Re: Stat Question on OLS assumptions

From: sinister (sinister_at_nospam.invalid)
Date: 07/28/04 

Date: Wed, 28 Jul 2004 11:26:57 GMT

"David Wright" <David_wright@spra.com> wrote in message
news:9533C4AAEDavidwrightspracom@64.164.98.29...

This is an OK place to post such a question, but you might have better luck
with sci.stat.*. I'm cross-posting to one of the groups.

> I am having a debate.
>
> In the standard model:
>
> y = a + bx + u
>
> y and u are defined as a random variables and x is a fixed variable.
>
>
> The first assumption given for OLS is often given:
>
> 1. E(u(i))=0
>
> Where u is defined as a random variable and i is given for 1 to N
> observations.
>
> Now, does this mean that the Expected value of each error is zero? Or that
> the expected value of all the errors is zero? That is, shouldnt rather be
> written:
>
> 2. E(u) = 0
>
> Since E(ui) makes no sense, if i is the realization of variable u, then
the
> expected value of ui is undefined, since u is already been realized?
>
> I have consulted many texts, and they all write E(ui)=0 and not E(u)=0.
>
> If the texts are correct, then I am led to conclude that each observation
has
> its own radom variable u(i), so u is really a vector of random variables
and
> not a scalar random variable.

In a practical sense, there isn't really much difference. There might be a
philosophical difference, in that if the entire "experiment" is done once,
each random variable should be realized once. If it were u, not ui, it
would be realized once per observation, not per the entire experiment.
There's probably an official definition of "random variable" that says your
"u" is incorrect for this reason.

In a formalistic, heuristic sense, ui is much better, because it allows you
to be explicit about the assumptions and to alter them. If you have ui, you
can explicitly state "where the ui are independent and identically
distributed". In the real world, this is often not the case; the ui might
be correlated, might have difference variances, etc.

Relevant Pages

  • Re: A realization of a random variable
    ... Kaba wrote: ... Let X be a continuous random variable in R (reals). ... A _realization_ of X is an element of such a sequence Q for which the pdf of X and the limit of the empirical pdf coincide. ... In discussion of stochastic processes (= families of random variables), a realization is the same as a sample path. ...
    (sci.math)
  • Re: Stat Question on OLS assumptions
    ... > y and u are defined as a random variables and x is a fixed variable. ... does this mean that the Expected value of each error is zero? ... > Since Emakes no sense, if i is the realization of variable u, then ... can explicitly state "where the ui are independent and identically ...
    (sci.econ)
  • Re: Stat Question on OLS assumptions
    ... >> y and u are defined as a random variables and x is a fixed variable. ... >> Since Emakes no sense, if i is the realization of variable u, then ... suppose one looks at a series of coin flips. ... variable (instead of sampling only once)---but it's still just a sampling. ...
    (sci.econ)
  • Re: Stat Question on OLS assumptions
    ... >> y and u are defined as a random variables and x is a fixed variable. ... >> Since Emakes no sense, if i is the realization of variable u, then ... suppose one looks at a series of coin flips. ... variable (instead of sampling only once)---but it's still just a sampling. ...
    (sci.stat.math)
  • Stat Question on OLS assumptions
    ... y and u are defined as a random variables and x is a fixed variable. ... does this mean that the Expected value of each error is zero? ... Since Emakes no sense, if i is the realization of variable u, then the ... If the texts are correct, then I am led to conclude that each observation has ...
    (sci.econ)
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