the assumption of fixed regressors (X) is the first to be relaxed
usually. There is no sence to assume fixed regressors unless your data is
coming from a controled expreriment. The model and estimation methods may
stay without change, only the interpretation of the model changes. Now
you can speak about conditional expectation of y (if regressors are fixed
there is nothing to condition on, remember: conditional expectation is
still a random variable, even if you call it expectation). Regarding
normality, you can keep it or drop it, it does not have any relation to
randomness of regressors. The same is correct regarding independence of
residuals. If you are interested in this topic seriously then get a good
Econometrics book. For example, A course in Econometrics by Goldberger is
a nice place to start.
Now to the second part. Prediction error = Y(future)-Y_hat is a random
variable, the problem is that that prediction error is unobservable random
variable, so yoou cannot treat it in the usual manner. Also, you have more
than just a single r.v., you have a series of such r.v. depending on how
far in the future you want to go (+1 period, +2, ... etc). For each period
there is a r.v. which has a distribution and hopefully finite mean and
variance.
On 12 Sep 2001, James Ankeny wrote:
> I have two questions regarding simple linear regression that I was hoping
> someone could help me with.
>
> 1) According to what I have learned so far, the levels of X are "fixed," so
> that only Y is the random variable ( error is random as well). My question
> is, what if X is a random variable as well? It seems like this could be the
> case with some of my textbook examples. Does simple model of y=a+bx+e still
> hold? Are assumptions the same, such as conditional distributions of Y are
> normal with same variance, E(Y) is a straight line function of X, and
> independence/normality of error terms? Also, in repeated sampling the sample
> slope is normal because Y is normal. However, if X also varies from sample
> to sample, is the sample slope still normally distributed (sampling
> distribution)?
>
> 2) My second question regards the prediction interval. I can perform this on
> a computer, but it is difficult for me to conceptualize. If you are using
> Y-hat (the mean of estimated regression function) to estimate a future
> response, does this mean that the difference,
> (Y(future response)-Y hat), is a statistic that has a sampling distribution,
> from which you can derive the standard error? It seems like this might be
> the case, but there is no parameter. I don't even know if what I just said
> makes any sense.
>
> I understand that my questions are long, and perhaps not in any logical
> order, but I would greatly appreciate any help with these conceptual
> matters.
>
> Thank you
>
>
>
>
>
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