1
$\begingroup$

Often it is said that heteroskedasticty could be assessed in a graphical way: for instance it can be inferred by looking at the residuals of a regression. However, this seems to me quite discretionary. For instance, in the following pictures:

http://www.hosting.universalsite.org/image-alpha-E3E5_58CE5E81.png

How would you say if residuals are homo- / heteroskedastic and to what extent? And based on what exactly?

Is there a best practice for this kind of analysis?

$\endgroup$
  • 2
    $\begingroup$ Possible duplicate of Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model $\endgroup$ – gung Mar 19 '17 at 12:59
  • 2
    $\begingroup$ I think you will find the information you need in the linked thread. Please read it. If it isn't what you want / you still have a question afterwards, come back here & edit your question to state what you learned & what you still need to know. Then we can provide the information you need without just duplicating material elsewhere that already didn't help you. $\endgroup$ – gung Mar 19 '17 at 12:59
4
$\begingroup$

As a complement to the plots that you show (and some others shown in the answers linked on the right-hand-side of this page), a range-mean plot is easy to get and is often informative.

The idea is to split the series of residuals into blocks of length, say, $k=\sqrt n$, where $n$ is the number of residuals; i.e, the first block contains the residuals from 1 to $k$, the second one contains the residuals from $k+1$ to $2k$, and so on.

The mean and the range are obtained for each block and displayed in a graphic. If the variance is homogeneous throughout time, the points will be located around an horizontal line; otherwise, an increasing or decreasing (or a more complex) pattern will be observed.

As the residuals will, in principle, have a constant mean. It is better to display the means of the times at which the observations are observed.


Example (taken from documentation of R package lmtest).

# Residuals of 'dy' in data set 'jocci' regressed on six lags
require("lmtest")
data(jocci)
fit <- lm(dy ~ dy1 + dy2 + dy3 + dy4 + dy5 +dy6, data=jocci)
e   <- residuals(fit)

As I said, as there is no trend in the residuals, the mean of the blocks of residuals is not informative.

k  <- floor(sqrt(length(e)))
le <- split(e, gl(ceiling(length(e)/k), k)[seq_along(e)])
r  <- unlist(lapply(le, FUN=function(x) diff(range(x))))
m  <- unlist(lapply(le, FUN=mean))
plot(m, r, ylab="range (residuals)", xlab="mean (residuals)", 
     main="range against mean of residuals")
abline(lm(r ~ m))

Alternatively, take blocks of the times of observations. It is observed that, as we advance in the series of residuals, the range increases. A regression line shows a significant trend. This suggests therefore a heteroskedastic pattern.

par(mfrow=c(2,1), mar=c(4,4,3,3))
plot(e, type="h", main="residuals")
r   <- unlist(lapply(le, FUN=function(x) diff(range(x))))
lid <- split(seq_along(e), gl(ceiling(length(e)/k), k)[seq_along(e)])
m   <- unlist(lapply(lid, FUN=mean))
plot(m, r, ylab="range (residuals)", xlab="mean (times)",
     main="range of residuals against time")
fit <- lm(r ~ m)
abline(fit)
summary(fit)

range-mean plot

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.