There are three car brands A, B, and C. An analyst has set up a model...

 

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There are three car brands A, B, and C. An analyst has set up a model to predict consumers' choice probabilities between those brands. In particular, she thinks that the relative prices between the brands are the key determinants. Her model is given by i 1 2 Pr (Y₁ = A) 3 4 Pr (Y₁ = B) = = Pr (Y= C) = 1 1+exp(ag+BPB. i) +exp(ac+BPc.;); exp(a +3PB.i) 1+exp(aB+BPB) + exp(ac+3Pc.i) where y, is the brand choice, PB, is the price difference between B and A, and Pc is the price difference between C and A in the ith observation. The parameters in this model are ag, ac, and 3. The analyst has the following four observations: B exp(ac+BPci) 1+ explaB+3PB)+exp(ac+3Pci)" Y₁ C B A B PB.i -1 0 2 -2 The two candidate parametric values are (ag, a c, 3) = (1, 0, -1) and (ag, ac, B) = (0, -1, -2). Pci -2 -1 0 -2 Q2-1. Compute the probability that we observe the choices Y₁ = C, Y₂ = B, Y₂ = A, Y₁ = B given that the true parameters are (ap, ac, B) = (1,0,-1). Q2-2. Compute the probability that we observe the choices Y₁ = C, Y₂ = B, Y₁ =A, Y₁ = B given that the true parameters are (aв, ac, B) = (0, -1, -2). Q2-3. Based on Q2-1 and Q2-3, which of the two candidate values is the maximum likelihood estimates' of (a, ac, B)? There are three car brands A, B, and C. An analyst has set up a model to predict consumers' choice probabilities between those brands. In particular, she thinks that the relative prices between the brands are the key determinants. Her model is given by i 1 2 Pr (Y₁ = A) 3 4 Pr (Y₁ = B) = = Pr (Y= C) = 1 1+exp(ag+BPB. i) +exp(ac+BPc.;); exp(a +3PB.i) 1+exp(aB+BPB) + exp(ac+3Pc.i) where y, is the brand choice, PB, is the price difference between B and A, and Pc is the price difference between C and A in the ith observation. The parameters in this model are ag, ac, and 3. The analyst has the following four observations: B exp(ac+BPci) 1+ explaB+3PB)+exp(ac+3Pci)" Y₁ C B A B PB.i -1 0 2 -2 The two candidate parametric values are (ag, a c, 3) = (1, 0, -1) and (ag, ac, B) = (0, -1, -2). Pci -2 -1 0 -2 Q2-1. Compute the probability that we observe the choices Y₁ = C, Y₂ = B, Y₂ = A, Y₁ = B given that the true parameters are (ap, ac, B) = (1,0,-1). Q2-2. Compute the probability that we observe the choices Y₁ = C, Y₂ = B, Y₁ =A, Y₁ = B given that the true parameters are (aв, ac, B) = (0, -1, -2). Q2-3. Based on Q2-1 and Q2-3, which of the two candidate values is the maximum likelihood estimates' of (a, ac, B)?

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This question involves computing the probabilities of observing certain brand choices given a multinomial logistic regression model with parameters B ... View the full answer