Should I convert variables that don't have units into a z-score In the regression analysis, some variables don't have units. How to interpret the coefficients of such variables? Should I convert them into z z-score before run regress? Should converting be done after taking log of variables? I'm quite new to regression and hope someone can help me.
 A: In a regression model the expectation of a response, $\mathbb{E}[Y]$, is equated to a linear combination of variable values $x_i$ with coefficients $\beta_i$:
$$\mathbb{E}[Y] = \cdots + \beta_i x_i + \cdots$$
If the units in which $Y$ is expressed are, for example, "meters", then its expectation will have the same units.  If the units in which $x_i$ is expressed are, for example, "hours", then this relationship can be more fully written as 
$$\mathbb{E}[Y]\,\left(\text{meters}\right) =  \cdots + \beta_i x_i\,\left(\text{hours}\right) + \cdots$$
For this to make sense, it is necessary that the units of $\beta_i$ be meters per hour:
$$\mathbb{E}[Y]\,\left(\text{meters}\right) =  \cdots + \beta_i \left(\frac{\text{meters}}{\text{hour}}\right)x_i\,\left(\text{hours}\right) + \cdots$$
The roles of "meters" and "hours" may be replaced by any units whatsoever, demonstrating a general conclusion: 

The units of the coefficients in a linear regression model are the units of the response per (that is, divided by) the units of the corresponding regressors.

This holds even for variables that are unitless.  For instance, if $x_i$ is a pure number (such as a fraction or a logarithm), then $\beta_i$ must have the same units as $Y$.  If both $x_i$ and $Y$ are logarithms, then $\beta_i$ is unitless.
