Linear regression terminology question — Beta (β)

Question

I was a bit confused with the meaning of $\beta$, and thought its usage was rather loose. In fact, it seems that $\beta$ is used to express two distinct concepts:

1. The generalisation of the sample "b coefficient" to the population concerned.
2. The standardized regression coefficients (regression coefficients obtained when all variables are standardized with a sd of 1).

Would there be an alternative symbol to any one of the two significations above, to avoid this confusion?

Answer 1

You're right. Most texts I've seen write a regression model as $$Y = \beta_0 + \beta_1 X_1 + \ldots + \beta_{p-1} X_{p-1} + \epsilon,$$ and the second usage of "beta" or "beta weight" to mean "standardised regession coefficient" is also relatively common (and is used in some statistical software).

I avoid this ambiguity by saying/writing "standardised regression coefficient" rather than "beta" in the second situation. I would also only say/write $\beta_i$ in the first situation if I've defined it, otherwise I'd say something like "true regression coefficient of $X_i$".

Answer 2

Assuming the linear model is correct, $\hat{\beta}$ (this is sometimes also called $b$ in elementary texts), the coefficient estimated from the data set you have, is an estimate of the true slope, $\beta$. To my knowledge there is no standard notation for the standardized coefficient, although some simple algebra will give you the relationship between the standardized coefficient and the un-standardized one.