8/20/2019 Discrete Choice Model Notes
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Notes to Memorize
March 4, 2015
• Density of a Truncated Random Variable (from
below)
f (x|x > a) = f (x)
Prob(x > a)
• Truncated Normal Prob(x > a) =
1 − Φ(a−µσ
) = 1 − Φ(α)
• Truncated Normal Density
f (x|x > a) =1
σ
x−µ
σ
1 − Φ(α)
• Moments of the Truncated Normal Distribution x
∼ N (µ, σ2) and constanta
E [x| truncation] = µ + σλ(α)V
ar[x| truncation] = σ2[1 − δ (α)]
α = a − µ
σ
λ(α) = φ(α)
1 − Φ(α) if truncation is x > a
λ(α) = −φ(α)
Φ(α) if truncation is x < a
δ (α) = λ(α)[λ(α) − α]
• Truncated Regression Model Given yi
= x
iβ + εi and εi|xi
∼ N (0, σ2) implies
yi|xi ∼ N (x
iβ, σ2).
E [yi|yi > a] = xi/β + σ
φ[(a − xiβ )/σ]
1 − Φ[(a − xiβ )/σ]
= xiβ + σλ(αi)
∂E [yi|yi > a]
∂xi= β (1 − λ2 + αλ)
= β (1 − δ i)
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8/20/2019 Discrete Choice Model Notes
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• Moments of Censored Normal Variable(From below)
y∗ ∼ N (µ, σ2) and cen-soring below by a
E [y] = Φa + (1 − Φ)(µ + σλ)
V ar[y] = σ2(1 − Φ)[(1 − δ ) +
(α − λ)2Φ]
Φ[(a − µ)/σ] = Φ(α) = P rob(y∗ ≤ a) = Φ,
λ = φ
1 − Φ
δ = λ2 − λα
(1)
• Censored Regression Model y∗i = x
iβ + εi.
E [y∗i |xi] = x
iβ
E [yi|xi] = [1 − Φ(α)](µ + σλ(α)) =
[1 − Φ(α)](x
iβ + σλ(α))
E [yi|xi, yi > 0] = x
β + σλ(α)∂E [y∗i |xi]
∂xi= β
∂E [yi|xi]
∂xi= β · P rob[a < y∗ < b]
= β Φ
xiβ
σ
when a = 0
E [yi|xi, yi > 0]
∂xi= β [1 − αλ − λ2]
• Moments of the Incidentally Truncated Bivariate Normal
Distributiony, z ∼ N [(µy, µz), (σ
2
y, σ2
z)] and ρ = Corr(y, z ).
E [y|z > a] = µy + ρσyλ(αz)
V ar[y|z > a] = σ2y[1 − ρ2δ (αz)]
αz = a − µz
σz
λ(αz) = φ(αz)
1 − Φ(αz)
δ (αa) = λ(αz)[λ(αz) − αz]
• Incidental Truncation Regression Model
z ∗i = w
iγ + ui
yi = x
iβ + εi
E [yi|yi is observed] = E [yi|z ∗
i > 0]
= xiβ + ρσελi(αu)
= xiβ + β λλi(αu)
∂E [yi|z ∗
i > 0]
∂xik= β k − γ k
ρσεσu
δ i(αu)
δ i = λ2 − αiλi
E [yi|z i = 1, xi, wi]
= xiβ + ρσελ(wiγ )
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