As the title says, I have some confusions about instrumental variable.
According to my lecture note (which is supposed to be reliable), instrumental variable has two big properties:
Validity: an instrumental variable and error term are not correlated and it implies that the instrumental variable does not directly lead to changes in dependent variable.
Relevance: an instrumental variable and endogenous variable are correlated.
So here is my confusion. (let's denote z is an instrumental variable)
Endogenous variables are correlated with the error terms and z is correlated with endogenous variable.
Doesn't this imply that z is correlated with error terms? If so, it violates the Validity (mentioned above).
In case my question isn't clear enough. Here is the case where I am stuck: $$ score_i = \beta_0 + \beta_1 course_i + u_i $$ where $course_i$ is a binary variable of taking a preparatory course and $u_i$ is an error term.
Quite obviously, $course_i$ is correlated with $u_i$ since the parental income is one of the unobserved variables (so in $u_i$) and $course_i$ and parental income are correlated (I think this is also quite obvious).
So far we know that parental income (I think this is an instrumental variable) is correlated with $course_i$. But since parental income was belonged to error term, it violates Validity.
Any help is welcome.