First, thanks for your reply!
Bob Hayden schrieb:
> This may not fully answer all your questions, but the various formulas
> for inference for regression have a place where you plug in THE
> variance of the points around the regression model, i.e., they treat
> this as a constant. If it is not a constant, the results of the
> formulas will not be correct. Specific ramifications would depend on
> what formual you are interested in, the exact manner in which the
> variance varies, etc.
Yes, I'm aware that non-constant variances let the usual computational rules break
down if you really want to
incorporate them in the model. But my problem is: Why do I want to consider them?
As stated below, there are really two questions:
1) Why do the standard errors *increase* -- is there a intuitive explanation (more
intuitive than the proof
that the estimate of b is not efficient)? So, I think the specific formula of interest
is either the
calculation of b or -- more probably -- of its SE. Subquestion: Why does OLS fail to
reflect this tendency to
have larger SEs?
2) Regarding Kmenta's argumentation: Inefficiency of OLS is proved by relating it to
WLS' better performance.
This *assumes* that weighting down the observations with higher residuals is
legitimate. How do I justify this
strong assumption, apart from the case when I know that the larger residuals result
from higher *true* random
disturbances? In other words: If residual diagnostics show uneven distribution of
residuals, shouldn't be the
first step to search for unobserved predictors? And if I cannot measure these, isn't
OLS as good a guess at the
true relationships of the observed variables as WLS would be?
(Because Mr. Hayden started a new thread, I do not delete his quotation of my original
message, in order to
retain the context.)
Regards, MQ
> ----- Forwarded message from Markus Quandt -----
>
> Hello all,
>
> when discussing linear regression assumptions with a colleague, we
> noticed that we were unable to explain WHY heteroscedasticity has
> the well known ill effects on the estimators' properties. I know
> WHAT the consequences are (loss of efficiency, tendency to
> underestimate the standard errors) and I also know why these
> consequences are undesirable. What I'm lacking is a substantial
> understanding of HOW the presence of inhomogeneous error variances
> increases the variability of the coefficients, and HOW the
> estimation of the standard errors fails to reflect this.
> I consulted a number of (obviously too basic) textbooks, all but
> one only state the problems that arise from het.sc. The one that
> isn't a total blank (Kmenta's Elements of Econometrics, 1986) tries
> to give an intuitive explanation (along with a proof of the
> inefficiency of the estimators with het.sc.), but I don't fully
> understand that.
> Kmenta writes:
> "The standard least squares principle involves minimizing
> [equation: sum of squared errors], which means that each squared
> disturbance is given equal weight. This is justifiable wehn each
> disturbance comes from the same distribution. Under het.sc.,
> however, different disturbances come from different distributions
> with different variances. Clearly, those disturbances that come
> from distributions with a smaller variance give more precise
> information about the regression line than those coming from
> distributions with a larger variance. To use sample information
> efficiently, one should give more weight to the observations with
> less dispersed disturbances than to those with more dispersed
> disturbances." p. 272
>
> I see that the conditional distributions of the disturbances
> obviously differ if het.sc. is present (well, this is the
> definition of het.sc., right?), and that, IF I want to compensate
> for this, I can weight the data accordingly (Kmenta goes on to
> explain WLS estimation). But firstly, I still don't see why
> standard errors increased in the first place... And secondly, is it
> really legitimate to claim that OLS is 'wrong', if it treats
> differing conditional disturbances with equal weight?
>
> Assume the simple case of increasing variances of Y with increasing
> values of X, and therefore het.sc. present. With differing
> precision of prediction for different X values, the standard error
> (SE) of the regression coefficient (b) should become conditional on
> the value of X, the higher X, the higher SE, with E(b) constant
> over all values of X - correct? Then, isn't the standard error as
> estimated by OLS implicitly an _average_ over all these conditional
> SEs (just following intuition here)? How can we claim that the
> specific SE at the X value with the lowest disturbance is the
> 'true' one? (Exception: het.sc. is due to uneven measurement error
> for Y - I can see that the respective data points are less
> reliable.)
>
> Regarding the first question: Can this be answered at all without
> the formal proof?
>
> ----- End of forwarded message from Markus Quandt -----
--
________________________________________________________________
Markus Quandt
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