#StackBounty: #regression Constant Variance restrictive as a general rule?

Bounty: 50

Think of a linear regression model:
$$
Y=Xbeta+epsilon
$$
where $epsilon|Xsim Nleft(0,sigma^{2}right)$. The vector of
parameters $beta$ can be consistenly estimated by OLS. I have one
conceptual question here: first, $epsilon$ has units of $Y.$ However,
when we minimize the (unweighted) squared sum of residuals, we treat
each observation the same. Conceptually, think of two different values
of $X.$ If the true conditional mean function is linear, that is
in the population:
$$
mathbb{E}left(Y|Xright)=Xbeta
$$
then the conditional mean of $Y$ changes with values of $X.$ If
$beta>0$ , then the value of the conditional mean grows with the
values of $X.$ However, under homoskedasticity, the variance of the
error term is assumed to be constant. However, for large values of
the conditional mean, the percentage deviation of the disturbance
decreases and tends to 0. In other words, the error dispersion becomes
more and more miniscule. If $X$ is non-stochastic, then the conditional
variance of $Y$ is $sigma^{2}$ as well. As such, the conditional
coefficient of variaion of $Y$ is:
$$
frac{sigma^{2}}{Xbeta}
$$
which goes to $0,$ simply as $X$ increases.. I guess my question
is isn’t the heteroskedastic assumption restrictive in its basic principle.
In other words, even if heteroscedasticity is not related to the value
of the regressors, isn’t this conceptually incorrect as we are imposing
that the average percentage deviation of the error term from
the conditional mean decreases without bound? Should the error term
not also be scaled by the value of $Y?$ Another way of stating is: an error of an equal magnitude means different things for different values of the regressand/regressors. An error of 1 when Y=10 is much more than an error of 1 when Y=100000; but OLS treats these as symmetric.


Get this bounty!!!

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.