Difference in statistical powers between two nested models An basic question came to my mind while I was working on sample size calculation. I have some possible explanation and I would like to compare it to yours.
Let us say I would like to explain some response variable $Y$ from explanatory variables $A$ and $B$.
Further, let us assume that $B$ has no effect on $Y$; the corresponding regression coefficient being $0$.
Two possible models are:
$E(Y) = \mu + \beta_{1} A$,
and
$E(Y) = \mu + \beta_{1} A + \beta_{2} B$.
Which of these two should I choose in order to have more power to detect some effect of $A$. 
What if $B$ has an effect ($\beta_{2} \neq 0$)? 
 A: In the case you know $B_2$ to be zero, and the covariance between $A$ and $B$ does not equal zero, the reduced model will have more power. The standard error of $B_1$ increases as a function of the shared variance between $A$ and the other independent variables in the regression (see this document I found while trying to google for a formula for the standard error of a regression coefficient). I italicize know because frequently you don't know A priori that the effect is zero (unless it is theoretically non-sense), and hence it would require being estimated to prevent a biased estimate of $B_1$ (although this is not directly pertinent to the power of the test). Does the power of a biased test make any sense?
I've seen this referred to as "effective sample size". I know the software such as G*Power 3 allows you to enter the $R^{2}$ from the auxiliary regression equation (that is in this example the regression of $A$ on $B$).
As I stated in reply to Henrik's answer, the loss of the 1 degree of freedom would be rather trivial in most practical sample sizes. The correlation between $A$ and $B$ is of much more pertinence. In the case that $A$ and $B$ aren't correlated this is all moot (except in regards to the degrees of freedom).
A: I would mainly argue with the degrees of freedom. In multiple regression the degrees of freedom are $N-k-1$. With $N$ being the overall sample size and $k$ the number of parameters. 
Therefore, adding a parameter with no explanatory power will reduce the overall degrees of freedom. As the degrees of freedom are directly proportional to your power, adding another explanatory variable with no explanatory power will reduce your overall power.
See also this link (first one on google: "degree of freedom multiple regression")
