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I am new to this form. I am going to run an OLS regression with dependent variable as a correlation coefficient (between -1 and 1). I am aware that this dependent variable may be problematic in the OLS regression. But what can I do in order to implement the regression? Thanks!

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  • $\begingroup$ A correlation coefficient between which two variables? How do they enter into the regression equation? $\endgroup$ – Herr K. Sep 27 '18 at 15:47
  • $\begingroup$ Thanks for your reply. I use the correlation coefficient between firm size and productivity as a measure of allocative efficiency of a industry s. Then I want to investigate the correlation of this dependent variable and other industry level variables. Is there any suggestion how should I deal with this dependent variables? $\endgroup$ – gxxjjj Sep 27 '18 at 17:02
  • $\begingroup$ Could you clarify your goal here? "Implementing" a regression is usually not the goal, unless in coursework just for practice. You presumably want some result/information from the data and care about whether the result is valid if you use a given method (like OLS). The goal determines the method. $\endgroup$ – Sander Heinsalu Sep 28 '18 at 0:59
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Welcome. Many things can be done. Some simple ones are:

Option 1: Do not care and just do OLS.

Option 2: Do some transformation such as $\ln (1+y) - \ln (1-y)$, logit of $\frac12 (y+1)$, to change the range from $(-1,1)$ to $(-\infty, \infty)$ and do OLS. I assumed that there are no $\pm 1$.

Option 3: Change your model. Let $y_{1,ij}$ and $y_{2,ij}$ be the firm-level ($j$=firm) variables from which you obtain the correlation coefficient $y_i$. As $y_i$ is also obtained by regressing $y_{1,ij}$ on $y_{2,ij}$ (up to a scaling factor), you might want to regress $y_{1,ij}$ on $y_{2,ij}$ including other industry-level variables $x_i$ as regressors. But in that case you need to think about which one of $y_{1,ij}$ and $y_{2,ij}$ to use as the dependent variable. If you are worried about endogeneity, you can also consider instrumental variable estimation, but that's a different story.

I would choose Option 1 unless many correlation coefficients in the sample are very close to $\pm 1$. Note that interpretation changes in Option 2.

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