I am reviewing some quant topics and I need some help understanding something. If a data set has conditional heteroskedasticity or positive serial correlation, why are the standard errors smaller? I think I understand heteroskedasticity and serial correlation but my mind is just not making the jump to see why the standard errors are smaller.
Where did you read this?
In schweser book 1 on page 194 it states" There are four effects of heteroskedasticity you need to be aware of: … “The standard errors are usually smaller than they would be in the absence of heteroskedasticity.” … And on page 197: “Positive serial correlation typically has the same effect as heteroskedasticity. The coefficient standard errors are too small, even though the estimated coefficients are accurate.”
i remember reading that awhile back, it has to do with the fact that heteroskedasticity makes the regression LOOK better at first glance than it actually is. so, the standard error is smaller than it should be.
is it akin to a smaller number of observations? the observations aren’t independent so there isn’t as much independent info as the statistical model assumes. of course, this explanation is more like a description of what you’re asking about than an explanation…
I think it’s somewhere along the lines mike mentions but to be honest… I haven’t got a clue. I’d like to have one of our quant-experts do some explaining!
There is no systematic relationship between the error terms and the independent variables. Generally, with the incr. in the value of the independent variable, the value of the error term should also increase. This is, however, not the case in heteroskedasticity. So the standard error is smaller.
heteroskedastic -> volatility of errors changes, sometimes it’s higher than estimated standard error, sometimes it’s lower -> there is a higher chance of Type I error (because of large deviations) positive correlation also leads to higher chance of Type I error
Ruhi, you’re my hero.
That makes sense…thanks Ruhi!
Whenever you generate a t-stat you are using the formula t= (bj^-bj)/Sbj The denominator in this equation is the standard error of the coefficient. As I understand it, every major problem in regression comes from the fact that the standard error of the coefficent gets messed up. For example if you have serial correlation, or conditional heteroskedacity the error terms are not constant. This affects the strd error of the coefficeint, which in turns affects the t-stat, which in turn affects your hypothesis test of whether the coefficient is statistically significant or not. So now the questions is, in what way do each these violations of affect the standard error? Heteroskedacity= stardard errors are too small-> t-stat is too large (because standard error is in the denom) = Type 1 error Positive Serial correlation= same as above (negative is the opposite) Multicollinerarity= The black sheep, it is the opposite of those above, the standard error is too big-> t-stat too small = Type II error. This is my understanding anyway. I don’t have a firm grasp on how each violation affects the standard error of the coefficent though. Maybe you guys can help me with that…
ruhi22 Wrote: ------------------------------------------------------- > There is no systematic relationship between the > error terms and the independent variables. > Generally, with the incr. in the value of the > independent variable, the value of the error term > should also increase. This is, however, not the > case in heteroskedasticity. So the standard error > is smaller. maratikus Wrote: ------------------------------------------------------- > heteroskedastic -> volatility of errors changes, > sometimes it’s higher than estimated standard > error, sometimes it’s lower -> there is a higher > chance of Type I error (because of large > deviations) > > positive correlation also leads to higher chance > of Type I error Could you guys elaborate a little. Everybody else seems to get it, but I am still confused.
that’s a pretty good explanation, mwvt9.
mwvt9 Wrote: ------------------------------------------------------- > Whenever you generate a t-stat you are using the > formula t= (bj^-bj)/Sbj > > The denominator in this equation is the standard > error of the coefficient. As I understand it, > every major problem in regression comes from the > fact that the standard error of the coefficent > gets messed up. For example if you have serial > correlation, or conditional heteroskedacity the > error terms are not constant. This affects the > strd error of the coefficeint, which in turns > affects the t-stat, which in turn affects your > hypothesis test of whether the coefficient is > statistically significant or not. > > So now the questions is, in what way do each these > violations of affect the standard error? > > Heteroskedacity= stardard errors are too small-> > t-stat is too large (because standard error is in > the denom) = Type 1 error > > Positive Serial correlation= same as above > (negative is the opposite) > > Multicollinerarity= The black sheep, it is the > opposite of those above, the standard error is too > big-> t-stat too small = Type II error. > > This is my understanding anyway. I don’t have a > firm grasp on how each violation affects the > standard error of the coefficent though. Maybe > you guys can help me with that… We are talking about the standard error of the co-efficient here. We also need to consider the F-stat. For both heteroskedasticity and serial correlation, the F-stats are unreliable because of the standard errors. And in the case of multicollinearity, F-stats are very high, even though th t-stat may be too low. Take a look at p. 293 onwards in the CFAI text. Everything is explained in a lot of details and is an excellent read. Edit: If you remember, t stat is calculated by dividing the difference between the estimated and the hypothesized slope coefficient by the co-efficient standard error. So, any effect in the co-efficient standard error will affect the t-stat too.
ruhi, Your edit is what I was explaining above in my last full post. Take conditional heteroskadacity for example, why is the standard errors of the coeff too small because the variance of the error terms increase with an increase in the independant variable? I think you answered this above, but I don’t quite get it.
mwvt9 Wrote: ------------------------------------------------------- > ruhi, > > Your edit is what I was explaining above in my > last full post. > > Take conditional heteroskadacity for example, why > is the standard errors of the coeff too small > because the variance of the error terms increase > with an increase in the independant variable? > > I think you answered this above, but I don’t quite > get it. From what I can understand, if the variance of the error terms incr. with the increase in the value of the independent variable, this means that there is a constant variance. If on the other hand, this doesn’t happen, then there is no constant variance (as in the case of heteroskedasticity). So, some regression residuals might be larger in value even though the corresponding independent variable might have a smaller value or vice versa. Because of this inconsistency, the overall value of the standard error terms would be smaller. This is what I understood after taking a look at the graph in the curriculum and re-reading whatever is there. It hasn’t really been explained in more details in the book. Probably Joey could give us some extra information here.
ruhi22 Wrote: ------------------------------------------------------- > From what I can understand, if the variance of the > error terms incr. with the increase in the value > of the independent variable, this means that there > is a constant variance. I don’t think this is correct. I thought constant variance meant the errors would NOT increase as the independant variable increased. So the graph would looks like the error terms are constant around the regression line througout the full range of values for X,Y. EDIT: I do think the predicted value of Y will increase if the slope is positive, but I didn’t think the errors should increase. I could be wrong though. If on the other hand, this > doesn’t happen, then there is no constant variance > (as in the case of heteroskedasticity). So, some > regression residuals might be larger in value even > though the corresponding independent variable > might have a smaller value or vice versa. Because > of this inconsistency, the overall value of the > standard error terms would be smaller. > > This is what I understood after taking a look at > the graph in the curriculum and re-reading > whatever is there. It hasn’t really been explained > in more details in the book. Probably Joey could > give us some extra information here.
mwvt9, i think you’re correct. I probably misunderstood that part. thanks for the clarification!
If I am correct then I still don’t understand WHY the standard error of the coefficent would be too small…which eventually leads to a type I error.
On second thoughts, I think what I said was correct. Take a look at p. 293, Book 1 Edit: I’m tossing like a ping pong ball. I need to get off AF.