I want to know if there is any literature about the following regression problem: $$ Y=X\beta +\epsilon$$ where $Y$ is unknown. But, i know $X$ and the OLS estimator of $\beta$ $$ \hat{\beta}=(X^\top X)^{-1}X^\top Y$$ because i know the value of $X^\top Y$ In fact i have an inverse problem of quadratic programming where i know the solution and i want the regression problem associated to this solution.
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1You just wrote down the estimator of $\beta$. What else do you want to know? – Aniko Feb 15 '12 at 20:11
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i want to know if there any theories to find $Y$, to know the variance of $Y-X\hat{\beta}$ or the mean squared error associated. – david Feb 15 '12 at 20:20
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Do you also know the sample covariance matrix of $\hat{\beta}$? – jbowman Feb 16 '12 at 14:44
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how to know the covariance of the beta without $y$ ? – grant Feb 16 '12 at 18:41
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Is Y a latent variable? If so, then something like structural equation modelling may be more appropriate. – Michelle Feb 16 '12 at 19:44
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yes i $Y$ is a latent variable – david Feb 16 '12 at 20:00
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Perhaps 'calibration models' would give you some insights. – Papo Feb 16 '12 at 13:45
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There will be an infinite number of Y vectors that would lead to the same $\beta$ vector for the given $x$ matrix. One option for further studying the relationship would be to generate a random vector of "errors" from a normal distribution with mean exactly equal to 0 and orthogonal to $x\beta$ (you could do this by generating a random vector, regressing this onto $x\beta$ and taking the residuals) then adding those "errors" to $x\beta$. This will give you one of the $y$ vectors that would give the results you have. Now you can use this simulated $y$ with standard routines to find other values of interest. You can repeate this with multiple simulations, many of the standard results will be identical (provided the "errors" have the exact same variance).
Greg Snow
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I don't know of a specific paper about this, but a decent book on linear models should contain the theory needed to justify it. – Greg Snow Feb 17 '12 at 18:25