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I am reading up on linear regression from 18.650 MIT. In the way of explaining it, the professor derives a few results and I have attached them in the image enter image description here

  1. The first result gets used along with the theorem's mentioned towards the end in deriving the distribution of $\hat{\beta}$ which consequently gets used in testing the significance of the parameters
  2. The unbiased estimator of $\sigma^2$, the $\hat{\sigma}^2$ gets used in creating the prediction intervals (I have doubts about this. The professor did not mention this explicitly but, I think that is its purpose)

Questions:

  1. What is the significance of the remaining three results? As one can see they depend upon $\sigma^2$ which is not known hence I cannot use them in their current form. These results were not used subsequently in deriving any other results as well
  2. Is the claim I have made above about the significance of the unbiased estimator of the variance true? (The unbiased estimator of $\sigma$, the $\hat{\sigma}$ gets used in creating the prediction intervals)
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1 Answer 1

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  1. $\sigma^2 tr((X^TX)^{-1}$ is telling us something about how difficult the regression problem is. First, $tr((X^TX)^{-1})$ is equal to the sum of the reciprocals of the eigenvalues of $X^TX$. So, if $X^TX$ is close to singular ($1/\lambda_i\rightarrow \infty$), we might have problems. This can occur with collinear columns in $X$. With this fixed design problem, we should choose a $X$ to avoid this. So, the difficulty of this problem is a function of the design of $X$ and the observation variance.

  2. $\sigma^2(n-p)$ is telling us how the problem is affected by the dimensionality of the task. Consider $n=p$ though, for each point, you may simply pass your regression through each point. This may minimize prediction error on your observations at the cost of overfitting. In a sense, we may always minimize the observed prediction error by increasing the number of predictors. Again, if $\sigma^2$ is large, then we will on average make more errors in our prediction.

  3. The theorem you have not discussed is used to derive the t-distribution for inference on $\hat{\beta}$, when $\sigma^2$ is unknown.

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