How to interpret the results of multiple regression when two dummy coded predictor variables are depended? I am would like to get the estimated influence of two conditions (variable A and variable B).
When I dummy-coded variable A and B and carried out a multiple regression analysis, both of them were significant predictors. 
However, things are little complicated. 
Variable B is independent, but Variable A depends on B; Variable A can be present (i.e., 1) only when Variable B is present (1). 
I am wondering how I can interpret the estimated coefficient for variable B. I thought I could run a simple regression analysis, but I would also like to get an estimated effect of variable A while B is zero (which is what the coefficient for variable A represents, if I am understanding correctly).
Hence, I started wondering this if I can interpret in the way that the estimated influence of B is the coefficient for B plus coefficient for A. However, I am not 100% confident if I am right.
I would really appreciate it if someone could confirm or correct my understanding. Thank you for your help!
 A: The way you are interpreting the coefficients is not quite right.  The general interpretation of the coefficient on a dummy variable in a multiple regression is "the expected (or average) difference in the dependent variable between those with $1$ and those with $0$ values of that dummy variable, holding other independent variables constant.
If you, for example, only have these two $A$ and $B$ variables as predictors, then the interpretation of the coefficient on variable $B$ is "the expected difference in the dependent variable between someone with the value of $1$ and someone with the value of $0$ of variable $B$, if they both had the same value of variable $A$."  That same value of variable $A$ can be anything, not just $0$.

Variable A can be present (i.e., 1) only when Variable B is present (1). I am wondering how I can interpret the estimated coefficient for variable B, because the coefficient for B represent the presence of B while A is 0, which logically does not make sense.

I think in these two sentences you may have meant to say that you cannot interpret the effect of $A$ while $B$ is $0$, because $A$ should be non-existent when $B$ is $0$.  But the regression model does not know that and the $\beta$'s (marginal effects) it provides apply regardless of what the value of the second variable is.  So you cannot just take interpret those coefficients as effects of the first variable for a specific value of the second variable.
It is a pretty simple exercise to figure out what each coefficient represents in this kind of regression:
$$
Y_i = \beta_0 + \beta_1A_i + \beta_2B_i + \epsilon_i
$$
where both $A$ and $B$ are binary dummy variables.  You can just plug in the possible values of those variables and see what you get, like this:
If $A = 0$ and $B = 0$: The expected value of $Y$ will be $\beta_0$.
If $A = 0$ and $B = 1$: The expected value of $Y$ will be $\beta_0 + \beta_2$.
If $A = 1$ and $B = 0$: The expected value of $Y$ will be $\beta_0 + \beta_1$.
If $A = 1$ and $B = 1$: The expected value of $Y$ will be $\beta_0 + \beta_1 + \beta_2$.
So, we see that $\beta_2$ is the average difference in $Y$ between individuals with $B = 0$ and $B = 1$, regardless of whether $A = 0$ or $A = 1$.
If you actually want to incorporate the knowledge that $A$ can be present only when $B$ is present into your model, to allow yourself to interpret results the way you want, you can modify the regression to something like this:
$$
Y_i = \beta_0 + \beta_1A_i + \beta_2B_i + \beta_3A_iB_i + \epsilon_i
$$
If you perform the same exercise as above with this equation, you'll see that different combinations of coefficients will represent the average difference in $Y$ between someone with the value of $1$ and someone with the value of $0$ of variable $B$ if they both had the value of $0$ for variable $A$ and the average difference in $Y$ between someone with the value of $1$ and someone with the value of $0$ of variable $B$ if they both had the value of $1$ for variable $A$.
