calculate a variance for a given R-squared

Asked Aug 21 '19 at 05:34

Active Aug 21 '19 at 20:37

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I am trying to replicate a study that has generating simulated data with the following variables and assumptions $x_j\sim U[0,1]$, and $j = (1,2,\dots,N)$ with

$$z_j = x_j + \sigma_z \epsilon_j$$

where, $\epsilon_j\sim N(0,1)$ and then state "we select the value $\sigma_z$ so that $R^2 = 0.75$", I believe $R^2 = 1 - \frac{\sum_j (z_j - x_j)^2}{\sum_j (z_j - \bar{z})^2}$

But I am struggling to figure how to address this problem.

edited Aug 21 '19 at 20:37

asked Aug 21 '19 at 05:34

Cyrillm_44

What values does $j$ take? – PascalIv Aug 21 '19 at 08:51
@PascalIv I have updated, its just an index – Cyrillm_44 Aug 21 '19 at 20:38
I answered this question at https://stats.stackexchange.com/a/152034/919, which includes code to simulate data based on the regression parameters $R^2,$ and the patterns you want to see in the residuals. – whuber Aug 21 '19 at 22:18
Thank you very much @whuber – Cyrillm_44 Aug 21 '19 at 22:45

calculate a variance for a given R-squared

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