Definition of validity of an instrumental variable What does "validity of an instrument" mean exactly? 
In my econometrics course we have just defined instrument validity as $E[Z|u]=0$, where $Z$ is the instrumental variable and $u$ is the error term of a univariate regression model. Then, we also talked about strength of an instrument, but I am pretty sure I have understood correctly that it is a different requirement than validity. 
In applications, I often find the definition of validity as ${\rm corr}(Z,X) \neq 0$, where $Z$ is the instrument and $X$ is the endogenous explanatory variable, plus the requirement that $E[Z|u]=0$ (as above), which is usually defined as exclusion restriction.
I am a bit confused and it is not so easy to find the kind of a primer on IV approaches I need.  Is anyone able to unravel these issues?
 A: Following Hernán and Robins' Causal Inference, Chapter 16: Instrumental variable estimation, instrumental variables have four assumptions/requirements:


*

*$Z$ must be associated with $X$.

*$Z$ must causally affect $Y$ only through $X$

*There must not be any prior causes of both $Y$ and $Z$.

*The effect of $X$ on $Y$ must be homogeneous. This assumption/requirement has two forms, weak and strong:


*

*Weak homogeneity of the effect of $X$ on $Y$: The effect of $X$ on $Y$ does not vary by the levels of $Z$ (i.e. $Z$ cannot modify the effect of $X$ on $Y$).

*Strong homogeneity of the effect of $X$ on $Y$: The effect of $X$ on $Y$ is constant across all individuals (or whatever your unit of analysis is).



Instruments that do not meet these assumptions are generally invalid. (2) and (3) are generally difficult to provide strong evidence for (hence assumptions).
The strong version of condition (4) may be a very unreasonable assumption to make depending on the nature of the phenomena being studied (e.g. the effects of drugs on individuals' health generally varies from individual to individual). The weak version of condition (4) may require the use of atypical IV estimators, depending on the circumstance.
The weakness of the effect of $Z$ on $X$ does not really have a formal definition. Certainly IV estimation produces biased results when the effect of $Z$ on $X$ is small relative to the effect of $U$ (unmeasured confounder) on $X$, but there's no hard and fast point, and the bias depends on sample size. Hernán and Robins are (respectfully and constructively) critical of the utility of IV regression relative to estimates based on formal causal reasoning of their approach (that is, the formal causal reasoning approach of the counterfactual causality folks like Pearl, etc.).
Hernán, M. A. and Robins, J. M. (2017). Causal Inference. Chapman & Hall/CRC.
A: Requirements for Z to be a valid instrument for X are:


*

*Relevance = Z needs to highly correlated with X 

*Exogenous = Z is correlated with Y solely through its correlation with X; so Z is uncorrelated with the error in the outcome equation


The main idea behind IV is that when Z changes, it should also alter X, but not the troublesome part of X that is correlated with the error. To get the effect of X on Y we are only using part of the variation in X, the part that's driven by variation in Z. 
A: Both assumptions can be seen by looking at the system of equations:
\begin{align}
x=&\gamma_1+\gamma_2 z+\epsilon\\
y=&\beta_1+\beta_2 x+\gamma_3 z+u
\end{align}


*

*The strength of the instrument relates to the coefficient
$\gamma_2\neq 0$ and to the $R^2$ of this equation (both should be high enough)

*The validity relates to the assumption that $\gamma_3=0$, i.e. $z$ has no direct effect on $y$.
Note that we cannot test $\gamma_3=0$, only assume it, which explains why it is called an identifying (=untestable) assumption.
