2sls - instrumental variables mean and variance of exogenous variable I do not usually use two-stage-least-square regression technique and I am not a theorist of econometrics; thus, I hope you will pardon me for this (possibly) clumsy post.
Introduction
Let's start from the second stage.
Here, I regress the outcome variable on the control variables plus the predicted covariate of interest, which comes from the first stage.
At the first stage, I regress my endogenous variable on at least one instrument, plus all of the control variables used in the second stage.
The instrument has to fulfill two assumptions: i) INDEPENDENCE, that is, it cannot directly affect the outcome variable of the second stage (i.e. it has to be valid), and it has to be as good as randomly assigned; ii) MONOTONICITY, for all of the units of observation, the instrument has to affect the endogenous variable in the same direction (i.e. there are no defiers).
Moreover, the instrument cannot only weakly affect the endogenous variable.
After I have computed the estimates from this regression, I compute the predicted covariate of interest, which will be used as a covariate in the second stage (see above).
First, I describe the situation, pose my question, and then provide an example:
I do not have a proper instrumental variable in my dataset, so I have to create one.
My question is:
can I compute the mean and the variance of the endogenous variable by group (i.e. by age group and by nationality) and use them as instrumental variables?
I have never seen done it, and there must be a good and banal answer for that--which I cannot see at this time of the day.
For instance, assume that my endogenous variable is x3, but this variable is endogenous only for a few people (i.e. most of the people are compliers, although there are a few never-takers and always-takers) and thus I will have to implement a 2SLS to gain a causal interpretation of b3_hat.
These are the variables:
Y = salary; x1 = age; x2 = gender; x3 = education; x4 = parents' country of birth; x5 = first language
I compute the mean and the variance of education (x3) by age (x1) and gender (x2).
Then I proceed as usual:
Stage 1 (endogenous variable = x3)

*

*x3 = z0 + z1mean_x3 + z2var_x3 + z3x1 + z4x2 + z5x4 + z6x5 + error

*compute x3_hat to be used in Stage 2

Stage 2

*

*Y = b0 + b1x1 + b2x2 + b3x3_hat + b4x4 + b5*x5 + error

z1 and z2 do not directly affect y.
z1 and z2 return the expected level of x3--i.e. as if it was exogenous, z3-z5 tell us the individual deviation from that expected level.
 A: A relatively recent way to create instrumental variables--when your data set does not include instruments yet--is the method proposed in the following paper:

Lewbel, A, 2012.  Using Heteroscedasticity to Identify and Estimate Mismeasured and Endogenous Regressor Models.  Journal of Business and Economic 
  Statistics, 30:1, 67-80. http://fmwww.bc.edu/EC-P/wp587.pdf.

In short, potential instruments are created exploiting the existence of heteroskedastic error; thus, the greater the degree of heteroskedasticity, the higher is the correlation between the generated instruments and the 
endogenous variable(s).
Lewbel's approach may be applied when: (i) no (or too few) external instruments are available; (ii) to supplement external instruments to improve the efficiency of the IV estimator (you have the right amount of instruments, but you want more).
Point (ii) is relevant also because it allows the researcher to conduct Sargan-Hansen tests on the orthogonality conditions or overidentifying restrictions--the latter test is not feasible when the number of instrumented variables is equal to the number of instrumental variables.
[For interested Stata users, the user-written xtivreg2 and the ivreg2h routines allow the researchers to construct these instruments automatically]
A: Recently, I have found one additional alternative, that is, the use of imperfect instruments.
In the case of imperfect instruments, the INDEPENDENCE assumption is weakened.

Nevo, A., Rosen, A. M. (2008) Identification with Imperfect Instruments. NBER Working Paper No. 14434. https://www.nber.org/papers/w14434.pdf

The authors assume that

(i) the correlation between the instrument and the error term has the same sign as the correlation between the endogenous regressor and the error term, and (ii) that the instrument is less correlated with the error term than is the endogenous regressor.

I have not read any application of this method; however, several studies in very good journals use it: https://scholar.google.com/scholar?cites=3308366933447025703&as_sdt=8000005&sciodt=0,19&hl=it
