The first method is simpler for the student project; the second method is often preferred in practice, when we know that certain explanatory variables are important but we don’t know if others are.

What are the answers to the bullet points? a) If two explanatory variables are highly correlated, does adding the second explanatory variable raise or lower the estimated sigma^2 of the regression? and b) If two explanatory variables are uncorrelated and each is correlated with the dependent variable, does adding the second variable raise or lower the estimated sigma^2?

[NEAS: The bullet points encourage you to reason through the intuition, examine the formulas, and read Fox's explanation. If you are unsure, explain what you infer from the textbook and the formulas and why you are not sure of the answers.]

How is sigma^2 calculated in multiple-regression?

I am not sure at all, but if a second closely correlated variable adds to the regression, I think it doesn't change sigma^2 much. May only increase it slight.

When a second uncorrelated variable adds to the regression, it increases sigma^2 by a decent amount.

This is all intuition because I can't find how variance of the regression is determined.

[NEAS: Compute the sum of squared residuals as the least squares estimate of sigma^2. Be sure to divide by the degrees of freedom, which depend on the data points and the number of parameters.]

Sigma^2 = RSS/(n-k-1) where RSS = Sum(yi- yi^).

If another parameter is added, then this should decrease the RSS because a greater percentage of the error is 'explained by' or allocated to the new varaible. I do not believe that this would depend on the correlation between the two variables.

If another parameter is added, then the denominator decreases by one to account for the change in degrees of freedom. When a denominator decreases, the entire expression will increase.

These two factors seem to offset each other. NEAS - can you provide any insight? 

[NEAS: Consider the extreme cases. If the two explanatory variables are orthogonal and each explains half the fluctuation in the response variable, the estimated sigma² decreases to zero. If the two explanatory variables are perfectly correlated, the RSS does not change, and the estimated sigma² increases because of the added variable. The effects in real studies fall between these two extremes.]