Alastair Hall ECON61001: Semester 1 2020-21 Econometric Methods
Problem Set for Tutorial 9
In this question, you examine the properties of the error in the linear probability model.
1. Let {yi, x′i}Ni=1 be a sequence of iid random vectors. Suppose that yi is a dummy variable and so has a sample space of {0, 1} with P (yi = 1|xi) = x′iβ0. Now consider the Linear Probability Model (LPM),
Show that
(a) E[ui|xi] = 0.
(b) V ar[ui|xi] = x′iβ0(1 − x′iβ0).
y i = x ′i β 0 + u i .
(c) Can ui have a normal distribution conditional on xi? In this question you consider the logit model.
2. Consider the logit model in which P (yi = 1|xi; β) = Λ(x′iβ) where xi is a (k × 1) vector with lth element xi,l and
Λ(z) = exp(z) . 1 + exp(z)
Assume we have a sample of size N and that {yi,x′i}Ni=1 are independently and identically distributed.
(a) Derive the (conditional) likelihood function.
(b) Write down the (conditional) log likelihood function.
(c) Assuming xi,l is a continuous variable, derive ∂Λ(x′iβ0)/∂xi,l.
In this question you consider ML estimation of the linear regression model under Assumptions CA1-CA6. Recall that under these assumptions, y ∼ N(Xβ0, σ02IT ). Recall from Lecture 9 that in the case of continuous random variables the likelihood function is the joint pdf of the random variables in question.
3. Consider the linear regression model
y = Xβ0 + u
where y and u are T ×1 vectors, and X is T ×k matrix. Suppose that Assumptions CA1-CA6 (from Lecture 1) hold. Let θ be the (k + 1) × 1 vector consisting of the parameters of this model that is, θ = (β′, σ2)′.
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(a) Show that the log likelihood function is given by
LLFT (θ) = −T ln[2π] − T ln[σ2] − (y − Xβ)′(y − Xβ).
2 2 2σ2 (b) Show that the score equations imply:
X ′ ( y − X βˆ T ) = 0 −TσˆT2 +(y−XβˆT)′(y−XβˆT)
(c) Show that the MLE’s are:
βˆ T = ( X ′ X ) − 1 X ′ y
σˆ T2 = ( y − X βˆ T ) ′ ( y − X βˆ T ) / T
(d) Compare the MLE’s to the OLS estimators of θ0.
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