 # Heteroskedasticity and Standard Errors

Maybe a simple question, but I just can’t visualize this. Why does the presence of heteroskedasticity (or positive serial correlation for that matter) create *smaller* standard errors? Anyone have an intuitive answer? I figure when compared to a homoskedastic scenario, the residuals are larger and therefore standard errors should be larger? I’m probably missing something very obvious here.

Asked myself the same question, dont have a answer. Your explanation will work for me on the exam though you can think of heteroskedasticity in terms of temperature during the year- standard error is average temperature. since there are large fluctuations of temperature (generally very hot in the summer, very cold in the winter) standard error (average temperature) will understate deviations during high volatility (summer) causing Type I errors.

brilliant ^^.i’ll never forget now!.thanks maratikus

I don’t get it. Anyway… There is the odd statement in the book that “standard errors are smaller than they would be in absence of heteroscedasticity”. Hmmm… Not really clear what that means and I’ve parsed it every way I can think of. The question is really whether the estimated s.e. for a regression coefficient is a good estimate for the std. dev. of the sampling distribution of the estimated coefficient. If there is heteroscedaticity, it’s not. The reason is that the standard error estimator treats all residuals equally. However, the regression coefficient estimators do not treat all residuals equally. Thus an observation near the mean of the X’s has no impact on the estimated slope of the regression line, but the largest and smallest observations have a big impact on the estimated regression line slope (in statistics lingo they are “influential”). That means that variability out on the frontier causes lots of variability in your coefficient estimator while variability near the mean does not cause much variability in your coefficient estimators. Thus, if you have a “bowtie” your s.e. estimate will underestimate the true variability of the coefficients and if you have a “football” it will overestimate it.

***Shakes head and blinks eyes then stares blankly at hands**** I just forgot everything in quant.

JoeyDVivre Wrote: ------------------------------------------------------- > I don’t get it. Anyway… > > There is the odd statement in the book that > “standard errors are smaller than they would be in > absence of heteroscedasticity”. Hmmm… Not > really clear what that means and I’ve parsed it > every way I can think of. > > The question is really whether the estimated s.e. > for a regression coefficient is a good estimate > for the std. dev. of the sampling distribution of > the estimated coefficient. If there is > heteroscedaticity, it’s not. > > The reason is that the standard error estimator > treats all residuals equally. However, the > regression coefficient estimators do not treat all > residuals equally. Thus an observation near the > mean of the X’s has no impact on the estimated > slope of the regression line, but the largest and > smallest observations have a big impact on the > estimated regression line slope (in statistics > lingo they are “influential”). > > That means that variability out on the frontier > causes lots of variability in your coefficient > estimator while variability near the mean does not > cause much variability in your coefficient > estimators. Thus, if you have a “bowtie” your > s.e. estimate will underestimate the true > variability of the coefficients and if you have a > “football” it will overestimate it. JDV, I think we have talked about this before. It is kind of like the regression line pivots on the intersection of the mean of X and Y right (as by definition the line will have to pass through those points)? The amount of “pivot” is influenced by the error terms.

JoeyDVivre Wrote: ------------------------------------------------------- > That means that variability out on the frontier > causes lots of variability in your coefficient > estimator while variability near the mean does not > cause much variability in your coefficient > estimators. Thus, if you have a “bowtie” your > s.e. estimate will underestimate the true > variability of the coefficients and if you have a > “football” it will overestimate it. I actually think I understand this! Because conditional heteroskedacity and autocorrelation are manifestations of the residuals in a systematic way, we can make blanket statement about the, such as the s.e. are smaller, based on the example above. Is that correct?

Sure - see from “staring blankly” to “I actually think I understand all this!” in 17 minutes. That’s what’s supposed to happen.

My brain was just rebooting.

mwvt9 Wrote: ------------------------------------------------------- > My brain was just rebooting. I think mine just crashed. maratikus, thanks for the great analogy. It’ll definitely help with understanding this stuff. Joey, I’m sure you’re right, but I just don’t have enough RAM at the moment to process your response. I will definitely give it another look when I get home.