# Can heteroskedastic residuals be justified by variance in dependent variable?

This is a very basic question and I hope it is not a duplicate.
Im using a pooled regression model with a log-transformed dependent variable (electricity consumption meter values).
The variance of $log(y)$ is not constant over the course of the year and neither are the residuals (see figure). The model includes month as a categorical variable. Model results (parameter estimates) seem feasible and I think the model could be useful (mainly for prediction).

Questions:
1) Is it trivial that residuals exhibit a similar seasonal variance as the dependent variable or does it reveal model shortcomings?
2) If it causes problems with respect to model validity, how can I avoid this?
3) Is it ok to just report robust standard errors and ignore the heteroskedastic residuals?

Edit:
The variance in $y$ is shaped differently compared to the variance in $log(y)$ (see figure) • Depending on the number of observations you have, it might be wise to model the seasonality. This will give smaller residuals, hence better predictions. One way of doing this could be to enter the months as dummy variables. If you do so, take care that a categorical variable with 12 levels results in 11 dummy variables. If you do not have enough observations, this could result in over-fitting. – spdrnl Jun 2 '15 at 13:08
• Month is already included as categorical variable but probably this is not enough ;) – Aki Jun 4 '15 at 11:29

It appears this is panel data, and so you have many cross-sectional observations per time period. Obviously, you are making an assumption that in the cross sectional dimension the variance is the same for all cross-sections, and this is why you use the sample variance as a valid estimate of the true variance that characterizes each time period.

You then plot the estimated cross-sectional sample variance over time, and you get the above picture.

Before thinking in "technical" terms, try to understand what this might reveal for the behavior in the sample. We expect that during the summer people will use Air Conditioning for cooling purposes while during the winter heating is taken care of by means other than electricity consumption. So we expect a shift in the mean electricity consumption. But why a shift in the variance of it, which literally sky-rockets during the hot months?

My guess is the following: a visible fraction of your sample do not use Air-Conditioning.
These entities therefore will have the same mean electricity consumption as during the other months, and their actual consumption will hover around the "non-hot-months" mean electricity consumption.
Technically, your cross-sectional sample for these months stops being "identically distributed".

The rest will consume more electricity due to A/C use, shifting the sample-wide mean up, but not as much as they will if they were alone in the sample. Nevertheless, their actual consumption will hover around the group-specific mean value, which will be higher than the sample-wide mean.

So here you have it: during the hot months, your sample is not identically distributed. You have two groups. The consumption for the non- A/C group hovers around the "old" mean, and so below the sample-wide mean for these months, while the consumption for the A/C group hovers above it. While during the other months, both groups' consumption hovers around the common mean. This is a reasonable explanation for the observed sample-wide increase in the variance of the mean electricity consumption. Schematically, Obviously the sample-wide-variance will be higher during hot months. Note that the inclusion of monthly dummies in the regression, will not affect the variance of the dependent variable itself.

Note also that this is not a case of "group-wise heteroskedasticity". It is a consequence of pooling together non-identically distributed RVs, and so, if the hypothesis I make is supported by your sample, you might want to dig on what to do when we face non-identically distributed RV's.

• Thank you for your answer! Unfortunately I made a mistake by confusing electricity and district heat (I am working with meter data of both kinds), so the plotted data refers to heat consumption, not electricity. However, your explanation seems reasonable to me. What strikes me is, that the variance in $y$ (untransformed variable) is lowest during summer while the variance in $log(y)$ is highest during summer months. I will try to add one more figure. – Aki Jun 3 '15 at 7:15