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I was reading an article about the polar motion of the Earth: https://www.researchgate.net/publication/241368199_The_Earth's_variable_Chandler_wobble. A regression is performed between observed and modelled complex time series of data - the type of regression is not specified. The regression coefficient is then calculated for different values of the estimated parameter (quality factor); its amplitude and phase are shown in Figure 2 (I don't know if I am allowed to post it directly here).

My question is why can the amplitude of the regression coefficient be as high as 2.3? I thought that the absolute value of the regression coefficient, for real series, had to be in the range 0-1. Therefore I was expecting a maximum amplitude of $\sqrt{2}$ for a complex series.

The definition of regression coefficient is not mentioned in the article. At first I thought it may be a linear regression since if the model is correct, the modelled and observed series should be very close; however, this seems incompatible with a value of 2.3.

Is there a more general definition that I'm missing?

physics20
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    Check the definition of the RC? –  Feb 04 '21 at 15:53
  • @Gert The article does not mention a specific definition of RC. I thought it may be the linear regression coefficient since ideally, if the model is correct, the modelled series should be close to the observed series. However, since it is as high as 2.3 there must be a different definition I am not aware of. –  Feb 04 '21 at 16:24
  • For standard regression problems, the $R^2$ value can't be greater than 1, by definition. Your source either made an error, or they are using regression statistics that very few people have heard of. –  Feb 04 '21 at 16:24
  • A regression coefficient can be literally any value. I'm sure you can find thousands of examples here on CV tagged with [tag:regression]. If you were expecting a maximum amplitude, that must be due to something you haven't yet explained: what is the basis of that expectation? Is the maximum a theoretical maximum? (If so, don't forget about the effects of variability, including measurement error and model mis-specification, on the *estimate.*) – whuber Feb 04 '21 at 17:40

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You are expecting the (absolute value of the) 'regression coefficient' to not exceed 1. This might be because you are confusing this term with 'coefficient of determination', '(squared) correlation coefficient' or $R^2$?

It is difficult to say what they did exactly since the article does not describe the method in much detail. But the regression coefficient used here may easily not be coefficient of determination $R^2$ (the square of the coefficient of correlation between the observed and modeled) and instead it seems to be the parameter in a linear model.

the regression coefficient between the observed and modeled Chandler modes

Note that also $R^2$ can be greater than 1, see: Can $R^2$ be greater than 1?

Sextus Empiricus
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  • I see, thank you very much! – physics20 Feb 04 '21 at 16:53
  • I thought that SSA was used just to extract a specific periodic component (CW) from the observed series, for comparison - because it is the only component to be modelled that is analysed in the article. But I'll check if it has a special meaning in SSA, thank you for the suggestion. – physics20 Feb 04 '21 at 17:14
  • $R^2$ is not a "regression coefficient," leading me to suspect this post does not answer the question. – whuber Feb 04 '21 at 17:42
  • @whuber *"$R^2$ is not a regression coefficient"*, that is what I am saying. And it is an answer to the question if the problem is that a 'regression coefficient' is confused with 'coefficient of determination'. – Sextus Empiricus Feb 04 '21 at 18:05
  • I see no reference at all to the coefficient of determination in the question. The language in the final two paragraphs of the question suggests to me that your post has just further confused the OP by introducing $R^2.$ – whuber Feb 04 '21 at 18:10
  • @whuber, I agree that I had made some assumptions in answering the confusing question. This was based on the fact that in most cases when people wonder about the regression coefficient exceeding the value of 1, the situation is that they confuse it with the coefficient of determination (for which indeed the upper limit is 1 when it is computed as a correlation). To be honest, I wanted to delete my answer after having doubts, but this became impossible after it got an accepted answer. – Sextus Empiricus Feb 04 '21 at 18:23