Timeline for For a linear regression of Y on X, when is the regression of X on Y linear? When is it non-linear?
Current License: CC BY-SA 4.0
15 events
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| Jun 30, 2021 at 13:28 | vote | accept | user62421 | ||
| Jun 25, 2021 at 17:48 | answer | added | user62421 | timeline score: 1 | |
| Jun 24, 2021 at 17:36 | comment | added | whuber♦ | Sorry, I was thinking of something else. | |
| Jun 24, 2021 at 16:09 | comment | added | user62421 | @whuber what is that modification? Every distribution I've tried (e.g., Exponential, Uniform, Gamma, even Cauchy!) gives a straight line for $E[X_1|Y=y]$ when $X_1$ and $X_2$ are iid. When the limits of the uniform distributions are the same (e.g., if you change $X_2 \sim Unifrom[-2,2]$ in my example) $E[X_1|Y=y]$ is a straight line. | |
| Jun 24, 2021 at 15:54 | comment | added | whuber♦ | A slight modification of your own example shows iid does not suffice. | |
| Jun 24, 2021 at 15:42 | comment | added | user62421 | @whuber yes, I agree that the regression of $X_1$ on $Y$ can be almost arbitrarily complicated. Thus my question: when is $E[X_1|Y=y]$ a straight line? It is a straight line when $X_1$ and $X_2$ are iid Normal. What other conditions? Is iid $X_1$ and $X_2$ also sufficient? | |
| Jun 24, 2021 at 15:32 | comment | added | whuber♦ | The linear relation to which you refer is between $X_1$ and $Y,$ not between $X_1$ and $X_2.$ If you were to plot the curve $y=ax_1$ on $(x_1,y)$ axes, then adding $bX_2$ will alter the heights of that graph randomly according to the distribution of $X_2.$ Since you allow $X_1$ and $X_2$ to be correlated (you don't even require the mean of $X_2$ conditional on $X_1$ be zero), those alterations may follow literally any path--even a deterministic one. Drawing a few such plots will reveal why the regression of $X_1$ against $Y$ can be almost arbitrarily complicated. | |
| Jun 24, 2021 at 15:01 | history | edited | user62421 | CC BY-SA 4.0 |
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| Jun 24, 2021 at 14:36 | comment | added | user62421 | @whuber ok, figures added. How should I phrase the linearity part to make it more accurate? | |
| Jun 24, 2021 at 14:34 | history | edited | user62421 | CC BY-SA 4.0 |
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| Jun 24, 2021 at 14:28 | history | edited | user62421 | CC BY-SA 4.0 |
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| Jun 24, 2021 at 14:21 | history | edited | user62421 | CC BY-SA 4.0 |
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| Jun 24, 2021 at 14:18 | comment | added | user62421 | @whuber I am indeed thinking of bX2 as an error term. I am trying to figure out how to post images so that I can post images of the numerical and simulation output to demonstrate what I mean . . . | |
| Jun 24, 2021 at 13:57 | comment | added | whuber♦ | The question is clear, but your terminology might confuse readers. Your initial definition of "linearly related" is without content: all pairs of random variables $(X_1,X_2)$ enjoy such a relation. See stats.stackexchange.com/questions/257779 for a somewhat related question. For insight into all three of your questions, think of $X_1$ as an "independent variable," $bX_2$ as an "error term," and draw pictures. | |
| Jun 24, 2021 at 13:32 | history | asked | user62421 | CC BY-SA 4.0 |