Let $X, Y, Z$ be r.v's. Suppose $X$ has some influence on $Y$ and $Z$, and $Y$ has some influence on $Z$. Can we quantify the influence $X$ has on $Z$, which is not due to the influence $X \rightarrow Y \rightarrow Z$?

Regress $Y = \beta_1 X + \alpha_1 + \epsilon_1$, then $Y_X = \beta_1 X + \alpha_1$ is the part of $Y$ explained by $X$. Next, regress $Z = \beta_2 X + \alpha_2 + \epsilon_2$, expanding to get $Z = \beta_2 Y_X + (\beta_2\epsilon_1 + \alpha_2 + \epsilon_2)$.

Since the former term is the part of $Z$ explained by $X \rightarrow Y \rightarrow Z$, the latter term must be the part of $Z$ which isn't. Therefore we want $\text{Cor}(X, \beta_2\epsilon_1 + \alpha_2 + \epsilon_2) = \text{Cor}(X, \beta_2\epsilon_1+\epsilon_2)$.

What is the name of this type of correlation? (I am pretty sure it isn't just the partial correlation). Also what is the corresponding notion in information theory?

Let $X, Y, Z$ be r.v's. Suppose $X$ has some influence on $Y$ and $Z$, and $Y$ has some influence on $Z$. Can we quantify the influence $X$ has on $Z$, which is not due to the influence $X \rightarrow Y \rightarrow Z$?

Regress $Y = \beta_1 X + \alpha_1 + \epsilon_1$, then $Y_X = \beta_1 X + \alpha_1$ is the part of $Y$ explained by $X$. Next, regress $Z = \beta_2 Y + \alpha_2 + \epsilon_2$, expanding to get $Z = \beta_2 Y_X + (\beta_2\epsilon_1 + \alpha_2 + \epsilon_2)$.

Since the former term is the part of $Z$ explained by $X \rightarrow Y \rightarrow Z$, the latter term must be the part of $Z$ which isn't. Therefore we want $\text{Cor}(X, \beta_2\epsilon_1 + \alpha_2 + \epsilon_2) = \text{Cor}(X, \beta_2\epsilon_1+\epsilon_2)$.

What is the name of this type of correlation? (I am pretty sure it isn't just the partial correlation). Also what is the corresponding notion in information theory?