Mị : y = Bo + B1x1 + B2x2 + e, M2 : y = Bo + B1x1 + B2x2 + Bzx² + B4x3 + B5x1 x2 + €. Assume that we fit the two models to the same data. (a) Can we compare the R2 of the two models without looking at the results of the fittings? Justify your answer.

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Chapter1: Combinatorial Analysis
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Consider the following two regression models (at the population level):
M1 : y = Bo + B1x1 + B2x2 +ɛ,
M2 : y = Bo + B1x1 + B2x2 + B3x² + B4x3 + Bzx1x2+ E.
Assume that we fit the two models to the same data.
(a) Can we compare the R2 of the two models without looking at the results of the fittings? Justify your
answer.
(b) Can we say that the residual vector of one model is orthogonal to the fitted value vector of the other
model? If so specify the details, if not argue why not.
(c) Assume that the data is actually generated from the following model (at the population level)
y = Bö + Bix1 + Bžx2 + Bž¤1¤2 + E,
with Bo, Bi, B, and B all nonzero. Which of the two models M1 and M2 will produce an unbiased
LSE of B*? Justify your answer.
(d) Now assume the data of sample size n = 100 is actually generated from model M1 with noise vector
following our standard Gaussian assumption ɛ ~
and M2. What is the probability that the p-value of that test is > 0.05? Justify your answer.
N (0, o?In). Consider an F-test for comparing M1
Transcribed Image Text:Consider the following two regression models (at the population level): M1 : y = Bo + B1x1 + B2x2 +ɛ, M2 : y = Bo + B1x1 + B2x2 + B3x² + B4x3 + Bzx1x2+ E. Assume that we fit the two models to the same data. (a) Can we compare the R2 of the two models without looking at the results of the fittings? Justify your answer. (b) Can we say that the residual vector of one model is orthogonal to the fitted value vector of the other model? If so specify the details, if not argue why not. (c) Assume that the data is actually generated from the following model (at the population level) y = Bö + Bix1 + Bžx2 + Bž¤1¤2 + E, with Bo, Bi, B, and B all nonzero. Which of the two models M1 and M2 will produce an unbiased LSE of B*? Justify your answer. (d) Now assume the data of sample size n = 100 is actually generated from model M1 with noise vector following our standard Gaussian assumption ɛ ~ and M2. What is the probability that the p-value of that test is > 0.05? Justify your answer. N (0, o?In). Consider an F-test for comparing M1
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