ECON6300/7320/8300 Advanced Microeconometrics Conditional Quantile Regressions
Christiern Rose 1University of Queensland
Lecture 8
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This lecture
What are quantiles
Types of regression models
What is quantile regression?
Optimality properties of QR
Computational aspects of QR
Interpreting QR
Asymptotic variance and bootstrap computations
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Quantiles and Distribution Function
Assuming a right-continuous distribution function for a scalar valued continuous random variable X
F(x) = 0 ≤ F(x) = x = F−1(q) = med[X] =
Pr[X≤x]
F(x)≤1, F(−∞)=0;F(+∞)=1
U, 0≤U≤1
F−1(U), inverse prob. transform inf[x:F(x)≥q]forany0
SoqN oftheyi arelessthanβq!
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Intuition: The B matrix
Given Q(β) = Ni=1 qi (β) and
∂qi (β)/∂β = −[q − 1 + 1(yi − x′i β)]xi , by the properties of m-estimators
B = plim 1 N ∂qi(β) ∂qi(β) N i=1 ∂β ∂β′
= plim 1 N [q − 1 + 1(yi − x′iβ)]2xix′i N i=1
= plim1N q(1−q)xix′i N i=1
Intuition:FromtheFOCNi=11(yi −x′iβ)=N(1−q)soif we only have an intercept (xi = 1) we obtain
Ni=1[q−1+1(yi −x′iβ)]2
= Ni=1(q−1)2+2(q−1)1(yi−x′iβ)+1(yi−x′iβ) = N(q−1)2 +2(q−1)N(1−q)+N(1−q)
= Nq(1−q)=Ni=1q(1−q)xix′i
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Intuition: The A Matrix
The indicator function 1(yi − x′i β)
changes sign only if yi − x′i β = 0 with derivative 1 and
probability fyi −x′i β (0|xi ).
At all points other than yi − x′i β = 0, the derivative is zero.
plim1N ∂∂qi(β) N i=1∂β∂β′
A =
= −plim1 N ∂ [q−1+1(yi −x′iβ)]x′i
N i=1 ∂β
= −plim1 N ∂ 1(yi −x′iβ)x′i
N i=1 ∂β
= −plim1 N ∂ 1(u)(−xi)x′i
N i=1 ∂u
= plim1 N [1×fyi−x′β(0|xi)+0×(1−fyi−x′β(0|xi))]xix′i
Ni=1i i
= plim 1 N fyi−x′β(0|xi)xix′i. N i=1 i
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