How would heteroskedasticity look like with negative correlation?

I hope I formulated the question correctly, but I am purely interested in heteroskedasticity: cov($$e_i$$,$$e_j$$) and not in autocorrelation with the dependent variables.

So positive correlation of the error terms will give you the standard graphs, in which large errors follow up on each other and you get these kind of typical time series where large volatility will stay for a while and periods of low volatility are observed, right?

But what about negative correlation? Then large errors will be quickly followed by small errors and vice versa right? How would that typically look like?

Also small side question, the left graph I would depict as positively correlated in its residuals, but someone told me that the right graph could be a case of negative correlation. But this is incorrect right? To me that seems more like a classic case of autocorrelation.

• The two are not directly related, so I don't think you can infer anything about the other. – user2974951 Feb 15 '19 at 8:47

It seems to me you misunderstood some concepts, so I'll try to be more thorough than usual:

1. heteroskedasticity happens when subsamples have different variances than others. In the case of time-series, it happens when the variance you observe varies if you look at different periods/lengths of time. Correlation is another concept, that in principle has nothing to do with heteroskedasticity. In the image below, the plot on the left corresponds to a homoscedastic variable (for any length of window you look, the variability is the same). On the other hand, the plot on the right is heteroskedastic, the variability increases as, say, time passes, and it won't be the same for any window period you consider.

1. The signal of the the correlation coefficient does not solely define the behavior you'll see in the plot, what you describe can happen with a positive or negative correlation value.

2. Say you have two random variables, $$X$$ and $$Y$$. Take a moment to analyse the formula for their (linear) correlation below. You see it's just the expected value of the product of each variable divergence from its mean value. It can only be positive if the difference is, on average, positive, (i.e. when they move the same direction). On the other hand it will be negative if the differences are, on average, of different signs (when they move on opposite directions).

$$\sigma_{xy} = \mathbb{E}\left[(x_i-\bar{x})(y_i-\bar{y})\right]= \frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})$$

1. Now, if we're talking about heteroskedasticity, we're considering the variation behavior of one random variable; in our case a time-series. In this sense correlation is autocorrelation, after all we are computing the correlation of a variable with itself, but in different points in time.

2. For autocorrelation, if the sign is negative it means that, on average, positive differences (from the mean value) are preceded by negative differences, and vice-verse. If it's positive, then differences of the same sign, on average, precede one another:

Finally, it's not that easy to inspect the signal of a correlation coefficient just by looking at its graph. For your first graph (left) it might be that the correlation is close to zero, for example.