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I face the following practically relevant problem. In a publication I read about a regression model $$E(Y|X)=f(X\beta)$$. This publication shows coefficient estimates $\hat{\beta}$ and their standard errors $\hat{V}(\hat{\beta})^{\frac{1}{2}}$. Let's say I want to apply this model for prediction.

However, the publications in my field (I guess most fields) commonly do not report the covariances between regression coefficient estimates. However, to make a confidence interval for a prediction $$\hat{E}(Y|x_o;\hat{\beta})$$ these covariances are needed (here, $x_0$ is a new observation). For the case of linear regression the confidence interval for a prediction $x_o\hat{\beta}$ is given, for example, by $$x_0 \hat{V}(\hat{\beta}) x_0^T = \sigma^2 x_0 (X^TX)^{-1} x_0^T$$ where $\sigma$ the error variance.

My question is: are there techniques to still obtain a confidence bound for the mean prediction in absence of covariance estimates or is this a hopeless endeavor?

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    $\begingroup$ Scientifically, the answer is no, if you cannot get estimate of the covariance of design matrix. The best and correct way to resolve this problem is to ask the author(s) providing the more information, but the chance of getting them is low (my experience). $\endgroup$ – user158565 May 27 '17 at 17:04
  • $\begingroup$ @a_statistician My intuition too, but I was thinking cross validation on a new data set (if available) to get an estimate of error variance, maybe? $\endgroup$ – tomka May 27 '17 at 17:10
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    $\begingroup$ If you have new data, you can 1) verify published model by plugin the your X and get predicted Y, then compare with your observed Y. 2) Fit a new model, and comparing the regression coefficients between your new model and published old model. But you cannot use your $(X'X)$ or your new covariance to replace old one, because it depends on the design of the studies. $\endgroup$ – user158565 May 27 '17 at 17:36
  • $\begingroup$ @a_statistician Hmm it feels that we could make sufficient assumptions to fix this... Let's say we assume the two studies both got random samples from the same population. Wouldn't it then be admissible to use $X^TX$ from the new data to estimate confidence for the parameter estimates from the first? I think we need to assume equal sample size on top of this but then it might work. $\endgroup$ – tomka May 27 '17 at 17:45
  • $\begingroup$ I saw a lecture video given by a professor in MIT years ago. He said if the conclusion is based on assumptions that cannot be verified, it is a myth, not science. If you have enough reasons to use your $X'X$ to replace old one, just do it, do not need to worry about the difference of sample sizes. Sample size has effect on the estimate of $\sigma^2$. Of course, the reliability of your $X'X$ depends on your sample size. $\endgroup$ – user158565 May 27 '17 at 18:19

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