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~
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$\begingroup$ 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) $var(\epsilon_{i} \mid x_{i}) = \sigma^2 x_{i}^2$ $\endgroup$– Eric ChenFeb 11, 2017 at 21:10
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$\begingroup$ Why do I even need those assumptions in the first place? $x_{i}$ seems irrelevant with $\mu$ $\endgroup$– Eric ChenFeb 11, 2017 at 21:13
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$\begingroup$ 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. $\endgroup$– GiskardFeb 11, 2017 at 22:52
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1$\begingroup$ @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. $\endgroup$– Eric ChenFeb 12, 2017 at 4:34
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2$\begingroup$ Please edit your question instead of add more comments. Then, we can better judge if the question is to be closed or not. $\endgroup$– luchonachoFeb 12, 2017 at 10:41
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2 Answers
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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 $y_i$.
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$\begingroup$ 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 $x_{i}$? $\endgroup$ Feb 11, 2017 at 21:06
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$\begingroup$ 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. $\endgroup$– GiskardFeb 11, 2017 at 22:50
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$\begingroup$ Indeed this is a very clear explanation. Thx! $\endgroup$ Feb 11, 2017 at 23:25
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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 $P_{\iota}y=\bar{y}$.
