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Simple Linear Regression. You regress $Y \sim X$, and You know $\beta_{y,x}$.

Now, if you regress $X \sim Y$, what can you say about $\beta_{x,y}$, what about its range?

$\beta_{x,y} = \beta_{y,x}\frac{s_{x,x}}{s_{y,y}}$.

How can I infer the range?

kuku
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    What do you mean by "range" in this context? There isn't any fixed range for regression coefficients, it's a function of the units of the data. – gung - Reinstate Monica Nov 12 '20 at 23:08
  • $\beta_{y,x} = 3.0$, the question is, can you give numerical value about the range of $\beta_{x,y}?$ – kuku Nov 12 '20 at 23:22
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    Does this answer your question? [Effect of switching response and explanatory variable in simple linear regression](https://stats.stackexchange.com/questions/20553/effect-of-switching-response-and-explanatory-variable-in-simple-linear-regressio) – Sextus Empiricus Nov 12 '20 at 23:46
  • Other related https://stats.stackexchange.com/questions/458928/ and https://stats.stackexchange.com/questions/385812 – Sextus Empiricus Nov 12 '20 at 23:47
  • @kuku, what do you mean, "the question is"? Is this a question from a course or a textbook? Also, I'm still not sure what you mean by "range" here. Are you asking if there is some maximum (minimum) number that $\hat{\beta}_{x,y}$ can take? – gung - Reinstate Monica Nov 13 '20 at 01:45
  • yes, maximum and minimum number. @gung-ReinstateMonica – kuku Nov 13 '20 at 01:54
  • There is no maximum (minimum) number; it's keyed to the units of your variables. Consider regressing distance on time, your beta can be in units of miles per hour. Now switch the units of the distance variable to metric, it will be in kilometers per hour, & be a higher number (about 1.6 times higher). From there, change the units of distance to meters. The beta will be 1k times higher, all with exactly the same relationship in the data. You can keep going further: change the units to millimeters and weeks: at the exact same speed, the number of mm / week will be truly astronomical. – gung - Reinstate Monica Nov 14 '20 at 06:58

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