Test for bias in the residuals of regressions

Most of the answers I found say "look at the residual plots". Which is great!

But I have a large set of curves, and would like to "distinguish" time series curves that fit a model without bias in the residuals. For example, testing whether curves are mostly "white noise" and fit a model like y = constant_mean + random_noise .

Examples

In the following example time series, a polynomial fit does not give biased residuals:

And, as expected, a linear fit does have biased residuals:

How can i test for bias in the residuals?

Is there a correct way of doing this? Or any thoughts?

Mine so far include:

Looking for a test similar to heteroscedasticity, but for the mean of the residuals.

Maybe some kind of sliding window scan, of the residuals of a linear fit with null slope. In that case perhaps the "distributions of means" over all time windows (continuous subsets) must be the same across all windows then there is no bias. For example, that should be the case in curves of this appearance:

I've also seen some suggestions for using auto-correlation, "serial correlation" or a "white noise test" for the residuals may work (see comments).

Use case

I have many curves, and i would like to find which of them fit noisy (flat|straight) lines and which do not. Or at least rank my curves in some way.

Thanks!

• what does 'mostly white noise" mean.:) There are tests if a model is white noise or not. The issue here btw is serial correlation not bias as I understand that term. I don't think that something that is not white noise is biased. – user54285 Aug 26 at 23:53
• Hi thanks for replying! by white noise i mean "random fluctuations around a mean value". What are the tests for that? – Naiky Aug 27 at 14:59
• I had not heard of "serial correlation" but it seems that it is the same as autocorrelation. How would you use that to approach the question? – Naiky Aug 27 at 15:01
• "not white noise" will have biased residuals if the model is linear and the data is not, right? If i tried to fit a straight line to data 3000000230, the residuals of the fit would be biased. – Naiky Aug 27 at 15:03
• serial correlation includes MA patterns I believe while I think autocorrelation is only AR patterns - although I could be wrong at that. There are many tests for white noise - which is the absence of AR and MA effects. In ARIMA Ljung Box tests for this. In regression I think Breusch-Godfrey is preferred because their are doubts about of the validity of Ljung Box. This link from this board is useful in that light. stats.stackexchange.com/questions/148004/… – user54285 Aug 27 at 20:55