st: Perfect fit, perfect collinearity and small OLS standard errors

[Stefano Lombardi <ste85@fastwebnet.it>]

Dear all,

First of all, please do not blame the model I am going to introduce, 
because actually I would have never fitted a model like that (it is a 
famous "case-study" on which I must write a report..).
I have just regressed a (log-linear) wage equation in order to predict 
Consumption of a good.
By adopting a model with 6 covariates (plus the constant), I have fitted 
a model with adjusted R^2 = 0.98
(I already know that the model is "useless" as "ex ante" the dependent 
variable is almost a linear combination of the regressors).
In addition, I have also found perfect collinearity between each 
regressors and all the others (in some cases R^2_k is close to 0.99)
The question concerns the partial coefficients estimates and their 
standard errors in particular. In fact, none of the standard errors of 
the coefficients is even major than 1, even if we would expect them to 
tend to infinity because of collinearity. I interpreted the result this 
way: the collinearity effect is "masked" by the really small MSE of the 
model (0.001), which of course is very small because of the (almost) 
perfect fit.
Is that sensible? Has this kind of problem (the "masked" collinearity 
effect) a specific name in literature?
Thank you very much for your attention,

Stefano Lombardi
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