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For the simple linear regression model, show that the hat matrix H has the following property: The text states that There is a small amount

For the simple linear regression model, show that the hat matrix H has the following property:

The text states that "There is a small amount of correlation present in standardized residuals, even if the errors are independent." Comment on when the covariances of the residuals are close to zero, for a fixed sample size. Why does it make sense that the covariances are close to zero in those situations?

Please explain in detail. Thank you so much.

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