# Does this graph show heteroscedasticity or homoscedasticity?

So I am using stock data for certain stocks over a period of about 13 years and now i want to check for heteroscedasticity and auto-correlation on stata. residual vs. fitted value is shown below.

• Does the plot hint heteroscedasticity?

• Should the plot be insufficient, what test may I use? Potential autocorrelation may impact the results of a Beusch-pagan or white test to for heteroscedasticity

1. There's no indication of changing variance there. If anything, it looks somewhat more consistent with constant variance than you typically see with data simulated under the assumption of constant variance, per the plot.

Nine example residual plots under homoskedasticity
(residuals vs fitted)

I strongly advise trying such an exercise on data generated from the model you're actually fitting to see what sorts of things you could reasonably expect to see. Then try it with heteroskedasticity added in and see what that does.

Here's a few examples where the conditional variance isn't constant (with shape of population spread function indicated):

I'd suggest trying many possibilities (more than once each) and seeing what they might look like.

Secondly, given the assumption is very likely false (assumptions will nearly always be false†), that knowledge doesn't tell you what the impact on your inference would be. Again, simulation is an easy way to investigate how much impact it will have on standard errors, on rejection rates under H0, on power at various effect sizes, and so on. I won't pursue that in this answer but I strongly encourage you to investigate this.

2. Changes in variance need to be reasonably large for there to be much impact on significance levels.

[Note that the Type I error rate is the rejection rate under $$H_0$$ -- i.e. under the condition that the populations were such that $$H_0$$ was true; with equality nulls $$H_0$$ is nearly always false, so the residuals only contain direct information about the situation under $$H_0$$ if you add some further assumptions connecting what you have sampled under $$H_1$$ to what would be the case under $$H_0$$ - assumptions that you don't necessarily need to hold for the test to work quite well.]

3. Any effect on $$\alpha$$ depends on how the variance is changing.

† You hope that they're not so far from true that your inferences are substantively impacted. Which is to say your problem is not really one of significance but a specific kind of 'effect size'.

* If that's fitted values on the x-axis, which is a diagnostic you'd often use with a regression, it looks fine. With stocks, however, you should worry about variance changing over time.

• Thank you very much for the input, it is very appreciated. Regarding your comment on how variance may change over time, I did plot the the returns (Dependent variable) against time, and saw significant deviations in terms of the variance. I am not know entirely certain how to consolidate these two results (i.e residuals against fitted values imply homosked. but plotting the dependent variable across time shows large changes in volatility/variance, which indicates that variance varies as time progressses and hence heterosked.). Again, thank you for taking time to help Commented Aug 2 at 11:11
• Your regression model's fitted values (if that's what you're plotting against) aren't in the "time" direction so you can't see heteroskedasticity over time in such a plot; unless fit and time were very strongly related you would see it all jumbled up. Consider an analogy; let's say you were plotting temperature of a motor against the load across multiple experiments with different loads. You might not see any differences in variability. But now imagine that as the motor wears out over time, the temperature becomes more variable. If you look at the residuals vs fitted, you won't see that ...ctd Commented Aug 2 at 12:39
• because the loads you modelled were not strongly related to time; any fitted value has some values from when it was newer and values from when it was more worn. There's no 'trend' in the spread in the plot. If you looked at it vs fitted values and vs time (a 3D plot), as you turn the 3D cloud so you're looking "into" time as you look into the plot (earlier values being closer to you and more recent ones behind), you don't see time changing so you don't see the changing variance, but if you turn it 90 degrees so you're looking across time as you look from left to right, it would become clear. Commented Aug 2 at 12:39