As I guess we all know, $\beta$ is the standardized version of $b$, which is the estimated slope parameter. However, how do I calculate it? I assume I have to divide $b$ by some kind of standard deviation. Any useful reference? I know there are other formulas, but I want to know how to get it from $b$.
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2$\begingroup$ I've removed the tag "beta distribution", which doesn't apply at all. $\endgroup$– Nick CoxCommented Nov 17, 2015 at 19:51
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4$\begingroup$ Just a note to say that "we all know" varies by discipline; for example, $\beta$ is sometimes used to refer to what you term $b$, whether standardized or not. $\endgroup$– AlexisCommented Nov 17, 2015 at 19:51
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1 Answer
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Actually, it is much simpler. To get $\beta$ instead of $b$, all you have to do is standardize your $x$ and your $y$ are run a regression of the standardized $y$ on the standardized $x$. Consider a simple bivariate model: $$y=x\beta+u$$
Here, $\beta$ measures the marginal effect of x on the conditional mean of y . Now, let us standardize both $x$ and $y$ as :$$\tilde{y}=\frac{y-\mu_{y}}{\sigma_{y}}$$ where $\mu_{y}$ denotes the mean of y and $\sigma_{y}$ denotes the standard deviation of $y$. Do a similar transformation for $x$. Then run the regression:$$\tilde{y}=\tilde{x}b+u$$ and you will estimate the slope paramter in standard deviation units.
If you want to use the OLS estimate directly, just multiply $\hat{\beta}$ with $\frac{\sigma_{x}}{\sigma_{y}}$.
