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Multiple Regression - Bivariate confidence intervals - implications of positive and negative correlation

Parsimony

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Hi everyone,

I do apologise if this has been asked before but I'm failing to get my head around the attached page 132 from Book 2. The implications of positive and negative correltion on the confidence intervals of the parameters.

I have experience with multiple regression but something just isn't "clicking" for me with this right now.

There are also practice questions 8.5 and 8.6, which ask:

8.5 For a regression Y = a + B1X1 + B2X2 + e. Using the F-stat you reject the null H0: B1 = B2 = 0 but fail to reject either of the nulls H0: B1 = 0 or H2: B2 = 0 using the t-stat of the coefficient. Which values of p = correlation (X1, X2) make this scenario more likely?

8.6 Suppose both t-stats are exactly +1. Could an F-statistic lead to rejection in this circumstance? Use a diagram to explain why not.

I attach the answer page for this.

I'm more interested in an intuitive explanation rather than hard proof but would love to hear any explanations for page 132 and these practice questions.

Thank you
 

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