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I'm preparing an exam in econometrics, and I have some trouble answering one question.

It writes:

"One of your graduate students, working on a joint empirical project with you, comes to you and says : “You’ll be happy professor : our estimations lacked precision, so I did the estimation again after having duplicated the sample so that its size is now 2N. Clever idea, as the estimated variance of my OLS estimator on this sample is now much lower !” What do you answer ?"

It seems obvious that doubling artificially the size of the sample is stupid, but I have trouble finding why. Could you help me with this?

Mr. Fafa
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    Think of the correlation structure of the errors of the sample with duplicates, then work through the expression for the variance of the OLS estimator with errors having a covariance matrix other than $\sigma^2\text{I}$. – jbowman Jan 04 '19 at 18:00
  • It can help to consider the smallest possible example you can think of. For instance, what is the OLS estimator of $\mu$ for two independent observations of the model $Y=\mu+\varepsilon$ What is its variance? How should you compute its variance when those data are artificially doubled? How would you compute its variance if you didn't know they had been doubled but instead supposed you had four independent observations? – whuber Jan 04 '19 at 18:46
  • You CAN double your sample, but you need to incorporate the fact that correlation coefficient is 1 between the original one and its copy when you analyze the data. The results will be the same between doubled sample and original sample. – user158565 Jan 05 '19 at 03:30
  • Thanks for the answer, the link posed by jbowman included the right answer – Mr. Fafa Jan 05 '19 at 11:51

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