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### Evaluating covariance terms for variance of residual in simple linear regression [duplicate]

I am trying to work through calculating the variance of a residual for simple linear regression not using vectors or matrices. I am having trouble with calculating two different covariance expressions ...
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### Understanding shape and calculation of confidence bands in linear regression

I am trying to understand the origin of the curved shaped of confidence bands associated with an OLS linear regression and how it relates to the confidence intervals of the regression parameters (...
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### Are the estimates of the intercept and slope in simple linear regression independent?

Consider a linear model $y_i= \alpha + \beta x_i + \epsilon_i$ and estimates for the slope and intercept $\hat{\alpha}$ and $\hat{\beta}$ using ordinary least squares. This reference for a ...
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### Residual variance formulas difference

There is a bi-dimensional table of frequencies: Doing the regression analysis with the fit formula being $\hat y=a+bx^2$, where $\hat y$ is the same as $y^{est}$, the filled table looks like this: ...
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### evaluating out of sample accuracy

I estimate a linear regression model and compute the variance of residuals in both the training-set and also on an additional test set. Ideally these should not be very different. Does it make sense ...
Problem Given a linear model $y_i = \beta_1 + \beta_2 x_i +\epsilon_i, \quad i = 1, \dots, n$ I need to compare the variance ordinary least squares estimator of $\beta_2$ without the restrictions and ...
### Proving that $V(\hat{y}_{x_0}) = \sigma^2\bigg[\frac{1}{n}+\frac{(x_0-\bar{x})^2}{S_{xx}}\bigg]$ [duplicate]
Exercise : Prove that the variance of $\hat{y}_{x_0} = \hat{b_0} + \hat{b_1}x_0$ is : \text{Var}(\hat{y}_{x_0}) = \frac{\sigma^2\sum x_i^2}{n\sum(x_i-\bar{x})^2}+\frac{\sigma^2x_0^2}{\sum(x_i-\...