is: Do I really need information about $x_{i}$ to estimate $\mu$? Or, does the formula of my estimator $\hat{\mu}$ include $x_{i}$?
Eric Chen
CC BY-SA 3.0
2017-02-11T22:50:42.153Z
Do you need to know how many people were born ($x_i$) each day to know  the average ($\hat{\mu}$) of the daily temperatures ($y_i$)? No, you do not need that.
Giskard
CC BY-SA 3.0
2017-02-11T23:25:55.847Z
Indeed this is a very clear explanation. Thx!
Eric Chen
CC BY-SA 3.0
2017-02-11T17:20:49.107Z
2017-02-11T17:20:49.107Z
Giskard
<p>I don't think it will have anything to do with $x_i$. Here is my thought:</p>

<p>Given your setup, in order to find $\hat{\mu}$, we regress $y$ on an $n\times1$ vector of ones, $\begin{bmatrix}1\\1\\ \vdots \\1\end{bmatrix}$ ,which we shall call $\iota$ (<em>iota</em>). Then we will have $\hat{\mu}=(\iota'\iota)^{-1}\iota'y=\frac{1}{n}\iota'y=\bar{y}$. So $x$ doesn't play a role here.</p>

<p>Note the projection matrix $P_{\iota} = \iota(\iota'\iota)^{-1}\iota'=\frac{1}{n}\iota\iota'$, and $P_{\iota}y=\bar{y}$.</p>

CC BY-SA 3.0
2017-02-12T21:15:09.560Z
2017-02-13T09:33:20.507Z
2017-02-13T09:33:20.507Z
SunDragon
<p>If I have observations of $y_{i}$ and $x_{i}$ which are i.i.d. I also have OLS assumptions such as $E(\epsilon_{i} \mid X_{i})= 0$, my qustion is:
If I project $y_{i}$ onto a constant $\mu$, that is, we have model
$y_{i} = \mu + \epsilon_{i}$. Does finding the OLS estimator $\hat\mu$ has anything to do with $x_{i}$? Because in my opinions, $x_{i}$ never emerges.
Thanks~</p>

CC BY-SA 3.0
2017-02-11T21:10:12.880Z
So this is actually a problem of my econometrics class. The problem asks me to write down the OLS estimator of $\mu$, while it gives me assumptions as following: (1)$y_{i}$ and $x_{i}$ are i.i.d. (2)$E(\epsilon_{i} \mid X_{i})= 0$ (3) $var(\epsilon_{i} \mid x_{i}) = \sigma^2 x_{i}^2$
Eric Chen
CC BY-SA 3.0
2017-02-11T21:13:57.817Z
Why do I even need those assumptions in the first place? $x_{i}$ seems irrelevant with $\mu$
Eric Chen
CC BY-SA 3.0
2017-02-11T22:52:41.103Z
I am voting to close this question as unclear because there is one question in the body of the question and two more in the comments. Each depict different scenarios. It is impossible to guess which one is the real one as we are not in your class.
Giskard
CC BY-SA 3.0
2017-02-12T04:34:30.407Z
@denesp I was simply trying to give more details and I'm actually asking the same question. Anyway, you have the right to vote for that.
Eric Chen
CC BY-SA 3.0
2017-02-12T10:41:12.873Z
Please edit your question instead of add more comments. Then, we can better judge if the question is to be closed or not.
luchonacho
<econometrics><regression>
Regression on a constant
Eric Chen
# Regression on a constant

(Asked by: username_0 on 2017-02-11)

If I have observations of $y\_{i}$ and $x\_{i}$ which are i.i.d. I also have OLS assumptions such as $E(\epsilon\_{i} \mid X\_{i})= 0$, my qustion is:
If I project $y\_{i}$ onto a constant $\mu$, that is, we have model
$y\_{i} = \mu + \epsilon\_{i}$. Does finding the OLS estimator $\hat\mu$ has anything to do with $x\_{i}$? Because in my opinions, $x\_{i}$ never emerges.
Thanks~

**Tags:** econometrics, regression

> **Comments:**
> **username_0**: So this is actually a problem of my econometrics class. The problem asks me to write down the OLS estimator of $\mu$, while it gives me assumptions as following: (1)$y_{i}$ and $x_{i}$ are i.i.d. (2)$E(\epsilon_{i} \mid X_{i})= 0$ (3) $var(\epsilon_{i} \mid x_{i}) = \sigma^2 x_{i}^2$  
> **username_0**: Why do I even need those assumptions in the first place? $x_{i}$ seems irrelevant with $\mu$  
> **username_1**: I am voting to close this question as unclear because there is one question in the body of the question and two more in the comments. Each depict different scenarios. It is impossible to guess which one is the real one as we are not in your class.  
> **username_0**: @denesp I was simply trying to give more details and I'm actually asking the same question. Anyway, you have the right to vote for that.  
> **username_2**: Please edit your question instead of add more comments. Then, we can better judge if the question is to be closed or not.  

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## 2 answers

### Answer by username_1 on 2017-02-11. Score: 5

So basically the question is:   
If I know the average ($\hat{\mu}$) of the daily temperatures ($y\_i$) of last year, does that tell me anything about how many people were born ($x\_i$) each day?

Unsurprisingly the answer is no.

The most you can get is the average of the $x\_i$ series if you have the parameters of an unbiased regression between $x\_i$ and $y\_i$.

> **Comments:**
> **username_0**: I have expressed my question in an improper way. What I really want to ask is: Do I really need information about $x_{i}$ to estimate $\mu$? Or, does the formula of my estimator $\hat{\mu}$ include $x_{i}$?  
> **username_1**: Do you need to know how many people were born ($x_i$) each day to know  the average ($\hat{\mu}$) of the daily temperatures ($y_i$)? No, you do not need that.  
> **username_0**: Indeed this is a very clear explanation. Thx!  

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### Answer by username_3 on 2017-02-12. Score: 2

I don't think it will have anything to do with $x\_i$. Here is my thought:

Given your setup, in order to find $\hat{\mu}$, we regress $y$ on an $n\times1$ vector of ones, $\begin{bmatrix}1\\1\\ \vdots \\1\end{bmatrix}$ ,which we shall call $\iota$ (*iota*). Then we will have $\hat{\mu}=(\iota'\iota)^{-1}\iota'y=\frac{1}{n}\iota'y=\bar{y}$. So $x$ doesn't play a role here.

Note the projection matrix $P\_{\iota} = \iota(\iota'\iota)^{-1}\iota'=\frac{1}{n}\iota\iota'$, and $P\_{\iota}y=\bar{y}$.


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