[Gretl-users] strange R^2 with TSLS

[Sven Schreiber]

Allin Cottrell schrieb:
> On Thu, 6 Dec 2007, Sven Schreiber wrote:
>
>   
>>> On the original issue of R^2, I'll file the bug report soon.
>>>
>>>       
>> "Delay that order"... Actually for my real-world cases it turns 
>> out there isn't anything (obviously) wrong. However, I'm still 
>> puzzled by the 3-liner results I posted earlier. The point 
>> estimates are quite different between ols and tsls -- then how 
>> come the correlation between fitted and observed is the same to 
>> five or six digits of precision? Hm.
>>     
>
> "Strange but true".  It seems to be in the arithmetic for the case 
> of one independent variable and one instrument.
>
> nulldata 50
> genr x = normal()
> genr y = normal()
> genr z = normal()
> ols y 0 x
> ols x 0 z --quiet
> genr xhat = $coeff(const) + $coeff(z)*z
> ols y 0 xhat
> genr yhat = $coeff(const) + $coeff(xhat)*x
> R2 = corr(y, yhat)^2
>
> "R2" is numerically identical to the R^2 from the first OLS.  
> Left as an exercise: prove that this is always the case.
>
>   

Sorry to leave that unclarified, Jack already pointed out the (simple) 
solution of this pseudo-mystery directly to me. (Trying to not embarrass 
me publicly I guess...) Any linear transformation (or some may call it 
affine) of a single variable will leave the linear correlation with 
another variable unchanged of course. So the estimated coefficients are 
really totally irrelevant in this case.

cheers,
sven