Comments:

Date: 20160320 15:21 Sender: Liviu AndronicThanks for looking into this!
As you mention, it seems like the matrix package works very hard around rounding issues and ends up believing in nearsingularity when it deals in fact with exact singularity (when accounting for rounding quirks). I agree that it's not a good idea to silently remove a regressor in case of exact singularity, but perhaps doing so with a warning should prove more useful in many cases.
As mentioned before, lm() silently drops regressors (but clearly identifies undefined coefficients in coef() and summary() output). The plm() fun will also silently drop regressors, although my local version issues a warning and I'll probably reenable it in SVN plm, too:
# also 2SLS IV
# my local version issues a warning when dropping coefficients because of singularities
require(plm)
oursample.p < pdata.frame(oursample, index="city")
plm(log(wage)~educ+exper+I(exper^2)+fact1+fact2
 motheduc+fatheduc+exper+I(exper^2)+fact1+fact2
, model="pooling", data=oursample.p)
## Model Formula: log(wage) ~ educ + exper + I(exper^2) + fact1 + fact2  motheduc +
## fatheduc + exper + I(exper^2) + fact1 + fact2
## <environment: 0xd198288>
##
## Coefficients:
## (Intercept) educ exper I(exper^2) fact1
## 0.03244628 0.06087732 0.04407168 0.00089489 0.04803179
##
## Warning message:
## In mylm(y, X, W) :
## Coefficient(s) 'fact2TRUE' could not be estimated and is (are) dropped.
The next version of AER, kindly fixed by Achim, will also work around singularity issues in ivreg() in the same style as lm():
require(AER)
ivreg(log(wage)~educ+exper+I(exper^2)+fact1+fact2
 motheduc+fatheduc+exper+I(exper^2)+fact1+fact2 , data=oursample)
## Call:
## ivreg(formula = log(wage) ~ educ + exper + I(exper^2) + fact1 + fact2  motheduc + fatheduc + exper + I(exper^2) + fact1 + fact2, data = oursample)
##
## Coefficients:
## (Intercept) educ exper I(exper^2) fact1 fact2TRUE
## 0.0324463 0.0608773 0.0440717 0.0008949 0.0480318 NA
IMO the most useful way to address this would be to mirror lm(): add a `singular.ok` argument which when FALSE outputs an error in case of exact singularity, and when TRUE drops a coefficient while issuing a warning. This way the user has a clear idea on why the estimation has failed, or where a potential issue lies when the estimation proceeds. In my experience perfect collinearity issues can prove very confusing to endusers (and indeed, very hard to track down unless you're very experienced), and the lm() way to dealing with thiscoupled with a warningseems to me like the optimal approach.  Date: 20160320 14:15 Sender: Arne HenningsenThanks a lot for reporting this problem and for providing a reproducible example!
The huge or even NA standard errors of the intercept and the regressors fact1 and fact2 indicate that there is very high multicollinearity between these three regressors. The matrix of the regressors should be exactly singular but due to rounding errors it seems to be not exactly singular so that it can be inverted and the estimates of the coefficients can be obtained.
When estimating the model by OLS, the estimation terminates with an error message that (correctly) claims that the matrix of the regressors is (nearly) singular:
systemfit(log(wage)~educ+exper+I(exper^2)+fact1+fact2, "OLS", data=oursample)
Error in LU.dgC(a) : cs_lu(A) failed: nearsingular A (or out of memory)
When instructing systemfit() not to use the "matrix" package to estimate the model by 2SLS, the estimation terminates with an error message that (correctly) claims that the matrix of the regressors is (exactly) singular:
systemfit(log(wage)~educ+exper+I(exper^2)+fact1+fact2, "2SLS",~motheduc+fatheduc+exper+I(exper^2)+fact1+fact2 , data=oursample, useMatrix = FALSE )
Error in solve.default(crossprod(zMatEq[[i]]), crossprod(zMatEq[[i]], :
Lapack routine dgesv: system is exactly singular: U[7,7] = 0
I don't think that it is a good idea to silently remove some of the regressors in case of too high multicollinearity, particularly in case of a system estimation. Therefore, I suggest that systemfit() returns an informative error message if there is "too high" multicollinearity, where the question remains how to define "too high" multicollinearity. Any thoughts?

