# Some confusions about instrumental variable (econometrics)

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:

1. 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.

2. 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.

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?

No it doesn't. For mean-centered variables for simplicity, we have for linear models,

ENDOGENEITY: $$E(xu) \neq 0$$

RELEVANCE : $$E(xz) \neq 0$$

The above conditions do not necessarily imply non-Validity ($$E(zu) \neq 0$$).

A very simple example to show that we may have a valid instrument. Assume $$x, u$$ are centered on their mean, $$E(x) = E(u) = 0$$, and that $$x$$ has a symmetric distribution, $$E(x^3) = 0$$. Assume that $$E(u \mid x) = ax$$. Then

ENDOGENEITY

$$E(xu) = E\big[E(xu\mid x)\big] = E\big[xE(u\mid x)\big] = aE(x^2) \neq 0.$$

Consider the candidate instrument $$z = x^2 + v$$, with $$v$$ independent from $$u$$, but $$E(xv) \neq 0$$. We have

RELEVANCE $$E(xz) = E(x^3) + E(xv) = 0 + E(xv) \neq 0$$.

VALIDITY

$$E(zu) = E(x^2u) + E(vu) = E\big[E(x^2u\mid x)\big] + E(v)E(u)$$

$$= E\big[x^2E(u\mid x)\big] + 0 = E\big[x^2a x\big] = aE(x^3) = 0.$$

Adding to the excellent answer by @Alecos Papadopoulos, here are two simple numerical examples with $$z=x^2+v$$, $$z^*$$ is mean-centred $$z$$ and $$v$$ is independent from $$u$$, in which $$E(xu)\neq 0$$, $$E(xz^*)\neq 0$$, but $$E(z^*u)=0$$.

Example 1 (with $$x$$ having a symmetric distribution)

$$E(xu)=1.5$$, $$E(xz^*)=-0.5$$, $$E(z^*u)=0$$

Example 2 (showing that the distribution of $$x$$ need not be symmetric)

$$E(xu)=2$$, $$E(xz^*)=-3$$, $$E(z^*u)=0$$